dlmfSource/dlmf-complete.xml
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<title>DLMF: 1.11 Zeros of Polynomials</title>
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</dl>
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<div id="Px1.p1" class="ltx_para">
<p class="ltx_p">Let</p>
<table id="E1" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.11.1</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E1.png" altimg-height="28px" altimg-valign="-7px" altimg-width="317px" alttext="f(z)=a_{n}z^{n}+a_{n-1}z^{n-1}+\dots+a_{0}." display="block"><mrow><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mrow><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><msup><mi href="./1.1#p2.t1.r2">z</mi><mi href="./1.1#p2.t1.r5">n</mi></msup></mrow><mo>+</mo><mrow><msub><mi>a</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow></msub><mo></mo><msup><mi href="./1.1#p2.t1.r2">z</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow></msup></mrow><mo>+</mo><mi mathvariant="normal">…</mi><mo>+</mo><msub><mi>a</mi><mn>0</mn></msub></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m131.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a> and
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m120.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Then</p>
<table id="E2" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.11.2</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E2.png" altimg-height="28px" altimg-valign="-7px" altimg-width="452px" alttext="f(z)=(z-\alpha)(b_{n}z^{n-1}+b_{n-1}z^{n-2}+\dots+b_{1})+b_{0}," display="block"><mrow><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>-</mo><mi>α</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><msub><mi>b</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><msup><mi href="./1.1#p2.t1.r2">z</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow></msup></mrow><mo>+</mo><mrow><msub><mi>b</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow></msub><mo></mo><msup><mi href="./1.1#p2.t1.r2">z</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>2</mn></mrow></msup></mrow><mo>+</mo><mi mathvariant="normal">…</mi><mo>+</mo><msub><mi>b</mi><mn>1</mn></msub></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>+</mo><msub><mi>b</mi><mn>0</mn></msub></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m131.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a> and
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m120.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m90.png" altimg-height="21px" altimg-valign="-5px" altimg-width="72px" alttext="b_{n}=a_{n}" display="inline"><mrow><msub><mi>b</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo>=</mo><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></mrow></math>,
</p>
<table id="E3" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.11.3</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E3.png" altimg-height="24px" altimg-valign="-7px" altimg-width="153px" alttext="b_{k}=\alpha b_{k+1}+a_{k}," display="block"><mrow><mrow><msub><mi>b</mi><mi href="./1.1#p2.t1.r4">k</mi></msub><mo>=</mo><mrow><mrow><mi>α</mi><mo></mo><msub><mi>b</mi><mrow><mi href="./1.1#p2.t1.r4">k</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><mo>+</mo><msub><mi>a</mi><mi href="./1.1#p2.t1.r4">k</mi></msub></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m117.png" altimg-height="21px" altimg-valign="-6px" altimg-width="198px" alttext="k=n-1,n-2,\dots,0" display="inline"><mrow><mi href="./1.1#p2.t1.r4">k</mi><mo>=</mo><mrow><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow><mo>,</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>2</mn></mrow><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><mn>0</mn></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.1#p2.t1.r4" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m119.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./1.1#p2.t1.r4">k</mi></math>: integer</a> and
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m120.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E4" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.11.4</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E4.png" altimg-height="25px" altimg-valign="-7px" altimg-width="96px" alttext="f(\alpha)=b_{0}." display="block"><mrow><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><msub><mi>b</mi><mn>0</mn></msub></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E4.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px2.p1" class="ltx_para">
<p class="ltx_p">With <math class="ltx_Math" altimg="m89.png" altimg-height="21px" altimg-valign="-5px" altimg-width="23px" alttext="b_{k}" display="inline"><msub><mi>b</mi><mi href="./1.1#p2.t1.r4">k</mi></msub></math> as in () let
<math class="ltx_Math" altimg="m92.png" altimg-height="16px" altimg-valign="-5px" altimg-width="72px" alttext="c_{n}=a_{n}" display="inline"><mrow><msub><mi>c</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo>=</mo><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></mrow></math> and
</p>
<table id="E5" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.11.5</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E5.png" altimg-height="24px" altimg-valign="-7px" altimg-width="151px" alttext="c_{k}=\alpha c_{k+1}+b_{k}," display="block"><mrow><mrow><msub><mi>c</mi><mi href="./1.1#p2.t1.r4">k</mi></msub><mo>=</mo><mrow><mrow><mi>α</mi><mo></mo><msub><mi>c</mi><mrow><mi href="./1.1#p2.t1.r4">k</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><mo>+</mo><msub><mi>b</mi><mi href="./1.1#p2.t1.r4">k</mi></msub></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m118.png" altimg-height="21px" altimg-valign="-6px" altimg-width="198px" alttext="k=n-1,n-2,\dots,1" display="inline"><mrow><mi href="./1.1#p2.t1.r4">k</mi><mo>=</mo><mrow><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow><mo>,</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>2</mn></mrow><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><mn>1</mn></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E5.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.1#p2.t1.r4" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m119.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./1.1#p2.t1.r4">k</mi></math>: integer</a> and
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m120.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Then
</p>
<table id="E6" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.11.6</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E6.png" altimg-height="26px" altimg-valign="-7px" altimg-width="102px" alttext="f^{\prime}(\alpha)=c_{1}." display="block"><mrow><mrow><mrow><msup><mi>f</mi><mo>′</mo></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><msub><mi>c</mi><mn>1</mn></msub></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E6.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="Px2.p2" class="ltx_para">
<p class="ltx_p">More generally, for polynomials <math class="ltx_Math" altimg="m103.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(z)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> and <math class="ltx_Math" altimg="m109.png" altimg-height="23px" altimg-valign="-7px" altimg-width="40px" alttext="g(z)" display="inline"><mrow><mi>g</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math>, there are polynomials <math class="ltx_Math" altimg="m123.png" altimg-height="23px" altimg-valign="-7px" altimg-width="40px" alttext="q(z)" display="inline"><mrow><mi href="./1.11#Px2.p2">q</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math>
and <math class="ltx_Math" altimg="m126.png" altimg-height="23px" altimg-valign="-7px" altimg-width="39px" alttext="r(z)" display="inline"><mrow><mi href="./1.11#Px2.p2">r</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math>, found by equating coefficients, such that</p>
<table id="E7" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.11.7</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E7.png" altimg-height="25px" altimg-valign="-7px" altimg-width="206px" alttext="f(z)=g(z)q(z)+r(z)," display="block"><mrow><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mrow><mi>g</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mi href="./1.11#Px2.p2">q</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mi href="./1.11#Px2.p2">r</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E7.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m131.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>,
<a href="./1.11#Px2.p2" title="Extended Horner Scheme ‣ §1.11(i) Division Algorithm ‣ §1.11 Zeros of Polynomials ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m123.png" altimg-height="23px" altimg-valign="-7px" altimg-width="40px" alttext="q(z)" display="inline"><mrow><mi href="./1.11#Px2.p2">q</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: polynomial</a> and
<a href="./1.11#Px2.p2" title="Extended Horner Scheme ‣ §1.11(i) Division Algorithm ‣ §1.11 Zeros of Polynomials ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m126.png" altimg-height="23px" altimg-valign="-7px" altimg-width="39px" alttext="r(z)" display="inline"><mrow><mi href="./1.11#Px2.p2">r</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: polynomial</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m46.png" altimg-height="23px" altimg-valign="-7px" altimg-width="205px" alttext="0\leq\deg r(z)<\deg g(z)" display="inline"><mrow><mn>0</mn><mo>≤</mo><mrow><mi>deg</mi><mo></mo><mrow><mi href="./1.11#Px2.p2">r</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo><</mo><mrow><mi>deg</mi><mo></mo><mrow><mi>g</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow></math>.</p>
</div>
</section>
</section>
<section id="ii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§1.11(ii) </span>Elementary Properties</h2>
<div id="SS2.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="SS2.p1" class="ltx_para">
<p class="ltx_p">A polynomial of degree <math class="ltx_Math" altimg="m120.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math> with real or complex coefficients has exactly <math class="ltx_Math" altimg="m120.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>
real or complex zeros counting multiplicity. Every <em class="ltx_emph ltx_font_italic">monic</em> (coefficient of
highest power is one) polynomial of odd degree with real coefficients has at
least one real zero with sign opposite to that of the constant term. A monic
polynomial of even degree with real coefficients has at least two zeros of
opposite signs when the constant term is negative.</p>
</div>
<section id="Px3" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Descartes’ Rule of Signs</h3>
<div id="Px3.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px3.p1" class="ltx_para">
<p class="ltx_p">The number of positive zeros of a polynomial with real coefficients cannot
exceed the number of times the coefficients change sign, and the two numbers
have same parity. A similar relation holds for the changes in sign of the
coefficients of <math class="ltx_Math" altimg="m96.png" altimg-height="23px" altimg-valign="-7px" altimg-width="57px" alttext="f(-z)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mi href="./1.1#p2.t1.r2">z</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></math>, and hence for the number of negative zeros of <math class="ltx_Math" altimg="m103.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(z)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math>.</p>
</div>
</section>
<section id="Px4" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Example</h3>
<div id="Px4.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px4.p1" class="ltx_para">
<table id="E8" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex1" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">1.11.8</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m36.png" altimg-height="25px" altimg-valign="-7px" altimg-width="44px" alttext="\displaystyle f(z)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m24.png" altimg-height="27px" altimg-valign="-6px" altimg-width="184px" alttext="\displaystyle=z^{8}+10z^{3}+z-4," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><msup><mi href="./1.1#p2.t1.r2">z</mi><mn>8</mn></msup><mo>+</mo><mrow><mn>10</mn><mo></mo><msup><mi href="./1.1#p2.t1.r2">z</mi><mn>3</mn></msup></mrow><mo>+</mo><mi href="./1.1#p2.t1.r2">z</mi></mrow><mo>-</mo><mn>4</mn></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex2" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m35.png" altimg-height="25px" altimg-valign="-7px" altimg-width="59px" alttext="\displaystyle f(-z)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mi href="./1.1#p2.t1.r2">z</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m25.png" altimg-height="25px" altimg-valign="-4px" altimg-width="184px" alttext="\displaystyle=z^{8}-10z^{3}-z-4." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msup><mi href="./1.1#p2.t1.r2">z</mi><mn>8</mn></msup><mo>-</mo><mrow><mn>10</mn><mo></mo><msup><mi href="./1.1#p2.t1.r2">z</mi><mn>3</mn></msup></mrow><mo>-</mo><mi href="./1.1#p2.t1.r2">z</mi><mo>-</mo><mn>4</mn></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E8.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m131.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a></dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Both polynomials have one change of sign; hence for each polynomial there is
one positive zero, one negative zero, and six complex zeros.</p>
</div>
<div id="Px4.p2" class="ltx_para">
<p class="ltx_p">Next, let <math class="ltx_Math" altimg="m98.png" altimg-height="25px" altimg-valign="-7px" altimg-width="310px" alttext="f(z)=a_{n}z^{n}+a_{n-1}z^{n-1}+\dots+a_{0}" display="inline"><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mrow><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><msup><mi href="./1.1#p2.t1.r2">z</mi><mi href="./1.1#p2.t1.r5">n</mi></msup></mrow><mo>+</mo><mrow><msub><mi>a</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow></msub><mo></mo><msup><mi href="./1.1#p2.t1.r2">z</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow></msup></mrow><mo>+</mo><mi mathvariant="normal">…</mi><mo>+</mo><msub><mi>a</mi><mn>0</mn></msub></mrow></mrow></math>. The zeros of
<math class="ltx_Math" altimg="m134.png" altimg-height="25px" altimg-valign="-7px" altimg-width="328px" alttext="z^{n}f(1/z)=a_{0}z^{n}+a_{1}z^{n-1}+\dots+a_{n}" display="inline"><mrow><mrow><msup><mi href="./1.1#p2.t1.r2">z</mi><mi href="./1.1#p2.t1.r5">n</mi></msup><mo></mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>/</mo><mi href="./1.1#p2.t1.r2">z</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>=</mo><mrow><mrow><msub><mi>a</mi><mn>0</mn></msub><mo></mo><msup><mi href="./1.1#p2.t1.r2">z</mi><mi href="./1.1#p2.t1.r5">n</mi></msup></mrow><mo>+</mo><mrow><msub><mi>a</mi><mn>1</mn></msub><mo></mo><msup><mi href="./1.1#p2.t1.r2">z</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow></msup></mrow><mo>+</mo><mi mathvariant="normal">…</mi><mo>+</mo><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></mrow></mrow></math> are reciprocals of the zeros
of <math class="ltx_Math" altimg="m103.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(z)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math>.</p>
</div>
<div id="Px4.p3" class="ltx_para">
<p class="ltx_p">The <em class="ltx_emph ltx_font_italic">discriminant</em> of <math class="ltx_Math" altimg="m103.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(z)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> is defined by
</p>
<table id="E9" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.11.9</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E9.png" altimg-height="55px" altimg-valign="-31px" altimg-width="222px" alttext="D=a_{n}^{2n-2}\prod_{j<k}(z_{j}-z_{k})^{2}," display="block"><mrow><mrow><mi href="./1.11#E9">D</mi><mo>=</mo><mrow><msubsup><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./1.1#p2.t1.r5">n</mi></mrow><mo>-</mo><mn>2</mn></mrow></msubsup><mo></mo><mrow><munder><mo largeop="true" movablelimits="false" symmetric="true">∏</mo><mrow><mi href="./1.1#p2.t1.r4">j</mi><mo><</mo><mi href="./1.1#p2.t1.r4">k</mi></mrow></munder><msup><mrow><mo stretchy="false">(</mo><mrow><msub><mi href="./1.1#p2.t1.r2">z</mi><mi href="./1.1#p2.t1.r4">j</mi></msub><mo>-</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mi href="./1.1#p2.t1.r4">k</mi></msub></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E9.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text"><math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.11#E9">D</mi></math>: discriminant of <math class="ltx_Math" altimg="m103.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(z)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> (locally)</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m131.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>,
<a href="./1.1#p2.t1.r4" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m113.png" altimg-height="20px" altimg-valign="-6px" altimg-width="14px" alttext="j" display="inline"><mi href="./1.1#p2.t1.r4">j</mi></math>: integer</a>,
<a href="./1.1#p2.t1.r4" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m119.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./1.1#p2.t1.r4">k</mi></math>: integer</a> and
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m120.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m135.png" altimg-height="16px" altimg-valign="-6px" altimg-width="114px" alttext="z_{1},z_{2},\dots,z_{n}" display="inline"><mrow><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>1</mn></msub><mo>,</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>2</mn></msub><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></mrow></math> are the zeros of <math class="ltx_Math" altimg="m103.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(z)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math>. The <em class="ltx_emph ltx_font_italic">elementary
symmetric functions</em>
of the zeros are (with <math class="ltx_Math" altimg="m87.png" altimg-height="21px" altimg-valign="-6px" altimg-width="62px" alttext="a_{n}\not=0" display="inline"><mrow><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo>≠</mo><mn>0</mn></mrow></math>)
</p>
</div>
<div id="Px4.p4" class="ltx_para">
<table id="E10" class="ltx_equationgroup ltx_eqn_table">
<tr id="E10X" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="4" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">1.11.10</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m41.png" altimg-height="20px" altimg-valign="-5px" altimg-width="159px" alttext="\displaystyle z_{1}+z_{2}+\dots+z_{n}" display="inline"><mrow><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>1</mn></msub><mo>+</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>2</mn></msub><mo>+</mo><mi mathvariant="normal">…</mi><mo>+</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m13.png" altimg-height="25px" altimg-valign="-7px" altimg-width="121px" alttext="\displaystyle=-a_{n-1}/a_{n}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo>-</mo><mrow><msub><mi>a</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow></msub><mo>/</mo><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="E10Xa" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m34.png" altimg-height="55px" altimg-valign="-31px" altimg-width="122px" alttext="\displaystyle\sum_{1\leq j<k\leq n}z_{j}z_{k}" display="inline"><mrow><mstyle displaystyle="true"><munder><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mn>1</mn><mo>≤</mo><mi href="./1.1#p2.t1.r4">j</mi><mo><</mo><mi href="./1.1#p2.t1.r4">k</mi><mo>≤</mo><mi href="./1.1#p2.t1.r5">n</mi></mrow></munder></mstyle><mrow><msub><mi href="./1.1#p2.t1.r2">z</mi><mi href="./1.1#p2.t1.r4">j</mi></msub><mo></mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mi href="./1.1#p2.t1.r4">k</mi></msub></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m20.png" altimg-height="25px" altimg-valign="-7px" altimg-width="106px" alttext="\displaystyle=a_{n-2}/a_{n}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msub><mi>a</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>2</mn></mrow></msub><mo>/</mo><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="E10Xb" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_eqn_cell"></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="E10c.png" altimg-height="36px" altimg-valign="-2px" altimg-width="12px" alttext="\displaystyle\mathrel{\vdots}" display="inline"><mo>⋮</mo></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="E10Xc" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m42.png" altimg-height="17px" altimg-valign="-5px" altimg-width="92px" alttext="\displaystyle z_{1}z_{2}\cdots z_{n}" display="inline"><mrow><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>1</mn></msub><mo></mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>2</mn></msub><mo></mo><mi mathvariant="normal">⋯</mi><mo></mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m5.png" altimg-height="25px" altimg-valign="-7px" altimg-width="135px" alttext="\displaystyle=(-1)^{n}a_{0}/a_{n}." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./1.1#p2.t1.r5">n</mi></msup><mo></mo><msub><mi>a</mi><mn>0</mn></msub></mrow><mo>/</mo><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E10.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m131.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>,
<a href="./1.1#p2.t1.r4" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m113.png" altimg-height="20px" altimg-valign="-6px" altimg-width="14px" alttext="j" display="inline"><mi href="./1.1#p2.t1.r4">j</mi></math>: integer</a>,
<a href="./1.1#p2.t1.r4" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m119.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./1.1#p2.t1.r4">k</mi></math>: integer</a> and
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m120.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
</section>
<section id="iii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§1.11(iii) </span>Polynomials of Degrees Two, Three, and Four</h2>
<div id="SS3.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px5.p1" class="ltx_para">
<p class="ltx_p">The roots of <math class="ltx_Math" altimg="m88.png" altimg-height="21px" altimg-valign="-4px" altimg-width="146px" alttext="az^{2}+bz+c=0" display="inline"><mrow><mrow><mrow><mi>a</mi><mo></mo><msup><mi href="./1.1#p2.t1.r2">z</mi><mn>2</mn></msup></mrow><mo>+</mo><mrow><mi>b</mi><mo></mo><mi href="./1.1#p2.t1.r2">z</mi></mrow><mo>+</mo><mi>c</mi></mrow><mo>=</mo><mn>0</mn></mrow></math> are</p>
<table id="E11" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex3" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">1.11.11</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center" colspan="2"><math class="ltx_Math" altimg="E11a.png" altimg-height="31px" altimg-valign="-9px" altimg-width="73px" alttext="\frac{-b\pm\sqrt{D}}{2a}," display="inline"><mrow><mstyle displaystyle="true"><mfrac><mrow><mrow><mo>-</mo><mi>b</mi></mrow><mo>±</mo><msqrt><mi href="./1.11#E12">D</mi></msqrt></mrow><mrow><mn>2</mn><mo></mo><mi>a</mi></mrow></mfrac></mstyle><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex4" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m28.png" altimg-height="19px" altimg-valign="-2px" altimg-width="23px" alttext="\displaystyle D" display="inline"><mi href="./1.11#E12">D</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m21.png" altimg-height="25px" altimg-valign="-4px" altimg-width="104px" alttext="\displaystyle=b^{2}-4ac." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msup><mi>b</mi><mn>2</mn></msup><mo>-</mo><mrow><mn>4</mn><mo></mo><mi>a</mi><mo></mo><mi>c</mi></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E11.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./1.11#E12" title="(1.11.12) ‣ Cubic Equations ‣ §1.11(iii) Polynomials of Degrees Two, Three, and Four ‣ §1.11 Zeros of Polynomials ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.11#E12">D</mi></math>: discriminant</a></dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">The sum and product of the roots are respectively <math class="ltx_Math" altimg="m45.png" altimg-height="23px" altimg-valign="-7px" altimg-width="49px" alttext="-b/a" display="inline"><mrow><mo>-</mo><mrow><mi>b</mi><mo>/</mo><mi>a</mi></mrow></mrow></math> and <math class="ltx_Math" altimg="m91.png" altimg-height="23px" altimg-valign="-7px" altimg-width="33px" alttext="c/a" display="inline"><mrow><mi>c</mi><mo>/</mo><mi>a</mi></mrow></math>.</p>
</div>
</section>
<section id="Px6" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Cubic Equations</h3>
<div id="Px6.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px6.p1" class="ltx_para">
<p class="ltx_p">Set <math class="ltx_Math" altimg="m129.png" altimg-height="27px" altimg-valign="-9px" altimg-width="103px" alttext="z=w-\tfrac{1}{3}a" display="inline"><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>=</mo><mrow><mi href="./1.1#p2.t1.r3">w</mi><mo>-</mo><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac><mo></mo><mi>a</mi></mrow></mrow></mrow></math> to reduce <math class="ltx_Math" altimg="m99.png" altimg-height="25px" altimg-valign="-7px" altimg-width="217px" alttext="f(z)=z^{3}+az^{2}+bz+c" display="inline"><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><msup><mi href="./1.1#p2.t1.r2">z</mi><mn>3</mn></msup><mo>+</mo><mrow><mi>a</mi><mo></mo><msup><mi href="./1.1#p2.t1.r2">z</mi><mn>2</mn></msup></mrow><mo>+</mo><mrow><mi>b</mi><mo></mo><mi href="./1.1#p2.t1.r2">z</mi></mrow><mo>+</mo><mi>c</mi></mrow></mrow></math> to
<math class="ltx_Math" altimg="m105.png" altimg-height="25px" altimg-valign="-7px" altimg-width="178px" alttext="g(w)=w^{3}+pw+q" display="inline"><mrow><mrow><mi>g</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r3">w</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><msup><mi href="./1.1#p2.t1.r3">w</mi><mn>3</mn></msup><mo>+</mo><mrow><mi href="./1.11#Px8.p1">p</mi><mo></mo><mi href="./1.1#p2.t1.r3">w</mi></mrow><mo>+</mo><mi href="./1.11#Px8.p1">q</mi></mrow></mrow></math>, with <math class="ltx_Math" altimg="m121.png" altimg-height="25px" altimg-valign="-7px" altimg-width="139px" alttext="p=(3b-a^{2})/3" display="inline"><mrow><mi href="./1.11#Px8.p1">p</mi><mo>=</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>3</mn><mo></mo><mi>b</mi></mrow><mo>-</mo><msup><mi>a</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mo>/</mo><mn>3</mn></mrow></mrow></math>, <math class="ltx_Math" altimg="m124.png" altimg-height="25px" altimg-valign="-7px" altimg-width="222px" alttext="q=(2a^{3}-9ab+27c)/27" display="inline"><mrow><mi href="./1.11#Px8.p1">q</mi><mo>=</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mrow><mn>2</mn><mo></mo><msup><mi>a</mi><mn>3</mn></msup></mrow><mo>-</mo><mrow><mn>9</mn><mo></mo><mi>a</mi><mo></mo><mi>b</mi></mrow></mrow><mo>+</mo><mrow><mn>27</mn><mo></mo><mi>c</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mo>/</mo><mn>27</mn></mrow></mrow></math>. The
discriminant of <math class="ltx_Math" altimg="m108.png" altimg-height="23px" altimg-valign="-7px" altimg-width="45px" alttext="g(w)" display="inline"><mrow><mi>g</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r3">w</mi><mo stretchy="false">)</mo></mrow></mrow></math> is</p>
<table id="E12" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.11.12</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E12.png" altimg-height="27px" altimg-valign="-6px" altimg-width="162px" alttext="D=-4p^{3}-27q^{2}." display="block"><mrow><mrow><mi href="./1.11#E12">D</mi><mo>=</mo><mrow><mrow><mo>-</mo><mrow><mn>4</mn><mo></mo><msup><mi href="./1.11#Px8.p1">p</mi><mn>3</mn></msup></mrow></mrow><mo>-</mo><mrow><mn>27</mn><mo></mo><msup><mi href="./1.11#Px8.p1">q</mi><mn>2</mn></msup></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E12.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text"><math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.11#E12">D</mi></math>: discriminant (locally)</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./1.11#Px8.p1" title="Quartic Equations ‣ §1.11(iii) Polynomials of Degrees Two, Three, and Four ‣ §1.11 Zeros of Polynomials ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m122.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="p" display="inline"><mi href="./1.11#Px8.p1">p</mi></math></a> and
<a href="./1.11#Px8.p1" title="Quartic Equations ‣ §1.11(iii) Polynomials of Degrees Two, Three, and Four ‣ §1.11 Zeros of Polynomials ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m125.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="q" display="inline"><mi href="./1.11#Px8.p1">q</mi></math></a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">3.8.1</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Let</p>
<table id="E13" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex5" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">1.11.13</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m26.png" altimg-height="19px" altimg-valign="-2px" altimg-width="21px" alttext="\displaystyle A" display="inline"><mi href="./1.11#E13">A</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m18.png" altimg-height="42px" altimg-valign="-12px" altimg-width="196px" alttext="\displaystyle=\sqrt[3]{-\tfrac{27}{2}q+\tfrac{3}{2}\sqrt{-3D}}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mroot><mrow><mrow><mo>-</mo><mrow><mfrac><mn>27</mn><mn>2</mn></mfrac><mo></mo><mi href="./1.11#Px8.p1">q</mi></mrow></mrow><mo>+</mo><mrow><mfrac><mn>3</mn><mn>2</mn></mfrac><mo></mo><msqrt><mrow><mo>-</mo><mrow><mn>3</mn><mo></mo><mi href="./1.11#E12">D</mi></mrow></mrow></msqrt></mrow></mrow><mn>3</mn></mroot></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex6" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m27.png" altimg-height="19px" altimg-valign="-2px" altimg-width="22px" alttext="\displaystyle B" display="inline"><mi href="./1.11#E13">B</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m9.png" altimg-height="25px" altimg-valign="-7px" altimg-width="92px" alttext="\displaystyle=-3p/A." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo>-</mo><mrow><mrow><mn>3</mn><mo></mo><mi href="./1.11#Px8.p1">p</mi></mrow><mo>/</mo><mi href="./1.11#E13">A</mi></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E13.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd>
<span class="ltx_text"><math class="ltx_Math" altimg="m56.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="A" display="inline"><mi href="./1.11#E13">A</mi></math> (locally)</span> and
<span class="ltx_text"><math class="ltx_Math" altimg="m58.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="B" display="inline"><mi href="./1.11#E13">B</mi></math> (locally)</span>
</dd>
<dt>Symbols:</dt>
<dd>
<a href="./1.11#Px8.p1" title="Quartic Equations ‣ §1.11(iii) Polynomials of Degrees Two, Three, and Four ‣ §1.11 Zeros of Polynomials ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m122.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="p" display="inline"><mi href="./1.11#Px8.p1">p</mi></math></a>,
<a href="./1.11#Px8.p1" title="Quartic Equations ‣ §1.11(iii) Polynomials of Degrees Two, Three, and Four ‣ §1.11 Zeros of Polynomials ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m125.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="q" display="inline"><mi href="./1.11#Px8.p1">q</mi></math></a> and
<a href="./1.11#E12" title="(1.11.12) ‣ Cubic Equations ‣ §1.11(iii) Polynomials of Degrees Two, Three, and Four ‣ §1.11 Zeros of Polynomials ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.11#E12">D</mi></math>: discriminant</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="Px6.p2" class="ltx_para">
<p class="ltx_p">The roots of <math class="ltx_Math" altimg="m104.png" altimg-height="23px" altimg-valign="-7px" altimg-width="81px" alttext="g(w)=0" display="inline"><mrow><mrow><mi>g</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r3">w</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mn>0</mn></mrow></math> are</p>
<table id="E14" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex7" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="3" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">1.11.14</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E14a.png" altimg-height="27px" altimg-valign="-9px" altimg-width="93px" alttext="\tfrac{1}{3}(A+B)," display="inline"><mrow><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./1.11#E13">A</mi><mo>+</mo><mi href="./1.11#E13">B</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex8" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E14b.png" altimg-height="27px" altimg-valign="-9px" altimg-width="123px" alttext="\tfrac{1}{3}(\rho A+\rho^{2}B)," display="inline"><mrow><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mi href="./1.11#Px6.p2">ρ</mi><mo></mo><mi href="./1.11#E13">A</mi></mrow><mo>+</mo><mrow><msup><mi href="./1.11#Px6.p2">ρ</mi><mn>2</mn></msup><mo></mo><mi href="./1.11#E13">B</mi></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex9" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E14c.png" altimg-height="27px" altimg-valign="-9px" altimg-width="123px" alttext="\tfrac{1}{3}(\rho^{2}A+\rho B)," display="inline"><mrow><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><msup><mi href="./1.11#Px6.p2">ρ</mi><mn>2</mn></msup><mo></mo><mi href="./1.11#E13">A</mi></mrow><mo>+</mo><mrow><mi href="./1.11#Px6.p2">ρ</mi><mo></mo><mi href="./1.11#E13">B</mi></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E14.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.11#Px6.p2" title="Cubic Equations ‣ §1.11(iii) Polynomials of Degrees Two, Three, and Four ‣ §1.11 Zeros of Polynomials ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m71.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="\rho" display="inline"><mi href="./1.11#Px6.p2">ρ</mi></math></a>,
<a href="./1.11#E13" title="(1.11.13) ‣ Cubic Equations ‣ §1.11(iii) Polynomials of Degrees Two, Three, and Four ‣ §1.11 Zeros of Polynomials ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="A" display="inline"><mi href="./1.11#E13">A</mi></math></a> and
<a href="./1.11#E13" title="(1.11.13) ‣ Cubic Equations ‣ §1.11(iii) Polynomials of Degrees Two, Three, and Four ‣ §1.11 Zeros of Polynomials ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="B" display="inline"><mi href="./1.11#E13">B</mi></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">with
</p>
<table id="E15" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex10" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">1.11.15</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m32.png" altimg-height="18px" altimg-valign="-6px" altimg-width="16px" alttext="\displaystyle\rho" display="inline"><mi href="./1.11#Px6.p2">ρ</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m12.png" altimg-height="31px" altimg-valign="-9px" altimg-width="216px" alttext="\displaystyle=-\tfrac{1}{2}+\tfrac{1}{2}\sqrt{-3}=e^{2\pi i/3}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>+</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><msqrt><mrow><mo>-</mo><mn>3</mn></mrow></msqrt></mrow></mrow><mo>=</mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi></mrow><mo>/</mo><mn>3</mn></mrow></msup></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex11" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m33.png" altimg-height="27px" altimg-valign="-6px" altimg-width="25px" alttext="\displaystyle\rho^{2}" display="inline"><msup><mi href="./1.11#Px6.p2">ρ</mi><mn>2</mn></msup></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m22.png" altimg-height="24px" altimg-valign="-2px" altimg-width="95px" alttext="\displaystyle=e^{-2\pi i/3}." display="inline"><mrow><mrow><mi></mi><mo>=</mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi></mrow><mo>/</mo><mn>3</mn></mrow></mrow></msup></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E15.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m68.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a> and
<a href="./1.11#Px6.p2" title="Cubic Equations ‣ §1.11(iii) Polynomials of Degrees Two, Three, and Four ‣ §1.11 Zeros of Polynomials ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m71.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="\rho" display="inline"><mi href="./1.11#Px6.p2">ρ</mi></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Addition of <math class="ltx_Math" altimg="m43.png" altimg-height="27px" altimg-valign="-9px" altimg-width="43px" alttext="-\frac{1}{3}a" display="inline"><mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac><mo></mo><mi>a</mi></mrow></mrow></math> to each of these roots gives the roots of
<math class="ltx_Math" altimg="m97.png" altimg-height="23px" altimg-valign="-7px" altimg-width="78px" alttext="f(z)=0" display="inline"><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mn>0</mn></mrow></math>.</p>
</div>
</section>
<section id="Px7" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Example</h3>
<div id="Px7.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px7.p1" class="ltx_para">
<p class="ltx_p"><math class="ltx_Math" altimg="m100.png" altimg-height="25px" altimg-valign="-7px" altimg-width="220px" alttext="f(z)=z^{3}-6z^{2}+6z-2" display="inline"><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mrow><msup><mi href="./1.1#p2.t1.r2">z</mi><mn>3</mn></msup><mo>-</mo><mrow><mn>6</mn><mo></mo><msup><mi href="./1.1#p2.t1.r2">z</mi><mn>2</mn></msup></mrow></mrow><mo>+</mo><mrow><mn>6</mn><mo></mo><mi href="./1.1#p2.t1.r2">z</mi></mrow></mrow><mo>-</mo><mn>2</mn></mrow></mrow></math>, <math class="ltx_Math" altimg="m106.png" altimg-height="25px" altimg-valign="-7px" altimg-width="179px" alttext="g(w)=w^{3}-6w-6" display="inline"><mrow><mrow><mi>g</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r3">w</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><msup><mi href="./1.1#p2.t1.r3">w</mi><mn>3</mn></msup><mo>-</mo><mrow><mn>6</mn><mo></mo><mi href="./1.1#p2.t1.r3">w</mi></mrow><mo>-</mo><mn>6</mn></mrow></mrow></math>, <math class="ltx_Math" altimg="m55.png" altimg-height="24px" altimg-valign="-5px" altimg-width="84px" alttext="A=3\sqrt[3]{4}" display="inline"><mrow><mi href="./1.11#E13">A</mi><mo>=</mo><mrow><mn>3</mn><mo></mo><mroot><mn>4</mn><mn>3</mn></mroot></mrow></mrow></math>,
<math class="ltx_Math" altimg="m57.png" altimg-height="24px" altimg-valign="-5px" altimg-width="85px" alttext="B=3\sqrt[3]{2}" display="inline"><mrow><mi href="./1.11#E13">B</mi><mo>=</mo><mrow><mn>3</mn><mo></mo><mroot><mn>2</mn><mn>3</mn></mroot></mrow></mrow></math>. Roots of <math class="ltx_Math" altimg="m97.png" altimg-height="23px" altimg-valign="-7px" altimg-width="78px" alttext="f(z)=0" display="inline"><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mn>0</mn></mrow></math> are <math class="ltx_Math" altimg="m48.png" altimg-height="24px" altimg-valign="-5px" altimg-width="119px" alttext="2+\sqrt[3]{4}+\sqrt[3]{2}" display="inline"><mrow><mn>2</mn><mo>+</mo><mroot><mn>4</mn><mn>3</mn></mroot><mo>+</mo><mroot><mn>2</mn><mn>3</mn></mroot></mrow></math>,
<math class="ltx_Math" altimg="m49.png" altimg-height="26px" altimg-valign="-6px" altimg-width="148px" alttext="2+\sqrt[3]{4}\rho+\sqrt[3]{2}\rho^{2}" display="inline"><mrow><mn>2</mn><mo>+</mo><mrow><mroot><mn>4</mn><mn>3</mn></mroot><mo></mo><mi href="./1.11#Px6.p2">ρ</mi></mrow><mo>+</mo><mrow><mroot><mn>2</mn><mn>3</mn></mroot><mo></mo><msup><mi href="./1.11#Px6.p2">ρ</mi><mn>2</mn></msup></mrow></mrow></math>,
<math class="ltx_Math" altimg="m50.png" altimg-height="26px" altimg-valign="-6px" altimg-width="148px" alttext="2+\sqrt[3]{4}\rho^{2}+\sqrt[3]{2}\rho" display="inline"><mrow><mn>2</mn><mo>+</mo><mrow><mroot><mn>4</mn><mn>3</mn></mroot><mo></mo><msup><mi href="./1.11#Px6.p2">ρ</mi><mn>2</mn></msup></mrow><mo>+</mo><mrow><mroot><mn>2</mn><mn>3</mn></mroot><mo></mo><mi href="./1.11#Px6.p2">ρ</mi></mrow></mrow></math>.</p>
</div>
<div id="Px7.p2" class="ltx_para">
<p class="ltx_p">For another method see §</dd>
</dl>
</div>
</div>
<div id="Px8.p1" class="ltx_para">
<p class="ltx_p">Set <math class="ltx_Math" altimg="m130.png" altimg-height="27px" altimg-valign="-9px" altimg-width="103px" alttext="z=w-\tfrac{1}{4}a" display="inline"><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>=</mo><mrow><mi href="./1.1#p2.t1.r3">w</mi><mo>-</mo><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac><mo></mo><mi>a</mi></mrow></mrow></mrow></math> to reduce <math class="ltx_Math" altimg="m101.png" altimg-height="25px" altimg-valign="-7px" altimg-width="271px" alttext="f(z)=z^{4}+az^{3}+bz^{2}+cz+d" display="inline"><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><msup><mi href="./1.1#p2.t1.r2">z</mi><mn>4</mn></msup><mo>+</mo><mrow><mi>a</mi><mo></mo><msup><mi href="./1.1#p2.t1.r2">z</mi><mn>3</mn></msup></mrow><mo>+</mo><mrow><mi>b</mi><mo></mo><msup><mi href="./1.1#p2.t1.r2">z</mi><mn>2</mn></msup></mrow><mo>+</mo><mrow><mi>c</mi><mo></mo><mi href="./1.1#p2.t1.r2">z</mi></mrow><mo>+</mo><mi>d</mi></mrow></mrow></math> to
</p>
<table id="E16" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex12" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="4" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">1.11.16</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m37.png" altimg-height="25px" altimg-valign="-7px" altimg-width="47px" alttext="\displaystyle g(w)" display="inline"><mrow><mi>g</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r3">w</mi><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m23.png" altimg-height="27px" altimg-valign="-6px" altimg-width="196px" alttext="\displaystyle=w^{4}+pw^{2}+qw+r," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msup><mi href="./1.1#p2.t1.r3">w</mi><mn>4</mn></msup><mo>+</mo><mrow><mi href="./1.11#Px8.p1">p</mi><mo></mo><msup><mi href="./1.1#p2.t1.r3">w</mi><mn>2</mn></msup></mrow><mo>+</mo><mrow><mi href="./1.11#Px8.p1">q</mi><mo></mo><mi href="./1.1#p2.t1.r3">w</mi></mrow><mo>+</mo><mi href="./1.11#Px8.p1">r</mi></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex13" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m38.png" altimg-height="18px" altimg-valign="-6px" altimg-width="16px" alttext="\displaystyle p" display="inline"><mi href="./1.11#Px8.p1">p</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m6.png" altimg-height="28px" altimg-valign="-7px" altimg-width="156px" alttext="\displaystyle=(-3a^{2}+8b)/8," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mo>-</mo><mrow><mn>3</mn><mo></mo><msup><mi>a</mi><mn>2</mn></msup></mrow></mrow><mo>+</mo><mrow><mn>8</mn><mo></mo><mi>b</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mo>/</mo><mn>8</mn></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex14" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m39.png" altimg-height="18px" altimg-valign="-6px" altimg-width="16px" alttext="\displaystyle q" display="inline"><mi href="./1.11#Px8.p1">q</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m8.png" altimg-height="28px" altimg-valign="-7px" altimg-width="184px" alttext="\displaystyle=(a^{3}-4ab+8c)/8," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><msup><mi>a</mi><mn>3</mn></msup><mo>-</mo><mrow><mn>4</mn><mo></mo><mi>a</mi><mo></mo><mi>b</mi></mrow></mrow><mo>+</mo><mrow><mn>8</mn><mo></mo><mi>c</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mo>/</mo><mn>8</mn></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex15" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m40.png" altimg-height="14px" altimg-valign="-2px" altimg-width="16px" alttext="\displaystyle r" display="inline"><mi href="./1.11#Px8.p1">r</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m7.png" altimg-height="28px" altimg-valign="-7px" altimg-width="333px" alttext="\displaystyle=(-3a^{4}+16a^{2}b-64ac+256d)/256." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mrow><mrow><mo>-</mo><mrow><mn>3</mn><mo></mo><msup><mi>a</mi><mn>4</mn></msup></mrow></mrow><mo>+</mo><mrow><mn>16</mn><mo></mo><msup><mi>a</mi><mn>2</mn></msup><mo></mo><mi>b</mi></mrow></mrow><mo>-</mo><mrow><mn>64</mn><mo></mo><mi>a</mi><mo></mo><mi>c</mi></mrow></mrow><mo>+</mo><mrow><mn>256</mn><mo></mo><mi>d</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mo>/</mo><mn>256</mn></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E16.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.1#p2.t1.r3" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m128.png" altimg-height="13px" altimg-valign="-2px" altimg-width="19px" alttext="w" display="inline"><mi href="./1.1#p2.t1.r3">w</mi></math>: variable</a>,
<a href="./1.11#Px8.p1" title="Quartic Equations ‣ §1.11(iii) Polynomials of Degrees Two, Three, and Four ‣ §1.11 Zeros of Polynomials ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m122.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="p" display="inline"><mi href="./1.11#Px8.p1">p</mi></math></a>,
<a href="./1.11#Px8.p1" title="Quartic Equations ‣ §1.11(iii) Polynomials of Degrees Two, Three, and Four ‣ §1.11 Zeros of Polynomials ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m125.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="q" display="inline"><mi href="./1.11#Px8.p1">q</mi></math></a> and
<a href="./1.11#Px8.p1" title="Quartic Equations ‣ §1.11(iii) Polynomials of Degrees Two, Three, and Four ‣ §1.11 Zeros of Polynomials ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="r" display="inline"><mi href="./1.11#Px8.p1">r</mi></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="Px8.p2" class="ltx_para">
<p class="ltx_p">The discriminant of <math class="ltx_Math" altimg="m108.png" altimg-height="23px" altimg-valign="-7px" altimg-width="45px" alttext="g(w)" display="inline"><mrow><mi>g</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r3">w</mi><mo stretchy="false">)</mo></mrow></mrow></math> is</p>
<table id="E17" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.11.17</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E17.png" altimg-height="27px" altimg-valign="-6px" altimg-width="495px" alttext="D=16p^{4}r-4p^{3}q^{2}-128p^{2}r^{2}+144pq^{2}r-27q^{4}+256r^{3}." display="block"><mrow><mrow><mi href="./1.11#E17">D</mi><mo>=</mo><mrow><mrow><mrow><mrow><mrow><mn>16</mn><mo></mo><msup><mi href="./1.11#Px8.p1">p</mi><mn>4</mn></msup><mo></mo><mi href="./1.11#Px8.p1">r</mi></mrow><mo>-</mo><mrow><mn>4</mn><mo></mo><msup><mi href="./1.11#Px8.p1">p</mi><mn>3</mn></msup><mo></mo><msup><mi href="./1.11#Px8.p1">q</mi><mn>2</mn></msup></mrow><mo>-</mo><mrow><mn>128</mn><mo></mo><msup><mi href="./1.11#Px8.p1">p</mi><mn>2</mn></msup><mo></mo><msup><mi href="./1.11#Px8.p1">r</mi><mn>2</mn></msup></mrow></mrow><mo>+</mo><mrow><mn>144</mn><mo></mo><mi href="./1.11#Px8.p1">p</mi><mo></mo><msup><mi href="./1.11#Px8.p1">q</mi><mn>2</mn></msup><mo></mo><mi href="./1.11#Px8.p1">r</mi></mrow></mrow><mo>-</mo><mrow><mn>27</mn><mo></mo><msup><mi href="./1.11#Px8.p1">q</mi><mn>4</mn></msup></mrow></mrow><mo>+</mo><mrow><mn>256</mn><mo></mo><msup><mi href="./1.11#Px8.p1">r</mi><mn>3</mn></msup></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E17.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text"><math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.11#E17">D</mi></math>: discriminant (locally)</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./1.11#Px8.p1" title="Quartic Equations ‣ §1.11(iii) Polynomials of Degrees Two, Three, and Four ‣ §1.11 Zeros of Polynomials ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m122.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="p" display="inline"><mi href="./1.11#Px8.p1">p</mi></math></a>,
<a href="./1.11#Px8.p1" title="Quartic Equations ‣ §1.11(iii) Polynomials of Degrees Two, Three, and Four ‣ §1.11 Zeros of Polynomials ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m125.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="q" display="inline"><mi href="./1.11#Px8.p1">q</mi></math></a> and
<a href="./1.11#Px8.p1" title="Quartic Equations ‣ §1.11(iii) Polynomials of Degrees Two, Three, and Four ‣ §1.11 Zeros of Polynomials ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="r" display="inline"><mi href="./1.11#Px8.p1">r</mi></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">For the roots <math class="ltx_Math" altimg="m65.png" altimg-height="16px" altimg-valign="-6px" altimg-width="118px" alttext="\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}" display="inline"><mrow><msub><mi>α</mi><mn>1</mn></msub><mo>,</mo><msub><mi>α</mi><mn>2</mn></msub><mo>,</mo><msub><mi>α</mi><mn>3</mn></msub><mo>,</mo><msub><mi>α</mi><mn>4</mn></msub></mrow></math> of <math class="ltx_Math" altimg="m104.png" altimg-height="23px" altimg-valign="-7px" altimg-width="81px" alttext="g(w)=0" display="inline"><mrow><mrow><mi>g</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r3">w</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mn>0</mn></mrow></math> and the
roots <math class="ltx_Math" altimg="m78.png" altimg-height="21px" altimg-valign="-6px" altimg-width="77px" alttext="\theta_{1},\theta_{2},\theta_{3}" display="inline"><mrow><msub><mi href="./1.11#E19">θ</mi><mn>1</mn></msub><mo>,</mo><msub><mi href="./1.11#E19">θ</mi><mn>2</mn></msub><mo>,</mo><msub><mi href="./1.11#E19">θ</mi><mn>3</mn></msub></mrow></math> of the <em class="ltx_emph ltx_font_italic">resolvent cubic equation</em>
</p>
<table id="E18" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.11.18</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E18.png" altimg-height="28px" altimg-valign="-7px" altimg-width="286px" alttext="z^{3}-2pz^{2}+(p^{2}-4r)z+q^{2}=0," display="block"><mrow><mrow><mrow><mrow><msup><mi href="./1.1#p2.t1.r2">z</mi><mn>3</mn></msup><mo>-</mo><mrow><mn>2</mn><mo></mo><mi href="./1.11#Px8.p1">p</mi><mo></mo><msup><mi href="./1.1#p2.t1.r2">z</mi><mn>2</mn></msup></mrow></mrow><mo>+</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><msup><mi href="./1.11#Px8.p1">p</mi><mn>2</mn></msup><mo>-</mo><mrow><mn>4</mn><mo></mo><mi href="./1.11#Px8.p1">r</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi href="./1.1#p2.t1.r2">z</mi></mrow><mo>+</mo><msup><mi href="./1.11#Px8.p1">q</mi><mn>2</mn></msup></mrow><mo>=</mo><mn>0</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E18.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m131.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>,
<a href="./1.11#Px8.p1" title="Quartic Equations ‣ §1.11(iii) Polynomials of Degrees Two, Three, and Four ‣ §1.11 Zeros of Polynomials ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m122.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="p" display="inline"><mi href="./1.11#Px8.p1">p</mi></math></a>,
<a href="./1.11#Px8.p1" title="Quartic Equations ‣ §1.11(iii) Polynomials of Degrees Two, Three, and Four ‣ §1.11 Zeros of Polynomials ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m125.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="q" display="inline"><mi href="./1.11#Px8.p1">q</mi></math></a> and
<a href="./1.11#Px8.p1" title="Quartic Equations ‣ §1.11(iii) Polynomials of Degrees Two, Three, and Four ‣ §1.11 Zeros of Polynomials ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="r" display="inline"><mi href="./1.11#Px8.p1">r</mi></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">we have</p>
<table id="E19" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex16" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="4" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">1.11.19</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m1.png" altimg-height="21px" altimg-valign="-5px" altimg-width="38px" alttext="\displaystyle 2\alpha_{1}" display="inline"><mrow><mn>2</mn><mo></mo><msub><mi>α</mi><mn>1</mn></msub></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m16.png" altimg-height="30px" altimg-valign="-7px" altimg-width="258px" alttext="\displaystyle=\phantom{+}\sqrt{-\theta_{1}}+\sqrt{-\theta_{2}}+\sqrt{-\theta_{%
3}}," display="inline"><mrow><mrow><mi></mi><mo rspace="10pt">=</mo><mrow><msqrt><mrow><mo>-</mo><msub><mi href="./1.11#E19">θ</mi><mn>1</mn></msub></mrow></msqrt><mo>+</mo><msqrt><mrow><mo>-</mo><msub><mi href="./1.11#E19">θ</mi><mn>2</mn></msub></mrow></msqrt><mo>+</mo><msqrt><mrow><mo>-</mo><msub><mi href="./1.11#E19">θ</mi><mn>3</mn></msub></mrow></msqrt></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex17" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m2.png" altimg-height="21px" altimg-valign="-5px" altimg-width="38px" alttext="\displaystyle 2\alpha_{2}" display="inline"><mrow><mn>2</mn><mo></mo><msub><mi>α</mi><mn>2</mn></msub></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m17.png" altimg-height="30px" altimg-valign="-7px" altimg-width="258px" alttext="\displaystyle=\phantom{+}\sqrt{-\theta_{1}}-\sqrt{-\theta_{2}}-\sqrt{-\theta_{%
3}}," display="inline"><mrow><mrow><mi></mi><mo rspace="10pt">=</mo><mrow><msqrt><mrow><mo>-</mo><msub><mi href="./1.11#E19">θ</mi><mn>1</mn></msub></mrow></msqrt><mo>-</mo><msqrt><mrow><mo>-</mo><msub><mi href="./1.11#E19">θ</mi><mn>2</mn></msub></mrow></msqrt><mo>-</mo><msqrt><mrow><mo>-</mo><msub><mi href="./1.11#E19">θ</mi><mn>3</mn></msub></mrow></msqrt></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex18" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m3.png" altimg-height="21px" altimg-valign="-5px" altimg-width="38px" alttext="\displaystyle 2\alpha_{3}" display="inline"><mrow><mn>2</mn><mo></mo><msub><mi>α</mi><mn>3</mn></msub></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m10.png" altimg-height="30px" altimg-valign="-7px" altimg-width="258px" alttext="\displaystyle=-\sqrt{-\theta_{1}}+\sqrt{-\theta_{2}}-\sqrt{-\theta_{3}}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mrow><mo>-</mo><msqrt><mrow><mo>-</mo><msub><mi href="./1.11#E19">θ</mi><mn>1</mn></msub></mrow></msqrt></mrow><mo>+</mo><msqrt><mrow><mo>-</mo><msub><mi href="./1.11#E19">θ</mi><mn>2</mn></msub></mrow></msqrt></mrow><mo>-</mo><msqrt><mrow><mo>-</mo><msub><mi href="./1.11#E19">θ</mi><mn>3</mn></msub></mrow></msqrt></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex19" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m4.png" altimg-height="21px" altimg-valign="-5px" altimg-width="38px" alttext="\displaystyle 2\alpha_{4}" display="inline"><mrow><mn>2</mn><mo></mo><msub><mi>α</mi><mn>4</mn></msub></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m11.png" altimg-height="30px" altimg-valign="-7px" altimg-width="258px" alttext="\displaystyle=-\sqrt{-\theta_{1}}-\sqrt{-\theta_{2}}+\sqrt{-\theta_{3}}." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mrow><mo>-</mo><msqrt><mrow><mo>-</mo><msub><mi href="./1.11#E19">θ</mi><mn>1</mn></msub></mrow></msqrt></mrow><mo>-</mo><msqrt><mrow><mo>-</mo><msub><mi href="./1.11#E19">θ</mi><mn>2</mn></msub></mrow></msqrt></mrow><mo>+</mo><msqrt><mrow><mo>-</mo><msub><mi href="./1.11#E19">θ</mi><mn>3</mn></msub></mrow></msqrt></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E19.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text"><math class="ltx_Math" altimg="m82.png" altimg-height="23px" altimg-valign="-8px" altimg-width="22px" alttext="\theta_{j}" display="inline"><msub><mi href="./1.11#E19">θ</mi><mi href="./1.1#p2.t1.r4">j</mi></msub></math>: cubic roots (locally)</span></dd>
<dt>Symbols:</dt>
<dd><a href="./1.1#p2.t1.r4" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m113.png" altimg-height="20px" altimg-valign="-6px" altimg-width="14px" alttext="j" display="inline"><mi href="./1.1#p2.t1.r4">j</mi></math>: integer</a></dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">The square roots are chosen so that</p>
<table id="E20" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.11.20</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E20.png" altimg-height="30px" altimg-valign="-7px" altimg-width="235px" alttext="\sqrt{-\theta_{1}}\;\sqrt{-\theta_{2}}\;\sqrt{-\theta_{3}}=-q." display="block"><mrow><mrow><mrow><mpadded width="+2.8pt"><msqrt><mrow><mo>-</mo><msub><mi href="./1.11#E19">θ</mi><mn>1</mn></msub></mrow></msqrt></mpadded><mo></mo><mpadded width="+2.8pt"><msqrt><mrow><mo>-</mo><msub><mi href="./1.11#E19">θ</mi><mn>2</mn></msub></mrow></msqrt></mpadded><mo></mo><msqrt><mrow><mo>-</mo><msub><mi href="./1.11#E19">θ</mi><mn>3</mn></msub></mrow></msqrt></mrow><mo>=</mo><mrow><mo>-</mo><mi href="./1.11#Px8.p1">q</mi></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E20.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.11#Px8.p1" title="Quartic Equations ‣ §1.11(iii) Polynomials of Degrees Two, Three, and Four ‣ §1.11 Zeros of Polynomials ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m125.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="q" display="inline"><mi href="./1.11#Px8.p1">q</mi></math></a> and
<a href="./1.11#E19" title="(1.11.19) ‣ Quartic Equations ‣ §1.11(iii) Polynomials of Degrees Two, Three, and Four ‣ §1.11 Zeros of Polynomials ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="23px" altimg-valign="-8px" altimg-width="22px" alttext="\theta_{j}" display="inline"><msub><mi href="./1.11#E19">θ</mi><mi href="./1.1#p2.t1.r4">j</mi></msub></math>: cubic roots</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Add <math class="ltx_Math" altimg="m44.png" altimg-height="27px" altimg-valign="-9px" altimg-width="43px" alttext="-\tfrac{1}{4}a" display="inline"><mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac><mo></mo><mi>a</mi></mrow></mrow></math> to the roots of <math class="ltx_Math" altimg="m104.png" altimg-height="23px" altimg-valign="-7px" altimg-width="81px" alttext="g(w)=0" display="inline"><mrow><mrow><mi>g</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r3">w</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mn>0</mn></mrow></math> to get those of <math class="ltx_Math" altimg="m97.png" altimg-height="23px" altimg-valign="-7px" altimg-width="78px" alttext="f(z)=0" display="inline"><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mn>0</mn></mrow></math>.</p>
</div>
</section>
<section id="Px9" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Example</h3>
<div id="Px9.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px9.p1" class="ltx_para">
<p class="ltx_p"><math class="ltx_Math" altimg="m102.png" altimg-height="25px" altimg-valign="-7px" altimg-width="220px" alttext="f(z)=z^{4}-4z^{3}+5z+2" display="inline"><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mrow><msup><mi href="./1.1#p2.t1.r2">z</mi><mn>4</mn></msup><mo>-</mo><mrow><mn>4</mn><mo></mo><msup><mi href="./1.1#p2.t1.r2">z</mi><mn>3</mn></msup></mrow></mrow><mo>+</mo><mrow><mn>5</mn><mo></mo><mi href="./1.1#p2.t1.r2">z</mi></mrow><mo>+</mo><mn>2</mn></mrow></mrow></math>, <math class="ltx_Math" altimg="m107.png" altimg-height="25px" altimg-valign="-7px" altimg-width="237px" alttext="g(w)=w^{4}-6w^{2}-3w+4" display="inline"><mrow><mrow><mi>g</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r3">w</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mrow><msup><mi href="./1.1#p2.t1.r3">w</mi><mn>4</mn></msup><mo>-</mo><mrow><mn>6</mn><mo></mo><msup><mi href="./1.1#p2.t1.r3">w</mi><mn>2</mn></msup></mrow><mo>-</mo><mrow><mn>3</mn><mo></mo><mi href="./1.1#p2.t1.r3">w</mi></mrow></mrow><mo>+</mo><mn>4</mn></mrow></mrow></math>. Resolvent cubic
is <math class="ltx_Math" altimg="m132.png" altimg-height="21px" altimg-valign="-4px" altimg-width="212px" alttext="z^{3}+12z^{2}+20z+9=0" display="inline"><mrow><mrow><msup><mi href="./1.1#p2.t1.r2">z</mi><mn>3</mn></msup><mo>+</mo><mrow><mn>12</mn><mo></mo><msup><mi href="./1.1#p2.t1.r2">z</mi><mn>2</mn></msup></mrow><mo>+</mo><mrow><mn>20</mn><mo></mo><mi href="./1.1#p2.t1.r2">z</mi></mrow><mo>+</mo><mn>9</mn></mrow><mo>=</mo><mn>0</mn></mrow></math> with roots <math class="ltx_Math" altimg="m79.png" altimg-height="21px" altimg-valign="-5px" altimg-width="75px" alttext="\theta_{1}=-1" display="inline"><mrow><msub><mi href="./1.11#E19">θ</mi><mn>1</mn></msub><mo>=</mo><mrow><mo>-</mo><mn>1</mn></mrow></mrow></math>, <math class="ltx_Math" altimg="m80.png" altimg-height="29px" altimg-valign="-9px" altimg-width="174px" alttext="\theta_{2}=-\tfrac{1}{2}(11+\sqrt{85})" display="inline"><mrow><msub><mi href="./1.11#E19">θ</mi><mn>2</mn></msub><mo>=</mo><mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mn>11</mn><mo>+</mo><msqrt><mn>85</mn></msqrt></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow></math>, <math class="ltx_Math" altimg="m81.png" altimg-height="29px" altimg-valign="-9px" altimg-width="174px" alttext="\theta_{3}=-\tfrac{1}{2}(11-\sqrt{85})" display="inline"><mrow><msub><mi href="./1.11#E19">θ</mi><mn>3</mn></msub><mo>=</mo><mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mn>11</mn><mo>-</mo><msqrt><mn>85</mn></msqrt></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow></math>, and
<math class="ltx_Math" altimg="m73.png" altimg-height="24px" altimg-valign="-6px" altimg-width="91px" alttext="\sqrt{-\theta_{1}}=1" display="inline"><mrow><msqrt><mrow><mo>-</mo><msub><mi href="./1.11#E19">θ</mi><mn>1</mn></msub></mrow></msqrt><mo>=</mo><mn>1</mn></mrow></math>, <math class="ltx_Math" altimg="m74.png" altimg-height="29px" altimg-valign="-9px" altimg-width="197px" alttext="\sqrt{-\theta_{2}}=\tfrac{1}{2}(\sqrt{17}+\sqrt{5})" display="inline"><mrow><msqrt><mrow><mo>-</mo><msub><mi href="./1.11#E19">θ</mi><mn>2</mn></msub></mrow></msqrt><mo>=</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mrow><mo stretchy="false">(</mo><mrow><msqrt><mn>17</mn></msqrt><mo>+</mo><msqrt><mn>5</mn></msqrt></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></math>,
<math class="ltx_Math" altimg="m75.png" altimg-height="29px" altimg-valign="-9px" altimg-width="197px" alttext="\sqrt{-\theta_{3}}=\tfrac{1}{2}(\sqrt{17}-\sqrt{5})" display="inline"><mrow><msqrt><mrow><mo>-</mo><msub><mi href="./1.11#E19">θ</mi><mn>3</mn></msub></mrow></msqrt><mo>=</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mrow><mo stretchy="false">(</mo><mrow><msqrt><mn>17</mn></msqrt><mo>-</mo><msqrt><mn>5</mn></msqrt></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></math>. So <math class="ltx_Math" altimg="m51.png" altimg-height="25px" altimg-valign="-5px" altimg-width="133px" alttext="2\alpha_{1}=1+\sqrt{17}" display="inline"><mrow><mrow><mn>2</mn><mo></mo><msub><mi>α</mi><mn>1</mn></msub></mrow><mo>=</mo><mrow><mn>1</mn><mo>+</mo><msqrt><mn>17</mn></msqrt></mrow></mrow></math>, <math class="ltx_Math" altimg="m52.png" altimg-height="25px" altimg-valign="-5px" altimg-width="133px" alttext="2\alpha_{2}=1-\sqrt{17}" display="inline"><mrow><mrow><mn>2</mn><mo></mo><msub><mi>α</mi><mn>2</mn></msub></mrow><mo>=</mo><mrow><mn>1</mn><mo>-</mo><msqrt><mn>17</mn></msqrt></mrow></mrow></math>, <math class="ltx_Math" altimg="m53.png" altimg-height="25px" altimg-valign="-5px" altimg-width="139px" alttext="2\alpha_{3}=-1+\sqrt{5}" display="inline"><mrow><mrow><mn>2</mn><mo></mo><msub><mi>α</mi><mn>3</mn></msub></mrow><mo>=</mo><mrow><mrow><mo>-</mo><mn>1</mn></mrow><mo>+</mo><msqrt><mn>5</mn></msqrt></mrow></mrow></math>, <math class="ltx_Math" altimg="m54.png" altimg-height="25px" altimg-valign="-5px" altimg-width="139px" alttext="2\alpha_{4}=-1-\sqrt{5}" display="inline"><mrow><mrow><mn>2</mn><mo></mo><msub><mi>α</mi><mn>4</mn></msub></mrow><mo>=</mo><mrow><mrow><mo>-</mo><mn>1</mn></mrow><mo>-</mo><msqrt><mn>5</mn></msqrt></mrow></mrow></math>, and the roots of <math class="ltx_Math" altimg="m97.png" altimg-height="23px" altimg-valign="-7px" altimg-width="78px" alttext="f(z)=0" display="inline"><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mn>0</mn></mrow></math> are <math class="ltx_Math" altimg="m77.png" altimg-height="29px" altimg-valign="-9px" altimg-width="103px" alttext="\tfrac{1}{2}(3\pm\sqrt{17})" display="inline"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mn>3</mn><mo>±</mo><msqrt><mn>17</mn></msqrt></mrow><mo stretchy="false">)</mo></mrow></mrow></math>,
<math class="ltx_Math" altimg="m76.png" altimg-height="29px" altimg-valign="-9px" altimg-width="93px" alttext="\tfrac{1}{2}(1\pm\sqrt{5})" display="inline"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>±</mo><msqrt><mn>5</mn></msqrt></mrow><mo stretchy="false">)</mo></mrow></mrow></math>.</p>
</div>
</section>
</section>
<section id="iv" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§1.11(iv) </span>Roots of Unity and of Other Constants</h2>
<div id="SS4.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="SS4.p1" class="ltx_para">
<p class="ltx_p">The roots of</p>
<table id="E21" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.11.21</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E21.png" altimg-height="28px" altimg-valign="-7px" altimg-width="423px" alttext="z^{n}-1=(z-1)(z^{n-1}+z^{n-2}+\dots+z+1)=0" display="block"><mrow><mrow><msup><mi href="./1.1#p2.t1.r2">z</mi><mi href="./1.1#p2.t1.r5">n</mi></msup><mo>-</mo><mn>1</mn></mrow><mo>=</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><msup><mi href="./1.1#p2.t1.r2">z</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow></msup><mo>+</mo><msup><mi href="./1.1#p2.t1.r2">z</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>2</mn></mrow></msup><mo>+</mo><mi mathvariant="normal">…</mi><mo>+</mo><mi href="./1.1#p2.t1.r2">z</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mn>0</mn></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E21.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m131.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a> and
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m120.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">are <math class="ltx_Math" altimg="m47.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="1" display="inline"><mn>1</mn></math>, <math class="ltx_Math" altimg="m93.png" altimg-height="21px" altimg-valign="-2px" altimg-width="56px" alttext="e^{2\pi i/n}" display="inline"><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi></mrow><mo>/</mo><mi href="./1.1#p2.t1.r5">n</mi></mrow></msup></math>, <math class="ltx_Math" altimg="m94.png" altimg-height="25px" altimg-valign="-6px" altimg-width="195px" alttext="e^{4\pi i/n},\dots,e^{(2n-2)\pi i/n}" display="inline"><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mrow><mn>4</mn><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi></mrow><mo>/</mo><mi href="./1.1#p2.t1.r5">n</mi></mrow></msup><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>2</mn><mo></mo><mi href="./1.1#p2.t1.r5">n</mi></mrow><mo>-</mo><mn>2</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi></mrow><mo>/</mo><mi href="./1.1#p2.t1.r5">n</mi></mrow></msup></mrow></math>, and of
<math class="ltx_Math" altimg="m133.png" altimg-height="19px" altimg-valign="-4px" altimg-width="96px" alttext="z^{n}+1=0" display="inline"><mrow><mrow><msup><mi href="./1.1#p2.t1.r2">z</mi><mi href="./1.1#p2.t1.r5">n</mi></msup><mo>+</mo><mn>1</mn></mrow><mo>=</mo><mn>0</mn></mrow></math> they are <math class="ltx_Math" altimg="m95.png" altimg-height="25px" altimg-valign="-6px" altimg-width="247px" alttext="e^{\pi i/n},e^{3\pi i/n},\dots,e^{(2n-1)\pi i/n}" display="inline"><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi></mrow><mo>/</mo><mi href="./1.1#p2.t1.r5">n</mi></mrow></msup><mo>,</mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mrow><mn>3</mn><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi></mrow><mo>/</mo><mi href="./1.1#p2.t1.r5">n</mi></mrow></msup><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>2</mn><mo></mo><mi href="./1.1#p2.t1.r5">n</mi></mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi></mrow><mo>/</mo><mi href="./1.1#p2.t1.r5">n</mi></mrow></msup></mrow></math>.</p>
</div>
<div id="SS4.p2" class="ltx_para">
<p class="ltx_p">The roots of</p>
<table id="E22" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.11.22</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E22.png" altimg-height="24px" altimg-valign="-6px" altimg-width="109px" alttext="z^{n}=a+ib," display="block"><mrow><mrow><msup><mi href="./1.1#p2.t1.r2">z</mi><mi href="./1.1#p2.t1.r5">n</mi></msup><mo>=</mo><mrow><mi>a</mi><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi>b</mi></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m83.png" altimg-height="21px" altimg-valign="-6px" altimg-width="32px" alttext="a,b" display="inline"><mrow><mi>a</mi><mo>,</mo><mi>b</mi></mrow></math> real,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E22.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m131.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a> and
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m120.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">are
</p>
<table id="E23" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.11.23</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E23.png" altimg-height="53px" altimg-valign="-21px" altimg-width="384px" alttext="\sqrt[n]{R}\left(\cos\left(\frac{\alpha+2k\pi}{n}\right)+i\sin\left(\frac{%
\alpha+2k\pi}{n}\right)\right)," display="block"><mrow><mrow><mroot><mi href="./1.11#SS4.p2">R</mi><mi href="./1.1#p2.t1.r5">n</mi></mroot><mo></mo><mrow><mo>(</mo><mrow><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mfrac><mrow><mi>α</mi><mo>+</mo><mrow><mn>2</mn><mo></mo><mi href="./1.1#p2.t1.r4">k</mi><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow><mi href="./1.1#p2.t1.r5">n</mi></mfrac><mo>)</mo></mrow></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mfrac><mrow><mi>α</mi><mo>+</mo><mrow><mn>2</mn><mo></mo><mi href="./1.1#p2.t1.r4">k</mi><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow><mi href="./1.1#p2.t1.r5">n</mi></mfrac><mo>)</mo></mrow></mrow></mrow></mrow><mo>)</mo></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E23.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./1.1#p2.t1.r4" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m119.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./1.1#p2.t1.r4">k</mi></math>: integer</a>,
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m120.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./1.11#SS4.p2" title="§1.11(iv) Roots of Unity and of Other Constants ‣ §1.11 Zeros of Polynomials ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="R" display="inline"><mi href="./1.11#SS4.p2">R</mi></math>: radius</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m62.png" altimg-height="26px" altimg-valign="-7px" altimg-width="148px" alttext="R=(a^{2}+b^{2})^{1/2}" display="inline"><mrow><mi href="./1.11#SS4.p2">R</mi><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></math>, <math class="ltx_Math" altimg="m64.png" altimg-height="23px" altimg-valign="-7px" altimg-width="135px" alttext="\alpha=\operatorname{ph}\left(a+ib\right)" display="inline"><mrow><mi>α</mi><mo>=</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mrow><mo>(</mo><mrow><mi>a</mi><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi>b</mi></mrow></mrow><mo>)</mo></mrow></mrow></mrow></math>, with the principal value
of phase (§), and <math class="ltx_Math" altimg="m115.png" altimg-height="21px" altimg-valign="-6px" altimg-width="161px" alttext="k=0,1,\dots,n-1" display="inline"><mrow><mi href="./1.1#p2.t1.r4">k</mi><mo>=</mo><mrow><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow></mrow></mrow></math>.</p>
</div>
</section>
<section id="v" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§1.11(v) </span>Stable Polynomials</h2>
<div id="SS5.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="SS5.p1" class="ltx_para">
<table id="E24" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.11.24</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E24.png" altimg-height="25px" altimg-valign="-7px" altimg-width="263px" alttext="f(z)=a_{0}+a_{1}z+\dots+a_{n}z^{n}," display="block"><mrow><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><msub><mi>a</mi><mn>0</mn></msub><mo>+</mo><mrow><msub><mi>a</mi><mn>1</mn></msub><mo></mo><mi href="./1.1#p2.t1.r2">z</mi></mrow><mo>+</mo><mi mathvariant="normal">…</mi><mo>+</mo><mrow><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><msup><mi href="./1.1#p2.t1.r2">z</mi><mi href="./1.1#p2.t1.r5">n</mi></msup></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E24.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m131.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a> and
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m120.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">with real coefficients, is called <em class="ltx_emph ltx_font_italic">stable</em> if the real parts of all the
zeros are strictly negative.
</p>
</div>
<section id="Px10" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Hurwitz Criterion</h3>
<div id="Px10.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px10.p1" class="ltx_para">
<p class="ltx_p">Let
</p>
<table id="E25" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex20" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="3" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">1.11.25</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m29.png" altimg-height="22px" altimg-valign="-5px" altimg-width="31px" alttext="\displaystyle D_{1}" display="inline"><msub><mi href="./1.11#Px10.p1">D</mi><mn>1</mn></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m19.png" altimg-height="18px" altimg-valign="-6px" altimg-width="52px" alttext="\displaystyle=a_{1}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><msub><mi>a</mi><mn>1</mn></msub></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex21" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m30.png" altimg-height="22px" altimg-valign="-5px" altimg-width="31px" alttext="\displaystyle D_{2}" display="inline"><msub><mi href="./1.11#Px10.p1">D</mi><mn>2</mn></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m15.png" altimg-height="53px" altimg-valign="-21px" altimg-width="108px" alttext="\displaystyle=\begin{vmatrix}a_{1}&a_{3}\\
a_{0}&a_{2}\end{vmatrix}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo href="./1.3#i">|</mo><mtable columnspacing="5pt" rowspacing="0pt"><mtr><mtd columnalign="center"><msub><mi>a</mi><mn>1</mn></msub></mtd><mtd columnalign="center"><msub><mi>a</mi><mn>3</mn></msub></mtd></mtr><mtr><mtd columnalign="center"><msub><mi>a</mi><mn>0</mn></msub></mtd><mtd columnalign="center"><msub><mi>a</mi><mn>2</mn></msub></mtd></mtr></mtable><mo href="./1.3#i">|</mo></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex22" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m31.png" altimg-height="22px" altimg-valign="-5px" altimg-width="31px" alttext="\displaystyle D_{3}" display="inline"><msub><mi href="./1.11#Px10.p1">D</mi><mn>3</mn></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m14.png" altimg-height="77px" altimg-valign="-33px" altimg-width="147px" alttext="\displaystyle=\begin{vmatrix}a_{1}&a_{3}&a_{5}\\
a_{0}&a_{2}&a_{4}\\
0&a_{1}&a_{3}\end{vmatrix}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo href="./1.3#i">|</mo><mtable columnspacing="5pt" rowspacing="0pt"><mtr><mtd columnalign="center"><msub><mi>a</mi><mn>1</mn></msub></mtd><mtd columnalign="center"><msub><mi>a</mi><mn>3</mn></msub></mtd><mtd columnalign="center"><msub><mi>a</mi><mn>5</mn></msub></mtd></mtr><mtr><mtd columnalign="center"><msub><mi>a</mi><mn>0</mn></msub></mtd><mtd columnalign="center"><msub><mi>a</mi><mn>2</mn></msub></mtd><mtd columnalign="center"><msub><mi>a</mi><mn>4</mn></msub></mtd></mtr><mtr><mtd columnalign="center"><mn>0</mn></mtd><mtd columnalign="center"><msub><mi>a</mi><mn>1</mn></msub></mtd><mtd columnalign="center"><msub><mi>a</mi><mn>3</mn></msub></mtd></mtr></mtable><mo href="./1.3#i">|</mo></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E25.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.3#i" title="§1.3(i) Definitions and Elementary Properties ‣ §1.3 Determinants ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m67.png" altimg-height="18px" altimg-valign="-2px" altimg-width="32px" alttext="\det" display="inline"><mo href="./1.3#i">det</mo></math>: determinant</a> and
<a href="./1.11#Px10.p1" title="Hurwitz Criterion ‣ §1.11(v) Stable Polynomials ‣ §1.11 Zeros of Polynomials ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m61.png" altimg-height="23px" altimg-valign="-8px" altimg-width="29px" alttext="D_{j}" display="inline"><msub><mi href="./1.11#Px10.p1">D</mi><mi href="./1.1#p2.t1.r4">j</mi></msub></math>: quantities</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and</p>
<table id="E26" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.11.26</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E26.png" altimg-height="32px" altimg-valign="-8px" altimg-width="284px" alttext="D_{k}=\det[h_{k}^{(1)},h_{k}^{(3)},\dots,h_{k}^{(2k-1)}]," display="block"><mrow><mrow><msub><mi href="./1.11#Px10.p1">D</mi><mi href="./1.1#p2.t1.r4">k</mi></msub><mo>=</mo><mrow><mo href="./1.3#i" movablelimits="false">det</mo><mo></mo><mrow><mo stretchy="false">[</mo><msubsup><mi>h</mi><mi href="./1.1#p2.t1.r4">k</mi><mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msubsup><mo>,</mo><msubsup><mi>h</mi><mi href="./1.1#p2.t1.r4">k</mi><mrow><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></mrow></msubsup><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><msubsup><mi>h</mi><mi href="./1.1#p2.t1.r4">k</mi><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>2</mn><mo></mo><mi href="./1.1#p2.t1.r4">k</mi></mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">]</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E26.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.3#i" title="§1.3(i) Definitions and Elementary Properties ‣ §1.3 Determinants ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m67.png" altimg-height="18px" altimg-valign="-2px" altimg-width="32px" alttext="\det" display="inline"><mo href="./1.3#i">det</mo></math>: determinant</a>,
<a href="./1.1#p2.t1.r4" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m119.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./1.1#p2.t1.r4">k</mi></math>: integer</a> and
<a href="./1.11#Px10.p1" title="Hurwitz Criterion ‣ §1.11(v) Stable Polynomials ‣ §1.11 Zeros of Polynomials ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m61.png" altimg-height="23px" altimg-valign="-8px" altimg-width="29px" alttext="D_{j}" display="inline"><msub><mi href="./1.11#Px10.p1">D</mi><mi href="./1.1#p2.t1.r4">j</mi></msub></math>: quantities</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where the column vector <math class="ltx_Math" altimg="m110.png" altimg-height="30px" altimg-valign="-8px" altimg-width="43px" alttext="h_{k}^{(m)}" display="inline"><msubsup><mi>h</mi><mi href="./1.1#p2.t1.r4">k</mi><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r5">m</mi><mo stretchy="false">)</mo></mrow></msubsup></math> consists of the first <math class="ltx_Math" altimg="m119.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./1.1#p2.t1.r4">k</mi></math> members of the
sequence <math class="ltx_Math" altimg="m86.png" altimg-height="17px" altimg-valign="-7px" altimg-width="172px" alttext="a_{m},a_{m-1},a_{m-2},\dots" display="inline"><mrow><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">m</mi></msub><mo>,</mo><msub><mi>a</mi><mrow><mi href="./1.1#p2.t1.r5">m</mi><mo>-</mo><mn>1</mn></mrow></msub><mo>,</mo><msub><mi>a</mi><mrow><mi href="./1.1#p2.t1.r5">m</mi><mo>-</mo><mn>2</mn></mrow></msub><mo>,</mo><mi mathvariant="normal">…</mi></mrow></math> with <math class="ltx_Math" altimg="m85.png" altimg-height="22px" altimg-valign="-8px" altimg-width="60px" alttext="a_{j}=0" display="inline"><mrow><msub><mi>a</mi><mi href="./1.1#p2.t1.r4">j</mi></msub><mo>=</mo><mn>0</mn></mrow></math> if <math class="ltx_Math" altimg="m111.png" altimg-height="20px" altimg-valign="-6px" altimg-width="50px" alttext="j<0" display="inline"><mrow><mi href="./1.1#p2.t1.r4">j</mi><mo><</mo><mn>0</mn></mrow></math> or <math class="ltx_Math" altimg="m112.png" altimg-height="20px" altimg-valign="-6px" altimg-width="52px" alttext="j>n" display="inline"><mrow><mi href="./1.1#p2.t1.r4">j</mi><mo>></mo><mi href="./1.1#p2.t1.r5">n</mi></mrow></math>.</p>
</div>
<div id="Px10.p2" class="ltx_para">
<p class="ltx_p">Then <math class="ltx_Math" altimg="m103.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(z)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math>, with <math class="ltx_Math" altimg="m87.png" altimg-height="21px" altimg-valign="-6px" altimg-width="62px" alttext="a_{n}\not=0" display="inline"><mrow><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo>≠</mo><mn>0</mn></mrow></math>, is stable iff <math class="ltx_Math" altimg="m84.png" altimg-height="21px" altimg-valign="-6px" altimg-width="60px" alttext="a_{0}\not=0" display="inline"><mrow><msub><mi>a</mi><mn>0</mn></msub><mo>≠</mo><mn>0</mn></mrow></math>; <math class="ltx_Math" altimg="m60.png" altimg-height="21px" altimg-valign="-5px" altimg-width="75px" alttext="D_{2k}>0" display="inline"><mrow><msub><mi href="./1.11#Px10.p1">D</mi><mrow><mn>2</mn><mo></mo><mi href="./1.1#p2.t1.r4">k</mi></mrow></msub><mo>></mo><mn>0</mn></mrow></math>,
<math class="ltx_Math" altimg="m116.png" altimg-height="27px" altimg-valign="-9px" altimg-width="140px" alttext="k=1,\dots,\left\lfloor\frac{1}{2}n\right\rfloor" display="inline"><mrow><mi href="./1.1#p2.t1.r4">k</mi><mo>=</mo><mrow><mn>1</mn><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r16">⌊</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./1.1#p2.t1.r5">n</mi></mrow><mo href="./front/introduction#Sx4.p1.t1.r16">⌋</mo></mrow></mrow></mrow></math>; <math class="ltx_Math" altimg="m69.png" altimg-height="22px" altimg-valign="-7px" altimg-width="180px" alttext="\operatorname{sign}D_{2k+1}=\operatorname{sign}a_{0}" display="inline"><mrow><mrow><mi href="./front/introduction#Sx4.p2.t1.r18">sign</mi><mo></mo><msub><mi href="./1.11#Px10.p1">D</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./1.1#p2.t1.r4">k</mi></mrow><mo>+</mo><mn>1</mn></mrow></msub></mrow><mo>=</mo><mrow><mi href="./front/introduction#Sx4.p2.t1.r18">sign</mi><mo></mo><msub><mi>a</mi><mn>0</mn></msub></mrow></mrow></math>,
<math class="ltx_Math" altimg="m114.png" altimg-height="27px" altimg-valign="-9px" altimg-width="195px" alttext="k=0,1,\dots,\left\lfloor\frac{1}{2}n-\frac{1}{2}\right\rfloor" display="inline"><mrow><mi href="./1.1#p2.t1.r4">k</mi><mo>=</mo><mrow><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r16">⌊</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./1.1#p2.t1.r5">n</mi></mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo href="./front/introduction#Sx4.p1.t1.r16">⌋</mo></mrow></mrow></mrow></math>.</p>
</div>
</section>
</section>
</section>
</div>
<div class="ltx_page_footer">
<span></div>
</div>
</body></text>
</html>
</page>
<page>
<!DOCTYPE html><html>
<head>
<title>DLMF: 1.12 Continued Fractions</title>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<!--GOOGLE BOOTSTRAP--></head>
<text><body class="color_default textfont_default titlefont_default fontsize_default navbar_default">
<div class="ltx_page_navbar">
<div class="ltx_page_navlogo"></dd>
</dl>
</div>
</div>
<div id="SS1.p1" class="ltx_para">
<p class="ltx_p">The notation used throughout
the DLMF for the continued fraction</p>
<table id="E1" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.12.1</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E1.png" altimg-height="95px" altimg-valign="-59px" altimg-width="160px" alttext="\cfracstyle{d}b_{0}+\cfrac{a_{1}}{b_{1}+\cfrac{a_{2}}{b_{2}+\raisebox{-6.0pt}{%
$\ddots$}}}" display="block"><mrow><msub><mi>b</mi><mn>0</mn></msub><mo>+</mo><mfrac><msub><mi>a</mi><mn>1</mn></msub><mrow><msub><mi>b</mi><mn>1</mn></msub><mo>+</mo><mstyle displaystyle="true"><mfrac><msub><mi>a</mi><mn>2</mn></msub><mrow><msub><mi>b</mi><mn>2</mn></msub><mo>+</mo><mi mathvariant="normal">⋱</mi></mrow></mfrac></mstyle></mrow></mfrac></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E1.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">is</p>
<table id="E2" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.12.2</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E2.png" altimg-height="55px" altimg-valign="-19px" altimg-width="173px" alttext="b_{0}+\cfrac{a_{1}}{b_{1}+\cfrac{a_{2}}{b_{2}+}}\cdots." display="block"><mrow><mrow><msub><mi>b</mi><mn>0</mn></msub><mo>+</mo><mrow><mrow><mfrac><msub><mi>a</mi><mn>1</mn></msub><mrow><msub><mi>b</mi><mn>1</mn></msub><mo>+</mo></mrow></mfrac><mfrac><msub><mi>a</mi><mn>2</mn></msub><mrow><msub><mi>b</mi><mn>2</mn></msub><mo>+</mo></mrow></mfrac></mrow><mo></mo><mi mathvariant="normal">⋯</mi></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E2.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="SS2.p1" class="ltx_para">
<table id="E3" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.12.3</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E3.png" altimg-height="55px" altimg-valign="-19px" altimg-width="230px" alttext="C=b_{0}+\cfrac{a_{1}}{b_{1}+\cfrac{a_{2}}{b_{2}+\cdots}}," display="block"><mrow><mrow><mi href="./1.12#E3">C</mi><mo>=</mo><mrow><msub><mi>b</mi><mn>0</mn></msub><mo>+</mo><mrow><mfrac><msub><mi>a</mi><mn>1</mn></msub><mrow><msub><mi>b</mi><mn>1</mn></msub><mo>+</mo></mrow></mfrac><mfrac><msub><mi>a</mi><mn>2</mn></msub><mrow><msub><mi>b</mi><mn>2</mn></msub><mo>+</mo></mrow></mfrac><mi mathvariant="normal">⋯</mi></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m54.png" altimg-height="21px" altimg-valign="-6px" altimg-width="62px" alttext="a_{n}\not=0" display="inline"><mrow><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo>≠</mo><mn>0</mn></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text"><math class="ltx_Math" altimg="m39.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.12#E3">C</mi></math>: continued fraction (locally)</span></dd>
<dt>Symbols:</dt>
<dd><a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a></dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">3.10.1</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E4" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.12.4</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E4.png" altimg-height="55px" altimg-valign="-19px" altimg-width="328px" alttext="C_{n}=b_{0}+\cfrac{a_{1}}{b_{1}+\cfrac{a_{2}}{b_{2}+\cdots\cfrac{a_{n}}{b_{n}}%
}}=\frac{A_{n}}{B_{n}}." display="block"><mrow><mrow><msub><mi href="./1.12#Px5.p1">C</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo>=</mo><mrow><msub><mi>b</mi><mn>0</mn></msub><mo>+</mo><mrow><mfrac><msub><mi>a</mi><mn>1</mn></msub><mrow><msub><mi>b</mi><mn>1</mn></msub><mo>+</mo></mrow></mfrac><mfrac><msub><mi>a</mi><mn>2</mn></msub><mrow><msub><mi>b</mi><mn>2</mn></msub><mo>+</mo></mrow></mfrac><mi mathvariant="normal">⋯</mi><mstyle displaystyle="true"><mfrac><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><msub><mi>b</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></mfrac></mstyle></mrow></mrow><mo>=</mo><mfrac><msub><mi href="./1.12#Px1">A</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><msub><mi href="./1.12#Px1">B</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></mfrac></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E4.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a>,
<a href="./1.12#Px1" title="Recurrence Relations ‣ §1.12(ii) Convergents ‣ §1.12 Continued Fractions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="21px" altimg-valign="-5px" altimg-width="30px" alttext="A_{n}" display="inline"><msub><mi href="./1.12#Px1">A</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></math>: <math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>th numerator</a>,
<a href="./1.12#Px1" title="Recurrence Relations ‣ §1.12(ii) Convergents ‣ §1.12 Continued Fractions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="21px" altimg-valign="-5px" altimg-width="30px" alttext="B_{n}" display="inline"><msub><mi href="./1.12#Px1">B</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></math>: <math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>th denominator</a> and
<a href="./1.12#Px5.p1" title="Fractional Transformations ‣ §1.12(ii) Convergents ‣ §1.12 Continued Fractions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="C_{n}(w)" display="inline"><mrow><msub><mi href="./1.12#Px5.p1">C</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r3">w</mi><mo stretchy="false">)</mo></mrow></mrow></math>: continued fraction</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">3.10.1</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p"><math class="ltx_Math" altimg="m44.png" altimg-height="21px" altimg-valign="-5px" altimg-width="29px" alttext="C_{n}" display="inline"><msub><mi href="./1.12#Px5.p1">C</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></math> is called the <math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>th <em class="ltx_emph ltx_font_italic">approximant</em>
or <em class="ltx_emph ltx_font_italic">convergent to <math class="ltx_Math" altimg="m39.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.12#E3">C</mi></math></em>. <math class="ltx_Math" altimg="m37.png" altimg-height="21px" altimg-valign="-5px" altimg-width="30px" alttext="A_{n}" display="inline"><msub><mi href="./1.12#Px1">A</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></math> and <math class="ltx_Math" altimg="m38.png" altimg-height="21px" altimg-valign="-5px" altimg-width="30px" alttext="B_{n}" display="inline"><msub><mi href="./1.12#Px1">B</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></math> are called the <math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>th
<em class="ltx_emph ltx_font_italic">(canonical) numerator</em> and <em class="ltx_emph ltx_font_italic">denominator</em> respectively.
</p>
</div>
<section id="Px1" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Recurrence Relations</h3>
<div id="Px1.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px1.p1" class="ltx_para">
<table id="E5" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex1" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="3" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">1.12.5</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m19.png" altimg-height="22px" altimg-valign="-5px" altimg-width="31px" alttext="\displaystyle A_{k}" display="inline"><msub><mi href="./1.12#Px1">A</mi><mi href="./1.1#p2.t1.r4">k</mi></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell">
<math class="ltx_Math" altimg="m15.png" altimg-height="24px" altimg-valign="-7px" altimg-width="180px" alttext="\displaystyle=b_{k}A_{k-1}+a_{k}A_{k-2}" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mrow><msub><mi>b</mi><mi href="./1.1#p2.t1.r4">k</mi></msub><mo></mo><msub><mi href="./1.12#Px1">A</mi><mrow><mi href="./1.1#p2.t1.r4">k</mi><mo>-</mo><mn>1</mn></mrow></msub></mrow><mo>+</mo><mrow><msub><mi>a</mi><mi href="./1.1#p2.t1.r4">k</mi></msub><mo></mo><msub><mi href="./1.12#Px1">A</mi><mrow><mi href="./1.1#p2.t1.r4">k</mi><mo>-</mo><mn>2</mn></mrow></msub></mrow></mrow></mrow></math>,</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex2" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m22.png" altimg-height="22px" altimg-valign="-5px" altimg-width="31px" alttext="\displaystyle B_{k}" display="inline"><msub><mi href="./1.12#Px1">B</mi><mi href="./1.1#p2.t1.r4">k</mi></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell">
<math class="ltx_Math" altimg="m16.png" altimg-height="24px" altimg-valign="-7px" altimg-width="181px" alttext="\displaystyle=b_{k}B_{k-1}+a_{k}B_{k-2}" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mrow><msub><mi>b</mi><mi href="./1.1#p2.t1.r4">k</mi></msub><mo></mo><msub><mi href="./1.12#Px1">B</mi><mrow><mi href="./1.1#p2.t1.r4">k</mi><mo>-</mo><mn>1</mn></mrow></msub></mrow><mo>+</mo><mrow><msub><mi>a</mi><mi href="./1.1#p2.t1.r4">k</mi></msub><mo></mo><msub><mi href="./1.12#Px1">B</mi><mrow><mi href="./1.1#p2.t1.r4">k</mi><mo>-</mo><mn>2</mn></mrow></msub></mrow></mrow></mrow></math>,
</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m63.png" altimg-height="21px" altimg-valign="-6px" altimg-width="122px" alttext="k=1,2,3,\dots" display="inline"><mrow><mi href="./1.1#p2.t1.r4">k</mi><mo>=</mo><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E5.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.1#p2.t1.r4" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m65.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./1.1#p2.t1.r4">k</mi></math>: integer</a>,
<a href="./1.12#Px1" title="Recurrence Relations ‣ §1.12(ii) Convergents ‣ §1.12 Continued Fractions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="21px" altimg-valign="-5px" altimg-width="30px" alttext="A_{n}" display="inline"><msub><mi href="./1.12#Px1">A</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></math>: <math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>th numerator</a> and
<a href="./1.12#Px1" title="Recurrence Relations ‣ §1.12(ii) Convergents ‣ §1.12 Continued Fractions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="21px" altimg-valign="-5px" altimg-width="30px" alttext="B_{n}" display="inline"><msub><mi href="./1.12#Px1">B</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></math>: <math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>th denominator</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">3.10.1</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E6" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex3" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="4" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">1.12.6</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m17.png" altimg-height="24px" altimg-valign="-7px" altimg-width="42px" alttext="\displaystyle A_{-1}" display="inline"><msub><mi href="./1.12#Px1">A</mi><mrow><mo>-</mo><mn>1</mn></mrow></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m2.png" altimg-height="22px" altimg-valign="-6px" altimg-width="43px" alttext="\displaystyle=1," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mn>1</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex4" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m18.png" altimg-height="22px" altimg-valign="-5px" altimg-width="30px" alttext="\displaystyle A_{0}" display="inline"><msub><mi href="./1.12#Px1">A</mi><mn>0</mn></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m14.png" altimg-height="23px" altimg-valign="-6px" altimg-width="50px" alttext="\displaystyle=b_{0}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><msub><mi>b</mi><mn>0</mn></msub></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex5" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m20.png" altimg-height="24px" altimg-valign="-7px" altimg-width="43px" alttext="\displaystyle B_{-1}" display="inline"><msub><mi href="./1.12#Px1">B</mi><mrow><mo>-</mo><mn>1</mn></mrow></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m1.png" altimg-height="22px" altimg-valign="-6px" altimg-width="43px" alttext="\displaystyle=0," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mn>0</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex6" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m21.png" altimg-height="22px" altimg-valign="-5px" altimg-width="30px" alttext="\displaystyle B_{0}" display="inline"><msub><mi href="./1.12#Px1">B</mi><mn>0</mn></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m3.png" altimg-height="18px" altimg-valign="-2px" altimg-width="43px" alttext="\displaystyle=1." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mn>1</mn></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E6.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.12#Px1" title="Recurrence Relations ‣ §1.12(ii) Convergents ‣ §1.12 Continued Fractions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="21px" altimg-valign="-5px" altimg-width="30px" alttext="A_{n}" display="inline"><msub><mi href="./1.12#Px1">A</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></math>: <math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>th numerator</a> and
<a href="./1.12#Px1" title="Recurrence Relations ‣ §1.12(ii) Convergents ‣ §1.12 Continued Fractions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="21px" altimg-valign="-5px" altimg-width="30px" alttext="B_{n}" display="inline"><msub><mi href="./1.12#Px1">B</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></math>: <math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>th denominator</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">3.10.1</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px2" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Determinant Formula</h3>
<div id="Px2.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px2.p1" class="ltx_para">
<table id="E7" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.12.7</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E7.png" altimg-height="64px" altimg-valign="-28px" altimg-width="335px" alttext="A_{n}B_{n-1}-B_{n}A_{n-1}=(-1)^{n-1}\prod^{n}_{k=1}a_{k}," display="block"><mrow><mrow><mrow><mrow><msub><mi href="./1.12#Px1">A</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><msub><mi href="./1.12#Px1">B</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow></msub></mrow><mo>-</mo><mrow><msub><mi href="./1.12#Px1">B</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><msub><mi href="./1.12#Px1">A</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow></msub></mrow></mrow><mo>=</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∏</mo><mrow><mi href="./1.1#p2.t1.r4">k</mi><mo>=</mo><mn>1</mn></mrow><mi href="./1.1#p2.t1.r5">n</mi></munderover><msub><mi>a</mi><mi href="./1.1#p2.t1.r4">k</mi></msub></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m66.png" altimg-height="20px" altimg-valign="-6px" altimg-width="123px" alttext="n=0,1,2,\dots" display="inline"><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mrow><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E7.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.1#p2.t1.r4" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m65.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./1.1#p2.t1.r4">k</mi></math>: integer</a>,
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a>,
<a href="./1.12#Px1" title="Recurrence Relations ‣ §1.12(ii) Convergents ‣ §1.12 Continued Fractions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="21px" altimg-valign="-5px" altimg-width="30px" alttext="A_{n}" display="inline"><msub><mi href="./1.12#Px1">A</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></math>: <math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>th numerator</a> and
<a href="./1.12#Px1" title="Recurrence Relations ‣ §1.12(ii) Convergents ‣ §1.12 Continued Fractions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="21px" altimg-valign="-5px" altimg-width="30px" alttext="B_{n}" display="inline"><msub><mi href="./1.12#Px1">B</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></math>: <math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>th denominator</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E8" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.12.8</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E8.png" altimg-height="55px" altimg-valign="-20px" altimg-width="286px" alttext="C_{n}-C_{n-1}=\frac{(-1)^{n-1}\prod^{n}_{k=1}a_{k}}{B_{n-1}B_{n}}," display="block"><mrow><mrow><mrow><msub><mi href="./1.12#Px5.p1">C</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo>-</mo><msub><mi href="./1.12#Px5.p1">C</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow></msub></mrow><mo>=</mo><mfrac><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mrow><msubsup><mo largeop="true" symmetric="true">∏</mo><mrow><mi href="./1.1#p2.t1.r4">k</mi><mo>=</mo><mn>1</mn></mrow><mi href="./1.1#p2.t1.r5">n</mi></msubsup><msub><mi>a</mi><mi href="./1.1#p2.t1.r4">k</mi></msub></mrow></mrow><mrow><msub><mi href="./1.12#Px1">B</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow></msub><mo></mo><msub><mi href="./1.12#Px1">B</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></mrow></mfrac></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m67.png" altimg-height="20px" altimg-valign="-6px" altimg-width="123px" alttext="n=1,2,3,\dots" display="inline"><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E8.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.1#p2.t1.r4" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m65.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./1.1#p2.t1.r4">k</mi></math>: integer</a>,
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a>,
<a href="./1.12#Px1" title="Recurrence Relations ‣ §1.12(ii) Convergents ‣ §1.12 Continued Fractions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="21px" altimg-valign="-5px" altimg-width="30px" alttext="B_{n}" display="inline"><msub><mi href="./1.12#Px1">B</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></math>: <math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>th denominator</a> and
<a href="./1.12#Px5.p1" title="Fractional Transformations ‣ §1.12(ii) Convergents ‣ §1.12 Continued Fractions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="C_{n}(w)" display="inline"><mrow><msub><mi href="./1.12#Px5.p1">C</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r3">w</mi><mo stretchy="false">)</mo></mrow></mrow></math>: continued fraction</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E9" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.12.9</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E9.png" altimg-height="54px" altimg-valign="-20px" altimg-width="379px" alttext="C_{n}=b_{0}+\frac{a_{1}}{B_{0}B_{1}}-\dots+(-1)^{n-1}\frac{\prod^{n}_{k=1}a_{k%
}}{B_{n-1}B_{n}}." display="block"><mrow><mrow><msub><mi href="./1.12#Px5.p1">C</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo>=</mo><mrow><mrow><mrow><msub><mi>b</mi><mn>0</mn></msub><mo>+</mo><mfrac><msub><mi>a</mi><mn>1</mn></msub><mrow><msub><mi href="./1.12#Px1">B</mi><mn>0</mn></msub><mo></mo><msub><mi href="./1.12#Px1">B</mi><mn>1</mn></msub></mrow></mfrac></mrow><mo>-</mo><mi mathvariant="normal">…</mi></mrow><mo>+</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mfrac><mrow><msubsup><mo largeop="true" symmetric="true">∏</mo><mrow><mi href="./1.1#p2.t1.r4">k</mi><mo>=</mo><mn>1</mn></mrow><mi href="./1.1#p2.t1.r5">n</mi></msubsup><msub><mi>a</mi><mi href="./1.1#p2.t1.r4">k</mi></msub></mrow><mrow><msub><mi href="./1.12#Px1">B</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow></msub><mo></mo><msub><mi href="./1.12#Px1">B</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></mrow></mfrac></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E9.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.1#p2.t1.r4" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m65.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./1.1#p2.t1.r4">k</mi></math>: integer</a>,
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a>,
<a href="./1.12#Px1" title="Recurrence Relations ‣ §1.12(ii) Convergents ‣ §1.12 Continued Fractions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="21px" altimg-valign="-5px" altimg-width="30px" alttext="B_{n}" display="inline"><msub><mi href="./1.12#Px1">B</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></math>: <math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>th denominator</a> and
<a href="./1.12#Px5.p1" title="Fractional Transformations ‣ §1.12(ii) Convergents ‣ §1.12 Continued Fractions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="C_{n}(w)" display="inline"><mrow><msub><mi href="./1.12#Px5.p1">C</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r3">w</mi><mo stretchy="false">)</mo></mrow></mrow></math>: continued fraction</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="EGx1" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E10">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.12.10</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m31.png" altimg-height="17px" altimg-valign="-5px" altimg-width="27px" alttext="\displaystyle a_{n}" display="inline"><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m9.png" altimg-height="51px" altimg-valign="-20px" altimg-width="247px" alttext="\displaystyle=\frac{A_{n-1}B_{n}-A_{n}B_{n-1}}{A_{n-1}B_{n-2}-A_{n-2}B_{n-1}}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mstyle displaystyle="true"><mfrac><mrow><mrow><msub><mi href="./1.12#Px1">A</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow></msub><mo></mo><msub><mi href="./1.12#Px1">B</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></mrow><mo>-</mo><mrow><msub><mi href="./1.12#Px1">A</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><msub><mi href="./1.12#Px1">B</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow></msub></mrow></mrow><mrow><mrow><msub><mi href="./1.12#Px1">A</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow></msub><mo></mo><msub><mi href="./1.12#Px1">B</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>2</mn></mrow></msub></mrow><mo>-</mo><mrow><msub><mi href="./1.12#Px1">A</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>2</mn></mrow></msub><mo></mo><msub><mi href="./1.12#Px1">B</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow></msub></mrow></mrow></mfrac></mstyle></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m67.png" altimg-height="20px" altimg-valign="-6px" altimg-width="123px" alttext="n=1,2,3,\dots" display="inline"><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E10.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a>,
<a href="./1.12#Px1" title="Recurrence Relations ‣ §1.12(ii) Convergents ‣ §1.12 Continued Fractions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="21px" altimg-valign="-5px" altimg-width="30px" alttext="A_{n}" display="inline"><msub><mi href="./1.12#Px1">A</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></math>: <math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>th numerator</a> and
<a href="./1.12#Px1" title="Recurrence Relations ‣ §1.12(ii) Convergents ‣ §1.12 Continued Fractions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="21px" altimg-valign="-5px" altimg-width="30px" alttext="B_{n}" display="inline"><msub><mi href="./1.12#Px1">B</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></math>: <math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>th denominator</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E11">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.12.11</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m31.png" altimg-height="17px" altimg-valign="-5px" altimg-width="27px" alttext="\displaystyle a_{n}" display="inline"><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m13.png" altimg-height="51px" altimg-valign="-20px" altimg-width="204px" alttext="\displaystyle=\frac{B_{n}}{B_{n-2}}\frac{C_{n-1}-C_{n}}{C_{n-1}-C_{n-2}}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><mfrac><msub><mi href="./1.12#Px1">B</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><msub><mi href="./1.12#Px1">B</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>2</mn></mrow></msub></mfrac></mstyle><mo></mo><mstyle displaystyle="true"><mfrac><mrow><msub><mi href="./1.12#Px5.p1">C</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow></msub><mo>-</mo><msub><mi href="./1.12#Px5.p1">C</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></mrow><mrow><msub><mi href="./1.12#Px5.p1">C</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow></msub><mo>-</mo><msub><mi href="./1.12#Px5.p1">C</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>2</mn></mrow></msub></mrow></mfrac></mstyle></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m68.png" altimg-height="20px" altimg-valign="-6px" altimg-width="123px" alttext="n=2,3,4,\dots" display="inline"><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mrow><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E11.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a>,
<a href="./1.12#Px1" title="Recurrence Relations ‣ §1.12(ii) Convergents ‣ §1.12 Continued Fractions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="21px" altimg-valign="-5px" altimg-width="30px" alttext="B_{n}" display="inline"><msub><mi href="./1.12#Px1">B</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></math>: <math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>th denominator</a> and
<a href="./1.12#Px5.p1" title="Fractional Transformations ‣ §1.12(ii) Convergents ‣ §1.12 Continued Fractions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="C_{n}(w)" display="inline"><mrow><msub><mi href="./1.12#Px5.p1">C</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r3">w</mi><mo stretchy="false">)</mo></mrow></mrow></math>: continued fraction</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
<table id="EGx2" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E12">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.12.12</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m34.png" altimg-height="22px" altimg-valign="-5px" altimg-width="25px" alttext="\displaystyle b_{n}" display="inline"><msub><mi>b</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m11.png" altimg-height="51px" altimg-valign="-20px" altimg-width="247px" alttext="\displaystyle=\frac{A_{n}B_{n-2}-A_{n-2}B_{n}}{A_{n-1}B_{n-2}-A_{n-2}B_{n-1}}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mstyle displaystyle="true"><mfrac><mrow><mrow><msub><mi href="./1.12#Px1">A</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><msub><mi href="./1.12#Px1">B</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>2</mn></mrow></msub></mrow><mo>-</mo><mrow><msub><mi href="./1.12#Px1">A</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>2</mn></mrow></msub><mo></mo><msub><mi href="./1.12#Px1">B</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></mrow></mrow><mrow><mrow><msub><mi href="./1.12#Px1">A</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow></msub><mo></mo><msub><mi href="./1.12#Px1">B</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>2</mn></mrow></msub></mrow><mo>-</mo><mrow><msub><mi href="./1.12#Px1">A</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>2</mn></mrow></msub><mo></mo><msub><mi href="./1.12#Px1">B</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow></msub></mrow></mrow></mfrac></mstyle></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m67.png" altimg-height="20px" altimg-valign="-6px" altimg-width="123px" alttext="n=1,2,3,\dots" display="inline"><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E12.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a>,
<a href="./1.12#Px1" title="Recurrence Relations ‣ §1.12(ii) Convergents ‣ §1.12 Continued Fractions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="21px" altimg-valign="-5px" altimg-width="30px" alttext="A_{n}" display="inline"><msub><mi href="./1.12#Px1">A</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></math>: <math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>th numerator</a> and
<a href="./1.12#Px1" title="Recurrence Relations ‣ §1.12(ii) Convergents ‣ §1.12 Continued Fractions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="21px" altimg-valign="-5px" altimg-width="30px" alttext="B_{n}" display="inline"><msub><mi href="./1.12#Px1">B</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></math>: <math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>th denominator</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E13">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.12.13</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m34.png" altimg-height="22px" altimg-valign="-5px" altimg-width="25px" alttext="\displaystyle b_{n}" display="inline"><msub><mi>b</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m12.png" altimg-height="51px" altimg-valign="-20px" altimg-width="204px" alttext="\displaystyle=\frac{B_{n}}{B_{n-1}}\frac{C_{n}-C_{n-2}}{C_{n-1}-C_{n-2}}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><mfrac><msub><mi href="./1.12#Px1">B</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><msub><mi href="./1.12#Px1">B</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow></msub></mfrac></mstyle><mo></mo><mstyle displaystyle="true"><mfrac><mrow><msub><mi href="./1.12#Px5.p1">C</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo>-</mo><msub><mi href="./1.12#Px5.p1">C</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>2</mn></mrow></msub></mrow><mrow><msub><mi href="./1.12#Px5.p1">C</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow></msub><mo>-</mo><msub><mi href="./1.12#Px5.p1">C</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>2</mn></mrow></msub></mrow></mfrac></mstyle></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m68.png" altimg-height="20px" altimg-valign="-6px" altimg-width="123px" alttext="n=2,3,4,\dots" display="inline"><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mrow><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E13.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a>,
<a href="./1.12#Px1" title="Recurrence Relations ‣ §1.12(ii) Convergents ‣ §1.12 Continued Fractions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="21px" altimg-valign="-5px" altimg-width="30px" alttext="B_{n}" display="inline"><msub><mi href="./1.12#Px1">B</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></math>: <math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>th denominator</a> and
<a href="./1.12#Px5.p1" title="Fractional Transformations ‣ §1.12(ii) Convergents ‣ §1.12 Continued Fractions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="C_{n}(w)" display="inline"><mrow><msub><mi href="./1.12#Px5.p1">C</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r3">w</mi><mo stretchy="false">)</mo></mrow></mrow></math>: continued fraction</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
<table id="E14" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex7" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="3" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">1.12.14</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m32.png" altimg-height="22px" altimg-valign="-5px" altimg-width="24px" alttext="\displaystyle b_{0}" display="inline"><msub><mi>b</mi><mn>0</mn></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m4.png" altimg-height="23px" altimg-valign="-6px" altimg-width="106px" alttext="\displaystyle=A_{0}=C_{0}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><msub><mi href="./1.12#Px1">A</mi><mn>0</mn></msub><mo>=</mo><msub><mi href="./1.12#Px5.p1">C</mi><mn>0</mn></msub></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex8" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m33.png" altimg-height="22px" altimg-valign="-5px" altimg-width="24px" alttext="\displaystyle b_{1}" display="inline"><msub><mi>b</mi><mn>1</mn></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m6.png" altimg-height="23px" altimg-valign="-6px" altimg-width="57px" alttext="\displaystyle=B_{1}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><msub><mi href="./1.12#Px1">B</mi><mn>1</mn></msub></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex9" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m30.png" altimg-height="17px" altimg-valign="-5px" altimg-width="26px" alttext="\displaystyle a_{1}" display="inline"><msub><mi>a</mi><mn>1</mn></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m5.png" altimg-height="22px" altimg-valign="-5px" altimg-width="129px" alttext="\displaystyle=A_{1}-A_{0}B_{1}." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msub><mi href="./1.12#Px1">A</mi><mn>1</mn></msub><mo>-</mo><mrow><msub><mi href="./1.12#Px1">A</mi><mn>0</mn></msub><mo></mo><msub><mi href="./1.12#Px1">B</mi><mn>1</mn></msub></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E14.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.12#Px1" title="Recurrence Relations ‣ §1.12(ii) Convergents ‣ §1.12 Continued Fractions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="21px" altimg-valign="-5px" altimg-width="30px" alttext="A_{n}" display="inline"><msub><mi href="./1.12#Px1">A</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></math>: <math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>th numerator</a>,
<a href="./1.12#Px1" title="Recurrence Relations ‣ §1.12(ii) Convergents ‣ §1.12 Continued Fractions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="21px" altimg-valign="-5px" altimg-width="30px" alttext="B_{n}" display="inline"><msub><mi href="./1.12#Px1">B</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></math>: <math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>th denominator</a> and
<a href="./1.12#Px5.p1" title="Fractional Transformations ‣ §1.12(ii) Convergents ‣ §1.12 Continued Fractions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="C_{n}(w)" display="inline"><mrow><msub><mi href="./1.12#Px5.p1">C</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r3">w</mi><mo stretchy="false">)</mo></mrow></mrow></math>: continued fraction</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px3" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Equivalence</h3>
<div id="Px3.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px3.p1" class="ltx_para">
<p class="ltx_p">Two continued fractions are <em class="ltx_emph ltx_font_italic">equivalent</em> if they have the same
convergents.
</p>
</div>
<div id="Px3.p2" class="ltx_para">
<p class="ltx_p"><math class="ltx_Math" altimg="m56.png" altimg-height="52px" altimg-valign="-19px" altimg-width="177px" alttext="b_{0}+\displaystyle{\cfrac{a_{1}}{b_{1}+\cfrac{a_{2}}{b_{2}+\cdots}}}" display="inline"><mrow><msub><mi>b</mi><mn>0</mn></msub><mo>+</mo><mstyle displaystyle="true"><mrow><mfrac><msub><mi>a</mi><mn>1</mn></msub><mrow><msub><mi>b</mi><mn>1</mn></msub><mo>+</mo></mrow></mfrac><mfrac><msub><mi>a</mi><mn>2</mn></msub><mrow><msub><mi>b</mi><mn>2</mn></msub><mo>+</mo></mrow></mfrac><mi mathvariant="normal">⋯</mi></mrow></mstyle></mrow></math> is equivalent
to <math class="ltx_Math" altimg="m55.png" altimg-height="54px" altimg-valign="-21px" altimg-width="177px" alttext="b^{\prime}_{0}+\displaystyle{\cfrac{a^{\prime}_{1}}{b^{\prime}_{1}+\cfrac{a^{%
\prime}_{2}}{b^{\prime}_{2}+\cdots}}}" display="inline"><mrow><msubsup><mi>b</mi><mn>0</mn><mo>′</mo></msubsup><mo>+</mo><mstyle displaystyle="true"><mrow><mfrac><msubsup><mi>a</mi><mn>1</mn><mo>′</mo></msubsup><mrow><msubsup><mi>b</mi><mn>1</mn><mo>′</mo></msubsup><mo>+</mo></mrow></mfrac><mfrac><msubsup><mi>a</mi><mn>2</mn><mo>′</mo></msubsup><mrow><msubsup><mi>b</mi><mn>2</mn><mo>′</mo></msubsup><mo>+</mo></mrow></mfrac><mi mathvariant="normal">⋯</mi></mrow></mstyle></mrow></math> if
there is a sequence <math class="ltx_Math" altimg="m53.png" altimg-height="23px" altimg-valign="-7px" altimg-width="76px" alttext="\{d_{n}\}^{\infty}_{n=0}" display="inline"><msubsup><mrow><mo stretchy="false">{</mo><msub><mi>d</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo stretchy="false">}</mo></mrow><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></msubsup></math>, <math class="ltx_Math" altimg="m60.png" altimg-height="21px" altimg-valign="-5px" altimg-width="60px" alttext="d_{0}=1" display="inline"><mrow><msub><mi>d</mi><mn>0</mn></msub><mo>=</mo><mn>1</mn></mrow></math>,
<br class="ltx_break">
<math class="ltx_Math" altimg="m61.png" altimg-height="21px" altimg-valign="-6px" altimg-width="62px" alttext="d_{n}\neq 0" display="inline"><mrow><msub><mi>d</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo>≠</mo><mn>0</mn></mrow></math>, such
that</p>
<table id="E15" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.12.15</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E15.png" altimg-height="26px" altimg-valign="-7px" altimg-width="144px" alttext="a^{\prime}_{n}=d_{n}d_{n-1}a_{n}," display="block"><mrow><mrow><msubsup><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi><mo>′</mo></msubsup><mo>=</mo><mrow><msub><mi>d</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><msub><mi>d</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow></msub><mo></mo><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m67.png" altimg-height="20px" altimg-valign="-6px" altimg-width="123px" alttext="n=1,2,3,\dots" display="inline"><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E15.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a></dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and</p>
<table id="E16" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.12.16</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E16.png" altimg-height="26px" altimg-valign="-7px" altimg-width="98px" alttext="b^{\prime}_{n}=d_{n}b_{n}," display="block"><mrow><mrow><msubsup><mi>b</mi><mi href="./1.1#p2.t1.r5">n</mi><mo>′</mo></msubsup><mo>=</mo><mrow><msub><mi>d</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><msub><mi>b</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m66.png" altimg-height="20px" altimg-valign="-6px" altimg-width="123px" alttext="n=0,1,2,\dots" display="inline"><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mrow><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E16.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a></dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="Px3.p3" class="ltx_para">
<p class="ltx_p">Formally,
</p>
<table id="E17" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.12.17</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_Math ltx_eqn_cell ltx_align_center"><math class="ltx_Math" alttext="b_{0}+\cfrac{a_{1}}{b_{1}+\cfrac{a_{2}}{b_{2}+\cfrac{a_{3}}{b_{3}+\cdots}}}={b%
_{0}+\cfrac{a_{1}/b_{1}}{1+\cfrac{a_{2}/(b_{1}b_{2})}{1+\cfrac{a_{3}/(b_{2}b_{%
3})}{1+\cdots\cfrac{a_{n}/(b_{n-1}b_{n})}{1+\cdots}}}}}={b_{0}+\cfrac{1}{(%
\ifrac{1}{a_{1}})b_{1}+\cfrac{1}{(\ifrac{a_{1}}{a_{2}})b_{2}+\cfrac{1}{(\ifrac%
{a_{2}}{(a_{1}a_{3})})b_{3}+\cfrac{1}{(\ifrac{a_{1}a_{3}}{(a_{2}a_{4})})b_{4}+%
\cdots}}}}}." display="block"><mrow><mtable align="baseline 1" columnalign="left" columnspacing="0.1em" displaystyle="true"><mtr><mtd><mrow><msub><mi>b</mi><mn>0</mn></msub><mo>+</mo><mrow><mfrac><msub><mi>a</mi><mn>1</mn></msub><mrow><msub><mi>b</mi><mn>1</mn></msub><mo>+</mo></mrow></mfrac><mfrac><msub><mi>a</mi><mn>2</mn></msub><mrow><msub><mi>b</mi><mn>2</mn></msub><mo>+</mo></mrow></mfrac><mfrac><msub><mi>a</mi><mn>3</mn></msub><mrow><msub><mi>b</mi><mn>3</mn></msub><mo>+</mo></mrow></mfrac><mi mathvariant="normal">⋯</mi></mrow></mrow></mtd><mtd><mo>=</mo><mrow><msub><mi>b</mi><mn>0</mn></msub><mo>+</mo><mrow><mfrac><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>/</mo><msub><mi>b</mi><mn>1</mn></msub></mrow><mrow><mn>1</mn><mo>+</mo></mrow></mfrac><mfrac><mrow><msub><mi>a</mi><mn>2</mn></msub><mo>/</mo><mrow><mo stretchy="false">(</mo><mrow><msub><mi>b</mi><mn>1</mn></msub><mo></mo><msub><mi>b</mi><mn>2</mn></msub></mrow><mo stretchy="false">)</mo></mrow></mrow><mrow><mn>1</mn><mo>+</mo></mrow></mfrac><mfrac><mrow><msub><mi>a</mi><mn>3</mn></msub><mo>/</mo><mrow><mo stretchy="false">(</mo><mrow><msub><mi>b</mi><mn>2</mn></msub><mo></mo><msub><mi>b</mi><mn>3</mn></msub></mrow><mo stretchy="false">)</mo></mrow></mrow><mrow><mn>1</mn><mo>+</mo></mrow></mfrac><mi mathvariant="normal">⋯</mi><mstyle displaystyle="true"><mrow><mfrac><mrow><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo>/</mo><mrow><mo stretchy="false">(</mo><mrow><msub><mi>b</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow></msub><mo></mo><msub><mi>b</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></mrow><mo stretchy="false">)</mo></mrow></mrow><mrow><mn>1</mn><mo>+</mo></mrow></mfrac><mi mathvariant="normal">⋯</mi></mrow></mstyle></mrow></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo><mrow><msub><mi>b</mi><mn>0</mn></msub><mo>+</mo><mrow><mfrac><mn>1</mn><mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>/</mo><msub><mi>a</mi><mn>1</mn></msub></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msub><mi>b</mi><mn>1</mn></msub></mrow><mo>+</mo></mrow></mfrac><mfrac><mn>1</mn><mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>/</mo><msub><mi>a</mi><mn>2</mn></msub></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msub><mi>b</mi><mn>2</mn></msub></mrow><mo>+</mo></mrow></mfrac><mfrac><mn>1</mn><mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><msub><mi>a</mi><mn>2</mn></msub><mo>/</mo><mrow><mo stretchy="false">(</mo><mrow><msub><mi>a</mi><mn>1</mn></msub><mo></mo><msub><mi>a</mi><mn>3</mn></msub></mrow><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msub><mi>b</mi><mn>3</mn></msub></mrow><mo>+</mo></mrow></mfrac><mfrac><mn>1</mn><mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><msub><mi>a</mi><mn>1</mn></msub><mo></mo><msub><mi>a</mi><mn>3</mn></msub></mrow><mo>/</mo><mrow><mo stretchy="false">(</mo><mrow><msub><mi>a</mi><mn>2</mn></msub><mo></mo><msub><mi>a</mi><mn>4</mn></msub></mrow><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msub><mi>b</mi><mn>4</mn></msub></mrow><mo>+</mo></mrow></mfrac><mi mathvariant="normal">⋯</mi></mrow></mrow><mo>.</mo></mtd></mtr></mtable></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E17.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a></dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px4" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Series</h3>
<div id="Px4.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px4.p1" class="ltx_para">
<table id="E18" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.12.18</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E18.png" altimg-height="64px" altimg-valign="-28px" altimg-width="565px" alttext="p_{0}+\sum^{n}_{k=1}p_{1}p_{2}\cdots p_{k}=p_{0}+\cfrac{p_{1}}{1-\cfrac{p_{2}}%
{1+p_{2}-\cfrac{p_{3}}{1+p_{3}-\cdots\cfrac{p_{n}}{1+p_{n}}}}}," display="block"><mrow><mrow><mrow><msub><mi href="./1.12#Px4">p</mi><mn>0</mn></msub><mo>+</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r4">k</mi><mo>=</mo><mn>1</mn></mrow><mi href="./1.1#p2.t1.r5">n</mi></munderover><mrow><msub><mi href="./1.12#Px4">p</mi><mn>1</mn></msub><mo></mo><msub><mi href="./1.12#Px4">p</mi><mn>2</mn></msub><mo></mo><mi mathvariant="normal">⋯</mi><mo></mo><msub><mi href="./1.12#Px4">p</mi><mi href="./1.1#p2.t1.r4">k</mi></msub></mrow></mrow></mrow><mo>=</mo><mrow><msub><mi href="./1.12#Px4">p</mi><mn>0</mn></msub><mo>+</mo><mrow><mfrac><msub><mi href="./1.12#Px4">p</mi><mn>1</mn></msub><mrow><mn>1</mn><mo>-</mo></mrow></mfrac><mfrac><msub><mi href="./1.12#Px4">p</mi><mn>2</mn></msub><mrow><mrow><mn>1</mn><mo>+</mo><msub><mi href="./1.12#Px4">p</mi><mn>2</mn></msub></mrow><mo>-</mo></mrow></mfrac><mfrac><msub><mi href="./1.12#Px4">p</mi><mn>3</mn></msub><mrow><mrow><mn>1</mn><mo>+</mo><msub><mi href="./1.12#Px4">p</mi><mn>3</mn></msub></mrow><mo>-</mo></mrow></mfrac><mi mathvariant="normal">⋯</mi><mstyle displaystyle="true"><mfrac><msub><mi href="./1.12#Px4">p</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mrow><mn>1</mn><mo>+</mo><msub><mi href="./1.12#Px4">p</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></mrow></mfrac></mstyle></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m66.png" altimg-height="20px" altimg-valign="-6px" altimg-width="123px" alttext="n=0,1,2,\dots" display="inline"><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mrow><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E18.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.1#p2.t1.r4" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m65.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./1.1#p2.t1.r4">k</mi></math>: integer</a>,
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./1.12#Px4" title="Series ‣ §1.12(ii) Convergents ‣ §1.12 Continued Fractions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m71.png" altimg-height="16px" altimg-valign="-6px" altimg-width="24px" alttext="p_{k}" display="inline"><msub><mi href="./1.12#Px4">p</mi><mi href="./1.1#p2.t1.r4">k</mi></msub></math>: coefficients</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">when <math class="ltx_Math" altimg="m72.png" altimg-height="21px" altimg-valign="-6px" altimg-width="61px" alttext="p_{k}\not=0" display="inline"><mrow><msub><mi href="./1.12#Px4">p</mi><mi href="./1.1#p2.t1.r4">k</mi></msub><mo>≠</mo><mn>0</mn></mrow></math>, <math class="ltx_Math" altimg="m63.png" altimg-height="21px" altimg-valign="-6px" altimg-width="122px" alttext="k=1,2,3,\dots" display="inline"><mrow><mi href="./1.1#p2.t1.r4">k</mi><mo>=</mo><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>.</p>
<table id="E19" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.12.19</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E19.png" altimg-height="64px" altimg-valign="-28px" altimg-width="668px" alttext="\sum^{n}_{k=0}c_{k}x^{k}=c_{0}+\cfrac{c_{1}x}{1-\cfrac{(\ifrac{c_{2}}{c_{1}})x%
}{1+(\ifrac{c_{2}}{c_{1}})x-\cfrac{(\ifrac{c_{3}}{c_{2}})x}{1+(\ifrac{c_{3}}{c%
_{2}})x-\cdots\cfrac{(\ifrac{c_{n}}{c_{n-1}})x}{1+(\ifrac{c_{n}}{c_{n-1}})x}}}}," display="block"><mrow><mrow><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r4">k</mi><mo>=</mo><mn>0</mn></mrow><mi href="./1.1#p2.t1.r5">n</mi></munderover><mrow><msub><mi>c</mi><mi href="./1.1#p2.t1.r4">k</mi></msub><mo></mo><msup><mi>x</mi><mi href="./1.1#p2.t1.r4">k</mi></msup></mrow></mrow><mo>=</mo><mrow><msub><mi>c</mi><mn>0</mn></msub><mo>+</mo><mrow><mfrac><mrow><msub><mi>c</mi><mn>1</mn></msub><mo></mo><mi>x</mi></mrow><mrow><mn>1</mn><mo>-</mo></mrow></mfrac><mfrac><mrow><mrow><mo stretchy="false">(</mo><mrow><msub><mi>c</mi><mn>2</mn></msub><mo>/</mo><msub><mi>c</mi><mn>1</mn></msub></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi>x</mi></mrow><mrow><mrow><mn>1</mn><mo>+</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><msub><mi>c</mi><mn>2</mn></msub><mo>/</mo><msub><mi>c</mi><mn>1</mn></msub></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi>x</mi></mrow></mrow><mo>-</mo></mrow></mfrac><mfrac><mrow><mrow><mo stretchy="false">(</mo><mrow><msub><mi>c</mi><mn>3</mn></msub><mo>/</mo><msub><mi>c</mi><mn>2</mn></msub></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi>x</mi></mrow><mrow><mrow><mn>1</mn><mo>+</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><msub><mi>c</mi><mn>3</mn></msub><mo>/</mo><msub><mi>c</mi><mn>2</mn></msub></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi>x</mi></mrow></mrow><mo>-</mo></mrow></mfrac><mi mathvariant="normal">⋯</mi><mstyle displaystyle="true"><mfrac><mrow><mrow><mo stretchy="false">(</mo><mrow><msub><mi>c</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo>/</mo><msub><mi>c</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow></msub></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi>x</mi></mrow><mrow><mn>1</mn><mo>+</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><msub><mi>c</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo>/</mo><msub><mi>c</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow></msub></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi>x</mi></mrow></mrow></mfrac></mstyle></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m66.png" altimg-height="20px" altimg-valign="-6px" altimg-width="123px" alttext="n=0,1,2,\dots" display="inline"><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mrow><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E19.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.1#p2.t1.r4" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m65.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./1.1#p2.t1.r4">k</mi></math>: integer</a> and
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">when <math class="ltx_Math" altimg="m59.png" altimg-height="21px" altimg-valign="-6px" altimg-width="59px" alttext="c_{k}\not=0" display="inline"><mrow><msub><mi>c</mi><mi href="./1.1#p2.t1.r4">k</mi></msub><mo>≠</mo><mn>0</mn></mrow></math>, <math class="ltx_Math" altimg="m63.png" altimg-height="21px" altimg-valign="-6px" altimg-width="122px" alttext="k=1,2,3,\dots" display="inline"><mrow><mi href="./1.1#p2.t1.r4">k</mi><mo>=</mo><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>.
</p>
</div>
</section>
<section id="Px5" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Fractional Transformations</h3>
<div id="Px5.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px5.p1" class="ltx_para">
<p class="ltx_p">Define
</p>
<table id="E20" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.12.20</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E20.png" altimg-height="55px" altimg-valign="-19px" altimg-width="338px" alttext="C_{n}(w)=b_{0}+\cfrac{a_{1}}{b_{1}+\cfrac{a_{2}}{b_{2}+\cdots\frac{a_{n}}{b_{n%
}+w}}}." display="block"><mrow><mrow><mrow><msub><mi href="./1.12#Px5.p1">C</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r3">w</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><msub><mi>b</mi><mn>0</mn></msub><mo>+</mo><mrow><mfrac><msub><mi>a</mi><mn>1</mn></msub><mrow><msub><mi>b</mi><mn>1</mn></msub><mo>+</mo></mrow></mfrac><mfrac><msub><mi>a</mi><mn>2</mn></msub><mrow><msub><mi>b</mi><mn>2</mn></msub><mo>+</mo></mrow></mfrac><mi mathvariant="normal">⋯</mi><mfrac><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mrow><msub><mi>b</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo>+</mo><mi href="./1.1#p2.t1.r3">w</mi></mrow></mfrac></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E20.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.1#p2.t1.r3" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m73.png" altimg-height="13px" altimg-valign="-2px" altimg-width="19px" alttext="w" display="inline"><mi href="./1.1#p2.t1.r3">w</mi></math>: variable</a>,
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./1.12#Px5.p1" title="Fractional Transformations ‣ §1.12(ii) Convergents ‣ §1.12 Continued Fractions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="C_{n}(w)" display="inline"><mrow><msub><mi href="./1.12#Px5.p1">C</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r3">w</mi><mo stretchy="false">)</mo></mrow></mrow></math>: continued fraction</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Then</p>
<table id="E21" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex10" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="3" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">1.12.21</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m26.png" altimg-height="25px" altimg-valign="-7px" altimg-width="61px" alttext="\displaystyle C_{n}(w)" display="inline"><mrow><msub><mi href="./1.12#Px5.p1">C</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r3">w</mi><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m10.png" altimg-height="51px" altimg-valign="-20px" altimg-width="149px" alttext="\displaystyle=\frac{A_{n}+A_{n-1}w}{B_{n}+B_{n-1}w}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mstyle displaystyle="true"><mfrac><mrow><msub><mi href="./1.12#Px1">A</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo>+</mo><mrow><msub><mi href="./1.12#Px1">A</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow></msub><mo></mo><mi href="./1.1#p2.t1.r3">w</mi></mrow></mrow><mrow><msub><mi href="./1.12#Px1">B</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo>+</mo><mrow><msub><mi href="./1.12#Px1">B</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow></msub><mo></mo><mi href="./1.1#p2.t1.r3">w</mi></mrow></mrow></mfrac></mstyle></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex11" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m24.png" altimg-height="25px" altimg-valign="-7px" altimg-width="57px" alttext="\displaystyle C_{n}(0)" display="inline"><mrow><msub><mi href="./1.12#Px5.p1">C</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m8.png" altimg-height="23px" altimg-valign="-6px" altimg-width="58px" alttext="\displaystyle=C_{n}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><msub><mi href="./1.12#Px5.p1">C</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex12" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m25.png" altimg-height="25px" altimg-valign="-7px" altimg-width="67px" alttext="\displaystyle C_{n}(\infty)" display="inline"><mrow><msub><mi href="./1.12#Px5.p1">C</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi mathvariant="normal">∞</mi><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m7.png" altimg-height="51px" altimg-valign="-20px" altimg-width="156px" alttext="\displaystyle=C_{n-1}=\frac{A_{n-1}}{B_{n-1}}." display="inline"><mrow><mrow><mi></mi><mo>=</mo><msub><mi href="./1.12#Px5.p1">C</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow></msub><mo>=</mo><mstyle displaystyle="true"><mfrac><msub><mi href="./1.12#Px1">A</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow></msub><msub><mi href="./1.12#Px1">B</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow></msub></mfrac></mstyle></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E21.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.1#p2.t1.r3" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m73.png" altimg-height="13px" altimg-valign="-2px" altimg-width="19px" alttext="w" display="inline"><mi href="./1.1#p2.t1.r3">w</mi></math>: variable</a>,
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a>,
<a href="./1.12#Px1" title="Recurrence Relations ‣ §1.12(ii) Convergents ‣ §1.12 Continued Fractions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="21px" altimg-valign="-5px" altimg-width="30px" alttext="A_{n}" display="inline"><msub><mi href="./1.12#Px1">A</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></math>: <math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>th numerator</a>,
<a href="./1.12#Px1" title="Recurrence Relations ‣ §1.12(ii) Convergents ‣ §1.12 Continued Fractions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="21px" altimg-valign="-5px" altimg-width="30px" alttext="B_{n}" display="inline"><msub><mi href="./1.12#Px1">B</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></math>: <math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>th denominator</a> and
<a href="./1.12#Px5.p1" title="Fractional Transformations ‣ §1.12(ii) Convergents ‣ §1.12 Continued Fractions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="C_{n}(w)" display="inline"><mrow><msub><mi href="./1.12#Px5.p1">C</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r3">w</mi><mo stretchy="false">)</mo></mrow></mrow></math>: continued fraction</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
</section>
<section id="iii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§1.12(iii) </span>Existence of Convergents</h2>
<div id="SS3.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="SS3.p1" class="ltx_para">
<p class="ltx_p">A sequence <math class="ltx_Math" altimg="m52.png" altimg-height="23px" altimg-valign="-7px" altimg-width="49px" alttext="\{C_{n}\}" display="inline"><mrow><mo stretchy="false">{</mo><msub><mi href="./1.12#E4">C</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo stretchy="false">}</mo></mrow></math> in the extended complex plane, <math class="ltx_Math" altimg="m47.png" altimg-height="23px" altimg-valign="-7px" altimg-width="81px" alttext="\mathbb{C}\cup\{\infty\}" display="inline"><mrow><mi href="./front/introduction#Sx4.p1.t1.r1">ℂ</mi><mo href="./front/introduction#Sx4.p1.t1.r27">∪</mo><mrow><mo stretchy="false">{</mo><mi mathvariant="normal">∞</mi><mo stretchy="false">}</mo></mrow></mrow></math>, can be a sequence of convergents of the continued fraction
() iff</p>
<table id="E22" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex13" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="3" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">1.12.22</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m23.png" altimg-height="22px" altimg-valign="-5px" altimg-width="29px" alttext="\displaystyle C_{0}" display="inline"><msub><mi href="./1.12#E4">C</mi><mn>0</mn></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m29.png" altimg-height="23px" altimg-valign="-6px" altimg-width="53px" alttext="\displaystyle\not=\infty," display="inline"><mrow><mrow><mi></mi><mo>≠</mo><mi mathvariant="normal">∞</mi></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex14" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m27.png" altimg-height="22px" altimg-valign="-5px" altimg-width="31px" alttext="\displaystyle C_{n}" display="inline"><msub><mi href="./1.12#E4">C</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m28.png" altimg-height="24px" altimg-valign="-7px" altimg-width="78px" alttext="\displaystyle\not=C_{n-1}," display="inline"><mrow><mrow><mi></mi><mo>≠</mo><msub><mi href="./1.12#E4">C</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow></msub></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m67.png" altimg-height="20px" altimg-valign="-6px" altimg-width="123px" alttext="n=1,2,3,\dots" display="inline"><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E22.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./1.12#E4" title="(1.12.4) ‣ §1.12(ii) Convergents ‣ §1.12 Continued Fractions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="21px" altimg-valign="-5px" altimg-width="29px" alttext="C_{n}" display="inline"><msub><mi href="./1.12#Px5.p1">C</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></math>: approximant</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="iv" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§1.12(iv) </span>Contraction and Extension</h2>
<div id="SS4.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="SS4.p1" class="ltx_para">
<p class="ltx_p">A <em class="ltx_emph ltx_font_italic">contraction</em> of a continued fraction <math class="ltx_Math" altimg="m39.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.12#E3">C</mi></math> is a continued fraction <math class="ltx_Math" altimg="m40.png" altimg-height="19px" altimg-valign="-2px" altimg-width="26px" alttext="C^{\prime}" display="inline"><msup><mi href="./1.12#E3">C</mi><mo>′</mo></msup></math>
whose convergents <math class="ltx_Math" altimg="m51.png" altimg-height="24px" altimg-valign="-7px" altimg-width="49px" alttext="\{C^{\prime}_{n}\}" display="inline"><mrow><mo stretchy="false">{</mo><msubsup><mi href="./1.12#E3">C</mi><mi href="./1.1#p2.t1.r5">n</mi><mo>′</mo></msubsup><mo stretchy="false">}</mo></mrow></math> form a subsequence of the convergents
<math class="ltx_Math" altimg="m52.png" altimg-height="23px" altimg-valign="-7px" altimg-width="49px" alttext="\{C_{n}\}" display="inline"><mrow><mo stretchy="false">{</mo><msub><mi href="./1.12#E4">C</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo stretchy="false">}</mo></mrow></math> of <math class="ltx_Math" altimg="m39.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.12#E3">C</mi></math>. Conversely, <math class="ltx_Math" altimg="m39.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.12#E3">C</mi></math> is called an <em class="ltx_emph ltx_font_italic">extension</em> of <math class="ltx_Math" altimg="m40.png" altimg-height="19px" altimg-valign="-2px" altimg-width="26px" alttext="C^{\prime}" display="inline"><msup><mi href="./1.12#E3">C</mi><mo>′</mo></msup></math>.
If <math class="ltx_Math" altimg="m42.png" altimg-height="23px" altimg-valign="-7px" altimg-width="89px" alttext="C^{\prime}_{n}=C_{2n}" display="inline"><mrow><msubsup><mi href="./1.12#E3">C</mi><mi href="./1.1#p2.t1.r5">n</mi><mo>′</mo></msubsup><mo>=</mo><msub><mi href="./1.12#E4">C</mi><mrow><mn>2</mn><mo></mo><mi href="./1.1#p2.t1.r5">n</mi></mrow></msub></mrow></math>, <math class="ltx_Math" altimg="m66.png" altimg-height="20px" altimg-valign="-6px" altimg-width="123px" alttext="n=0,1,2,\dots" display="inline"><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mrow><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>, then <math class="ltx_Math" altimg="m40.png" altimg-height="19px" altimg-valign="-2px" altimg-width="26px" alttext="C^{\prime}" display="inline"><msup><mi href="./1.12#E3">C</mi><mo>′</mo></msup></math> is called the <em class="ltx_emph ltx_font_italic">even part</em>
of <math class="ltx_Math" altimg="m39.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.12#E3">C</mi></math>. The even part of <math class="ltx_Math" altimg="m39.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.12#E3">C</mi></math> exists iff <math class="ltx_Math" altimg="m58.png" altimg-height="21px" altimg-valign="-6px" altimg-width="67px" alttext="b_{2k}\not=0" display="inline"><mrow><msub><mi>b</mi><mrow><mn>2</mn><mo></mo><mi href="./1.1#p2.t1.r4">k</mi></mrow></msub><mo>≠</mo><mn>0</mn></mrow></math>, <math class="ltx_Math" altimg="m64.png" altimg-height="21px" altimg-valign="-6px" altimg-width="103px" alttext="k=1,2,\dots" display="inline"><mrow><mi href="./1.1#p2.t1.r4">k</mi><mo>=</mo><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>, and up
to equivalence is given by
</p>
<table id="E23" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.12.23</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E23.png" altimg-height="57px" altimg-valign="-21px" altimg-width="820px" alttext="b_{0}+\cfrac{a_{1}b_{2}}{a_{2}+b_{1}b_{2}-\cfrac{a_{2}a_{3}b_{4}}{a_{3}b_{4}+b%
_{2}(a_{4}+b_{3}b_{4})-\cfrac{a_{4}a_{5}b_{2}b_{6}}{a_{5}b_{6}+b_{4}(a_{6}+b_{%
5}b_{6})-\cfrac{a_{6}a_{7}b_{4}b_{8}}{a_{7}b_{8}+b_{6}(a_{8}+b_{7}b_{8})-%
\cdots}}}}." display="block"><mrow><mrow><msub><mi>b</mi><mn>0</mn></msub><mo>+</mo><mrow><mfrac><mrow><msub><mi>a</mi><mn>1</mn></msub><mo></mo><msub><mi>b</mi><mn>2</mn></msub></mrow><mrow><mrow><msub><mi>a</mi><mn>2</mn></msub><mo>+</mo><mrow><msub><mi>b</mi><mn>1</mn></msub><mo></mo><msub><mi>b</mi><mn>2</mn></msub></mrow></mrow><mo>-</mo></mrow></mfrac><mfrac><mrow><msub><mi>a</mi><mn>2</mn></msub><mo></mo><msub><mi>a</mi><mn>3</mn></msub><mo></mo><msub><mi>b</mi><mn>4</mn></msub></mrow><mrow><mrow><mrow><msub><mi>a</mi><mn>3</mn></msub><mo></mo><msub><mi>b</mi><mn>4</mn></msub></mrow><mo>+</mo><mrow><msub><mi>b</mi><mn>2</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mrow><msub><mi>a</mi><mn>4</mn></msub><mo>+</mo><mrow><msub><mi>b</mi><mn>3</mn></msub><mo></mo><msub><mi>b</mi><mn>4</mn></msub></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>-</mo></mrow></mfrac><mfrac><mrow><msub><mi>a</mi><mn>4</mn></msub><mo></mo><msub><mi>a</mi><mn>5</mn></msub><mo></mo><msub><mi>b</mi><mn>2</mn></msub><mo></mo><msub><mi>b</mi><mn>6</mn></msub></mrow><mrow><mrow><mrow><msub><mi>a</mi><mn>5</mn></msub><mo></mo><msub><mi>b</mi><mn>6</mn></msub></mrow><mo>+</mo><mrow><msub><mi>b</mi><mn>4</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mrow><msub><mi>a</mi><mn>6</mn></msub><mo>+</mo><mrow><msub><mi>b</mi><mn>5</mn></msub><mo></mo><msub><mi>b</mi><mn>6</mn></msub></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>-</mo></mrow></mfrac><mfrac><mrow><msub><mi>a</mi><mn>6</mn></msub><mo></mo><msub><mi>a</mi><mn>7</mn></msub><mo></mo><msub><mi>b</mi><mn>4</mn></msub><mo></mo><msub><mi>b</mi><mn>8</mn></msub></mrow><mrow><mrow><mrow><msub><mi>a</mi><mn>7</mn></msub><mo></mo><msub><mi>b</mi><mn>8</mn></msub></mrow><mo>+</mo><mrow><msub><mi>b</mi><mn>6</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mrow><msub><mi>a</mi><mn>8</mn></msub><mo>+</mo><mrow><msub><mi>b</mi><mn>7</mn></msub><mo></mo><msub><mi>b</mi><mn>8</mn></msub></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>-</mo></mrow></mfrac><mi mathvariant="normal">⋯</mi></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E23.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">If <math class="ltx_Math" altimg="m41.png" altimg-height="23px" altimg-valign="-7px" altimg-width="109px" alttext="C^{\prime}_{n}=C_{2n+1}" display="inline"><mrow><msubsup><mi href="./1.12#E3">C</mi><mi href="./1.1#p2.t1.r5">n</mi><mo>′</mo></msubsup><mo>=</mo><msub><mi href="./1.12#E4">C</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./1.1#p2.t1.r5">n</mi></mrow><mo>+</mo><mn>1</mn></mrow></msub></mrow></math>, <math class="ltx_Math" altimg="m66.png" altimg-height="20px" altimg-valign="-6px" altimg-width="123px" alttext="n=0,1,2,\dots" display="inline"><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mrow><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>, then <math class="ltx_Math" altimg="m40.png" altimg-height="19px" altimg-valign="-2px" altimg-width="26px" alttext="C^{\prime}" display="inline"><msup><mi href="./1.12#E3">C</mi><mo>′</mo></msup></math> is called the <em class="ltx_emph ltx_font_italic">odd part</em>
of <math class="ltx_Math" altimg="m39.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.12#E3">C</mi></math>. The odd part
of <math class="ltx_Math" altimg="m39.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.12#E3">C</mi></math> exists iff <math class="ltx_Math" altimg="m57.png" altimg-height="22px" altimg-valign="-7px" altimg-width="87px" alttext="b_{2k+1}\not=0" display="inline"><mrow><msub><mi>b</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./1.1#p2.t1.r4">k</mi></mrow><mo>+</mo><mn>1</mn></mrow></msub><mo>≠</mo><mn>0</mn></mrow></math>, <math class="ltx_Math" altimg="m62.png" altimg-height="21px" altimg-valign="-6px" altimg-width="122px" alttext="k=0,1,2,\dots" display="inline"><mrow><mi href="./1.1#p2.t1.r4">k</mi><mo>=</mo><mrow><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>, and up to equivalence is
given by
</p>
<table id="E24" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.12.24</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E24.png" altimg-height="57px" altimg-valign="-21px" altimg-width="773px" alttext="\frac{a_{1}+b_{0}b_{1}}{b_{1}}-\cfrac{a_{1}a_{2}b_{3}/b_{1}}{a_{2}b_{3}+b_{1}(%
a_{3}+b_{2}b_{3})-\cfrac{a_{3}a_{4}b_{1}b_{5}}{a_{4}b_{5}+b_{3}(a_{5}+b_{4}b_{%
5})-\cfrac{a_{5}a_{6}b_{3}b_{7}}{a_{6}b_{7}+b_{5}(a_{7}+b_{6}b_{7})-\cdots}}}." display="block"><mrow><mrow><mfrac><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>+</mo><mrow><msub><mi>b</mi><mn>0</mn></msub><mo></mo><msub><mi>b</mi><mn>1</mn></msub></mrow></mrow><msub><mi>b</mi><mn>1</mn></msub></mfrac><mo>-</mo><mrow><mfrac><mrow><mrow><msub><mi>a</mi><mn>1</mn></msub><mo></mo><msub><mi>a</mi><mn>2</mn></msub><mo></mo><msub><mi>b</mi><mn>3</mn></msub></mrow><mo>/</mo><msub><mi>b</mi><mn>1</mn></msub></mrow><mrow><mrow><mrow><msub><mi>a</mi><mn>2</mn></msub><mo></mo><msub><mi>b</mi><mn>3</mn></msub></mrow><mo>+</mo><mrow><msub><mi>b</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mrow><msub><mi>a</mi><mn>3</mn></msub><mo>+</mo><mrow><msub><mi>b</mi><mn>2</mn></msub><mo></mo><msub><mi>b</mi><mn>3</mn></msub></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>-</mo></mrow></mfrac><mfrac><mrow><msub><mi>a</mi><mn>3</mn></msub><mo></mo><msub><mi>a</mi><mn>4</mn></msub><mo></mo><msub><mi>b</mi><mn>1</mn></msub><mo></mo><msub><mi>b</mi><mn>5</mn></msub></mrow><mrow><mrow><mrow><msub><mi>a</mi><mn>4</mn></msub><mo></mo><msub><mi>b</mi><mn>5</mn></msub></mrow><mo>+</mo><mrow><msub><mi>b</mi><mn>3</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mrow><msub><mi>a</mi><mn>5</mn></msub><mo>+</mo><mrow><msub><mi>b</mi><mn>4</mn></msub><mo></mo><msub><mi>b</mi><mn>5</mn></msub></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>-</mo></mrow></mfrac><mfrac><mrow><msub><mi>a</mi><mn>5</mn></msub><mo></mo><msub><mi>a</mi><mn>6</mn></msub><mo></mo><msub><mi>b</mi><mn>3</mn></msub><mo></mo><msub><mi>b</mi><mn>7</mn></msub></mrow><mrow><mrow><mrow><msub><mi>a</mi><mn>6</mn></msub><mo></mo><msub><mi>b</mi><mn>7</mn></msub></mrow><mo>+</mo><mrow><msub><mi>b</mi><mn>5</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mrow><msub><mi>a</mi><mn>7</mn></msub><mo>+</mo><mrow><msub><mi>b</mi><mn>6</mn></msub><mo></mo><msub><mi>b</mi><mn>7</mn></msub></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>-</mo></mrow></mfrac><mi mathvariant="normal">⋯</mi></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E24.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="SS5.p1" class="ltx_para">
<p class="ltx_p">A continued fraction <em class="ltx_emph ltx_font_italic">converges</em> if the convergents <math class="ltx_Math" altimg="m44.png" altimg-height="21px" altimg-valign="-5px" altimg-width="29px" alttext="C_{n}" display="inline"><msub><mi href="./1.12#E4">C</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></math> tend to a finite
limit as <math class="ltx_Math" altimg="m70.png" altimg-height="13px" altimg-valign="-2px" altimg-width="67px" alttext="n\to\infty" display="inline"><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></math>.</p>
</div>
<section id="Px6" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Pringsheim’s Theorem</h3>
<div id="Px6.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px6.p1" class="ltx_para">
<p class="ltx_p">The continued fraction
<math class="ltx_Math" altimg="m36.png" altimg-height="55px" altimg-valign="-19px" altimg-width="137px" alttext="\displaystyle{\cfrac{a_{1}}{b_{1}+\cfrac{a_{2}}{b_{2}+\cdots}}}" display="inline"><mstyle displaystyle="true"><mrow><mfrac><msub><mi>a</mi><mn>1</mn></msub><mrow><msub><mi>b</mi><mn>1</mn></msub><mo>+</mo></mrow></mfrac><mfrac><msub><mi>a</mi><mn>2</mn></msub><mrow><msub><mi>b</mi><mn>2</mn></msub><mo>+</mo></mrow></mfrac><mi mathvariant="normal">⋯</mi></mrow></mstyle></math> converges when</p>
<table id="E25" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.12.25</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E25.png" altimg-height="25px" altimg-valign="-7px" altimg-width="135px" alttext="|b_{n}|\geq|a_{n}|+1," display="block"><mrow><mrow><mrow><mo stretchy="false">|</mo><msub><mi>b</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo stretchy="false">|</mo></mrow><mo>≥</mo><mrow><mrow><mo stretchy="false">|</mo><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo stretchy="false">|</mo></mrow><mo>+</mo><mn>1</mn></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m67.png" altimg-height="20px" altimg-valign="-6px" altimg-width="123px" alttext="n=1,2,3,\dots" display="inline"><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E25.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a></dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">With these conditions the convergents <math class="ltx_Math" altimg="m44.png" altimg-height="21px" altimg-valign="-5px" altimg-width="29px" alttext="C_{n}" display="inline"><msub><mi href="./1.12#E4">C</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></math> satisfy <math class="ltx_Math" altimg="m74.png" altimg-height="23px" altimg-valign="-7px" altimg-width="77px" alttext="|C_{n}|<1" display="inline"><mrow><mrow><mo stretchy="false">|</mo><msub><mi href="./1.12#E4">C</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo stretchy="false">|</mo></mrow><mo><</mo><mn>1</mn></mrow></math> and <math class="ltx_Math" altimg="m45.png" altimg-height="21px" altimg-valign="-5px" altimg-width="76px" alttext="C_{n}\to C" display="inline"><mrow><msub><mi href="./1.12#E4">C</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo>→</mo><mi href="./1.12#E3">C</mi></mrow></math>
with <math class="ltx_Math" altimg="m75.png" altimg-height="23px" altimg-valign="-7px" altimg-width="68px" alttext="|C|\leq 1" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mi href="./1.12#E3">C</mi><mo stretchy="false">|</mo></mrow><mo>≤</mo><mn>1</mn></mrow></math>.</p>
</div>
</section>
<section id="Px7" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Van Vleck’s Theorem</h3>
<div id="Px7.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px7.p1" class="ltx_para">
<p class="ltx_p">Let the elements of the continued fraction
<math class="ltx_Math" altimg="m35.png" altimg-height="55px" altimg-valign="-19px" altimg-width="137px" alttext="\displaystyle{\cfrac{1}{b_{1}+\cfrac{1}{b_{2}+\cdots}}}" display="inline"><mstyle displaystyle="true"><mrow><mfrac><mn>1</mn><mrow><msub><mi>b</mi><mn>1</mn></msub><mo>+</mo></mrow></mfrac><mfrac><mn>1</mn><mrow><msub><mi>b</mi><mn>2</mn></msub><mo>+</mo></mrow></mfrac><mi mathvariant="normal">⋯</mi></mrow></mstyle></math> satisfy</p>
<table id="E26" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.12.26</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E26.png" altimg-height="29px" altimg-valign="-9px" altimg-width="241px" alttext="-\tfrac{1}{2}\pi+\delta<\operatorname{ph}b_{n}<\tfrac{1}{2}\pi-\delta," display="block"><mrow><mrow><mrow><mrow><mo>-</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow><mo>+</mo><mi>δ</mi></mrow><mo><</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><msub><mi>b</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></mrow><mo><</mo><mrow><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo>-</mo><mi>δ</mi></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m67.png" altimg-height="20px" altimg-valign="-6px" altimg-width="123px" alttext="n=1,2,3,\dots" display="inline"><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E26.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.9#E7" title="(1.9.7) ‣ Modulus and Phase ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="21px" altimg-valign="-6px" altimg-width="26px" alttext="\operatorname{ph}" display="inline"><mi href="./1.9#E7">ph</mi></math>: phase</a> and
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m46.png" altimg-height="18px" altimg-valign="-2px" altimg-width="14px" alttext="\delta" display="inline"><mi>δ</mi></math> is an arbitrary small positive constant. Then the convergents
<math class="ltx_Math" altimg="m44.png" altimg-height="21px" altimg-valign="-5px" altimg-width="29px" alttext="C_{n}" display="inline"><msub><mi href="./1.12#E4">C</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></math> satisfy</p>
<table id="E27" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.12.27</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E27.png" altimg-height="29px" altimg-valign="-9px" altimg-width="247px" alttext="-\tfrac{1}{2}\pi+\delta<\operatorname{ph}C_{n}<\tfrac{1}{2}\pi-\delta," display="block"><mrow><mrow><mrow><mrow><mo>-</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow><mo>+</mo><mi>δ</mi></mrow><mo><</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><msub><mi href="./1.12#E4">C</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></mrow><mo><</mo><mrow><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo>-</mo><mi>δ</mi></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m67.png" altimg-height="20px" altimg-valign="-6px" altimg-width="123px" alttext="n=1,2,3,\dots" display="inline"><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E27.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.9#E7" title="(1.9.7) ‣ Modulus and Phase ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="21px" altimg-valign="-6px" altimg-width="26px" alttext="\operatorname{ph}" display="inline"><mi href="./1.9#E7">ph</mi></math>: phase</a>,
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./1.12#E4" title="(1.12.4) ‣ §1.12(ii) Convergents ‣ §1.12 Continued Fractions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="21px" altimg-valign="-5px" altimg-width="29px" alttext="C_{n}" display="inline"><msub><mi href="./1.12#Px5.p1">C</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></math>: approximant</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and the even and odd parts of the continued fraction converge to finite values.
The continued fraction converges iff, in addition,</p>
<table id="E28" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.12.28</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E28.png" altimg-height="64px" altimg-valign="-27px" altimg-width="126px" alttext="\sum^{\infty}_{n=1}|b_{n}|=\infty." display="block"><mrow><mrow><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><mo stretchy="false">|</mo><msub><mi>b</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo stretchy="false">|</mo></mrow></mrow><mo>=</mo><mi mathvariant="normal">∞</mi></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E28.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a></dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">In this case <math class="ltx_Math" altimg="m76.png" altimg-height="27px" altimg-valign="-9px" altimg-width="111px" alttext="|\operatorname{ph}C|\leq\tfrac{1}{2}\pi" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mi href="./1.12#E3">C</mi></mrow><mo stretchy="false">|</mo></mrow><mo>≤</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow></math>.</p>
</div>
</section>
</section>
<section id="vi" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§1.12(vi) </span>Applications</h2>
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<title>DLMF: 1.5 Calculus of Two or More Variables</title>
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<div id="SS1.p1" class="ltx_para">
<p class="ltx_p">A function <math class="ltx_Math" altimg="m125.png" altimg-height="23px" altimg-valign="-7px" altimg-width="62px" alttext="f(x,y)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></math> is <em class="ltx_emph ltx_font_italic">continuous at a point</em> <math class="ltx_Math" altimg="m52.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="(a,b)" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi>a</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi>b</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math> if</p>
<table id="E1" class="ltx_equation ltx_eqn_table">
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<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.5.1</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E1.png" altimg-height="40px" altimg-valign="-21px" altimg-width="243px" alttext="\lim_{(x,y)\to(a,b)}f(x,y)=f(a,b)," display="block"><mrow><mrow><mrow><munder><mo movablelimits="false">lim</mo><mrow><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi>x</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi>y</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow><mo>→</mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi>a</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi>b</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></mrow></munder><mo></mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>=</mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
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<div id="E1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./front/introduction#Sx4.p1.t1.r28" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="(\NVar{a},\NVar{b})" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi class="ltx_nvar">a</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi class="ltx_nvar">b</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math>: open interval</a></dd>
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<p class="ltx_p">that is, for every arbitrarily small positive constant <math class="ltx_Math" altimg="m88.png" altimg-height="13px" altimg-valign="-2px" altimg-width="12px" alttext="\epsilon" display="inline"><mi>ϵ</mi></math> there exists
<math class="ltx_Math" altimg="m86.png" altimg-height="18px" altimg-valign="-2px" altimg-width="14px" alttext="\delta" display="inline"><mi>δ</mi></math> (<math class="ltx_Math" altimg="m62.png" altimg-height="17px" altimg-valign="-3px" altimg-width="35px" alttext=">0" display="inline"><mrow><mi></mi><mo>></mo><mn>0</mn></mrow></math>) such that</p>
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<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.5.2</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E2.png" altimg-height="25px" altimg-valign="-7px" altimg-width="266px" alttext="|f(a+\alpha,b+\beta)-f(a,b)|<\epsilon," display="block"><mrow><mrow><mrow><mo stretchy="false">|</mo><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>a</mi><mo>+</mo><mi>α</mi></mrow><mo>,</mo><mrow><mi>b</mi><mo>+</mo><mi>β</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>-</mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo stretchy="false">|</mo></mrow><mo><</mo><mi>ϵ</mi></mrow><mo>,</mo></mrow></math></td>
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<p class="ltx_p">for all <math class="ltx_Math" altimg="m82.png" altimg-height="13px" altimg-valign="-2px" altimg-width="17px" alttext="\alpha" display="inline"><mi>α</mi></math> and <math class="ltx_Math" altimg="m84.png" altimg-height="21px" altimg-valign="-6px" altimg-width="17px" alttext="\beta" display="inline"><mi>β</mi></math> that satisfy <math class="ltx_Math" altimg="m147.png" altimg-height="23px" altimg-valign="-7px" altimg-width="97px" alttext="|\alpha|,|\beta|<\delta" display="inline"><mrow><mrow><mrow><mo stretchy="false">|</mo><mi>α</mi><mo stretchy="false">|</mo></mrow><mo>,</mo><mrow><mo stretchy="false">|</mo><mi>β</mi><mo stretchy="false">|</mo></mrow></mrow><mo><</mo><mi>δ</mi></mrow></math>.</p>
</div>
<div id="SS1.p2" class="ltx_para">
<p class="ltx_p">A function is <em class="ltx_emph ltx_font_italic">continuous on a point set</em> <math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.5#SS1.p2">D</mi></math> if it is continuous at all
points of <math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.5#SS1.p2">D</mi></math>. A function <math class="ltx_Math" altimg="m125.png" altimg-height="23px" altimg-valign="-7px" altimg-width="62px" alttext="f(x,y)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></math> is <em class="ltx_emph ltx_font_italic">piecewise continuous</em> on
<math class="ltx_Math" altimg="m71.png" altimg-height="21px" altimg-valign="-5px" altimg-width="64px" alttext="I_{1}\times I_{2}" display="inline"><mrow><msub><mi href="./1.5#SS1.p2">I</mi><mn>1</mn></msub><mo>×</mo><msub><mi href="./1.5#SS1.p2">I</mi><mn>2</mn></msub></mrow></math>, where <math class="ltx_Math" altimg="m70.png" altimg-height="21px" altimg-valign="-5px" altimg-width="22px" alttext="I_{1}" display="inline"><msub><mi href="./1.5#SS1.p2">I</mi><mn>1</mn></msub></math> and <math class="ltx_Math" altimg="m72.png" altimg-height="21px" altimg-valign="-5px" altimg-width="22px" alttext="I_{2}" display="inline"><msub><mi href="./1.5#SS1.p2">I</mi><mn>2</mn></msub></math> are intervals, if it is piecewise
continuous in <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math> for each <math class="ltx_Math" altimg="m143.png" altimg-height="21px" altimg-valign="-6px" altimg-width="57px" alttext="y\in I_{2}" display="inline"><mrow><mi>y</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><msub><mi href="./1.5#SS1.p2">I</mi><mn>2</mn></msub></mrow></math> and piecewise continuous in <math class="ltx_Math" altimg="m142.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi>y</mi></math> for each
<math class="ltx_Math" altimg="m139.png" altimg-height="21px" altimg-valign="-5px" altimg-width="58px" alttext="x\in I_{1}" display="inline"><mrow><mi>x</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><msub><mi href="./1.5#SS1.p2">I</mi><mn>1</mn></msub></mrow></math>.
</p>
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<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.5.3</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m23.png" altimg-height="47px" altimg-valign="-16px" altimg-width="34px" alttext="\displaystyle\frac{\partial f}{\partial x}" display="inline"><mstyle displaystyle="true"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow></mfrac></mstyle></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m1.png" altimg-height="49px" altimg-valign="-17px" altimg-width="362px" alttext="\displaystyle=D_{x}f=f_{x}=\lim_{h\to 0}\frac{f(x+h,y)-f(x,y)}{h}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msub><mo href="./1.5#E3" mathvariant="italic">D</mo><mi>x</mi></msub><mi>f</mi></mrow><mo>=</mo><msub><mi>f</mi><mi>x</mi></msub><mo>=</mo><mrow><munder><mo movablelimits="false">lim</mo><mrow><mi>h</mi><mo>→</mo><mn>0</mn></mrow></munder><mo></mo><mstyle displaystyle="true"><mfrac><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>x</mi><mo>+</mo><mi>h</mi></mrow><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow><mo>-</mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mi>h</mi></mfrac></mstyle></mrow></mrow><mo>,</mo></mrow></math></td>
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<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd>
<span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m91.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m126.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></span>,
<span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m104.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></span> and
<span class="ltx_text"><math class="ltx_Math" altimg="m65.png" altimg-height="21px" altimg-valign="-5px" altimg-width="31px" alttext="D_{x}" display="inline"><msub><mo href="./1.5#E3" mathvariant="italic">D</mo><mi>x</mi></msub></math>: differential operator (locally)</span>
</dd>
</dl>
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<tbody id="E4">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.5.4</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m24.png" altimg-height="50px" altimg-valign="-20px" altimg-width="34px" alttext="\displaystyle\frac{\partial f}{\partial y}" display="inline"><mstyle displaystyle="true"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow></mfrac></mstyle></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m2.png" altimg-height="49px" altimg-valign="-17px" altimg-width="361px" alttext="\displaystyle=D_{y}f=f_{y}=\lim_{h\to 0}\frac{f(x,y+h)-f(x,y)}{h}." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msub><mo href="./1.5#E3" mathvariant="italic">D</mo><mi>y</mi></msub><mi>f</mi></mrow><mo>=</mo><msub><mi>f</mi><mi>y</mi></msub><mo>=</mo><mrow><munder><mo movablelimits="false">lim</mo><mrow><mi>h</mi><mo>→</mo><mn>0</mn></mrow></munder><mo></mo><mstyle displaystyle="true"><mfrac><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mrow><mi>y</mi><mo>+</mo><mi>h</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>-</mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mi>h</mi></mfrac></mstyle></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E4.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m91.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m126.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m104.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m65.png" altimg-height="21px" altimg-valign="-5px" altimg-width="31px" alttext="D_{x}" display="inline"><msub><mo href="./1.5#E3" mathvariant="italic">D</mo><mi>x</mi></msub></math>: differential operator</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
<table id="E5" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex1" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">1.5.5</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m33.png" altimg-height="53px" altimg-valign="-20px" altimg-width="56px" alttext="\displaystyle\frac{{\partial}^{2}f}{\partial x\partial y}" display="inline"><mstyle displaystyle="true"><mfrac><mrow><mpadded width="-1.1pt"><msup><mo href="./1.5#E3">∂</mo><mn>2</mn></msup></mpadded><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mrow><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow><mo></mo><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow></mrow></mfrac></mstyle></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m9.png" altimg-height="53px" altimg-valign="-21px" altimg-width="125px" alttext="\displaystyle=\frac{\partial}{\partial x}\left(\frac{\partial f}{\partial y}%
\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><mfrac><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow></mfrac></mstyle><mo></mo><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow></mfrac></mstyle><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex2" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m34.png" altimg-height="53px" altimg-valign="-20px" altimg-width="56px" alttext="\displaystyle\frac{{\partial}^{2}f}{\partial y\partial x}" display="inline"><mstyle displaystyle="true"><mfrac><mrow><mpadded width="-1.1pt"><msup><mo href="./1.5#E3">∂</mo><mn>2</mn></msup></mpadded><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mrow><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow><mo></mo><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow></mrow></mfrac></mstyle></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m10.png" altimg-height="53px" altimg-valign="-21px" altimg-width="124px" alttext="\displaystyle=\frac{\partial}{\partial y}\left(\frac{\partial f}{\partial x}%
\right)." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><mfrac><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow></mfrac></mstyle><mo></mo><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow></mfrac></mstyle><mo>)</mo></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E5.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m91.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m126.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m104.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">The function <math class="ltx_Math" altimg="m125.png" altimg-height="23px" altimg-valign="-7px" altimg-width="62px" alttext="f(x,y)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></math> is <em class="ltx_emph ltx_font_italic">continuously differentiable</em> if <math class="ltx_Math" altimg="m126.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math>,
<math class="ltx_Math" altimg="m93.png" altimg-height="23px" altimg-valign="-7px" altimg-width="66px" alttext="\ifrac{\partial f}{\partial x}" display="inline"><mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mo href="./1.5#E3">/</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow></mrow></math>, and <math class="ltx_Math" altimg="m94.png" altimg-height="23px" altimg-valign="-7px" altimg-width="65px" alttext="\ifrac{\partial f}{\partial y}" display="inline"><mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mo href="./1.5#E3">/</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow></mrow></math> are continuous, <em class="ltx_emph ltx_font_italic">and
twice-continuously differentiable</em> if also <math class="ltx_Math" altimg="m95.png" altimg-height="28px" altimg-valign="-9px" altimg-width="85px" alttext="\ifrac{{\partial}^{2}f}{{\partial x}^{2}}" display="inline"><mrow><mrow><mpadded width="-1.7pt"><msup><mo href="./1.5#E3">∂</mo><mn>2</mn></msup></mpadded><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mo href="./1.5#E3">/</mo><msup><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow><mn>2</mn></msup></mrow></math>,
<math class="ltx_Math" altimg="m96.png" altimg-height="28px" altimg-valign="-9px" altimg-width="84px" alttext="\ifrac{{\partial}^{2}f}{{\partial y}^{2}}" display="inline"><mrow><mrow><mpadded width="-1.7pt"><msup><mo href="./1.5#E3">∂</mo><mn>2</mn></msup></mpadded><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mo href="./1.5#E3">/</mo><msup><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow><mn>2</mn></msup></mrow></math>, <math class="ltx_Math" altimg="m145.png" altimg-height="25px" altimg-valign="-7px" altimg-width="92px" alttext="{\partial}^{2}f/\partial x\partial y" display="inline"><mrow><mrow><mrow><mpadded width="-1.1pt"><msup><mo href="./1.5#E3">∂</mo><mn>2</mn></msup></mpadded><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mo>/</mo><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow></mrow><mo></mo><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow></mrow></math>, and
<math class="ltx_Math" altimg="m146.png" altimg-height="25px" altimg-valign="-7px" altimg-width="92px" alttext="{\partial}^{2}f/\partial y\partial x" display="inline"><mrow><mrow><mrow><mpadded width="-1.1pt"><msup><mo href="./1.5#E3">∂</mo><mn>2</mn></msup></mpadded><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mo>/</mo><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow></mrow><mo></mo><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow></mrow></math> are continuous. In the latter event</p>
<table id="E6" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.5.6</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E6.png" altimg-height="53px" altimg-valign="-20px" altimg-width="138px" alttext="\frac{{\partial}^{2}f}{\partial x\partial y}=\frac{{\partial}^{2}f}{\partial y%
\partial x}." display="block"><mrow><mrow><mfrac><mrow><mpadded width="-1.1pt"><msup><mo href="./1.5#E3">∂</mo><mn>2</mn></msup></mpadded><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mrow><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow><mo></mo><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow></mrow></mfrac><mo>=</mo><mfrac><mrow><mpadded width="-1.1pt"><msup><mo href="./1.5#E3">∂</mo><mn>2</mn></msup></mpadded><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mrow><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow><mo></mo><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow></mrow></mfrac></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E6.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m104.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a></dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<section id="Px1" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Chain Rule</h3>
<div id="Px1.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px1.p1" class="ltx_para">
<table id="EGx2" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E7">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.5.7</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m22.png" altimg-height="47px" altimg-valign="-16px" altimg-width="133px" alttext="\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}f(x(t),y(t))" display="inline"><mrow><mstyle displaystyle="true"><mfrac><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi>t</mi></mrow></mfrac></mstyle><mo></mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>x</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo><mrow><mi>y</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m6.png" altimg-height="50px" altimg-valign="-20px" altimg-width="167px" alttext="\displaystyle=\frac{\partial f}{\partial x}\frac{\mathrm{d}x}{\mathrm{d}t}+%
\frac{\partial f}{\partial y}\frac{\mathrm{d}y}{\mathrm{d}t}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mstyle displaystyle="true"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow></mfrac></mstyle><mo></mo><mstyle displaystyle="true"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi>x</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi>t</mi></mrow></mfrac></mstyle></mrow><mo>+</mo><mrow><mstyle displaystyle="true"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow></mfrac></mstyle><mo></mo><mstyle displaystyle="true"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi>y</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi>t</mi></mrow></mfrac></mstyle></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E7.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#E4" title="(1.4.4) ‣ §1.4(iii) Derivatives ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m89.png" altimg-height="29px" altimg-valign="-9px" altimg-width="27px" alttext="\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: derivative of <math class="ltx_Math" altimg="m126.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m91.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m126.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m104.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E8">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.5.8</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m29.png" altimg-height="47px" altimg-valign="-16px" altimg-width="184px" alttext="\displaystyle\frac{\partial}{\partial u}f(x(u,v),y(u,v))" display="inline"><mrow><mstyle displaystyle="true"><mfrac><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>u</mi></mrow></mfrac></mstyle><mo></mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>x</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo><mrow><mi>y</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m7.png" altimg-height="50px" altimg-valign="-20px" altimg-width="169px" alttext="\displaystyle=\frac{\partial f}{\partial x}\frac{\partial x}{\partial u}+\frac%
{\partial f}{\partial y}\frac{\partial y}{\partial u}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mstyle displaystyle="true"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow></mfrac></mstyle><mo></mo><mstyle displaystyle="true"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>u</mi></mrow></mfrac></mstyle></mrow><mo>+</mo><mrow><mstyle displaystyle="true"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow></mfrac></mstyle><mo></mo><mstyle displaystyle="true"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>u</mi></mrow></mfrac></mstyle></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E8.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m91.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m126.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m104.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E9">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.5.9</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m30.png" altimg-height="47px" altimg-valign="-16px" altimg-width="248px" alttext="\displaystyle\frac{\partial}{\partial v}f(x(u,v),y(u,v),z(u,v))" display="inline"><mrow><mstyle displaystyle="true"><mfrac><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>v</mi></mrow></mfrac></mstyle><mo></mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>x</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo><mrow><mi>y</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo><mrow><mi>z</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m8.png" altimg-height="50px" altimg-valign="-20px" altimg-width="248px" alttext="\displaystyle=\frac{\partial f}{\partial x}\frac{\partial x}{\partial v}+\frac%
{\partial f}{\partial y}\frac{\partial y}{\partial v}+\frac{\partial f}{%
\partial z}\frac{\partial z}{\partial v}." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mstyle displaystyle="true"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow></mfrac></mstyle><mo></mo><mstyle displaystyle="true"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>v</mi></mrow></mfrac></mstyle></mrow><mo>+</mo><mrow><mstyle displaystyle="true"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow></mfrac></mstyle><mo></mo><mstyle displaystyle="true"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>v</mi></mrow></mfrac></mstyle></mrow><mo>+</mo><mrow><mstyle displaystyle="true"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>z</mi></mrow></mfrac></mstyle><mo></mo><mstyle displaystyle="true"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>z</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>v</mi></mrow></mfrac></mstyle></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E9.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m91.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m126.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m104.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
</div>
</section>
<section id="Px2" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Implicit Function Theorem</h3>
<div id="Px2.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px2.p1" class="ltx_para">
<p class="ltx_p">If <math class="ltx_Math" altimg="m69.png" altimg-height="23px" altimg-valign="-7px" altimg-width="66px" alttext="F(x,y)" display="inline"><mrow><mi>F</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></math> is continuously differentiable, <math class="ltx_Math" altimg="m66.png" altimg-height="23px" altimg-valign="-7px" altimg-width="100px" alttext="F(a,b)=0" display="inline"><mrow><mrow><mi>F</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mn>0</mn></mrow></math>, and
<math class="ltx_Math" altimg="m92.png" altimg-height="23px" altimg-valign="-7px" altimg-width="105px" alttext="\ifrac{\partial F}{\partial y}\not=0" display="inline"><mrow><mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>F</mi></mrow><mo href="./1.5#E3">/</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow></mrow><mo>≠</mo><mn>0</mn></mrow></math> at
<math class="ltx_Math" altimg="m52.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="(a,b)" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi>a</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi>b</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math>, then in a <em class="ltx_emph ltx_font_italic">neighborhood</em> of <math class="ltx_Math" altimg="m52.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="(a,b)" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi>a</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi>b</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math>, that is, an open disk
centered at <math class="ltx_Math" altimg="m117.png" altimg-height="21px" altimg-valign="-6px" altimg-width="32px" alttext="a,b" display="inline"><mrow><mi>a</mi><mo>,</mo><mi>b</mi></mrow></math>, the equation <math class="ltx_Math" altimg="m68.png" altimg-height="23px" altimg-valign="-7px" altimg-width="103px" alttext="F(x,y)=0" display="inline"><mrow><mrow><mi>F</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mn>0</mn></mrow></math> defines a continuously
differentiable function <math class="ltx_Math" altimg="m141.png" altimg-height="23px" altimg-valign="-7px" altimg-width="78px" alttext="y=g(x)" display="inline"><mrow><mi>y</mi><mo>=</mo><mrow><mi>g</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></math> such that <math class="ltx_Math" altimg="m67.png" altimg-height="23px" altimg-valign="-7px" altimg-width="129px" alttext="F(x,g(x))=0" display="inline"><mrow><mrow><mi>F</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mrow><mi>g</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mn>0</mn></mrow></math>, <math class="ltx_Math" altimg="m120.png" altimg-height="23px" altimg-valign="-7px" altimg-width="76px" alttext="b=g(a)" display="inline"><mrow><mi>b</mi><mo>=</mo><mrow><mi>g</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></math>, and
<math class="ltx_Math" altimg="m127.png" altimg-height="24px" altimg-valign="-8px" altimg-width="144px" alttext="g^{\prime}(x)=-F_{x}/F_{y}" display="inline"><mrow><mrow><msup><mi>g</mi><mo>′</mo></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mo>-</mo><mrow><msub><mi>F</mi><mi>x</mi></msub><mo>/</mo><msub><mi>F</mi><mi>y</mi></msub></mrow></mrow></mrow></math>.</p>
</div>
</section>
</section>
<section id="ii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§1.5(ii) </span>Coordinate Systems</h2>
<div id="SS2.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px3.p1" class="ltx_para">
<p class="ltx_p">The notations given in this subsection, and also in other coordinate
systems in the DLMF, are those generally used by physicists. For
mathematicians the symbols <math class="ltx_Math" altimg="m114.png" altimg-height="18px" altimg-valign="-2px" altimg-width="14px" alttext="\theta" display="inline"><mi href="./1.5#Px6.p1">θ</mi></math> and <math class="ltx_Math" altimg="m105.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="\phi" display="inline"><mi href="./1.5#Px6.p1">ϕ</mi></math> now are usually interchanged.</p>
</div>
</section>
<section id="Px4" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Polar Coordinates</h3>
<div id="Px4.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px4.p1" class="ltx_para">
<p class="ltx_p">With <math class="ltx_Math" altimg="m58.png" altimg-height="19px" altimg-valign="-5px" altimg-width="97px" alttext="0\leq r<\infty" display="inline"><mrow><mn>0</mn><mo>≤</mo><mi href="./1.5#Px5.p1">r</mi><mo><</mo><mi mathvariant="normal">∞</mi></mrow></math>, <math class="ltx_Math" altimg="m59.png" altimg-height="21px" altimg-valign="-6px" altimg-width="101px" alttext="0\leq\phi\leq 2\pi" display="inline"><mrow><mn>0</mn><mo>≤</mo><mi href="./1.5#Px6.p1">ϕ</mi><mo>≤</mo><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow></math>,
</p>
<table id="E10" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex3" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">1.5.10</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m45.png" altimg-height="14px" altimg-valign="-2px" altimg-width="17px" alttext="\displaystyle x" display="inline"><mi>x</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m16.png" altimg-height="23px" altimg-valign="-6px" altimg-width="87px" alttext="\displaystyle=r\cos\phi," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./1.5#Px5.p1">r</mi><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi href="./1.5#Px6.p1">ϕ</mi></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex4" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m46.png" altimg-height="18px" altimg-valign="-6px" altimg-width="17px" alttext="\displaystyle y" display="inline"><mi>y</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m18.png" altimg-height="23px" altimg-valign="-6px" altimg-width="85px" alttext="\displaystyle=r\sin\phi," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./1.5#Px5.p1">r</mi><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi href="./1.5#Px6.p1">ϕ</mi></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E10.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m113.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./1.5#Px6.p1" title="Spherical Coordinates ‣ §1.5(ii) Coordinate Systems ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m105.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="\phi" display="inline"><mi href="./1.5#Px6.p1">ϕ</mi></math>: longitude</a> and
<a href="./1.5#Px5.p1" title="Cylindrical Coordinates ‣ §1.5(ii) Coordinate Systems ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m136.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="r" display="inline"><mi href="./1.5#Px5.p1">r</mi></math>: radius</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="EGx3" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E11">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.5.11</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m31.png" altimg-height="47px" altimg-valign="-16px" altimg-width="34px" alttext="\displaystyle\frac{\partial}{\partial x}" display="inline"><mstyle displaystyle="true"><mfrac><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow></mfrac></mstyle></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m5.png" altimg-height="50px" altimg-valign="-20px" altimg-width="198px" alttext="\displaystyle=\cos\phi\frac{\partial}{\partial r}-\frac{\sin\phi}{r}\frac{%
\partial}{\partial\phi}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi href="./1.5#Px6.p1">ϕ</mi></mrow><mo></mo><mstyle displaystyle="true"><mfrac><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.5#Px5.p1">r</mi></mrow></mfrac></mstyle></mrow><mo>-</mo><mrow><mstyle displaystyle="true"><mfrac><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi href="./1.5#Px6.p1">ϕ</mi></mrow><mi href="./1.5#Px5.p1">r</mi></mfrac></mstyle><mo></mo><mstyle displaystyle="true"><mfrac><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.5#Px6.p1">ϕ</mi></mrow></mfrac></mstyle></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E11.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m91.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m126.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m104.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m113.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./1.5#Px6.p1" title="Spherical Coordinates ‣ §1.5(ii) Coordinate Systems ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m105.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="\phi" display="inline"><mi href="./1.5#Px6.p1">ϕ</mi></math>: longitude</a> and
<a href="./1.5#Px5.p1" title="Cylindrical Coordinates ‣ §1.5(ii) Coordinate Systems ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m136.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="r" display="inline"><mi href="./1.5#Px5.p1">r</mi></math>: radius</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E12">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.5.12</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m32.png" altimg-height="50px" altimg-valign="-20px" altimg-width="33px" alttext="\displaystyle\frac{\partial}{\partial y}" display="inline"><mstyle displaystyle="true"><mfrac><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow></mfrac></mstyle></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m15.png" altimg-height="50px" altimg-valign="-20px" altimg-width="198px" alttext="\displaystyle=\sin\phi\frac{\partial}{\partial r}+\frac{\cos\phi}{r}\frac{%
\partial}{\partial\phi}." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi href="./1.5#Px6.p1">ϕ</mi></mrow><mo></mo><mstyle displaystyle="true"><mfrac><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.5#Px5.p1">r</mi></mrow></mfrac></mstyle></mrow><mo>+</mo><mrow><mstyle displaystyle="true"><mfrac><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi href="./1.5#Px6.p1">ϕ</mi></mrow><mi href="./1.5#Px5.p1">r</mi></mfrac></mstyle><mo></mo><mstyle displaystyle="true"><mfrac><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.5#Px6.p1">ϕ</mi></mrow></mfrac></mstyle></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E12.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m91.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m126.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m104.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m113.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./1.5#Px6.p1" title="Spherical Coordinates ‣ §1.5(ii) Coordinate Systems ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m105.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="\phi" display="inline"><mi href="./1.5#Px6.p1">ϕ</mi></math>: longitude</a> and
<a href="./1.5#Px5.p1" title="Cylindrical Coordinates ‣ §1.5(ii) Coordinate Systems ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m136.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="r" display="inline"><mi href="./1.5#Px5.p1">r</mi></math>: radius</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
<p class="ltx_p">The <em class="ltx_emph ltx_font_italic">Laplacian</em> is given by</p>
<table id="E13" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.5.13</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E13.png" altimg-height="56px" altimg-valign="-22px" altimg-width="391px" alttext="\nabla^{2}f=\frac{{\partial}^{2}f}{{\partial x}^{2}}+\frac{{\partial}^{2}f}{{%
\partial y}^{2}}=\frac{{\partial}^{2}f}{{\partial r}^{2}}+\frac{1}{r}\frac{%
\partial f}{\partial r}+\frac{1}{r^{2}}\frac{{\partial}^{2}f}{{\partial\phi}^{%
2}}." display="block"><mrow><mrow><mrow><msup><mo>∇</mo><mn>2</mn></msup><mo></mo><mi>f</mi></mrow><mo>=</mo><mrow><mfrac><mrow><mpadded width="-1.7pt"><msup><mo href="./1.5#E3">∂</mo><mn>2</mn></msup></mpadded><mo href="./1.5#E3"></mo><mi>f</mi></mrow><msup><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow><mn>2</mn></msup></mfrac><mo>+</mo><mfrac><mrow><mpadded width="-1.7pt"><msup><mo href="./1.5#E3">∂</mo><mn>2</mn></msup></mpadded><mo href="./1.5#E3"></mo><mi>f</mi></mrow><msup><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow><mn>2</mn></msup></mfrac></mrow><mo>=</mo><mrow><mfrac><mrow><mpadded width="-1.7pt"><msup><mo href="./1.5#E3">∂</mo><mn>2</mn></msup></mpadded><mo href="./1.5#E3"></mo><mi>f</mi></mrow><msup><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.5#Px5.p1">r</mi></mrow><mn>2</mn></msup></mfrac><mo>+</mo><mrow><mfrac><mn>1</mn><mi href="./1.5#Px5.p1">r</mi></mfrac><mo></mo><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.5#Px5.p1">r</mi></mrow></mfrac></mrow><mo>+</mo><mrow><mfrac><mn>1</mn><msup><mi href="./1.5#Px5.p1">r</mi><mn>2</mn></msup></mfrac><mo></mo><mfrac><mrow><mpadded width="-1.7pt"><msup><mo href="./1.5#E3">∂</mo><mn>2</mn></msup></mpadded><mo href="./1.5#E3"></mo><mi>f</mi></mrow><msup><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.5#Px6.p1">ϕ</mi></mrow><mn>2</mn></msup></mfrac></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E13.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m91.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m126.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m104.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.5#Px6.p1" title="Spherical Coordinates ‣ §1.5(ii) Coordinate Systems ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m105.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="\phi" display="inline"><mi href="./1.5#Px6.p1">ϕ</mi></math>: longitude</a> and
<a href="./1.5#Px5.p1" title="Cylindrical Coordinates ‣ §1.5(ii) Coordinate Systems ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m136.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="r" display="inline"><mi href="./1.5#Px5.p1">r</mi></math>: radius</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px5" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Cylindrical Coordinates</h3>
<div id="Px5.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px5.p1" class="ltx_para">
<p class="ltx_p">With <math class="ltx_Math" altimg="m58.png" altimg-height="19px" altimg-valign="-5px" altimg-width="97px" alttext="0\leq r<\infty" display="inline"><mrow><mn>0</mn><mo>≤</mo><mi href="./1.5#Px5.p1">r</mi><mo><</mo><mi mathvariant="normal">∞</mi></mrow></math>, <math class="ltx_Math" altimg="m59.png" altimg-height="21px" altimg-valign="-6px" altimg-width="101px" alttext="0\leq\phi\leq 2\pi" display="inline"><mrow><mn>0</mn><mo>≤</mo><mi href="./1.5#Px6.p1">ϕ</mi><mo>≤</mo><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow></math>, <math class="ltx_Math" altimg="m57.png" altimg-height="17px" altimg-valign="-4px" altimg-width="123px" alttext="-\infty<z<\infty" display="inline"><mrow><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mo><</mo><mi href="./1.1#p2.t1.r2">z</mi><mo><</mo><mi mathvariant="normal">∞</mi></mrow></math>,
</p>
<table id="E14" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex5" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="3" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">1.5.14</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m45.png" altimg-height="14px" altimg-valign="-2px" altimg-width="17px" alttext="\displaystyle x" display="inline"><mi>x</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m16.png" altimg-height="23px" altimg-valign="-6px" altimg-width="87px" alttext="\displaystyle=r\cos\phi," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./1.5#Px5.p1">r</mi><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi href="./1.5#Px6.p1">ϕ</mi></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex6" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m46.png" altimg-height="18px" altimg-valign="-6px" altimg-width="17px" alttext="\displaystyle y" display="inline"><mi>y</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m18.png" altimg-height="23px" altimg-valign="-6px" altimg-width="85px" alttext="\displaystyle=r\sin\phi," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./1.5#Px5.p1">r</mi><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi href="./1.5#Px6.p1">ϕ</mi></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex7" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m47.png" altimg-height="14px" altimg-valign="-2px" altimg-width="16px" alttext="\displaystyle z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m21.png" altimg-height="14px" altimg-valign="-2px" altimg-width="43px" alttext="\displaystyle=z." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mi href="./1.1#p2.t1.r2">z</mi></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E14.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m113.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m144.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>,
<a href="./1.5#Px6.p1" title="Spherical Coordinates ‣ §1.5(ii) Coordinate Systems ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m105.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="\phi" display="inline"><mi href="./1.5#Px6.p1">ϕ</mi></math>: longitude</a> and
<a href="./1.5#Px5.p1" title="Cylindrical Coordinates ‣ §1.5(ii) Coordinate Systems ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m136.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="r" display="inline"><mi href="./1.5#Px5.p1">r</mi></math>: radius</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Equations () still apply, but</p>
<table id="E15" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.5.15</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E15.png" altimg-height="56px" altimg-valign="-22px" altimg-width="514px" alttext="\nabla^{2}f=\frac{{\partial}^{2}f}{{\partial x}^{2}}+\frac{{\partial}^{2}f}{{%
\partial y}^{2}}+\frac{{\partial}^{2}f}{{\partial z}^{2}}=\frac{{\partial}^{2}%
f}{{\partial r}^{2}}+\frac{1}{r}\frac{\partial f}{\partial r}+\frac{1}{r^{2}}%
\frac{{\partial}^{2}f}{{\partial\phi}^{2}}+\frac{{\partial}^{2}f}{{\partial z}%
^{2}}." display="block"><mrow><mrow><mrow><msup><mo>∇</mo><mn>2</mn></msup><mo></mo><mi>f</mi></mrow><mo>=</mo><mrow><mfrac><mrow><mpadded width="-1.7pt"><msup><mo href="./1.5#E3">∂</mo><mn>2</mn></msup></mpadded><mo href="./1.5#E3"></mo><mi>f</mi></mrow><msup><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow><mn>2</mn></msup></mfrac><mo>+</mo><mfrac><mrow><mpadded width="-1.7pt"><msup><mo href="./1.5#E3">∂</mo><mn>2</mn></msup></mpadded><mo href="./1.5#E3"></mo><mi>f</mi></mrow><msup><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow><mn>2</mn></msup></mfrac><mo>+</mo><mfrac><mrow><mpadded width="-1.7pt"><msup><mo href="./1.5#E3">∂</mo><mn>2</mn></msup></mpadded><mo href="./1.5#E3"></mo><mi>f</mi></mrow><msup><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.1#p2.t1.r2">z</mi></mrow><mn>2</mn></msup></mfrac></mrow><mo>=</mo><mrow><mfrac><mrow><mpadded width="-1.7pt"><msup><mo href="./1.5#E3">∂</mo><mn>2</mn></msup></mpadded><mo href="./1.5#E3"></mo><mi>f</mi></mrow><msup><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.5#Px5.p1">r</mi></mrow><mn>2</mn></msup></mfrac><mo>+</mo><mrow><mfrac><mn>1</mn><mi href="./1.5#Px5.p1">r</mi></mfrac><mo></mo><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.5#Px5.p1">r</mi></mrow></mfrac></mrow><mo>+</mo><mrow><mfrac><mn>1</mn><msup><mi href="./1.5#Px5.p1">r</mi><mn>2</mn></msup></mfrac><mo></mo><mfrac><mrow><mpadded width="-1.7pt"><msup><mo href="./1.5#E3">∂</mo><mn>2</mn></msup></mpadded><mo href="./1.5#E3"></mo><mi>f</mi></mrow><msup><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.5#Px6.p1">ϕ</mi></mrow><mn>2</mn></msup></mfrac></mrow><mo>+</mo><mfrac><mrow><mpadded width="-1.7pt"><msup><mo href="./1.5#E3">∂</mo><mn>2</mn></msup></mpadded><mo href="./1.5#E3"></mo><mi>f</mi></mrow><msup><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.1#p2.t1.r2">z</mi></mrow><mn>2</mn></msup></mfrac></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E15.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m91.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m126.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m104.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m144.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>,
<a href="./1.5#Px6.p1" title="Spherical Coordinates ‣ §1.5(ii) Coordinate Systems ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m105.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="\phi" display="inline"><mi href="./1.5#Px6.p1">ϕ</mi></math>: longitude</a> and
<a href="./1.5#Px5.p1" title="Cylindrical Coordinates ‣ §1.5(ii) Coordinate Systems ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m136.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="r" display="inline"><mi href="./1.5#Px5.p1">r</mi></math>: radius</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px6" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Spherical Coordinates</h3>
<div id="Px6.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px6.p1" class="ltx_para">
<p class="ltx_p">With <math class="ltx_Math" altimg="m60.png" altimg-height="20px" altimg-valign="-6px" altimg-width="98px" alttext="0\leq\rho<\infty" display="inline"><mrow><mn>0</mn><mo>≤</mo><mi href="./1.5#Px6.p1">ρ</mi><mo><</mo><mi mathvariant="normal">∞</mi></mrow></math>, <math class="ltx_Math" altimg="m59.png" altimg-height="21px" altimg-valign="-6px" altimg-width="101px" alttext="0\leq\phi\leq 2\pi" display="inline"><mrow><mn>0</mn><mo>≤</mo><mi href="./1.5#Px6.p1">ϕ</mi><mo>≤</mo><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow></math>, <math class="ltx_Math" altimg="m61.png" altimg-height="20px" altimg-valign="-5px" altimg-width="89px" alttext="0\leq\theta\leq\pi" display="inline"><mrow><mn>0</mn><mo>≤</mo><mi href="./1.5#Px6.p1">θ</mi><mo>≤</mo><mi href="./3.12#E1">π</mi></mrow></math>,
</p>
<table id="E16" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex8" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="3" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">1.5.16</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m45.png" altimg-height="14px" altimg-valign="-2px" altimg-width="17px" alttext="\displaystyle x" display="inline"><mi>x</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m12.png" altimg-height="23px" altimg-valign="-6px" altimg-width="129px" alttext="\displaystyle=\rho\sin\theta\cos\phi," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./1.5#Px6.p1">ρ</mi><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi href="./1.5#Px6.p1">θ</mi></mrow><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi href="./1.5#Px6.p1">ϕ</mi></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex9" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m46.png" altimg-height="18px" altimg-valign="-6px" altimg-width="17px" alttext="\displaystyle y" display="inline"><mi>y</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m13.png" altimg-height="23px" altimg-valign="-6px" altimg-width="127px" alttext="\displaystyle=\rho\sin\theta\sin\phi," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./1.5#Px6.p1">ρ</mi><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi href="./1.5#Px6.p1">θ</mi></mrow><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi href="./1.5#Px6.p1">ϕ</mi></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex10" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m47.png" altimg-height="14px" altimg-valign="-2px" altimg-width="16px" alttext="\displaystyle z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m11.png" altimg-height="23px" altimg-valign="-6px" altimg-width="86px" alttext="\displaystyle=\rho\cos\theta." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./1.5#Px6.p1">ρ</mi><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi href="./1.5#Px6.p1">θ</mi></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E16.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m113.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m144.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>,
<a href="./1.5#Px6.p1" title="Spherical Coordinates ‣ §1.5(ii) Coordinate Systems ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m105.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="\phi" display="inline"><mi href="./1.5#Px6.p1">ϕ</mi></math>: longitude</a>,
<a href="./1.5#Px6.p1" title="Spherical Coordinates ‣ §1.5(ii) Coordinate Systems ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m111.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="\rho" display="inline"><mi href="./1.5#Px6.p1">ρ</mi></math>: radius</a> and
<a href="./1.5#Px6.p1" title="Spherical Coordinates ‣ §1.5(ii) Coordinate Systems ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m114.png" altimg-height="18px" altimg-valign="-2px" altimg-width="14px" alttext="\theta" display="inline"><mi href="./1.5#Px6.p1">θ</mi></math>: azimuth</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">The Laplacian is given by
</p>
<table id="E17" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.5.17</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_Math ltx_eqn_cell ltx_align_center"><math class="ltx_Math" alttext="\nabla^{2}f=\frac{{\partial}^{2}f}{{\partial x}^{2}}+\frac{{\partial}^{2}f}{{%
\partial y}^{2}}+\frac{{\partial}^{2}f}{{\partial z}^{2}}={\frac{1}{\rho^{2}}%
\frac{\partial}{\partial\rho}\left(\rho^{2}\frac{\partial f}{\partial\rho}%
\right)+\frac{1}{\rho^{2}{\sin^{2}}\theta}\frac{{\partial}^{2}f}{{\partial\phi%
}^{2}}}+\frac{1}{\rho^{2}\sin\theta}\frac{\partial}{\partial\theta}\left(\sin%
\theta\frac{\partial f}{\partial\theta}\right)." display="block"><mrow><mtable align="baseline 1" columnalign="left" columnspacing="0.1em" displaystyle="true"><mtr><mtd><mrow><msup><mo>∇</mo><mn>2</mn></msup><mo></mo><mi>f</mi></mrow></mtd><mtd><mo>=</mo><mrow><mfrac><mrow><mpadded width="-1.7pt"><msup><mo href="./1.5#E3">∂</mo><mn>2</mn></msup></mpadded><mo href="./1.5#E3"></mo><mi>f</mi></mrow><msup><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow><mn>2</mn></msup></mfrac><mo>+</mo><mfrac><mrow><mpadded width="-1.7pt"><msup><mo href="./1.5#E3">∂</mo><mn>2</mn></msup></mpadded><mo href="./1.5#E3"></mo><mi>f</mi></mrow><msup><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow><mn>2</mn></msup></mfrac><mo>+</mo><mfrac><mrow><mpadded width="-1.7pt"><msup><mo href="./1.5#E3">∂</mo><mn>2</mn></msup></mpadded><mo href="./1.5#E3"></mo><mi>f</mi></mrow><msup><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.1#p2.t1.r2">z</mi></mrow><mn>2</mn></msup></mfrac></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo><mrow><mrow><mfrac><mn>1</mn><msup><mi href="./1.5#Px6.p1">ρ</mi><mn>2</mn></msup></mfrac><mo></mo><mrow><mfrac><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.5#Px6.p1">ρ</mi></mrow></mfrac><mo></mo><mrow><mo>(</mo><mrow><msup><mi href="./1.5#Px6.p1">ρ</mi><mn>2</mn></msup><mo></mo><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.5#Px6.p1">ρ</mi></mrow></mfrac></mrow><mo>)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mfrac><mn>1</mn><mrow><msup><mi href="./1.5#Px6.p1">ρ</mi><mn>2</mn></msup><mo></mo><mrow><msup><mi href="./4.14#E1">sin</mi><mn>2</mn></msup><mo></mo><mi href="./1.5#Px6.p1">θ</mi></mrow></mrow></mfrac><mo></mo><mfrac><mrow><mpadded width="-1.7pt"><msup><mo href="./1.5#E3">∂</mo><mn>2</mn></msup></mpadded><mo href="./1.5#E3"></mo><mi>f</mi></mrow><msup><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.5#Px6.p1">ϕ</mi></mrow><mn>2</mn></msup></mfrac></mrow><mo>+</mo><mrow><mfrac><mn>1</mn><mrow><msup><mi href="./1.5#Px6.p1">ρ</mi><mn>2</mn></msup><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi href="./1.5#Px6.p1">θ</mi></mrow></mrow></mfrac><mo></mo><mrow><mfrac><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.5#Px6.p1">θ</mi></mrow></mfrac><mo></mo><mrow><mo>(</mo><mrow><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi href="./1.5#Px6.p1">θ</mi></mrow><mo></mo><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.5#Px6.p1">θ</mi></mrow></mfrac></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>.</mo></mtd></mtr></mtable></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E17.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m91.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m126.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m104.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m113.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m144.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>,
<a href="./1.5#Px6.p1" title="Spherical Coordinates ‣ §1.5(ii) Coordinate Systems ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m105.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="\phi" display="inline"><mi href="./1.5#Px6.p1">ϕ</mi></math>: longitude</a>,
<a href="./1.5#Px6.p1" title="Spherical Coordinates ‣ §1.5(ii) Coordinate Systems ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m111.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="\rho" display="inline"><mi href="./1.5#Px6.p1">ρ</mi></math>: radius</a> and
<a href="./1.5#Px6.p1" title="Spherical Coordinates ‣ §1.5(ii) Coordinate Systems ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m114.png" altimg-height="18px" altimg-valign="-2px" altimg-width="14px" alttext="\theta" display="inline"><mi href="./1.5#Px6.p1">θ</mi></math>: azimuth</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="Px6.p2" class="ltx_para">
<p class="ltx_p">For applications and other coordinate systems see §§</dd>
</dl>
</div>
</div>
<div id="SS3.p1" class="ltx_para">
<p class="ltx_p">If <math class="ltx_Math" altimg="m126.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> is <math class="ltx_Math" altimg="m134.png" altimg-height="18px" altimg-valign="-4px" altimg-width="51px" alttext="n+1" display="inline"><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>+</mo><mn>1</mn></mrow></math> times continuously differentiable, then</p>
<table id="E18" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.5.18</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E18.png" altimg-height="54px" altimg-valign="-21px" altimg-width="649px" alttext="f(a+\lambda,b+\mu)=f+\left(\lambda\frac{\partial}{\partial x}+\mu\frac{%
\partial}{\partial y}\right)f+\dots+\frac{1}{n!}\left(\lambda\frac{\partial}{%
\partial x}+\mu\frac{\partial}{\partial y}\right)^{n}f+R_{n}," display="block"><mrow><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>a</mi><mo>+</mo><mi>λ</mi></mrow><mo>,</mo><mrow><mi>b</mi><mo>+</mo><mi>μ</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mi>f</mi><mo>+</mo><mrow><mrow><mo>(</mo><mrow><mrow><mi>λ</mi><mo></mo><mfrac><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow></mfrac></mrow><mo>+</mo><mrow><mi>μ</mi><mo></mo><mfrac><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow></mfrac></mrow></mrow><mo>)</mo></mrow><mo></mo><mi>f</mi></mrow><mo>+</mo><mi mathvariant="normal">…</mi><mo>+</mo><mrow><mfrac><mn>1</mn><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mfrac><mo></mo><msup><mrow><mo>(</mo><mrow><mrow><mi>λ</mi><mo></mo><mfrac><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow></mfrac></mrow><mo>+</mo><mrow><mi>μ</mi><mo></mo><mfrac><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow></mfrac></mrow></mrow><mo>)</mo></mrow><mi href="./1.1#p2.t1.r5">n</mi></msup><mo></mo><mi>f</mi></mrow><mo>+</mo><msub><mi href="./1.4#SS6.p1">R</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E18.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m133.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m91.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m126.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m104.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m135.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./1.4#SS6.p1" title="§1.4(vi) Taylor’s Theorem for Real Variables ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="21px" altimg-valign="-5px" altimg-width="30px" alttext="R_{n}" display="inline"><msub><mi href="./1.4#SS6.p1">R</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></math>: remainder</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m126.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> and its partial derivatives on the right-hand side are evaluated at
<math class="ltx_Math" altimg="m52.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="(a,b)" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi>a</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi>b</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math>, and <math class="ltx_Math" altimg="m76.png" altimg-height="26px" altimg-valign="-7px" altimg-width="189px" alttext="R_{n}/(\lambda^{2}+\mu^{2})^{n/2}\to 0" display="inline"><mrow><mrow><msub><mi href="./1.4#SS6.p1">R</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo>/</mo><msup><mrow><mo stretchy="false">(</mo><mrow><msup><mi>λ</mi><mn>2</mn></msup><mo>+</mo><msup><mi>μ</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>/</mo><mn>2</mn></mrow></msup></mrow><mo>→</mo><mn>0</mn></mrow></math> as <math class="ltx_Math" altimg="m50.png" altimg-height="23px" altimg-valign="-7px" altimg-width="128px" alttext="(\lambda,\mu)\to(0,0)" display="inline"><mrow><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi>λ</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi>μ</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow><mo>→</mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mn>0</mn><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mn>0</mn><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></mrow></math>.</p>
</div>
<div id="SS3.p2" class="ltx_para">
<p class="ltx_p"><math class="ltx_Math" altimg="m125.png" altimg-height="23px" altimg-valign="-7px" altimg-width="62px" alttext="f(x,y)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></math> has a <em class="ltx_emph ltx_font_italic">local minimum</em> (<em class="ltx_emph ltx_font_italic">maximum</em>)
at <math class="ltx_Math" altimg="m52.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="(a,b)" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi>a</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi>b</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math> if
</p>
<table id="E19" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.5.19</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E19.png" altimg-height="50px" altimg-valign="-20px" altimg-width="226px" alttext="\frac{\partial f}{\partial x}=\frac{\partial f}{\partial y}=0\quad\mbox{ at $(%
a,b)$,}" display="block"><mrow><mrow><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow></mfrac><mo>=</mo><mn>0</mn></mrow><mo mathvariant="italic" separator="true"> </mo><mrow><mtext> at </mtext><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi>a</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi>b</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow><mtext>,</mtext></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E19.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./front/introduction#Sx4.p1.t1.r28" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="(\NVar{a},\NVar{b})" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi class="ltx_nvar">a</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi class="ltx_nvar">b</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math>: open interval</a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m91.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m126.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m104.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and the second-order term in () is <em class="ltx_emph ltx_font_italic">positive definite
(negative definite)</em>, that is,
</p>
<table id="E20" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.5.20</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E20.png" altimg-height="51px" altimg-valign="-18px" altimg-width="243px" alttext="\frac{{\partial}^{2}f}{{\partial x}^{2}}>0\;\;\;\mbox{$(<0)$}\quad\mbox{ at $(%
a,b)$,}" display="block"><mrow><mfrac><mrow><mpadded width="-1.7pt"><msup><mo href="./1.5#E3">∂</mo><mn>2</mn></msup></mpadded><mo href="./1.5#E3"></mo><mi>f</mi></mrow><msup><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow><mn>2</mn></msup></mfrac><mo>></mo><mrow><mrow><mpadded width="+8.3pt"><mn>0</mn></mpadded><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi></mi><mo><</mo><mn>0</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mo mathvariant="italic" separator="true"> </mo><mrow><mtext> at </mtext><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi>a</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi>b</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow><mtext>,</mtext></mrow></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E20.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./front/introduction#Sx4.p1.t1.r28" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="(\NVar{a},\NVar{b})" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi class="ltx_nvar">a</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi class="ltx_nvar">b</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math>: open interval</a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m91.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m126.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m104.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and</p>
<table id="E21" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.5.21</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E21.png" altimg-height="60px" altimg-valign="-22px" altimg-width="323px" alttext="\frac{{\partial}^{2}f}{{\partial x}^{2}}\frac{{\partial}^{2}f}{{\partial y}^{2%
}}-\left(\frac{{\partial}^{2}f}{\partial x\partial y}\right)^{2}>0\quad\mbox{ %
at $(a,b)$}." display="block"><mrow><mrow><mrow><mrow><mfrac><mrow><mpadded width="-1.7pt"><msup><mo href="./1.5#E3">∂</mo><mn>2</mn></msup></mpadded><mo href="./1.5#E3"></mo><mi>f</mi></mrow><msup><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow><mn>2</mn></msup></mfrac><mo></mo><mfrac><mrow><mpadded width="-1.7pt"><msup><mo href="./1.5#E3">∂</mo><mn>2</mn></msup></mpadded><mo href="./1.5#E3"></mo><mi>f</mi></mrow><msup><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow><mn>2</mn></msup></mfrac></mrow><mo>-</mo><msup><mrow><mo>(</mo><mfrac><mrow><mpadded width="-1.1pt"><msup><mo href="./1.5#E3">∂</mo><mn>2</mn></msup></mpadded><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mrow><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow><mo></mo><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow></mrow></mfrac><mo>)</mo></mrow><mn>2</mn></msup></mrow><mo>></mo><mrow><mn>0</mn><mo mathvariant="italic" separator="true"> </mo><mrow><mtext> at </mtext><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi>a</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi>b</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E21.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./front/introduction#Sx4.p1.t1.r28" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="(\NVar{a},\NVar{b})" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi class="ltx_nvar">a</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi class="ltx_nvar">b</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math>: open interval</a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m91.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m126.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m104.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="iv" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§1.5(iv) </span>Leibniz’s Theorem for Differentiation of Integrals</h2>
<div id="SS4.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px7.p1" class="ltx_para">
<table id="E22" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.5.22</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E22.png" altimg-height="61px" altimg-valign="-24px" altimg-width="622px" alttext="\frac{\mathrm{d}}{\mathrm{d}x}\int^{\beta(x)}_{\alpha(x)}f(x,y)\mathrm{d}y={f(%
x,\beta(x))\beta^{\prime}(x)-f(x,\alpha(x))\alpha^{\prime}(x)}+\int^{\beta(x)}%
_{\alpha(x)}\frac{\partial f}{\partial x}\mathrm{d}y." display="block"><mrow><mrow><mrow><mfrac><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi>x</mi></mrow></mfrac><mo></mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><mi>α</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow><mrow><mi>β</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></msubsup><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>y</mi></mrow></mrow></mrow></mrow><mo>=</mo><mrow><mrow><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mrow><mi>β</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><msup><mi>β</mi><mo>′</mo></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>-</mo><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mrow><mi>α</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><msup><mi>α</mi><mo>′</mo></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow><mo>+</mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><mi>α</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow><mrow><mi>β</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></msubsup><mrow><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow></mfrac><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>y</mi></mrow></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E22.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#E4" title="(1.4.4) ‣ §1.4(iii) Derivatives ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m89.png" altimg-height="29px" altimg-valign="-9px" altimg-width="27px" alttext="\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: derivative of <math class="ltx_Math" altimg="m126.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m101.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m98.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m91.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m126.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m104.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">3.3.7</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Sufficient conditions for validity are: (a) <math class="ltx_Math" altimg="m126.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> and <math class="ltx_Math" altimg="m93.png" altimg-height="23px" altimg-valign="-7px" altimg-width="66px" alttext="\ifrac{\partial f}{\partial x}" display="inline"><mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mo href="./1.5#E3">/</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow></mrow></math> are
continuous on a rectangle <math class="ltx_Math" altimg="m119.png" altimg-height="20px" altimg-valign="-5px" altimg-width="88px" alttext="a\leq x\leq b" display="inline"><mrow><mi>a</mi><mo>≤</mo><mi>x</mi><mo>≤</mo><mi>b</mi></mrow></math>, <math class="ltx_Math" altimg="m121.png" altimg-height="21px" altimg-valign="-6px" altimg-width="87px" alttext="c\leq y\leq d" display="inline"><mrow><mi>c</mi><mo>≤</mo><mi>y</mi><mo>≤</mo><mi>d</mi></mrow></math>; (b) when
<math class="ltx_Math" altimg="m140.png" altimg-height="23px" altimg-valign="-7px" altimg-width="79px" alttext="x\in[a,b]" display="inline"><mrow><mi>x</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">[</mo><mi>a</mi><mo href="./front/introduction#Sx4.p1.t1.r29">,</mo><mi>b</mi><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">]</mo></mrow></mrow></math> both <math class="ltx_Math" altimg="m81.png" altimg-height="23px" altimg-valign="-7px" altimg-width="44px" alttext="\alpha(x)" display="inline"><mrow><mi>α</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math> and <math class="ltx_Math" altimg="m83.png" altimg-height="23px" altimg-valign="-7px" altimg-width="44px" alttext="\beta(x)" display="inline"><mrow><mi>β</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math> are continuously differentiable
and lie in <math class="ltx_Math" altimg="m79.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="[c,d]" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">[</mo><mi>c</mi><mo href="./front/introduction#Sx4.p1.t1.r29">,</mo><mi>d</mi><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">]</mo></mrow></math>.</p>
</div>
</section>
<section id="Px8" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Infinite Integrals</h3>
<div id="Px8.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px8.p1" class="ltx_para">
<p class="ltx_p">Suppose that <math class="ltx_Math" altimg="m116.png" altimg-height="21px" altimg-valign="-6px" altimg-width="50px" alttext="a,b,c" display="inline"><mrow><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi></mrow></math> are finite, <math class="ltx_Math" altimg="m124.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="d" display="inline"><mi>d</mi></math> is finite or <math class="ltx_Math" altimg="m56.png" altimg-height="17px" altimg-valign="-4px" altimg-width="40px" alttext="+\infty" display="inline"><mrow><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></math>, and <math class="ltx_Math" altimg="m125.png" altimg-height="23px" altimg-valign="-7px" altimg-width="62px" alttext="f(x,y)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></math>,
<math class="ltx_Math" altimg="m93.png" altimg-height="23px" altimg-valign="-7px" altimg-width="66px" alttext="\ifrac{\partial f}{\partial x}" display="inline"><mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mo href="./1.5#E3">/</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow></mrow></math> are continuous on the partly-closed rectangle or infinite
strip <math class="ltx_Math" altimg="m78.png" altimg-height="23px" altimg-valign="-7px" altimg-width="109px" alttext="[a,b]\times[c,d)" display="inline"><mrow><mrow><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">[</mo><mi>a</mi><mo href="./front/introduction#Sx4.p1.t1.r29">,</mo><mi>b</mi><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">]</mo></mrow><mo>×</mo><mrow><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">[</mo><mi>c</mi><mo href="./front/introduction#Sx4.p2.t1.r1">,</mo><mi>d</mi><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">)</mo></mrow></mrow></math>. Suppose also that <math class="ltx_Math" altimg="m100.png" altimg-height="31px" altimg-valign="-9px" altimg-width="110px" alttext="\int^{d}_{c}f(x,y)\mathrm{d}y" display="inline"><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi>c</mi><mi>d</mi></msubsup><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>y</mi></mrow></mrow></mrow></math>
converges and <math class="ltx_Math" altimg="m99.png" altimg-height="31px" altimg-valign="-9px" altimg-width="125px" alttext="\int^{d}_{c}(\ifrac{\partial f}{\partial x})\mathrm{d}y" display="inline"><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi>c</mi><mi>d</mi></msubsup><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mo href="./1.5#E3">/</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>y</mi></mrow></mrow></mrow></math> <em class="ltx_emph ltx_font_italic">converges uniformly</em>
on <math class="ltx_Math" altimg="m119.png" altimg-height="20px" altimg-valign="-5px" altimg-width="88px" alttext="a\leq x\leq b" display="inline"><mrow><mi>a</mi><mo>≤</mo><mi>x</mi><mo>≤</mo><mi>b</mi></mrow></math>, that is, given any positive number <math class="ltx_Math" altimg="m88.png" altimg-height="13px" altimg-valign="-2px" altimg-width="12px" alttext="\epsilon" display="inline"><mi href="./1.5#Px8.p1">ϵ</mi></math>, however
small, we can find a number <math class="ltx_Math" altimg="m122.png" altimg-height="23px" altimg-valign="-7px" altimg-width="87px" alttext="c_{0}\in[c,d)" display="inline"><mrow><msub><mi>c</mi><mn>0</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mrow><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">[</mo><mi>c</mi><mo href="./front/introduction#Sx4.p2.t1.r1">,</mo><mi>d</mi><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">)</mo></mrow></mrow></math> that is independent of <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math> and is
such that
</p>
<table id="E23" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.5.23</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E23.png" altimg-height="65px" altimg-valign="-27px" altimg-width="187px" alttext="\left|\int_{c_{1}}^{d}(\ifrac{\partial f}{\partial x})\mathrm{d}y\right|<\epsilon," display="block"><mrow><mrow><mrow><mo>|</mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><msub><mi>c</mi><mn>1</mn></msub><mi>d</mi></msubsup><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mo href="./1.5#E3">/</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>y</mi></mrow></mrow></mrow><mo>|</mo></mrow><mo><</mo><mi href="./1.5#Px8.p1">ϵ</mi></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E23.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m101.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m98.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m91.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m126.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m104.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./1.5#Px8.p1" title="Infinite Integrals ‣ §1.5(iv) Leibniz’s Theorem for Differentiation of Integrals ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m88.png" altimg-height="13px" altimg-valign="-2px" altimg-width="12px" alttext="\epsilon" display="inline"><mi href="./1.5#Px8.p1">ϵ</mi></math>: positive number</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">for all <math class="ltx_Math" altimg="m123.png" altimg-height="23px" altimg-valign="-7px" altimg-width="96px" alttext="c_{1}\in[c_{0},d)" display="inline"><mrow><msub><mi>c</mi><mn>1</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mrow><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">[</mo><msub><mi>c</mi><mn>0</mn></msub><mo href="./front/introduction#Sx4.p2.t1.r1">,</mo><mi>d</mi><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">)</mo></mrow></mrow></math> and all <math class="ltx_Math" altimg="m140.png" altimg-height="23px" altimg-valign="-7px" altimg-width="79px" alttext="x\in[a,b]" display="inline"><mrow><mi>x</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">[</mo><mi>a</mi><mo href="./front/introduction#Sx4.p1.t1.r29">,</mo><mi>b</mi><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">]</mo></mrow></mrow></math>. Then</p>
<table id="E24" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.5.24</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E24.png" altimg-height="56px" altimg-valign="-20px" altimg-width="263px" alttext="\frac{\mathrm{d}}{\mathrm{d}x}\int^{d}_{c}f(x,y)\mathrm{d}y=\int^{d}_{c}\frac{%
\partial f}{\partial x}\mathrm{d}y," display="block"><mrow><mrow><mrow><mfrac><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi>x</mi></mrow></mfrac><mo></mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi>c</mi><mi>d</mi></msubsup><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>y</mi></mrow></mrow></mrow></mrow><mo>=</mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi>c</mi><mi>d</mi></msubsup><mrow><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow></mfrac><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>y</mi></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m118.png" altimg-height="18px" altimg-valign="-3px" altimg-width="88px" alttext="a<x<b" display="inline"><mrow><mi>a</mi><mo><</mo><mi>x</mi><mo><</mo><mi>b</mi></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E24.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#E4" title="(1.4.4) ‣ §1.4(iii) Derivatives ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m89.png" altimg-height="29px" altimg-valign="-9px" altimg-width="27px" alttext="\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: derivative of <math class="ltx_Math" altimg="m126.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m101.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m98.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m91.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m126.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m104.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
</section>
<section id="v" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§1.5(v) </span>Multiple Integrals</h2>
<div id="SS5.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px9.p1" class="ltx_para">
<p class="ltx_p">Let <math class="ltx_Math" altimg="m125.png" altimg-height="23px" altimg-valign="-7px" altimg-width="62px" alttext="f(x,y)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></math> be defined on a closed rectangle <math class="ltx_Math" altimg="m74.png" altimg-height="23px" altimg-valign="-7px" altimg-width="148px" alttext="R=[a,b]\times[c,d]" display="inline"><mrow><mi href="./1.5#Px9.p1">R</mi><mo>=</mo><mrow><mrow><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">[</mo><mi>a</mi><mo href="./front/introduction#Sx4.p1.t1.r29">,</mo><mi>b</mi><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">]</mo></mrow><mo>×</mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">[</mo><mi>c</mi><mo href="./front/introduction#Sx4.p1.t1.r29">,</mo><mi>d</mi><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">]</mo></mrow></mrow></mrow></math>. For
</p>
<table id="EGx4" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E25">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.5.25</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m43.png" altimg-height="14px" altimg-valign="-2px" altimg-width="17px" alttext="\displaystyle a" display="inline"><mi>a</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m19.png" altimg-height="23px" altimg-valign="-6px" altimg-width="234px" alttext="\displaystyle=x_{0}<x_{1}<\cdots<x_{n}=b," display="inline"><mrow><mrow><mi></mi><mo>=</mo><msub><mi>x</mi><mn>0</mn></msub><mo><</mo><msub><mi>x</mi><mn>1</mn></msub><mo><</mo><mi mathvariant="normal">⋯</mi><mo><</mo><msub><mi>x</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo>=</mo><mi>b</mi></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E25.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m135.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a></dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E26">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.5.26</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m44.png" altimg-height="14px" altimg-valign="-2px" altimg-width="15px" alttext="\displaystyle c" display="inline"><mi>c</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m20.png" altimg-height="23px" altimg-valign="-6px" altimg-width="235px" alttext="\displaystyle=y_{0}<y_{1}<\cdots<y_{m}=d," display="inline"><mrow><mrow><mi></mi><mo>=</mo><msub><mi>y</mi><mn>0</mn></msub><mo><</mo><msub><mi>y</mi><mn>1</mn></msub><mo><</mo><mi mathvariant="normal">⋯</mi><mo><</mo><msub><mi>y</mi><mi href="./1.1#p2.t1.r5">m</mi></msub><mo>=</mo><mi>d</mi></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E26.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m132.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./1.1#p2.t1.r5">m</mi></math>: nonnegative integer</a></dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
<p class="ltx_p">let <math class="ltx_Math" altimg="m51.png" altimg-height="24px" altimg-valign="-8px" altimg-width="65px" alttext="(\xi_{j},\eta_{k})" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><msub><mi>ξ</mi><mi href="./1.1#p2.t1.r4">j</mi></msub><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><msub><mi>η</mi><mi href="./1.1#p2.t1.r4">k</mi></msub><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math> denote any point in the rectangle
<math class="ltx_Math" altimg="m80.png" altimg-height="24px" altimg-valign="-8px" altimg-width="188px" alttext="[x_{j},x_{j+1}]\times[y_{k},y_{k+1}]" display="inline"><mrow><mrow><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">[</mo><msub><mi>x</mi><mi href="./1.1#p2.t1.r4">j</mi></msub><mo href="./front/introduction#Sx4.p1.t1.r29">,</mo><msub><mi>x</mi><mrow><mi href="./1.1#p2.t1.r4">j</mi><mo>+</mo><mn>1</mn></mrow></msub><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">]</mo></mrow><mo>×</mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">[</mo><msub><mi>y</mi><mi href="./1.1#p2.t1.r4">k</mi></msub><mo href="./front/introduction#Sx4.p1.t1.r29">,</mo><msub><mi>y</mi><mrow><mi href="./1.1#p2.t1.r4">k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">]</mo></mrow></mrow></math>, <math class="ltx_Math" altimg="m128.png" altimg-height="20px" altimg-valign="-6px" altimg-width="141px" alttext="j=0,\dots,n-1" display="inline"><mrow><mi href="./1.1#p2.t1.r4">j</mi><mo>=</mo><mrow><mn>0</mn><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow></mrow></mrow></math>, <math class="ltx_Math" altimg="m130.png" altimg-height="21px" altimg-valign="-6px" altimg-width="148px" alttext="k=0,\dots,m-1" display="inline"><mrow><mi href="./1.1#p2.t1.r4">k</mi><mo>=</mo><mrow><mn>0</mn><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><mrow><mi href="./1.1#p2.t1.r5">m</mi><mo>-</mo><mn>1</mn></mrow></mrow></mrow></math>. Then
the <em class="ltx_emph ltx_font_italic">double integral</em> of <math class="ltx_Math" altimg="m125.png" altimg-height="23px" altimg-valign="-7px" altimg-width="62px" alttext="f(x,y)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></math> over <math class="ltx_Math" altimg="m75.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="R" display="inline"><mi href="./1.5#Px9.p1">R</mi></math> is defined by
</p>
<table id="E27" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.5.27</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E27.png" altimg-height="61px" altimg-valign="-31px" altimg-width="492px" alttext="\iint_{R}f(x,y)\mathrm{d}A={\lim\sum_{j,k}f(\xi_{j},\eta_{k})(x_{j+1}-x_{j})(y%
_{k+1}-y_{k})}" display="block"><mrow><mrow><msub><mo largeop="true" symmetric="true">∬</mo><mi href="./1.5#Px9.p1">R</mi></msub><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>A</mi></mrow></mrow></mrow><mo>=</mo><mrow><mo movablelimits="false">lim</mo><mo></mo><mrow><munder><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r4">j</mi><mo>,</mo><mi href="./1.1#p2.t1.r4">k</mi></mrow></munder><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><msub><mi>ξ</mi><mi href="./1.1#p2.t1.r4">j</mi></msub><mo>,</mo><msub><mi>η</mi><mi href="./1.1#p2.t1.r4">k</mi></msub><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><msub><mi>x</mi><mrow><mi href="./1.1#p2.t1.r4">j</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>-</mo><msub><mi>x</mi><mi href="./1.1#p2.t1.r4">j</mi></msub></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><msub><mi>y</mi><mrow><mi href="./1.1#p2.t1.r4">k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>-</mo><msub><mi>y</mi><mi href="./1.1#p2.t1.r4">k</mi></msub></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E27.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m101.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.1#p2.t1.r4" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m129.png" altimg-height="20px" altimg-valign="-6px" altimg-width="14px" alttext="j" display="inline"><mi href="./1.1#p2.t1.r4">j</mi></math>: integer</a>,
<a href="./1.1#p2.t1.r4" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m131.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./1.1#p2.t1.r4">k</mi></math>: integer</a> and
<a href="./1.5#Px9.p1" title="Double Integrals ‣ §1.5(v) Multiple Integrals ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m75.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="R" display="inline"><mi href="./1.5#Px9.p1">R</mi></math>: closed rectangle</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">as <math class="ltx_Math" altimg="m102.png" altimg-height="24px" altimg-valign="-8px" altimg-width="321px" alttext="\max((x_{j+1}-x_{j})+(y_{k+1}-y_{k}))\to 0" display="inline"><mrow><mrow><mi>max</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><msub><mi>x</mi><mrow><mi href="./1.1#p2.t1.r4">j</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>-</mo><msub><mi>x</mi><mi href="./1.1#p2.t1.r4">j</mi></msub></mrow><mo stretchy="false">)</mo></mrow><mo>+</mo><mrow><mo stretchy="false">(</mo><mrow><msub><mi>y</mi><mrow><mi href="./1.1#p2.t1.r4">k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>-</mo><msub><mi>y</mi><mi href="./1.1#p2.t1.r4">k</mi></msub></mrow><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>→</mo><mn>0</mn></mrow></math>. Sufficient conditions for the
limit to exist are that <math class="ltx_Math" altimg="m125.png" altimg-height="23px" altimg-valign="-7px" altimg-width="62px" alttext="f(x,y)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></math> is continuous, or piecewise continuous, on
<math class="ltx_Math" altimg="m75.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="R" display="inline"><mi href="./1.5#Px9.p1">R</mi></math>.</p>
</div>
<div id="Px9.p2" class="ltx_para">
<p class="ltx_p">For <math class="ltx_Math" altimg="m125.png" altimg-height="23px" altimg-valign="-7px" altimg-width="62px" alttext="f(x,y)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></math> defined on a point set <math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.5#Px9.p2">D</mi></math> contained in a rectangle <math class="ltx_Math" altimg="m75.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="R" display="inline"><mi href="./1.5#Px9.p1">R</mi></math>, let
</p>
<table id="E28" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.5.28</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E28.png" altimg-height="65px" altimg-valign="-27px" altimg-width="347px" alttext="f^{*}(x,y)=\begin{cases}f(x,y),&\mbox{if $(x,y)\in D$},\\
0,&\mbox{if $(x,y)\in R\setminus D$.}\end{cases}" display="block"><mrow><mrow><msup><mi>f</mi><mo>*</mo></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mo>{</mo><mtable columnspacing="5pt" displaystyle="true" rowspacing="0pt"><mtr><mtd columnalign="left"><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mrow><mtext>if </mtext><mrow><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi>x</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi>y</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mi href="./1.5#Px9.p2">D</mi></mrow></mrow><mo>,</mo></mrow></mtd></mtr><mtr><mtd columnalign="left"><mrow><mn>0</mn><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mtext>if </mtext><mrow><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi>x</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi>y</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mrow><mi href="./1.5#Px9.p1">R</mi><mo href="./front/introduction#Sx4.p2.t1.r19">∖</mo><mi href="./1.5#Px9.p2">D</mi></mrow></mrow><mtext>.</mtext></mrow></mtd></mtr></mtable></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E28.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./front/introduction#Sx4.p1.t1.r9" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m97.png" altimg-height="15px" altimg-valign="-3px" altimg-width="18px" alttext="\in" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo></math>: element of</a>,
<a href="./front/introduction#Sx4.p1.t1.r28" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="(\NVar{a},\NVar{b})" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi class="ltx_nvar">a</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi class="ltx_nvar">b</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math>: open interval</a>,
<a href="./front/introduction#Sx4.p2.t1.r19" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m112.png" altimg-height="23px" altimg-valign="-7px" altimg-width="14px" alttext="\setminus" display="inline"><mo href="./front/introduction#Sx4.p2.t1.r19">∖</mo></math>: set subtraction</a>,
<a href="./1.5#Px9.p1" title="Double Integrals ‣ §1.5(v) Multiple Integrals ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m75.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="R" display="inline"><mi href="./1.5#Px9.p1">R</mi></math>: closed rectangle</a> and
<a href="./1.5#Px9.p2" title="Double Integrals ‣ §1.5(v) Multiple Integrals ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.5#Px9.p2">D</mi></math>: region contained in <math class="ltx_Math" altimg="m75.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="R" display="inline"><mi href="./1.5#Px9.p1">R</mi></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Then
</p>
<table id="E29" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.5.29</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E29.png" altimg-height="52px" altimg-valign="-20px" altimg-width="298px" alttext="\iint_{D}f(x,y)\mathrm{d}A=\iint_{R}f^{*}(x,y)\mathrm{d}A," display="block"><mrow><mrow><mrow><msub><mo largeop="true" symmetric="true">∬</mo><mi href="./1.5#Px9.p2">D</mi></msub><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>A</mi></mrow></mrow></mrow><mo>=</mo><mrow><msub><mo largeop="true" symmetric="true">∬</mo><mi href="./1.5#Px9.p1">R</mi></msub><mrow><mrow><msup><mi>f</mi><mo>*</mo></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>A</mi></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E29.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m101.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.5#Px9.p1" title="Double Integrals ‣ §1.5(v) Multiple Integrals ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m75.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="R" display="inline"><mi href="./1.5#Px9.p1">R</mi></math>: closed rectangle</a> and
<a href="./1.5#Px9.p2" title="Double Integrals ‣ §1.5(v) Multiple Integrals ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.5#Px9.p2">D</mi></math>: region contained in <math class="ltx_Math" altimg="m75.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="R" display="inline"><mi href="./1.5#Px9.p1">R</mi></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">provided the latter integral exists.</p>
</div>
<div id="Px9.p3" class="ltx_para">
<p class="ltx_p">If <math class="ltx_Math" altimg="m125.png" altimg-height="23px" altimg-valign="-7px" altimg-width="62px" alttext="f(x,y)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></math> is continuous, and <math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.5#Px9.p2">D</mi></math> is the set</p>
<table id="E30" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex11" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">1.5.30</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m43.png" altimg-height="14px" altimg-valign="-2px" altimg-width="17px" alttext="\displaystyle a" display="inline"><mi>a</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m35.png" altimg-height="23px" altimg-valign="-6px" altimg-width="79px" alttext="\displaystyle\leq x\leq b," display="inline"><mrow><mrow><mi></mi><mo>≤</mo><mi>x</mi><mo>≤</mo><mi>b</mi></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex12" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m40.png" altimg-height="25px" altimg-valign="-7px" altimg-width="54px" alttext="\displaystyle\phi_{1}(x)" display="inline"><mrow><msub><mi>ϕ</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m38.png" altimg-height="25px" altimg-valign="-7px" altimg-width="117px" alttext="\displaystyle\leq y\leq\phi_{2}(x)," display="inline"><mrow><mrow><mi></mi><mo>≤</mo><mi>y</mi><mo>≤</mo><mrow><msub><mi>ϕ</mi><mn>2</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E30.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">with <math class="ltx_Math" altimg="m106.png" altimg-height="23px" altimg-valign="-7px" altimg-width="52px" alttext="\phi_{1}(x)" display="inline"><mrow><msub><mi>ϕ</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math> and <math class="ltx_Math" altimg="m107.png" altimg-height="23px" altimg-valign="-7px" altimg-width="52px" alttext="\phi_{2}(x)" display="inline"><mrow><msub><mi>ϕ</mi><mn>2</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math> continuous, then</p>
<table id="E31" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.5.31</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E31.png" altimg-height="61px" altimg-valign="-24px" altimg-width="361px" alttext="\iint_{D}f(x,y)\mathrm{d}A=\int^{b}_{a}\int^{\phi_{2}(x)}_{\phi_{1}(x)}f(x,y)%
\mathrm{d}y\mathrm{d}x," display="block"><mrow><mrow><mrow><msub><mo largeop="true" symmetric="true">∬</mo><mi href="./1.5#Px9.p2">D</mi></msub><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>A</mi></mrow></mrow></mrow><mo>=</mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi>a</mi><mi>b</mi></msubsup><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><msub><mi>ϕ</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow><mrow><msub><mi>ϕ</mi><mn>2</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></msubsup><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>y</mi></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>x</mi></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E31.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m101.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m98.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a> and
<a href="./1.5#Px9.p2" title="Double Integrals ‣ §1.5(v) Multiple Integrals ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.5#Px9.p2">D</mi></math>: region contained in <math class="ltx_Math" altimg="m75.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="R" display="inline"><mi href="./1.5#Px9.p1">R</mi></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where the right-hand side is interpreted as the repeated integral</p>
<table id="E32" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.5.32</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E32.png" altimg-height="65px" altimg-valign="-27px" altimg-width="243px" alttext="\int^{b}_{a}\left(\int^{\phi_{2}(x)}_{\phi_{1}(x)}f(x,y)\mathrm{d}y\right)%
\mathrm{d}x." display="block"><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi>a</mi><mi>b</mi></msubsup><mrow><mrow><mo>(</mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><msub><mi>ϕ</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow><mrow><msub><mi>ϕ</mi><mn>2</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></msubsup><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>y</mi></mrow></mrow></mrow><mo>)</mo></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>x</mi></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E32.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m101.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m98.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">In particular, <math class="ltx_Math" altimg="m106.png" altimg-height="23px" altimg-valign="-7px" altimg-width="52px" alttext="\phi_{1}(x)" display="inline"><mrow><msub><mi>ϕ</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math> and <math class="ltx_Math" altimg="m107.png" altimg-height="23px" altimg-valign="-7px" altimg-width="52px" alttext="\phi_{2}(x)" display="inline"><mrow><msub><mi>ϕ</mi><mn>2</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math> can be constants.</p>
</div>
<div id="Px9.p4" class="ltx_para">
<p class="ltx_p">Similarly, if <math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.5#Px9.p2">D</mi></math> is the set</p>
<table id="E33" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex13" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">1.5.33</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m44.png" altimg-height="14px" altimg-valign="-2px" altimg-width="15px" alttext="\displaystyle c" display="inline"><mi>c</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m37.png" altimg-height="23px" altimg-valign="-6px" altimg-width="80px" alttext="\displaystyle\leq y\leq d," display="inline"><mrow><mrow><mi></mi><mo>≤</mo><mi>y</mi><mo>≤</mo><mi>d</mi></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex14" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m42.png" altimg-height="25px" altimg-valign="-7px" altimg-width="54px" alttext="\displaystyle\psi_{1}(y)" display="inline"><mrow><msub><mi>ψ</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m36.png" altimg-height="25px" altimg-valign="-7px" altimg-width="118px" alttext="\displaystyle\leq x\leq\psi_{2}(y)," display="inline"><mrow><mrow><mi></mi><mo>≤</mo><mi>x</mi><mo>≤</mo><mrow><msub><mi>ψ</mi><mn>2</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E33.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">with <math class="ltx_Math" altimg="m109.png" altimg-height="23px" altimg-valign="-7px" altimg-width="52px" alttext="\psi_{1}(y)" display="inline"><mrow><msub><mi>ψ</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></math> and <math class="ltx_Math" altimg="m110.png" altimg-height="23px" altimg-valign="-7px" altimg-width="52px" alttext="\psi_{2}(y)" display="inline"><mrow><msub><mi>ψ</mi><mn>2</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></math> continuous, then</p>
<table id="E34" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.5.34</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E34.png" altimg-height="61px" altimg-valign="-24px" altimg-width="363px" alttext="\iint_{D}f(x,y)\mathrm{d}A=\int^{d}_{c}\int^{\psi_{2}(y)}_{\psi_{1}(y)}f(x,y)%
\mathrm{d}x\mathrm{d}y." display="block"><mrow><mrow><mrow><msub><mo largeop="true" symmetric="true">∬</mo><mi href="./1.5#Px9.p2">D</mi></msub><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>A</mi></mrow></mrow></mrow><mo>=</mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi>c</mi><mi>d</mi></msubsup><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><msub><mi>ψ</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow><mrow><msub><mi>ψ</mi><mn>2</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></msubsup><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>x</mi></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>y</mi></mrow></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E34.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m101.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m98.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a> and
<a href="./1.5#Px9.p2" title="Double Integrals ‣ §1.5(v) Multiple Integrals ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.5#Px9.p2">D</mi></math>: region contained in <math class="ltx_Math" altimg="m75.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="R" display="inline"><mi href="./1.5#Px9.p1">R</mi></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px10" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Change of Order of Integration</h3>
<div id="Px10.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px10.p1" class="ltx_para">
<p class="ltx_p">If <math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.5#Px9.p2">D</mi></math> can be represented in both forms (), and <math class="ltx_Math" altimg="m125.png" altimg-height="23px" altimg-valign="-7px" altimg-width="62px" alttext="f(x,y)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></math> is continuous on <math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.5#Px9.p2">D</mi></math>, then</p>
<table id="E35" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.5.35</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E35.png" altimg-height="61px" altimg-valign="-24px" altimg-width="433px" alttext="\int^{b}_{a}\int^{\phi_{2}(x)}_{\phi_{1}(x)}f(x,y)\mathrm{d}y\mathrm{d}x=\int^%
{d}_{c}\int^{\psi_{2}(y)}_{\psi_{1}(y)}f(x,y)\mathrm{d}x\mathrm{d}y." display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi>a</mi><mi>b</mi></msubsup><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><msub><mi>ϕ</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow><mrow><msub><mi>ϕ</mi><mn>2</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></msubsup><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>y</mi></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>x</mi></mrow></mrow></mrow></mrow><mo>=</mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi>c</mi><mi>d</mi></msubsup><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><msub><mi>ψ</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow><mrow><msub><mi>ψ</mi><mn>2</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></msubsup><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>x</mi></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>y</mi></mrow></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E35.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m101.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m98.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px11" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Infinite Double Integrals</h3>
<div id="Px11.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px11.p1" class="ltx_para">
<p class="ltx_p">Infinite double integrals occur when <math class="ltx_Math" altimg="m125.png" altimg-height="23px" altimg-valign="-7px" altimg-width="62px" alttext="f(x,y)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></math> becomes infinite at points in <math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.5#Px9.p2">D</mi></math>
or when <math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.5#Px9.p2">D</mi></math> is unbounded. In the cases ().</p>
</div>
<div id="Px11.p2" class="ltx_para">
<p class="ltx_p">Moreover, if <math class="ltx_Math" altimg="m115.png" altimg-height="21px" altimg-valign="-6px" altimg-width="69px" alttext="a,b,c,d" display="inline"><mrow><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>,</mo><mi>d</mi></mrow></math> are finite or infinite constants and <math class="ltx_Math" altimg="m125.png" altimg-height="23px" altimg-valign="-7px" altimg-width="62px" alttext="f(x,y)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></math> is
piecewise continuous on the set <math class="ltx_Math" altimg="m53.png" altimg-height="23px" altimg-valign="-7px" altimg-width="115px" alttext="(a,b)\times(c,d)" display="inline"><mrow><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi>a</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi>b</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow><mo>×</mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi>c</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi>d</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></mrow></math>, then</p>
<table id="E36" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.5.36</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E36.png" altimg-height="56px" altimg-valign="-20px" altimg-width="370px" alttext="\int^{b}_{a}\int^{d}_{c}f(x,y)\mathrm{d}y\mathrm{d}x=\int^{d}_{c}\int^{b}_{a}f%
(x,y)\mathrm{d}x\mathrm{d}y," display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi>a</mi><mi>b</mi></msubsup><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi>c</mi><mi>d</mi></msubsup><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>y</mi></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>x</mi></mrow></mrow></mrow></mrow><mo>=</mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi>c</mi><mi>d</mi></msubsup><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi>a</mi><mi>b</mi></msubsup><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>x</mi></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>y</mi></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E36.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m101.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m98.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">whenever both repeated integrals exist and at least one is absolutely
convergent.</p>
</div>
</section>
<section id="Px12" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Triple Integrals</h3>
<div id="Px12.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px12.p1" class="ltx_para">
<p class="ltx_p">Finite and infinite integrals can be defined in a similar way. Often the
<math class="ltx_Math" altimg="m55.png" altimg-height="23px" altimg-valign="-7px" altimg-width="70px" alttext="(x,y,z)" display="inline"><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></math> sets are of the form</p>
<table id="E37" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex15" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="3" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">1.5.37</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m43.png" altimg-height="14px" altimg-valign="-2px" altimg-width="17px" alttext="\displaystyle a" display="inline"><mi>a</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m35.png" altimg-height="23px" altimg-valign="-6px" altimg-width="79px" alttext="\displaystyle\leq x\leq b," display="inline"><mrow><mrow><mi></mi><mo>≤</mo><mi>x</mi><mo>≤</mo><mi>b</mi></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex16" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m40.png" altimg-height="25px" altimg-valign="-7px" altimg-width="54px" alttext="\displaystyle\phi_{1}(x)" display="inline"><mrow><msub><mi>ϕ</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m38.png" altimg-height="25px" altimg-valign="-7px" altimg-width="117px" alttext="\displaystyle\leq y\leq\phi_{2}(x)," display="inline"><mrow><mrow><mi></mi><mo>≤</mo><mi>y</mi><mo>≤</mo><mrow><msub><mi>ϕ</mi><mn>2</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex17" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m41.png" altimg-height="25px" altimg-valign="-7px" altimg-width="74px" alttext="\displaystyle\psi_{1}(x,y)" display="inline"><mrow><msub><mi>ψ</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m39.png" altimg-height="25px" altimg-valign="-7px" altimg-width="137px" alttext="\displaystyle\leq z\leq\psi_{2}(x,y)." display="inline"><mrow><mrow><mi></mi><mo>≤</mo><mi href="./1.1#p2.t1.r2">z</mi><mo>≤</mo><mrow><msub><mi>ψ</mi><mn>2</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E37.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m144.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a></dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
</section>
<section id="vi" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§1.5(vi) </span>Jacobians and Change of Variables</h2>
<div id="SS6.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px13.p1" class="ltx_para">
<table id="EGx5" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E38">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.5.38</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m25.png" altimg-height="53px" altimg-valign="-21px" altimg-width="69px" alttext="\displaystyle\frac{\partial(f,g)}{\partial(x,y)}" display="inline"><mstyle displaystyle="true"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi>f</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi>g</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi>x</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi>y</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></mrow></mfrac></mstyle></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m4.png" altimg-height="53px" altimg-valign="-21px" altimg-width="191px" alttext="\displaystyle=\begin{vmatrix}\ifrac{\partial f}{\partial x}&\ifrac{\partial f}%
{\partial y}\\
\ifrac{\partial g}{\partial x}&\ifrac{\partial g}{\partial y}\end{vmatrix}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo href="./1.3#i">|</mo><mtable columnspacing="5pt" rowspacing="0pt"><mtr><mtd columnalign="center"><mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mo href="./1.5#E3">/</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow></mrow></mtd><mtd columnalign="center"><mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mo href="./1.5#E3">/</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow></mrow></mtd></mtr><mtr><mtd columnalign="center"><mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>g</mi></mrow><mo href="./1.5#E3">/</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow></mrow></mtd><mtd columnalign="center"><mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>g</mi></mrow><mo href="./1.5#E3">/</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow></mrow></mtd></mtr></mtable><mo href="./1.3#i">|</mo></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E38.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.3#i" title="§1.3(i) Definitions and Elementary Properties ‣ §1.3 Determinants ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="18px" altimg-valign="-2px" altimg-width="32px" alttext="\det" display="inline"><mo href="./1.3#i">det</mo></math>: determinant</a>,
<a href="./front/introduction#Sx4.p1.t1.r28" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="(\NVar{a},\NVar{b})" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi class="ltx_nvar">a</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi class="ltx_nvar">b</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math>: open interval</a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m91.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m126.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m104.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E39">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.5.39</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m27.png" altimg-height="53px" altimg-valign="-21px" altimg-width="69px" alttext="\displaystyle\frac{\partial(x,y)}{\partial(r,\phi)}" display="inline"><mstyle displaystyle="true"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi>x</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi>y</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi href="./1.5#Px4.p1">r</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi href="./1.5#Px4.p1">ϕ</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></mrow></mfrac></mstyle></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m17.png" altimg-height="25px" altimg-valign="-7px" altimg-width="228px" alttext="\displaystyle=r\quad\text{(polar coordinates)}." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./1.5#Px4.p1">r</mi><mo separator="true"> </mo><mtext>(polar coordinates)</mtext></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E39.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./front/introduction#Sx4.p1.t1.r28" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="(\NVar{a},\NVar{b})" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi class="ltx_nvar">a</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi class="ltx_nvar">b</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math>: open interval</a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m91.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m126.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m104.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.5#Px4.p1" title="Polar Coordinates ‣ §1.5(ii) Coordinate Systems ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m136.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="r" display="inline"><mi href="./1.5#Px5.p1">r</mi></math>: radius</a> and
<a href="./1.5#Px4.p1" title="Polar Coordinates ‣ §1.5(ii) Coordinate Systems ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m105.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="\phi" display="inline"><mi href="./1.5#Px6.p1">ϕ</mi></math>: angle</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E40">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.5.40</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m26.png" altimg-height="53px" altimg-valign="-21px" altimg-width="88px" alttext="\displaystyle\frac{\partial(f,g,h)}{\partial(x,y,z)}" display="inline"><mstyle displaystyle="true"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mrow><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo>,</mo><mi>h</mi><mo stretchy="false">)</mo></mrow></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></mfrac></mstyle></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m3.png" altimg-height="77px" altimg-valign="-33px" altimg-width="271px" alttext="\displaystyle=\begin{vmatrix}\ifrac{\partial f}{\partial x}&\ifrac{\partial f}%
{\partial y}&\ifrac{\partial f}{\partial z}\\
\ifrac{\partial g}{\partial x}&\ifrac{\partial g}{\partial y}&\ifrac{\partial g%
}{\partial z}\\
\ifrac{\partial h}{\partial x}&\ifrac{\partial h}{\partial y}&\ifrac{\partial h%
}{\partial z}\end{vmatrix}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo href="./1.3#i">|</mo><mtable columnspacing="5pt" rowspacing="0pt"><mtr><mtd columnalign="center"><mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mo href="./1.5#E3">/</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow></mrow></mtd><mtd columnalign="center"><mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mo href="./1.5#E3">/</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow></mrow></mtd><mtd columnalign="center"><mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mo href="./1.5#E3">/</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>z</mi></mrow></mrow></mtd></mtr><mtr><mtd columnalign="center"><mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>g</mi></mrow><mo href="./1.5#E3">/</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow></mrow></mtd><mtd columnalign="center"><mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>g</mi></mrow><mo href="./1.5#E3">/</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow></mrow></mtd><mtd columnalign="center"><mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>g</mi></mrow><mo href="./1.5#E3">/</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>z</mi></mrow></mrow></mtd></mtr><mtr><mtd columnalign="center"><mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>h</mi></mrow><mo href="./1.5#E3">/</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow></mrow></mtd><mtd columnalign="center"><mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>h</mi></mrow><mo href="./1.5#E3">/</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow></mrow></mtd><mtd columnalign="center"><mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>h</mi></mrow><mo href="./1.5#E3">/</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>z</mi></mrow></mrow></mtd></mtr></mtable><mo href="./1.3#i">|</mo></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E40.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.3#i" title="§1.3(i) Definitions and Elementary Properties ‣ §1.3 Determinants ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="18px" altimg-valign="-2px" altimg-width="32px" alttext="\det" display="inline"><mo href="./1.3#i">det</mo></math>: determinant</a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m91.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m126.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m104.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E41">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.5.41</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m28.png" altimg-height="53px" altimg-valign="-21px" altimg-width="88px" alttext="\displaystyle\frac{\partial(x,y,z)}{\partial(\rho,\theta,\phi)}" display="inline"><mstyle displaystyle="true"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mrow><mo stretchy="false">(</mo><mi href="./1.5#Px6.p1">ρ</mi><mo>,</mo><mi href="./1.5#Px6.p1">θ</mi><mo>,</mo><mi href="./1.5#Px6.p1">ϕ</mi><mo stretchy="false">)</mo></mrow></mrow></mfrac></mstyle></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m14.png" altimg-height="28px" altimg-valign="-7px" altimg-width="311px" alttext="\displaystyle=\rho^{2}\sin\theta\quad\text{(spherical coordinates)}." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><msup><mi href="./1.5#Px6.p1">ρ</mi><mn>2</mn></msup><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi href="./1.5#Px6.p1">θ</mi></mrow></mrow><mo separator="true"> </mo><mtext>(spherical coordinates)</mtext></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E41.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m91.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m126.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m104.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m113.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./1.5#Px6.p1" title="Spherical Coordinates ‣ §1.5(ii) Coordinate Systems ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m105.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="\phi" display="inline"><mi href="./1.5#Px6.p1">ϕ</mi></math>: longitude</a>,
<a href="./1.5#Px6.p1" title="Spherical Coordinates ‣ §1.5(ii) Coordinate Systems ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m111.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="\rho" display="inline"><mi href="./1.5#Px6.p1">ρ</mi></math>: radius</a> and
<a href="./1.5#Px6.p1" title="Spherical Coordinates ‣ §1.5(ii) Coordinate Systems ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m114.png" altimg-height="18px" altimg-valign="-2px" altimg-width="14px" alttext="\theta" display="inline"><mi href="./1.5#Px6.p1">θ</mi></math>: azimuth</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
</div>
</section>
<section id="Px14" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Change of Variables</h3>
<div id="Px14.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px14.p1" class="ltx_para">
<table id="E42" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.5.42</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E42.png" altimg-height="53px" altimg-valign="-21px" altimg-width="510px" alttext="\iint_{D}f(x,y)\mathrm{d}x\mathrm{d}y=\iint_{D^{*}}f(x(u,v),y(u,v))\left|\frac%
{\partial(x,y)}{\partial(u,v)}\right|\mathrm{d}u\mathrm{d}v," display="block"><mrow><mrow><mrow><msub><mo largeop="true" symmetric="true">∬</mo><mi>D</mi></msub><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>x</mi></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>y</mi></mrow></mrow></mrow><mo>=</mo><mrow><msub><mo largeop="true" symmetric="true">∬</mo><msup><mi>D</mi><mo>*</mo></msup></msub><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>x</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo><mrow><mi>y</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo>|</mo><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi>x</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi>y</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi>u</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi>v</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></mrow></mfrac><mo>|</mo></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>u</mi></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>v</mi></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E42.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m101.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./front/introduction#Sx4.p1.t1.r28" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="(\NVar{a},\NVar{b})" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi class="ltx_nvar">a</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi class="ltx_nvar">b</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math>: open interval</a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m91.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m126.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m104.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi>D</mi></math> is the image of <math class="ltx_Math" altimg="m64.png" altimg-height="18px" altimg-valign="-2px" altimg-width="31px" alttext="D^{*}" display="inline"><msup><mi>D</mi><mo>*</mo></msup></math> under a mapping <math class="ltx_Math" altimg="m54.png" altimg-height="23px" altimg-valign="-7px" altimg-width="220px" alttext="(u,v)\to(x(u,v),y(u,v))" display="inline"><mrow><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi>u</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi>v</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow><mo>→</mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mrow><mi>x</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></mrow><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mrow><mi>y</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></mrow></math>
which is one-to-one except perhaps for a set of points of area zero.</p>
<table id="E43" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.5.43</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_Math ltx_eqn_cell ltx_align_center"><math class="ltx_Math" alttext="\iiint_{D}f(x,y,z)\mathrm{d}x\mathrm{d}y\mathrm{d}z=\iiint_{D^{*}}f(x(u,v,w),y%
(u,v,w),z(u,v,w))\*\left|\frac{\partial(x,y,z)}{\partial(u,v,w)}\right|\mathrm%
{d}u\mathrm{d}v\mathrm{d}w." display="block"><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mrow><msub><mo largeop="true" symmetric="true">∭</mo><mi>D</mi></msub><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>x</mi></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>y</mi></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>z</mi></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>=</mo><mrow><msub><mo largeop="true" symmetric="true">∭</mo><msup><mi>D</mi><mo>*</mo></msup></msub><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>x</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>,</mo><mi href="./1.1#p2.t1.r3">w</mi><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo><mrow><mi>y</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>,</mo><mi href="./1.1#p2.t1.r3">w</mi><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo><mrow><mi>z</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>,</mo><mi href="./1.1#p2.t1.r3">w</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo>|</mo><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mrow><mo stretchy="false">(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>,</mo><mi href="./1.1#p2.t1.r3">w</mi><mo stretchy="false">)</mo></mrow></mrow></mfrac><mo>|</mo></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>u</mi></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>v</mi></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./1.1#p2.t1.r3">w</mi></mrow></mrow></mrow><mo>.</mo></mtd></mtr></mtable></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E43.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m101.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m91.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m126.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m104.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./1.1#p2.t1.r3" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m137.png" altimg-height="13px" altimg-valign="-2px" altimg-width="19px" alttext="w" display="inline"><mi href="./1.1#p2.t1.r3">w</mi></math>: variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Again the mapping is one-to-one except perhaps for a set of points of volume
zero.
</p>
</div>
</section>
</section>
</section>
</div>
<div class="ltx_page_footer">
<span></div>
</div>
</body></text>
</html>
</page>
<page>
<!DOCTYPE html><html>
<head>
<title>DLMF: 1.10 Functions of a Complex Variable</title>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<!--GOOGLE BOOTSTRAP--></head>
<text><body class="color_default textfont_default titlefont_default fontsize_default navbar_default">
<div class="ltx_page_navbar">
<div class="ltx_page_navlogo"></dd>
</dl>
</div>
</div>
<div id="SS1.p1" class="ltx_para">
<p class="ltx_p">Let <math class="ltx_Math" altimg="m120.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(z)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> be analytic on the disk
<math class="ltx_Math" altimg="m191.png" altimg-height="23px" altimg-valign="-7px" altimg-width="110px" alttext="|z-z_{0}|<R" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>-</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></mrow><mo stretchy="false">|</mo></mrow><mo><</mo><mi href="./1.10#SS1.p1">R</mi></mrow></math>. Then</p>
<table id="E1" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.10.1</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E1.png" altimg-height="64px" altimg-valign="-27px" altimg-width="262px" alttext="f(z)=\sum^{\infty}_{n=0}\frac{f^{(n)}(z_{0})}{n!}(z-z_{0})^{n}." display="block"><mrow><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><mfrac><mrow><msup><mi>f</mi><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r5">n</mi><mo stretchy="false">)</mo></mrow></msup><mo></mo><mrow><mo stretchy="false">(</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mrow><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mfrac><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>-</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></mrow><mo stretchy="false">)</mo></mrow><mi href="./1.1#p2.t1.r5">n</mi></msup></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m1.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m139.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m177.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a> and
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m143.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">The right-hand side is the <em class="ltx_emph ltx_font_italic">Taylor series for</em> <math class="ltx_Math" altimg="m120.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(z)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> <em class="ltx_emph ltx_font_italic">at</em> <math class="ltx_Math" altimg="m175.png" altimg-height="16px" altimg-valign="-5px" altimg-width="59px" alttext="z=z_{0}" display="inline"><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>=</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></mrow></math>,
and its radius of convergence is at least <math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="R" display="inline"><mi href="./1.10#SS1.p1">R</mi></math>.
</p>
</div>
<section id="Px1" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Examples</h3>
<div id="Px1.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px1.p1" class="ltx_para">
<table id="E2" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.10.2</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E2.png" altimg-height="50px" altimg-valign="-16px" altimg-width="210px" alttext="e^{z}=1+\frac{z}{1!}+\frac{z^{2}}{2!}+\cdots," display="block"><mrow><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mi href="./1.1#p2.t1.r2">z</mi></msup><mo>=</mo><mrow><mn>1</mn><mo>+</mo><mfrac><mi href="./1.1#p2.t1.r2">z</mi><mrow><mn>1</mn><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mfrac><mo>+</mo><mfrac><msup><mi href="./1.1#p2.t1.r2">z</mi><mn>2</mn></msup><mrow><mn>2</mn><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mfrac><mo>+</mo><mi mathvariant="normal">⋯</mi></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m194.png" altimg-height="23px" altimg-valign="-7px" altimg-width="72px" alttext="|z|<\infty" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">|</mo></mrow><mo><</mo><mi mathvariant="normal">∞</mi></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m75.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m1.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m139.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a> and
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m177.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E3" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.10.3</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E3.png" altimg-height="50px" altimg-valign="-16px" altimg-width="276px" alttext="\ln\left(1+z\right)=z-\frac{z^{2}}{2}+\frac{z^{3}}{3}-\cdots," display="block"><mrow><mrow><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mi href="./1.1#p2.t1.r2">z</mi></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>-</mo><mfrac><msup><mi href="./1.1#p2.t1.r2">z</mi><mn>2</mn></msup><mn>2</mn></mfrac></mrow><mo>+</mo><mfrac><msup><mi href="./1.1#p2.t1.r2">z</mi><mn>3</mn></msup><mn>3</mn></mfrac></mrow><mo>-</mo><mi mathvariant="normal">⋯</mi></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m192.png" altimg-height="23px" altimg-valign="-7px" altimg-width="62px" alttext="|z|<1" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">|</mo></mrow><mo><</mo><mn>1</mn></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a> and
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m177.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E4" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.10.4</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E4.png" altimg-height="48px" altimg-valign="-16px" altimg-width="540px" alttext="(1-z)^{-\alpha}=1+\alpha z+\frac{\alpha(\alpha+1)}{2!}z^{2}+\frac{\alpha(%
\alpha+1)(\alpha+2)}{3!}z^{3}+\cdots," display="block"><mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><mi href="./1.1#p2.t1.r2">z</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mi>α</mi></mrow></msup><mo>=</mo><mrow><mn>1</mn><mo>+</mo><mrow><mi>α</mi><mo></mo><mi href="./1.1#p2.t1.r2">z</mi></mrow><mo>+</mo><mrow><mfrac><mrow><mi>α</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>α</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mrow><mn>2</mn><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mfrac><mo></mo><msup><mi href="./1.1#p2.t1.r2">z</mi><mn>2</mn></msup></mrow><mo>+</mo><mrow><mfrac><mrow><mi>α</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>α</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>α</mi><mo>+</mo><mn>2</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mrow><mn>3</mn><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mfrac><mo></mo><msup><mi href="./1.1#p2.t1.r2">z</mi><mn>3</mn></msup></mrow><mo>+</mo><mi mathvariant="normal">⋯</mi></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m192.png" altimg-height="23px" altimg-valign="-7px" altimg-width="62px" alttext="|z|<1" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">|</mo></mrow><mo><</mo><mn>1</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E4.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m1.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m139.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a> and
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m177.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Note that ().
Again, in these examples <math class="ltx_Math" altimg="m71.png" altimg-height="23px" altimg-valign="-7px" altimg-width="84px" alttext="\ln\left(1+z\right)" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mi href="./1.1#p2.t1.r2">z</mi></mrow><mo>)</mo></mrow></mrow></math> and <math class="ltx_Math" altimg="m4.png" altimg-height="24px" altimg-valign="-7px" altimg-width="88px" alttext="(1-z)^{-\alpha}" display="inline"><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><mi href="./1.1#p2.t1.r2">z</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mi>α</mi></mrow></msup></math> have their principal
values; see §§</dd>
</dl>
</div>
</div>
<div id="Px2.p1" class="ltx_para">
<p class="ltx_p">An analytic function <math class="ltx_Math" altimg="m120.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(z)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> has a <em class="ltx_emph ltx_font_italic">zero of order</em> (or <em class="ltx_emph ltx_font_italic">multiplicity</em>)
<math class="ltx_Math" altimg="m136.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./1.1#p2.t1.r5">m</mi></math> (<math class="ltx_Math" altimg="m63.png" altimg-height="19px" altimg-valign="-5px" altimg-width="32px" alttext="\geq\!1" display="inline"><mrow><mi></mi><mo rspace="0.8pt">≥</mo><mn>1</mn></mrow></math>)
at <math class="ltx_Math" altimg="m181.png" altimg-height="16px" altimg-valign="-5px" altimg-width="23px" alttext="z_{0}" display="inline"><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></math> if the first nonzero coefficient in its Taylor series at <math class="ltx_Math" altimg="m181.png" altimg-height="16px" altimg-valign="-5px" altimg-width="23px" alttext="z_{0}" display="inline"><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></math> is that
of <math class="ltx_Math" altimg="m14.png" altimg-height="23px" altimg-valign="-7px" altimg-width="88px" alttext="(z-z_{0})^{m}" display="inline"><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>-</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></mrow><mo stretchy="false">)</mo></mrow><mi href="./1.1#p2.t1.r5">m</mi></msup></math>. When <math class="ltx_Math" altimg="m135.png" altimg-height="17px" altimg-valign="-2px" altimg-width="58px" alttext="m=1" display="inline"><mrow><mi href="./1.1#p2.t1.r5">m</mi><mo>=</mo><mn>1</mn></mrow></math> the zero is <em class="ltx_emph ltx_font_italic">simple</em>.
</p>
</div>
</section>
</section>
<section id="ii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§1.10(ii) </span>Analytic Continuation</h2>
<div id="SS2.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="SS2.p1" class="ltx_para">
<p class="ltx_p">Let <math class="ltx_Math" altimg="m126.png" altimg-height="23px" altimg-valign="-7px" altimg-width="49px" alttext="f_{1}(z)" display="inline"><mrow><msub><mi>f</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math> be analytic in a domain <math class="ltx_Math" altimg="m34.png" altimg-height="21px" altimg-valign="-5px" altimg-width="30px" alttext="D_{1}" display="inline"><msub><mi href="./1.10#SS2.p1">D</mi><mn>1</mn></msub></math>. If <math class="ltx_Math" altimg="m127.png" altimg-height="23px" altimg-valign="-7px" altimg-width="49px" alttext="f_{2}(z)" display="inline"><mrow><msub><mi>f</mi><mn>2</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math>, analytic in <math class="ltx_Math" altimg="m35.png" altimg-height="21px" altimg-valign="-5px" altimg-width="30px" alttext="D_{2}" display="inline"><msub><mi href="./1.10#SS2.p1">D</mi><mn>2</mn></msub></math>,
equals <math class="ltx_Math" altimg="m126.png" altimg-height="23px" altimg-valign="-7px" altimg-width="49px" alttext="f_{1}(z)" display="inline"><mrow><msub><mi>f</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math> on an arc in <math class="ltx_Math" altimg="m30.png" altimg-height="21px" altimg-valign="-5px" altimg-width="121px" alttext="D=D_{1}\cap D_{2}" display="inline"><mrow><mi href="./1.10#SS2.p1">D</mi><mo>=</mo><mrow><msub><mi href="./1.10#SS2.p1">D</mi><mn>1</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r26">∩</mo><msub><mi href="./1.10#SS2.p1">D</mi><mn>2</mn></msub></mrow></mrow></math>, or on just an infinite number
of points with a limit point in <math class="ltx_Math" altimg="m33.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.10#SS2.p1">D</mi></math>, then they are equal throughout <math class="ltx_Math" altimg="m33.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.10#SS2.p1">D</mi></math> and
<math class="ltx_Math" altimg="m127.png" altimg-height="23px" altimg-valign="-7px" altimg-width="49px" alttext="f_{2}(z)" display="inline"><mrow><msub><mi>f</mi><mn>2</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math> is called an <em class="ltx_emph ltx_font_italic">analytic continuation</em> of <math class="ltx_Math" altimg="m126.png" altimg-height="23px" altimg-valign="-7px" altimg-width="49px" alttext="f_{1}(z)" display="inline"><mrow><msub><mi>f</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math>. We write
<math class="ltx_Math" altimg="m7.png" altimg-height="23px" altimg-valign="-7px" altimg-width="73px" alttext="(f_{1},D_{1})" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><msub><mi>f</mi><mn>1</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><msub><mi href="./1.10#SS2.p1">D</mi><mn>1</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math>, <math class="ltx_Math" altimg="m9.png" altimg-height="23px" altimg-valign="-7px" altimg-width="73px" alttext="(f_{2},D_{2})" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><msub><mi>f</mi><mn>2</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><msub><mi href="./1.10#SS2.p1">D</mi><mn>2</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math> to signify this continuation.</p>
</div>
<div id="SS2.p2" class="ltx_para">
<p class="ltx_p">Suppose <math class="ltx_Math" altimg="m167.png" altimg-height="23px" altimg-valign="-7px" altimg-width="162px" alttext="z(t)=x(t)+iy(t)" display="inline"><mrow><mrow><mi>z</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mi>x</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mrow><mi>y</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow></mrow></math>, <math class="ltx_Math" altimg="m100.png" altimg-height="20px" altimg-valign="-5px" altimg-width="84px" alttext="a\leq t\leq b" display="inline"><mrow><mi>a</mi><mo>≤</mo><mi>t</mi><mo>≤</mo><mi>b</mi></mrow></math>, is an arc and <math class="ltx_Math" altimg="m98.png" altimg-height="21px" altimg-valign="-5px" altimg-width="230px" alttext="a=t_{0}<t_{1}<\cdots<t_{n}=b" display="inline"><mrow><mi>a</mi><mo>=</mo><msub><mi>t</mi><mn>0</mn></msub><mo><</mo><msub><mi>t</mi><mn>1</mn></msub><mo><</mo><mi mathvariant="normal">⋯</mi><mo><</mo><msub><mi>t</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo>=</mo><mi>b</mi></mrow></math>. Suppose the subarc <math class="ltx_Math" altimg="m168.png" altimg-height="23px" altimg-valign="-7px" altimg-width="37px" alttext="z(t)" display="inline"><mrow><mi>z</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math>, <math class="ltx_Math" altimg="m159.png" altimg-height="24px" altimg-valign="-8px" altimg-width="107px" alttext="t\in[t_{j-1},t_{j}]" display="inline"><mrow><mi>t</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">[</mo><msub><mi>t</mi><mrow><mi href="./1.1#p2.t1.r4">j</mi><mo>-</mo><mn>1</mn></mrow></msub><mo href="./front/introduction#Sx4.p1.t1.r29">,</mo><msub><mi>t</mi><mi href="./1.1#p2.t1.r4">j</mi></msub><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">]</mo></mrow></mrow></math>
is contained in a domain <math class="ltx_Math" altimg="m36.png" altimg-height="23px" altimg-valign="-8px" altimg-width="29px" alttext="D_{j}" display="inline"><msub><mi href="./1.10#SS2.p1">D</mi><mi href="./1.1#p2.t1.r4">j</mi></msub></math>, <math class="ltx_Math" altimg="m132.png" altimg-height="20px" altimg-valign="-6px" altimg-width="106px" alttext="j=1,\dots,n" display="inline"><mrow><mi href="./1.1#p2.t1.r4">j</mi><mo>=</mo><mrow><mn>1</mn><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><mi href="./1.1#p2.t1.r5">n</mi></mrow></mrow></math>. The function <math class="ltx_Math" altimg="m126.png" altimg-height="23px" altimg-valign="-7px" altimg-width="49px" alttext="f_{1}(z)" display="inline"><mrow><msub><mi>f</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math> on <math class="ltx_Math" altimg="m34.png" altimg-height="21px" altimg-valign="-5px" altimg-width="30px" alttext="D_{1}" display="inline"><msub><mi href="./1.10#SS2.p1">D</mi><mn>1</mn></msub></math>
is said to be <em class="ltx_emph ltx_font_italic">analytically continued along the path</em> <math class="ltx_Math" altimg="m168.png" altimg-height="23px" altimg-valign="-7px" altimg-width="37px" alttext="z(t)" display="inline"><mrow><mi>z</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math>,
<math class="ltx_Math" altimg="m100.png" altimg-height="20px" altimg-valign="-5px" altimg-width="84px" alttext="a\leq t\leq b" display="inline"><mrow><mi>a</mi><mo>≤</mo><mi>t</mi><mo>≤</mo><mi>b</mi></mrow></math>, if there is a chain <math class="ltx_Math" altimg="m7.png" altimg-height="23px" altimg-valign="-7px" altimg-width="73px" alttext="(f_{1},D_{1})" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><msub><mi>f</mi><mn>1</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><msub><mi href="./1.10#SS2.p1">D</mi><mn>1</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math>, <math class="ltx_Math" altimg="m8.png" altimg-height="23px" altimg-valign="-7px" altimg-width="189px" alttext="(f_{2},D_{2}),\dots,(f_{n},D_{n})" display="inline"><mrow><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><msub><mi>f</mi><mn>2</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><msub><mi href="./1.10#SS2.p1">D</mi><mn>2</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><msub><mi>f</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><msub><mi href="./1.10#SS2.p1">D</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></mrow></math>.
</p>
</div>
<div id="SS2.p3" class="ltx_para">
<p class="ltx_p">Analytic continuation is a powerful aid in establishing transformations or
functional equations for complex variables, because it enables the problem to
be reduced to: (a) deriving the transformation (or functional equation) with
real variables; followed by (b) finding the domain on which the transformed
function is analytic.
</p>
</div>
<section id="Px3" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Schwarz Reflection Principle</h3>
<div id="Px3.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px3.p1" class="ltx_para">
<p class="ltx_p">Let <math class="ltx_Math" altimg="m29.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.10#Px3.p1">C</mi></math> be a simple closed contour consisting of a segment <math class="ltx_Math" altimg="m73.png" altimg-height="18px" altimg-valign="-2px" altimg-width="35px" alttext="\mathit{AB}" display="inline"><mi mathvariant="italic">AB</mi></math> of the real
axis and a contour in the upper half-plane joining the ends of <math class="ltx_Math" altimg="m73.png" altimg-height="18px" altimg-valign="-2px" altimg-width="35px" alttext="\mathit{AB}" display="inline"><mi mathvariant="italic">AB</mi></math>. Also, let
<math class="ltx_Math" altimg="m120.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(z)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math> be analytic within <math class="ltx_Math" altimg="m29.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.10#Px3.p1">C</mi></math>, continuous within and on <math class="ltx_Math" altimg="m29.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.10#Px3.p1">C</mi></math>, and real on <math class="ltx_Math" altimg="m73.png" altimg-height="18px" altimg-valign="-2px" altimg-width="35px" alttext="\mathit{AB}" display="inline"><mi mathvariant="italic">AB</mi></math>.
Then <math class="ltx_Math" altimg="m120.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(z)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math> can be continued analytically across <math class="ltx_Math" altimg="m73.png" altimg-height="18px" altimg-valign="-2px" altimg-width="35px" alttext="\mathit{AB}" display="inline"><mi mathvariant="italic">AB</mi></math> by <em class="ltx_emph ltx_font_italic">reflection</em>,
that is,</p>
<table id="E5" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.10.5</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E5.png" altimg-height="29px" altimg-valign="-7px" altimg-width="113px" alttext="f(\overline{z})=\overline{f(z)}." display="block"><mrow><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mover accent="true"><mi>z</mi><mo href="./1.9#E11">¯</mo></mover><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mover accent="true"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow><mo href="./1.9#E11">¯</mo></mover></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E5.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./1.9#E11" title="(1.9.11) ‣ Complex Conjugate ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="\overline{\NVar{z}}" display="inline"><mover accent="true"><mi class="ltx_nvar" href="./1.1#p2.t1.r2">z</mi><mo href="./1.9#E11">¯</mo></mover></math>: complex conjugate</a></dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
</section>
<section id="iii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§1.10(iii) </span>Laurent Series</h2>
<div id="SS3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m57.png" altimg-height="13px" altimg-valign="-2px" altimg-width="29px" alttext="\Residue" display="inline"><mo href="./1.10#SS3.p5">res</mo></math>: residue</span></dd>
<dt>Keywords:</dt>
<dd>
</dd>
</dl>
</div>
</div>
<div id="SS3.p1" class="ltx_para">
<p class="ltx_p">Suppose <math class="ltx_Math" altimg="m120.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(z)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> is analytic in the <em class="ltx_emph ltx_font_italic">annulus</em>
<math class="ltx_Math" altimg="m152.png" altimg-height="23px" altimg-valign="-7px" altimg-width="157px" alttext="r_{1}<|z-z_{0}|<r_{2}" display="inline"><mrow><msub><mi href="./1.10#SS3.p1">r</mi><mn>1</mn></msub><mo><</mo><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>-</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></mrow><mo stretchy="false">|</mo></mrow><mo><</mo><msub><mi href="./1.10#SS3.p1">r</mi><mn>2</mn></msub></mrow></math>, <math class="ltx_Math" altimg="m22.png" altimg-height="20px" altimg-valign="-5px" altimg-width="150px" alttext="0\leq r_{1}<r_{2}\leq\infty" display="inline"><mrow><mn>0</mn><mo>≤</mo><msub><mi href="./1.10#SS3.p1">r</mi><mn>1</mn></msub><mo><</mo><msub><mi href="./1.10#SS3.p1">r</mi><mn>2</mn></msub><mo>≤</mo><mi mathvariant="normal">∞</mi></mrow></math>, and <math class="ltx_Math" altimg="m151.png" altimg-height="23px" altimg-valign="-7px" altimg-width="98px" alttext="r\in(r_{1},r_{2})" display="inline"><mrow><mi href="./1.10#SS3.p1">r</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><msub><mi href="./1.10#SS3.p1">r</mi><mn>1</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><msub><mi href="./1.10#SS3.p1">r</mi><mn>2</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></mrow></math>.
Then
</p>
<table id="E6" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.10.6</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E6.png" altimg-height="65px" altimg-valign="-29px" altimg-width="230px" alttext="f(z)=\sum^{\infty}_{n=-\infty}a_{n}(z-z_{0})^{n}," display="block"><mrow><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>-</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></mrow><mo stretchy="false">)</mo></mrow><mi href="./1.1#p2.t1.r5">n</mi></msup></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E6.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m177.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a> and
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m143.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where</p>
<table id="E7" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.10.7</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E7.png" altimg-height="56px" altimg-valign="-24px" altimg-width="303px" alttext="a_{n}=\frac{1}{2\pi i}\int_{|z-z_{0}|=r}\frac{f(z)}{(z-z_{0})^{n+1}}\mathrm{d}z," display="block"><mrow><mrow><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo>=</mo><mrow><mfrac><mn>1</mn><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi></mrow></mfrac><mo></mo><mrow><msub><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>-</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></mrow><mo stretchy="false">|</mo></mrow><mo>=</mo><mi href="./1.10#SS3.p1">r</mi></mrow></msub><mrow><mfrac><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>-</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></mrow><mo stretchy="false">)</mo></mrow><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>+</mo><mn>1</mn></mrow></msup></mfrac><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./1.1#p2.t1.r2">z</mi></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E7.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m84.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m166.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m177.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>,
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m143.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./1.10#SS3.p1" title="§1.10(iii) Laurent Series ‣ §1.10 Functions of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m154.png" altimg-height="16px" altimg-valign="-5px" altimg-width="22px" alttext="r_{1}" display="inline"><msub><mi href="./1.10#SS3.p1">r</mi><mn>1</mn></msub></math>, <math class="ltx_Math" altimg="m155.png" altimg-height="16px" altimg-valign="-5px" altimg-width="22px" alttext="r_{2}" display="inline"><msub><mi href="./1.10#SS3.p1">r</mi><mn>2</mn></msub></math>: annulus radii</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and the integration contour is described once in the positive sense. The
series () converges uniformly and absolutely on compact sets
in the annulus.</p>
</div>
<div id="SS3.p2" class="ltx_para">
<p class="ltx_p">Let <math class="ltx_Math" altimg="m153.png" altimg-height="20px" altimg-valign="-5px" altimg-width="59px" alttext="r_{1}=0" display="inline"><mrow><msub><mi href="./1.10#SS3.p1">r</mi><mn>1</mn></msub><mo>=</mo><mn>0</mn></mrow></math>, so that the annulus becomes the <em class="ltx_emph ltx_font_italic">punctured neighborhood</em>
<math class="ltx_Math" altimg="m50.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="N" display="inline"><mi>N</mi></math>: <math class="ltx_Math" altimg="m21.png" altimg-height="23px" altimg-valign="-7px" altimg-width="149px" alttext="0<|z-z_{0}|<r_{2}" display="inline"><mrow><mn>0</mn><mo><</mo><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>-</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></mrow><mo stretchy="false">|</mo></mrow><mo><</mo><msub><mi href="./1.10#SS3.p1">r</mi><mn>2</mn></msub></mrow></math>, and assume that <math class="ltx_Math" altimg="m120.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(z)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> is analytic in <math class="ltx_Math" altimg="m50.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="N" display="inline"><mi>N</mi></math>, but not at
<math class="ltx_Math" altimg="m181.png" altimg-height="16px" altimg-valign="-5px" altimg-width="23px" alttext="z_{0}" display="inline"><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></math>. Then <math class="ltx_Math" altimg="m175.png" altimg-height="16px" altimg-valign="-5px" altimg-width="59px" alttext="z=z_{0}" display="inline"><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>=</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></mrow></math> is an <em class="ltx_emph ltx_font_italic">isolated singularity</em>
of <math class="ltx_Math" altimg="m120.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(z)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math>. This singularity is <em class="ltx_emph ltx_font_italic">removable</em>
if <math class="ltx_Math" altimg="m107.png" altimg-height="20px" altimg-valign="-5px" altimg-width="62px" alttext="a_{n}=0" display="inline"><mrow><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo>=</mo><mn>0</mn></mrow></math> for all <math class="ltx_Math" altimg="m140.png" altimg-height="17px" altimg-valign="-3px" altimg-width="53px" alttext="n<0" display="inline"><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo><</mo><mn>0</mn></mrow></math>, and in this case the Laurent series becomes the
Taylor series. Next, <math class="ltx_Math" altimg="m181.png" altimg-height="16px" altimg-valign="-5px" altimg-width="23px" alttext="z_{0}" display="inline"><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></math> is a <em class="ltx_emph ltx_font_italic">pole</em>
if <math class="ltx_Math" altimg="m111.png" altimg-height="21px" altimg-valign="-6px" altimg-width="62px" alttext="a_{n}\not=0" display="inline"><mrow><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo>≠</mo><mn>0</mn></mrow></math> for at least one, but only finitely many, negative <math class="ltx_Math" altimg="m143.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>. If
<math class="ltx_Math" altimg="m19.png" altimg-height="17px" altimg-valign="-4px" altimg-width="32px" alttext="-n" display="inline"><mrow><mo>-</mo><mi href="./1.1#p2.t1.r5">n</mi></mrow></math> is the first negative integer (counting from <math class="ltx_Math" altimg="m16.png" altimg-height="17px" altimg-valign="-4px" altimg-width="40px" alttext="-\infty" display="inline"><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow></math>) with
<math class="ltx_Math" altimg="m102.png" altimg-height="22px" altimg-valign="-7px" altimg-width="75px" alttext="a_{-n}\not=0" display="inline"><mrow><msub><mi>a</mi><mrow><mo>-</mo><mi href="./1.1#p2.t1.r5">n</mi></mrow></msub><mo>≠</mo><mn>0</mn></mrow></math>, then <math class="ltx_Math" altimg="m181.png" altimg-height="16px" altimg-valign="-5px" altimg-width="23px" alttext="z_{0}" display="inline"><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></math> is a <em class="ltx_emph ltx_font_italic">pole of order</em> (or <em class="ltx_emph ltx_font_italic">multiplicity</em>)
<math class="ltx_Math" altimg="m143.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>. Lastly, if <math class="ltx_Math" altimg="m111.png" altimg-height="21px" altimg-valign="-6px" altimg-width="62px" alttext="a_{n}\not=0" display="inline"><mrow><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo>≠</mo><mn>0</mn></mrow></math> for infinitely many negative <math class="ltx_Math" altimg="m143.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>, then <math class="ltx_Math" altimg="m181.png" altimg-height="16px" altimg-valign="-5px" altimg-width="23px" alttext="z_{0}" display="inline"><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></math> is
an <em class="ltx_emph ltx_font_italic">isolated essential singularity</em>.</p>
</div>
<div id="SS3.p3" class="ltx_para">
<p class="ltx_p">The singularities of <math class="ltx_Math" altimg="m120.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(z)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> at infinity are classified in the same way as the
singularities of <math class="ltx_Math" altimg="m116.png" altimg-height="23px" altimg-valign="-7px" altimg-width="62px" alttext="f(1/z)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>/</mo><mi href="./1.1#p2.t1.r2">z</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></math> at <math class="ltx_Math" altimg="m170.png" altimg-height="17px" altimg-valign="-2px" altimg-width="51px" alttext="z=0" display="inline"><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>=</mo><mn>0</mn></mrow></math>.
</p>
</div>
<div id="SS3.p4" class="ltx_para">
<p class="ltx_p">An isolated singularity <math class="ltx_Math" altimg="m181.png" altimg-height="16px" altimg-valign="-5px" altimg-width="23px" alttext="z_{0}" display="inline"><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></math> is always removable when <math class="ltx_Math" altimg="m69.png" altimg-height="23px" altimg-valign="-7px" altimg-width="113px" alttext="\lim_{z\to z_{0}}f(z)" display="inline"><mrow><msub><mo>lim</mo><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>→</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></mrow></msub><mo></mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></math>
exists, for example <math class="ltx_Math" altimg="m6.png" altimg-height="23px" altimg-valign="-7px" altimg-width="78px" alttext="(\sin z)/z" display="inline"><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi href="./1.1#p2.t1.r2">z</mi></mrow><mo stretchy="false">)</mo></mrow><mo>/</mo><mi href="./1.1#p2.t1.r2">z</mi></mrow></math> at <math class="ltx_Math" altimg="m170.png" altimg-height="17px" altimg-valign="-2px" altimg-width="51px" alttext="z=0" display="inline"><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>=</mo><mn>0</mn></mrow></math>.</p>
</div>
<div id="SS3.p5" class="ltx_para">
<p class="ltx_p">The coefficient <math class="ltx_Math" altimg="m101.png" altimg-height="17px" altimg-valign="-7px" altimg-width="36px" alttext="a_{-1}" display="inline"><msub><mi>a</mi><mrow><mo>-</mo><mn>1</mn></mrow></msub></math> of <math class="ltx_Math" altimg="m13.png" altimg-height="25px" altimg-valign="-7px" altimg-width="94px" alttext="(z-z_{0})^{-1}" display="inline"><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>-</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mn>1</mn></mrow></msup></math> in the Laurent series for <math class="ltx_Math" altimg="m120.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(z)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math>
is called the <em class="ltx_emph ltx_font_italic">residue</em>
of <math class="ltx_Math" altimg="m120.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(z)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> at <math class="ltx_Math" altimg="m181.png" altimg-height="16px" altimg-valign="-5px" altimg-width="23px" alttext="z_{0}" display="inline"><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></math>, and denoted by <math class="ltx_Math" altimg="m59.png" altimg-height="23px" altimg-valign="-7px" altimg-width="114px" alttext="\Residue_{z=z_{0}}[f(z)]" display="inline"><mrow><msub><mo href="./1.10#SS3.p5">res</mo><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>=</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></mrow></msub><mo></mo><mrow><mo stretchy="false">[</mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">]</mo></mrow></mrow></math>,
<math class="ltx_Math" altimg="m58.png" altimg-height="34px" altimg-valign="-18px" altimg-width="89px" alttext="\Residue\limits_{z=z_{0}}[f(z)]" display="inline"><mrow><munder><mo href="./1.10#SS3.p5" movablelimits="false">res</mo><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>=</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></mrow></munder><mo></mo><mrow><mo stretchy="false">[</mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">]</mo></mrow></mrow></math>, or (when there
is no ambiguity) <math class="ltx_Math" altimg="m56.png" altimg-height="23px" altimg-valign="-7px" altimg-width="77px" alttext="\Residue[f(z)]" display="inline"><mrow><mo href="./1.10#SS3.p5">res</mo><mo></mo><mrow><mo stretchy="false">[</mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">]</mo></mrow></mrow></math>.
</p>
</div>
<div id="SS3.p6" class="ltx_para">
<p class="ltx_p">A function whose only singularities, other than the point at infinity, are
poles is called a <em class="ltx_emph ltx_font_italic">meromorphic function</em>.
If the poles are infinite in number, then the point at infinity is called an
<em class="ltx_emph ltx_font_italic">essential singularity</em>:
it is the limit point of the poles.
</p>
</div>
<section id="Px4" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Picard’s Theorem</h3>
<div id="Px4.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px4.p1" class="ltx_para">
<p class="ltx_p">In any neighborhood of an isolated essential singularity, however small, an
analytic function assumes every value in <math class="ltx_Math" altimg="m72.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\mathbb{C}" display="inline"><mi href="./front/introduction#Sx4.p1.t1.r1">ℂ</mi></math> with at most one exception.</p>
</div>
</section>
</section>
<section id="iv" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§1.10(iv) </span>Residue Theorem</h2>
<div id="SS4.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="SS4.p1" class="ltx_para">
<p class="ltx_p">If <math class="ltx_Math" altimg="m120.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(z)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> is analytic within a simple closed contour <math class="ltx_Math" altimg="m29.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.10#SS4.p1">C</mi></math>, and continuous within
and on <math class="ltx_Math" altimg="m29.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.10#SS4.p1">C</mi></math>—except in both instances for a finite number of singularities
within <math class="ltx_Math" altimg="m29.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.10#SS4.p1">C</mi></math>—then</p>
<table id="E8" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.10.8</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E8.png" altimg-height="52px" altimg-valign="-20px" altimg-width="481px" alttext="\frac{1}{2\pi i}\int_{C}f(z)\mathrm{d}z=\mbox{sum of the residues of $f(z)$ %
within $C$}." display="block"><mrow><mrow><mrow><mfrac><mn>1</mn><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi></mrow></mfrac><mo></mo><mrow><msub><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi href="./1.10#SS4.p1">C</mi></msub><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./1.1#p2.t1.r2">z</mi></mrow></mrow></mrow></mrow><mo>=</mo><mrow><mtext>sum of the residues of </mtext><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mtext> within </mtext><mi href="./1.10#SS4.p1">C</mi></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E8.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m84.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m166.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m177.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a> and
<a href="./1.10#SS4.p1" title="§1.10(iv) Residue Theorem ‣ §1.10 Functions of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.10#SS4.p1">C</mi></math>: closed contour</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Here and elsewhere in this subsection the path <math class="ltx_Math" altimg="m29.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.10#SS4.p1">C</mi></math> is described in the positive
sense.</p>
</div>
<section id="Px5" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Phase (or Argument) Principle</h3>
<div id="Px5.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px5.p1" class="ltx_para">
<p class="ltx_p">If the singularities within <math class="ltx_Math" altimg="m29.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.10#SS4.p1">C</mi></math> are poles and <math class="ltx_Math" altimg="m120.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(z)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> is analytic and
nonvanishing on <math class="ltx_Math" altimg="m29.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.10#SS4.p1">C</mi></math>, then</p>
<table id="E9" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.10.9</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E9.png" altimg-height="53px" altimg-valign="-21px" altimg-width="392px" alttext="N-P=\frac{1}{2\pi i}\int_{C}\frac{f^{\prime}(z)}{f(z)}\mathrm{d}z=\frac{1}{2%
\pi}\Delta_{C}(\operatorname{ph}f(z))," display="block"><mrow><mrow><mrow><mi href="./1.10#Px5.p1">N</mi><mo>-</mo><mi href="./1.10#Px5.p1">P</mi></mrow><mo>=</mo><mrow><mfrac><mn>1</mn><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi></mrow></mfrac><mo></mo><mrow><msub><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi href="./1.10#SS4.p1">C</mi></msub><mrow><mfrac><mrow><msup><mi>f</mi><mo>′</mo></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></mfrac><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./1.1#p2.t1.r2">z</mi></mrow></mrow></mrow></mrow><mo>=</mo><mrow><mfrac><mn>1</mn><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi></mrow></mfrac><mo></mo><mrow><msub><mi mathvariant="normal">Δ</mi><mi href="./1.10#SS4.p1">C</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E9.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m84.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m166.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.9#E7" title="(1.9.7) ‣ Modulus and Phase ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="21px" altimg-valign="-6px" altimg-width="26px" alttext="\operatorname{ph}" display="inline"><mi href="./1.9#E7">ph</mi></math>: phase</a>,
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m177.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>,
<a href="./1.10#SS4.p1" title="§1.10(iv) Residue Theorem ‣ §1.10 Functions of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.10#SS4.p1">C</mi></math>: closed contour</a>,
<a href="./1.10#Px5.p1" title="Phase (or Argument) Principle ‣ §1.10(iv) Residue Theorem ‣ §1.10 Functions of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="N" display="inline"><mi href="./1.10#Px5.p1">N</mi></math>: number of zeros</a> and
<a href="./1.10#Px5.p1" title="Phase (or Argument) Principle ‣ §1.10(iv) Residue Theorem ‣ §1.10 Functions of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="P" display="inline"><mi href="./1.10#Px5.p1">P</mi></math>: number of zeros</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m50.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="N" display="inline"><mi href="./1.10#Px5.p1">N</mi></math> and <math class="ltx_Math" altimg="m51.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="P" display="inline"><mi href="./1.10#Px5.p1">P</mi></math> are respectively the numbers of zeros and poles, counting
multiplicity, of <math class="ltx_Math" altimg="m123.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> within <math class="ltx_Math" altimg="m29.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.10#SS4.p1">C</mi></math>, and <math class="ltx_Math" altimg="m55.png" altimg-height="23px" altimg-valign="-7px" altimg-width="113px" alttext="\Delta_{C}(\operatorname{ph}f(z))" display="inline"><mrow><msub><mi mathvariant="normal">Δ</mi><mi href="./1.10#SS4.p1">C</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></math> is the change in any
continuous branch of <math class="ltx_Math" altimg="m79.png" altimg-height="23px" altimg-valign="-7px" altimg-width="83px" alttext="\operatorname{ph}\left(f(z)\right)" display="inline"><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mrow><mo>(</mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>)</mo></mrow></mrow></math> as <math class="ltx_Math" altimg="m177.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math> passes once around <math class="ltx_Math" altimg="m29.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.10#SS4.p1">C</mi></math> in the
positive sense. For examples of applications see
<cite class="ltx_cite ltx_citemacro_citet">Olver (, pp. 252–254)</cite>.</p>
</div>
<div id="Px5.p2" class="ltx_para">
<p class="ltx_p">In addition,</p>
<table id="E10" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.10.10</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_Math ltx_eqn_cell ltx_align_center"><math class="ltx_Math" alttext="\frac{1}{2\pi i}\int_{C}\frac{zf^{\prime}(z)}{f(z)}\mathrm{d}z=\mbox{(sum of %
locations of zeros)}-\mbox{(sum of locations of poles)}," display="block"><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mrow><mfrac><mn>1</mn><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi></mrow></mfrac><mo></mo><mrow><msub><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi href="./1.10#SS4.p1">C</mi></msub><mrow><mfrac><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo></mo><mrow><msup><mi>f</mi><mo>′</mo></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></mfrac><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./1.1#p2.t1.r2">z</mi></mrow></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>=</mo><mrow><mtext>(sum of locations of zeros)</mtext><mo>-</mo><mtext>(sum of locations of poles)</mtext></mrow><mo>,</mo></mtd></mtr></mtable></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E10.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m84.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m166.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m177.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a> and
<a href="./1.10#SS4.p1" title="§1.10(iv) Residue Theorem ‣ §1.10 Functions of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.10#SS4.p1">C</mi></math>: closed contour</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">each location again being counted with multiplicity equal to that of the
corresponding zero or pole.</p>
</div>
</section>
<section id="Px6" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Rouché’s Theorem</h3>
<div id="Px6.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px6.p1" class="ltx_para">
<p class="ltx_p">If <math class="ltx_Math" altimg="m120.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(z)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> and <math class="ltx_Math" altimg="m129.png" altimg-height="23px" altimg-valign="-7px" altimg-width="40px" alttext="g(z)" display="inline"><mrow><mi>g</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> are analytic on and inside a simple closed contour <math class="ltx_Math" altimg="m29.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.10#SS4.p1">C</mi></math>,
and <math class="ltx_Math" altimg="m190.png" altimg-height="23px" altimg-valign="-7px" altimg-width="126px" alttext="|g(z)|<|f(z)|" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mrow><mi>g</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow><mo><</mo><mrow><mo stretchy="false">|</mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow></mrow></math> on <math class="ltx_Math" altimg="m29.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.10#SS4.p1">C</mi></math>, then <math class="ltx_Math" altimg="m120.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(z)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> and <math class="ltx_Math" altimg="m117.png" altimg-height="23px" altimg-valign="-7px" altimg-width="102px" alttext="f(z)+g(z)" display="inline"><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>+</mo><mrow><mi>g</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></math> have the same
number of zeros inside <math class="ltx_Math" altimg="m29.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.10#SS4.p1">C</mi></math>.</p>
</div>
</section>
</section>
<section id="v" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§1.10(v) </span>Maximum-Modulus Principle</h2>
<div id="SS5.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px7.p1" class="ltx_para">
<p class="ltx_p">If <math class="ltx_Math" altimg="m120.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(z)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> is analytic in a domain <math class="ltx_Math" altimg="m33.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.10#Px7.p1">D</mi></math>, <math class="ltx_Math" altimg="m182.png" altimg-height="21px" altimg-valign="-5px" altimg-width="64px" alttext="z_{0}\in D" display="inline"><mrow><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mi href="./1.10#Px7.p1">D</mi></mrow></math> and <math class="ltx_Math" altimg="m188.png" altimg-height="23px" altimg-valign="-7px" altimg-width="136px" alttext="|f(z)|\leq|f(z_{0})|" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow><mo>≤</mo><mrow><mo stretchy="false">|</mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow></mrow></math>
for all <math class="ltx_Math" altimg="m178.png" altimg-height="18px" altimg-valign="-3px" altimg-width="56px" alttext="z\in D" display="inline"><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mi href="./1.10#Px7.p1">D</mi></mrow></math>, then <math class="ltx_Math" altimg="m120.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(z)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> is a constant in <math class="ltx_Math" altimg="m33.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.10#Px7.p1">D</mi></math>.
</p>
</div>
<div id="Px7.p2" class="ltx_para">
<p class="ltx_p">Let <math class="ltx_Math" altimg="m33.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.10#Px7.p1">D</mi></math> be a bounded domain with boundary <math class="ltx_Math" altimg="m83.png" altimg-height="18px" altimg-valign="-2px" altimg-width="33px" alttext="\partial D" display="inline"><mrow><mo href="./1.5#E3">∂</mo><mo></mo><mi href="./1.10#Px7.p1">D</mi></mrow></math> and let <math class="ltx_Math" altimg="m80.png" altimg-height="21px" altimg-valign="-2px" altimg-width="116px" alttext="\overline{D}=D\cup\partial D" display="inline"><mrow><mover accent="true"><mi href="./1.10#Px7.p1">D</mi><mo>¯</mo></mover><mo>=</mo><mrow><mi href="./1.10#Px7.p1">D</mi><mo href="./front/introduction#Sx4.p1.t1.r27">∪</mo><mrow><mo href="./1.5#E3">∂</mo><mo></mo><mi href="./1.10#Px7.p1">D</mi></mrow></mrow></mrow></math>. If <math class="ltx_Math" altimg="m120.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(z)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> is continuous on <math class="ltx_Math" altimg="m81.png" altimg-height="21px" altimg-valign="-2px" altimg-width="21px" alttext="\overline{D}" display="inline"><mover accent="true"><mi href="./1.10#Px7.p1">D</mi><mo>¯</mo></mover></math> and analytic in
<math class="ltx_Math" altimg="m33.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.10#Px7.p1">D</mi></math>, then <math class="ltx_Math" altimg="m186.png" altimg-height="23px" altimg-valign="-7px" altimg-width="53px" alttext="|f(z)|" display="inline"><mrow><mo stretchy="false">|</mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow></math> attains its maximum on <math class="ltx_Math" altimg="m83.png" altimg-height="18px" altimg-valign="-2px" altimg-width="33px" alttext="\partial D" display="inline"><mrow><mo href="./1.5#E3">∂</mo><mo></mo><mi href="./1.10#Px7.p1">D</mi></mrow></math>.</p>
</div>
</section>
<section id="Px8" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Harmonic Functions</h3>
<div id="Px8.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px8.p1" class="ltx_para">
<p class="ltx_p">If <math class="ltx_Math" altimg="m160.png" altimg-height="23px" altimg-valign="-7px" altimg-width="41px" alttext="u(z)" display="inline"><mrow><mi href="./1.10#Px8">u</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> is harmonic in <math class="ltx_Math" altimg="m33.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.10#Px7.p1">D</mi></math>, <math class="ltx_Math" altimg="m182.png" altimg-height="21px" altimg-valign="-5px" altimg-width="64px" alttext="z_{0}\in D" display="inline"><mrow><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mi href="./1.10#Px7.p1">D</mi></mrow></math>, and <math class="ltx_Math" altimg="m161.png" altimg-height="23px" altimg-valign="-7px" altimg-width="113px" alttext="u(z)\leq u(z_{0})" display="inline"><mrow><mrow><mi href="./1.10#Px8">u</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>≤</mo><mrow><mi href="./1.10#Px8">u</mi><mo></mo><mrow><mo stretchy="false">(</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mrow></mrow></math> for all
<math class="ltx_Math" altimg="m178.png" altimg-height="18px" altimg-valign="-3px" altimg-width="56px" alttext="z\in D" display="inline"><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mi href="./1.10#Px7.p1">D</mi></mrow></math>, then <math class="ltx_Math" altimg="m160.png" altimg-height="23px" altimg-valign="-7px" altimg-width="41px" alttext="u(z)" display="inline"><mrow><mi href="./1.10#Px8">u</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> is constant in <math class="ltx_Math" altimg="m33.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.10#Px7.p1">D</mi></math>. Moreover, if <math class="ltx_Math" altimg="m33.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.10#Px7.p1">D</mi></math> is bounded and
<math class="ltx_Math" altimg="m160.png" altimg-height="23px" altimg-valign="-7px" altimg-width="41px" alttext="u(z)" display="inline"><mrow><mi href="./1.10#Px8">u</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> is continuous on <math class="ltx_Math" altimg="m81.png" altimg-height="21px" altimg-valign="-2px" altimg-width="21px" alttext="\overline{D}" display="inline"><mover accent="true"><mi href="./1.10#Px7.p1">D</mi><mo>¯</mo></mover></math> and harmonic in <math class="ltx_Math" altimg="m33.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.10#Px7.p1">D</mi></math>, then <math class="ltx_Math" altimg="m160.png" altimg-height="23px" altimg-valign="-7px" altimg-width="41px" alttext="u(z)" display="inline"><mrow><mi href="./1.10#Px8">u</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> is
maximum at some point on <math class="ltx_Math" altimg="m83.png" altimg-height="18px" altimg-valign="-2px" altimg-width="33px" alttext="\partial D" display="inline"><mrow><mo href="./1.5#E3">∂</mo><mo></mo><mi href="./1.10#Px7.p1">D</mi></mrow></math>.</p>
</div>
</section>
<section id="Px9" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Schwarz’s Lemma</h3>
<div id="Px9.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px9.p1" class="ltx_para">
<p class="ltx_p">In <math class="ltx_Math" altimg="m193.png" altimg-height="23px" altimg-valign="-7px" altimg-width="67px" alttext="|z|<R" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">|</mo></mrow><mo><</mo><mi href="./1.10#Px9.p1">R</mi></mrow></math>, if <math class="ltx_Math" altimg="m120.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(z)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> is analytic, <math class="ltx_Math" altimg="m187.png" altimg-height="23px" altimg-valign="-7px" altimg-width="101px" alttext="|f(z)|\leq M" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow><mo>≤</mo><mi>M</mi></mrow></math>, and <math class="ltx_Math" altimg="m115.png" altimg-height="23px" altimg-valign="-7px" altimg-width="78px" alttext="f(0)=0" display="inline"><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mn>0</mn></mrow></math>, then
</p>
<table id="E11" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.10.11</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E11.png" altimg-height="48px" altimg-valign="-16px" altimg-width="298px" alttext="|f(z)|\leq\frac{M|z|}{R}\;\mbox{ and }\;|f^{\prime}(0)|\leq\frac{M}{R}." display="block"><mrow><mrow><mrow><mo stretchy="false">|</mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow><mo>≤</mo><mrow><mpadded width="+2.8pt"><mfrac><mrow><mi>M</mi><mo></mo><mrow><mo stretchy="false">|</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">|</mo></mrow></mrow><mi href="./1.10#Px9.p1">R</mi></mfrac></mpadded><mo></mo><mpadded width="+2.8pt"><mtext> and </mtext></mpadded><mo></mo><mrow><mo stretchy="false">|</mo><mrow><msup><mi>f</mi><mo>′</mo></msup><mo></mo><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow></mrow><mo>≤</mo><mfrac><mi>M</mi><mi href="./1.10#Px9.p1">R</mi></mfrac></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E11.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m177.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a> and
<a href="./1.10#Px9.p1" title="Schwarz’s Lemma ‣ §1.10(v) Maximum-Modulus Principle ‣ §1.10 Functions of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="R" display="inline"><mi href="./1.10#Px9.p1">R</mi></math>: radius</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Equalities hold iff <math class="ltx_Math" altimg="m119.png" altimg-height="23px" altimg-valign="-7px" altimg-width="93px" alttext="f(z)=Az" display="inline"><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mi>A</mi><mo></mo><mi href="./1.1#p2.t1.r2">z</mi></mrow></mrow></math>, where <math class="ltx_Math" altimg="m28.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="A" display="inline"><mi>A</mi></math> is a constant such that
<math class="ltx_Math" altimg="m68.png" altimg-height="23px" altimg-valign="-7px" altimg-width="103px" alttext="\left|A\right|=M/R" display="inline"><mrow><mrow><mo>|</mo><mi>A</mi><mo>|</mo></mrow><mo>=</mo><mrow><mi>M</mi><mo>/</mo><mi href="./1.10#Px9.p1">R</mi></mrow></mrow></math>.</p>
</div>
</section>
</section>
<section id="vi" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§1.10(vi) </span>Multivalued Functions</h2>
<div id="SS6.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="SS6.p1" class="ltx_para">
<p class="ltx_p">Functions which have more than one value at a given point <math class="ltx_Math" altimg="m177.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math> are called
<em class="ltx_emph ltx_font_italic">multivalued</em> (or <em class="ltx_emph ltx_font_italic">many-valued</em>) functions. Let <math class="ltx_Math" altimg="m42.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="F(z)" display="inline"><mrow><mi href="./1.10#SS6.p1">F</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> be a
multivalued function and <math class="ltx_Math" altimg="m33.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.10#SS6.p1">D</mi></math> be a domain. If we can assign a unique value
<math class="ltx_Math" altimg="m120.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(z)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> to <math class="ltx_Math" altimg="m42.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="F(z)" display="inline"><mrow><mi href="./1.10#SS6.p1">F</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> at each point of <math class="ltx_Math" altimg="m33.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.10#SS6.p1">D</mi></math>, and <math class="ltx_Math" altimg="m120.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(z)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> is analytic on <math class="ltx_Math" altimg="m33.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.10#SS6.p1">D</mi></math>, then
<math class="ltx_Math" altimg="m120.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(z)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> is a <em class="ltx_emph ltx_font_italic">branch</em> of <math class="ltx_Math" altimg="m42.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="F(z)" display="inline"><mrow><mi href="./1.10#SS6.p1">F</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math>.</p>
</div>
<section id="Px10" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Example</h3>
<div id="Px10.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px10.p1" class="ltx_para">
<p class="ltx_p"><math class="ltx_Math" altimg="m41.png" altimg-height="24px" altimg-valign="-7px" altimg-width="99px" alttext="F(z)=\sqrt{z}" display="inline"><mrow><mrow><mi href="./1.10#SS6.p1">F</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><msqrt><mi href="./1.1#p2.t1.r2">z</mi></msqrt></mrow></math> is two-valued for <math class="ltx_Math" altimg="m180.png" altimg-height="21px" altimg-valign="-6px" altimg-width="51px" alttext="z\not=0" display="inline"><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>≠</mo><mn>0</mn></mrow></math>. If <math class="ltx_Math" altimg="m31.png" altimg-height="23px" altimg-valign="-7px" altimg-width="149px" alttext="D=\mathbb{C}\setminus(-\infty,0]" display="inline"><mrow><mi href="./1.10#SS6.p1">D</mi><mo>=</mo><mrow><mi href="./front/introduction#Sx4.p1.t1.r1">ℂ</mi><mo href="./front/introduction#Sx4.p2.t1.r19">∖</mo><mrow><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">(</mo><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mo href="./front/introduction#Sx4.p2.t1.r1">,</mo><mn>0</mn><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">]</mo></mrow></mrow></mrow></math> and <math class="ltx_Math" altimg="m174.png" altimg-height="20px" altimg-valign="-2px" altimg-width="74px" alttext="z=re^{i\theta}" display="inline"><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>=</mo><mrow><mi>r</mi><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mo></mo><mi>θ</mi></mrow></msup></mrow></mrow></math>, then one branch is
<math class="ltx_Math" altimg="m89.png" altimg-height="26px" altimg-valign="-7px" altimg-width="70px" alttext="\sqrt{r}e^{i\theta/2}" display="inline"><mrow><msqrt><mi>r</mi></msqrt><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mrow><mi mathvariant="normal">i</mi><mo></mo><mi>θ</mi></mrow><mo>/</mo><mn>2</mn></mrow></msup></mrow></math>, the other branch is <math class="ltx_Math" altimg="m18.png" altimg-height="26px" altimg-valign="-7px" altimg-width="86px" alttext="-\sqrt{r}e^{i\theta/2}" display="inline"><mrow><mo>-</mo><mrow><msqrt><mi>r</mi></msqrt><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mrow><mi mathvariant="normal">i</mi><mo></mo><mi>θ</mi></mrow><mo>/</mo><mn>2</mn></mrow></msup></mrow></mrow></math>,
with <math class="ltx_Math" altimg="m17.png" altimg-height="19px" altimg-valign="-4px" altimg-width="107px" alttext="-\pi<\theta<\pi" display="inline"><mrow><mrow><mo>-</mo><mi href="./3.12#E1">π</mi></mrow><mo><</mo><mi>θ</mi><mo><</mo><mi href="./3.12#E1">π</mi></mrow></math> in both cases.
Similarly if <math class="ltx_Math" altimg="m32.png" altimg-height="23px" altimg-valign="-7px" altimg-width="133px" alttext="D=\mathbb{C}\setminus[0,\infty)" display="inline"><mrow><mi href="./1.10#SS6.p1">D</mi><mo>=</mo><mrow><mi href="./front/introduction#Sx4.p1.t1.r1">ℂ</mi><mo href="./front/introduction#Sx4.p2.t1.r19">∖</mo><mrow><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">[</mo><mn>0</mn><mo href="./front/introduction#Sx4.p2.t1.r1">,</mo><mi mathvariant="normal">∞</mi><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">)</mo></mrow></mrow></mrow></math>,
then one branch is <math class="ltx_Math" altimg="m89.png" altimg-height="26px" altimg-valign="-7px" altimg-width="70px" alttext="\sqrt{r}e^{i\theta/2}" display="inline"><mrow><msqrt><mi>r</mi></msqrt><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mrow><mi mathvariant="normal">i</mi><mo></mo><mi>θ</mi></mrow><mo>/</mo><mn>2</mn></mrow></msup></mrow></math>, the other branch is <math class="ltx_Math" altimg="m18.png" altimg-height="26px" altimg-valign="-7px" altimg-width="86px" alttext="-\sqrt{r}e^{i\theta/2}" display="inline"><mrow><mo>-</mo><mrow><msqrt><mi>r</mi></msqrt><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mrow><mi mathvariant="normal">i</mi><mo></mo><mi>θ</mi></mrow><mo>/</mo><mn>2</mn></mrow></msup></mrow></mrow></math>,
with <math class="ltx_Math" altimg="m20.png" altimg-height="18px" altimg-valign="-3px" altimg-width="99px" alttext="0<\theta<2\pi" display="inline"><mrow><mn>0</mn><mo><</mo><mi>θ</mi><mo><</mo><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow></math> in both cases.</p>
</div>
<div id="Px10.p2" class="ltx_para">
<p class="ltx_p">A <em class="ltx_emph ltx_font_italic">cut domain</em>
is one from which the points on finitely many nonintersecting simple contours
(§) have been
removed. Each contour is called a <em class="ltx_emph ltx_font_italic">cut</em>. A <em class="ltx_emph ltx_font_italic">cut neighborhood</em> is
formed by deleting a ray emanating from the center. (Or more generally, a
simple contour that starts at the center and terminates on the boundary.)
</p>
</div>
<div id="Px10.p3" class="ltx_para">
<p class="ltx_p">Suppose <math class="ltx_Math" altimg="m42.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="F(z)" display="inline"><mrow><mi href="./1.10#SS6.p1">F</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> is multivalued and <math class="ltx_Math" altimg="m99.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi>a</mi></math> is a point such that there exists a
branch of <math class="ltx_Math" altimg="m42.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="F(z)" display="inline"><mrow><mi href="./1.10#SS6.p1">F</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> in a cut neighborhood of <math class="ltx_Math" altimg="m99.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi>a</mi></math>, but there does not exist a
branch of <math class="ltx_Math" altimg="m42.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="F(z)" display="inline"><mrow><mi href="./1.10#SS6.p1">F</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> in any punctured neighborhood of <math class="ltx_Math" altimg="m99.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi>a</mi></math>. Then <math class="ltx_Math" altimg="m99.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi>a</mi></math> is a
<em class="ltx_emph ltx_font_italic">branch point</em>
of <math class="ltx_Math" altimg="m42.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="F(z)" display="inline"><mrow><mi href="./1.10#SS6.p1">F</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math>. For example, <math class="ltx_Math" altimg="m170.png" altimg-height="17px" altimg-valign="-2px" altimg-width="51px" alttext="z=0" display="inline"><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>=</mo><mn>0</mn></mrow></math> is a branch point of <math class="ltx_Math" altimg="m90.png" altimg-height="24px" altimg-valign="-7px" altimg-width="31px" alttext="\sqrt{z}" display="inline"><msqrt><mi href="./1.1#p2.t1.r2">z</mi></msqrt></math>.
</p>
</div>
<div id="Px10.p4" class="ltx_para">
<p class="ltx_p">Branches can be constructed in two ways:
</p>
</div>
<div id="Px10.p5" class="ltx_para">
<p class="ltx_p">(a) By introducing appropriate cuts from the branch points and restricting
<math class="ltx_Math" altimg="m42.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="F(z)" display="inline"><mrow><mi href="./1.10#SS6.p1">F</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> to be single-valued in the cut plane (or domain).</p>
</div>
<div id="Px10.p6" class="ltx_para">
<p class="ltx_p">(b) By specifying the value of <math class="ltx_Math" altimg="m42.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="F(z)" display="inline"><mrow><mi href="./1.10#SS6.p1">F</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> at a point <math class="ltx_Math" altimg="m181.png" altimg-height="16px" altimg-valign="-5px" altimg-width="23px" alttext="z_{0}" display="inline"><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></math> (not a branch point),
and requiring <math class="ltx_Math" altimg="m42.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="F(z)" display="inline"><mrow><mi href="./1.10#SS6.p1">F</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> to be continuous on any path that begins at <math class="ltx_Math" altimg="m181.png" altimg-height="16px" altimg-valign="-5px" altimg-width="23px" alttext="z_{0}" display="inline"><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></math> and does
not pass through any branch points or other singularities of <math class="ltx_Math" altimg="m42.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="F(z)" display="inline"><mrow><mi href="./1.10#SS6.p1">F</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math>.</p>
</div>
<div id="Px10.p7" class="ltx_para">
<p class="ltx_p">If the path circles a branch point at <math class="ltx_Math" altimg="m173.png" altimg-height="13px" altimg-valign="-2px" altimg-width="52px" alttext="z=a" display="inline"><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>=</mo><mi>a</mi></mrow></math>, <math class="ltx_Math" altimg="m133.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./1.1#p2.t1.r4">k</mi></math> times in the positive sense,
and returns to <math class="ltx_Math" altimg="m181.png" altimg-height="16px" altimg-valign="-5px" altimg-width="23px" alttext="z_{0}" display="inline"><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></math> without encircling any other branch point, then its value
is denoted conventionally as <math class="ltx_Math" altimg="m38.png" altimg-height="25px" altimg-valign="-7px" altimg-width="181px" alttext="F((z_{0}-a)e^{2k\pi i}+a)" display="inline"><mrow><mi href="./1.10#SS6.p1">F</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub><mo>-</mo><mi>a</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mn>2</mn><mo></mo><mi href="./1.1#p2.t1.r4">k</mi><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi></mrow></msup></mrow><mo>+</mo><mi>a</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></math>.
</p>
</div>
</section>
<section id="Px11" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Example</h3>
<div id="Px11.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px11.p1" class="ltx_para">
<p class="ltx_p">Let <math class="ltx_Math" altimg="m60.png" altimg-height="13px" altimg-valign="-2px" altimg-width="17px" alttext="\alpha" display="inline"><mi>α</mi></math> and <math class="ltx_Math" altimg="m61.png" altimg-height="21px" altimg-valign="-6px" altimg-width="17px" alttext="\beta" display="inline"><mi>β</mi></math> be real or complex numbers that are not integers. The
function <math class="ltx_Math" altimg="m40.png" altimg-height="25px" altimg-valign="-7px" altimg-width="214px" alttext="F(z)=(1-z)^{\alpha}(1+z)^{\beta}" display="inline"><mrow><mrow><mi href="./1.10#SS6.p1">F</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><mi href="./1.1#p2.t1.r2">z</mi></mrow><mo stretchy="false">)</mo></mrow><mi>α</mi></msup><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>+</mo><mi href="./1.1#p2.t1.r2">z</mi></mrow><mo stretchy="false">)</mo></mrow><mi>β</mi></msup></mrow></mrow></math> is many-valued with branch points at
<math class="ltx_Math" altimg="m85.png" altimg-height="18px" altimg-valign="-4px" altimg-width="30px" alttext="\pm 1" display="inline"><mrow><mo>±</mo><mn>1</mn></mrow></math>. Branches of <math class="ltx_Math" altimg="m42.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="F(z)" display="inline"><mrow><mi href="./1.10#SS6.p1">F</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> can be defined, for example, in the cut plane <math class="ltx_Math" altimg="m33.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.10#SS6.p1">D</mi></math>
obtained from <math class="ltx_Math" altimg="m72.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\mathbb{C}" display="inline"><mi href="./front/introduction#Sx4.p1.t1.r1">ℂ</mi></math> by removing the real axis from <math class="ltx_Math" altimg="m26.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="1" display="inline"><mn>1</mn></math> to <math class="ltx_Math" altimg="m65.png" altimg-height="13px" altimg-valign="-2px" altimg-width="24px" alttext="\infty" display="inline"><mi mathvariant="normal">∞</mi></math> and
from <math class="ltx_Math" altimg="m15.png" altimg-height="18px" altimg-valign="-4px" altimg-width="30px" alttext="-1" display="inline"><mrow><mo>-</mo><mn>1</mn></mrow></math> to <math class="ltx_Math" altimg="m16.png" altimg-height="17px" altimg-valign="-4px" altimg-width="40px" alttext="-\infty" display="inline"><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow></math>; see Figure . One such branch is
obtained by assigning <math class="ltx_Math" altimg="m5.png" altimg-height="23px" altimg-valign="-7px" altimg-width="76px" alttext="(1-z)^{\alpha}" display="inline"><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><mi href="./1.1#p2.t1.r2">z</mi></mrow><mo stretchy="false">)</mo></mrow><mi>α</mi></msup></math> and <math class="ltx_Math" altimg="m3.png" altimg-height="25px" altimg-valign="-7px" altimg-width="75px" alttext="(1+z)^{\beta}" display="inline"><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>+</mo><mi href="./1.1#p2.t1.r2">z</mi></mrow><mo stretchy="false">)</mo></mrow><mi>β</mi></msup></math> their principal values
(§<figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 1.10.1: </span>Domain <math class="ltx_Math" altimg="m33.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.10#SS6.p1">D</mi></math>.
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./1.10#SS6.p1" title="§1.10(vi) Multivalued Functions ‣ §1.10 Functions of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m33.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.10#SS6.p1">D</mi></math>: domain</a></dd>
</dl>
</div>
</div>
</figure>
<div id="Px11.p2" class="ltx_para">
<p class="ltx_p">Alternatively, take <math class="ltx_Math" altimg="m181.png" altimg-height="16px" altimg-valign="-5px" altimg-width="23px" alttext="z_{0}" display="inline"><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></math> to be any point in <math class="ltx_Math" altimg="m33.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.10#SS6.p1">D</mi></math> and set <math class="ltx_Math" altimg="m44.png" altimg-height="26px" altimg-valign="-7px" altimg-width="249px" alttext="F(z_{0})=e^{\alpha\ln\left(1-z_{0}\right)}e^{\beta\ln\left(1+z_{0}\right)}" display="inline"><mrow><mrow><mi href="./1.10#SS6.p1">F</mi><mo></mo><mrow><mo stretchy="false">(</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mi>α</mi><mo></mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></mrow><mo>)</mo></mrow></mrow></mrow></msup><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mi>β</mi><mo></mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></mrow><mo>)</mo></mrow></mrow></mrow></msup></mrow></mrow></math> where the logarithms assume their
principal values. (Thus if <math class="ltx_Math" altimg="m181.png" altimg-height="16px" altimg-valign="-5px" altimg-width="23px" alttext="z_{0}" display="inline"><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></math> is in the interval <math class="ltx_Math" altimg="m2.png" altimg-height="23px" altimg-valign="-7px" altimg-width="64px" alttext="(-1,1)" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mn>1</mn><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math>, then the
logarithms are real.) Then the value of <math class="ltx_Math" altimg="m42.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="F(z)" display="inline"><mrow><mi href="./1.10#SS6.p1">F</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> at any other point is obtained
by analytic continuation.</p>
</div>
<div id="Px11.p3" class="ltx_para">
<p class="ltx_p">Thus if <math class="ltx_Math" altimg="m42.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="F(z)" display="inline"><mrow><mi href="./1.10#SS6.p1">F</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> is continued along a path that circles <math class="ltx_Math" altimg="m171.png" altimg-height="17px" altimg-valign="-2px" altimg-width="51px" alttext="z=1" display="inline"><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>=</mo><mn>1</mn></mrow></math> <math class="ltx_Math" altimg="m136.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./1.1#p2.t1.r5">m</mi></math> times in the
positive sense and returns to <math class="ltx_Math" altimg="m181.png" altimg-height="16px" altimg-valign="-5px" altimg-width="23px" alttext="z_{0}" display="inline"><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></math> without circling <math class="ltx_Math" altimg="m169.png" altimg-height="18px" altimg-valign="-4px" altimg-width="66px" alttext="z=-1" display="inline"><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>=</mo><mrow><mo>-</mo><mn>1</mn></mrow></mrow></math>, then
<math class="ltx_Math" altimg="m37.png" altimg-height="26px" altimg-valign="-7px" altimg-width="439px" alttext="F((z_{0}-1)e^{2m\pi i}+1)=e^{\alpha\ln\left(1-z_{0}\right)}e^{\beta\ln\left(1+%
z_{0}\right)}e^{2\pi im\alpha}" display="inline"><mrow><mrow><mi href="./1.10#SS6.p1">F</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mn>2</mn><mo></mo><mi href="./1.1#p2.t1.r5">m</mi><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi></mrow></msup></mrow><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mi>α</mi><mo></mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></mrow><mo>)</mo></mrow></mrow></mrow></msup><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mi>β</mi><mo></mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></mrow><mo>)</mo></mrow></mrow></mrow></msup><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi><mo></mo><mi href="./1.1#p2.t1.r5">m</mi><mo></mo><mi>α</mi></mrow></msup></mrow></mrow></math>. If the path
also circles <math class="ltx_Math" altimg="m169.png" altimg-height="18px" altimg-valign="-4px" altimg-width="66px" alttext="z=-1" display="inline"><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>=</mo><mrow><mo>-</mo><mn>1</mn></mrow></mrow></math> <math class="ltx_Math" altimg="m143.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math> times in the clockwise or negative sense before
returning to <math class="ltx_Math" altimg="m181.png" altimg-height="16px" altimg-valign="-5px" altimg-width="23px" alttext="z_{0}" display="inline"><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></math>, then the value of <math class="ltx_Math" altimg="m45.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="F(z_{0})" display="inline"><mrow><mi href="./1.10#SS6.p1">F</mi><mo></mo><mrow><mo stretchy="false">(</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mrow></math> becomes <math class="ltx_Math" altimg="m114.png" altimg-height="21px" altimg-valign="-2px" altimg-width="297px" alttext="e^{\alpha\ln\left(1-z_{0}\right)}e^{\beta\ln\left(1+z_{0}\right)}e^{2\pi im%
\alpha}e^{-2\pi in\beta}" display="inline"><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mi>α</mi><mo></mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></mrow><mo>)</mo></mrow></mrow></mrow></msup><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mi>β</mi><mo></mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></mrow><mo>)</mo></mrow></mrow></mrow></msup><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi><mo></mo><mi href="./1.1#p2.t1.r5">m</mi><mo></mo><mi>α</mi></mrow></msup><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi><mo></mo><mi href="./1.1#p2.t1.r5">n</mi><mo></mo><mi>β</mi></mrow></mrow></msup></mrow></math>.</p>
</div>
</section>
</section>
<section id="vii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§1.10(vii) </span>Inverse Functions</h2>
<div id="SS7.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px12.p1" class="ltx_para">
<p class="ltx_p">Suppose <math class="ltx_Math" altimg="m120.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(z)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> is analytic at <math class="ltx_Math" altimg="m175.png" altimg-height="16px" altimg-valign="-5px" altimg-width="59px" alttext="z=z_{0}" display="inline"><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>=</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></mrow></math>, <math class="ltx_Math" altimg="m124.png" altimg-height="24px" altimg-valign="-7px" altimg-width="92px" alttext="f^{\prime}(z_{0})\not=0" display="inline"><mrow><mrow><msup><mi>f</mi><mo>′</mo></msup><mo></mo><mrow><mo stretchy="false">(</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mrow><mo>≠</mo><mn>0</mn></mrow></math>, and <math class="ltx_Math" altimg="m122.png" altimg-height="23px" altimg-valign="-7px" altimg-width="100px" alttext="f(z_{0})=w_{0}" display="inline"><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><msub><mi href="./1.1#p2.t1.r3">w</mi><mn>0</mn></msub></mrow></math>.
Then the equation</p>
<table id="E12" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.10.12</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E12.png" altimg-height="25px" altimg-valign="-7px" altimg-width="85px" alttext="f(z)=w" display="block"><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mi href="./1.1#p2.t1.r3">w</mi></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E12.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m177.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a> and
<a href="./1.1#p2.t1.r3" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m163.png" altimg-height="13px" altimg-valign="-2px" altimg-width="19px" alttext="w" display="inline"><mi href="./1.1#p2.t1.r3">w</mi></math>: variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">has a unique solution <math class="ltx_Math" altimg="m172.png" altimg-height="23px" altimg-valign="-7px" altimg-width="87px" alttext="z=F(w)" display="inline"><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>=</mo><mrow><mi href="./1.10#E13">F</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r3">w</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></math> analytic at <math class="ltx_Math" altimg="m162.png" altimg-height="16px" altimg-valign="-5px" altimg-width="69px" alttext="w=w_{0}" display="inline"><mrow><mi href="./1.1#p2.t1.r3">w</mi><mo>=</mo><msub><mi href="./1.1#p2.t1.r3">w</mi><mn>0</mn></msub></mrow></math>, and</p>
<table id="E13" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.10.13</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E13.png" altimg-height="64px" altimg-valign="-27px" altimg-width="267px" alttext="F(w)=z_{0}+\sum^{\infty}_{n=1}F_{n}(w-w_{0})^{n}" display="block"><mrow><mrow><mi href="./1.10#E13">F</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r3">w</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub><mo>+</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><msub><mi href="./1.10#Px12.p1">F</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./1.1#p2.t1.r3">w</mi><mo>-</mo><msub><mi href="./1.1#p2.t1.r3">w</mi><mn>0</mn></msub></mrow><mo stretchy="false">)</mo></mrow><mi href="./1.1#p2.t1.r5">n</mi></msup></mrow></mrow></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E13.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text"><math class="ltx_Math" altimg="m39.png" altimg-height="23px" altimg-valign="-7px" altimg-width="50px" alttext="F(w)" display="inline"><mrow><mi href="./1.10#E13">F</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r3">w</mi><mo stretchy="false">)</mo></mrow></mrow></math>: inverse function of <math class="ltx_Math" altimg="m120.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(z)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> (locally)</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m177.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>,
<a href="./1.1#p2.t1.r3" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m163.png" altimg-height="13px" altimg-valign="-2px" altimg-width="19px" alttext="w" display="inline"><mi href="./1.1#p2.t1.r3">w</mi></math>: variable</a>,
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m143.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./1.10#Px12.p1" title="Lagrange Inversion Theorem ‣ §1.10(vii) Inverse Functions ‣ §1.10 Functions of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="21px" altimg-valign="-5px" altimg-width="28px" alttext="F_{n}" display="inline"><msub><mi href="./1.10#Px12.p1">F</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></math>: residue</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">in a neighborhood of <math class="ltx_Math" altimg="m165.png" altimg-height="16px" altimg-valign="-5px" altimg-width="28px" alttext="w_{0}" display="inline"><msub><mi href="./1.1#p2.t1.r3">w</mi><mn>0</mn></msub></math>, where <math class="ltx_Math" altimg="m141.png" altimg-height="21px" altimg-valign="-5px" altimg-width="40px" alttext="nF_{n}" display="inline"><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo></mo><msub><mi href="./1.10#Px12.p1">F</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></mrow></math> is the residue of <math class="ltx_Math" altimg="m25.png" altimg-height="23px" altimg-valign="-7px" altimg-width="158px" alttext="1/(f(z)-f(z_{0}))^{n}" display="inline"><mrow><mn>1</mn><mo>/</mo><msup><mrow><mo stretchy="false">(</mo><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>-</mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mrow></mrow><mo stretchy="false">)</mo></mrow><mi href="./1.1#p2.t1.r5">n</mi></msup></mrow></math>
at <math class="ltx_Math" altimg="m175.png" altimg-height="16px" altimg-valign="-5px" altimg-width="59px" alttext="z=z_{0}" display="inline"><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>=</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></mrow></math>. (In other words <math class="ltx_Math" altimg="m141.png" altimg-height="21px" altimg-valign="-5px" altimg-width="40px" alttext="nF_{n}" display="inline"><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo></mo><msub><mi href="./1.10#Px12.p1">F</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></mrow></math> is the coefficient of <math class="ltx_Math" altimg="m13.png" altimg-height="25px" altimg-valign="-7px" altimg-width="94px" alttext="(z-z_{0})^{-1}" display="inline"><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>-</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mn>1</mn></mrow></msup></math> in
the Laurent expansion of <math class="ltx_Math" altimg="m25.png" altimg-height="23px" altimg-valign="-7px" altimg-width="158px" alttext="1/(f(z)-f(z_{0}))^{n}" display="inline"><mrow><mn>1</mn><mo>/</mo><msup><mrow><mo stretchy="false">(</mo><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>-</mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mrow></mrow><mo stretchy="false">)</mo></mrow><mi href="./1.1#p2.t1.r5">n</mi></msup></mrow></math> in powers of <math class="ltx_Math" altimg="m12.png" altimg-height="23px" altimg-valign="-7px" altimg-width="72px" alttext="(z-z_{0})" display="inline"><mrow><mo stretchy="false">(</mo><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>-</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></mrow><mo stretchy="false">)</mo></mrow></math>;
compare §.)</p>
</div>
<div id="Px12.p2" class="ltx_para">
<p class="ltx_p">Furthermore, if <math class="ltx_Math" altimg="m129.png" altimg-height="23px" altimg-valign="-7px" altimg-width="40px" alttext="g(z)" display="inline"><mrow><mi>g</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> is analytic at <math class="ltx_Math" altimg="m181.png" altimg-height="16px" altimg-valign="-5px" altimg-width="23px" alttext="z_{0}" display="inline"><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></math>, then</p>
<table id="E14" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.10.14</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E14.png" altimg-height="64px" altimg-valign="-27px" altimg-width="327px" alttext="g(F(w))=g(z_{0})+\sum^{\infty}_{n=1}G_{n}(w-w_{0})^{n}," display="block"><mrow><mrow><mrow><mi>g</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./1.10#E13">F</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r3">w</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mi>g</mi><mo></mo><mrow><mo stretchy="false">(</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mrow><mo>+</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><msub><mi href="./1.10#Px13.p2">G</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./1.1#p2.t1.r3">w</mi><mo>-</mo><msub><mi href="./1.1#p2.t1.r3">w</mi><mn>0</mn></msub></mrow><mo stretchy="false">)</mo></mrow><mi href="./1.1#p2.t1.r5">n</mi></msup></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E14.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m177.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>,
<a href="./1.1#p2.t1.r3" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m163.png" altimg-height="13px" altimg-valign="-2px" altimg-width="19px" alttext="w" display="inline"><mi href="./1.1#p2.t1.r3">w</mi></math>: variable</a>,
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m143.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a>,
<a href="./1.10#E13" title="(1.10.13) ‣ Lagrange Inversion Theorem ‣ §1.10(vii) Inverse Functions ‣ §1.10 Functions of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="23px" altimg-valign="-7px" altimg-width="50px" alttext="F(w)" display="inline"><mrow><mi href="./1.10#E13">F</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r3">w</mi><mo stretchy="false">)</mo></mrow></mrow></math>: inverse function of <math class="ltx_Math" altimg="m120.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(z)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math></a> and
<a href="./1.10#Px13.p2" title="Extended Inversion Theorem ‣ §1.10(vii) Inverse Functions ‣ §1.10 Functions of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="21px" altimg-valign="-5px" altimg-width="31px" alttext="G_{n}" display="inline"><msub><mi href="./1.10#Px13.p2">G</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></math>: residue</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m142.png" altimg-height="21px" altimg-valign="-5px" altimg-width="43px" alttext="nG_{n}" display="inline"><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo></mo><msub><mi href="./1.10#Px13.p2">G</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></mrow></math> is the residue of <math class="ltx_Math" altimg="m131.png" altimg-height="24px" altimg-valign="-7px" altimg-width="190px" alttext="g^{\prime}(z)/(f(z)-f(z_{0}))^{n}" display="inline"><mrow><mrow><msup><mi>g</mi><mo>′</mo></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>/</mo><msup><mrow><mo stretchy="false">(</mo><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>-</mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mrow></mrow><mo stretchy="false">)</mo></mrow><mi href="./1.1#p2.t1.r5">n</mi></msup></mrow></math> at <math class="ltx_Math" altimg="m175.png" altimg-height="16px" altimg-valign="-5px" altimg-width="59px" alttext="z=z_{0}" display="inline"><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>=</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></mrow></math>.</p>
</div>
</section>
<section id="Px13" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Extended Inversion Theorem</h3>
<div id="Px13.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px13.p1" class="ltx_para">
<p class="ltx_p">Suppose that
</p>
<table id="E15" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.10.15</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E15.png" altimg-height="64px" altimg-valign="-27px" altimg-width="301px" alttext="f(z)=f(z_{0})+\sum^{\infty}_{n=0}f_{n}(z-z_{0})^{\mu+n}," display="block"><mrow><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mrow><mo>+</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><msub><mi href="./1.10#Px13.p1">f</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>-</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></mrow><mo stretchy="false">)</mo></mrow><mrow><mi href="./1.10#Px13.p1">μ</mi><mo>+</mo><mi href="./1.1#p2.t1.r5">n</mi></mrow></msup></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E15.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m177.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>,
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m143.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a>,
<a href="./1.10#Px13.p1" title="Extended Inversion Theorem ‣ §1.10(vii) Inverse Functions ‣ §1.10 Functions of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m128.png" altimg-height="21px" altimg-valign="-6px" altimg-width="25px" alttext="f_{n}" display="inline"><msub><mi href="./1.10#Px13.p1">f</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></math>: residue</a> and
<a href="./1.10#Px13.p1" title="Extended Inversion Theorem ‣ §1.10(vii) Inverse Functions ‣ §1.10 Functions of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\mu" display="inline"><mi href="./1.10#Px13.p1">μ</mi></math>: parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m76.png" altimg-height="20px" altimg-valign="-6px" altimg-width="53px" alttext="\mu>0" display="inline"><mrow><mi href="./1.10#Px13.p1">μ</mi><mo>></mo><mn>0</mn></mrow></math>, <math class="ltx_Math" altimg="m125.png" altimg-height="21px" altimg-valign="-6px" altimg-width="60px" alttext="f_{0}\not=0" display="inline"><mrow><msub><mi href="./1.10#Px13.p1">f</mi><mn>0</mn></msub><mo>≠</mo><mn>0</mn></mrow></math>, and the series converges in a neighborhood of
<math class="ltx_Math" altimg="m181.png" altimg-height="16px" altimg-valign="-5px" altimg-width="23px" alttext="z_{0}" display="inline"><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></math>. (For example, when <math class="ltx_Math" altimg="m77.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\mu" display="inline"><mi href="./1.10#Px13.p1">μ</mi></math> is an integer <math class="ltx_Math" altimg="m118.png" altimg-height="23px" altimg-valign="-7px" altimg-width="112px" alttext="f(z)-f(z_{0})" display="inline"><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>-</mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mrow></mrow></math> has a zero of
order <math class="ltx_Math" altimg="m77.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\mu" display="inline"><mi href="./1.10#Px13.p1">μ</mi></math> at <math class="ltx_Math" altimg="m181.png" altimg-height="16px" altimg-valign="-5px" altimg-width="23px" alttext="z_{0}" display="inline"><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></math>.) Let <math class="ltx_Math" altimg="m164.png" altimg-height="23px" altimg-valign="-7px" altimg-width="100px" alttext="w_{0}=f(z_{0})" display="inline"><mrow><msub><mi href="./1.1#p2.t1.r3">w</mi><mn>0</mn></msub><mo>=</mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mrow></mrow></math>. Then () has a
solution <math class="ltx_Math" altimg="m172.png" altimg-height="23px" altimg-valign="-7px" altimg-width="87px" alttext="z=F(w)" display="inline"><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>=</mo><mrow><mi href="./1.10#E13">F</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r3">w</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></math>, where</p>
<table id="E16" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.10.16</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E16.png" altimg-height="64px" altimg-valign="-27px" altimg-width="285px" alttext="F(w)=z_{0}+\sum^{\infty}_{n=1}F_{n}(w-w_{0})^{n/\mu}" display="block"><mrow><mrow><mi href="./1.10#E13">F</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r3">w</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub><mo>+</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><msub><mi href="./1.10#Px12.p1">F</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./1.1#p2.t1.r3">w</mi><mo>-</mo><msub><mi href="./1.1#p2.t1.r3">w</mi><mn>0</mn></msub></mrow><mo stretchy="false">)</mo></mrow><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>/</mo><mi href="./1.10#Px13.p1">μ</mi></mrow></msup></mrow></mrow></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E16.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m177.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>,
<a href="./1.1#p2.t1.r3" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m163.png" altimg-height="13px" altimg-valign="-2px" altimg-width="19px" alttext="w" display="inline"><mi href="./1.1#p2.t1.r3">w</mi></math>: variable</a>,
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m143.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a>,
<a href="./1.10#E13" title="(1.10.13) ‣ Lagrange Inversion Theorem ‣ §1.10(vii) Inverse Functions ‣ §1.10 Functions of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="23px" altimg-valign="-7px" altimg-width="50px" alttext="F(w)" display="inline"><mrow><mi href="./1.10#E13">F</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r3">w</mi><mo stretchy="false">)</mo></mrow></mrow></math>: inverse function of <math class="ltx_Math" altimg="m120.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(z)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math></a>,
<a href="./1.10#Px12.p1" title="Lagrange Inversion Theorem ‣ §1.10(vii) Inverse Functions ‣ §1.10 Functions of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="21px" altimg-valign="-5px" altimg-width="28px" alttext="F_{n}" display="inline"><msub><mi href="./1.10#Px12.p1">F</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></math>: residue</a> and
<a href="./1.10#Px13.p1" title="Extended Inversion Theorem ‣ §1.10(vii) Inverse Functions ‣ §1.10 Functions of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\mu" display="inline"><mi href="./1.10#Px13.p1">μ</mi></math>: parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">in a neighborhood of <math class="ltx_Math" altimg="m165.png" altimg-height="16px" altimg-valign="-5px" altimg-width="28px" alttext="w_{0}" display="inline"><msub><mi href="./1.1#p2.t1.r3">w</mi><mn>0</mn></msub></math>, <math class="ltx_Math" altimg="m141.png" altimg-height="21px" altimg-valign="-5px" altimg-width="40px" alttext="nF_{n}" display="inline"><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo></mo><msub><mi href="./1.10#Px12.p1">F</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></mrow></math> being the residue of
<math class="ltx_Math" altimg="m24.png" altimg-height="26px" altimg-valign="-7px" altimg-width="176px" alttext="1/(f(z)-f(z_{0}))^{n/\mu}" display="inline"><mrow><mn>1</mn><mo>/</mo><msup><mrow><mo stretchy="false">(</mo><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>-</mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mrow></mrow><mo stretchy="false">)</mo></mrow><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>/</mo><mi href="./1.10#Px13.p1">μ</mi></mrow></msup></mrow></math> at <math class="ltx_Math" altimg="m175.png" altimg-height="16px" altimg-valign="-5px" altimg-width="59px" alttext="z=z_{0}" display="inline"><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>=</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></mrow></math>.</p>
</div>
<div id="Px13.p2" class="ltx_para">
<p class="ltx_p">It should be noted that different branches of <math class="ltx_Math" altimg="m10.png" altimg-height="26px" altimg-valign="-7px" altimg-width="109px" alttext="(w-w_{0})^{1/\mu}" display="inline"><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./1.1#p2.t1.r3">w</mi><mo>-</mo><msub><mi href="./1.1#p2.t1.r3">w</mi><mn>0</mn></msub></mrow><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mi href="./1.10#Px13.p1">μ</mi></mrow></msup></math> used in forming
<math class="ltx_Math" altimg="m11.png" altimg-height="26px" altimg-valign="-7px" altimg-width="111px" alttext="(w-w_{0})^{n/\mu}" display="inline"><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./1.1#p2.t1.r3">w</mi><mo>-</mo><msub><mi href="./1.1#p2.t1.r3">w</mi><mn>0</mn></msub></mrow><mo stretchy="false">)</mo></mrow><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>/</mo><mi href="./1.10#Px13.p1">μ</mi></mrow></msup></math> in (). Also, if in addition <math class="ltx_Math" altimg="m129.png" altimg-height="23px" altimg-valign="-7px" altimg-width="40px" alttext="g(z)" display="inline"><mrow><mi>g</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> is analytic at <math class="ltx_Math" altimg="m181.png" altimg-height="16px" altimg-valign="-5px" altimg-width="23px" alttext="z_{0}" display="inline"><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></math>, then</p>
<table id="E17" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.10.17</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E17.png" altimg-height="64px" altimg-valign="-27px" altimg-width="344px" alttext="g(F(w))=g(z_{0})+\sum^{\infty}_{n=1}G_{n}(w-w_{0})^{n/\mu}," display="block"><mrow><mrow><mrow><mi>g</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./1.10#E13">F</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r3">w</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mi>g</mi><mo></mo><mrow><mo stretchy="false">(</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mrow><mo>+</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><msub><mi href="./1.10#Px13.p2">G</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./1.1#p2.t1.r3">w</mi><mo>-</mo><msub><mi href="./1.1#p2.t1.r3">w</mi><mn>0</mn></msub></mrow><mo stretchy="false">)</mo></mrow><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>/</mo><mi href="./1.10#Px13.p1">μ</mi></mrow></msup></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E17.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m177.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>,
<a href="./1.1#p2.t1.r3" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m163.png" altimg-height="13px" altimg-valign="-2px" altimg-width="19px" alttext="w" display="inline"><mi href="./1.1#p2.t1.r3">w</mi></math>: variable</a>,
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m143.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a>,
<a href="./1.10#E13" title="(1.10.13) ‣ Lagrange Inversion Theorem ‣ §1.10(vii) Inverse Functions ‣ §1.10 Functions of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="23px" altimg-valign="-7px" altimg-width="50px" alttext="F(w)" display="inline"><mrow><mi href="./1.10#E13">F</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r3">w</mi><mo stretchy="false">)</mo></mrow></mrow></math>: inverse function of <math class="ltx_Math" altimg="m120.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(z)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math></a>,
<a href="./1.10#Px13.p1" title="Extended Inversion Theorem ‣ §1.10(vii) Inverse Functions ‣ §1.10 Functions of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\mu" display="inline"><mi href="./1.10#Px13.p1">μ</mi></math>: parameter</a> and
<a href="./1.10#Px13.p2" title="Extended Inversion Theorem ‣ §1.10(vii) Inverse Functions ‣ §1.10 Functions of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="21px" altimg-valign="-5px" altimg-width="31px" alttext="G_{n}" display="inline"><msub><mi href="./1.10#Px13.p2">G</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></math>: residue</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m142.png" altimg-height="21px" altimg-valign="-5px" altimg-width="43px" alttext="nG_{n}" display="inline"><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo></mo><msub><mi href="./1.10#Px13.p2">G</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></mrow></math> is the residue of <math class="ltx_Math" altimg="m130.png" altimg-height="26px" altimg-valign="-7px" altimg-width="207px" alttext="g^{\prime}(z)/(f(z)-f(z_{0}))^{n/\mu}" display="inline"><mrow><mrow><msup><mi>g</mi><mo>′</mo></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>/</mo><msup><mrow><mo stretchy="false">(</mo><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>-</mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mrow></mrow><mo stretchy="false">)</mo></mrow><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>/</mo><mi href="./1.10#Px13.p1">μ</mi></mrow></msup></mrow></math> at <math class="ltx_Math" altimg="m175.png" altimg-height="16px" altimg-valign="-5px" altimg-width="59px" alttext="z=z_{0}" display="inline"><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>=</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></mrow></math>.</p>
</div>
</section>
</section>
<section id="viii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§1.10(viii) </span>Functions Defined by Contour Integrals</h2>
<div id="SS8.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="SS8.p1" class="ltx_para">
<p class="ltx_p">Let <math class="ltx_Math" altimg="m33.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.10#SS8.p1">D</mi></math> be a domain and <math class="ltx_Math" altimg="m54.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="[a,b]" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">[</mo><mi>a</mi><mo href="./front/introduction#Sx4.p1.t1.r29">,</mo><mi>b</mi><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">]</mo></mrow></math> be a closed finite segment of the real axis.
Assume that for each <math class="ltx_Math" altimg="m158.png" altimg-height="23px" altimg-valign="-7px" altimg-width="75px" alttext="t\in[a,b]" display="inline"><mrow><mi>t</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">[</mo><mi>a</mi><mo href="./front/introduction#Sx4.p1.t1.r29">,</mo><mi>b</mi><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">]</mo></mrow></mrow></math>, <math class="ltx_Math" altimg="m121.png" altimg-height="23px" altimg-valign="-7px" altimg-width="58px" alttext="f(z,t)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math> is an analytic function of <math class="ltx_Math" altimg="m177.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math> in
<math class="ltx_Math" altimg="m33.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.10#SS8.p1">D</mi></math>, and also that <math class="ltx_Math" altimg="m121.png" altimg-height="23px" altimg-valign="-7px" altimg-width="58px" alttext="f(z,t)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math> is a continuous function of both variables. Then</p>
<table id="E18" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.10.18</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E18.png" altimg-height="56px" altimg-valign="-20px" altimg-width="177px" alttext="F(z)=\int^{b}_{a}f(z,t)\mathrm{d}t" display="block"><mrow><mrow><mi>F</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi>a</mi><mi>b</mi></msubsup><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E18.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m166.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a> and
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m177.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">is analytic in <math class="ltx_Math" altimg="m33.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.10#SS8.p1">D</mi></math> and its derivatives of all orders can be found by
differentiating under the sign of integration.
</p>
</div>
<div id="SS8.p2" class="ltx_para">
<p class="ltx_p">This result is also true when <math class="ltx_Math" altimg="m112.png" altimg-height="18px" altimg-valign="-2px" altimg-width="59px" alttext="b=\infty" display="inline"><mrow><mi>b</mi><mo>=</mo><mi mathvariant="normal">∞</mi></mrow></math>, or when <math class="ltx_Math" altimg="m121.png" altimg-height="23px" altimg-valign="-7px" altimg-width="58px" alttext="f(z,t)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math> has a singularity
at <math class="ltx_Math" altimg="m156.png" altimg-height="18px" altimg-valign="-2px" altimg-width="47px" alttext="t=b" display="inline"><mrow><mi>t</mi><mo>=</mo><mi>b</mi></mrow></math>, with the following conditions. For each <math class="ltx_Math" altimg="m157.png" altimg-height="23px" altimg-valign="-7px" altimg-width="77px" alttext="t\in[a,b)" display="inline"><mrow><mi>t</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mrow><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">[</mo><mi>a</mi><mo href="./front/introduction#Sx4.p2.t1.r1">,</mo><mi>b</mi><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">)</mo></mrow></mrow></math>, <math class="ltx_Math" altimg="m121.png" altimg-height="23px" altimg-valign="-7px" altimg-width="58px" alttext="f(z,t)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math> is
analytic in <math class="ltx_Math" altimg="m33.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.10#SS8.p1">D</mi></math>; <math class="ltx_Math" altimg="m121.png" altimg-height="23px" altimg-valign="-7px" altimg-width="58px" alttext="f(z,t)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math> is a continuous function of both variables when
<math class="ltx_Math" altimg="m178.png" altimg-height="18px" altimg-valign="-3px" altimg-width="56px" alttext="z\in D" display="inline"><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mi href="./1.10#SS8.p1">D</mi></mrow></math> and <math class="ltx_Math" altimg="m157.png" altimg-height="23px" altimg-valign="-7px" altimg-width="77px" alttext="t\in[a,b)" display="inline"><mrow><mi>t</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mrow><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">[</mo><mi>a</mi><mo href="./front/introduction#Sx4.p2.t1.r1">,</mo><mi>b</mi><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">)</mo></mrow></mrow></math>; the integral () converges at
<math class="ltx_Math" altimg="m113.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="b" display="inline"><mi>b</mi></math>, and this convergence is uniform with respect to <math class="ltx_Math" altimg="m177.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math> in every compact
subset <math class="ltx_Math" altimg="m53.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi href="./1.10#SS8.p2">S</mi></math> of <math class="ltx_Math" altimg="m33.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.10#SS8.p1">D</mi></math>.
</p>
</div>
<div id="SS8.p3" class="ltx_para">
<p class="ltx_p">The last condition means that given <math class="ltx_Math" altimg="m62.png" altimg-height="13px" altimg-valign="-2px" altimg-width="12px" alttext="\epsilon" display="inline"><mi href="./1.10#SS8.p3">ϵ</mi></math> (<math class="ltx_Math" altimg="m27.png" altimg-height="17px" altimg-valign="-3px" altimg-width="35px" alttext=">0" display="inline"><mrow><mi></mi><mo>></mo><mn>0</mn></mrow></math>) there exists a number
<math class="ltx_Math" altimg="m103.png" altimg-height="23px" altimg-valign="-7px" altimg-width="89px" alttext="a_{0}\in[a,b)" display="inline"><mrow><msub><mi>a</mi><mn>0</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mrow><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">[</mo><mi>a</mi><mo href="./front/introduction#Sx4.p2.t1.r1">,</mo><mi>b</mi><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">)</mo></mrow></mrow></math> that is independent of <math class="ltx_Math" altimg="m177.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math> and is such that</p>
<table id="E19" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.10.19</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E19.png" altimg-height="65px" altimg-valign="-27px" altimg-width="163px" alttext="\left|\int_{a_{1}}^{b}f(z,t)\mathrm{d}t\right|<\epsilon," display="block"><mrow><mrow><mrow><mo>|</mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><msub><mi>a</mi><mn>1</mn></msub><mi>b</mi></msubsup><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow><mo>|</mo></mrow><mo><</mo><mi href="./1.10#SS8.p3">ϵ</mi></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E19.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m166.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m177.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a> and
<a href="./1.10#SS8.p3" title="§1.10(viii) Functions Defined by Contour Integrals ‣ §1.10 Functions of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m62.png" altimg-height="13px" altimg-valign="-2px" altimg-width="12px" alttext="\epsilon" display="inline"><mi href="./1.10#SS8.p3">ϵ</mi></math>: positive number</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">for all <math class="ltx_Math" altimg="m104.png" altimg-height="23px" altimg-valign="-7px" altimg-width="98px" alttext="a_{1}\in[a_{0},b)" display="inline"><mrow><msub><mi>a</mi><mn>1</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mrow><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">[</mo><msub><mi>a</mi><mn>0</mn></msub><mo href="./front/introduction#Sx4.p2.t1.r1">,</mo><mi>b</mi><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">)</mo></mrow></mrow></math> and all <math class="ltx_Math" altimg="m179.png" altimg-height="18px" altimg-valign="-3px" altimg-width="52px" alttext="z\in S" display="inline"><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mi href="./1.10#SS8.p2">S</mi></mrow></math>; compare §.</p>
</div>
<section id="Px14" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">
<math class="ltx_Math" altimg="m49.png" altimg-height="18px" altimg-valign="-2px" altimg-width="26px" alttext="M" display="inline"><mi>M</mi></math>-test</h3>
<div id="Px14.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px14.p1" class="ltx_para">
<p class="ltx_p">If <math class="ltx_Math" altimg="m189.png" altimg-height="23px" altimg-valign="-7px" altimg-width="140px" alttext="|f(z,t)|\leq M(t)" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow><mo>≤</mo><mrow><mi>M</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></math> for <math class="ltx_Math" altimg="m179.png" altimg-height="18px" altimg-valign="-3px" altimg-width="52px" alttext="z\in S" display="inline"><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mi href="./1.10#SS8.p2">S</mi></mrow></math> and <math class="ltx_Math" altimg="m67.png" altimg-height="31px" altimg-valign="-9px" altimg-width="91px" alttext="\int^{b}_{a}M(t)\mathrm{d}t" display="inline"><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi>a</mi><mi>b</mi></msubsup><mrow><mrow><mi>M</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></math> converges,
then the integral () converges uniformly and absolutely in
<math class="ltx_Math" altimg="m53.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi href="./1.10#SS8.p2">S</mi></math>.</p>
</div>
</section>
</section>
<section id="ix" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§1.10(ix) </span>Infinite Products</h2>
<div id="SS9.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="SS9.p1" class="ltx_para">
<p class="ltx_p">Let <math class="ltx_Math" altimg="m148.png" altimg-height="26px" altimg-valign="-8px" altimg-width="191px" alttext="p_{k,m}=\prod_{n=k}^{m}(1+a_{n})" display="inline"><mrow><msub><mi>p</mi><mrow><mi href="./1.1#p2.t1.r4">k</mi><mo>,</mo><mi href="./1.1#p2.t1.r5">m</mi></mrow></msub><mo>=</mo><mrow><msubsup><mo largeop="true" symmetric="true">∏</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mi href="./1.1#p2.t1.r4">k</mi></mrow><mi href="./1.1#p2.t1.r5">m</mi></msubsup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>+</mo><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></math>. If for some <math class="ltx_Math" altimg="m134.png" altimg-height="20px" altimg-valign="-5px" altimg-width="52px" alttext="k\geq 1" display="inline"><mrow><mi href="./1.1#p2.t1.r4">k</mi><mo>≥</mo><mn>1</mn></mrow></math>, <math class="ltx_Math" altimg="m150.png" altimg-height="23px" altimg-valign="-8px" altimg-width="130px" alttext="p_{k,m}\to p_{k}\not=0" display="inline"><mrow><msub><mi>p</mi><mrow><mi href="./1.1#p2.t1.r4">k</mi><mo>,</mo><mi href="./1.1#p2.t1.r5">m</mi></mrow></msub><mo>→</mo><msub><mi>p</mi><mi href="./1.1#p2.t1.r4">k</mi></msub><mo>≠</mo><mn>0</mn></mrow></math> as <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="73px" alttext="m\to\infty" display="inline"><mrow><mi href="./1.1#p2.t1.r5">m</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></math>, then we say that the infinite product
<math class="ltx_Math" altimg="m87.png" altimg-height="26px" altimg-valign="-8px" altimg-width="125px" alttext="\prod^{\infty}_{n=1}(1+a_{n})" display="inline"><mrow><msubsup><mo largeop="true" symmetric="true">∏</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></msubsup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>+</mo><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></mrow><mo stretchy="false">)</mo></mrow></mrow></math> <em class="ltx_emph ltx_font_italic">converges</em>. (The integer <math class="ltx_Math" altimg="m133.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./1.1#p2.t1.r4">k</mi></math> may be greater
than one to allow for a finite number of zero factors.) The convergence of the
product is <em class="ltx_emph ltx_font_italic">absolute</em> if <math class="ltx_Math" altimg="m88.png" altimg-height="26px" altimg-valign="-8px" altimg-width="136px" alttext="\prod^{\infty}_{n=1}(1+|a_{n}|)" display="inline"><mrow><msubsup><mo largeop="true" symmetric="true">∏</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></msubsup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>+</mo><mrow><mo stretchy="false">|</mo><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo stretchy="false">|</mo></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></math> converges. The
product <math class="ltx_Math" altimg="m87.png" altimg-height="26px" altimg-valign="-8px" altimg-width="125px" alttext="\prod^{\infty}_{n=1}(1+a_{n})" display="inline"><mrow><msubsup><mo largeop="true" symmetric="true">∏</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></msubsup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>+</mo><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></mrow><mo stretchy="false">)</mo></mrow></mrow></math>, with <math class="ltx_Math" altimg="m110.png" altimg-height="21px" altimg-valign="-6px" altimg-width="78px" alttext="a_{n}\not=-1" display="inline"><mrow><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo>≠</mo><mrow><mo>-</mo><mn>1</mn></mrow></mrow></math> for all <math class="ltx_Math" altimg="m143.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>,
converges iff <math class="ltx_Math" altimg="m92.png" altimg-height="26px" altimg-valign="-8px" altimg-width="151px" alttext="\sum^{\infty}_{n=1}\ln\left(1+a_{n}\right)" display="inline"><mrow><msubsup><mo largeop="true" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></msubsup><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></mrow><mo>)</mo></mrow></mrow></mrow></math> converges; and it converges
absolutely iff <math class="ltx_Math" altimg="m94.png" altimg-height="26px" altimg-valign="-8px" altimg-width="92px" alttext="\sum^{\infty}_{n=1}|a_{n}|" display="inline"><mrow><msubsup><mo largeop="true" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></msubsup><mrow><mo stretchy="false">|</mo><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo stretchy="false">|</mo></mrow></mrow></math> converges.</p>
</div>
<div id="SS9.p2" class="ltx_para">
<p class="ltx_p">Suppose <math class="ltx_Math" altimg="m108.png" altimg-height="23px" altimg-valign="-7px" altimg-width="99px" alttext="a_{n}=a_{n}(z)" display="inline"><mrow><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo>=</mo><mrow><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></math>, <math class="ltx_Math" altimg="m178.png" altimg-height="18px" altimg-valign="-3px" altimg-width="56px" alttext="z\in D" display="inline"><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mi href="./1.10#SS9.p2">D</mi></mrow></math>, a domain. The convergence of the infinite
product is <em class="ltx_emph ltx_font_italic">uniform</em>
if the sequence of partial products converges uniformly.
</p>
</div>
<section id="Px15" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">
<math class="ltx_Math" altimg="m49.png" altimg-height="18px" altimg-valign="-2px" altimg-width="26px" alttext="M" display="inline"><mi>M</mi></math>-test</h3>
<div id="Px15.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Keywords:</dt>
<dd>
<a class="ltx_keyword" href="./idx/M#Mtestforuniformconvergence"><math class="ltx_Math" altimg="m49.png" altimg-height="18px" altimg-valign="-2px" altimg-width="26px" alttext="M" display="inline"><mi>M</mi></math>-test for uniform convergence</a>, <a class="ltx_keyword" href="./idx/W#WeierstrassMtest">Weierstrass <math class="ltx_Math" altimg="m49.png" altimg-height="18px" altimg-valign="-2px" altimg-width="26px" alttext="M" display="inline"><mi>M</mi></math>-test</a>, </dd>
</dl>
</div>
</div>
<div id="Px15.p1" class="ltx_para">
<p class="ltx_p">Suppose that <math class="ltx_Math" altimg="m105.png" altimg-height="23px" altimg-valign="-7px" altimg-width="51px" alttext="a_{n}(z)" display="inline"><mrow><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> are analytic functions in <math class="ltx_Math" altimg="m33.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.10#SS9.p2">D</mi></math>. If there is an <math class="ltx_Math" altimg="m50.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="N" display="inline"><mi>N</mi></math>,
independent of <math class="ltx_Math" altimg="m178.png" altimg-height="18px" altimg-valign="-3px" altimg-width="56px" alttext="z\in D" display="inline"><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mi href="./1.10#SS9.p2">D</mi></mrow></math>, such that</p>
<table id="E20" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.10.20</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E20.png" altimg-height="25px" altimg-valign="-7px" altimg-width="203px" alttext="|\ln\left(1+a_{n}(z)\right)|\leq M_{n}," display="block"><mrow><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mrow><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow><mo>≤</mo><msub><mi>M</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m145.png" altimg-height="20px" altimg-valign="-5px" altimg-width="61px" alttext="n\geq N" display="inline"><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>≥</mo><mi>N</mi></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E20.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a>,
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m177.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a> and
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m143.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and</p>
<table id="E21" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.10.21</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E21.png" altimg-height="64px" altimg-valign="-27px" altimg-width="126px" alttext="\sum^{\infty}_{n=1}M_{n}<\infty," display="block"><mrow><mrow><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover><msub><mi>M</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></mrow><mo><</mo><mi mathvariant="normal">∞</mi></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E21.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m143.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a></dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">then the product <math class="ltx_Math" altimg="m86.png" altimg-height="26px" altimg-valign="-8px" altimg-width="151px" alttext="\prod^{\infty}_{n=1}(1+a_{n}(z))" display="inline"><mrow><msubsup><mo largeop="true" symmetric="true">∏</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></msubsup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>+</mo><mrow><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></math> converges uniformly to an
analytic
function <math class="ltx_Math" altimg="m147.png" altimg-height="23px" altimg-valign="-7px" altimg-width="40px" alttext="p(z)" display="inline"><mrow><mi>p</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> in <math class="ltx_Math" altimg="m33.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.10#SS9.p2">D</mi></math>, and <math class="ltx_Math" altimg="m146.png" altimg-height="23px" altimg-valign="-7px" altimg-width="76px" alttext="p(z)=0" display="inline"><mrow><mrow><mi>p</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mn>0</mn></mrow></math> only when at least one of the factors
<math class="ltx_Math" altimg="m23.png" altimg-height="23px" altimg-valign="-7px" altimg-width="86px" alttext="1+a_{n}(z)" display="inline"><mrow><mn>1</mn><mo>+</mo><mrow><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></math> is zero in <math class="ltx_Math" altimg="m33.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.10#SS9.p2">D</mi></math>. This conclusion remains true if, in place of
(), <math class="ltx_Math" altimg="m185.png" altimg-height="23px" altimg-valign="-7px" altimg-width="119px" alttext="|a_{n}(z)|\leq M_{n}" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mrow><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow><mo>≤</mo><msub><mi>M</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></mrow></math> for all <math class="ltx_Math" altimg="m143.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>, and again
<math class="ltx_Math" altimg="m91.png" altimg-height="26px" altimg-valign="-8px" altimg-width="136px" alttext="\sum^{\infty}_{n=1}M_{n}<\infty" display="inline"><mrow><mrow><msubsup><mo largeop="true" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></msubsup><msub><mi>M</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></mrow><mo><</mo><mi mathvariant="normal">∞</mi></mrow></math>.</p>
</div>
</section>
<section id="Px16" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Weierstrass Product</h3>
<div id="Px16.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px16.p1" class="ltx_para">
<p class="ltx_p">If <math class="ltx_Math" altimg="m97.png" altimg-height="23px" altimg-valign="-7px" altimg-width="44px" alttext="\{z_{n}\}" display="inline"><mrow><mo stretchy="false">{</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo stretchy="false">}</mo></mrow></math> is a sequence such that <math class="ltx_Math" altimg="m95.png" altimg-height="26px" altimg-valign="-8px" altimg-width="102px" alttext="\sum^{\infty}_{n=1}|z_{n}^{-2}|" display="inline"><mrow><msubsup><mo largeop="true" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></msubsup><mrow><mo stretchy="false">|</mo><msubsup><mi href="./1.1#p2.t1.r2">z</mi><mi href="./1.1#p2.t1.r5">n</mi><mrow><mo>-</mo><mn>2</mn></mrow></msubsup><mo stretchy="false">|</mo></mrow></mrow></math> is
convergent, then</p>
<table id="E22" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.10.22</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E22.png" altimg-height="64px" altimg-valign="-27px" altimg-width="243px" alttext="P(z)=\prod^{\infty}_{n=1}\left(1-\frac{z}{z_{n}}\right)e^{z/z_{n}}" display="block"><mrow><mrow><mi>P</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∏</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mfrac><mi href="./1.1#p2.t1.r2">z</mi><msub><mi href="./1.1#p2.t1.r2">z</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></mfrac></mrow><mo>)</mo></mrow><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>/</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></mrow></msup></mrow></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E22.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m75.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m177.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a> and
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m143.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">is an entire function with zeros at <math class="ltx_Math" altimg="m184.png" altimg-height="16px" altimg-valign="-5px" altimg-width="24px" alttext="z_{n}" display="inline"><msub><mi href="./1.1#p2.t1.r2">z</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></math>.</p>
</div>
</section>
</section>
<section id="x" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§1.10(x) </span>Infinite Partial Fractions</h2>
<div id="SS10.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="SS10.p1" class="ltx_para">
<p class="ltx_p">Suppose <math class="ltx_Math" altimg="m33.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.10#SS10.p1">D</mi></math> is a domain, and
</p>
<table id="E23" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.10.23</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E23.png" altimg-height="64px" altimg-valign="-27px" altimg-width="160px" alttext="F(z)=\prod^{\infty}_{n=1}a_{n}(z)," display="block"><mrow><mrow><mrow><mi href="./1.10#E23">F</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∏</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m178.png" altimg-height="18px" altimg-valign="-3px" altimg-width="56px" alttext="z\in D" display="inline"><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mi href="./1.10#SS10.p1">D</mi></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E23.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text"><math class="ltx_Math" altimg="m46.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="F" display="inline"><mi href="./1.10#E23">F</mi></math>: function (locally)</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./front/introduction#Sx4.p1.t1.r9" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m64.png" altimg-height="15px" altimg-valign="-3px" altimg-width="18px" alttext="\in" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo></math>: element of</a>,
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m177.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>,
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m143.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./1.10#SS10.p1" title="§1.10(x) Infinite Partial Fractions ‣ §1.10 Functions of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m33.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.10#SS10.p1">D</mi></math>: domain</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m105.png" altimg-height="23px" altimg-valign="-7px" altimg-width="51px" alttext="a_{n}(z)" display="inline"><mrow><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> is analytic for all <math class="ltx_Math" altimg="m144.png" altimg-height="19px" altimg-valign="-5px" altimg-width="53px" alttext="n\geq 1" display="inline"><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>≥</mo><mn>1</mn></mrow></math>, and the convergence of the
product is uniform in any compact subset of <math class="ltx_Math" altimg="m33.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.10#SS10.p1">D</mi></math>. Then <math class="ltx_Math" altimg="m42.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="F(z)" display="inline"><mrow><mi href="./1.10#E23">F</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> is analytic in
<math class="ltx_Math" altimg="m33.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.10#SS10.p1">D</mi></math>.</p>
</div>
<div id="SS10.p2" class="ltx_para">
<p class="ltx_p">If, also, <math class="ltx_Math" altimg="m106.png" altimg-height="23px" altimg-valign="-7px" altimg-width="88px" alttext="a_{n}(z)\neq 0" display="inline"><mrow><mrow><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>≠</mo><mn>0</mn></mrow></math> when <math class="ltx_Math" altimg="m144.png" altimg-height="19px" altimg-valign="-5px" altimg-width="53px" alttext="n\geq 1" display="inline"><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>≥</mo><mn>1</mn></mrow></math> and <math class="ltx_Math" altimg="m178.png" altimg-height="18px" altimg-valign="-3px" altimg-width="56px" alttext="z\in D" display="inline"><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mi href="./1.10#SS10.p1">D</mi></mrow></math>, then <math class="ltx_Math" altimg="m43.png" altimg-height="23px" altimg-valign="-7px" altimg-width="82px" alttext="F(z)\neq 0" display="inline"><mrow><mrow><mi href="./1.10#E23">F</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>≠</mo><mn>0</mn></mrow></math> on <math class="ltx_Math" altimg="m33.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.10#SS10.p1">D</mi></math>
and</p>
<table id="E24" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.10.24</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E24.png" altimg-height="64px" altimg-valign="-27px" altimg-width="175px" alttext="\frac{F^{\prime}(z)}{F(z)}=\sum^{\infty}_{n=1}\frac{a_{n}^{\prime}(z)}{a_{n}(z%
)}." display="block"><mrow><mrow><mfrac><mrow><msup><mi href="./1.10#E23">F</mi><mo>′</mo></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mrow><mi href="./1.10#E23">F</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></mfrac><mo>=</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mfrac><mrow><msubsup><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi><mo>′</mo></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mrow><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></mfrac></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E24.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m177.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>,
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m143.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./1.10#E23" title="(1.10.23) ‣ §1.10(x) Infinite Partial Fractions ‣ §1.10 Functions of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m46.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="F" display="inline"><mi href="./1.10#E23">F</mi></math>: function</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<section id="Px17" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Mittag-Leffler’s Expansion</h3>
<div id="Px17.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px17.p1" class="ltx_para">
<p class="ltx_p">If <math class="ltx_Math" altimg="m96.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\{a_{n}\}" display="inline"><mrow><mo stretchy="false">{</mo><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo stretchy="false">}</mo></mrow></math> and <math class="ltx_Math" altimg="m97.png" altimg-height="23px" altimg-valign="-7px" altimg-width="44px" alttext="\{z_{n}\}" display="inline"><mrow><mo stretchy="false">{</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo stretchy="false">}</mo></mrow></math> are sequences such that <math class="ltx_Math" altimg="m183.png" altimg-height="21px" altimg-valign="-6px" altimg-width="75px" alttext="z_{m}\neq z_{n}" display="inline"><mrow><msub><mi href="./1.1#p2.t1.r2">z</mi><mi href="./1.1#p2.t1.r5">m</mi></msub><mo>≠</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></mrow></math>
(<math class="ltx_Math" altimg="m137.png" altimg-height="21px" altimg-valign="-6px" altimg-width="60px" alttext="m\neq n" display="inline"><mrow><mi href="./1.1#p2.t1.r5">m</mi><mo>≠</mo><mi href="./1.1#p2.t1.r5">n</mi></mrow></math>) and <math class="ltx_Math" altimg="m93.png" altimg-height="26px" altimg-valign="-8px" altimg-width="124px" alttext="\sum^{\infty}_{n=1}|a_{n}z_{n}^{-2}|" display="inline"><mrow><msubsup><mo largeop="true" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></msubsup><mrow><mo stretchy="false">|</mo><mrow><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><msubsup><mi href="./1.1#p2.t1.r2">z</mi><mi href="./1.1#p2.t1.r5">n</mi><mrow><mo>-</mo><mn>2</mn></mrow></msubsup></mrow><mo stretchy="false">|</mo></mrow></mrow></math> is convergent, then</p>
<table id="E25" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.10.25</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E25.png" altimg-height="64px" altimg-valign="-27px" altimg-width="266px" alttext="f(z)=\sum^{\infty}_{n=1}a_{n}\left(\frac{1}{z-z_{n}}+\frac{1}{z_{n}}\right)" display="block"><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>-</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></mrow></mfrac><mo>+</mo><mfrac><mn>1</mn><msub><mi href="./1.1#p2.t1.r2">z</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></mfrac></mrow><mo>)</mo></mrow></mrow></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E25.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m177.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a> and
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m143.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">is analytic in <math class="ltx_Math" altimg="m72.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\mathbb{C}" display="inline"><mi href="./front/introduction#Sx4.p1.t1.r1">ℂ</mi></math>, except for simple poles at <math class="ltx_Math" altimg="m176.png" altimg-height="16px" altimg-valign="-5px" altimg-width="61px" alttext="z=z_{n}" display="inline"><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>=</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></mrow></math> of residue <math class="ltx_Math" altimg="m109.png" altimg-height="16px" altimg-valign="-5px" altimg-width="26px" alttext="a_{n}" display="inline"><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></math>.
</p>
</div>
</section>
</section>
</section>
</div>
<div class="ltx_page_footer">
<span></div>
</div>
</body></text>
</html>
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<page>
<!DOCTYPE html><html>
<head>
<title>DLMF: 1.6 Vectors and Vector-Valued Functions</title>
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<text><body class="color_default textfont_default titlefont_default fontsize_default navbar_default">
<div class="ltx_page_navbar">
<div class="ltx_page_navlogo"></dd>
</dl>
</div>
</div>
<div id="SS1.p1" class="ltx_para">
<table id="E1" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex1" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">1.6.1</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m22.png" altimg-height="14px" altimg-valign="-2px" altimg-width="17px" alttext="\displaystyle\mathbf{a}" display="inline"><mi mathvariant="bold">a</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m6.png" altimg-height="25px" altimg-valign="-7px" altimg-width="124px" alttext="\displaystyle=(a_{1},a_{2},a_{3})," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo stretchy="false">(</mo><msub><mi>a</mi><mn>1</mn></msub><mo>,</mo><msub><mi>a</mi><mn>2</mn></msub><mo>,</mo><msub><mi>a</mi><mn>3</mn></msub><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex2" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m24.png" altimg-height="19px" altimg-valign="-2px" altimg-width="19px" alttext="\displaystyle\mathbf{b}" display="inline"><mi mathvariant="bold">b</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m7.png" altimg-height="25px" altimg-valign="-7px" altimg-width="118px" alttext="\displaystyle=(b_{1},b_{2},b_{3})." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo stretchy="false">(</mo><msub><mi>b</mi><mn>1</mn></msub><mo>,</mo><msub><mi>b</mi><mn>2</mn></msub><mo>,</mo><msub><mi>b</mi><mn>3</mn></msub><mo stretchy="false">)</mo></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E1.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px1.p1" class="ltx_para">
<table id="E2" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.2</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E2.png" altimg-height="22px" altimg-valign="-5px" altimg-width="236px" alttext="\mathbf{a}\cdot\mathbf{b}=a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}." display="block"><mrow><mrow><mrow><mi mathvariant="bold">a</mi><mo>⋅</mo><mi mathvariant="bold">b</mi></mrow><mo>=</mo><mrow><mrow><msub><mi>a</mi><mn>1</mn></msub><mo></mo><msub><mi>b</mi><mn>1</mn></msub></mrow><mo>+</mo><mrow><msub><mi>a</mi><mn>2</mn></msub><mo></mo><msub><mi>b</mi><mn>2</mn></msub></mrow><mo>+</mo><mrow><msub><mi>a</mi><mn>3</mn></msub><mo></mo><msub><mi>b</mi><mn>3</mn></msub></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E2.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px2" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Magnitude and Angle of Vector <math class="ltx_Math" altimg="m75.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\mathbf{a}" display="inline"><mi mathvariant="bold">a</mi></math>
</h3>
<div id="Px2.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px2.p1" class="ltx_para">
<table id="E3" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.3</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E3.png" altimg-height="27px" altimg-valign="-7px" altimg-width="122px" alttext="\|\mathbf{a}\|=\sqrt{\mathbf{a}\cdot\mathbf{a}}," display="block"><mrow><mrow><mrow><mo>∥</mo><mi mathvariant="bold">a</mi><mo>∥</mo></mrow><mo>=</mo><msqrt><mrow><mi mathvariant="bold">a</mi><mo>⋅</mo><mi mathvariant="bold">a</mi></mrow></msqrt></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E3.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E4" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.4</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E4.png" altimg-height="51px" altimg-valign="-21px" altimg-width="152px" alttext="\cos\theta=\frac{\mathbf{a}\cdot\mathbf{b}}{\|\mathbf{a}\|\;\|\mathbf{b}\|};" display="block"><mrow><mrow><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi href="./1.6#Px2.p1">θ</mi></mrow><mo>=</mo><mfrac><mrow><mi mathvariant="bold">a</mi><mo>⋅</mo><mi mathvariant="bold">b</mi></mrow><mrow><mrow><mo>∥</mo><mi mathvariant="bold">a</mi><mo rspace="5.3pt">∥</mo></mrow><mo></mo><mrow><mo>∥</mo><mi mathvariant="bold">b</mi><mo>∥</mo></mrow></mrow></mfrac></mrow><mo>;</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E4.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a> and
<a href="./1.6#Px2.p1" title="Magnitude and Angle of Vector a ‣ §1.6(i) Vectors ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m104.png" altimg-height="18px" altimg-valign="-2px" altimg-width="14px" alttext="\theta" display="inline"><mi href="./1.6#Px2.p1">θ</mi></math>: angle between <math class="ltx_Math" altimg="m75.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\mathbf{a}" display="inline"><mi mathvariant="bold">a</mi></math> and <math class="ltx_Math" altimg="m77.png" altimg-height="18px" altimg-valign="-2px" altimg-width="17px" alttext="\mathbf{b}" display="inline"><mi mathvariant="bold">b</mi></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p"><math class="ltx_Math" altimg="m104.png" altimg-height="18px" altimg-valign="-2px" altimg-width="14px" alttext="\theta" display="inline"><mi href="./1.6#Px2.p1">θ</mi></math> is the angle between <math class="ltx_Math" altimg="m75.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\mathbf{a}" display="inline"><mi mathvariant="bold">a</mi></math> and <math class="ltx_Math" altimg="m77.png" altimg-height="18px" altimg-valign="-2px" altimg-width="17px" alttext="\mathbf{b}" display="inline"><mi mathvariant="bold">b</mi></math>.</p>
</div>
</section>
<section id="Px3" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Unit Vectors</h3>
<div id="Px3.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px3.p1" class="ltx_para">
<table id="E5" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex3" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="3" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">1.6.5</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m28.png" altimg-height="19px" altimg-valign="-2px" altimg-width="12px" alttext="\displaystyle\mathbf{i}" display="inline"><mi href="./1.6#E5" mathvariant="bold">i</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m5.png" altimg-height="25px" altimg-valign="-7px" altimg-width="96px" alttext="\displaystyle=(1,0,0)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex4" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m31.png" altimg-height="23px" altimg-valign="-6px" altimg-width="13px" alttext="\displaystyle\mathbf{j}" display="inline"><mi href="./1.6#E5" mathvariant="bold">j</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m4.png" altimg-height="25px" altimg-valign="-7px" altimg-width="96px" alttext="\displaystyle=(0,1,0)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex5" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m34.png" altimg-height="19px" altimg-valign="-2px" altimg-width="18px" alttext="\displaystyle\mathbf{k}" display="inline"><mi href="./1.6#E5" mathvariant="bold">k</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m2.png" altimg-height="25px" altimg-valign="-7px" altimg-width="96px" alttext="\displaystyle=(0,0,1)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E5.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd>
<span class="ltx_text"><math class="ltx_Math" altimg="m85.png" altimg-height="18px" altimg-valign="-2px" altimg-width="11px" alttext="\mathbf{i}" display="inline"><mi href="./1.6#E5" mathvariant="bold">i</mi></math>: unit vector (locally)</span>,
<span class="ltx_text"><math class="ltx_Math" altimg="m86.png" altimg-height="21px" altimg-valign="-6px" altimg-width="11px" alttext="\mathbf{j}" display="inline"><mi href="./1.6#E5" mathvariant="bold">j</mi></math>: unit vector (locally)</span> and
<span class="ltx_text"><math class="ltx_Math" altimg="m87.png" altimg-height="18px" altimg-valign="-2px" altimg-width="16px" alttext="\mathbf{k}" display="inline"><mi href="./1.6#E5" mathvariant="bold">k</mi></math>: unit vector (locally)</span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E6" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.6</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E6.png" altimg-height="23px" altimg-valign="-6px" altimg-width="182px" alttext="\mathbf{a}=a_{1}\mathbf{i}+a_{2}\mathbf{j}+a_{3}\mathbf{k}." display="block"><mrow><mrow><mi mathvariant="bold">a</mi><mo>=</mo><mrow><mrow><msub><mi>a</mi><mn>1</mn></msub><mo></mo><mi href="./1.6#E5" mathvariant="bold">i</mi></mrow><mo>+</mo><mrow><msub><mi>a</mi><mn>2</mn></msub><mo></mo><mi href="./1.6#E5" mathvariant="bold">j</mi></mrow><mo>+</mo><mrow><msub><mi>a</mi><mn>3</mn></msub><mo></mo><mi href="./1.6#E5" mathvariant="bold">k</mi></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E6.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.6#E5" title="(1.6.5) ‣ Unit Vectors ‣ §1.6(i) Vectors ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="18px" altimg-valign="-2px" altimg-width="11px" alttext="\mathbf{i}" display="inline"><mi href="./1.6#E5" mathvariant="bold">i</mi></math>: unit vector</a>,
<a href="./1.6#E5" title="(1.6.5) ‣ Unit Vectors ‣ §1.6(i) Vectors ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m86.png" altimg-height="21px" altimg-valign="-6px" altimg-width="11px" alttext="\mathbf{j}" display="inline"><mi href="./1.6#E5" mathvariant="bold">j</mi></math>: unit vector</a> and
<a href="./1.6#E5" title="(1.6.5) ‣ Unit Vectors ‣ §1.6(i) Vectors ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="18px" altimg-valign="-2px" altimg-width="16px" alttext="\mathbf{k}" display="inline"><mi href="./1.6#E5" mathvariant="bold">k</mi></math>: unit vector</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px4" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Cross Product (or Vector Product)</h3>
<div id="Px4.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px4.p1" class="ltx_para">
<table id="E7" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex6" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="3" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">1.6.7</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m29.png" altimg-height="23px" altimg-valign="-6px" altimg-width="44px" alttext="\displaystyle\mathbf{i}\times\mathbf{j}" display="inline"><mrow><mi href="./1.6#E5" mathvariant="bold">i</mi><mo>×</mo><mi href="./1.6#E5" mathvariant="bold">j</mi></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m18.png" altimg-height="23px" altimg-valign="-6px" altimg-width="45px" alttext="\displaystyle=\mathbf{k}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mi href="./1.6#E5" mathvariant="bold">k</mi></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex7" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m33.png" altimg-height="23px" altimg-valign="-6px" altimg-width="49px" alttext="\displaystyle\mathbf{j}\times\mathbf{k}" display="inline"><mrow><mi href="./1.6#E5" mathvariant="bold">j</mi><mo>×</mo><mi href="./1.6#E5" mathvariant="bold">k</mi></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m16.png" altimg-height="23px" altimg-valign="-6px" altimg-width="39px" alttext="\displaystyle=\mathbf{i}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mi href="./1.6#E5" mathvariant="bold">i</mi></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex8" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m35.png" altimg-height="21px" altimg-valign="-4px" altimg-width="49px" alttext="\displaystyle\mathbf{k}\times\mathbf{i}" display="inline"><mrow><mi href="./1.6#E5" mathvariant="bold">k</mi><mo>×</mo><mi href="./1.6#E5" mathvariant="bold">i</mi></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m17.png" altimg-height="23px" altimg-valign="-6px" altimg-width="40px" alttext="\displaystyle=\mathbf{j}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mi href="./1.6#E5" mathvariant="bold">j</mi></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E7.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.6#E5" title="(1.6.5) ‣ Unit Vectors ‣ §1.6(i) Vectors ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="18px" altimg-valign="-2px" altimg-width="11px" alttext="\mathbf{i}" display="inline"><mi href="./1.6#E5" mathvariant="bold">i</mi></math>: unit vector</a>,
<a href="./1.6#E5" title="(1.6.5) ‣ Unit Vectors ‣ §1.6(i) Vectors ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m86.png" altimg-height="21px" altimg-valign="-6px" altimg-width="11px" alttext="\mathbf{j}" display="inline"><mi href="./1.6#E5" mathvariant="bold">j</mi></math>: unit vector</a> and
<a href="./1.6#E5" title="(1.6.5) ‣ Unit Vectors ‣ §1.6(i) Vectors ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="18px" altimg-valign="-2px" altimg-width="16px" alttext="\mathbf{k}" display="inline"><mi href="./1.6#E5" mathvariant="bold">k</mi></math>: unit vector</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E8" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex9" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="3" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">1.6.8</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m32.png" altimg-height="23px" altimg-valign="-6px" altimg-width="44px" alttext="\displaystyle\mathbf{j}\times\mathbf{i}" display="inline"><mrow><mi href="./1.6#E5" mathvariant="bold">j</mi><mo>×</mo><mi href="./1.6#E5" mathvariant="bold">i</mi></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m10.png" altimg-height="23px" altimg-valign="-6px" altimg-width="60px" alttext="\displaystyle=-\mathbf{k}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo>-</mo><mi href="./1.6#E5" mathvariant="bold">k</mi></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex10" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m36.png" altimg-height="23px" altimg-valign="-6px" altimg-width="49px" alttext="\displaystyle\mathbf{k}\times\mathbf{j}" display="inline"><mrow><mi href="./1.6#E5" mathvariant="bold">k</mi><mo>×</mo><mi href="./1.6#E5" mathvariant="bold">j</mi></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m8.png" altimg-height="23px" altimg-valign="-6px" altimg-width="54px" alttext="\displaystyle=-\mathbf{i}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo>-</mo><mi href="./1.6#E5" mathvariant="bold">i</mi></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex11" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m30.png" altimg-height="21px" altimg-valign="-4px" altimg-width="49px" alttext="\displaystyle\mathbf{i}\times\mathbf{k}" display="inline"><mrow><mi href="./1.6#E5" mathvariant="bold">i</mi><mo>×</mo><mi href="./1.6#E5" mathvariant="bold">k</mi></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m9.png" altimg-height="23px" altimg-valign="-6px" altimg-width="55px" alttext="\displaystyle=-\mathbf{j}." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo>-</mo><mi href="./1.6#E5" mathvariant="bold">j</mi></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E8.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.6#E5" title="(1.6.5) ‣ Unit Vectors ‣ §1.6(i) Vectors ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="18px" altimg-valign="-2px" altimg-width="11px" alttext="\mathbf{i}" display="inline"><mi href="./1.6#E5" mathvariant="bold">i</mi></math>: unit vector</a>,
<a href="./1.6#E5" title="(1.6.5) ‣ Unit Vectors ‣ §1.6(i) Vectors ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m86.png" altimg-height="21px" altimg-valign="-6px" altimg-width="11px" alttext="\mathbf{j}" display="inline"><mi href="./1.6#E5" mathvariant="bold">j</mi></math>: unit vector</a> and
<a href="./1.6#E5" title="(1.6.5) ‣ Unit Vectors ‣ §1.6(i) Vectors ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="18px" altimg-valign="-2px" altimg-width="16px" alttext="\mathbf{k}" display="inline"><mi href="./1.6#E5" mathvariant="bold">k</mi></math>: unit vector</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E9" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.9</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_Math ltx_eqn_cell ltx_align_center"><math class="ltx_Math" alttext="\mathbf{a}\times\mathbf{b}=\begin{vmatrix}\mathbf{i}&\mathbf{j}&\mathbf{k}\\
a_{1}&a_{2}&a_{3}\\
b_{1}&b_{2}&b_{3}\end{vmatrix}\\
=(a_{2}b_{3}-a_{3}b_{2})\mathbf{i}+(a_{3}b_{1}-a_{1}b_{3})\mathbf{j}+(a_{1}b_{%
2}-a_{2}b_{1})\mathbf{k}\\
=\|\mathbf{a}\|\|\mathbf{b}\|(\sin\theta)\mathbf{n}," display="block"><mrow><mtable align="baseline 1" columnalign="left" columnspacing="0.1em" displaystyle="true"><mtr><mtd><mrow><mi mathvariant="bold">a</mi><mo>×</mo><mi mathvariant="bold">b</mi></mrow></mtd><mtd><mo>=</mo><mrow><mo href="./1.3#i">|</mo><mtable columnspacing="5pt" displaystyle="true" rowspacing="0pt"><mtr><mtd columnalign="center"><mi href="./1.6#E5" mathvariant="bold">i</mi></mtd><mtd columnalign="center"><mi href="./1.6#E5" mathvariant="bold">j</mi></mtd><mtd columnalign="center"><mi href="./1.6#E5" mathvariant="bold">k</mi></mtd></mtr><mtr><mtd columnalign="center"><msub><mi>a</mi><mn>1</mn></msub></mtd><mtd columnalign="center"><msub><mi>a</mi><mn>2</mn></msub></mtd><mtd columnalign="center"><msub><mi>a</mi><mn>3</mn></msub></mtd></mtr><mtr><mtd columnalign="center"><msub><mi>b</mi><mn>1</mn></msub></mtd><mtd columnalign="center"><msub><mi>b</mi><mn>2</mn></msub></mtd><mtd columnalign="center"><msub><mi>b</mi><mn>3</mn></msub></mtd></mtr></mtable><mo href="./1.3#i">|</mo></mrow><mo>=</mo><mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><msub><mi>a</mi><mn>2</mn></msub><mo></mo><msub><mi>b</mi><mn>3</mn></msub></mrow><mo>-</mo><mrow><msub><mi>a</mi><mn>3</mn></msub><mo></mo><msub><mi>b</mi><mn>2</mn></msub></mrow></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi href="./1.6#E5" mathvariant="bold">i</mi></mrow><mo>+</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><msub><mi>a</mi><mn>3</mn></msub><mo></mo><msub><mi>b</mi><mn>1</mn></msub></mrow><mo>-</mo><mrow><msub><mi>a</mi><mn>1</mn></msub><mo></mo><msub><mi>b</mi><mn>3</mn></msub></mrow></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi href="./1.6#E5" mathvariant="bold">j</mi></mrow><mo>+</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><msub><mi>a</mi><mn>1</mn></msub><mo></mo><msub><mi>b</mi><mn>2</mn></msub></mrow><mo>-</mo><mrow><msub><mi>a</mi><mn>2</mn></msub><mo></mo><msub><mi>b</mi><mn>1</mn></msub></mrow></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi href="./1.6#E5" mathvariant="bold">k</mi></mrow></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo><mrow><mrow><mo>∥</mo><mi mathvariant="bold">a</mi><mo>∥</mo></mrow><mo></mo><mrow><mo>∥</mo><mi mathvariant="bold">b</mi><mo>∥</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi href="./1.6#Px2.p1">θ</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi mathvariant="bold">n</mi></mrow><mo>,</mo></mtd></mtr></mtable></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E9.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.3#i" title="§1.3(i) Definitions and Elementary Properties ‣ §1.3 Determinants ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="32px" alttext="\det" display="inline"><mo href="./1.3#i">det</mo></math>: determinant</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./1.6#Px2.p1" title="Magnitude and Angle of Vector a ‣ §1.6(i) Vectors ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m104.png" altimg-height="18px" altimg-valign="-2px" altimg-width="14px" alttext="\theta" display="inline"><mi href="./1.6#Px2.p1">θ</mi></math>: angle between <math class="ltx_Math" altimg="m75.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\mathbf{a}" display="inline"><mi mathvariant="bold">a</mi></math> and <math class="ltx_Math" altimg="m77.png" altimg-height="18px" altimg-valign="-2px" altimg-width="17px" alttext="\mathbf{b}" display="inline"><mi mathvariant="bold">b</mi></math></a>,
<a href="./1.6#E5" title="(1.6.5) ‣ Unit Vectors ‣ §1.6(i) Vectors ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="18px" altimg-valign="-2px" altimg-width="11px" alttext="\mathbf{i}" display="inline"><mi href="./1.6#E5" mathvariant="bold">i</mi></math>: unit vector</a>,
<a href="./1.6#E5" title="(1.6.5) ‣ Unit Vectors ‣ §1.6(i) Vectors ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m86.png" altimg-height="21px" altimg-valign="-6px" altimg-width="11px" alttext="\mathbf{j}" display="inline"><mi href="./1.6#E5" mathvariant="bold">j</mi></math>: unit vector</a> and
<a href="./1.6#E5" title="(1.6.5) ‣ Unit Vectors ‣ §1.6(i) Vectors ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="18px" altimg-valign="-2px" altimg-width="16px" alttext="\mathbf{k}" display="inline"><mi href="./1.6#E5" mathvariant="bold">k</mi></math>: unit vector</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m88.png" altimg-height="13px" altimg-valign="-2px" altimg-width="17px" alttext="\mathbf{n}" display="inline"><mi mathvariant="bold">n</mi></math> is the unit vector normal to <math class="ltx_Math" altimg="m75.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\mathbf{a}" display="inline"><mi mathvariant="bold">a</mi></math> and <math class="ltx_Math" altimg="m77.png" altimg-height="18px" altimg-valign="-2px" altimg-width="17px" alttext="\mathbf{b}" display="inline"><mi mathvariant="bold">b</mi></math>
whose direction is determined by the right-hand rule; see
Figure </dd>
</dl>
</div>
</div>
</figure>
<div id="Px4.p2" class="ltx_para">
<p class="ltx_p">Area of parallelogram with vectors <math class="ltx_Math" altimg="m75.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\mathbf{a}" display="inline"><mi mathvariant="bold">a</mi></math> and <math class="ltx_Math" altimg="m77.png" altimg-height="18px" altimg-valign="-2px" altimg-width="17px" alttext="\mathbf{b}" display="inline"><mi mathvariant="bold">b</mi></math> as sides
<math class="ltx_Math" altimg="m43.png" altimg-height="23px" altimg-valign="-7px" altimg-width="93px" alttext="=\|\mathbf{a}\times\mathbf{b}\|" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mo>∥</mo><mrow><mi mathvariant="bold">a</mi><mo>×</mo><mi mathvariant="bold">b</mi></mrow><mo>∥</mo></mrow></mrow></math>.
</p>
</div>
<div id="Px4.p3" class="ltx_para">
<p class="ltx_p">Volume of a parallelepiped with vectors <math class="ltx_Math" altimg="m75.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\mathbf{a}" display="inline"><mi mathvariant="bold">a</mi></math>, <math class="ltx_Math" altimg="m77.png" altimg-height="18px" altimg-valign="-2px" altimg-width="17px" alttext="\mathbf{b}" display="inline"><mi mathvariant="bold">b</mi></math>, and
<math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="\mathbf{c}" display="inline"><mi mathvariant="bold">c</mi></math> as edges
<math class="ltx_Math" altimg="m42.png" altimg-height="23px" altimg-valign="-7px" altimg-width="125px" alttext="=\left|\mathbf{a}\cdot(\mathbf{b}\times\mathbf{c})\right|" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mo>|</mo><mrow><mi mathvariant="bold">a</mi><mo>⋅</mo><mrow><mo stretchy="false">(</mo><mrow><mi mathvariant="bold">b</mi><mo>×</mo><mi mathvariant="bold">c</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>|</mo></mrow></mrow></math>.
</p>
<table id="EGx1" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E10">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.10</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m23.png" altimg-height="25px" altimg-valign="-7px" altimg-width="104px" alttext="\displaystyle\mathbf{a}\times(\mathbf{b}\times\mathbf{c})" display="inline"><mrow><mi mathvariant="bold">a</mi><mo>×</mo><mrow><mo stretchy="false">(</mo><mrow><mi mathvariant="bold">b</mi><mo>×</mo><mi mathvariant="bold">c</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m15.png" altimg-height="25px" altimg-valign="-7px" altimg-width="185px" alttext="\displaystyle=\mathbf{b}(\mathbf{a}\cdot\mathbf{c})-\mathbf{c}(\mathbf{a}\cdot%
\mathbf{b})," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mi mathvariant="bold">b</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi mathvariant="bold">a</mi><mo>⋅</mo><mi mathvariant="bold">c</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>-</mo><mrow><mi mathvariant="bold">c</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi mathvariant="bold">a</mi><mo>⋅</mo><mi mathvariant="bold">b</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E10.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E11">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.11</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m1.png" altimg-height="25px" altimg-valign="-7px" altimg-width="104px" alttext="\displaystyle(\mathbf{a}\times\mathbf{b})\times\mathbf{c}" display="inline"><mrow><mrow><mo stretchy="false">(</mo><mrow><mi mathvariant="bold">a</mi><mo>×</mo><mi mathvariant="bold">b</mi></mrow><mo stretchy="false">)</mo></mrow><mo>×</mo><mi mathvariant="bold">c</mi></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m14.png" altimg-height="25px" altimg-valign="-7px" altimg-width="185px" alttext="\displaystyle=\mathbf{b}(\mathbf{a}\cdot\mathbf{c})-\mathbf{a}(\mathbf{b}\cdot%
\mathbf{c})." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mi mathvariant="bold">b</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi mathvariant="bold">a</mi><mo>⋅</mo><mi mathvariant="bold">c</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>-</mo><mrow><mi mathvariant="bold">a</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi mathvariant="bold">b</mi><mo>⋅</mo><mi mathvariant="bold">c</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E11.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px6.p1" class="ltx_para">
<table id="E12" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.12</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E12.png" altimg-height="70px" altimg-valign="-30px" altimg-width="207px" alttext="a_{j}b_{j}=\sum_{j=1}^{3}a_{j}b_{j}=\mathbf{a}\cdot\mathbf{b}." display="block"><mrow><mrow><mrow><msub><mi>a</mi><mi href="./1.1#p2.t1.r4">j</mi></msub><mo></mo><msub><mi>b</mi><mi href="./1.1#p2.t1.r4">j</mi></msub></mrow><mo>=</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r4">j</mi><mo>=</mo><mn>1</mn></mrow><mn>3</mn></munderover><mrow><msub><mi>a</mi><mi href="./1.1#p2.t1.r4">j</mi></msub><mo></mo><msub><mi>b</mi><mi href="./1.1#p2.t1.r4">j</mi></msub></mrow></mrow><mo>=</mo><mrow><mi mathvariant="bold">a</mi><mo>⋅</mo><mi mathvariant="bold">b</mi></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E12.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./1.1#p2.t1.r4" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m117.png" altimg-height="20px" altimg-valign="-6px" altimg-width="14px" alttext="j" display="inline"><mi href="./1.1#p2.t1.r4">j</mi></math>: integer</a></dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="Px6.p2" class="ltx_para">
<p class="ltx_p">Next,</p>
<table id="E13" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex12" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="3" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">1.6.13</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m25.png" altimg-height="17px" altimg-valign="-5px" altimg-width="25px" alttext="\displaystyle\mathbf{e}_{1}" display="inline"><msub><mi href="./1.6#E13" mathvariant="bold">e</mi><mn>1</mn></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m5.png" altimg-height="25px" altimg-valign="-7px" altimg-width="96px" alttext="\displaystyle=(1,0,0)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex13" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m26.png" altimg-height="17px" altimg-valign="-5px" altimg-width="25px" alttext="\displaystyle\mathbf{e}_{2}" display="inline"><msub><mi href="./1.6#E13" mathvariant="bold">e</mi><mn>2</mn></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m4.png" altimg-height="25px" altimg-valign="-7px" altimg-width="96px" alttext="\displaystyle=(0,1,0)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex14" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m27.png" altimg-height="17px" altimg-valign="-5px" altimg-width="25px" alttext="\displaystyle\mathbf{e}_{3}" display="inline"><msub><mi href="./1.6#E13" mathvariant="bold">e</mi><mn>3</mn></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m3.png" altimg-height="25px" altimg-valign="-7px" altimg-width="96px" alttext="\displaystyle=(0,0,1);" display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mrow><mo>;</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E13.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text"><math class="ltx_Math" altimg="m84.png" altimg-height="18px" altimg-valign="-8px" altimg-width="23px" alttext="\mathbf{e}_{j}" display="inline"><msub><mi href="./1.6#E13" mathvariant="bold">e</mi><mi href="./1.1#p2.t1.r4">j</mi></msub></math>: unit vectors (locally)</span></dd>
<dt>Symbols:</dt>
<dd><a href="./1.1#p2.t1.r4" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m117.png" altimg-height="20px" altimg-valign="-6px" altimg-width="14px" alttext="j" display="inline"><mi href="./1.1#p2.t1.r4">j</mi></math>: integer</a></dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">compare (). Thus <math class="ltx_Math" altimg="m108.png" altimg-height="18px" altimg-valign="-8px" altimg-width="80px" alttext="a_{j}\mathbf{e}_{j}=\mathbf{a}" display="inline"><mrow><mrow><msub><mi>a</mi><mi href="./1.1#p2.t1.r4">j</mi></msub><mo></mo><msub><mi href="./1.6#E13" mathvariant="bold">e</mi><mi href="./1.1#p2.t1.r4">j</mi></msub></mrow><mo>=</mo><mi mathvariant="bold">a</mi></mrow></math>.</p>
</div>
</section>
<section id="Px7" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Levi-Civita Symbol</h3>
<div id="Px7.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px7.p1" class="ltx_para">
<table id="E14" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.14</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E14.png" altimg-height="92px" altimg-valign="-40px" altimg-width="461px" alttext="\epsilon_{jk\ell}=\begin{cases}+1,&\text{if }j,k,\ell\text{ is even %
permutation of }1,2,3,\\
-1,&\text{if }j,k,\ell\text{ is odd permutation of }1,2,3,\\
\phantom{-}0,&\text{otherwise}.\end{cases}" display="block"><mrow><msub><mi href="./1.6#E14">ϵ</mi><mrow><mi href="./1.1#p2.t1.r4">j</mi><mo href="./1.6#E14"></mo><mi href="./1.1#p2.t1.r4">k</mi><mo href="./1.6#E14"></mo><mi mathvariant="normal">ℓ</mi></mrow></msub><mo>=</mo><mrow><mo>{</mo><mtable columnspacing="5pt" displaystyle="true" rowspacing="0pt"><mtr><mtd columnalign="left"><mrow><mrow><mo>+</mo><mn>1</mn></mrow><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mrow><mrow><mtext>if </mtext><mo></mo><mi href="./1.1#p2.t1.r4">j</mi></mrow><mo>,</mo><mi href="./1.1#p2.t1.r4">k</mi><mo>,</mo><mrow><mi mathvariant="normal">ℓ</mi><mo></mo><mtext> is even permutation of </mtext><mo></mo><mn>1</mn></mrow><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn></mrow><mo>,</mo></mrow></mtd></mtr><mtr><mtd columnalign="left"><mrow><mrow><mo>-</mo><mn>1</mn></mrow><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mrow><mrow><mtext>if </mtext><mo></mo><mi href="./1.1#p2.t1.r4">j</mi></mrow><mo>,</mo><mi href="./1.1#p2.t1.r4">k</mi><mo>,</mo><mrow><mi mathvariant="normal">ℓ</mi><mo></mo><mtext> is odd permutation of </mtext><mo></mo><mn>1</mn></mrow><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn></mrow><mo>,</mo></mrow></mtd></mtr><mtr><mtd columnalign="left"><mrow><mpadded lspace="7.5pt" width="+7.5pt"><mn>0</mn></mpadded><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mtext>otherwise</mtext><mo>.</mo></mrow></mtd></mtr></mtable></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E14.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m64.png" altimg-height="18px" altimg-valign="-8px" altimg-width="36px" alttext="\epsilon_{\NVar{j}\NVar{k}\NVar{\ell}}" display="inline"><msub><mi href="./1.6#E14">ϵ</mi><mrow><mi class="ltx_nvar" href="./1.1#p2.t1.r4">j</mi><mo href="./1.6#E14"></mo><mi class="ltx_nvar" href="./1.1#p2.t1.r4">k</mi><mo href="./1.6#E14"></mo><mi class="ltx_nvar" mathvariant="normal">ℓ</mi></mrow></msub></math>: Levi-Civita symbol</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./1.1#p2.t1.r4" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m117.png" altimg-height="20px" altimg-valign="-6px" altimg-width="14px" alttext="j" display="inline"><mi href="./1.1#p2.t1.r4">j</mi></math>: integer</a> and
<a href="./1.1#p2.t1.r4" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m118.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./1.1#p2.t1.r4">k</mi></math>: integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px8" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Examples</h3>
<div id="Px8.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px8.p1" class="ltx_para">
<table id="E15" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex15" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="3" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">1.6.15</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m19.png" altimg-height="17px" altimg-valign="-5px" altimg-width="39px" alttext="\displaystyle\epsilon_{123}" display="inline"><msub><mi href="./1.6#E14">ϵ</mi><mrow><mn>1</mn><mo href="./1.6#E14"></mo><mn>2</mn><mo href="./1.6#E14"></mo><mn>3</mn></mrow></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m12.png" altimg-height="22px" altimg-valign="-6px" altimg-width="102px" alttext="\displaystyle=\epsilon_{312}=1," display="inline"><mrow><mrow><mi></mi><mo>=</mo><msub><mi href="./1.6#E14">ϵ</mi><mrow><mn>3</mn><mo href="./1.6#E14"></mo><mn>1</mn><mo href="./1.6#E14"></mo><mn>2</mn></mrow></msub><mo>=</mo><mn>1</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex16" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m20.png" altimg-height="17px" altimg-valign="-5px" altimg-width="39px" alttext="\displaystyle\epsilon_{213}" display="inline"><msub><mi href="./1.6#E14">ϵ</mi><mrow><mn>2</mn><mo href="./1.6#E14"></mo><mn>1</mn><mo href="./1.6#E14"></mo><mn>3</mn></mrow></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m13.png" altimg-height="22px" altimg-valign="-6px" altimg-width="118px" alttext="\displaystyle=\epsilon_{321}=-1," display="inline"><mrow><mrow><mi></mi><mo>=</mo><msub><mi href="./1.6#E14">ϵ</mi><mrow><mn>3</mn><mo href="./1.6#E14"></mo><mn>2</mn><mo href="./1.6#E14"></mo><mn>1</mn></mrow></msub><mo>=</mo><mrow><mo>-</mo><mn>1</mn></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex17" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m21.png" altimg-height="17px" altimg-valign="-5px" altimg-width="39px" alttext="\displaystyle\epsilon_{221}" display="inline"><msub><mi href="./1.6#E14">ϵ</mi><mrow><mn>2</mn><mo href="./1.6#E14"></mo><mn>2</mn><mo href="./1.6#E14"></mo><mn>1</mn></mrow></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m11.png" altimg-height="18px" altimg-valign="-2px" altimg-width="43px" alttext="\displaystyle=0." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mn>0</mn></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E15.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./1.6#E14" title="(1.6.14) ‣ Levi-Civita Symbol ‣ §1.6(ii) Vectors: Alternative Notations ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m64.png" altimg-height="18px" altimg-valign="-8px" altimg-width="36px" alttext="\epsilon_{\NVar{j}\NVar{k}\NVar{\ell}}" display="inline"><msub><mi href="./1.6#E14">ϵ</mi><mrow><mi class="ltx_nvar" href="./1.1#p2.t1.r4">j</mi><mo href="./1.6#E14"></mo><mi class="ltx_nvar" href="./1.1#p2.t1.r4">k</mi><mo href="./1.6#E14"></mo><mi class="ltx_nvar" mathvariant="normal">ℓ</mi></mrow></msub></math>: Levi-Civita symbol</a></dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E16" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.16</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E16.png" altimg-height="25px" altimg-valign="-8px" altimg-width="271px" alttext="\epsilon_{jk\ell}\epsilon_{\ell mn}=\delta_{j,m}\delta_{k,n}-\delta_{j,n}%
\delta_{k,m}," display="block"><mrow><mrow><mrow><msub><mi href="./1.6#E14">ϵ</mi><mrow><mi href="./1.1#p2.t1.r4">j</mi><mo href="./1.6#E14"></mo><mi href="./1.1#p2.t1.r4">k</mi><mo href="./1.6#E14"></mo><mi mathvariant="normal">ℓ</mi></mrow></msub><mo></mo><msub><mi href="./1.6#E14">ϵ</mi><mrow><mi mathvariant="normal">ℓ</mi><mo href="./1.6#E14"></mo><mi href="./1.1#p2.t1.r5">m</mi><mo href="./1.6#E14"></mo><mi href="./1.1#p2.t1.r5">n</mi></mrow></msub></mrow><mo>=</mo><mrow><mrow><msub><mi href="./front/introduction#Sx4.p1.t1.r4">δ</mi><mrow><mi href="./1.1#p2.t1.r4">j</mi><mo href="./front/introduction#Sx4.p1.t1.r4">,</mo><mi href="./1.1#p2.t1.r5">m</mi></mrow></msub><mo></mo><msub><mi href="./front/introduction#Sx4.p1.t1.r4">δ</mi><mrow><mi href="./1.1#p2.t1.r4">k</mi><mo href="./front/introduction#Sx4.p1.t1.r4">,</mo><mi href="./1.1#p2.t1.r5">n</mi></mrow></msub></mrow><mo>-</mo><mrow><msub><mi href="./front/introduction#Sx4.p1.t1.r4">δ</mi><mrow><mi href="./1.1#p2.t1.r4">j</mi><mo href="./front/introduction#Sx4.p1.t1.r4">,</mo><mi href="./1.1#p2.t1.r5">n</mi></mrow></msub><mo></mo><msub><mi href="./front/introduction#Sx4.p1.t1.r4">δ</mi><mrow><mi href="./1.1#p2.t1.r4">k</mi><mo href="./front/introduction#Sx4.p1.t1.r4">,</mo><mi href="./1.1#p2.t1.r5">m</mi></mrow></msub></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E16.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./front/introduction#Sx4.p1.t1.r4" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m61.png" altimg-height="23px" altimg-valign="-8px" altimg-width="35px" alttext="\delta_{\NVar{j},\NVar{k}}" display="inline"><msub><mi href="./front/introduction#Sx4.p1.t1.r4">δ</mi><mrow><mi class="ltx_nvar">j</mi><mo href="./front/introduction#Sx4.p1.t1.r4">,</mo><mi class="ltx_nvar">k</mi></mrow></msub></math>: Kronecker delta</a>,
<a href="./1.6#E14" title="(1.6.14) ‣ Levi-Civita Symbol ‣ §1.6(ii) Vectors: Alternative Notations ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m64.png" altimg-height="18px" altimg-valign="-8px" altimg-width="36px" alttext="\epsilon_{\NVar{j}\NVar{k}\NVar{\ell}}" display="inline"><msub><mi href="./1.6#E14">ϵ</mi><mrow><mi class="ltx_nvar" href="./1.1#p2.t1.r4">j</mi><mo href="./1.6#E14"></mo><mi class="ltx_nvar" href="./1.1#p2.t1.r4">k</mi><mo href="./1.6#E14"></mo><mi class="ltx_nvar" mathvariant="normal">ℓ</mi></mrow></msub></math>: Levi-Civita symbol</a>,
<a href="./1.1#p2.t1.r4" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m117.png" altimg-height="20px" altimg-valign="-6px" altimg-width="14px" alttext="j" display="inline"><mi href="./1.1#p2.t1.r4">j</mi></math>: integer</a>,
<a href="./1.1#p2.t1.r4" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m118.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./1.1#p2.t1.r4">k</mi></math>: integer</a>,
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m119.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./1.1#p2.t1.r5">m</mi></math>: nonnegative integer</a> and
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m120.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m62.png" altimg-height="23px" altimg-valign="-8px" altimg-width="34px" alttext="\delta_{j,k}" display="inline"><msub><mi href="./front/introduction#Sx4.p1.t1.r4">δ</mi><mrow><mi href="./1.1#p2.t1.r4">j</mi><mo href="./front/introduction#Sx4.p1.t1.r4">,</mo><mi href="./1.1#p2.t1.r4">k</mi></mrow></msub></math> is the Kronecker delta.</p>
<table id="E17" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.17</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E17.png" altimg-height="23px" altimg-valign="-8px" altimg-width="152px" alttext="\mathbf{e}_{j}\times\mathbf{e}_{k}=\epsilon_{jk\ell}\mathbf{e}_{\ell};" display="block"><mrow><mrow><mrow><msub><mi href="./1.6#E13" mathvariant="bold">e</mi><mi href="./1.1#p2.t1.r4">j</mi></msub><mo>×</mo><msub><mi href="./1.6#E13" mathvariant="bold">e</mi><mi href="./1.1#p2.t1.r4">k</mi></msub></mrow><mo>=</mo><mrow><msub><mi href="./1.6#E14">ϵ</mi><mrow><mi href="./1.1#p2.t1.r4">j</mi><mo href="./1.6#E14"></mo><mi href="./1.1#p2.t1.r4">k</mi><mo href="./1.6#E14"></mo><mi mathvariant="normal">ℓ</mi></mrow></msub><mo></mo><msub><mi href="./1.6#E13" mathvariant="bold">e</mi><mi mathvariant="normal">ℓ</mi></msub></mrow></mrow><mo>;</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E17.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.6#E14" title="(1.6.14) ‣ Levi-Civita Symbol ‣ §1.6(ii) Vectors: Alternative Notations ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m64.png" altimg-height="18px" altimg-valign="-8px" altimg-width="36px" alttext="\epsilon_{\NVar{j}\NVar{k}\NVar{\ell}}" display="inline"><msub><mi href="./1.6#E14">ϵ</mi><mrow><mi class="ltx_nvar" href="./1.1#p2.t1.r4">j</mi><mo href="./1.6#E14"></mo><mi class="ltx_nvar" href="./1.1#p2.t1.r4">k</mi><mo href="./1.6#E14"></mo><mi class="ltx_nvar" mathvariant="normal">ℓ</mi></mrow></msub></math>: Levi-Civita symbol</a>,
<a href="./1.1#p2.t1.r4" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m117.png" altimg-height="20px" altimg-valign="-6px" altimg-width="14px" alttext="j" display="inline"><mi href="./1.1#p2.t1.r4">j</mi></math>: integer</a>,
<a href="./1.1#p2.t1.r4" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m118.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./1.1#p2.t1.r4">k</mi></math>: integer</a> and
<a href="./1.6#E13" title="(1.6.13) ‣ Example ‣ §1.6(ii) Vectors: Alternative Notations ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m84.png" altimg-height="18px" altimg-valign="-8px" altimg-width="23px" alttext="\mathbf{e}_{j}" display="inline"><msub><mi href="./1.6#E13" mathvariant="bold">e</mi><mi href="./1.1#p2.t1.r4">j</mi></msub></math>: unit vectors</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">compare ().
</p>
<table id="E18" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.18</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E18.png" altimg-height="25px" altimg-valign="-8px" altimg-width="226px" alttext="a_{j}\mathbf{e}_{j}\times b_{k}\mathbf{e}_{k}=\epsilon_{jk\ell}a_{j}b_{k}%
\mathbf{e}_{\ell};" display="block"><mrow><mrow><mrow><mrow><mrow><msub><mi>a</mi><mi href="./1.1#p2.t1.r4">j</mi></msub><mo></mo><msub><mi href="./1.6#E13" mathvariant="bold">e</mi><mi href="./1.1#p2.t1.r4">j</mi></msub></mrow><mo>×</mo><msub><mi>b</mi><mi href="./1.1#p2.t1.r4">k</mi></msub></mrow><mo></mo><msub><mi href="./1.6#E13" mathvariant="bold">e</mi><mi href="./1.1#p2.t1.r4">k</mi></msub></mrow><mo>=</mo><mrow><msub><mi href="./1.6#E14">ϵ</mi><mrow><mi href="./1.1#p2.t1.r4">j</mi><mo href="./1.6#E14"></mo><mi href="./1.1#p2.t1.r4">k</mi><mo href="./1.6#E14"></mo><mi mathvariant="normal">ℓ</mi></mrow></msub><mo></mo><msub><mi>a</mi><mi href="./1.1#p2.t1.r4">j</mi></msub><mo></mo><msub><mi>b</mi><mi href="./1.1#p2.t1.r4">k</mi></msub><mo></mo><msub><mi href="./1.6#E13" mathvariant="bold">e</mi><mi mathvariant="normal">ℓ</mi></msub></mrow></mrow><mo>;</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E18.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.6#E14" title="(1.6.14) ‣ Levi-Civita Symbol ‣ §1.6(ii) Vectors: Alternative Notations ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m64.png" altimg-height="18px" altimg-valign="-8px" altimg-width="36px" alttext="\epsilon_{\NVar{j}\NVar{k}\NVar{\ell}}" display="inline"><msub><mi href="./1.6#E14">ϵ</mi><mrow><mi class="ltx_nvar" href="./1.1#p2.t1.r4">j</mi><mo href="./1.6#E14"></mo><mi class="ltx_nvar" href="./1.1#p2.t1.r4">k</mi><mo href="./1.6#E14"></mo><mi class="ltx_nvar" mathvariant="normal">ℓ</mi></mrow></msub></math>: Levi-Civita symbol</a>,
<a href="./1.1#p2.t1.r4" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m117.png" altimg-height="20px" altimg-valign="-6px" altimg-width="14px" alttext="j" display="inline"><mi href="./1.1#p2.t1.r4">j</mi></math>: integer</a>,
<a href="./1.1#p2.t1.r4" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m118.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./1.1#p2.t1.r4">k</mi></math>: integer</a> and
<a href="./1.6#E13" title="(1.6.13) ‣ Example ‣ §1.6(ii) Vectors: Alternative Notations ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m84.png" altimg-height="18px" altimg-valign="-8px" altimg-width="23px" alttext="\mathbf{e}_{j}" display="inline"><msub><mi href="./1.6#E13" mathvariant="bold">e</mi><mi href="./1.1#p2.t1.r4">j</mi></msub></math>: unit vectors</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">compare ().</p>
</div>
<div id="Px8.p2" class="ltx_para">
<p class="ltx_p">Lastly, the volume of a parallelepiped with vectors <math class="ltx_Math" altimg="m75.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\mathbf{a}" display="inline"><mi mathvariant="bold">a</mi></math>, <math class="ltx_Math" altimg="m77.png" altimg-height="18px" altimg-valign="-2px" altimg-width="17px" alttext="\mathbf{b}" display="inline"><mi mathvariant="bold">b</mi></math>,
and <math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="\mathbf{c}" display="inline"><mi mathvariant="bold">c</mi></math> as edges is <math class="ltx_Math" altimg="m136.png" altimg-height="24px" altimg-valign="-8px" altimg-width="101px" alttext="|\epsilon_{jk\ell}a_{j}b_{k}c_{\ell}|" display="inline"><mrow><mo stretchy="false">|</mo><mrow><msub><mi href="./1.6#E14">ϵ</mi><mrow><mi href="./1.1#p2.t1.r4">j</mi><mo href="./1.6#E14"></mo><mi href="./1.1#p2.t1.r4">k</mi><mo href="./1.6#E14"></mo><mi mathvariant="normal">ℓ</mi></mrow></msub><mo></mo><msub><mi>a</mi><mi href="./1.1#p2.t1.r4">j</mi></msub><mo></mo><msub><mi>b</mi><mi href="./1.1#p2.t1.r4">k</mi></msub><mo></mo><msub><mi>c</mi><mi mathvariant="normal">ℓ</mi></msub></mrow><mo stretchy="false">|</mo></mrow></math>.
</p>
</div>
</section>
</section>
<section id="iii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§1.6(iii) </span>Vector-Valued Functions</h2>
<div id="SS3.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px9.p1" class="ltx_para">
<table id="E19" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.19</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E19.png" altimg-height="50px" altimg-valign="-20px" altimg-width="210px" alttext="\nabla=\mathbf{i}\frac{\partial}{\partial x}+\mathbf{j}\frac{\partial}{%
\partial y}+\mathbf{k}\frac{\partial}{\partial z}." display="block"><mrow><mrow><mo>∇</mo><mo>=</mo><mrow><mrow><mi href="./1.6#E5" mathvariant="bold">i</mi><mo></mo><mfrac><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow></mfrac></mrow><mo>+</mo><mrow><mi href="./1.6#E5" mathvariant="bold">j</mi><mo></mo><mfrac><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow></mfrac></mrow><mo>+</mo><mrow><mi href="./1.6#E5" mathvariant="bold">k</mi><mo></mo><mfrac><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.1#p2.t1.r2">z</mi></mrow></mfrac></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E19.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m110.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m99.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m135.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>,
<a href="./1.6#E5" title="(1.6.5) ‣ Unit Vectors ‣ §1.6(i) Vectors ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="18px" altimg-valign="-2px" altimg-width="11px" alttext="\mathbf{i}" display="inline"><mi href="./1.6#E5" mathvariant="bold">i</mi></math>: unit vector</a>,
<a href="./1.6#E5" title="(1.6.5) ‣ Unit Vectors ‣ §1.6(i) Vectors ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m86.png" altimg-height="21px" altimg-valign="-6px" altimg-width="11px" alttext="\mathbf{j}" display="inline"><mi href="./1.6#E5" mathvariant="bold">j</mi></math>: unit vector</a> and
<a href="./1.6#E5" title="(1.6.5) ‣ Unit Vectors ‣ §1.6(i) Vectors ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="18px" altimg-valign="-2px" altimg-width="16px" alttext="\mathbf{k}" display="inline"><mi href="./1.6#E5" mathvariant="bold">k</mi></math>: unit vector</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="Px9.p2" class="ltx_para">
<p class="ltx_p">The <em class="ltx_emph ltx_font_italic">gradient</em>
of a differentiable scalar function <math class="ltx_Math" altimg="m109.png" altimg-height="23px" altimg-valign="-7px" altimg-width="81px" alttext="f(x,y,z)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> is</p>
<table id="E20" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.20</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E20.png" altimg-height="50px" altimg-valign="-20px" altimg-width="306px" alttext="\operatorname{grad}f=\nabla f=\frac{\partial f}{\partial x}\mathbf{i}+\frac{%
\partial f}{\partial y}\mathbf{j}+\frac{\partial f}{\partial z}\mathbf{k}." display="block"><mrow><mrow><mrow><mo href="./1.6#E20">grad</mo><mi>f</mi></mrow><mo>=</mo><mrow><mo>∇</mo><mo></mo><mi>f</mi></mrow><mo>=</mo><mrow><mrow><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow></mfrac><mo></mo><mi href="./1.6#E5" mathvariant="bold">i</mi></mrow><mo>+</mo><mrow><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow></mfrac><mo></mo><mi href="./1.6#E5" mathvariant="bold">j</mi></mrow><mo>+</mo><mrow><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.1#p2.t1.r2">z</mi></mrow></mfrac><mo></mo><mi href="./1.6#E5" mathvariant="bold">k</mi></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E20.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m97.png" altimg-height="21px" altimg-valign="-6px" altimg-width="43px" alttext="\operatorname{grad}" display="inline"><mo href="./1.6#E20">grad</mo></math>: gradient of differentiable scalar function</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m110.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m99.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m135.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>,
<a href="./1.6#E5" title="(1.6.5) ‣ Unit Vectors ‣ §1.6(i) Vectors ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="18px" altimg-valign="-2px" altimg-width="11px" alttext="\mathbf{i}" display="inline"><mi href="./1.6#E5" mathvariant="bold">i</mi></math>: unit vector</a>,
<a href="./1.6#E5" title="(1.6.5) ‣ Unit Vectors ‣ §1.6(i) Vectors ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m86.png" altimg-height="21px" altimg-valign="-6px" altimg-width="11px" alttext="\mathbf{j}" display="inline"><mi href="./1.6#E5" mathvariant="bold">j</mi></math>: unit vector</a> and
<a href="./1.6#E5" title="(1.6.5) ‣ Unit Vectors ‣ §1.6(i) Vectors ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="18px" altimg-valign="-2px" altimg-width="16px" alttext="\mathbf{k}" display="inline"><mi href="./1.6#E5" mathvariant="bold">k</mi></math>: unit vector</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="Px9.p3" class="ltx_para">
<p class="ltx_p">The <em class="ltx_emph ltx_font_italic">divergence</em>
of a differentiable vector-valued function
<math class="ltx_Math" altimg="m70.png" altimg-height="21px" altimg-valign="-6px" altimg-width="185px" alttext="\mathbf{F}=F_{1}\mathbf{i}+F_{2}\mathbf{j}+F_{3}\mathbf{k}" display="inline"><mrow><mi mathvariant="bold">F</mi><mo>=</mo><mrow><mrow><msub><mi href="./1.6#Px9.p3">F</mi><mn>1</mn></msub><mo></mo><mi href="./1.6#E5" mathvariant="bold">i</mi></mrow><mo>+</mo><mrow><msub><mi href="./1.6#Px9.p3">F</mi><mn>2</mn></msub><mo></mo><mi href="./1.6#E5" mathvariant="bold">j</mi></mrow><mo>+</mo><mrow><msub><mi href="./1.6#Px9.p3">F</mi><mn>3</mn></msub><mo></mo><mi href="./1.6#E5" mathvariant="bold">k</mi></mrow></mrow></mrow></math> is
</p>
<table id="E21" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.21</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E21.png" altimg-height="50px" altimg-valign="-20px" altimg-width="319px" alttext="\operatorname{div}\mathbf{F}=\nabla\cdot\mathbf{F}=\frac{\partial F_{1}}{%
\partial x}+\frac{\partial F_{2}}{\partial y}+\frac{\partial F_{3}}{\partial z}." display="block"><mrow><mrow><mrow><mo href="./1.6#E21">div</mo><mi mathvariant="bold">F</mi></mrow><mo>=</mo><mrow><mo>∇</mo><mo>⋅</mo><mi mathvariant="bold">F</mi></mrow><mo>=</mo><mrow><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><msub><mi href="./1.6#Px9.p3">F</mi><mn>1</mn></msub></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow></mfrac><mo>+</mo><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><msub><mi href="./1.6#Px9.p3">F</mi><mn>2</mn></msub></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow></mfrac><mo>+</mo><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><msub><mi href="./1.6#Px9.p3">F</mi><mn>3</mn></msub></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.1#p2.t1.r2">z</mi></mrow></mfrac></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E21.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m96.png" altimg-height="18px" altimg-valign="-2px" altimg-width="32px" alttext="\operatorname{div}" display="inline"><mo href="./1.6#E21">div</mo></math>: divergence of vector-valued function</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m110.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m99.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m135.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a> and
<a href="./1.6#Px9.p3" title="Del Operator ‣ §1.6(iii) Vector-Valued Functions ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="23px" altimg-valign="-8px" altimg-width="26px" alttext="F_{j}" display="inline"><msub><mi href="./1.6#Px9.p3">F</mi><mi href="./1.1#p2.t1.r4">j</mi></msub></math>: vector function components</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="Px9.p4" class="ltx_para">
<p class="ltx_p">The <em class="ltx_emph ltx_font_italic">curl</em> of
<math class="ltx_Math" altimg="m71.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\mathbf{F}" display="inline"><mi mathvariant="bold">F</mi></math> is
</p>
<table id="E22" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.22</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E22.png" altimg-height="152px" altimg-valign="-43px" altimg-width="564px" alttext="\operatorname{curl}\mathbf{F}=\nabla\times\mathbf{F}=\begin{vmatrix}\mathbf{i}%
&\mathbf{j}&\mathbf{k}\\
\displaystyle{\frac{\partial}{\partial x}}&\displaystyle{\frac{\partial}{%
\partial y}}&\displaystyle{\frac{\partial}{\partial z}}\\
F_{1}&F_{2}&F_{3}\end{vmatrix}\\
=\left(\frac{\partial F_{3}}{\partial y}-\frac{\partial F_{2}}{\partial z}%
\right)\mathbf{i}+\left(\frac{\partial F_{1}}{\partial z}-\frac{\partial F_{3}%
}{\partial x}\right)\mathbf{j}+\left(\frac{\partial F_{2}}{\partial x}-\frac{%
\partial F_{1}}{\partial y}\right)\mathbf{k}." display="block"><mrow><mrow><mrow><mo href="./1.6#E22">curl</mo><mi mathvariant="bold">F</mi></mrow><mo>=</mo><mrow><mo>∇</mo><mo>×</mo><mi mathvariant="bold">F</mi></mrow><mo>=</mo><mrow><mo href="./1.3#i">|</mo><mtable columnspacing="5pt" displaystyle="true" rowspacing="0pt"><mtr><mtd columnalign="center"><mi href="./1.6#E5" mathvariant="bold">i</mi></mtd><mtd columnalign="center"><mi href="./1.6#E5" mathvariant="bold">j</mi></mtd><mtd columnalign="center"><mi href="./1.6#E5" mathvariant="bold">k</mi></mtd></mtr><mtr><mtd columnalign="center"><mfrac><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow></mfrac></mtd><mtd columnalign="center"><mfrac><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow></mfrac></mtd><mtd columnalign="center"><mfrac><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.1#p2.t1.r2">z</mi></mrow></mfrac></mtd></mtr><mtr><mtd columnalign="center"><msub><mi href="./1.6#Px9.p3">F</mi><mn>1</mn></msub></mtd><mtd columnalign="center"><msub><mi href="./1.6#Px9.p3">F</mi><mn>2</mn></msub></mtd><mtd columnalign="center"><msub><mi href="./1.6#Px9.p3">F</mi><mn>3</mn></msub></mtd></mtr></mtable><mo href="./1.3#i">|</mo></mrow><mo>=</mo><mrow><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><msub><mi href="./1.6#Px9.p3">F</mi><mn>3</mn></msub></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow></mfrac><mo>-</mo><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><msub><mi href="./1.6#Px9.p3">F</mi><mn>2</mn></msub></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.1#p2.t1.r2">z</mi></mrow></mfrac></mrow><mo>)</mo></mrow><mo></mo><mi href="./1.6#E5" mathvariant="bold">i</mi></mrow><mo>+</mo><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><msub><mi href="./1.6#Px9.p3">F</mi><mn>1</mn></msub></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.1#p2.t1.r2">z</mi></mrow></mfrac><mo>-</mo><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><msub><mi href="./1.6#Px9.p3">F</mi><mn>3</mn></msub></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow></mfrac></mrow><mo>)</mo></mrow><mo></mo><mi href="./1.6#E5" mathvariant="bold">j</mi></mrow><mo>+</mo><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><msub><mi href="./1.6#Px9.p3">F</mi><mn>2</mn></msub></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow></mfrac><mo>-</mo><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><msub><mi href="./1.6#Px9.p3">F</mi><mn>1</mn></msub></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow></mfrac></mrow><mo>)</mo></mrow><mo></mo><mi href="./1.6#E5" mathvariant="bold">k</mi></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E22.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m95.png" altimg-height="18px" altimg-valign="-2px" altimg-width="38px" alttext="\operatorname{curl}" display="inline"><mo href="./1.6#E22">curl</mo></math>: of vector-valued function</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./1.3#i" title="§1.3(i) Definitions and Elementary Properties ‣ §1.3 Determinants ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="32px" alttext="\det" display="inline"><mo href="./1.3#i">det</mo></math>: determinant</a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m110.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m99.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m135.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>,
<a href="./1.6#E5" title="(1.6.5) ‣ Unit Vectors ‣ §1.6(i) Vectors ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="18px" altimg-valign="-2px" altimg-width="11px" alttext="\mathbf{i}" display="inline"><mi href="./1.6#E5" mathvariant="bold">i</mi></math>: unit vector</a>,
<a href="./1.6#E5" title="(1.6.5) ‣ Unit Vectors ‣ §1.6(i) Vectors ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m86.png" altimg-height="21px" altimg-valign="-6px" altimg-width="11px" alttext="\mathbf{j}" display="inline"><mi href="./1.6#E5" mathvariant="bold">j</mi></math>: unit vector</a>,
<a href="./1.6#E5" title="(1.6.5) ‣ Unit Vectors ‣ §1.6(i) Vectors ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="18px" altimg-valign="-2px" altimg-width="16px" alttext="\mathbf{k}" display="inline"><mi href="./1.6#E5" mathvariant="bold">k</mi></math>: unit vector</a> and
<a href="./1.6#Px9.p3" title="Del Operator ‣ §1.6(iii) Vector-Valued Functions ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="23px" altimg-valign="-8px" altimg-width="26px" alttext="F_{j}" display="inline"><msub><mi href="./1.6#Px9.p3">F</mi><mi href="./1.1#p2.t1.r4">j</mi></msub></math>: vector function components</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E23" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.23</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E23.png" altimg-height="25px" altimg-valign="-7px" altimg-width="193px" alttext="\nabla(fg)=f\nabla g+g\nabla f," display="block"><mrow><mrow><mrow><mo>∇</mo><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>f</mi><mo></mo><mi>g</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mi>f</mi><mo></mo><mrow><mo>∇</mo><mo></mo><mi>g</mi></mrow></mrow><mo>+</mo><mrow><mi>g</mi><mo></mo><mrow><mo>∇</mo><mo></mo><mi>f</mi></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E23.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E24" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.24</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E24.png" altimg-height="28px" altimg-valign="-7px" altimg-width="249px" alttext="\nabla(f/g)=(g\nabla f-f\nabla g)/g^{2}," display="block"><mrow><mrow><mrow><mo>∇</mo><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>f</mi><mo>/</mo><mi>g</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mi>g</mi><mo></mo><mrow><mo>∇</mo><mo></mo><mi>f</mi></mrow></mrow><mo>-</mo><mrow><mi>f</mi><mo></mo><mrow><mo>∇</mo><mo></mo><mi>g</mi></mrow></mrow></mrow><mo stretchy="false">)</mo></mrow><mo>/</mo><msup><mi>g</mi><mn>2</mn></msup></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E24.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E25" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.25</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E25.png" altimg-height="25px" altimg-valign="-7px" altimg-width="264px" alttext="\nabla\cdot(f\mathbf{F})=f(\nabla\cdot\mathbf{F})+\mathbf{F}\cdot\nabla f," display="block"><mrow><mrow><mrow><mo>∇</mo><mo>⋅</mo><mrow><mo stretchy="false">(</mo><mrow><mi>f</mi><mo></mo><mi mathvariant="bold">F</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mo>∇</mo><mo>⋅</mo><mi mathvariant="bold">F</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>+</mo><mrow><mi mathvariant="bold">F</mi><mo>⋅</mo><mrow><mo>∇</mo><mo></mo><mi>f</mi></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E25.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E26" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.26</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E26.png" altimg-height="25px" altimg-valign="-7px" altimg-width="372px" alttext="\nabla\cdot(\mathbf{F}\times\mathbf{G})=\mathbf{G}\cdot(\nabla\times\mathbf{F}%
)-\mathbf{F}\cdot(\nabla\times\mathbf{G})," display="block"><mrow><mrow><mrow><mo>∇</mo><mo>⋅</mo><mrow><mo stretchy="false">(</mo><mrow><mi mathvariant="bold">F</mi><mo>×</mo><mi mathvariant="bold">G</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mi mathvariant="bold">G</mi><mo>⋅</mo><mrow><mo stretchy="false">(</mo><mrow><mo>∇</mo><mo>×</mo><mi mathvariant="bold">F</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>-</mo><mrow><mi mathvariant="bold">F</mi><mo>⋅</mo><mrow><mo stretchy="false">(</mo><mrow><mo>∇</mo><mo>×</mo><mi mathvariant="bold">G</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E26.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E27" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.27</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E27.png" altimg-height="25px" altimg-valign="-7px" altimg-width="258px" alttext="\nabla\cdot(\nabla\times\mathbf{F})=\operatorname{div}\operatorname{curl}%
\mathbf{F}=0," display="block"><mrow><mrow><mrow><mo>∇</mo><mo>⋅</mo><mrow><mo stretchy="false">(</mo><mrow><mo>∇</mo><mo>×</mo><mi mathvariant="bold">F</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mo href="./1.6#E21">div</mo><mrow><mo href="./1.6#E22">curl</mo><mi mathvariant="bold">F</mi></mrow></mrow><mo>=</mo><mn>0</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E27.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.6#E22" title="(1.6.22) ‣ Del Operator ‣ §1.6(iii) Vector-Valued Functions ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m95.png" altimg-height="18px" altimg-valign="-2px" altimg-width="38px" alttext="\operatorname{curl}" display="inline"><mo href="./1.6#E22">curl</mo></math>: of vector-valued function</a> and
<a href="./1.6#E21" title="(1.6.21) ‣ Del Operator ‣ §1.6(iii) Vector-Valued Functions ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m96.png" altimg-height="18px" altimg-valign="-2px" altimg-width="32px" alttext="\operatorname{div}" display="inline"><mo href="./1.6#E21">div</mo></math>: divergence of vector-valued function</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E28" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.28</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E28.png" altimg-height="25px" altimg-valign="-7px" altimg-width="311px" alttext="\nabla\times(f\mathbf{F})=f(\nabla\times\mathbf{F})+(\nabla f)\times\mathbf{F}," display="block"><mrow><mrow><mrow><mo>∇</mo><mo>×</mo><mrow><mo stretchy="false">(</mo><mrow><mi>f</mi><mo></mo><mi mathvariant="bold">F</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mo>∇</mo><mo>×</mo><mi mathvariant="bold">F</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>+</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mo>∇</mo><mo></mo><mi>f</mi></mrow><mo stretchy="false">)</mo></mrow><mo>×</mo><mi mathvariant="bold">F</mi></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E28.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E29" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.29</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E29.png" altimg-height="25px" altimg-valign="-7px" altimg-width="250px" alttext="\nabla\times(\nabla f)=\operatorname{curl}\operatorname{grad}f=0," display="block"><mrow><mrow><mrow><mo>∇</mo><mo>×</mo><mrow><mo stretchy="false">(</mo><mrow><mo>∇</mo><mo></mo><mi>f</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mo href="./1.6#E22">curl</mo><mrow><mo href="./1.6#E20">grad</mo><mi>f</mi></mrow></mrow><mo>=</mo><mn>0</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E29.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.6#E22" title="(1.6.22) ‣ Del Operator ‣ §1.6(iii) Vector-Valued Functions ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m95.png" altimg-height="18px" altimg-valign="-2px" altimg-width="38px" alttext="\operatorname{curl}" display="inline"><mo href="./1.6#E22">curl</mo></math>: of vector-valued function</a> and
<a href="./1.6#E20" title="(1.6.20) ‣ Del Operator ‣ §1.6(iii) Vector-Valued Functions ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m97.png" altimg-height="21px" altimg-valign="-6px" altimg-width="43px" alttext="\operatorname{grad}" display="inline"><mo href="./1.6#E20">grad</mo></math>: gradient of differentiable scalar function</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E30" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.30</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E30.png" altimg-height="28px" altimg-valign="-7px" altimg-width="151px" alttext="\nabla^{2}f=\nabla\cdot(\nabla f)," display="block"><mrow><mrow><mrow><msup><mo>∇</mo><mn>2</mn></msup><mo></mo><mi>f</mi></mrow><mo>=</mo><mrow><mo>∇</mo><mo>⋅</mo><mrow><mo stretchy="false">(</mo><mrow><mo>∇</mo><mo></mo><mi>f</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E30.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E31" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.31</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E31.png" altimg-height="28px" altimg-valign="-7px" altimg-width="341px" alttext="\nabla^{2}(fg)=f\nabla^{2}g+g\nabla^{2}f+2(\nabla f\cdot\nabla g)," display="block"><mrow><mrow><mrow><msup><mo>∇</mo><mn>2</mn></msup><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>f</mi><mo></mo><mi>g</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mi>f</mi><mo></mo><mrow><msup><mo>∇</mo><mn>2</mn></msup><mo></mo><mi>g</mi></mrow></mrow><mo>+</mo><mrow><mi>g</mi><mo></mo><mrow><msup><mo>∇</mo><mn>2</mn></msup><mo></mo><mi>f</mi></mrow></mrow><mo>+</mo><mrow><mn>2</mn><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mo>∇</mo><mo></mo><mi>f</mi></mrow><mo>⋅</mo><mrow><mo>∇</mo><mo></mo><mi>g</mi></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E31.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E32" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.32</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E32.png" altimg-height="25px" altimg-valign="-7px" altimg-width="174px" alttext="\nabla\cdot(\nabla f\times\nabla g)=0," display="block"><mrow><mrow><mrow><mo>∇</mo><mo>⋅</mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mo>∇</mo><mo></mo><mi>f</mi></mrow><mo>×</mo><mrow><mo>∇</mo><mo></mo><mi>g</mi></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mn>0</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E32.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E33" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.33</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E33.png" altimg-height="28px" altimg-valign="-7px" altimg-width="305px" alttext="\nabla\cdot(f\nabla g-g\nabla f)=f\nabla^{2}g-g\nabla^{2}f," display="block"><mrow><mrow><mrow><mo>∇</mo><mo>⋅</mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mi>f</mi><mo></mo><mrow><mo>∇</mo><mo></mo><mi>g</mi></mrow></mrow><mo>-</mo><mrow><mi>g</mi><mo></mo><mrow><mo>∇</mo><mo></mo><mi>f</mi></mrow></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mi>f</mi><mo></mo><mrow><msup><mo>∇</mo><mn>2</mn></msup><mo></mo><mi>g</mi></mrow></mrow><mo>-</mo><mrow><mi>g</mi><mo></mo><mrow><msup><mo>∇</mo><mn>2</mn></msup><mo></mo><mi>f</mi></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E33.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E34" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.34</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E34.png" altimg-height="28px" altimg-valign="-7px" altimg-width="406px" alttext="\nabla\times(\nabla\times\mathbf{F})=\operatorname{curl}\operatorname{curl}%
\mathbf{F}=\nabla(\nabla\cdot\mathbf{F})-\nabla^{2}\mathbf{F}." display="block"><mrow><mrow><mrow><mo>∇</mo><mo>×</mo><mrow><mo stretchy="false">(</mo><mrow><mo>∇</mo><mo>×</mo><mi mathvariant="bold">F</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mo href="./1.6#E22">curl</mo><mrow><mo href="./1.6#E22">curl</mo><mi mathvariant="bold">F</mi></mrow></mrow><mo>=</mo><mrow><mrow><mo>∇</mo><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mo>∇</mo><mo>⋅</mo><mi mathvariant="bold">F</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>-</mo><mrow><msup><mo>∇</mo><mn>2</mn></msup><mo></mo><mi mathvariant="bold">F</mi></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E34.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./1.6#E22" title="(1.6.22) ‣ Del Operator ‣ §1.6(iii) Vector-Valued Functions ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m95.png" altimg-height="18px" altimg-valign="-2px" altimg-width="38px" alttext="\operatorname{curl}" display="inline"><mo href="./1.6#E22">curl</mo></math>: of vector-valued function</a></dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
</section>
<section id="iv" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§1.6(iv) </span>Path and Line Integrals</h2>
<div id="SS4.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="SS4.p1" class="ltx_para">
<p class="ltx_p">Note: The terminology <em class="ltx_emph ltx_font_italic">open</em> and <em class="ltx_emph ltx_font_italic">closed sets</em> and <em class="ltx_emph ltx_font_italic">boundary
points</em> in the <math class="ltx_Math" altimg="m40.png" altimg-height="23px" altimg-valign="-7px" altimg-width="51px" alttext="(x,y)" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi>x</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi>y</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math> plane that is used in this subsection and
§.
</p>
</div>
<div id="SS4.p2" class="ltx_para">
<p class="ltx_p"><math class="ltx_Math" altimg="m79.png" altimg-height="23px" altimg-valign="-7px" altimg-width="197px" alttext="\mathbf{c}(t)=(x(t),y(t),z(t))" display="inline"><mrow><mrow><mi mathvariant="bold">c</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mo stretchy="false">(</mo><mrow><mi>x</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo><mrow><mi>y</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo><mrow><mi>z</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></math>, with <math class="ltx_Math" altimg="m121.png" altimg-height="17px" altimg-valign="-2px" altimg-width="11px" alttext="t" display="inline"><mi>t</mi></math> ranging
over an interval and <math class="ltx_Math" altimg="m125.png" altimg-height="23px" altimg-valign="-7px" altimg-width="122px" alttext="x(t),y(t),z(t)" display="inline"><mrow><mrow><mi>x</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo><mrow><mi>y</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo><mrow><mi>z</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></math> differentiable, defines a <em class="ltx_emph ltx_font_italic">path</em>.</p>
<table id="E35" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.35</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E35.png" altimg-height="26px" altimg-valign="-7px" altimg-width="227px" alttext="\mathbf{c}^{\prime}(t)=(x^{\prime}(t),y^{\prime}(t),z^{\prime}(t))." display="block"><mrow><mrow><mrow><msup><mi mathvariant="bold">c</mi><mo>′</mo></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mo stretchy="false">(</mo><mrow><msup><mi>x</mi><mo>′</mo></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo><mrow><msup><mi>y</mi><mo>′</mo></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo><mrow><msup><mi>z</mi><mo>′</mo></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E35.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">The <em class="ltx_emph ltx_font_italic">length</em> of a path for <math class="ltx_Math" altimg="m107.png" altimg-height="20px" altimg-valign="-5px" altimg-width="84px" alttext="a\leq t\leq b" display="inline"><mrow><mi>a</mi><mo>≤</mo><mi>t</mi><mo>≤</mo><mi>b</mi></mrow></math> is
</p>
<table id="E36" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.36</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E36.png" altimg-height="56px" altimg-valign="-20px" altimg-width="119px" alttext="\int_{a}^{b}\|\mathbf{c}^{\prime}(t)\|\mathrm{d}t." display="block"><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi>a</mi><mi>b</mi></msubsup><mrow><mrow><mo>∥</mo><mrow><msup><mi mathvariant="bold">c</mi><mo>′</mo></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo>∥</mo></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E36.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m89.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m68.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">The <em class="ltx_emph ltx_font_italic">path integral</em> of a continuous function <math class="ltx_Math" altimg="m109.png" altimg-height="23px" altimg-valign="-7px" altimg-width="81px" alttext="f(x,y,z)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math> is
</p>
<table id="E37" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.37</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E37.png" altimg-height="56px" altimg-valign="-20px" altimg-width="347px" alttext="\int_{\mathbf{c}}f\mathrm{d}s=\int^{b}_{a}f(x(t),y(t),z(t))\|\mathbf{c}^{%
\prime}(t)\|\mathrm{d}t." display="block"><mrow><mrow><mrow><msub><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi mathvariant="bold">c</mi></msub><mrow><mi>f</mi><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>s</mi></mrow></mrow></mrow><mo>=</mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi>a</mi><mi>b</mi></msubsup><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>x</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo><mrow><mi>y</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo><mrow><mi>z</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo>∥</mo><mrow><msup><mi mathvariant="bold">c</mi><mo>′</mo></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo>∥</mo></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E37.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m89.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m68.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">The <em class="ltx_emph ltx_font_italic">line integral</em> of a vector-valued function <math class="ltx_Math" altimg="m70.png" altimg-height="21px" altimg-valign="-6px" altimg-width="185px" alttext="\mathbf{F}=F_{1}\mathbf{i}+F_{2}\mathbf{j}+F_{3}\mathbf{k}" display="inline"><mrow><mi mathvariant="bold">F</mi><mo>=</mo><mrow><mrow><msub><mi href="./1.6#SS4.p2">F</mi><mn>1</mn></msub><mo></mo><mi href="./1.6#E5" mathvariant="bold">i</mi></mrow><mo>+</mo><mrow><msub><mi href="./1.6#SS4.p2">F</mi><mn>2</mn></msub><mo></mo><mi href="./1.6#E5" mathvariant="bold">j</mi></mrow><mo>+</mo><mrow><msub><mi href="./1.6#SS4.p2">F</mi><mn>3</mn></msub><mo></mo><mi href="./1.6#E5" mathvariant="bold">k</mi></mrow></mrow></mrow></math> along <math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="\mathbf{c}" display="inline"><mi mathvariant="bold">c</mi></math> is given
by
</p>
<table id="E38" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.38</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E38.png" altimg-height="56px" altimg-valign="-21px" altimg-width="807px" alttext="\int_{\mathbf{c}}\mathbf{F}\cdot\mathrm{d}\mathbf{s}=\int^{b}_{a}\mathbf{F}(%
\mathbf{c}(t))\cdot\mathbf{c}^{\prime}(t)\mathrm{d}t=\int^{b}_{a}\left(F_{1}%
\frac{\mathrm{d}x}{\mathrm{d}t}+F_{2}\frac{\mathrm{d}y}{\mathrm{d}t}+F_{3}%
\frac{\mathrm{d}z}{\mathrm{d}t}\right)\mathrm{d}t=\int_{\mathbf{c}}F_{1}%
\mathrm{d}x+F_{2}\mathrm{d}y+F_{3}\mathrm{d}z." display="block"><mrow><mrow><mrow><msub><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi mathvariant="bold">c</mi></msub><mrow><mi mathvariant="bold">F</mi><mo>⋅</mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi mathvariant="bold">s</mi></mrow></mrow></mrow><mo>=</mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi>a</mi><mi>b</mi></msubsup><mrow><mrow><mrow><mi mathvariant="bold">F</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi mathvariant="bold">c</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>⋅</mo><mrow><msup><mi mathvariant="bold">c</mi><mo>′</mo></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow><mo>=</mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi>a</mi><mi>b</mi></msubsup><mrow><mrow><mo>(</mo><mrow><mrow><msub><mi href="./1.6#SS4.p2">F</mi><mn>1</mn></msub><mo></mo><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi>x</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi>t</mi></mrow></mfrac></mrow><mo>+</mo><mrow><msub><mi href="./1.6#SS4.p2">F</mi><mn>2</mn></msub><mo></mo><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi>y</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi>t</mi></mrow></mfrac></mrow><mo>+</mo><mrow><msub><mi href="./1.6#SS4.p2">F</mi><mn>3</mn></msub><mo></mo><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi>z</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi>t</mi></mrow></mfrac></mrow></mrow><mo>)</mo></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow><mo>=</mo><mrow><mrow><msub><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi mathvariant="bold">c</mi></msub><mrow><msub><mi href="./1.6#SS4.p2">F</mi><mn>1</mn></msub><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>x</mi></mrow></mrow></mrow><mo>+</mo><mrow><msub><mi href="./1.6#SS4.p2">F</mi><mn>2</mn></msub><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>y</mi></mrow></mrow><mo>+</mo><mrow><msub><mi href="./1.6#SS4.p2">F</mi><mn>3</mn></msub><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>z</mi></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E38.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#E4" title="(1.4.4) ‣ §1.4(iii) Derivatives ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m65.png" altimg-height="29px" altimg-valign="-9px" altimg-width="27px" alttext="\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: derivative of <math class="ltx_Math" altimg="m110.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m89.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m68.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a> and
<a href="./1.6#SS4.p2" title="§1.6(iv) Path and Line Integrals ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="23px" altimg-valign="-8px" altimg-width="26px" alttext="F_{j}" display="inline"><msub><mi href="./1.6#SS4.p2">F</mi><mi href="./1.1#p2.t1.r4">j</mi></msub></math>: vector function components</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">A path <math class="ltx_Math" altimg="m83.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\mathbf{c}_{1}(t)" display="inline"><mrow><msub><mi mathvariant="bold">c</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math>, <math class="ltx_Math" altimg="m122.png" altimg-height="23px" altimg-valign="-7px" altimg-width="75px" alttext="t\in[a,b]" display="inline"><mrow><mi>t</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">[</mo><mi>a</mi><mo href="./front/introduction#Sx4.p1.t1.r29">,</mo><mi>b</mi><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">]</mo></mrow></mrow></math>, is a <em class="ltx_emph ltx_font_italic">reparametrization</em> of
<math class="ltx_Math" altimg="m80.png" altimg-height="24px" altimg-valign="-7px" altimg-width="43px" alttext="\mathbf{c}(t^{\prime})" display="inline"><mrow><mi mathvariant="bold">c</mi><mo></mo><mrow><mo stretchy="false">(</mo><msup><mi>t</mi><mo>′</mo></msup><mo stretchy="false">)</mo></mrow></mrow></math>, <math class="ltx_Math" altimg="m124.png" altimg-height="24px" altimg-valign="-7px" altimg-width="92px" alttext="t^{\prime}\in[a^{\prime},b^{\prime}]" display="inline"><mrow><msup><mi>t</mi><mo>′</mo></msup><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">[</mo><msup><mi>a</mi><mo>′</mo></msup><mo href="./front/introduction#Sx4.p1.t1.r29">,</mo><msup><mi>b</mi><mo>′</mo></msup><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">]</mo></mrow></mrow></math>, if <math class="ltx_Math" altimg="m82.png" altimg-height="24px" altimg-valign="-7px" altimg-width="111px" alttext="\mathbf{c}_{1}(t)=\mathbf{c}(t^{\prime})" display="inline"><mrow><mrow><msub><mi mathvariant="bold">c</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mi mathvariant="bold">c</mi><mo></mo><mrow><mo stretchy="false">(</mo><msup><mi>t</mi><mo>′</mo></msup><mo stretchy="false">)</mo></mrow></mrow></mrow></math> and
<math class="ltx_Math" altimg="m123.png" altimg-height="24px" altimg-valign="-7px" altimg-width="78px" alttext="t^{\prime}=h(t)" display="inline"><mrow><msup><mi>t</mi><mo>′</mo></msup><mo>=</mo><mrow><mi href="./1.6#SS4.p2">h</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></math> with <math class="ltx_Math" altimg="m116.png" altimg-height="23px" altimg-valign="-7px" altimg-width="38px" alttext="h(t)" display="inline"><mrow><mi href="./1.6#SS4.p2">h</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math> differentiable and monotonic. If <math class="ltx_Math" altimg="m112.png" altimg-height="24px" altimg-valign="-7px" altimg-width="84px" alttext="h(a)=a^{\prime}" display="inline"><mrow><mrow><mi href="./1.6#SS4.p2">h</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><msup><mi>a</mi><mo>′</mo></msup></mrow></math> and
<math class="ltx_Math" altimg="m115.png" altimg-height="24px" altimg-valign="-7px" altimg-width="81px" alttext="h(b)=b^{\prime}" display="inline"><mrow><mrow><mi href="./1.6#SS4.p2">h</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><msup><mi>b</mi><mo>′</mo></msup></mrow></math>, then the
reparametrization is called <em class="ltx_emph ltx_font_italic">orientation-preserving</em>, and
</p>
<table id="E39" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.39</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E39.png" altimg-height="54px" altimg-valign="-22px" altimg-width="191px" alttext="\int_{\mathbf{c}}\mathbf{F}\cdot\mathrm{d}\mathbf{s}=\int_{\mathbf{c}_{1}}%
\mathbf{F}\cdot\mathrm{d}\mathbf{s}." display="block"><mrow><mrow><mrow><msub><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi mathvariant="bold">c</mi></msub><mrow><mi mathvariant="bold">F</mi><mo>⋅</mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi mathvariant="bold">s</mi></mrow></mrow></mrow><mo>=</mo><mrow><msub><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><msub><mi mathvariant="bold">c</mi><mn>1</mn></msub></msub><mrow><mi mathvariant="bold">F</mi><mo>⋅</mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi mathvariant="bold">s</mi></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E39.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m89.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m68.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">If <math class="ltx_Math" altimg="m113.png" altimg-height="24px" altimg-valign="-7px" altimg-width="83px" alttext="h(a)=b^{\prime}" display="inline"><mrow><mrow><mi href="./1.6#SS4.p2">h</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><msup><mi>b</mi><mo>′</mo></msup></mrow></math> and <math class="ltx_Math" altimg="m114.png" altimg-height="24px" altimg-valign="-7px" altimg-width="83px" alttext="h(b)=a^{\prime}" display="inline"><mrow><mrow><mi href="./1.6#SS4.p2">h</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><msup><mi>a</mi><mo>′</mo></msup></mrow></math>, then the reparametrization is
<em class="ltx_emph ltx_font_italic">orientation-reversing</em> and
</p>
<table id="E40" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.40</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E40.png" altimg-height="54px" altimg-valign="-22px" altimg-width="209px" alttext="\int_{\mathbf{c}}\mathbf{F}\cdot\mathrm{d}\mathbf{s}=-\int_{\mathbf{c}_{1}}%
\mathbf{F}\cdot\mathrm{d}\mathbf{s}." display="block"><mrow><mrow><mrow><msub><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi mathvariant="bold">c</mi></msub><mrow><mi mathvariant="bold">F</mi><mo>⋅</mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi mathvariant="bold">s</mi></mrow></mrow></mrow><mo>=</mo><mrow><mo>-</mo><mrow><msub><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><msub><mi mathvariant="bold">c</mi><mn>1</mn></msub></msub><mrow><mi mathvariant="bold">F</mi><mo>⋅</mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi mathvariant="bold">s</mi></mrow></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E40.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m89.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m68.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">In either case</p>
<table id="E41" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.41</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E41.png" altimg-height="54px" altimg-valign="-22px" altimg-width="157px" alttext="\int_{\mathbf{c}}f\mathrm{d}s=\int_{\mathbf{c}_{1}}f\mathrm{d}s," display="block"><mrow><mrow><mrow><msub><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi mathvariant="bold">c</mi></msub><mrow><mi>f</mi><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>s</mi></mrow></mrow></mrow><mo>=</mo><mrow><msub><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><msub><mi mathvariant="bold">c</mi><mn>1</mn></msub></msub><mrow><mi>f</mi><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>s</mi></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E41.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m89.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m68.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">when <math class="ltx_Math" altimg="m110.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> is continuous, and</p>
<table id="E42" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.42</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E42.png" altimg-height="52px" altimg-valign="-20px" altimg-width="274px" alttext="\int_{\mathbf{c}}\nabla f\cdot\mathrm{d}\mathbf{s}=f(\mathbf{c}(b))-f(\mathbf{%
c}(a))," display="block"><mrow><mrow><mrow><msub><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi mathvariant="bold">c</mi></msub><mrow><mrow><mo>∇</mo><mo></mo><mi>f</mi></mrow><mo>⋅</mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi mathvariant="bold">s</mi></mrow></mrow></mrow><mo>=</mo><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi mathvariant="bold">c</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>-</mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi mathvariant="bold">c</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E42.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m89.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m68.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">when <math class="ltx_Math" altimg="m110.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> is continuously differentiable.</p>
</div>
<div id="SS4.p3" class="ltx_para">
<p class="ltx_p">The geometrical image <math class="ltx_Math" altimg="m45.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.6#Px10.p1">C</mi></math> of a path <math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="\mathbf{c}" display="inline"><mi mathvariant="bold">c</mi></math> is called a <em class="ltx_emph ltx_font_italic">simple
closed curve</em>
if <math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="\mathbf{c}" display="inline"><mi mathvariant="bold">c</mi></math> is one-to-one, with the exception <math class="ltx_Math" altimg="m78.png" altimg-height="23px" altimg-valign="-7px" altimg-width="101px" alttext="\mathbf{c}(a)=\mathbf{c}(b)" display="inline"><mrow><mrow><mi mathvariant="bold">c</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mi mathvariant="bold">c</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></math>. The curve <math class="ltx_Math" altimg="m45.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.6#Px10.p1">C</mi></math> is <em class="ltx_emph ltx_font_italic">piecewise differentiable</em>
if <math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="\mathbf{c}" display="inline"><mi mathvariant="bold">c</mi></math> is piecewise differentiable. Note that <math class="ltx_Math" altimg="m45.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.6#Px10.p1">C</mi></math> can be given an
orientation by means of <math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="\mathbf{c}" display="inline"><mi mathvariant="bold">c</mi></math>.
</p>
</div>
<section id="Px10" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Green’s Theorem</h3>
<div id="Px10.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px10.p1" class="ltx_para">
<p class="ltx_p">Let</p>
<table id="E43" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.43</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E43.png" altimg-height="25px" altimg-valign="-7px" altimg-width="267px" alttext="\mathbf{F}(x,y)=F_{1}(x,y)\mathbf{i}+F_{2}(x,y)\mathbf{j}" display="block"><mrow><mrow><mi mathvariant="bold">F</mi><mo></mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi>x</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi>y</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mrow><msub><mi href="./1.6#SS4.p2">F</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mi href="./1.6#E5" mathvariant="bold">i</mi></mrow><mo>+</mo><mrow><mrow><msub><mi href="./1.6#SS4.p2">F</mi><mn>2</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mi href="./1.6#E5" mathvariant="bold">j</mi></mrow></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E43.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./front/introduction#Sx4.p1.t1.r28" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="(\NVar{a},\NVar{b})" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi class="ltx_nvar">a</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi class="ltx_nvar">b</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math>: open interval</a>,
<a href="./1.6#E5" title="(1.6.5) ‣ Unit Vectors ‣ §1.6(i) Vectors ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="18px" altimg-valign="-2px" altimg-width="11px" alttext="\mathbf{i}" display="inline"><mi href="./1.6#E5" mathvariant="bold">i</mi></math>: unit vector</a>,
<a href="./1.6#E5" title="(1.6.5) ‣ Unit Vectors ‣ §1.6(i) Vectors ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m86.png" altimg-height="21px" altimg-valign="-6px" altimg-width="11px" alttext="\mathbf{j}" display="inline"><mi href="./1.6#E5" mathvariant="bold">j</mi></math>: unit vector</a> and
<a href="./1.6#SS4.p2" title="§1.6(iv) Path and Line Integrals ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="23px" altimg-valign="-8px" altimg-width="26px" alttext="F_{j}" display="inline"><msub><mi href="./1.6#SS4.p2">F</mi><mi href="./1.1#p2.t1.r4">j</mi></msub></math>: vector function components</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and <math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi href="./1.6#Px10.p1">S</mi></math> be the closed and bounded point set in the <math class="ltx_Math" altimg="m40.png" altimg-height="23px" altimg-valign="-7px" altimg-width="51px" alttext="(x,y)" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi>x</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi>y</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math> plane having a
simple closed curve <math class="ltx_Math" altimg="m45.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.6#Px10.p1">C</mi></math>
as boundary. If <math class="ltx_Math" altimg="m45.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.6#Px10.p1">C</mi></math> is oriented in the positive (anticlockwise) sense, then</p>
<table id="E44" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.44</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E44.png" altimg-height="53px" altimg-valign="-21px" altimg-width="480px" alttext="\iint_{S}\left(\frac{\partial F_{2}}{\partial x}-\frac{\partial F_{1}}{%
\partial y}\right)\mathrm{d}A=\int_{C}\mathbf{F}\cdot\mathrm{d}\mathbf{s}=\int%
_{C}F_{1}\mathrm{d}x+F_{2}\mathrm{d}y." display="block"><mrow><mrow><mrow><msub><mo largeop="true" symmetric="true">∬</mo><mi href="./1.6#Px10.p1">S</mi></msub><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><msub><mi href="./1.6#SS4.p2">F</mi><mn>2</mn></msub></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow></mfrac><mo>-</mo><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><msub><mi href="./1.6#SS4.p2">F</mi><mn>1</mn></msub></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow></mfrac></mrow><mo>)</mo></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>A</mi></mrow></mrow></mrow><mo>=</mo><mrow><msub><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi href="./1.6#Px10.p1">C</mi></msub><mrow><mi mathvariant="bold">F</mi><mo>⋅</mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi mathvariant="bold">s</mi></mrow></mrow></mrow><mo>=</mo><mrow><mrow><msub><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi href="./1.6#Px10.p1">C</mi></msub><mrow><msub><mi href="./1.6#SS4.p2">F</mi><mn>1</mn></msub><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>x</mi></mrow></mrow></mrow><mo>+</mo><mrow><msub><mi href="./1.6#SS4.p2">F</mi><mn>2</mn></msub><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>y</mi></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E44.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m89.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m68.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m110.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m99.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.6#Px10.p1" title="Green’s Theorem ‣ §1.6(iv) Path and Line Integrals ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi href="./1.6#Px10.p1">S</mi></math>: closed region</a>,
<a href="./1.6#Px10.p1" title="Green’s Theorem ‣ §1.6(iv) Path and Line Integrals ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m45.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.6#Px10.p1">C</mi></math>: closed curve</a> and
<a href="./1.6#SS4.p2" title="§1.6(iv) Path and Line Integrals ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="23px" altimg-valign="-8px" altimg-width="26px" alttext="F_{j}" display="inline"><msub><mi href="./1.6#SS4.p2">F</mi><mi href="./1.1#p2.t1.r4">j</mi></msub></math>: vector function components</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Sufficient conditions for this result to hold are that <math class="ltx_Math" altimg="m49.png" altimg-height="23px" altimg-valign="-7px" altimg-width="72px" alttext="F_{1}(x,y)" display="inline"><mrow><msub><mi href="./1.6#SS4.p2">F</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></math> and
<math class="ltx_Math" altimg="m50.png" altimg-height="23px" altimg-valign="-7px" altimg-width="72px" alttext="F_{2}(x,y)" display="inline"><mrow><msub><mi href="./1.6#SS4.p2">F</mi><mn>2</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></math> are continuously differentiable on <math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi href="./1.6#Px10.p1">S</mi></math>, and <math class="ltx_Math" altimg="m45.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.6#Px10.p1">C</mi></math> is piecewise
differentiable.</p>
</div>
<div id="Px10.p2" class="ltx_para">
<p class="ltx_p">The area of <math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi href="./1.6#Px10.p1">S</mi></math> can be found from () by taking
<math class="ltx_Math" altimg="m69.png" altimg-height="23px" altimg-valign="-7px" altimg-width="124px" alttext="\mathbf{F}(x,y)=-y\mathbf{i}" display="inline"><mrow><mrow><mi mathvariant="bold">F</mi><mo></mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi>x</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi>y</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mo>-</mo><mrow><mi>y</mi><mo></mo><mi href="./1.6#E5" mathvariant="bold">i</mi></mrow></mrow></mrow></math>, <math class="ltx_Math" altimg="m129.png" altimg-height="21px" altimg-valign="-6px" altimg-width="23px" alttext="x\mathbf{j}" display="inline"><mrow><mi>x</mi><mo></mo><mi href="./1.6#E5" mathvariant="bold">j</mi></mrow></math>, or
<math class="ltx_Math" altimg="m41.png" altimg-height="27px" altimg-valign="-9px" altimg-width="105px" alttext="-\frac{1}{2}y\mathbf{i}+\frac{1}{2}x\mathbf{j}" display="inline"><mrow><mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi>y</mi><mo></mo><mi href="./1.6#E5" mathvariant="bold">i</mi></mrow></mrow><mo>+</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi>x</mi><mo></mo><mi href="./1.6#E5" mathvariant="bold">j</mi></mrow></mrow></math>.</p>
</div>
</section>
</section>
<section id="v" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§1.6(v) </span>Surfaces and Integrals over Surfaces</h2>
<div id="SS5.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="SS5.p1" class="ltx_para">
<p class="ltx_p">A <em class="ltx_emph ltx_font_italic">parametrized surface</em> <math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi href="./1.6#SS5.p1">S</mi></math> is defined by
</p>
<table id="E45" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.45</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E45.png" altimg-height="25px" altimg-valign="-7px" altimg-width="299px" alttext="\boldsymbol{{\Phi}}(u,v)=(x(u,v),y(u,v),z(u,v))" display="block"><mrow><mrow><mi href="./1.6#E45" mathvariant="bold">Φ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mo stretchy="false">(</mo><mrow><mi>x</mi><mo></mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi>u</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi>v</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></mrow><mo>,</mo><mrow><mi>y</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo><mrow><mi>z</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E45.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text"><math class="ltx_Math" altimg="m56.png" altimg-height="23px" altimg-valign="-7px" altimg-width="86px" alttext="\boldsymbol{{\Phi}}(x,y,z)" display="inline"><mrow><mi href="./1.6#E45" mathvariant="bold">Φ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: parameterization (locally)</span></dd>
<dt>Symbols:</dt>
<dd><a href="./front/introduction#Sx4.p1.t1.r28" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="(\NVar{a},\NVar{b})" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi class="ltx_nvar">a</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi class="ltx_nvar">b</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math>: open interval</a></dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">with <math class="ltx_Math" altimg="m38.png" altimg-height="23px" altimg-valign="-7px" altimg-width="92px" alttext="(u,v)\in D" display="inline"><mrow><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi>u</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi>v</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mi href="./1.6#SS5.p1">D</mi></mrow></math>, an open set in the plane.</p>
</div>
<div id="SS5.p2" class="ltx_para">
<p class="ltx_p">For <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math>, <math class="ltx_Math" altimg="m132.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi>y</mi></math>, and <math class="ltx_Math" altimg="m135.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi>z</mi></math> continuously differentiable, the vectors</p>
<table id="E46" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.46</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E46.png" altimg-height="47px" altimg-valign="-16px" altimg-width="407px" alttext="\mathbf{T}_{u}=\frac{\partial x}{\partial u}(u_{0},v_{0})\mathbf{i}+\frac{%
\partial y}{\partial u}(u_{0},v_{0})\mathbf{j}+\frac{\partial z}{\partial u}(u%
_{0},v_{0})\mathbf{k}" display="block"><mrow><msub><mi mathvariant="bold">T</mi><mi>u</mi></msub><mo>=</mo><mrow><mrow><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>u</mi></mrow></mfrac><mo></mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><msub><mi>u</mi><mn>0</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><msub><mi>v</mi><mn>0</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow><mo></mo><mi href="./1.6#E5" mathvariant="bold">i</mi></mrow><mo>+</mo><mrow><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>u</mi></mrow></mfrac><mo></mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><msub><mi>u</mi><mn>0</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><msub><mi>v</mi><mn>0</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow><mo></mo><mi href="./1.6#E5" mathvariant="bold">j</mi></mrow><mo>+</mo><mrow><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>z</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>u</mi></mrow></mfrac><mo></mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><msub><mi>u</mi><mn>0</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><msub><mi>v</mi><mn>0</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow><mo></mo><mi href="./1.6#E5" mathvariant="bold">k</mi></mrow></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E46.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./front/introduction#Sx4.p1.t1.r28" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="(\NVar{a},\NVar{b})" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi class="ltx_nvar">a</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi class="ltx_nvar">b</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math>: open interval</a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m110.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m99.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.6#E5" title="(1.6.5) ‣ Unit Vectors ‣ §1.6(i) Vectors ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="18px" altimg-valign="-2px" altimg-width="11px" alttext="\mathbf{i}" display="inline"><mi href="./1.6#E5" mathvariant="bold">i</mi></math>: unit vector</a>,
<a href="./1.6#E5" title="(1.6.5) ‣ Unit Vectors ‣ §1.6(i) Vectors ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m86.png" altimg-height="21px" altimg-valign="-6px" altimg-width="11px" alttext="\mathbf{j}" display="inline"><mi href="./1.6#E5" mathvariant="bold">j</mi></math>: unit vector</a> and
<a href="./1.6#E5" title="(1.6.5) ‣ Unit Vectors ‣ §1.6(i) Vectors ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="18px" altimg-valign="-2px" altimg-width="16px" alttext="\mathbf{k}" display="inline"><mi href="./1.6#E5" mathvariant="bold">k</mi></math>: unit vector</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and</p>
<table id="E47" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.47</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E47.png" altimg-height="47px" altimg-valign="-16px" altimg-width="404px" alttext="\mathbf{T}_{v}=\frac{\partial x}{\partial v}(u_{0},v_{0})\mathbf{i}+\frac{%
\partial y}{\partial v}(u_{0},v_{0})\mathbf{j}+\frac{\partial z}{\partial v}(u%
_{0},v_{0})\mathbf{k}" display="block"><mrow><msub><mi mathvariant="bold">T</mi><mi>v</mi></msub><mo>=</mo><mrow><mrow><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>v</mi></mrow></mfrac><mo></mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><msub><mi>u</mi><mn>0</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><msub><mi>v</mi><mn>0</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow><mo></mo><mi href="./1.6#E5" mathvariant="bold">i</mi></mrow><mo>+</mo><mrow><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>v</mi></mrow></mfrac><mo></mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><msub><mi>u</mi><mn>0</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><msub><mi>v</mi><mn>0</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow><mo></mo><mi href="./1.6#E5" mathvariant="bold">j</mi></mrow><mo>+</mo><mrow><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>z</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>v</mi></mrow></mfrac><mo></mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><msub><mi>u</mi><mn>0</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><msub><mi>v</mi><mn>0</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow><mo></mo><mi href="./1.6#E5" mathvariant="bold">k</mi></mrow></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E47.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./front/introduction#Sx4.p1.t1.r28" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="(\NVar{a},\NVar{b})" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi class="ltx_nvar">a</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi class="ltx_nvar">b</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math>: open interval</a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m110.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m99.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.6#E5" title="(1.6.5) ‣ Unit Vectors ‣ §1.6(i) Vectors ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="18px" altimg-valign="-2px" altimg-width="11px" alttext="\mathbf{i}" display="inline"><mi href="./1.6#E5" mathvariant="bold">i</mi></math>: unit vector</a>,
<a href="./1.6#E5" title="(1.6.5) ‣ Unit Vectors ‣ §1.6(i) Vectors ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m86.png" altimg-height="21px" altimg-valign="-6px" altimg-width="11px" alttext="\mathbf{j}" display="inline"><mi href="./1.6#E5" mathvariant="bold">j</mi></math>: unit vector</a> and
<a href="./1.6#E5" title="(1.6.5) ‣ Unit Vectors ‣ §1.6(i) Vectors ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="18px" altimg-valign="-2px" altimg-width="16px" alttext="\mathbf{k}" display="inline"><mi href="./1.6#E5" mathvariant="bold">k</mi></math>: unit vector</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">are tangent to the surface at <math class="ltx_Math" altimg="m55.png" altimg-height="23px" altimg-valign="-7px" altimg-width="84px" alttext="\boldsymbol{{\Phi}}(u_{0},v_{0})" display="inline"><mrow><mi href="./1.6#E45" mathvariant="bold">Φ</mi><mo></mo><mrow><mo stretchy="false">(</mo><msub><mi>u</mi><mn>0</mn></msub><mo>,</mo><msub><mi>v</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mrow></math>. The surface is
<em class="ltx_emph ltx_font_italic">smooth</em>
at this point if <math class="ltx_Math" altimg="m73.png" altimg-height="21px" altimg-valign="-6px" altimg-width="117px" alttext="\mathbf{T}_{u}\times\mathbf{T}_{v}\not=0" display="inline"><mrow><mrow><msub><mi mathvariant="bold">T</mi><mi>u</mi></msub><mo>×</mo><msub><mi mathvariant="bold">T</mi><mi>v</mi></msub></mrow><mo>≠</mo><mn>0</mn></mrow></math>. A surface is
<em class="ltx_emph ltx_font_italic">smooth</em> if it is smooth at every point. The vector
<math class="ltx_Math" altimg="m72.png" altimg-height="21px" altimg-valign="-5px" altimg-width="80px" alttext="\mathbf{T}_{u}\times\mathbf{T}_{v}" display="inline"><mrow><msub><mi mathvariant="bold">T</mi><mi>u</mi></msub><mo>×</mo><msub><mi mathvariant="bold">T</mi><mi>v</mi></msub></mrow></math> at <math class="ltx_Math" altimg="m39.png" altimg-height="23px" altimg-valign="-7px" altimg-width="68px" alttext="(u_{0},v_{0})" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><msub><mi>u</mi><mn>0</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><msub><mi>v</mi><mn>0</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math> is normal to the surface at
<math class="ltx_Math" altimg="m55.png" altimg-height="23px" altimg-valign="-7px" altimg-width="84px" alttext="\boldsymbol{{\Phi}}(u_{0},v_{0})" display="inline"><mrow><mi href="./1.6#E45" mathvariant="bold">Φ</mi><mo></mo><mrow><mo stretchy="false">(</mo><msub><mi>u</mi><mn>0</mn></msub><mo>,</mo><msub><mi>v</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mrow></math>.
</p>
</div>
<div id="SS5.p3" class="ltx_para">
<p class="ltx_p">The <em class="ltx_emph ltx_font_italic">area</em> <math class="ltx_Math" altimg="m44.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="A(S)" display="inline"><mrow><mi href="./1.6#E48">A</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.6#SS5.p1">S</mi><mo stretchy="false">)</mo></mrow></mrow></math> of a parametrized smooth surface is given by
</p>
<table id="E48" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.48</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E48.png" altimg-height="52px" altimg-valign="-20px" altimg-width="264px" alttext="A(S)=\iint_{D}\|\mathbf{T}_{u}\times\mathbf{T}_{v}\|\mathrm{d}u\mathrm{d}v," display="block"><mrow><mrow><mrow><mi href="./1.6#E48">A</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.6#SS5.p1">S</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><msub><mo largeop="true" symmetric="true">∬</mo><mi href="./1.6#SS5.p1">D</mi></msub><mrow><mrow><mo>∥</mo><mrow><msub><mi mathvariant="bold">T</mi><mi>u</mi></msub><mo>×</mo><msub><mi mathvariant="bold">T</mi><mi>v</mi></msub></mrow><mo>∥</mo></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>u</mi></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>v</mi></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E48.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text"><math class="ltx_Math" altimg="m44.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="A(S)" display="inline"><mrow><mi href="./1.6#E48">A</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.6#SS5.p1">S</mi><mo stretchy="false">)</mo></mrow></mrow></math>: area of a parameterized smooth surface <math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi href="./1.6#SS5.p1">S</mi></math> (locally)</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m89.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.6#SS5.p1" title="§1.6(v) Surfaces and Integrals over Surfaces ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi href="./1.6#SS5.p1">S</mi></math>: parameterized surface</a> and
<a href="./1.6#SS5.p1" title="§1.6(v) Surfaces and Integrals over Surfaces ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m46.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.6#SS5.p1">D</mi></math>: open set in the plane</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and</p>
<table id="E49" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.49</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E49.png" altimg-height="66px" altimg-valign="-23px" altimg-width="506px" alttext="\|\mathbf{T}_{u}\times\mathbf{T}_{v}\|=\sqrt{\left(\frac{\partial(x,y)}{%
\partial(u,v)}\right)^{2}+\left(\frac{\partial(y,z)}{\partial(u,v)}\right)^{2}%
+\left(\frac{\partial(x,z)}{\partial(u,v)}\right)^{2}}." display="block"><mrow><mrow><mrow><mo>∥</mo><mrow><msub><mi mathvariant="bold">T</mi><mi>u</mi></msub><mo>×</mo><msub><mi mathvariant="bold">T</mi><mi>v</mi></msub></mrow><mo>∥</mo></mrow><mo>=</mo><msqrt><mrow><msup><mrow><mo>(</mo><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi>x</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi>y</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi>u</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi>v</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></mrow></mfrac><mo>)</mo></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow><mo>(</mo><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi>y</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi>z</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi>u</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi>v</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></mrow></mfrac><mo>)</mo></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow><mo>(</mo><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi>x</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi>z</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi>u</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi>v</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></mrow></mfrac><mo>)</mo></mrow><mn>2</mn></msup></mrow></msqrt></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E49.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./front/introduction#Sx4.p1.t1.r28" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="(\NVar{a},\NVar{b})" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi class="ltx_nvar">a</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi class="ltx_nvar">b</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math>: open interval</a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m110.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m99.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">The area is independent of the parametrizations.</p>
</div>
<div id="SS5.p4" class="ltx_para">
<p class="ltx_p">For a sphere <math class="ltx_Math" altimg="m126.png" altimg-height="21px" altimg-valign="-6px" altimg-width="139px" alttext="x=\rho\sin\theta\cos\phi" display="inline"><mrow><mi>x</mi><mo>=</mo><mrow><mi href="./1.6#SS5.p4">ρ</mi><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi href="./1.6#SS5.p4">θ</mi></mrow><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi href="./1.6#SS5.p4">ϕ</mi></mrow></mrow></mrow></math>,
<math class="ltx_Math" altimg="m130.png" altimg-height="21px" altimg-valign="-6px" altimg-width="136px" alttext="y=\rho\sin\theta\sin\phi" display="inline"><mrow><mi>y</mi><mo>=</mo><mrow><mi href="./1.6#SS5.p4">ρ</mi><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi href="./1.6#SS5.p4">θ</mi></mrow><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi href="./1.6#SS5.p4">ϕ</mi></mrow></mrow></mrow></math>, <math class="ltx_Math" altimg="m133.png" altimg-height="21px" altimg-valign="-6px" altimg-width="95px" alttext="z=\rho\cos\theta" display="inline"><mrow><mi>z</mi><mo>=</mo><mrow><mi href="./1.6#SS5.p4">ρ</mi><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi href="./1.6#SS5.p4">θ</mi></mrow></mrow></mrow></math>,
</p>
<table id="E50" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.50</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E50.png" altimg-height="29px" altimg-valign="-8px" altimg-width="209px" alttext="\|\mathbf{T}_{\theta}\times\mathbf{T}_{\phi}\|=\rho^{2}\left|\sin\theta\right|." display="block"><mrow><mrow><mrow><mo>∥</mo><mrow><msub><mi mathvariant="bold">T</mi><mi href="./1.6#SS5.p4">θ</mi></msub><mo>×</mo><msub><mi mathvariant="bold">T</mi><mi href="./1.6#SS5.p4">ϕ</mi></msub></mrow><mo>∥</mo></mrow><mo>=</mo><mrow><msup><mi href="./1.6#SS5.p4">ρ</mi><mn>2</mn></msup><mo></mo><mrow><mo>|</mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi href="./1.6#SS5.p4">θ</mi></mrow><mo>|</mo></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E50.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./1.6#SS5.p4" title="§1.6(v) Surfaces and Integrals over Surfaces ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m102.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="\rho" display="inline"><mi href="./1.6#SS5.p4">ρ</mi></math>: radius</a>,
<a href="./1.6#SS5.p4" title="§1.6(v) Surfaces and Integrals over Surfaces ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m104.png" altimg-height="18px" altimg-valign="-2px" altimg-width="14px" alttext="\theta" display="inline"><mi href="./1.6#SS5.p4">θ</mi></math>: angle</a> and
<a href="./1.6#SS5.p4" title="§1.6(v) Surfaces and Integrals over Surfaces ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m100.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="\phi" display="inline"><mi href="./1.6#SS5.p4">ϕ</mi></math>: angle</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS5.p5" class="ltx_para">
<p class="ltx_p">For a surface <math class="ltx_Math" altimg="m134.png" altimg-height="23px" altimg-valign="-7px" altimg-width="99px" alttext="z=f(x,y)" display="inline"><mrow><mi>z</mi><mo>=</mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></math>,</p>
<table id="E51" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.51</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E51.png" altimg-height="66px" altimg-valign="-23px" altimg-width="362px" alttext="A(S)=\iint_{D}\sqrt{1+\left(\frac{\partial f}{\partial x}\right)^{2}+\left(%
\frac{\partial f}{\partial y}\right)^{2}}\mathrm{d}A." display="block"><mrow><mrow><mrow><mi href="./1.6#E48">A</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.6#SS5.p1">S</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><msub><mo largeop="true" symmetric="true">∬</mo><mi href="./1.6#SS5.p1">D</mi></msub><mrow><msqrt><mrow><mn>1</mn><mo>+</mo><msup><mrow><mo>(</mo><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow></mfrac><mo>)</mo></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow><mo>(</mo><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow></mfrac><mo>)</mo></mrow><mn>2</mn></msup></mrow></msqrt><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./1.6#E48">A</mi></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E51.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m89.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m110.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m99.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.6#SS5.p1" title="§1.6(v) Surfaces and Integrals over Surfaces ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi href="./1.6#SS5.p1">S</mi></math>: parameterized surface</a>,
<a href="./1.6#SS5.p1" title="§1.6(v) Surfaces and Integrals over Surfaces ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m46.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.6#SS5.p1">D</mi></math>: open set in the plane</a> and
<a href="./1.6#E48" title="(1.6.48) ‣ §1.6(v) Surfaces and Integrals over Surfaces ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="A(S)" display="inline"><mrow><mi href="./1.6#E48">A</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.6#SS5.p1">S</mi><mo stretchy="false">)</mo></mrow></mrow></math>: area of a parameterized smooth surface <math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi href="./1.6#SS5.p1">S</mi></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS5.p6" class="ltx_para">
<p class="ltx_p">For a surface of revolution, <math class="ltx_Math" altimg="m131.png" altimg-height="23px" altimg-valign="-7px" altimg-width="80px" alttext="y=f(x)" display="inline"><mrow><mi>y</mi><mo>=</mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></math>, <math class="ltx_Math" altimg="m128.png" altimg-height="23px" altimg-valign="-7px" altimg-width="79px" alttext="x\in[a,b]" display="inline"><mrow><mi>x</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">[</mo><mi>a</mi><mo href="./front/introduction#Sx4.p1.t1.r29">,</mo><mi>b</mi><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">]</mo></mrow></mrow></math>, about the <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math>-axis,
</p>
<table id="E52" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.52</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E52.png" altimg-height="56px" altimg-valign="-20px" altimg-width="334px" alttext="A(S)=2\pi\int^{b}_{a}|f(x)|\sqrt{1+(f^{\prime}(x))^{2}}\mathrm{d}x," display="block"><mrow><mrow><mrow><mi href="./1.6#E48">A</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.6#SS5.p1">S</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi>a</mi><mi>b</mi></msubsup><mrow><mrow><mo stretchy="false">|</mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow><mo></mo><msqrt><mrow><mn>1</mn><mo>+</mo><msup><mrow><mo stretchy="false">(</mo><mrow><msup><mi>f</mi><mo>′</mo></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow></msqrt><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>x</mi></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E52.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m101.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m89.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m68.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.6#SS5.p1" title="§1.6(v) Surfaces and Integrals over Surfaces ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi href="./1.6#SS5.p1">S</mi></math>: parameterized surface</a> and
<a href="./1.6#E48" title="(1.6.48) ‣ §1.6(v) Surfaces and Integrals over Surfaces ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="A(S)" display="inline"><mrow><mi href="./1.6#E48">A</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.6#SS5.p1">S</mi><mo stretchy="false">)</mo></mrow></mrow></math>: area of a parameterized smooth surface <math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi href="./1.6#SS5.p1">S</mi></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and about the <math class="ltx_Math" altimg="m132.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi>y</mi></math>-axis,</p>
<table id="E53" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.53</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E53.png" altimg-height="56px" altimg-valign="-20px" altimg-width="307px" alttext="A(S)=2\pi\int^{b}_{a}|x|\sqrt{1+(f^{\prime}(x))^{2}}\mathrm{d}x." display="block"><mrow><mrow><mrow><mi href="./1.6#E48">A</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.6#SS5.p1">S</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi>a</mi><mi>b</mi></msubsup><mrow><mrow><mo stretchy="false">|</mo><mi>x</mi><mo stretchy="false">|</mo></mrow><mo></mo><msqrt><mrow><mn>1</mn><mo>+</mo><msup><mrow><mo stretchy="false">(</mo><mrow><msup><mi>f</mi><mo>′</mo></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow></msqrt><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>x</mi></mrow></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E53.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m101.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m89.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m68.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.6#SS5.p1" title="§1.6(v) Surfaces and Integrals over Surfaces ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi href="./1.6#SS5.p1">S</mi></math>: parameterized surface</a> and
<a href="./1.6#E48" title="(1.6.48) ‣ §1.6(v) Surfaces and Integrals over Surfaces ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="A(S)" display="inline"><mrow><mi href="./1.6#E48">A</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.6#SS5.p1">S</mi><mo stretchy="false">)</mo></mrow></mrow></math>: area of a parameterized smooth surface <math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi href="./1.6#SS5.p1">S</mi></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS5.p7" class="ltx_para">
<p class="ltx_p">The integral of a continuous function <math class="ltx_Math" altimg="m109.png" altimg-height="23px" altimg-valign="-7px" altimg-width="81px" alttext="f(x,y,z)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math> over a surface <math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi href="./1.6#SS5.p1">S</mi></math> is
</p>
<table id="E54" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.54</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E54.png" altimg-height="52px" altimg-valign="-20px" altimg-width="451px" alttext="\iint_{S}f(x,y,z)\mathrm{d}S=\iint_{D}f(\boldsymbol{{\Phi}}(u,v))\|\mathbf{T}_%
{u}\times\mathbf{T}_{v}\|\mathrm{d}u\mathrm{d}v." display="block"><mrow><mrow><mrow><msub><mo largeop="true" symmetric="true">∬</mo><mi href="./1.6#SS5.p1">S</mi></msub><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./1.6#SS5.p1">S</mi></mrow></mrow></mrow><mo>=</mo><mrow><msub><mo largeop="true" symmetric="true">∬</mo><mi href="./1.6#SS5.p1">D</mi></msub><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./1.6#E45" mathvariant="bold">Φ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo>∥</mo><mrow><msub><mi mathvariant="bold">T</mi><mi>u</mi></msub><mo>×</mo><msub><mi mathvariant="bold">T</mi><mi>v</mi></msub></mrow><mo>∥</mo></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>u</mi></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>v</mi></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E54.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m89.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.6#SS5.p1" title="§1.6(v) Surfaces and Integrals over Surfaces ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi href="./1.6#SS5.p1">S</mi></math>: parameterized surface</a>,
<a href="./1.6#E45" title="(1.6.45) ‣ §1.6(v) Surfaces and Integrals over Surfaces ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="23px" altimg-valign="-7px" altimg-width="86px" alttext="\boldsymbol{{\Phi}}(x,y,z)" display="inline"><mrow><mi href="./1.6#E45" mathvariant="bold">Φ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: parameterization</a> and
<a href="./1.6#SS5.p1" title="§1.6(v) Surfaces and Integrals over Surfaces ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m46.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.6#SS5.p1">D</mi></math>: open set in the plane</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">For a vector-valued function <math class="ltx_Math" altimg="m71.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\mathbf{F}" display="inline"><mi mathvariant="bold">F</mi></math>,</p>
<table id="E55" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.55</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E55.png" altimg-height="52px" altimg-valign="-20px" altimg-width="336px" alttext="\iint_{S}\mathbf{F}\cdot\mathrm{d}\mathbf{S}=\iint_{D}\mathbf{F}\cdot(\mathbf{%
T}_{u}\times\mathbf{T}_{v})\mathrm{d}u\mathrm{d}v," display="block"><mrow><mrow><mrow><msub><mo largeop="true" symmetric="true">∬</mo><mi href="./1.6#SS5.p1">S</mi></msub><mrow><mi mathvariant="bold">F</mi><mo>⋅</mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi mathvariant="bold">S</mi></mrow></mrow></mrow><mo>=</mo><mrow><msub><mo largeop="true" symmetric="true">∬</mo><mi href="./1.6#SS5.p1">D</mi></msub><mrow><mrow><mi mathvariant="bold">F</mi><mo>⋅</mo><mrow><mo stretchy="false">(</mo><mrow><msub><mi mathvariant="bold">T</mi><mi>u</mi></msub><mo>×</mo><msub><mi mathvariant="bold">T</mi><mi>v</mi></msub></mrow><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>u</mi></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>v</mi></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E55.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m89.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.6#SS5.p1" title="§1.6(v) Surfaces and Integrals over Surfaces ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi href="./1.6#SS5.p1">S</mi></math>: parameterized surface</a> and
<a href="./1.6#SS5.p1" title="§1.6(v) Surfaces and Integrals over Surfaces ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m46.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.6#SS5.p1">D</mi></math>: open set in the plane</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m90.png" altimg-height="18px" altimg-valign="-2px" altimg-width="28px" alttext="\mathrm{d}\mathbf{S}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi mathvariant="bold">S</mi></mrow></math> is the surface element with an attached normal
direction <math class="ltx_Math" altimg="m72.png" altimg-height="21px" altimg-valign="-5px" altimg-width="80px" alttext="\mathbf{T}_{u}\times\mathbf{T}_{v}" display="inline"><mrow><msub><mi mathvariant="bold">T</mi><mi>u</mi></msub><mo>×</mo><msub><mi mathvariant="bold">T</mi><mi>v</mi></msub></mrow></math>.</p>
</div>
<div id="SS5.p8" class="ltx_para">
<p class="ltx_p">A surface is <em class="ltx_emph ltx_font_italic">orientable</em>
if a continuously varying normal can be defined at all points of the surface.
An orientable surface is <em class="ltx_emph ltx_font_italic">oriented</em> if suitable normals have been chosen.
A parametrization <math class="ltx_Math" altimg="m54.png" altimg-height="23px" altimg-valign="-7px" altimg-width="67px" alttext="\boldsymbol{{\Phi}}(u,v)" display="inline"><mrow><mi href="./1.6#E45" mathvariant="bold">Φ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></mrow></math> of an oriented surface <math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi href="./1.6#SS5.p1">S</mi></math> is
<em class="ltx_emph ltx_font_italic">orientation preserving</em> if <math class="ltx_Math" altimg="m72.png" altimg-height="21px" altimg-valign="-5px" altimg-width="80px" alttext="\mathbf{T}_{u}\times\mathbf{T}_{v}" display="inline"><mrow><msub><mi mathvariant="bold">T</mi><mi>u</mi></msub><mo>×</mo><msub><mi mathvariant="bold">T</mi><mi>v</mi></msub></mrow></math> has the
same direction as the chosen normal at each point of <math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi href="./1.6#SS5.p1">S</mi></math>, otherwise it is
<em class="ltx_emph ltx_font_italic">orientation reversing</em>.</p>
</div>
<div id="SS5.p9" class="ltx_para">
<p class="ltx_p">If <math class="ltx_Math" altimg="m57.png" altimg-height="21px" altimg-valign="-5px" altimg-width="30px" alttext="\boldsymbol{{\Phi}}_{1}" display="inline"><msub><mi href="./1.6#E45" mathvariant="bold">Φ</mi><mn>1</mn></msub></math> and <math class="ltx_Math" altimg="m58.png" altimg-height="21px" altimg-valign="-5px" altimg-width="30px" alttext="\boldsymbol{{\Phi}}_{2}" display="inline"><msub><mi href="./1.6#E45" mathvariant="bold">Φ</mi><mn>2</mn></msub></math> are both orientation preserving or
both orientation reversing parametrizations of <math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi href="./1.6#SS5.p1">S</mi></math> defined on open sets <math class="ltx_Math" altimg="m47.png" altimg-height="21px" altimg-valign="-5px" altimg-width="30px" alttext="D_{1}" display="inline"><msub><mi href="./1.6#SS5.p1">D</mi><mn>1</mn></msub></math>
and <math class="ltx_Math" altimg="m48.png" altimg-height="21px" altimg-valign="-5px" altimg-width="30px" alttext="D_{2}" display="inline"><msub><mi href="./1.6#SS5.p1">D</mi><mn>2</mn></msub></math> respectively, then</p>
<table id="E56" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.56</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E56.png" altimg-height="54px" altimg-valign="-24px" altimg-width="309px" alttext="\iint_{\boldsymbol{{\Phi}}_{1}(D_{1})}\mathbf{F}\cdot\mathrm{d}\mathbf{S}=%
\iint_{\boldsymbol{{\Phi}}_{2}(D_{2})}\mathbf{F}\cdot\mathrm{d}\mathbf{S};" display="block"><mrow><mrow><mrow><msub><mo largeop="true" symmetric="true">∬</mo><mrow><msub><mi href="./1.6#E45" mathvariant="bold">Φ</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><msub><mi href="./1.6#SS5.p1">D</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mrow></mrow></msub><mrow><mi mathvariant="bold">F</mi><mo>⋅</mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi mathvariant="bold">S</mi></mrow></mrow></mrow><mo>=</mo><mrow><msub><mo largeop="true" symmetric="true">∬</mo><mrow><msub><mi href="./1.6#E45" mathvariant="bold">Φ</mi><mn>2</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><msub><mi href="./1.6#SS5.p1">D</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mrow></mrow></msub><mrow><mi mathvariant="bold">F</mi><mo>⋅</mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi mathvariant="bold">S</mi></mrow></mrow></mrow></mrow><mo>;</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E56.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m89.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.6#E45" title="(1.6.45) ‣ §1.6(v) Surfaces and Integrals over Surfaces ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="23px" altimg-valign="-7px" altimg-width="86px" alttext="\boldsymbol{{\Phi}}(x,y,z)" display="inline"><mrow><mi href="./1.6#E45" mathvariant="bold">Φ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: parameterization</a> and
<a href="./1.6#SS5.p1" title="§1.6(v) Surfaces and Integrals over Surfaces ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m46.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.6#SS5.p1">D</mi></math>: open set in the plane</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">otherwise, one is the negative of the other.</p>
</div>
<section id="Px11" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Stokes’s Theorem</h3>
<div id="Px11.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px11.p1" class="ltx_para">
<p class="ltx_p">Suppose <math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi href="./1.6#SS5.p1">S</mi></math> is an oriented surface with boundary <math class="ltx_Math" altimg="m98.png" altimg-height="18px" altimg-valign="-2px" altimg-width="29px" alttext="\partial S" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.6#SS5.p1">S</mi></mrow></math> which is oriented
so that its direction is clockwise relative to the normals of <math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi href="./1.6#SS5.p1">S</mi></math>. Then</p>
<table id="E57" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.57</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E57.png" altimg-height="52px" altimg-valign="-20px" altimg-width="267px" alttext="\iint_{S}(\nabla\times\mathbf{F})\cdot\mathrm{d}\mathbf{S}=\int_{\partial S}%
\mathbf{F}\cdot\mathrm{d}\mathbf{s}," display="block"><mrow><mrow><mrow><msub><mo largeop="true" symmetric="true">∬</mo><mi href="./1.6#SS5.p1">S</mi></msub><mrow><mrow><mo stretchy="false">(</mo><mrow><mo>∇</mo><mo>×</mo><mi mathvariant="bold">F</mi></mrow><mo stretchy="false">)</mo></mrow><mo>⋅</mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi mathvariant="bold">S</mi></mrow></mrow></mrow><mo>=</mo><mrow><msub><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.6#SS5.p1">S</mi></mrow></msub><mrow><mi mathvariant="bold">F</mi><mo>⋅</mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi mathvariant="bold">s</mi></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E57.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m89.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m68.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m99.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./1.6#SS5.p1" title="§1.6(v) Surfaces and Integrals over Surfaces ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi href="./1.6#SS5.p1">S</mi></math>: parameterized surface</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">when <math class="ltx_Math" altimg="m71.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\mathbf{F}" display="inline"><mi mathvariant="bold">F</mi></math> is a continuously differentiable vector-valued function.</p>
</div>
</section>
<section id="Px12" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Gauss’s (or Divergence) Theorem</h3>
<div id="Px12.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px12.p1" class="ltx_para">
<p class="ltx_p">Suppose <math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi href="./1.6#SS5.p1">S</mi></math> is a piecewise smooth surface which forms the complete boundary of
a bounded closed point set <math class="ltx_Math" altimg="m53.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="V" display="inline"><mi href="./1.6#Px12.p1">V</mi></math>, and <math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi href="./1.6#SS5.p1">S</mi></math> is oriented by its normal being
outwards
from <math class="ltx_Math" altimg="m53.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="V" display="inline"><mi href="./1.6#Px12.p1">V</mi></math>. Then</p>
<table id="E58" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.58</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E58.png" altimg-height="52px" altimg-valign="-20px" altimg-width="269px" alttext="\iiint_{V}(\nabla\cdot\mathbf{F})\mathrm{d}V=\iint_{S}\mathbf{F}\cdot\mathrm{d%
}\mathbf{S}," display="block"><mrow><mrow><mrow><msub><mo largeop="true" symmetric="true">∭</mo><mi href="./1.6#Px12.p1">V</mi></msub><mrow><mrow><mo stretchy="false">(</mo><mrow><mo>∇</mo><mo>⋅</mo><mi mathvariant="bold">F</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./1.6#Px12.p1">V</mi></mrow></mrow></mrow><mo>=</mo><mrow><msub><mo largeop="true" symmetric="true">∬</mo><mi href="./1.6#SS5.p1">S</mi></msub><mrow><mi mathvariant="bold">F</mi><mo>⋅</mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi mathvariant="bold">S</mi></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E58.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m89.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.6#SS5.p1" title="§1.6(v) Surfaces and Integrals over Surfaces ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi href="./1.6#SS5.p1">S</mi></math>: parameterized surface</a> and
<a href="./1.6#Px12.p1" title="Gauss’s (or Divergence) Theorem ‣ §1.6(v) Surfaces and Integrals over Surfaces ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="V" display="inline"><mi href="./1.6#Px12.p1">V</mi></math>: closed region</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">when <math class="ltx_Math" altimg="m71.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\mathbf{F}" display="inline"><mi mathvariant="bold">F</mi></math> is a continuously differentiable vector-valued function.</p>
</div>
</section>
<section id="Px13" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Green’s Theorem (for Volume)</h3>
<div id="Px13.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px13.p1" class="ltx_para">
<p class="ltx_p">For <math class="ltx_Math" altimg="m110.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> and <math class="ltx_Math" altimg="m111.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="g" display="inline"><mi>g</mi></math> twice-continuously differentiable functions
</p>
<table id="E59" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.59</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E59.png" altimg-height="51px" altimg-valign="-20px" altimg-width="380px" alttext="\iiint_{V}(f\nabla^{2}g+\nabla f\cdot\nabla g)\mathrm{d}V=\iint_{S}f\frac{%
\partial g}{\partial n}\mathrm{d}A," display="block"><mrow><mrow><mrow><msub><mo largeop="true" symmetric="true">∭</mo><mi href="./1.6#Px12.p1">V</mi></msub><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mi>f</mi><mo></mo><mrow><msup><mo>∇</mo><mn>2</mn></msup><mo></mo><mi>g</mi></mrow></mrow><mo>+</mo><mrow><mrow><mo>∇</mo><mo></mo><mi>f</mi></mrow><mo>⋅</mo><mrow><mo>∇</mo><mo></mo><mi>g</mi></mrow></mrow></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./1.6#Px12.p1">V</mi></mrow></mrow></mrow><mo>=</mo><mrow><msub><mo largeop="true" symmetric="true">∬</mo><mi href="./1.6#SS5.p1">S</mi></msub><mrow><mi>f</mi><mo></mo><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>g</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.1#p2.t1.r5">n</mi></mrow></mfrac><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./1.6#E48">A</mi></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E59.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m89.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m110.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m99.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m120.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a>,
<a href="./1.6#SS5.p1" title="§1.6(v) Surfaces and Integrals over Surfaces ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi href="./1.6#SS5.p1">S</mi></math>: parameterized surface</a>,
<a href="./1.6#E48" title="(1.6.48) ‣ §1.6(v) Surfaces and Integrals over Surfaces ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="A(S)" display="inline"><mrow><mi href="./1.6#E48">A</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.6#SS5.p1">S</mi><mo stretchy="false">)</mo></mrow></mrow></math>: area of a parameterized smooth surface <math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi href="./1.6#SS5.p1">S</mi></math></a> and
<a href="./1.6#Px12.p1" title="Gauss’s (or Divergence) Theorem ‣ §1.6(v) Surfaces and Integrals over Surfaces ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="V" display="inline"><mi href="./1.6#Px12.p1">V</mi></math>: closed region</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and</p>
<table id="E60" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.60</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E60.png" altimg-height="53px" altimg-valign="-21px" altimg-width="453px" alttext="\iiint_{V}(f\nabla^{2}g-g\nabla^{2}f)\mathrm{d}V=\iint_{S}\left(f\frac{%
\partial g}{\partial n}-g\frac{\partial f}{\partial n}\right)\mathrm{d}A," display="block"><mrow><mrow><mrow><msub><mo largeop="true" symmetric="true">∭</mo><mi href="./1.6#Px12.p1">V</mi></msub><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mi>f</mi><mo></mo><mrow><msup><mo>∇</mo><mn>2</mn></msup><mo></mo><mi>g</mi></mrow></mrow><mo>-</mo><mrow><mi>g</mi><mo></mo><mrow><msup><mo>∇</mo><mn>2</mn></msup><mo></mo><mi>f</mi></mrow></mrow></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./1.6#Px12.p1">V</mi></mrow></mrow></mrow><mo>=</mo><mrow><msub><mo largeop="true" symmetric="true">∬</mo><mi href="./1.6#SS5.p1">S</mi></msub><mrow><mrow><mo>(</mo><mrow><mrow><mi>f</mi><mo></mo><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>g</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.1#p2.t1.r5">n</mi></mrow></mfrac></mrow><mo>-</mo><mrow><mi>g</mi><mo></mo><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.1#p2.t1.r5">n</mi></mrow></mfrac></mrow></mrow><mo>)</mo></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./1.6#E48">A</mi></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E60.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m89.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m110.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m99.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m120.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a>,
<a href="./1.6#SS5.p1" title="§1.6(v) Surfaces and Integrals over Surfaces ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi href="./1.6#SS5.p1">S</mi></math>: parameterized surface</a>,
<a href="./1.6#E48" title="(1.6.48) ‣ §1.6(v) Surfaces and Integrals over Surfaces ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="A(S)" display="inline"><mrow><mi href="./1.6#E48">A</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.6#SS5.p1">S</mi><mo stretchy="false">)</mo></mrow></mrow></math>: area of a parameterized smooth surface <math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi href="./1.6#SS5.p1">S</mi></math></a> and
<a href="./1.6#Px12.p1" title="Gauss’s (or Divergence) Theorem ‣ §1.6(v) Surfaces and Integrals over Surfaces ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="V" display="inline"><mi href="./1.6#Px12.p1">V</mi></math>: closed region</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m67.png" altimg-height="23px" altimg-valign="-7px" altimg-width="145px" alttext="\ifrac{\partial g}{\partial n}=\nabla g\cdot\mathbf{n}" display="inline"><mrow><mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>g</mi></mrow><mo href="./1.5#E3">/</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.1#p2.t1.r5">n</mi></mrow></mrow><mo>=</mo><mrow><mrow><mo>∇</mo><mo></mo><mi>g</mi></mrow><mo>⋅</mo><mi mathvariant="bold">n</mi></mrow></mrow></math> is the derivative of <math class="ltx_Math" altimg="m111.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="g" display="inline"><mi>g</mi></math>
normal to the surface outwards from <math class="ltx_Math" altimg="m53.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="V" display="inline"><mi href="./1.6#Px12.p1">V</mi></math> and <math class="ltx_Math" altimg="m88.png" altimg-height="13px" altimg-valign="-2px" altimg-width="17px" alttext="\mathbf{n}" display="inline"><mi mathvariant="bold">n</mi></math> is the unit outer
normal vector.
</p>
</div>
</section>
</section>
</section>
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<title>DLMF: 1.9 Calculus of a Complex Variable</title>
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<div id="SS1.p1" class="ltx_para">
<table id="E1" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.1</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E1.png" altimg-height="22px" altimg-valign="-6px" altimg-width="101px" alttext="z=x+iy," display="block"><mrow><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>=</mo><mrow><mi>x</mi><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi>y</mi></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m204.png" altimg-height="21px" altimg-valign="-6px" altimg-width="74px" alttext="x,y\in\mathbb{R}" display="inline"><mrow><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mi href="./front/introduction#Sx4.p2.t1.r14">ℝ</mi></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./front/introduction#Sx4.p1.t1.r9" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m94.png" altimg-height="15px" altimg-valign="-3px" altimg-width="18px" alttext="\in" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo></math>: element of</a>,
<a href="./front/introduction#Sx4.p2.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m114.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\mathbb{R}" display="inline"><mi href="./front/introduction#Sx4.p2.t1.r14">ℝ</mi></math>: real line</a> and
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">3.7.1</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<section id="Px1" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Real and Imaginary Parts</h3>
<div id="Px1.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px1.p1" class="ltx_para">
<table id="E2" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex1" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">1.9.2</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m24.png" altimg-height="19px" altimg-valign="-2px" altimg-width="31px" alttext="\displaystyle\Re z" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./1.1#p2.t1.r2">z</mi></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m19.png" altimg-height="18px" altimg-valign="-6px" altimg-width="44px" alttext="\displaystyle=x," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mi>x</mi></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex2" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m23.png" altimg-height="19px" altimg-valign="-2px" altimg-width="31px" alttext="\displaystyle\Im z" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℑ</mi><mo></mo><mi href="./1.1#p2.t1.r2">z</mi></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m21.png" altimg-height="18px" altimg-valign="-6px" altimg-width="43px" alttext="\displaystyle=y." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mi>y</mi></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd>
<span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m79.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Im" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℑ</mi><mo></mo></mrow></math>: imaginary part</span> and
<span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m80.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</span>
</dd>
<dt>Symbols:</dt>
<dd><a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a></dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">3.7.5</span></span> <span class="ltx_origref"><span class="ltx_tag">3.7.6</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px2" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Polar Representation</h3>
<div id="Px2.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px2.p1" class="ltx_para">
<table id="E3" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex3" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">1.9.3</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m42.png" altimg-height="14px" altimg-valign="-2px" altimg-width="17px" alttext="\displaystyle x" display="inline"><mi>x</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m17.png" altimg-height="23px" altimg-valign="-6px" altimg-width="85px" alttext="\displaystyle=r\cos\theta," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./1.9#Px2.p1">r</mi><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi href="./1.9#Px2.p1">θ</mi></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex4" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m43.png" altimg-height="18px" altimg-valign="-6px" altimg-width="17px" alttext="\displaystyle y" display="inline"><mi>y</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m18.png" altimg-height="23px" altimg-valign="-6px" altimg-width="83px" alttext="\displaystyle=r\sin\theta," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./1.9#Px2.p1">r</mi><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi href="./1.9#Px2.p1">θ</mi></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m128.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./1.9#Px2.p1" title="Polar Representation ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m189.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="r" display="inline"><mi href="./1.9#Px2.p1">r</mi></math>: radius</a> and
<a href="./1.9#Px2.p1" title="Polar Representation ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m139.png" altimg-height="18px" altimg-valign="-2px" altimg-width="14px" alttext="\theta" display="inline"><mi href="./1.9#Px2.p1">θ</mi></math>: angle</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">3.7.2</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where
</p>
<table id="E4" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.4</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E4.png" altimg-height="29px" altimg-valign="-7px" altimg-width="152px" alttext="r=(x^{2}+y^{2})^{1/2}," display="block"><mrow><mrow><mi href="./1.9#Px2.p1">r</mi><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E4.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./1.9#Px2.p1" title="Polar Representation ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m189.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="r" display="inline"><mi href="./1.9#Px2.p1">r</mi></math>: radius</a></dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">3.7.3</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and when <math class="ltx_Math" altimg="m220.png" altimg-height="21px" altimg-valign="-6px" altimg-width="51px" alttext="z\neq 0" display="inline"><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>≠</mo><mn>0</mn></mrow></math>,
</p>
<table id="E5" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.5</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E5.png" altimg-height="23px" altimg-valign="-6px" altimg-width="293px" alttext="\theta=\omega,\;\;\pi-\omega,\;\;-\pi+\omega,\mbox{ or }-\omega," display="block"><mrow><mrow><mi href="./1.9#Px2.p1">θ</mi><mo>=</mo><mrow><mi>ω</mi><mo rspace="8.1pt">,</mo><mrow><mi href="./3.12#E1">π</mi><mo>-</mo><mi>ω</mi></mrow><mo rspace="8.1pt">,</mo><mrow><mrow><mo>-</mo><mi href="./3.12#E1">π</mi></mrow><mo>+</mo><mi>ω</mi></mrow><mo>,</mo><mrow><mtext> or </mtext><mo>-</mo><mi>ω</mi></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E5.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a> and
<a href="./1.9#Px2.p1" title="Polar Representation ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m139.png" altimg-height="18px" altimg-valign="-2px" altimg-width="14px" alttext="\theta" display="inline"><mi href="./1.9#Px2.p1">θ</mi></math>: angle</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">according as <math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math> lies in the 1st, 2nd, 3rd, or 4th quadrants. Here
</p>
<table id="E6" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.6</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E6.png" altimg-height="29px" altimg-valign="-9px" altimg-width="256px" alttext="\omega=\operatorname{arctan}\left(|y/x|\right)\in\left[0,\tfrac{1}{2}\pi\right]." display="block"><mrow><mrow><mi>ω</mi><mo>=</mo><mrow><mi href="./4.23#SS2.p1">arctan</mi><mo></mo><mrow><mo>(</mo><mrow><mo stretchy="false">|</mo><mrow><mi>y</mi><mo>/</mo><mi>x</mi></mrow><mo stretchy="false">|</mo></mrow><mo>)</mo></mrow></mrow><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r29">[</mo><mn>0</mn><mo href="./front/introduction#Sx4.p1.t1.r29">,</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo href="./front/introduction#Sx4.p1.t1.r29">]</mo></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E6.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./front/introduction#Sx4.p1.t1.r29" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="[\NVar{a},\NVar{b}]" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">[</mo><mi class="ltx_nvar">a</mi><mo href="./front/introduction#Sx4.p1.t1.r29">,</mo><mi class="ltx_nvar">b</mi><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">]</mo></mrow></math>: closed interval</a>,
<a href="./front/introduction#Sx4.p1.t1.r9" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m94.png" altimg-height="15px" altimg-valign="-3px" altimg-width="18px" alttext="\in" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo></math>: element of</a> and
<a href="./4.23#SS2.p1" title="§4.23(ii) Principal Values ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m122.png" altimg-height="17px" altimg-valign="-2px" altimg-width="73px" alttext="\operatorname{arctan}\NVar{z}" display="inline"><mrow><mi href="./4.23#SS2.p1">arctan</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: arctangent function</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">3.7.4</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px3" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Modulus and Phase</h3>
<div id="Px3.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px3.p1" class="ltx_para">
<table id="E7" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex5" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="3" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">1.9.7</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m47.png" altimg-height="25px" altimg-valign="-7px" altimg-width="27px" alttext="\displaystyle|z|" display="inline"><mrow><mo stretchy="false">|</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">|</mo></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m16.png" altimg-height="18px" altimg-valign="-6px" altimg-width="41px" alttext="\displaystyle=r," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mi href="./1.9#Px2.p1">r</mi></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex6" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m29.png" altimg-height="23px" altimg-valign="-6px" altimg-width="42px" alttext="\displaystyle\operatorname{ph}z" display="inline"><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mi href="./1.1#p2.t1.r2">z</mi></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m13.png" altimg-height="23px" altimg-valign="-6px" altimg-width="101px" alttext="\displaystyle=\theta+2n\pi," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./1.9#Px2.p1">θ</mi><mo>+</mo><mrow><mn>2</mn><mo></mo><mi href="./1.1#p2.t1.r5">n</mi><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m186.png" altimg-height="18px" altimg-valign="-3px" altimg-width="54px" alttext="n\in\mathbb{Z}" display="inline"><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mi href="./front/introduction#Sx4.p2.t1.r20">ℤ</mi></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E7.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m123.png" altimg-height="21px" altimg-valign="-6px" altimg-width="26px" alttext="\operatorname{ph}" display="inline"><mi href="./1.9#E7">ph</mi></math>: phase</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./front/introduction#Sx4.p1.t1.r9" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m94.png" altimg-height="15px" altimg-valign="-3px" altimg-width="18px" alttext="\in" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo></math>: element of</a>,
<a href="./front/introduction#Sx4.p2.t1.r20" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m115.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="\mathbb{Z}" display="inline"><mi href="./front/introduction#Sx4.p2.t1.r20">ℤ</mi></math>: set of all integers</a>,
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>,
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m181.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a>,
<a href="./1.9#Px2.p1" title="Polar Representation ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m189.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="r" display="inline"><mi href="./1.9#Px2.p1">r</mi></math>: radius</a> and
<a href="./1.9#Px2.p1" title="Polar Representation ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m139.png" altimg-height="18px" altimg-valign="-2px" altimg-width="14px" alttext="\theta" display="inline"><mi href="./1.9#Px2.p1">θ</mi></math>: angle</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">The <em class="ltx_emph ltx_font_italic">principal value</em>
of <math class="ltx_Math" altimg="m124.png" altimg-height="21px" altimg-valign="-6px" altimg-width="40px" alttext="\operatorname{ph}z" display="inline"><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mi href="./1.1#p2.t1.r2">z</mi></mrow></math> corresponds to <math class="ltx_Math" altimg="m178.png" altimg-height="17px" altimg-valign="-2px" altimg-width="53px" alttext="n=0" display="inline"><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>0</mn></mrow></math>, that is, <math class="ltx_Math" altimg="m55.png" altimg-height="21px" altimg-valign="-6px" altimg-width="133px" alttext="-\pi\leq\operatorname{ph}z\leq\pi" display="inline"><mrow><mrow><mo>-</mo><mi href="./3.12#E1">π</mi></mrow><mo>≤</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mi href="./1.1#p2.t1.r2">z</mi></mrow><mo>≤</mo><mi href="./3.12#E1">π</mi></mrow></math>. It
is single-valued on <math class="ltx_Math" altimg="m113.png" altimg-height="23px" altimg-valign="-7px" altimg-width="67px" alttext="\mathbb{C}\setminus\{0\}" display="inline"><mrow><mi href="./front/introduction#Sx4.p1.t1.r1">ℂ</mi><mo href="./front/introduction#Sx4.p2.t1.r19">∖</mo><mrow><mo stretchy="false">{</mo><mn>0</mn><mo stretchy="false">}</mo></mrow></mrow></math>, except on the interval
<math class="ltx_Math" altimg="m49.png" altimg-height="23px" altimg-valign="-7px" altimg-width="74px" alttext="(-\infty,0)" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mn>0</mn><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math> where it is discontinuous and two-valued. <em class="ltx_emph ltx_font_italic">Unless indicated
otherwise</em>, these principal values are assumed throughout
the DLMF. (However, if we
require a principal value to be single-valued, then we can restrict
<math class="ltx_Math" altimg="m54.png" altimg-height="21px" altimg-valign="-6px" altimg-width="133px" alttext="-\pi<\operatorname{ph}z\leq\pi" display="inline"><mrow><mrow><mo>-</mo><mi href="./3.12#E1">π</mi></mrow><mo><</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mi href="./1.1#p2.t1.r2">z</mi></mrow><mo>≤</mo><mi href="./3.12#E1">π</mi></mrow></math>.)
</p>
<table id="E8" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex7" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">1.9.8</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m45.png" altimg-height="25px" altimg-valign="-7px" altimg-width="42px" alttext="\displaystyle|\Re z|" display="inline"><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./1.1#p2.t1.r2">z</mi></mrow><mo stretchy="false">|</mo></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m27.png" altimg-height="25px" altimg-valign="-7px" altimg-width="54px" alttext="\displaystyle\leq|z|," display="inline"><mrow><mrow><mi></mi><mo>≤</mo><mrow><mo stretchy="false">|</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">|</mo></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex8" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m44.png" altimg-height="25px" altimg-valign="-7px" altimg-width="42px" alttext="\displaystyle|\Im z|" display="inline"><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℑ</mi><mo></mo><mi href="./1.1#p2.t1.r2">z</mi></mrow><mo stretchy="false">|</mo></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m27.png" altimg-height="25px" altimg-valign="-7px" altimg-width="54px" alttext="\displaystyle\leq|z|," display="inline"><mrow><mrow><mi></mi><mo>≤</mo><mrow><mo stretchy="false">|</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">|</mo></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E8.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m79.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Im" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℑ</mi><mo></mo></mrow></math>: imaginary part</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m80.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a> and
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E9" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.9</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E9.png" altimg-height="27px" altimg-valign="-6px" altimg-width="82px" alttext="z=re^{i\theta}," display="block"><mrow><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>=</mo><mrow><mi href="./1.9#Px2.p1">r</mi><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./1.9#Px2.p1">θ</mi></mrow></msup></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E9.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m119.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>,
<a href="./1.9#Px2.p1" title="Polar Representation ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m189.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="r" display="inline"><mi href="./1.9#Px2.p1">r</mi></math>: radius</a> and
<a href="./1.9#Px2.p1" title="Polar Representation ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m139.png" altimg-height="18px" altimg-valign="-2px" altimg-width="14px" alttext="\theta" display="inline"><mi href="./1.9#Px2.p1">θ</mi></math>: angle</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where
</p>
<table id="E10" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.10</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E10.png" altimg-height="27px" altimg-valign="-6px" altimg-width="174px" alttext="e^{i\theta}=\cos\theta+i\sin\theta;" display="block"><mrow><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./1.9#Px2.p1">θ</mi></mrow></msup><mo>=</mo><mrow><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi href="./1.9#Px2.p1">θ</mi></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi href="./1.9#Px2.p1">θ</mi></mrow></mrow></mrow></mrow><mo>;</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E10.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m119.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m128.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a> and
<a href="./1.9#Px2.p1" title="Polar Representation ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m139.png" altimg-height="18px" altimg-valign="-2px" altimg-width="14px" alttext="\theta" display="inline"><mi href="./1.9#Px2.p1">θ</mi></math>: angle</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">see §</dd>
</dl>
</div>
</div>
<div id="Px4.p1" class="ltx_para">
<table id="EGx1" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E11">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.11</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m30.png" altimg-height="18px" altimg-valign="-2px" altimg-width="16px" alttext="\displaystyle\overline{z}" display="inline"><mover accent="true"><mi href="./1.1#p2.t1.r2">z</mi><mo href="./1.9#E11">¯</mo></mover></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m20.png" altimg-height="22px" altimg-valign="-6px" altimg-width="86px" alttext="\displaystyle=x-iy," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi>x</mi><mo>-</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi>y</mi></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E11.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m125.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="\overline{\NVar{z}}" display="inline"><mover accent="true"><mi class="ltx_nvar" href="./1.1#p2.t1.r2">z</mi><mo href="./1.9#E11">¯</mo></mover></math>: complex conjugate</span></dd>
<dt>Symbols:</dt>
<dd><a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a></dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">3.7.7</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E12">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.12</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m46.png" altimg-height="25px" altimg-valign="-7px" altimg-width="27px" alttext="\displaystyle|\overline{z}|" display="inline"><mrow><mo stretchy="false">|</mo><mover accent="true"><mi href="./1.1#p2.t1.r2">z</mi><mo href="./1.9#E11">¯</mo></mover><mo stretchy="false">|</mo></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m22.png" altimg-height="25px" altimg-valign="-7px" altimg-width="54px" alttext="\displaystyle=|z|," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo stretchy="false">|</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">|</mo></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E12.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.9#E11" title="(1.9.11) ‣ Complex Conjugate ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m125.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="\overline{\NVar{z}}" display="inline"><mover accent="true"><mi class="ltx_nvar" href="./1.1#p2.t1.r2">z</mi><mo href="./1.9#E11">¯</mo></mover></math>: complex conjugate</a> and
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">3.7.8</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E13">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.13</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m28.png" altimg-height="23px" altimg-valign="-6px" altimg-width="42px" alttext="\displaystyle\operatorname{ph}\overline{z}" display="inline"><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mover accent="true"><mi href="./1.1#p2.t1.r2">z</mi><mo href="./1.9#E11">¯</mo></mover></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m5.png" altimg-height="23px" altimg-valign="-6px" altimg-width="87px" alttext="\displaystyle=-\operatorname{ph}z." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo>-</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mi href="./1.1#p2.t1.r2">z</mi></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E13.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.9#E11" title="(1.9.11) ‣ Complex Conjugate ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m125.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="\overline{\NVar{z}}" display="inline"><mover accent="true"><mi class="ltx_nvar" href="./1.1#p2.t1.r2">z</mi><mo href="./1.9#E11">¯</mo></mover></math>: complex conjugate</a>,
<a href="./1.9#E7" title="(1.9.7) ‣ Modulus and Phase ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m123.png" altimg-height="21px" altimg-valign="-6px" altimg-width="26px" alttext="\operatorname{ph}" display="inline"><mi href="./1.9#E7">ph</mi></math>: phase</a> and
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">3.7.9</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
</div>
</section>
<section id="Px5" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Arithmetic Operations</h3>
<div id="Px5.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px5.p1" class="ltx_para">
<p class="ltx_p">If <math class="ltx_Math" altimg="m227.png" altimg-height="20px" altimg-valign="-6px" altimg-width="119px" alttext="z_{1}=x_{1}+iy_{1}" display="inline"><mrow><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>1</mn></msub><mo>=</mo><mrow><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><msub><mi>y</mi><mn>1</mn></msub></mrow></mrow></mrow></math>, <math class="ltx_Math" altimg="m228.png" altimg-height="20px" altimg-valign="-6px" altimg-width="119px" alttext="z_{2}=x_{2}+iy_{2}" display="inline"><mrow><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>2</mn></msub><mo>=</mo><mrow><msub><mi>x</mi><mn>2</mn></msub><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><msub><mi>y</mi><mn>2</mn></msub></mrow></mrow></mrow></math>, then</p>
<table id="E14" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.14</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E14.png" altimg-height="25px" altimg-valign="-7px" altimg-width="271px" alttext="z_{1}\pm z_{2}=x_{1}\pm x_{2}+\mathrm{i}(y_{1}\pm y_{2})," display="block"><mrow><mrow><mrow><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>1</mn></msub><mo>±</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>2</mn></msub></mrow><mo>=</mo><mrow><mrow><msub><mi>x</mi><mn>1</mn></msub><mo>±</mo><msub><mi>x</mi><mn>2</mn></msub></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><msub><mi>y</mi><mn>1</mn></msub><mo>±</mo><msub><mi>y</mi><mn>2</mn></msub></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E14.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a></dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E15" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.15</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E15.png" altimg-height="25px" altimg-valign="-7px" altimg-width="326px" alttext="z_{1}z_{2}=x_{1}x_{2}-y_{1}y_{2}+i(x_{1}y_{2}+x_{2}y_{1})," display="block"><mrow><mrow><mrow><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>1</mn></msub><mo></mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>2</mn></msub></mrow><mo>=</mo><mrow><mrow><mrow><msub><mi>x</mi><mn>1</mn></msub><mo></mo><msub><mi>x</mi><mn>2</mn></msub></mrow><mo>-</mo><mrow><msub><mi>y</mi><mn>1</mn></msub><mo></mo><msub><mi>y</mi><mn>2</mn></msub></mrow></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><msub><mi>x</mi><mn>1</mn></msub><mo></mo><msub><mi>y</mi><mn>2</mn></msub></mrow><mo>+</mo><mrow><msub><mi>x</mi><mn>2</mn></msub><mo></mo><msub><mi>y</mi><mn>1</mn></msub></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E15.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a></dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">3.7.10</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E16" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.16</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E16.png" altimg-height="53px" altimg-valign="-21px" altimg-width="387px" alttext="\frac{z_{1}}{z_{2}}=\frac{z_{1}\overline{z}_{2}}{|z_{2}|^{2}}=\frac{x_{1}x_{2}%
+y_{1}y_{2}+i(x_{2}y_{1}-x_{1}y_{2})}{x_{2}^{2}+y_{2}^{2}}," display="block"><mrow><mrow><mfrac><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>1</mn></msub><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>2</mn></msub></mfrac><mo>=</mo><mfrac><mrow><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>1</mn></msub><mo></mo><msub><mover accent="true"><mi href="./1.1#p2.t1.r2">z</mi><mo href="./1.9#E11">¯</mo></mover><mn>2</mn></msub></mrow><msup><mrow><mo stretchy="false">|</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>2</mn></msub><mo stretchy="false">|</mo></mrow><mn>2</mn></msup></mfrac><mo>=</mo><mfrac><mrow><mrow><msub><mi>x</mi><mn>1</mn></msub><mo></mo><msub><mi>x</mi><mn>2</mn></msub></mrow><mo>+</mo><mrow><msub><mi>y</mi><mn>1</mn></msub><mo></mo><msub><mi>y</mi><mn>2</mn></msub></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><msub><mi>x</mi><mn>2</mn></msub><mo></mo><msub><mi>y</mi><mn>1</mn></msub></mrow><mo>-</mo><mrow><msub><mi>x</mi><mn>1</mn></msub><mo></mo><msub><mi>y</mi><mn>2</mn></msub></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mrow><msubsup><mi>x</mi><mn>2</mn><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>y</mi><mn>2</mn><mn>2</mn></msubsup></mrow></mfrac></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E16.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.9#E11" title="(1.9.11) ‣ Complex Conjugate ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m125.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="\overline{\NVar{z}}" display="inline"><mover accent="true"><mi class="ltx_nvar" href="./1.1#p2.t1.r2">z</mi><mo href="./1.9#E11">¯</mo></mover></math>: complex conjugate</a> and
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">3.7.13</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">provided that <math class="ltx_Math" altimg="m229.png" altimg-height="21px" altimg-valign="-6px" altimg-width="59px" alttext="z_{2}\neq 0" display="inline"><mrow><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>2</mn></msub><mo>≠</mo><mn>0</mn></mrow></math>. Also,</p>
<table id="E17" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.17</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E17.png" altimg-height="25px" altimg-valign="-7px" altimg-width="150px" alttext="|z_{1}z_{2}|=|z_{1}|\;|z_{2}|," display="block"><mrow><mrow><mrow><mo stretchy="false">|</mo><mrow><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>1</mn></msub><mo></mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>2</mn></msub></mrow><mo stretchy="false">|</mo></mrow><mo>=</mo><mrow><mrow><mo stretchy="false">|</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>1</mn></msub><mo rspace="5.3pt" stretchy="false">|</mo></mrow><mo></mo><mrow><mo stretchy="false">|</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>2</mn></msub><mo stretchy="false">|</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E17.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a></dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">3.7.11</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E18" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.18</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E18.png" altimg-height="25px" altimg-valign="-7px" altimg-width="227px" alttext="\operatorname{ph}\left(z_{1}z_{2}\right)=\operatorname{ph}z_{1}+\operatorname{%
ph}z_{2}," display="block"><mrow><mrow><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mrow><mo>(</mo><mrow><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>1</mn></msub><mo></mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>2</mn></msub></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mi href="./1.9#E7">ph</mi><mo></mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>1</mn></msub></mrow><mo>+</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>2</mn></msub></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E18.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.9#E7" title="(1.9.7) ‣ Modulus and Phase ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m123.png" altimg-height="21px" altimg-valign="-6px" altimg-width="26px" alttext="\operatorname{ph}" display="inline"><mi href="./1.9#E7">ph</mi></math>: phase</a> and
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">3.7.12</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E19" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.19</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E19.png" altimg-height="53px" altimg-valign="-21px" altimg-width="108px" alttext="\left|\frac{z_{1}}{z_{2}}\right|=\frac{|z_{1}|}{|z_{2}|}," display="block"><mrow><mrow><mrow><mo>|</mo><mfrac><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>1</mn></msub><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>2</mn></msub></mfrac><mo>|</mo></mrow><mo>=</mo><mfrac><mrow><mo stretchy="false">|</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>1</mn></msub><mo stretchy="false">|</mo></mrow><mrow><mo stretchy="false">|</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>2</mn></msub><mo stretchy="false">|</mo></mrow></mfrac></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E19.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a></dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">3.7.14</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E20" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.20</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E20.png" altimg-height="44px" altimg-valign="-19px" altimg-width="198px" alttext="\operatorname{ph}\frac{z_{1}}{z_{2}}=\operatorname{ph}z_{1}-\operatorname{ph}z%
_{2}." display="block"><mrow><mrow><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mfrac><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>1</mn></msub><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>2</mn></msub></mfrac></mrow><mo>=</mo><mrow><mrow><mi href="./1.9#E7">ph</mi><mo></mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>1</mn></msub></mrow><mo>-</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>2</mn></msub></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E20.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.9#E7" title="(1.9.7) ‣ Modulus and Phase ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m123.png" altimg-height="21px" altimg-valign="-6px" altimg-width="26px" alttext="\operatorname{ph}" display="inline"><mi href="./1.9#E7">ph</mi></math>: phase</a> and
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">3.7.15</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Equations (</dd>
</dl>
</div>
</div>
<div id="Px6.p1" class="ltx_para">
<table id="E21" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.21</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E21.png" altimg-height="53px" altimg-valign="-21px" altimg-width="751px" alttext="z^{n}=\left(x^{n}-\genfrac{(}{)}{0.0pt}{}{n}{2}x^{n-2}y^{2}+\genfrac{(}{)}{0.0%
pt}{}{n}{4}x^{n-4}y^{4}-\cdots\right)+i\left(\genfrac{(}{)}{0.0pt}{}{n}{1}x^{n%
-1}y-\genfrac{(}{)}{0.0pt}{}{n}{3}x^{n-3}y^{3}+\cdots\right)," display="block"><mrow><mrow><msup><mi href="./1.1#p2.t1.r2">z</mi><mi href="./1.1#p2.t1.r5">n</mi></msup><mo>=</mo><mrow><mrow><mo>(</mo><mrow><mrow><mrow><msup><mi>x</mi><mi href="./1.1#p2.t1.r5">n</mi></msup><mo>-</mo><mrow><mrow><mo href="./1.2#i">(</mo><mfrac linethickness="0.0pt"><mi href="./1.1#p2.t1.r5">n</mi><mn>2</mn></mfrac><mo href="./1.2#i">)</mo></mrow><mo></mo><msup><mi>x</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>2</mn></mrow></msup><mo></mo><msup><mi>y</mi><mn>2</mn></msup></mrow></mrow><mo>+</mo><mrow><mrow><mo href="./1.2#i">(</mo><mfrac linethickness="0.0pt"><mi href="./1.1#p2.t1.r5">n</mi><mn>4</mn></mfrac><mo href="./1.2#i">)</mo></mrow><mo></mo><msup><mi>x</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>4</mn></mrow></msup><mo></mo><msup><mi>y</mi><mn>4</mn></msup></mrow></mrow><mo>-</mo><mi mathvariant="normal">⋯</mi></mrow><mo>)</mo></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mrow><mrow><mo href="./1.2#i">(</mo><mfrac linethickness="0.0pt"><mi href="./1.1#p2.t1.r5">n</mi><mn>1</mn></mfrac><mo href="./1.2#i">)</mo></mrow><mo></mo><msup><mi>x</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mi>y</mi></mrow><mo>-</mo><mrow><mrow><mo href="./1.2#i">(</mo><mfrac linethickness="0.0pt"><mi href="./1.1#p2.t1.r5">n</mi><mn>3</mn></mfrac><mo href="./1.2#i">)</mo></mrow><mo></mo><msup><mi>x</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>3</mn></mrow></msup><mo></mo><msup><mi>y</mi><mn>3</mn></msup></mrow></mrow><mo>+</mo><mi mathvariant="normal">⋯</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m180.png" altimg-height="20px" altimg-valign="-6px" altimg-width="104px" alttext="n=1,2,\dots" display="inline"><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E21.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.2#i" title="§1.2(i) Binomial Coefficients ‣ §1.2 Elementary Algebra ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m89.png" altimg-height="27px" altimg-valign="-9px" altimg-width="37px" alttext="\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}" display="inline"><mrow><mo href="./1.2#i">(</mo><mfrac linethickness="0.0pt"><mi class="ltx_nvar" href="./1.1#p2.t1.r5">m</mi><mi class="ltx_nvar" href="./1.1#p2.t1.r5">n</mi></mfrac><mo href="./1.2#i">)</mo></mrow></math>: binomial coefficient</a>,
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a> and
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m181.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">3.7.22</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px7" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">DeMoivre’s Theorem</h3>
<div id="Px7.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px7.p1" class="ltx_para">
<table id="E22" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.22</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E22.png" altimg-height="25px" altimg-valign="-7px" altimg-width="313px" alttext="\cos n\theta+i\sin n\theta=(\cos\theta+i\sin\theta)^{n}," display="block"><mrow><mrow><mrow><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo></mo><mi href="./1.9#Px2.p1">θ</mi></mrow></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo></mo><mi href="./1.9#Px2.p1">θ</mi></mrow></mrow></mrow></mrow><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mrow><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi href="./1.9#Px2.p1">θ</mi></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi href="./1.9#Px2.p1">θ</mi></mrow></mrow></mrow><mo stretchy="false">)</mo></mrow><mi href="./1.1#p2.t1.r5">n</mi></msup></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m186.png" altimg-height="18px" altimg-valign="-3px" altimg-width="54px" alttext="n\in\mathbb{Z}" display="inline"><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mi href="./front/introduction#Sx4.p2.t1.r20">ℤ</mi></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E22.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./front/introduction#Sx4.p1.t1.r9" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m94.png" altimg-height="15px" altimg-valign="-3px" altimg-width="18px" alttext="\in" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo></math>: element of</a>,
<a href="./front/introduction#Sx4.p2.t1.r20" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m115.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="\mathbb{Z}" display="inline"><mi href="./front/introduction#Sx4.p2.t1.r20">ℤ</mi></math>: set of all integers</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m128.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m181.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./1.9#Px2.p1" title="Polar Representation ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m139.png" altimg-height="18px" altimg-valign="-2px" altimg-width="14px" alttext="\theta" display="inline"><mi href="./1.9#Px2.p1">θ</mi></math>: angle</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px8" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Triangle Inequality</h3>
<div id="Px8.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px8.p1" class="ltx_para">
<table id="E23" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.23</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E23.png" altimg-height="25px" altimg-valign="-7px" altimg-width="317px" alttext="\left|\left|z_{1}\right|-\left|z_{2}\right|\right|\leq\left|z_{1}+z_{2}\right|%
\leq\left|z_{1}\right|+\left|z_{2}\right|." display="block"><mrow><mrow><mrow><mo>|</mo><mrow><mrow><mo>|</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>1</mn></msub><mo>|</mo></mrow><mo>-</mo><mrow><mo>|</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>2</mn></msub><mo>|</mo></mrow></mrow><mo>|</mo></mrow><mo>≤</mo><mrow><mo>|</mo><mrow><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>1</mn></msub><mo>+</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>2</mn></msub></mrow><mo>|</mo></mrow><mo>≤</mo><mrow><mrow><mo>|</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>1</mn></msub><mo>|</mo></mrow><mo>+</mo><mrow><mo>|</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>2</mn></msub><mo>|</mo></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E23.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a></dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">3.7.29</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
</section>
<section id="ii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§1.9(ii) </span>Continuity, Point Sets, and Differentiation</h2>
<div id="SS2.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px9.p1" class="ltx_para">
<p class="ltx_p">A function <math class="ltx_Math" altimg="m161.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(z)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> is <em class="ltx_emph ltx_font_italic">continuous</em> at a point <math class="ltx_Math" altimg="m224.png" altimg-height="16px" altimg-valign="-5px" altimg-width="23px" alttext="z_{0}" display="inline"><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></math> if
<math class="ltx_Math" altimg="m106.png" altimg-height="34px" altimg-valign="-18px" altimg-width="157px" alttext="\lim\limits_{z\to z_{0}}f(z)=f(z_{0})" display="inline"><mrow><mrow><munder><mo movablelimits="false">lim</mo><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>→</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></mrow></munder><mo></mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>=</mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mrow></mrow></math>. That is, given any positive number
<math class="ltx_Math" altimg="m86.png" altimg-height="13px" altimg-valign="-2px" altimg-width="12px" alttext="\epsilon" display="inline"><mi>ϵ</mi></math>, however small, we can find a positive number <math class="ltx_Math" altimg="m83.png" altimg-height="18px" altimg-valign="-2px" altimg-width="14px" alttext="\delta" display="inline"><mi href="./1.9#Px9.p1">δ</mi></math> such that
<math class="ltx_Math" altimg="m234.png" altimg-height="23px" altimg-valign="-7px" altimg-width="158px" alttext="|f(z)-f(z_{0})|<\epsilon" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>-</mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mrow></mrow><mo stretchy="false">|</mo></mrow><mo><</mo><mi>ϵ</mi></mrow></math> for all <math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math> in the open disk <math class="ltx_Math" altimg="m238.png" altimg-height="23px" altimg-valign="-7px" altimg-width="104px" alttext="|z-z_{0}|<\delta" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>-</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></mrow><mo stretchy="false">|</mo></mrow><mo><</mo><mi href="./1.9#Px9.p1">δ</mi></mrow></math>.</p>
</div>
<div id="Px9.p2" class="ltx_para">
<p class="ltx_p">A function of two complex variables <math class="ltx_Math" altimg="m162.png" altimg-height="23px" altimg-valign="-7px" altimg-width="65px" alttext="f(z,w)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo>,</mo><mi href="./1.1#p2.t1.r3">w</mi><mo stretchy="false">)</mo></mrow></mrow></math> is <em class="ltx_emph ltx_font_italic">continuous</em> at
<math class="ltx_Math" altimg="m53.png" altimg-height="23px" altimg-valign="-7px" altimg-width="70px" alttext="(z_{0},w_{0})" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><msub><mi href="./1.1#p2.t1.r3">w</mi><mn>0</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math> if <math class="ltx_Math" altimg="m101.png" altimg-height="38px" altimg-valign="-21px" altimg-width="278px" alttext="\lim\limits_{(z,w)\to(z_{0},w_{0})}f(z,w)=f(z_{0},w_{0})" display="inline"><mrow><mrow><munder><mo movablelimits="false">lim</mo><mrow><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi href="./1.1#p2.t1.r3">w</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow><mo>→</mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><msub><mi href="./1.1#p2.t1.r3">w</mi><mn>0</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></mrow></munder><mo></mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo>,</mo><mi href="./1.1#p2.t1.r3">w</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>=</mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub><mo>,</mo><msub><mi href="./1.1#p2.t1.r3">w</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mrow></mrow></math>; compare
().</p>
</div>
</section>
<section id="Px10" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Point Sets in <math class="ltx_Math" altimg="m112.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\mathbb{C}" display="inline"><mi href="./front/introduction#Sx4.p1.t1.r1">ℂ</mi></math>
</h3>
<div id="Px10.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px10.p1" class="ltx_para">
<p class="ltx_p">A <em class="ltx_emph ltx_font_italic">neighborhood of a point</em> <math class="ltx_Math" altimg="m224.png" altimg-height="16px" altimg-valign="-5px" altimg-width="23px" alttext="z_{0}" display="inline"><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></math> is a disk <math class="ltx_Math" altimg="m100.png" altimg-height="23px" altimg-valign="-7px" altimg-width="104px" alttext="\left|z-z_{0}\right|<\delta" display="inline"><mrow><mrow><mo>|</mo><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>-</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></mrow><mo>|</mo></mrow><mo><</mo><mi href="./1.9#Px9.p1">δ</mi></mrow></math>. An
<em class="ltx_emph ltx_font_italic">open set</em> in <math class="ltx_Math" altimg="m112.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\mathbb{C}" display="inline"><mi href="./front/introduction#Sx4.p1.t1.r1">ℂ</mi></math> is one in which each point has a neighborhood
that is contained in the set.</p>
</div>
<div id="Px10.p2" class="ltx_para">
<p class="ltx_p">A point <math class="ltx_Math" altimg="m224.png" altimg-height="16px" altimg-valign="-5px" altimg-width="23px" alttext="z_{0}" display="inline"><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></math> is a <em class="ltx_emph ltx_font_italic">limit point</em> (<em class="ltx_emph ltx_font_italic">limiting point</em> or
<em class="ltx_emph ltx_font_italic">accumulation point</em>) of a set of points <math class="ltx_Math" altimg="m76.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi>S</mi></math> in <math class="ltx_Math" altimg="m112.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\mathbb{C}" display="inline"><mi href="./front/introduction#Sx4.p1.t1.r1">ℂ</mi></math> (or
<math class="ltx_Math" altimg="m111.png" altimg-height="18px" altimg-valign="-2px" altimg-width="61px" alttext="\mathbb{C}\cup\infty" display="inline"><mrow><mi href="./front/introduction#Sx4.p1.t1.r1">ℂ</mi><mo href="./front/introduction#Sx4.p1.t1.r27">∪</mo><mi mathvariant="normal">∞</mi></mrow></math>) if every neighborhood of <math class="ltx_Math" altimg="m224.png" altimg-height="16px" altimg-valign="-5px" altimg-width="23px" alttext="z_{0}" display="inline"><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></math> contains a point of <math class="ltx_Math" altimg="m76.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi>S</mi></math>
distinct from <math class="ltx_Math" altimg="m224.png" altimg-height="16px" altimg-valign="-5px" altimg-width="23px" alttext="z_{0}" display="inline"><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></math>. (<math class="ltx_Math" altimg="m224.png" altimg-height="16px" altimg-valign="-5px" altimg-width="23px" alttext="z_{0}" display="inline"><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></math> may or may not belong to <math class="ltx_Math" altimg="m76.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi>S</mi></math>.) As a consequence,
every neighborhood of a limit point of <math class="ltx_Math" altimg="m76.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi>S</mi></math> contains an infinite number of
points of <math class="ltx_Math" altimg="m76.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi>S</mi></math>.
Also, the union of <math class="ltx_Math" altimg="m76.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi>S</mi></math> and its limit points is the <em class="ltx_emph ltx_font_italic">closure</em> of <math class="ltx_Math" altimg="m76.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi>S</mi></math>.</p>
</div>
<div id="Px10.p3" class="ltx_para">
<p class="ltx_p">A <em class="ltx_emph ltx_font_italic">domain</em>
<math class="ltx_Math" altimg="m71.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi>D</mi></math>, say, is an open set in <math class="ltx_Math" altimg="m112.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\mathbb{C}" display="inline"><mi href="./front/introduction#Sx4.p1.t1.r1">ℂ</mi></math> that is <em class="ltx_emph ltx_font_italic">connected</em>,
that is, any two points can be joined by a polygonal arc (a finite chain of
straight-line segments) lying in the set. Any point whose neighborhoods always
contain members and nonmembers of <math class="ltx_Math" altimg="m71.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi>D</mi></math> is a <em class="ltx_emph ltx_font_italic">boundary point</em>
of <math class="ltx_Math" altimg="m71.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi>D</mi></math>. When its boundary points are added the domain is said to be
<em class="ltx_emph ltx_font_italic">closed</em>,
but unless specified otherwise a domain is assumed to be open.
</p>
</div>
<div id="Px10.p4" class="ltx_para">
<p class="ltx_p">A <em class="ltx_emph ltx_font_italic">region</em> is an open domain together with none, some, or all of its
boundary points. Points of a region that are not boundary points are called
<em class="ltx_emph ltx_font_italic">interior points</em>.
</p>
</div>
<div id="Px10.p5" class="ltx_para">
<p class="ltx_p">A function <math class="ltx_Math" altimg="m161.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(z)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> is <em class="ltx_emph ltx_font_italic">continuous on a region</em> <math class="ltx_Math" altimg="m75.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="R" display="inline"><mi>R</mi></math> if for each point <math class="ltx_Math" altimg="m224.png" altimg-height="16px" altimg-valign="-5px" altimg-width="23px" alttext="z_{0}" display="inline"><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></math>
in <math class="ltx_Math" altimg="m75.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="R" display="inline"><mi>R</mi></math> and any given number <math class="ltx_Math" altimg="m86.png" altimg-height="13px" altimg-valign="-2px" altimg-width="12px" alttext="\epsilon" display="inline"><mi>ϵ</mi></math> (<math class="ltx_Math" altimg="m61.png" altimg-height="17px" altimg-valign="-3px" altimg-width="35px" alttext=">0" display="inline"><mrow><mi></mi><mo>></mo><mn>0</mn></mrow></math>) we can find a neighborhood of
<math class="ltx_Math" altimg="m224.png" altimg-height="16px" altimg-valign="-5px" altimg-width="23px" alttext="z_{0}" display="inline"><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></math> such that <math class="ltx_Math" altimg="m97.png" altimg-height="23px" altimg-valign="-7px" altimg-width="158px" alttext="\left|f(z)-f(z_{0})\right|<\epsilon" display="inline"><mrow><mrow><mo>|</mo><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>-</mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>|</mo></mrow><mo><</mo><mi>ϵ</mi></mrow></math> for all points <math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math> in the
intersection of the neighborhood with <math class="ltx_Math" altimg="m75.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="R" display="inline"><mi>R</mi></math>.
</p>
</div>
</section>
<section id="Px11" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Differentiation</h3>
<div id="Px11.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px11.p1" class="ltx_para">
<p class="ltx_p">A function <math class="ltx_Math" altimg="m161.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(z)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> is <em class="ltx_emph ltx_font_italic">differentiable</em> at a point <math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math> if the following
limit exists:</p>
<table id="E24" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.24</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E24.png" altimg-height="49px" altimg-valign="-17px" altimg-width="312px" alttext="f^{\prime}(z)=\frac{\mathrm{d}f}{\mathrm{d}z}=\lim_{h\to 0}\frac{f(z+h)-f(z)}{%
h}." display="block"><mrow><mrow><mrow><msup><mi>f</mi><mo>′</mo></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi>f</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi href="./1.1#p2.t1.r2">z</mi></mrow></mfrac><mo>=</mo><mrow><munder><mo movablelimits="false">lim</mo><mrow><mi>h</mi><mo>→</mo><mn>0</mn></mrow></munder><mo></mo><mfrac><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>+</mo><mi>h</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>-</mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mi>h</mi></mfrac></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E24.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#E4" title="(1.4.4) ‣ §1.4(iii) Derivatives ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="29px" altimg-valign="-9px" altimg-width="27px" alttext="\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: derivative of <math class="ltx_Math" altimg="m163.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="Px11.p2" class="ltx_para">
<p class="ltx_p">Differentiability automatically implies continuity.</p>
</div>
</section>
<section id="Px12" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Cauchy–Riemann Equations</h3>
<div id="Px12.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px12.p1" class="ltx_para">
<p class="ltx_p">If <math class="ltx_Math" altimg="m167.png" altimg-height="24px" altimg-valign="-7px" altimg-width="47px" alttext="f^{\prime}(z)" display="inline"><mrow><msup><mi>f</mi><mo>′</mo></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> exists at <math class="ltx_Math" altimg="m216.png" altimg-height="20px" altimg-valign="-6px" altimg-width="94px" alttext="z=x+iy" display="inline"><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>=</mo><mrow><mi>x</mi><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi>y</mi></mrow></mrow></mrow></math> and <math class="ltx_Math" altimg="m160.png" altimg-height="23px" altimg-valign="-7px" altimg-width="214px" alttext="f(z)=u(x,y)+iv(x,y)" display="inline"><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mi href="./1.9#Px14.p1">u</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mrow><mi href="./1.9#Px14.p1">v</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow></mrow></math>, then
</p>
<table id="E25" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex9" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">1.9.25</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m25.png" altimg-height="47px" altimg-valign="-16px" altimg-width="34px" alttext="\displaystyle\frac{\partial u}{\partial x}" display="inline"><mstyle displaystyle="true"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.9#Px14.p1">u</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow></mfrac></mstyle></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m9.png" altimg-height="50px" altimg-valign="-20px" altimg-width="60px" alttext="\displaystyle=\frac{\partial v}{\partial y}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mstyle displaystyle="true"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.9#Px14.p1">v</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow></mfrac></mstyle></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex10" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m26.png" altimg-height="50px" altimg-valign="-20px" altimg-width="34px" alttext="\displaystyle\frac{\partial u}{\partial y}" display="inline"><mstyle displaystyle="true"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.9#Px14.p1">u</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow></mfrac></mstyle></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m4.png" altimg-height="47px" altimg-valign="-16px" altimg-width="70px" alttext="\displaystyle=-\frac{\partial v}{\partial x}" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mo>-</mo><mstyle displaystyle="true"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.9#Px14.p1">v</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow></mfrac></mstyle></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E25.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m88.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m163.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m126.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.9#Px14.p1" title="Harmonic Functions ‣ §1.9(ii) Continuity, Point Sets, and Differentiation ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m196.png" altimg-height="23px" altimg-valign="-7px" altimg-width="62px" alttext="u(x,y)" display="inline"><mrow><mi href="./1.9#Px14.p1">u</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function</a> and
<a href="./1.9#Px14.p1" title="Harmonic Functions ‣ §1.9(ii) Continuity, Point Sets, and Differentiation ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m199.png" altimg-height="23px" altimg-valign="-7px" altimg-width="61px" alttext="v(x,y)" display="inline"><mrow><mi href="./1.9#Px14.p1">v</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">3.7.30</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">at <math class="ltx_Math" altimg="m52.png" altimg-height="23px" altimg-valign="-7px" altimg-width="51px" alttext="(x,y)" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi>x</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi>y</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math>.</p>
</div>
<div id="Px12.p2" class="ltx_para">
<p class="ltx_p">Conversely, if at a given point <math class="ltx_Math" altimg="m52.png" altimg-height="23px" altimg-valign="-7px" altimg-width="51px" alttext="(x,y)" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi>x</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi>y</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math> the partial derivatives
<math class="ltx_Math" altimg="m90.png" altimg-height="23px" altimg-valign="-7px" altimg-width="65px" alttext="\ifrac{\partial u}{\partial x}" display="inline"><mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.9#Px14.p1">u</mi></mrow><mo href="./1.5#E3">/</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow></mrow></math>, <math class="ltx_Math" altimg="m91.png" altimg-height="23px" altimg-valign="-7px" altimg-width="64px" alttext="\ifrac{\partial u}{\partial y}" display="inline"><mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.9#Px14.p1">u</mi></mrow><mo href="./1.5#E3">/</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow></mrow></math>, <math class="ltx_Math" altimg="m92.png" altimg-height="23px" altimg-valign="-7px" altimg-width="64px" alttext="\ifrac{\partial v}{\partial x}" display="inline"><mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.9#Px14.p1">v</mi></mrow><mo href="./1.5#E3">/</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow></mrow></math>, and <math class="ltx_Math" altimg="m93.png" altimg-height="23px" altimg-valign="-7px" altimg-width="63px" alttext="\ifrac{\partial v}{\partial y}" display="inline"><mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.9#Px14.p1">v</mi></mrow><mo href="./1.5#E3">/</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow></mrow></math>
exist, are continuous, and satisfy (), then <math class="ltx_Math" altimg="m161.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(z)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> is
differentiable at <math class="ltx_Math" altimg="m216.png" altimg-height="20px" altimg-valign="-6px" altimg-width="94px" alttext="z=x+iy" display="inline"><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>=</mo><mrow><mi>x</mi><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi>y</mi></mrow></mrow></mrow></math>.</p>
</div>
</section>
<section id="Px13" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Analyticity</h3>
<div id="Px13.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px13.p1" class="ltx_para">
<p class="ltx_p">A function <math class="ltx_Math" altimg="m161.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(z)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> is said to be <em class="ltx_emph ltx_font_italic">analytic</em> (<em class="ltx_emph ltx_font_italic">holomorphic</em>) at
<math class="ltx_Math" altimg="m217.png" altimg-height="16px" altimg-valign="-5px" altimg-width="59px" alttext="z=z_{0}" display="inline"><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>=</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></mrow></math> if it is differentiable in a neighborhood of <math class="ltx_Math" altimg="m224.png" altimg-height="16px" altimg-valign="-5px" altimg-width="23px" alttext="z_{0}" display="inline"><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></math>.</p>
</div>
<div id="Px13.p2" class="ltx_para">
<p class="ltx_p">A function <math class="ltx_Math" altimg="m161.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(z)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> is <em class="ltx_emph ltx_font_italic">analytic in a domain</em>
<math class="ltx_Math" altimg="m71.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi>D</mi></math> if it is analytic at each point of <math class="ltx_Math" altimg="m71.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi>D</mi></math>. A function analytic at every point
of <math class="ltx_Math" altimg="m112.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\mathbb{C}" display="inline"><mi href="./front/introduction#Sx4.p1.t1.r1">ℂ</mi></math> is said to be <em class="ltx_emph ltx_font_italic">entire</em>.
</p>
</div>
<div id="Px13.p3" class="ltx_para">
<p class="ltx_p">If <math class="ltx_Math" altimg="m161.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(z)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> is analytic in an open domain <math class="ltx_Math" altimg="m71.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi>D</mi></math>, then each of its derivatives
<math class="ltx_Math" altimg="m167.png" altimg-height="24px" altimg-valign="-7px" altimg-width="47px" alttext="f^{\prime}(z)" display="inline"><mrow><msup><mi>f</mi><mo>′</mo></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math>, <math class="ltx_Math" altimg="m164.png" altimg-height="24px" altimg-valign="-7px" altimg-width="52px" alttext="f^{\prime\prime}(z)" display="inline"><mrow><msup><mi>f</mi><mo>′′</mo></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math>, <math class="ltx_Math" altimg="m84.png" altimg-height="6px" altimg-valign="-2px" altimg-width="28px" alttext="\dots" display="inline"><mi mathvariant="normal">…</mi></math> exists and is analytic in <math class="ltx_Math" altimg="m71.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi>D</mi></math>.
</p>
</div>
</section>
<section id="Px14" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Harmonic Functions</h3>
<div id="Px14.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px14.p1" class="ltx_para">
<p class="ltx_p">If <math class="ltx_Math" altimg="m160.png" altimg-height="23px" altimg-valign="-7px" altimg-width="214px" alttext="f(z)=u(x,y)+iv(x,y)" display="inline"><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mi href="./1.9#Px14.p1">u</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mrow><mi href="./1.9#Px14.p1">v</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow></mrow></math> is analytic in an open domain <math class="ltx_Math" altimg="m71.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi>D</mi></math>, then <math class="ltx_Math" altimg="m198.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="u" display="inline"><mi href="./1.9#Px14.p1">u</mi></math> and
<math class="ltx_Math" altimg="m200.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="v" display="inline"><mi href="./1.9#Px14.p1">v</mi></math> are
<em class="ltx_emph ltx_font_italic">harmonic</em> in <math class="ltx_Math" altimg="m71.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi>D</mi></math>, that is,</p>
<table id="E26" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.26</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E26.png" altimg-height="56px" altimg-valign="-22px" altimg-width="270px" alttext="\frac{{\partial}^{2}u}{{\partial x}^{2}}+\frac{{\partial}^{2}u}{{\partial y}^{%
2}}=\frac{{\partial}^{2}v}{{\partial x}^{2}}+\frac{{\partial}^{2}v}{{\partial y%
}^{2}}=0," display="block"><mrow><mrow><mrow><mfrac><mrow><mpadded width="-1.7pt"><msup><mo href="./1.5#E3">∂</mo><mn>2</mn></msup></mpadded><mo href="./1.5#E3"></mo><mi href="./1.9#Px14.p1">u</mi></mrow><msup><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow><mn>2</mn></msup></mfrac><mo>+</mo><mfrac><mrow><mpadded width="-1.7pt"><msup><mo href="./1.5#E3">∂</mo><mn>2</mn></msup></mpadded><mo href="./1.5#E3"></mo><mi href="./1.9#Px14.p1">u</mi></mrow><msup><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow><mn>2</mn></msup></mfrac></mrow><mo>=</mo><mrow><mfrac><mrow><mpadded width="-1.7pt"><msup><mo href="./1.5#E3">∂</mo><mn>2</mn></msup></mpadded><mo href="./1.5#E3"></mo><mi href="./1.9#Px14.p1">v</mi></mrow><msup><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow><mn>2</mn></msup></mfrac><mo>+</mo><mfrac><mrow><mpadded width="-1.7pt"><msup><mo href="./1.5#E3">∂</mo><mn>2</mn></msup></mpadded><mo href="./1.5#E3"></mo><mi href="./1.9#Px14.p1">v</mi></mrow><msup><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow><mn>2</mn></msup></mfrac></mrow><mo>=</mo><mn>0</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E26.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m88.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m163.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m126.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.9#Px14.p1" title="Harmonic Functions ‣ §1.9(ii) Continuity, Point Sets, and Differentiation ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m196.png" altimg-height="23px" altimg-valign="-7px" altimg-width="62px" alttext="u(x,y)" display="inline"><mrow><mi href="./1.9#Px14.p1">u</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function</a> and
<a href="./1.9#Px14.p1" title="Harmonic Functions ‣ §1.9(ii) Continuity, Point Sets, and Differentiation ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m199.png" altimg-height="23px" altimg-valign="-7px" altimg-width="61px" alttext="v(x,y)" display="inline"><mrow><mi href="./1.9#Px14.p1">v</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">3.7.32</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">or in polar form (()) <math class="ltx_Math" altimg="m198.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="u" display="inline"><mi href="./1.9#Px14.p1">u</mi></math> and <math class="ltx_Math" altimg="m200.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="v" display="inline"><mi href="./1.9#Px14.p1">v</mi></math> satisfy</p>
<table id="E27" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.27</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E27.png" altimg-height="51px" altimg-valign="-18px" altimg-width="231px" alttext="\frac{{\partial}^{2}u}{{\partial r}^{2}}+\frac{1}{r}\frac{\partial u}{\partial
r%
}+\frac{1}{r^{2}}\frac{{\partial}^{2}u}{{\partial\theta}^{2}}=0" display="block"><mrow><mrow><mfrac><mrow><mpadded width="-1.7pt"><msup><mo href="./1.5#E3">∂</mo><mn>2</mn></msup></mpadded><mo href="./1.5#E3"></mo><mi href="./1.9#Px14.p1">u</mi></mrow><msup><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.9#Px2.p1">r</mi></mrow><mn>2</mn></msup></mfrac><mo>+</mo><mrow><mfrac><mn>1</mn><mi href="./1.9#Px2.p1">r</mi></mfrac><mo></mo><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.9#Px14.p1">u</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.9#Px2.p1">r</mi></mrow></mfrac></mrow><mo>+</mo><mrow><mfrac><mn>1</mn><msup><mi href="./1.9#Px2.p1">r</mi><mn>2</mn></msup></mfrac><mo></mo><mfrac><mrow><mpadded width="-1.7pt"><msup><mo href="./1.5#E3">∂</mo><mn>2</mn></msup></mpadded><mo href="./1.5#E3"></mo><mi href="./1.9#Px14.p1">u</mi></mrow><msup><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.9#Px2.p1">θ</mi></mrow><mn>2</mn></msup></mfrac></mrow></mrow><mo>=</mo><mn>0</mn></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E27.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m88.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m163.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m126.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.9#Px2.p1" title="Polar Representation ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m189.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="r" display="inline"><mi href="./1.9#Px2.p1">r</mi></math>: radius</a>,
<a href="./1.9#Px2.p1" title="Polar Representation ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m139.png" altimg-height="18px" altimg-valign="-2px" altimg-width="14px" alttext="\theta" display="inline"><mi href="./1.9#Px2.p1">θ</mi></math>: angle</a> and
<a href="./1.9#Px14.p1" title="Harmonic Functions ‣ §1.9(ii) Continuity, Point Sets, and Differentiation ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m196.png" altimg-height="23px" altimg-valign="-7px" altimg-width="62px" alttext="u(x,y)" display="inline"><mrow><mi href="./1.9#Px14.p1">u</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">3.7.33</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">at all points of <math class="ltx_Math" altimg="m71.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi>D</mi></math>.</p>
</div>
</section>
</section>
<section id="iii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§1.9(iii) </span>Integration</h2>
<div id="SS3.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="SS3.p1" class="ltx_para">
<p class="ltx_p">An <em class="ltx_emph ltx_font_italic">arc</em> <math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.9#Px19.p1">C</mi></math> is given by <math class="ltx_Math" altimg="m211.png" altimg-height="23px" altimg-valign="-7px" altimg-width="162px" alttext="z(t)=x(t)+iy(t)" display="inline"><mrow><mrow><mi>z</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mi>x</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mrow><mi>y</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow></mrow></math>, <math class="ltx_Math" altimg="m149.png" altimg-height="20px" altimg-valign="-5px" altimg-width="84px" alttext="a\leq t\leq b" display="inline"><mrow><mi>a</mi><mo>≤</mo><mi>t</mi><mo>≤</mo><mi>b</mi></mrow></math>, where
<math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math> and <math class="ltx_Math" altimg="m208.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi>y</mi></math> are continuously differentiable. If <math class="ltx_Math" altimg="m203.png" altimg-height="23px" altimg-valign="-7px" altimg-width="38px" alttext="x(t)" display="inline"><mrow><mi>x</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math> and <math class="ltx_Math" altimg="m207.png" altimg-height="23px" altimg-valign="-7px" altimg-width="37px" alttext="y(t)" display="inline"><mrow><mi>y</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math> are
continuous and <math class="ltx_Math" altimg="m206.png" altimg-height="24px" altimg-valign="-7px" altimg-width="44px" alttext="x^{\prime}(t)" display="inline"><mrow><msup><mi>x</mi><mo>′</mo></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math> and <math class="ltx_Math" altimg="m209.png" altimg-height="24px" altimg-valign="-7px" altimg-width="43px" alttext="y^{\prime}(t)" display="inline"><mrow><msup><mi>y</mi><mo>′</mo></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math> are piecewise continuous, then <math class="ltx_Math" altimg="m212.png" altimg-height="23px" altimg-valign="-7px" altimg-width="37px" alttext="z(t)" display="inline"><mrow><mi>z</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math>
defines a <em class="ltx_emph ltx_font_italic">contour</em>.</p>
</div>
<div id="SS3.p2" class="ltx_para">
<p class="ltx_p">A contour is <em class="ltx_emph ltx_font_italic">simple</em> if it contains no multiple points, that is, for
every pair of distinct values <math class="ltx_Math" altimg="m195.png" altimg-height="20px" altimg-valign="-6px" altimg-width="45px" alttext="t_{1},t_{2}" display="inline"><mrow><msub><mi>t</mi><mn>1</mn></msub><mo>,</mo><msub><mi>t</mi><mn>2</mn></msub></mrow></math> of <math class="ltx_Math" altimg="m193.png" altimg-height="17px" altimg-valign="-2px" altimg-width="11px" alttext="t" display="inline"><mi>t</mi></math>, <math class="ltx_Math" altimg="m213.png" altimg-height="23px" altimg-valign="-7px" altimg-width="114px" alttext="z(t_{1})\neq z(t_{2})" display="inline"><mrow><mrow><mi>z</mi><mo></mo><mrow><mo stretchy="false">(</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mrow></mrow><mo>≠</mo><mrow><mi>z</mi><mo></mo><mrow><mo stretchy="false">(</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mrow></mrow></mrow></math>. A
<em class="ltx_emph ltx_font_italic">simple closed contour</em> is a simple contour, except that <math class="ltx_Math" altimg="m210.png" altimg-height="23px" altimg-valign="-7px" altimg-width="101px" alttext="z(a)=z(b)" display="inline"><mrow><mrow><mi>z</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mi>z</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></math>.
</p>
</div>
<div id="SS3.p3" class="ltx_para">
<p class="ltx_p">Next,</p>
<table id="E28" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.28</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E28.png" altimg-height="56px" altimg-valign="-20px" altimg-width="360px" alttext="\int_{C}f(z)\mathrm{d}z=\int_{a}^{b}f(z(t))(x^{\prime}(t)+iy^{\prime}(t))%
\mathrm{d}t," display="block"><mrow><mrow><mrow><msub><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi href="./1.9#Px19.p1">C</mi></msub><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>z</mi></mrow></mrow></mrow><mo>=</mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi>a</mi><mi>b</mi></msubsup><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>z</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><msup><mi>x</mi><mo>′</mo></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mrow><msup><mi>y</mi><mo>′</mo></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E28.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m118.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m96.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a> and
<a href="./1.9#Px19.p1" title="Winding Number ‣ §1.9(iii) Integration ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.9#Px19.p1">C</mi></math>: closed contour</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">for a contour <math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.9#Px19.p1">C</mi></math> and <math class="ltx_Math" altimg="m157.png" altimg-height="23px" altimg-valign="-7px" altimg-width="65px" alttext="f(z(t))" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>z</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></math> continuous, <math class="ltx_Math" altimg="m149.png" altimg-height="20px" altimg-valign="-5px" altimg-width="84px" alttext="a\leq t\leq b" display="inline"><mrow><mi>a</mi><mo>≤</mo><mi>t</mi><mo>≤</mo><mi>b</mi></mrow></math>. If <math class="ltx_Math" altimg="m158.png" altimg-height="23px" altimg-valign="-7px" altimg-width="120px" alttext="f(z(t_{0}))=\infty" display="inline"><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>z</mi><mo></mo><mrow><mo stretchy="false">(</mo><msub><mi>t</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mi mathvariant="normal">∞</mi></mrow></math>, <math class="ltx_Math" altimg="m150.png" altimg-height="21px" altimg-valign="-5px" altimg-width="93px" alttext="a\leq t_{0}\leq b" display="inline"><mrow><mi>a</mi><mo>≤</mo><msub><mi>t</mi><mn>0</mn></msub><mo>≤</mo><mi>b</mi></mrow></math>, then the integral is defined analogously to the
infinite integrals in §. Similarly when <math class="ltx_Math" altimg="m148.png" altimg-height="17px" altimg-valign="-4px" altimg-width="77px" alttext="a=-\infty" display="inline"><mrow><mi>a</mi><mo>=</mo><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow></mrow></math> or
<math class="ltx_Math" altimg="m155.png" altimg-height="19px" altimg-valign="-4px" altimg-width="75px" alttext="b=+\infty" display="inline"><mrow><mi>b</mi><mo>=</mo><mrow><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></mrow></math>.</p>
</div>
<section id="Px15" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Jordan Curve Theorem</h3>
<div id="Px15.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px15.p1" class="ltx_para">
<p class="ltx_p">Any simple closed contour <math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.9#Px19.p1">C</mi></math> divides <math class="ltx_Math" altimg="m112.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\mathbb{C}" display="inline"><mi href="./front/introduction#Sx4.p1.t1.r1">ℂ</mi></math> into two open domains that
have <math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.9#Px19.p1">C</mi></math> as common boundary. One of these domains is bounded and is called the
<em class="ltx_emph ltx_font_italic">interior domain of</em> <math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.9#Px19.p1">C</mi></math>; the other is unbounded and is called the
<em class="ltx_emph ltx_font_italic">exterior domain of</em> <math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.9#Px19.p1">C</mi></math>.</p>
</div>
</section>
<section id="Px16" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Cauchy’s Theorem</h3>
<div id="Px16.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px16.p1" class="ltx_para">
<p class="ltx_p">If <math class="ltx_Math" altimg="m161.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(z)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math> is continuous within and on a simple closed contour <math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.9#Px19.p1">C</mi></math> and analytic
within <math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.9#Px19.p1">C</mi></math>, then</p>
<table id="E29" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.29</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E29.png" altimg-height="52px" altimg-valign="-20px" altimg-width="135px" alttext="\int_{C}f(z)\mathrm{d}z=0." display="block"><mrow><mrow><mrow><msub><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi href="./1.9#Px19.p1">C</mi></msub><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>z</mi></mrow></mrow></mrow><mo>=</mo><mn>0</mn></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E29.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m118.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m96.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a> and
<a href="./1.9#Px19.p1" title="Winding Number ‣ §1.9(iii) Integration ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.9#Px19.p1">C</mi></math>: closed contour</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px17" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Cauchy’s Integral Formula</h3>
<div id="Px17.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px17.p1" class="ltx_para">
<p class="ltx_p">If <math class="ltx_Math" altimg="m161.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(z)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math> is continuous within and on a simple closed contour <math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.9#Px19.p1">C</mi></math> and analytic
within <math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.9#Px19.p1">C</mi></math>, and if <math class="ltx_Math" altimg="m224.png" altimg-height="16px" altimg-valign="-5px" altimg-width="23px" alttext="z_{0}" display="inline"><msub><mi>z</mi><mn>0</mn></msub></math> is a point within <math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.9#Px19.p1">C</mi></math>, then
</p>
<table id="E30" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.30</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E30.png" altimg-height="52px" altimg-valign="-20px" altimg-width="227px" alttext="f(z_{0})=\frac{1}{2\pi i}\int_{C}\frac{f(z)}{z-z_{0}}\mathrm{d}z," display="block"><mrow><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><msub><mi>z</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mfrac><mn>1</mn><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi></mrow></mfrac><mo></mo><mrow><msub><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi href="./1.9#Px19.p1">C</mi></msub><mrow><mfrac><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow><mrow><mi>z</mi><mo>-</mo><msub><mi>z</mi><mn>0</mn></msub></mrow></mfrac><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>z</mi></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E30.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m118.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m96.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a> and
<a href="./1.9#Px19.p1" title="Winding Number ‣ §1.9(iii) Integration ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.9#Px19.p1">C</mi></math>: closed contour</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and</p>
<table id="E31" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.31</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E31.png" altimg-height="53px" altimg-valign="-21px" altimg-width="297px" alttext="f^{(n)}(z_{0})=\frac{n!}{2\pi i}\int_{C}\frac{f(z)}{(z-z_{0})^{n+1}}\mathrm{d}z," display="block"><mrow><mrow><mrow><msup><mi>f</mi><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r5">n</mi><mo stretchy="false">)</mo></mrow></msup><mo></mo><mrow><mo stretchy="false">(</mo><msub><mi>z</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mfrac><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi></mrow></mfrac><mo></mo><mrow><msub><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi href="./1.9#Px19.p1">C</mi></msub><mrow><mfrac><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mi>z</mi><mo>-</mo><msub><mi>z</mi><mn>0</mn></msub></mrow><mo stretchy="false">)</mo></mrow><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>+</mo><mn>1</mn></mrow></msup></mfrac><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>z</mi></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m179.png" altimg-height="20px" altimg-valign="-6px" altimg-width="123px" alttext="n=1,2,3,\dots" display="inline"><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E31.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m118.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m177.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m96.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m181.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./1.9#Px19.p1" title="Winding Number ‣ §1.9(iii) Integration ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.9#Px19.p1">C</mi></math>: closed contour</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">provided that in both cases <math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.9#Px19.p1">C</mi></math> is described in the positive rotational
(anticlockwise) sense.</p>
</div>
</section>
<section id="Px18" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Liouville’s Theorem</h3>
<div id="Px18.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px19.p1" class="ltx_para">
<p class="ltx_p">If <math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.9#Px19.p1">C</mi></math> is a closed contour, and <math class="ltx_Math" altimg="m225.png" altimg-height="21px" altimg-valign="-6px" altimg-width="63px" alttext="z_{0}\not\in C" display="inline"><mrow><msub><mi>z</mi><mn>0</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r10">∉</mo><mi href="./1.9#Px19.p1">C</mi></mrow></math>, then
</p>
<table id="E32" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.32</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E32.png" altimg-height="52px" altimg-valign="-20px" altimg-width="258px" alttext="\frac{1}{2\pi i}\int_{C}\frac{1}{z-z_{0}}\mathrm{d}z=\mathcal{N}(C,z_{0})," display="block"><mrow><mrow><mrow><mfrac><mn>1</mn><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi></mrow></mfrac><mo></mo><mrow><msub><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi href="./1.9#Px19.p1">C</mi></msub><mrow><mfrac><mn>1</mn><mrow><mi>z</mi><mo>-</mo><msub><mi>z</mi><mn>0</mn></msub></mrow></mfrac><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>z</mi></mrow></mrow></mrow></mrow><mo>=</mo><mrow><mi class="ltx_font_mathcaligraphic" href="./1.9#E32">𝒩</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.9#Px19.p1">C</mi><mo>,</mo><msub><mi>z</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E32.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text"><math class="ltx_Math" altimg="m116.png" altimg-height="23px" altimg-valign="-7px" altimg-width="81px" alttext="\mathcal{N}(C,z_{0})" display="inline"><mrow><mi class="ltx_font_mathcaligraphic" href="./1.9#E32">𝒩</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.9#Px19.p1">C</mi><mo>,</mo><msub><mi>z</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mrow></math>: winding number of <math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.9#Px19.p1">C</mi></math> (locally)</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m118.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m96.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a> and
<a href="./1.9#Px19.p1" title="Winding Number ‣ §1.9(iii) Integration ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.9#Px19.p1">C</mi></math>: closed contour</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m116.png" altimg-height="23px" altimg-valign="-7px" altimg-width="81px" alttext="\mathcal{N}(C,z_{0})" display="inline"><mrow><mi class="ltx_font_mathcaligraphic" href="./1.9#E32">𝒩</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.9#Px19.p1">C</mi><mo>,</mo><msub><mi>z</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mrow></math> is an integer called the <em class="ltx_emph ltx_font_italic">winding number of <math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.9#Px19.p1">C</mi></math>
with respect to <math class="ltx_Math" altimg="m224.png" altimg-height="16px" altimg-valign="-5px" altimg-width="23px" alttext="z_{0}" display="inline"><msub><mi>z</mi><mn mathvariant="normal">0</mn></msub></math></em>. If <math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.9#Px19.p1">C</mi></math> is simple and oriented in the positive
rotational sense, then <math class="ltx_Math" altimg="m116.png" altimg-height="23px" altimg-valign="-7px" altimg-width="81px" alttext="\mathcal{N}(C,z_{0})" display="inline"><mrow><mi class="ltx_font_mathcaligraphic" href="./1.9#E32">𝒩</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.9#Px19.p1">C</mi><mo>,</mo><msub><mi>z</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mrow></math> is <math class="ltx_Math" altimg="m59.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="1" display="inline"><mn>1</mn></math> or <math class="ltx_Math" altimg="m57.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math> depending whether
<math class="ltx_Math" altimg="m224.png" altimg-height="16px" altimg-valign="-5px" altimg-width="23px" alttext="z_{0}" display="inline"><msub><mi>z</mi><mn>0</mn></msub></math> is inside or outside <math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.9#Px19.p1">C</mi></math>.</p>
</div>
</section>
<section id="Px20" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Mean Value Property</h3>
<div id="Px20.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px20.p1" class="ltx_para">
<p class="ltx_p">For <math class="ltx_Math" altimg="m197.png" altimg-height="23px" altimg-valign="-7px" altimg-width="41px" alttext="u(z)" display="inline"><mrow><mi href="./1.9#Px20.p1">u</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math> harmonic,
</p>
<table id="E33" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.33</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E33.png" altimg-height="56px" altimg-valign="-20px" altimg-width="267px" alttext="u(z)=\frac{1}{2\pi}\int^{2\pi}_{0}u(z+re^{i\phi})\mathrm{d}\phi." display="block"><mrow><mrow><mrow><mi href="./1.9#Px20.p1">u</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mfrac><mn>1</mn><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi></mrow></mfrac><mo></mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi></mrow></msubsup><mrow><mrow><mi href="./1.9#Px20.p1">u</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>z</mi><mo>+</mo><mrow><mi href="./1.9#Px2.p1">r</mi><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mo></mo><mi>ϕ</mi></mrow></msup></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>ϕ</mi></mrow></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E33.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m118.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m119.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m96.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.9#Px2.p1" title="Polar Representation ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m189.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="r" display="inline"><mi href="./1.9#Px2.p1">r</mi></math>: radius</a> and
<a href="./1.9#Px20.p1" title="Mean Value Property ‣ §1.9(iii) Integration ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m197.png" altimg-height="23px" altimg-valign="-7px" altimg-width="41px" alttext="u(z)" display="inline"><mrow><mi href="./1.9#Px20.p1">u</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: harmonic function</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px21" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Poisson Integral</h3>
<div id="Px21.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px21.p1" class="ltx_para">
<p class="ltx_p">If <math class="ltx_Math" altimg="m172.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="h(w)" display="inline"><mrow><mi>h</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r3">w</mi><mo stretchy="false">)</mo></mrow></mrow></math> is continuous on <math class="ltx_Math" altimg="m236.png" altimg-height="23px" altimg-valign="-7px" altimg-width="72px" alttext="|w|=R" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mi href="./1.1#p2.t1.r3">w</mi><mo stretchy="false">|</mo></mrow><mo>=</mo><mi href="./1.9#Px21.p1">R</mi></mrow></math>, then with <math class="ltx_Math" altimg="m215.png" altimg-height="20px" altimg-valign="-2px" altimg-width="74px" alttext="z=re^{i\theta}" display="inline"><mrow><mi>z</mi><mo>=</mo><mrow><mi href="./1.9#Px2.p1">r</mi><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mo></mo><mi>θ</mi></mrow></msup></mrow></mrow></math>
</p>
<table id="E34" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.34</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E34.png" altimg-height="55px" altimg-valign="-21px" altimg-width="391px" alttext="u(re^{i\theta})=\frac{1}{2\pi}\int^{2\pi}_{0}\frac{(R^{2}-r^{2})h(Re^{i\phi})%
\mathrm{d}\phi}{R^{2}-2Rr\cos\left(\phi-\theta\right)+r^{2}}" display="block"><mrow><mrow><mi href="./1.9#Px20.p1">u</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./1.9#Px2.p1">r</mi><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mo></mo><mi>θ</mi></mrow></msup></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mfrac><mn>1</mn><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi></mrow></mfrac><mo></mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi></mrow></msubsup><mfrac><mrow><mrow><mo stretchy="false">(</mo><mrow><msup><mi href="./1.9#Px21.p1">R</mi><mn>2</mn></msup><mo>-</mo><msup><mi href="./1.9#Px2.p1">r</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mi>h</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./1.9#Px21.p1">R</mi><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mo></mo><mi>ϕ</mi></mrow></msup></mrow><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>ϕ</mi></mrow></mrow><mrow><mrow><msup><mi href="./1.9#Px21.p1">R</mi><mn>2</mn></msup><mo>-</mo><mrow><mn>2</mn><mo></mo><mi href="./1.9#Px21.p1">R</mi><mo></mo><mi href="./1.9#Px2.p1">r</mi><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mi>ϕ</mi><mo>-</mo><mi>θ</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>+</mo><msup><mi href="./1.9#Px2.p1">r</mi><mn>2</mn></msup></mrow></mfrac></mrow></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E34.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m118.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m119.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m96.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.9#Px2.p1" title="Polar Representation ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m189.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="r" display="inline"><mi href="./1.9#Px2.p1">r</mi></math>: radius</a>,
<a href="./1.9#Px20.p1" title="Mean Value Property ‣ §1.9(iii) Integration ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m197.png" altimg-height="23px" altimg-valign="-7px" altimg-width="41px" alttext="u(z)" display="inline"><mrow><mi href="./1.9#Px20.p1">u</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: harmonic function</a> and
<a href="./1.9#Px21.p1" title="Poisson Integral ‣ §1.9(iii) Integration ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m75.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="R" display="inline"><mi href="./1.9#Px21.p1">R</mi></math>: radius</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">is harmonic in <math class="ltx_Math" altimg="m242.png" altimg-height="23px" altimg-valign="-7px" altimg-width="67px" alttext="|z|<R" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mi>z</mi><mo stretchy="false">|</mo></mrow><mo><</mo><mi href="./1.9#Px21.p1">R</mi></mrow></math>. Also with <math class="ltx_Math" altimg="m98.png" altimg-height="23px" altimg-valign="-7px" altimg-width="72px" alttext="\left|w\right|=R" display="inline"><mrow><mrow><mo>|</mo><mi href="./1.1#p2.t1.r3">w</mi><mo>|</mo></mrow><mo>=</mo><mi href="./1.9#Px21.p1">R</mi></mrow></math>,
<math class="ltx_Math" altimg="m105.png" altimg-height="32px" altimg-valign="-16px" altimg-width="149px" alttext="\lim\limits_{z\to w}u(z)=h(w)" display="inline"><mrow><mrow><munder><mo movablelimits="false">lim</mo><mrow><mi>z</mi><mo>→</mo><mi href="./1.1#p2.t1.r3">w</mi></mrow></munder><mo></mo><mrow><mi href="./1.9#Px20.p1">u</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>=</mo><mrow><mi>h</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r3">w</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></math> as <math class="ltx_Math" altimg="m223.png" altimg-height="13px" altimg-valign="-2px" altimg-width="60px" alttext="z\to w" display="inline"><mrow><mi>z</mi><mo>→</mo><mi href="./1.1#p2.t1.r3">w</mi></mrow></math> within <math class="ltx_Math" altimg="m242.png" altimg-height="23px" altimg-valign="-7px" altimg-width="67px" alttext="|z|<R" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mi>z</mi><mo stretchy="false">|</mo></mrow><mo><</mo><mi href="./1.9#Px21.p1">R</mi></mrow></math>.</p>
</div>
</section>
</section>
<section id="iv" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§1.9(iv) </span>Conformal Mapping</h2>
<div id="SS4.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="SS4.p1" class="ltx_para">
<p class="ltx_p">The <em class="ltx_emph ltx_font_italic">extended complex plane</em>,
<math class="ltx_Math" altimg="m110.png" altimg-height="23px" altimg-valign="-7px" altimg-width="87px" alttext="\mathbb{C}\,\cup\,\{\infty\}" display="inline"><mrow><mpadded width="+1.7pt"><mi href="./front/introduction#Sx4.p1.t1.r1">ℂ</mi></mpadded><mo href="./front/introduction#Sx4.p1.t1.r27" rspace="4.2pt">∪</mo><mrow><mo stretchy="false">{</mo><mi mathvariant="normal">∞</mi><mo stretchy="false">}</mo></mrow></mrow></math>, consists of the points of the complex plane
<math class="ltx_Math" altimg="m112.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\mathbb{C}" display="inline"><mi href="./front/introduction#Sx4.p1.t1.r1">ℂ</mi></math> together with an ideal point <math class="ltx_Math" altimg="m95.png" altimg-height="13px" altimg-valign="-2px" altimg-width="24px" alttext="\infty" display="inline"><mi mathvariant="normal">∞</mi></math> called the <em class="ltx_emph ltx_font_italic">point at
infinity</em>.
A system of <em class="ltx_emph ltx_font_italic">open disks around infinity</em> is given by
</p>
<table id="E35" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.35</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E35.png" altimg-height="25px" altimg-valign="-7px" altimg-width="245px" alttext="S_{r}=\{z\mid|z|>1/r\}\cup\{\infty\}," display="block"><mrow><mrow><msub><mi href="./1.9#SS4.p1">S</mi><mi href="./1.9#Px2.p1">r</mi></msub><mo>=</mo><mrow><mrow><mo stretchy="false">{</mo><mi href="./1.1#p2.t1.r2">z</mi><mo>∣</mo><mrow><mrow><mo stretchy="false">|</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">|</mo></mrow><mo>></mo><mrow><mn>1</mn><mo>/</mo><mi href="./1.9#Px2.p1">r</mi></mrow></mrow><mo stretchy="false">}</mo></mrow><mo href="./front/introduction#Sx4.p1.t1.r27">∪</mo><mrow><mo stretchy="false">{</mo><mi mathvariant="normal">∞</mi><mo stretchy="false">}</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m56.png" altimg-height="17px" altimg-valign="-3px" altimg-width="97px" alttext="0<r<\infty" display="inline"><mrow><mn>0</mn><mo><</mo><mi href="./1.9#Px2.p1">r</mi><mo><</mo><mi mathvariant="normal">∞</mi></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E35.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./front/introduction#Sx4.p1.t1.r27" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="15px" altimg-valign="-2px" altimg-width="18px" alttext="\cup" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r27">∪</mo></math>: union</a>,
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>,
<a href="./1.9#Px2.p1" title="Polar Representation ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m189.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="r" display="inline"><mi href="./1.9#Px2.p1">r</mi></math>: radius</a> and
<a href="./1.9#SS4.p1" title="§1.9(iv) Conformal Mapping ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="21px" altimg-valign="-5px" altimg-width="25px" alttext="S_{r}" display="inline"><msub><mi href="./1.9#SS4.p1">S</mi><mi href="./1.9#Px2.p1">r</mi></msub></math>: neighborhood</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Each <math class="ltx_Math" altimg="m77.png" altimg-height="21px" altimg-valign="-5px" altimg-width="25px" alttext="S_{r}" display="inline"><msub><mi href="./1.9#SS4.p1">S</mi><mi href="./1.9#Px2.p1">r</mi></msub></math> is a <em class="ltx_emph ltx_font_italic">neighborhood</em>
of <math class="ltx_Math" altimg="m95.png" altimg-height="13px" altimg-valign="-2px" altimg-width="24px" alttext="\infty" display="inline"><mi mathvariant="normal">∞</mi></math>. Also,
</p>
<table id="E36" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.36</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E36.png" altimg-height="21px" altimg-valign="-6px" altimg-width="193px" alttext="\infty\pm z=z\pm\infty=\infty," display="block"><mrow><mrow><mrow><mi mathvariant="normal">∞</mi><mo>±</mo><mi href="./1.1#p2.t1.r2">z</mi></mrow><mo>=</mo><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>±</mo><mi mathvariant="normal">∞</mi></mrow><mo>=</mo><mi mathvariant="normal">∞</mi></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E36.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a></dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E37" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.37</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E37.png" altimg-height="18px" altimg-valign="-6px" altimg-width="174px" alttext="\infty\cdot z=z\cdot\infty=\infty," display="block"><mrow><mrow><mrow><mi mathvariant="normal">∞</mi><mo>⋅</mo><mi href="./1.1#p2.t1.r2">z</mi></mrow><mo>=</mo><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>⋅</mo><mi mathvariant="normal">∞</mi></mrow><mo>=</mo><mi mathvariant="normal">∞</mi></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m222.png" altimg-height="21px" altimg-valign="-6px" altimg-width="51px" alttext="z\not=0" display="inline"><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>≠</mo><mn>0</mn></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E37.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a></dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E38" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.38</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E38.png" altimg-height="25px" altimg-valign="-7px" altimg-width="88px" alttext="z/\infty=0," display="block"><mrow><mrow><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>/</mo><mi mathvariant="normal">∞</mi></mrow><mo>=</mo><mn>0</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E38.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a></dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E39" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.39</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E39.png" altimg-height="25px" altimg-valign="-7px" altimg-width="88px" alttext="z/0=\infty," display="block"><mrow><mrow><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>/</mo><mn>0</mn></mrow><mo>=</mo><mi mathvariant="normal">∞</mi></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m220.png" altimg-height="21px" altimg-valign="-6px" altimg-width="51px" alttext="z\neq 0" display="inline"><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>≠</mo><mn>0</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E39.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a></dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">A function <math class="ltx_Math" altimg="m161.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(z)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> is <em class="ltx_emph ltx_font_italic">analytic at</em> <math class="ltx_Math" altimg="m95.png" altimg-height="13px" altimg-valign="-2px" altimg-width="24px" alttext="\infty" display="inline"><mi mathvariant="normal">∞</mi></math> if <math class="ltx_Math" altimg="m171.png" altimg-height="23px" altimg-valign="-7px" altimg-width="124px" alttext="g(z)=f(1/z)" display="inline"><mrow><mrow><mi>g</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>/</mo><mi href="./1.1#p2.t1.r2">z</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></math> is analytic
at <math class="ltx_Math" altimg="m214.png" altimg-height="17px" altimg-valign="-2px" altimg-width="51px" alttext="z=0" display="inline"><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>=</mo><mn>0</mn></mrow></math>, and we set <math class="ltx_Math" altimg="m165.png" altimg-height="24px" altimg-valign="-7px" altimg-width="125px" alttext="f^{\prime}(\infty)=g^{\prime}(0)" display="inline"><mrow><mrow><msup><mi>f</mi><mo>′</mo></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi mathvariant="normal">∞</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><msup><mi>g</mi><mo>′</mo></msup><mo></mo><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></mrow></mrow></math>.
</p>
</div>
<section id="Px22" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Conformal Transformation</h3>
<div id="Px22.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px22.p1" class="ltx_para">
<p class="ltx_p">Suppose <math class="ltx_Math" altimg="m161.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(z)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> is analytic in a domain <math class="ltx_Math" altimg="m71.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.9#Px22.p1">D</mi></math> and <math class="ltx_Math" altimg="m67.png" altimg-height="21px" altimg-valign="-6px" altimg-width="60px" alttext="C_{1},C_{2}" display="inline"><mrow><msub><mi href="./1.9#Px22.p1">C</mi><mn>1</mn></msub><mo>,</mo><msub><mi href="./1.9#Px22.p1">C</mi><mn>2</mn></msub></mrow></math> are two arcs in <math class="ltx_Math" altimg="m71.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.9#Px22.p1">D</mi></math>
passing through <math class="ltx_Math" altimg="m224.png" altimg-height="16px" altimg-valign="-5px" altimg-width="23px" alttext="z_{0}" display="inline"><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></math>. Let <math class="ltx_Math" altimg="m64.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="C^{\prime}_{1},C^{\prime}_{2}" display="inline"><mrow><msubsup><mi href="./1.9#Px22.p1">C</mi><mn>1</mn><mo>′</mo></msubsup><mo>,</mo><msubsup><mi href="./1.9#Px22.p1">C</mi><mn>2</mn><mo>′</mo></msubsup></mrow></math> be the images of <math class="ltx_Math" altimg="m68.png" altimg-height="21px" altimg-valign="-5px" altimg-width="27px" alttext="C_{1}" display="inline"><msub><mi href="./1.9#Px22.p1">C</mi><mn>1</mn></msub></math> and <math class="ltx_Math" altimg="m69.png" altimg-height="21px" altimg-valign="-5px" altimg-width="27px" alttext="C_{2}" display="inline"><msub><mi href="./1.9#Px22.p1">C</mi><mn>2</mn></msub></math> under
the mapping <math class="ltx_Math" altimg="m201.png" altimg-height="23px" altimg-valign="-7px" altimg-width="83px" alttext="w=f(z)" display="inline"><mrow><mi href="./1.1#p2.t1.r3">w</mi><mo>=</mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></math>. The <em class="ltx_emph ltx_font_italic">angle between</em> <math class="ltx_Math" altimg="m68.png" altimg-height="21px" altimg-valign="-5px" altimg-width="27px" alttext="C_{1}" display="inline"><msub><mi href="./1.9#Px22.p1">C</mi><mn>1</mn></msub></math> <em class="ltx_emph ltx_font_italic">and</em> <math class="ltx_Math" altimg="m69.png" altimg-height="21px" altimg-valign="-5px" altimg-width="27px" alttext="C_{2}" display="inline"><msub><mi href="./1.9#Px22.p1">C</mi><mn>2</mn></msub></math> at
<math class="ltx_Math" altimg="m224.png" altimg-height="16px" altimg-valign="-5px" altimg-width="23px" alttext="z_{0}" display="inline"><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></math> is the angle between the tangents to the two arcs at <math class="ltx_Math" altimg="m224.png" altimg-height="16px" altimg-valign="-5px" altimg-width="23px" alttext="z_{0}" display="inline"><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></math>, that is, the
difference of the signed angles that the tangents make with the positive
direction of the real axis. If <math class="ltx_Math" altimg="m168.png" altimg-height="24px" altimg-valign="-7px" altimg-width="92px" alttext="f^{\prime}(z_{0})\not=0" display="inline"><mrow><mrow><msup><mi>f</mi><mo>′</mo></msup><mo></mo><mrow><mo stretchy="false">(</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mrow><mo>≠</mo><mn>0</mn></mrow></math>, then the angle between <math class="ltx_Math" altimg="m68.png" altimg-height="21px" altimg-valign="-5px" altimg-width="27px" alttext="C_{1}" display="inline"><msub><mi href="./1.9#Px22.p1">C</mi><mn>1</mn></msub></math>
and <math class="ltx_Math" altimg="m69.png" altimg-height="21px" altimg-valign="-5px" altimg-width="27px" alttext="C_{2}" display="inline"><msub><mi href="./1.9#Px22.p1">C</mi><mn>2</mn></msub></math> equals the angle between <math class="ltx_Math" altimg="m65.png" altimg-height="23px" altimg-valign="-7px" altimg-width="27px" alttext="C^{\prime}_{1}" display="inline"><msubsup><mi href="./1.9#Px22.p1">C</mi><mn>1</mn><mo>′</mo></msubsup></math> and <math class="ltx_Math" altimg="m66.png" altimg-height="23px" altimg-valign="-7px" altimg-width="27px" alttext="C^{\prime}_{2}" display="inline"><msubsup><mi href="./1.9#Px22.p1">C</mi><mn>2</mn><mo>′</mo></msubsup></math> both in magnitude and
sense. We then say that the mapping <math class="ltx_Math" altimg="m201.png" altimg-height="23px" altimg-valign="-7px" altimg-width="83px" alttext="w=f(z)" display="inline"><mrow><mi href="./1.1#p2.t1.r3">w</mi><mo>=</mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></math> is <em class="ltx_emph ltx_font_italic">conformal</em>
(angle-preserving) at <math class="ltx_Math" altimg="m224.png" altimg-height="16px" altimg-valign="-5px" altimg-width="23px" alttext="z_{0}" display="inline"><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></math>.</p>
</div>
<div id="Px22.p2" class="ltx_para">
<p class="ltx_p">The <em class="ltx_emph ltx_font_italic">linear transformation</em> <math class="ltx_Math" altimg="m159.png" altimg-height="23px" altimg-valign="-7px" altimg-width="122px" alttext="f(z)=az+b" display="inline"><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mi>a</mi><mo></mo><mi href="./1.1#p2.t1.r2">z</mi></mrow><mo>+</mo><mi>b</mi></mrow></mrow></math>, <math class="ltx_Math" altimg="m151.png" altimg-height="21px" altimg-valign="-6px" altimg-width="51px" alttext="a\not=0" display="inline"><mrow><mi>a</mi><mo>≠</mo><mn>0</mn></mrow></math>, has <math class="ltx_Math" altimg="m166.png" altimg-height="24px" altimg-valign="-7px" altimg-width="85px" alttext="f^{\prime}(z)=a" display="inline"><mrow><mrow><msup><mi>f</mi><mo>′</mo></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mi>a</mi></mrow></math>
and <math class="ltx_Math" altimg="m201.png" altimg-height="23px" altimg-valign="-7px" altimg-width="83px" alttext="w=f(z)" display="inline"><mrow><mi href="./1.1#p2.t1.r3">w</mi><mo>=</mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></math> maps <math class="ltx_Math" altimg="m112.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\mathbb{C}" display="inline"><mi href="./front/introduction#Sx4.p1.t1.r1">ℂ</mi></math> conformally onto <math class="ltx_Math" altimg="m112.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\mathbb{C}" display="inline"><mi href="./front/introduction#Sx4.p1.t1.r1">ℂ</mi></math>.
</p>
</div>
</section>
<section id="Px23" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Bilinear Transformation</h3>
<div id="Px23.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px23.p1" class="ltx_para">
<table id="E40" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.40</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E40.png" altimg-height="48px" altimg-valign="-17px" altimg-width="175px" alttext="w=f(z)=\frac{az+b}{cz+d}," display="block"><mrow><mrow><mi href="./1.1#p2.t1.r3">w</mi><mo>=</mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mfrac><mrow><mrow><mi>a</mi><mo></mo><mi href="./1.1#p2.t1.r2">z</mi></mrow><mo>+</mo><mi>b</mi></mrow><mrow><mrow><mi>c</mi><mo></mo><mi href="./1.1#p2.t1.r2">z</mi></mrow><mo>+</mo><mi>d</mi></mrow></mfrac></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m154.png" altimg-height="21px" altimg-valign="-6px" altimg-width="103px" alttext="ad-bc\not=0" display="inline"><mrow><mrow><mrow><mi>a</mi><mo></mo><mi>d</mi></mrow><mo>-</mo><mrow><mi>b</mi><mo></mo><mi>c</mi></mrow></mrow><mo>≠</mo><mn>0</mn></mrow></math>, <math class="ltx_Math" altimg="m156.png" altimg-height="21px" altimg-valign="-6px" altimg-width="49px" alttext="c\not=0" display="inline"><mrow><mi>c</mi><mo>≠</mo><mn>0</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E40.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a> and
<a href="./1.1#p2.t1.r3" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m202.png" altimg-height="13px" altimg-valign="-2px" altimg-width="19px" alttext="w" display="inline"><mi href="./1.1#p2.t1.r3">w</mi></math>: variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E41" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex11" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">1.9.41</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m34.png" altimg-height="25px" altimg-valign="-7px" altimg-width="78px" alttext="\displaystyle f(-d/c)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mrow><mi>d</mi><mo>/</mo><mi>c</mi></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m10.png" altimg-height="18px" altimg-valign="-6px" altimg-width="53px" alttext="\displaystyle=\infty," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mi mathvariant="normal">∞</mi></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex12" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m35.png" altimg-height="25px" altimg-valign="-7px" altimg-width="53px" alttext="\displaystyle f(\infty)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi mathvariant="normal">∞</mi><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m14.png" altimg-height="25px" altimg-valign="-7px" altimg-width="62px" alttext="\displaystyle=a/c." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi>a</mi><mo>/</mo><mi>c</mi></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E41.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E42" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.42</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E42.png" altimg-height="51px" altimg-valign="-21px" altimg-width="164px" alttext="f^{\prime}(z)=\frac{ad-bc}{(cz+d)^{2}}," display="block"><mrow><mrow><mrow><msup><mi>f</mi><mo>′</mo></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mfrac><mrow><mrow><mi>a</mi><mo></mo><mi>d</mi></mrow><mo>-</mo><mrow><mi>b</mi><mo></mo><mi>c</mi></mrow></mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mrow><mi>c</mi><mo></mo><mi href="./1.1#p2.t1.r2">z</mi></mrow><mo>+</mo><mi>d</mi></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mfrac></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m221.png" altimg-height="23px" altimg-valign="-7px" altimg-width="85px" alttext="z\not=-d/c" display="inline"><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>≠</mo><mrow><mo>-</mo><mrow><mi>d</mi><mo>/</mo><mi>c</mi></mrow></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E42.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a></dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E43" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.43</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E43.png" altimg-height="47px" altimg-valign="-16px" altimg-width="158px" alttext="f^{\prime}(\infty)=\frac{bc-ad}{c^{2}}." display="block"><mrow><mrow><mrow><msup><mi>f</mi><mo>′</mo></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi mathvariant="normal">∞</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mfrac><mrow><mrow><mi>b</mi><mo></mo><mi>c</mi></mrow><mo>-</mo><mrow><mi>a</mi><mo></mo><mi>d</mi></mrow></mrow><msup><mi>c</mi><mn>2</mn></msup></mfrac></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E43.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E44" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.44</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E44.png" altimg-height="48px" altimg-valign="-17px" altimg-width="127px" alttext="z=\frac{dw-b}{-cw+a}." display="block"><mrow><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>=</mo><mfrac><mrow><mrow><mi>d</mi><mo></mo><mi href="./1.1#p2.t1.r3">w</mi></mrow><mo>-</mo><mi>b</mi></mrow><mrow><mrow><mo>-</mo><mrow><mi>c</mi><mo></mo><mi href="./1.1#p2.t1.r3">w</mi></mrow></mrow><mo>+</mo><mi>a</mi></mrow></mfrac></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E44.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a> and
<a href="./1.1#p2.t1.r3" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m202.png" altimg-height="13px" altimg-valign="-2px" altimg-width="19px" alttext="w" display="inline"><mi href="./1.1#p2.t1.r3">w</mi></math>: variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="Px23.p2" class="ltx_para">
<p class="ltx_p">The transformation () is a one-to-one conformal mapping
of <math class="ltx_Math" altimg="m110.png" altimg-height="23px" altimg-valign="-7px" altimg-width="87px" alttext="\mathbb{C}\,\cup\,\{\infty\}" display="inline"><mrow><mpadded width="+1.7pt"><mi href="./front/introduction#Sx4.p1.t1.r1">ℂ</mi></mpadded><mo href="./front/introduction#Sx4.p1.t1.r27" rspace="4.2pt">∪</mo><mrow><mo stretchy="false">{</mo><mi mathvariant="normal">∞</mi><mo stretchy="false">}</mo></mrow></mrow></math> onto itself.
</p>
</div>
<div id="Px23.p3" class="ltx_para">
<p class="ltx_p">The <em class="ltx_emph ltx_font_italic">cross ratio</em>
of <math class="ltx_Math" altimg="m226.png" altimg-height="23px" altimg-valign="-7px" altimg-width="204px" alttext="z_{1},z_{2},z_{3},z_{4}\in\mathbb{C}\cup\{\infty\}" display="inline"><mrow><mrow><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>1</mn></msub><mo>,</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>2</mn></msub><mo>,</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>3</mn></msub><mo>,</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>4</mn></msub></mrow><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mrow><mi href="./front/introduction#Sx4.p1.t1.r1">ℂ</mi><mo href="./front/introduction#Sx4.p1.t1.r27">∪</mo><mrow><mo stretchy="false">{</mo><mi mathvariant="normal">∞</mi><mo stretchy="false">}</mo></mrow></mrow></mrow></math> is defined by
</p>
<table id="E45" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.45</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E45.png" altimg-height="53px" altimg-valign="-21px" altimg-width="169px" alttext="\frac{(z_{1}-z_{2})(z_{3}-z_{4})}{(z_{1}-z_{4})(z_{3}-z_{2})}," display="block"><mrow><mfrac><mrow><mrow><mo stretchy="false">(</mo><mrow><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>1</mn></msub><mo>-</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>2</mn></msub></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>3</mn></msub><mo>-</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>4</mn></msub></mrow><mo stretchy="false">)</mo></mrow></mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>1</mn></msub><mo>-</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>4</mn></msub></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>3</mn></msub><mo>-</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>2</mn></msub></mrow><mo stretchy="false">)</mo></mrow></mrow></mfrac><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E45.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a></dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">or its limiting form, and is invariant under bilinear transformations.</p>
</div>
<div id="Px23.p4" class="ltx_para">
<p class="ltx_p">Other names for the bilinear transformation are <em class="ltx_emph ltx_font_italic">fractional linear
transformation</em>, <em class="ltx_emph ltx_font_italic">homographic transformation</em>, and <em class="ltx_emph ltx_font_italic">Möbius
transformation</em>.</p>
</div>
</section>
</section>
<section id="v" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§1.9(v) </span>Infinite Sequences and Series</h2>
<div id="SS5.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="SS5.p1" class="ltx_para">
<p class="ltx_p">A sequence <math class="ltx_Math" altimg="m147.png" altimg-height="23px" altimg-valign="-7px" altimg-width="44px" alttext="\{z_{n}\}" display="inline"><mrow><mo stretchy="false">{</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo stretchy="false">}</mo></mrow></math> <em class="ltx_emph ltx_font_italic">converges</em>
to <math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math> if <math class="ltx_Math" altimg="m103.png" altimg-height="32px" altimg-valign="-16px" altimg-width="106px" alttext="\lim\limits_{n\to\infty}z_{n}=z" display="inline"><mrow><mrow><munder><mo movablelimits="false">lim</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><mo></mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></mrow><mo>=</mo><mi href="./1.1#p2.t1.r2">z</mi></mrow></math>. For <math class="ltx_Math" altimg="m230.png" altimg-height="20px" altimg-valign="-6px" altimg-width="125px" alttext="z_{n}=x_{n}+iy_{n}" display="inline"><mrow><msub><mi href="./1.1#p2.t1.r2">z</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo>=</mo><mrow><msub><mi>x</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><msub><mi>y</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></mrow></mrow></mrow></math>, the
sequence <math class="ltx_Math" altimg="m147.png" altimg-height="23px" altimg-valign="-7px" altimg-width="44px" alttext="\{z_{n}\}" display="inline"><mrow><mo stretchy="false">{</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo stretchy="false">}</mo></mrow></math> converges iff the sequences <math class="ltx_Math" altimg="m144.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\{x_{n}\}" display="inline"><mrow><mo stretchy="false">{</mo><msub><mi>x</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo stretchy="false">}</mo></mrow></math> and <math class="ltx_Math" altimg="m145.png" altimg-height="23px" altimg-valign="-7px" altimg-width="45px" alttext="\{y_{n}\}" display="inline"><mrow><mo stretchy="false">{</mo><msub><mi>y</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo stretchy="false">}</mo></mrow></math>
separately converge. A <em class="ltx_emph ltx_font_italic">series</em> <math class="ltx_Math" altimg="m136.png" altimg-height="26px" altimg-valign="-8px" altimg-width="80px" alttext="\sum^{\infty}_{n=0}z_{n}" display="inline"><mrow><msubsup><mo largeop="true" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></msubsup><msub><mi href="./1.1#p2.t1.r2">z</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></mrow></math> <em class="ltx_emph ltx_font_italic">converges</em>
if the sequence <math class="ltx_Math" altimg="m191.png" altimg-height="26px" altimg-valign="-8px" altimg-width="124px" alttext="s_{n}=\sum^{n}_{k=0}z_{k}" display="inline"><mrow><msub><mi href="./1.9#SS5.p1">s</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo>=</mo><mrow><msubsup><mo largeop="true" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r4">k</mi><mo>=</mo><mn>0</mn></mrow><mi href="./1.1#p2.t1.r5">n</mi></msubsup><msub><mi href="./1.1#p2.t1.r2">z</mi><mi href="./1.1#p2.t1.r4">k</mi></msub></mrow></mrow></math> converges. The series is
<em class="ltx_emph ltx_font_italic">divergent</em> if <math class="ltx_Math" altimg="m192.png" altimg-height="16px" altimg-valign="-5px" altimg-width="24px" alttext="s_{n}" display="inline"><msub><mi href="./1.9#SS5.p1">s</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></math> does not converge. The series converges
<em class="ltx_emph ltx_font_italic">absolutely</em> if <math class="ltx_Math" altimg="m137.png" altimg-height="26px" altimg-valign="-8px" altimg-width="91px" alttext="\sum^{\infty}_{n=0}|z_{n}|" display="inline"><mrow><msubsup><mo largeop="true" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></msubsup><mrow><mo stretchy="false">|</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo stretchy="false">|</mo></mrow></mrow></math> converges. A series
<math class="ltx_Math" altimg="m136.png" altimg-height="26px" altimg-valign="-8px" altimg-width="80px" alttext="\sum^{\infty}_{n=0}z_{n}" display="inline"><mrow><msubsup><mo largeop="true" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></msubsup><msub><mi href="./1.1#p2.t1.r2">z</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></mrow></math> converges (diverges) absolutely when
<math class="ltx_Math" altimg="m104.png" altimg-height="35px" altimg-valign="-16px" altimg-width="144px" alttext="\lim\limits_{n\to\infty}|z_{n}|^{1/n}<1" display="inline"><mrow><mrow><munder><mo movablelimits="false">lim</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><mo></mo><msup><mrow><mo stretchy="false">|</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo stretchy="false">|</mo></mrow><mrow><mn>1</mn><mo>/</mo><mi href="./1.1#p2.t1.r5">n</mi></mrow></msup></mrow><mo><</mo><mn>1</mn></mrow></math> (<math class="ltx_Math" altimg="m62.png" altimg-height="17px" altimg-valign="-3px" altimg-width="35px" alttext=">1" display="inline"><mrow><mi></mi><mo>></mo><mn>1</mn></mrow></math>), or when
<math class="ltx_Math" altimg="m102.png" altimg-height="32px" altimg-valign="-16px" altimg-width="172px" alttext="\lim\limits_{n\to\infty}\left|\ifrac{z_{n+1}}{z_{n}}\right|<1" display="inline"><mrow><mrow><munder><mo movablelimits="false">lim</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><mo></mo><mrow><mo>|</mo><mrow><msub><mi href="./1.1#p2.t1.r2">z</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>/</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></mrow><mo>|</mo></mrow></mrow><mo><</mo><mn>1</mn></mrow></math> (<math class="ltx_Math" altimg="m62.png" altimg-height="17px" altimg-valign="-3px" altimg-width="35px" alttext=">1" display="inline"><mrow><mi></mi><mo>></mo><mn>1</mn></mrow></math>).
Absolutely convergent series are also convergent.</p>
</div>
<div id="SS5.p2" class="ltx_para">
<p class="ltx_p">Let <math class="ltx_Math" altimg="m143.png" altimg-height="23px" altimg-valign="-7px" altimg-width="70px" alttext="\{f_{n}(z)\}" display="inline"><mrow><mo stretchy="false">{</mo><mrow><msub><mi>f</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">}</mo></mrow></math> be a sequence of functions defined on a set <math class="ltx_Math" altimg="m76.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi href="./1.9#SS5.p2">S</mi></math>. This
sequence <em class="ltx_emph ltx_font_italic">converges pointwise</em>
to a function <math class="ltx_Math" altimg="m161.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(z)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> if</p>
<table id="E46" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.46</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E46.png" altimg-height="34px" altimg-valign="-16px" altimg-width="161px" alttext="f(z)=\lim_{n\to\infty}f_{n}(z)" display="block"><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><munder><mo movablelimits="false">lim</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><mo></mo><mrow><msub><mi>f</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E46.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a> and
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m181.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">for each <math class="ltx_Math" altimg="m219.png" altimg-height="18px" altimg-valign="-3px" altimg-width="52px" alttext="z\in S" display="inline"><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mi href="./1.9#SS5.p2">S</mi></mrow></math>. The sequence <em class="ltx_emph ltx_font_italic">converges uniformly</em>
on <math class="ltx_Math" altimg="m76.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi href="./1.9#SS5.p2">S</mi></math>, if for every <math class="ltx_Math" altimg="m85.png" altimg-height="17px" altimg-valign="-3px" altimg-width="49px" alttext="\epsilon>0" display="inline"><mrow><mi href="./1.9#SS5.p2">ϵ</mi><mo>></mo><mn>0</mn></mrow></math> there exists an integer <math class="ltx_Math" altimg="m74.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="N" display="inline"><mi>N</mi></math>, independent of
<math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>, such that</p>
<table id="E47" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.47</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E47.png" altimg-height="25px" altimg-valign="-7px" altimg-width="160px" alttext="|f_{n}(z)-f(z)|<\epsilon" display="block"><mrow><mrow><mo stretchy="false">|</mo><mrow><mrow><msub><mi>f</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>-</mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo stretchy="false">|</mo></mrow><mo><</mo><mi href="./1.9#SS5.p2">ϵ</mi></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E47.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>,
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m181.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./1.9#SS5.p2" title="§1.9(v) Infinite Sequences and Series ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m86.png" altimg-height="13px" altimg-valign="-2px" altimg-width="12px" alttext="\epsilon" display="inline"><mi href="./1.9#SS5.p2">ϵ</mi></math>: positive number</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">for all <math class="ltx_Math" altimg="m219.png" altimg-height="18px" altimg-valign="-3px" altimg-width="52px" alttext="z\in S" display="inline"><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mi href="./1.9#SS5.p2">S</mi></mrow></math> and <math class="ltx_Math" altimg="m185.png" altimg-height="20px" altimg-valign="-5px" altimg-width="61px" alttext="n\geq N" display="inline"><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>≥</mo><mi>N</mi></mrow></math>.</p>
</div>
<div id="SS5.p3" class="ltx_para">
<p class="ltx_p">A series
<math class="ltx_Math" altimg="m135.png" altimg-height="26px" altimg-valign="-8px" altimg-width="106px" alttext="\sum^{\infty}_{n=0}f_{n}(z)" display="inline"><mrow><msubsup><mo largeop="true" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></msubsup><mrow><msub><mi>f</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></math> <em class="ltx_emph ltx_font_italic">converges uniformly</em> on <math class="ltx_Math" altimg="m76.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi href="./1.9#SS5.p2">S</mi></math>, if the sequence
<math class="ltx_Math" altimg="m190.png" altimg-height="26px" altimg-valign="-8px" altimg-width="176px" alttext="s_{n}(z)=\sum^{n}_{k=0}f_{k}(z)" display="inline"><mrow><mrow><msub><mi href="./1.9#SS5.p1">s</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><msubsup><mo largeop="true" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r4">k</mi><mo>=</mo><mn>0</mn></mrow><mi href="./1.1#p2.t1.r5">n</mi></msubsup><mrow><msub><mi>f</mi><mi href="./1.1#p2.t1.r4">k</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow></math> converges uniformly on <math class="ltx_Math" altimg="m76.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi href="./1.9#SS5.p2">S</mi></math>.</p>
</div>
<section id="Px24" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Weierstrass <math class="ltx_Math" altimg="m72.png" altimg-height="18px" altimg-valign="-2px" altimg-width="26px" alttext="M" display="inline"><mi href="./1.9#Px24">M</mi></math>-test</h3>
<div id="Px24.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Keywords:</dt>
<dd>
<a class="ltx_keyword" href="./idx/M#Mtestforuniformconvergence"><math class="ltx_Math" altimg="m72.png" altimg-height="18px" altimg-valign="-2px" altimg-width="26px" alttext="M" display="inline"><mi href="./1.9#Px24">M</mi></math>-test for uniform convergence</a>, <a class="ltx_keyword" href="./idx/W#WeierstrassMtest">Weierstrass <math class="ltx_Math" altimg="m72.png" altimg-height="18px" altimg-valign="-2px" altimg-width="26px" alttext="M" display="inline"><mi href="./1.9#Px24">M</mi></math>-test</a>, </dd>
</dl>
</div>
</div>
<div id="Px24.p1" class="ltx_para">
<p class="ltx_p">Suppose <math class="ltx_Math" altimg="m142.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="\{M_{n}\}" display="inline"><mrow><mo stretchy="false">{</mo><msub><mi href="./1.9#Px24">M</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo stretchy="false">}</mo></mrow></math> is a sequence of real numbers such that
<math class="ltx_Math" altimg="m132.png" altimg-height="26px" altimg-valign="-8px" altimg-width="90px" alttext="\sum^{\infty}_{n=0}M_{n}" display="inline"><mrow><msubsup><mo largeop="true" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></msubsup><msub><mi href="./1.9#Px24">M</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></mrow></math> converges and <math class="ltx_Math" altimg="m235.png" altimg-height="23px" altimg-valign="-7px" altimg-width="118px" alttext="|f_{n}(z)|\leq M_{n}" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mrow><msub><mi>f</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow><mo>≤</mo><msub><mi href="./1.9#Px24">M</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></mrow></math> for all <math class="ltx_Math" altimg="m219.png" altimg-height="18px" altimg-valign="-3px" altimg-width="52px" alttext="z\in S" display="inline"><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mi href="./1.9#SS5.p2">S</mi></mrow></math>
and all <math class="ltx_Math" altimg="m182.png" altimg-height="19px" altimg-valign="-5px" altimg-width="53px" alttext="n\geq 0" display="inline"><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>≥</mo><mn>0</mn></mrow></math>. Then the series <math class="ltx_Math" altimg="m135.png" altimg-height="26px" altimg-valign="-8px" altimg-width="106px" alttext="\sum^{\infty}_{n=0}f_{n}(z)" display="inline"><mrow><msubsup><mo largeop="true" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></msubsup><mrow><msub><mi>f</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></math> converges
uniformly on <math class="ltx_Math" altimg="m76.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi href="./1.9#SS5.p2">S</mi></math>.
</p>
</div>
<div id="Px24.p2" class="ltx_para">
<p class="ltx_p">A doubly-infinite series
<math class="ltx_Math" altimg="m131.png" altimg-height="27px" altimg-valign="-10px" altimg-width="126px" alttext="\sum^{\infty}_{n=-\infty}f_{n}(z)" display="inline"><mrow><msubsup><mo largeop="true" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow></mrow><mi mathvariant="normal">∞</mi></msubsup><mrow><msub><mi>f</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></math> converges (uniformly) on <math class="ltx_Math" altimg="m76.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi href="./1.9#SS5.p2">S</mi></math> iff each of the
series <math class="ltx_Math" altimg="m135.png" altimg-height="26px" altimg-valign="-8px" altimg-width="106px" alttext="\sum^{\infty}_{n=0}f_{n}(z)" display="inline"><mrow><msubsup><mo largeop="true" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></msubsup><mrow><msub><mi>f</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></math> and <math class="ltx_Math" altimg="m138.png" altimg-height="26px" altimg-valign="-8px" altimg-width="118px" alttext="\sum^{\infty}_{n=1}f_{-n}(z)" display="inline"><mrow><msubsup><mo largeop="true" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></msubsup><mrow><msub><mi>f</mi><mrow><mo>-</mo><mi href="./1.1#p2.t1.r5">n</mi></mrow></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></math> converges
(uniformly) on <math class="ltx_Math" altimg="m76.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi href="./1.9#SS5.p2">S</mi></math>.
</p>
</div>
</section>
</section>
<section id="vi" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§1.9(vi) </span>Power Series</h2>
<div id="SS6.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="SS6.p1" class="ltx_para">
<p class="ltx_p">For a series <math class="ltx_Math" altimg="m133.png" altimg-height="26px" altimg-valign="-8px" altimg-width="160px" alttext="\sum^{\infty}_{n=0}a_{n}(z-z_{0})^{n}" display="inline"><mrow><msubsup><mo largeop="true" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></msubsup><mrow><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>-</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></mrow><mo stretchy="false">)</mo></mrow><mi href="./1.1#p2.t1.r5">n</mi></msup></mrow></mrow></math> there is a number <math class="ltx_Math" altimg="m75.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="R" display="inline"><mi href="./1.9#E49">R</mi></math>, <math class="ltx_Math" altimg="m58.png" altimg-height="20px" altimg-valign="-5px" altimg-width="103px" alttext="0\leq R\leq\infty" display="inline"><mrow><mn>0</mn><mo>≤</mo><mi href="./1.9#E49">R</mi><mo>≤</mo><mi mathvariant="normal">∞</mi></mrow></math>, such that the series converges for all <math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math> in <math class="ltx_Math" altimg="m237.png" altimg-height="23px" altimg-valign="-7px" altimg-width="110px" alttext="|z-z_{0}|<R" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>-</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></mrow><mo stretchy="false">|</mo></mrow><mo><</mo><mi href="./1.9#E49">R</mi></mrow></math> and
diverges for <math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math> in <math class="ltx_Math" altimg="m240.png" altimg-height="23px" altimg-valign="-7px" altimg-width="110px" alttext="|z-z_{0}|>R" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>-</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></mrow><mo stretchy="false">|</mo></mrow><mo>></mo><mi href="./1.9#E49">R</mi></mrow></math>. The circle <math class="ltx_Math" altimg="m239.png" altimg-height="23px" altimg-valign="-7px" altimg-width="110px" alttext="|z-z_{0}|=R" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>-</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></mrow><mo stretchy="false">|</mo></mrow><mo>=</mo><mi href="./1.9#E49">R</mi></mrow></math> is called the
<em class="ltx_emph ltx_font_italic">circle of convergence</em>
of the series, and <math class="ltx_Math" altimg="m75.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="R" display="inline"><mi href="./1.9#E49">R</mi></math> is the <em class="ltx_emph ltx_font_italic">radius of convergence</em>. Inside the circle
the sum of the series is an analytic function <math class="ltx_Math" altimg="m161.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(z)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math>. For <math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math> in
<math class="ltx_Math" altimg="m241.png" altimg-height="23px" altimg-valign="-7px" altimg-width="105px" alttext="|z-z_{0}|\leq\rho" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>-</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></mrow><mo stretchy="false">|</mo></mrow><mo>≤</mo><mi>ρ</mi></mrow></math> (<math class="ltx_Math" altimg="m60.png" altimg-height="18px" altimg-valign="-3px" altimg-width="41px" alttext="<R" display="inline"><mrow><mi></mi><mo><</mo><mi href="./1.9#E49">R</mi></mrow></math>), the convergence is absolute and
uniform. Moreover,</p>
<table id="E48" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.48</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E48.png" altimg-height="50px" altimg-valign="-16px" altimg-width="133px" alttext="a_{n}=\frac{f^{(n)}(z_{0})}{n!}," display="block"><mrow><mrow><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo>=</mo><mfrac><mrow><msup><mi>f</mi><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r5">n</mi><mo stretchy="false">)</mo></mrow></msup><mo></mo><mrow><mo stretchy="false">(</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mrow><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mfrac></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E48.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m177.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a> and
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m181.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and</p>
<table id="E49" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.49</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E49.png" altimg-height="38px" altimg-valign="-16px" altimg-width="184px" alttext="R=\liminf_{n\to\infty}|a_{n}|^{-1/n}." display="block"><mrow><mrow><mi href="./1.9#E49">R</mi><mo>=</mo><mrow><munder><mo href="./front/introduction#Sx4.p2.t1.r4" movablelimits="false">lim inf</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><mo></mo><msup><mrow><mo stretchy="false">|</mo><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo stretchy="false">|</mo></mrow><mrow><mo>-</mo><mrow><mn>1</mn><mo>/</mo><mi href="./1.1#p2.t1.r5">n</mi></mrow></mrow></msup></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E49.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text"><math class="ltx_Math" altimg="m75.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="R" display="inline"><mi href="./1.9#E49">R</mi></math>: radius of convergence (locally)</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./front/introduction#Sx4.p2.t1.r4" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m107.png" altimg-height="18px" altimg-valign="-2px" altimg-width="60px" alttext="\liminf" display="inline"><mo href="./front/introduction#Sx4.p2.t1.r4">lim inf</mo></math>: least limit point</a> and
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m181.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS6.p2" class="ltx_para">
<p class="ltx_p">For the converse of this result see §</dd>
</dl>
</div>
</div>
<div id="Px25.p1" class="ltx_para">
<p class="ltx_p">When <math class="ltx_Math" altimg="m129.png" altimg-height="23px" altimg-valign="-7px" altimg-width="71px" alttext="\sum a_{n}z^{n}" display="inline"><mrow><mo largeop="true" symmetric="true">∑</mo><mrow><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><msup><mi href="./1.1#p2.t1.r2">z</mi><mi href="./1.1#p2.t1.r5">n</mi></msup></mrow></mrow></math> and <math class="ltx_Math" altimg="m130.png" altimg-height="23px" altimg-valign="-7px" altimg-width="69px" alttext="\sum b_{n}z^{n}" display="inline"><mrow><mo largeop="true" symmetric="true">∑</mo><mrow><msub><mi>b</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><msup><mi href="./1.1#p2.t1.r2">z</mi><mi href="./1.1#p2.t1.r5">n</mi></msup></mrow></mrow></math> both converge</p>
<table id="E50" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.50</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E50.png" altimg-height="64px" altimg-valign="-27px" altimg-width="348px" alttext="\sum^{\infty}_{n=0}(a_{n}\pm b_{n})z^{n}=\sum^{\infty}_{n=0}a_{n}z^{n}\pm\sum^%
{\infty}_{n=0}b_{n}z^{n}," display="block"><mrow><mrow><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><mrow><mo stretchy="false">(</mo><mrow><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo>±</mo><msub><mi>b</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msup><mi href="./1.1#p2.t1.r2">z</mi><mi href="./1.1#p2.t1.r5">n</mi></msup></mrow></mrow><mo>=</mo><mrow><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><msup><mi href="./1.1#p2.t1.r2">z</mi><mi href="./1.1#p2.t1.r5">n</mi></msup></mrow></mrow><mo>±</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><msub><mi>b</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><msup><mi href="./1.1#p2.t1.r2">z</mi><mi href="./1.1#p2.t1.r5">n</mi></msup></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E50.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a> and
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m181.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and</p>
<table id="E51" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.51</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E51.png" altimg-height="66px" altimg-valign="-27px" altimg-width="328px" alttext="\left(\sum^{\infty}_{n=0}a_{n}z^{n}\right)\left(\sum^{\infty}_{n=0}b_{n}z^{n}%
\right)=\sum^{\infty}_{n=0}c_{n}z^{n}," display="block"><mrow><mrow><mrow><mrow><mo>(</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><msup><mi href="./1.1#p2.t1.r2">z</mi><mi href="./1.1#p2.t1.r5">n</mi></msup></mrow></mrow><mo>)</mo></mrow><mo></mo><mrow><mo>(</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><msub><mi>b</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><msup><mi href="./1.1#p2.t1.r2">z</mi><mi href="./1.1#p2.t1.r5">n</mi></msup></mrow></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><msub><mi>c</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><msup><mi href="./1.1#p2.t1.r2">z</mi><mi href="./1.1#p2.t1.r5">n</mi></msup></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E51.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a> and
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m181.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where</p>
<table id="E52" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.52</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E52.png" altimg-height="64px" altimg-valign="-28px" altimg-width="151px" alttext="c_{n}=\sum^{n}_{k=0}a_{k}b_{n-k}." display="block"><mrow><mrow><msub><mi>c</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo>=</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r4">k</mi><mo>=</mo><mn>0</mn></mrow><mi href="./1.1#p2.t1.r5">n</mi></munderover><mrow><msub><mi>a</mi><mi href="./1.1#p2.t1.r4">k</mi></msub><mo></mo><msub><mi>b</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mi href="./1.1#p2.t1.r4">k</mi></mrow></msub></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E52.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.1#p2.t1.r4" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m173.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./1.1#p2.t1.r4">k</mi></math>: integer</a> and
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m181.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="Px25.p2" class="ltx_para">
<p class="ltx_p">Next, let</p>
<table id="E53" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.53</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E53.png" altimg-height="28px" altimg-valign="-7px" altimg-width="263px" alttext="f(z)=a_{0}+a_{1}z+a_{2}z^{2}+\cdots," display="block"><mrow><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><msub><mi>a</mi><mn>0</mn></msub><mo>+</mo><mrow><msub><mi>a</mi><mn>1</mn></msub><mo></mo><mi href="./1.1#p2.t1.r2">z</mi></mrow><mo>+</mo><mrow><msub><mi>a</mi><mn>2</mn></msub><mo></mo><msup><mi href="./1.1#p2.t1.r2">z</mi><mn>2</mn></msup></mrow><mo>+</mo><mi mathvariant="normal">⋯</mi></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m153.png" altimg-height="21px" altimg-valign="-6px" altimg-width="60px" alttext="a_{0}\not=0" display="inline"><mrow><msub><mi>a</mi><mn>0</mn></msub><mo>≠</mo><mn>0</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E53.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a></dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Then the expansions () hold for all sufficiently small <math class="ltx_Math" altimg="m243.png" altimg-height="23px" altimg-valign="-7px" altimg-width="26px" alttext="|z|" display="inline"><mrow><mo stretchy="false">|</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">|</mo></mrow></math>.</p>
<table id="E54" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.54</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E54.png" altimg-height="51px" altimg-valign="-21px" altimg-width="262px" alttext="\frac{1}{f(z)}=b_{0}+b_{1}z+b_{2}z^{2}+\cdots," display="block"><mrow><mrow><mfrac><mn>1</mn><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></mfrac><mo>=</mo><mrow><msub><mi>b</mi><mn>0</mn></msub><mo>+</mo><mrow><msub><mi>b</mi><mn>1</mn></msub><mo></mo><mi href="./1.1#p2.t1.r2">z</mi></mrow><mo>+</mo><mrow><msub><mi>b</mi><mn>2</mn></msub><mo></mo><msup><mi href="./1.1#p2.t1.r2">z</mi><mn>2</mn></msup></mrow><mo>+</mo><mi mathvariant="normal">⋯</mi></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E54.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a></dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where</p>
<table id="E55" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex13" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="3" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">1.9.55</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m31.png" altimg-height="22px" altimg-valign="-5px" altimg-width="24px" alttext="\displaystyle b_{0}" display="inline"><msub><mi>b</mi><mn>0</mn></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m8.png" altimg-height="25px" altimg-valign="-7px" altimg-width="72px" alttext="\displaystyle=1/a_{0}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mn>1</mn><mo>/</mo><msub><mi>a</mi><mn>0</mn></msub></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex14" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m32.png" altimg-height="22px" altimg-valign="-5px" altimg-width="24px" alttext="\displaystyle b_{1}" display="inline"><msub><mi>b</mi><mn>1</mn></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m6.png" altimg-height="28px" altimg-valign="-7px" altimg-width="97px" alttext="\displaystyle=-a_{1}/a_{0}^{2}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo>-</mo><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>/</mo><msubsup><mi>a</mi><mn>0</mn><mn>2</mn></msubsup></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex15" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m33.png" altimg-height="22px" altimg-valign="-5px" altimg-width="24px" alttext="\displaystyle b_{2}" display="inline"><msub><mi>b</mi><mn>2</mn></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m3.png" altimg-height="28px" altimg-valign="-7px" altimg-width="160px" alttext="\displaystyle=(a_{1}^{2}-a_{0}a_{2})/a_{0}^{3}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><msubsup><mi>a</mi><mn>1</mn><mn>2</mn></msubsup><mo>-</mo><mrow><msub><mi>a</mi><mn>0</mn></msub><mo></mo><msub><mi>a</mi><mn>2</mn></msub></mrow></mrow><mo stretchy="false">)</mo></mrow><mo>/</mo><msubsup><mi>a</mi><mn>0</mn><mn>3</mn></msubsup></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E55.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E56" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.56</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E56.png" altimg-height="25px" altimg-valign="-7px" altimg-width="372px" alttext="b_{n}=-(a_{1}b_{n-1}+a_{2}b_{n-2}+\dots+a_{n}b_{0})/a_{0}," display="block"><mrow><mrow><msub><mi>b</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo>=</mo><mrow><mo>-</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><msub><mi>a</mi><mn>1</mn></msub><mo></mo><msub><mi>b</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow></msub></mrow><mo>+</mo><mrow><msub><mi>a</mi><mn>2</mn></msub><mo></mo><msub><mi>b</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>2</mn></mrow></msub></mrow><mo>+</mo><mi mathvariant="normal">…</mi><mo>+</mo><mrow><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><msub><mi>b</mi><mn>0</mn></msub></mrow></mrow><mo stretchy="false">)</mo></mrow><mo>/</mo><msub><mi>a</mi><mn>0</mn></msub></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m183.png" altimg-height="19px" altimg-valign="-5px" altimg-width="53px" alttext="n\geq 1" display="inline"><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>≥</mo><mn>1</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E56.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m181.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a></dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="Px25.p3" class="ltx_para">
<p class="ltx_p">With <math class="ltx_Math" altimg="m152.png" altimg-height="20px" altimg-valign="-5px" altimg-width="60px" alttext="a_{0}=1" display="inline"><mrow><msub><mi>a</mi><mn>0</mn></msub><mo>=</mo><mn>1</mn></mrow></math>,</p>
<table id="E57" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.57</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E57.png" altimg-height="28px" altimg-valign="-7px" altimg-width="297px" alttext="\ln f(z)=q_{1}z+q_{2}z^{2}+q_{3}z^{3}+\cdots," display="block"><mrow><mrow><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>=</mo><mrow><mrow><msub><mi href="./1.9#Px25.p3">q</mi><mn>1</mn></msub><mo></mo><mi href="./1.1#p2.t1.r2">z</mi></mrow><mo>+</mo><mrow><msub><mi href="./1.9#Px25.p3">q</mi><mn>2</mn></msub><mo></mo><msup><mi href="./1.1#p2.t1.r2">z</mi><mn>2</mn></msup></mrow><mo>+</mo><mrow><msub><mi href="./1.9#Px25.p3">q</mi><mn>3</mn></msub><mo></mo><msup><mi href="./1.1#p2.t1.r2">z</mi><mn>3</mn></msup></mrow><mo>+</mo><mi mathvariant="normal">⋯</mi></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E57.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m109.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a>,
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a> and
<a href="./1.9#Px25.p3" title="Operations ‣ §1.9(vi) Power Series ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m188.png" altimg-height="18px" altimg-valign="-8px" altimg-width="22px" alttext="q_{j}" display="inline"><msub><mi href="./1.9#Px25.p3">q</mi><mi href="./1.1#p2.t1.r4">j</mi></msub></math>: coefficients</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">(principal value), where
</p>
<table id="E58" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex16" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="3" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">1.9.58</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m39.png" altimg-height="18px" altimg-valign="-6px" altimg-width="24px" alttext="\displaystyle q_{1}" display="inline"><msub><mi href="./1.9#Px25.p3">q</mi><mn>1</mn></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m15.png" altimg-height="18px" altimg-valign="-6px" altimg-width="52px" alttext="\displaystyle=a_{1}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><msub><mi>a</mi><mn>1</mn></msub></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex17" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m40.png" altimg-height="18px" altimg-valign="-6px" altimg-width="24px" alttext="\displaystyle q_{2}" display="inline"><msub><mi href="./1.9#Px25.p3">q</mi><mn>2</mn></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m1.png" altimg-height="28px" altimg-valign="-7px" altimg-width="141px" alttext="\displaystyle=(2a_{2}-a_{1}^{2})/2," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>2</mn><mo></mo><msub><mi>a</mi><mn>2</mn></msub></mrow><mo>-</mo><msubsup><mi>a</mi><mn>1</mn><mn>2</mn></msubsup></mrow><mo stretchy="false">)</mo></mrow><mo>/</mo><mn>2</mn></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex18" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m41.png" altimg-height="18px" altimg-valign="-6px" altimg-width="24px" alttext="\displaystyle q_{3}" display="inline"><msub><mi href="./1.9#Px25.p3">q</mi><mn>3</mn></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m2.png" altimg-height="28px" altimg-valign="-7px" altimg-width="215px" alttext="\displaystyle=(3a_{3}-3a_{1}a_{2}+a_{1}^{3})/3," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mrow><mn>3</mn><mo></mo><msub><mi>a</mi><mn>3</mn></msub></mrow><mo>-</mo><mrow><mn>3</mn><mo></mo><msub><mi>a</mi><mn>1</mn></msub><mo></mo><msub><mi>a</mi><mn>2</mn></msub></mrow></mrow><mo>+</mo><msubsup><mi>a</mi><mn>1</mn><mn>3</mn></msubsup></mrow><mo stretchy="false">)</mo></mrow><mo>/</mo><mn>3</mn></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E58.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./1.9#Px25.p3" title="Operations ‣ §1.9(vi) Power Series ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m188.png" altimg-height="18px" altimg-valign="-8px" altimg-width="22px" alttext="q_{j}" display="inline"><msub><mi href="./1.9#Px25.p3">q</mi><mi href="./1.1#p2.t1.r4">j</mi></msub></math>: coefficients</a></dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and</p>
<table id="E59" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.59</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E59.png" altimg-height="25px" altimg-valign="-7px" altimg-width="552px" alttext="q_{n}=(na_{n}-(n-1)a_{1}q_{n-1}-(n-2)a_{2}q_{n-2}-\cdots-a_{n-1}q_{1})/n," display="block"><mrow><mrow><msub><mi href="./1.9#Px25.p3">q</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo>=</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo></mo><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></mrow><mo>-</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msub><mi>a</mi><mn>1</mn></msub><mo></mo><msub><mi href="./1.9#Px25.p3">q</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow></msub></mrow><mo>-</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>2</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msub><mi>a</mi><mn>2</mn></msub><mo></mo><msub><mi href="./1.9#Px25.p3">q</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>2</mn></mrow></msub></mrow><mo>-</mo><mi mathvariant="normal">⋯</mi><mo>-</mo><mrow><msub><mi>a</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow></msub><mo></mo><msub><mi href="./1.9#Px25.p3">q</mi><mn>1</mn></msub></mrow></mrow><mo stretchy="false">)</mo></mrow><mo>/</mo><mi href="./1.1#p2.t1.r5">n</mi></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m184.png" altimg-height="19px" altimg-valign="-5px" altimg-width="53px" alttext="n\geq 2" display="inline"><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>≥</mo><mn>2</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E59.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m181.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./1.9#Px25.p3" title="Operations ‣ §1.9(vi) Power Series ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m188.png" altimg-height="18px" altimg-valign="-8px" altimg-width="22px" alttext="q_{j}" display="inline"><msub><mi href="./1.9#Px25.p3">q</mi><mi href="./1.1#p2.t1.r4">j</mi></msub></math>: coefficients</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Also,</p>
<table id="E60" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.60</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E60.png" altimg-height="28px" altimg-valign="-7px" altimg-width="287px" alttext="(f(z))^{\nu}=p_{0}+p_{1}z+p_{2}z^{2}+\cdots," display="block"><mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow><mi href="./1.9#Px25.p3">ν</mi></msup><mo>=</mo><mrow><msub><mi href="./1.9#Px25.p3">p</mi><mn>0</mn></msub><mo>+</mo><mrow><msub><mi href="./1.9#Px25.p3">p</mi><mn>1</mn></msub><mo></mo><mi href="./1.1#p2.t1.r2">z</mi></mrow><mo>+</mo><mrow><msub><mi href="./1.9#Px25.p3">p</mi><mn>2</mn></msub><mo></mo><msup><mi href="./1.1#p2.t1.r2">z</mi><mn>2</mn></msup></mrow><mo>+</mo><mi mathvariant="normal">⋯</mi></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E60.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>,
<a href="./1.9#Px25.p3" title="Operations ‣ §1.9(vi) Power Series ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m120.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./1.9#Px25.p3">ν</mi></math>: complex</a> and
<a href="./1.9#Px25.p3" title="Operations ‣ §1.9(vi) Power Series ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m187.png" altimg-height="18px" altimg-valign="-8px" altimg-width="23px" alttext="p_{j}" display="inline"><msub><mi href="./1.9#Px25.p3">p</mi><mi href="./1.1#p2.t1.r4">j</mi></msub></math>: coefficients</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">(principal value), where <math class="ltx_Math" altimg="m121.png" altimg-height="18px" altimg-valign="-3px" altimg-width="54px" alttext="\nu\in\mathbb{C}" display="inline"><mrow><mi href="./1.9#Px25.p3">ν</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mi href="./front/introduction#Sx4.p1.t1.r1">ℂ</mi></mrow></math>,
</p>
<table id="E61" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex19" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="3" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">1.9.61</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m36.png" altimg-height="18px" altimg-valign="-6px" altimg-width="25px" alttext="\displaystyle p_{0}" display="inline"><msub><mi href="./1.9#Px25.p3">p</mi><mn>0</mn></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m7.png" altimg-height="22px" altimg-valign="-6px" altimg-width="43px" alttext="\displaystyle=1," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mn>1</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex20" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m37.png" altimg-height="18px" altimg-valign="-6px" altimg-width="25px" alttext="\displaystyle p_{1}" display="inline"><msub><mi href="./1.9#Px25.p3">p</mi><mn>1</mn></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m11.png" altimg-height="18px" altimg-valign="-6px" altimg-width="63px" alttext="\displaystyle=\nu a_{1}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./1.9#Px25.p3">ν</mi><mo></mo><msub><mi>a</mi><mn>1</mn></msub></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex21" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m38.png" altimg-height="18px" altimg-valign="-6px" altimg-width="25px" alttext="\displaystyle p_{2}" display="inline"><msub><mi href="./1.9#Px25.p3">p</mi><mn>2</mn></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m12.png" altimg-height="28px" altimg-valign="-7px" altimg-width="213px" alttext="\displaystyle=\nu((\nu-1)a_{1}^{2}+2a_{2})/2," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mi href="./1.9#Px25.p3">ν</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./1.9#Px25.p3">ν</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msubsup><mi>a</mi><mn>1</mn><mn>2</mn></msubsup></mrow><mo>+</mo><mrow><mn>2</mn><mo></mo><msub><mi>a</mi><mn>2</mn></msub></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>/</mo><mn>2</mn></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E61.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.9#Px25.p3" title="Operations ‣ §1.9(vi) Power Series ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m120.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./1.9#Px25.p3">ν</mi></math>: complex</a> and
<a href="./1.9#Px25.p3" title="Operations ‣ §1.9(vi) Power Series ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m187.png" altimg-height="18px" altimg-valign="-8px" altimg-width="23px" alttext="p_{j}" display="inline"><msub><mi href="./1.9#Px25.p3">p</mi><mi href="./1.1#p2.t1.r4">j</mi></msub></math>: coefficients</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and</p>
<table id="E62" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.62</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E62.png" altimg-height="25px" altimg-valign="-7px" altimg-width="771px" alttext="p_{n}=((\nu-n+1)a_{1}p_{n-1}+(2\nu-n+2)a_{2}p_{n-2}+\dots+((n-1)\nu-1)a_{n-1}p%
_{1}+n\nu a_{n})/n," display="block"><mrow><mrow><msub><mi href="./1.9#Px25.p3">p</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo>=</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mi href="./1.9#Px25.p3">ν</mi><mo>-</mo><mi href="./1.1#p2.t1.r5">n</mi></mrow><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msub><mi>a</mi><mn>1</mn></msub><mo></mo><msub><mi href="./1.9#Px25.p3">p</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow></msub></mrow><mo>+</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mrow><mn>2</mn><mo></mo><mi href="./1.9#Px25.p3">ν</mi></mrow><mo>-</mo><mi href="./1.1#p2.t1.r5">n</mi></mrow><mo>+</mo><mn>2</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msub><mi>a</mi><mn>2</mn></msub><mo></mo><msub><mi href="./1.9#Px25.p3">p</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>2</mn></mrow></msub></mrow><mo>+</mo><mi mathvariant="normal">…</mi><mo>+</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi href="./1.9#Px25.p3">ν</mi></mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msub><mi>a</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow></msub><mo></mo><msub><mi href="./1.9#Px25.p3">p</mi><mn>1</mn></msub></mrow><mo>+</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo></mo><mi href="./1.9#Px25.p3">ν</mi><mo></mo><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></mrow></mrow><mo stretchy="false">)</mo></mrow><mo>/</mo><mi href="./1.1#p2.t1.r5">n</mi></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m183.png" altimg-height="19px" altimg-valign="-5px" altimg-width="53px" alttext="n\geq 1" display="inline"><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>≥</mo><mn>1</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E62.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m181.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a>,
<a href="./1.9#Px25.p3" title="Operations ‣ §1.9(vi) Power Series ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m120.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./1.9#Px25.p3">ν</mi></math>: complex</a> and
<a href="./1.9#Px25.p3" title="Operations ‣ §1.9(vi) Power Series ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m187.png" altimg-height="18px" altimg-valign="-8px" altimg-width="23px" alttext="p_{j}" display="inline"><msub><mi href="./1.9#Px25.p3">p</mi><mi href="./1.1#p2.t1.r4">j</mi></msub></math>: coefficients</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">For the definitions of the principal values of <math class="ltx_Math" altimg="m108.png" altimg-height="23px" altimg-valign="-7px" altimg-width="62px" alttext="\ln f(z)" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></math> and <math class="ltx_Math" altimg="m51.png" altimg-height="23px" altimg-valign="-7px" altimg-width="67px" alttext="(f(z))^{\nu}" display="inline"><msup><mrow><mo stretchy="false">(</mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow><mi href="./1.9#Px25.p3">ν</mi></msup></math>
see §§.</p>
</div>
<div id="Px25.p4" class="ltx_para">
<p class="ltx_p">Lastly, a power series can be differentiated any number of times within its
circle of convergence:</p>
<table id="E63" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.63</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E63.png" altimg-height="64px" altimg-valign="-27px" altimg-width="340px" alttext="f^{(m)}(z)=\sum_{n=0}^{\infty}{\left(n+1\right)_{m}}a_{n+m}(z-z_{0})^{n}," display="block"><mrow><mrow><mrow><msup><mi>f</mi><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r5">m</mi><mo stretchy="false">)</mo></mrow></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><msub><mrow><mo href="./5.2#iii">(</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>+</mo><mn>1</mn></mrow><mo href="./5.2#iii">)</mo></mrow><mi href="./1.1#p2.t1.r5">m</mi></msub><mo></mo><msub><mi>a</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>+</mo><mi href="./1.1#p2.t1.r5">m</mi></mrow></msub><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>-</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></mrow><mo stretchy="false">)</mo></mrow><mi href="./1.1#p2.t1.r5">n</mi></msup></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m99.png" altimg-height="23px" altimg-valign="-7px" altimg-width="110px" alttext="\left|z-z_{0}\right|<R" display="inline"><mrow><mrow><mo>|</mo><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>-</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></mrow><mo>|</mo></mrow><mo><</mo><mi href="./1.9#E49">R</mi></mrow></math>, <math class="ltx_Math" altimg="m175.png" altimg-height="20px" altimg-valign="-6px" altimg-width="128px" alttext="m=0,1,2,\dots" display="inline"><mrow><mi href="./1.1#p2.t1.r5">m</mi><mo>=</mo><mrow><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E63.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#iii" title="§5.2(iii) Pochhammer’s Symbol ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m231.png" altimg-height="24px" altimg-valign="-8px" altimg-width="41px" alttext="{\left(\NVar{a}\right)_{\NVar{n}}}" display="inline"><msub><mrow><mo href="./5.2#iii">(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r5">a</mi><mo href="./5.2#iii">)</mo></mrow><mi class="ltx_nvar" href="./5.1#p2.t1.r1">n</mi></msub></math>: Pochhammer’s symbol (or shifted factorial)</a>,
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>,
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./1.1#p2.t1.r5">m</mi></math>: nonnegative integer</a>,
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m181.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./1.9#E49" title="(1.9.49) ‣ §1.9(vi) Power Series ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m75.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="R" display="inline"><mi href="./1.9#E49">R</mi></math>: radius of convergence</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
</section>
<section id="vii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§1.9(vii) </span>Inversion of Limits</h2>
<div id="SS7.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px26.p1" class="ltx_para">
<p class="ltx_p">A set of complex numbers <math class="ltx_Math" altimg="m146.png" altimg-height="24px" altimg-valign="-8px" altimg-width="63px" alttext="\{z_{m,n}\}" display="inline"><mrow><mo stretchy="false">{</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mrow><mi href="./1.1#p2.t1.r5">m</mi><mo>,</mo><mi href="./1.1#p2.t1.r5">n</mi></mrow></msub><mo stretchy="false">}</mo></mrow></math> where <math class="ltx_Math" altimg="m176.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./1.1#p2.t1.r5">m</mi></math> and <math class="ltx_Math" altimg="m181.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math> take all positive
integer values is called a <em class="ltx_emph ltx_font_italic">double sequence</em>. It <em class="ltx_emph ltx_font_italic">converges to</em> <math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>
if for every <math class="ltx_Math" altimg="m85.png" altimg-height="17px" altimg-valign="-3px" altimg-width="49px" alttext="\epsilon>0" display="inline"><mrow><mi href="./1.9#Px26.p1">ϵ</mi><mo>></mo><mn>0</mn></mrow></math>, there is an integer <math class="ltx_Math" altimg="m74.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="N" display="inline"><mi>N</mi></math> such that
</p>
<table id="E64" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.64</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E64.png" altimg-height="26px" altimg-valign="-8px" altimg-width="125px" alttext="|z_{m,n}-z|<\epsilon" display="block"><mrow><mrow><mo stretchy="false">|</mo><mrow><msub><mi href="./1.1#p2.t1.r2">z</mi><mrow><mi href="./1.1#p2.t1.r5">m</mi><mo>,</mo><mi href="./1.1#p2.t1.r5">n</mi></mrow></msub><mo>-</mo><mi href="./1.1#p2.t1.r2">z</mi></mrow><mo stretchy="false">|</mo></mrow><mo><</mo><mi href="./1.9#Px26.p1">ϵ</mi></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E64.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>,
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./1.1#p2.t1.r5">m</mi></math>: nonnegative integer</a>,
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m181.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./1.9#Px26.p1" title="Double Sequences and Series ‣ §1.9(vii) Inversion of Limits ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m86.png" altimg-height="13px" altimg-valign="-2px" altimg-width="12px" alttext="\epsilon" display="inline"><mi href="./1.9#Px26.p1">ϵ</mi></math>: positive number</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">for all <math class="ltx_Math" altimg="m174.png" altimg-height="21px" altimg-valign="-6px" altimg-width="87px" alttext="m,n\geq N" display="inline"><mrow><mrow><mi href="./1.1#p2.t1.r5">m</mi><mo>,</mo><mi href="./1.1#p2.t1.r5">n</mi></mrow><mo>≥</mo><mi>N</mi></mrow></math>. Suppose <math class="ltx_Math" altimg="m146.png" altimg-height="24px" altimg-valign="-8px" altimg-width="63px" alttext="\{z_{m,n}\}" display="inline"><mrow><mo stretchy="false">{</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mrow><mi href="./1.1#p2.t1.r5">m</mi><mo>,</mo><mi href="./1.1#p2.t1.r5">n</mi></mrow></msub><mo stretchy="false">}</mo></mrow></math> converges to <math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math> and the repeated
limits</p>
<table id="E65" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex22" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">1.9.65</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E65a.png" altimg-height="24px" altimg-valign="-8px" altimg-width="219px" alttext="\lim_{m\to\infty}\left(\lim_{n\to\infty}z_{m,n}\right)," display="inline"><mrow><mrow><munder><mo movablelimits="false">lim</mo><mrow><mi href="./1.1#p2.t1.r5">m</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><mo></mo><mrow><mo>(</mo><mrow><munder><mo movablelimits="false">lim</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><mo></mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mrow><mi href="./1.1#p2.t1.r5">m</mi><mo>,</mo><mi href="./1.1#p2.t1.r5">n</mi></mrow></msub></mrow><mo>)</mo></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex23" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E65b.png" altimg-height="24px" altimg-valign="-8px" altimg-width="210px" alttext="\lim_{n\to\infty}\left(\lim_{m\to\infty}z_{m,n}\right)" display="inline"><mrow><munder><mo movablelimits="false">lim</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><mo></mo><mrow><mo>(</mo><mrow><munder><mo movablelimits="false">lim</mo><mrow><mi href="./1.1#p2.t1.r5">m</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><mo></mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mrow><mi href="./1.1#p2.t1.r5">m</mi><mo>,</mo><mi href="./1.1#p2.t1.r5">n</mi></mrow></msub></mrow><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E65.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>,
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./1.1#p2.t1.r5">m</mi></math>: nonnegative integer</a> and
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m181.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">exist. Then both repeated limits equal <math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>.</p>
</div>
<div id="Px26.p2" class="ltx_para">
<p class="ltx_p">A <em class="ltx_emph ltx_font_italic">double series</em>
is the limit of the double sequence
</p>
<table id="E66" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.66</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E66.png" altimg-height="65px" altimg-valign="-27px" altimg-width="179px" alttext="z_{p,q}=\sum^{p}_{m=0}\sum^{q}_{n=0}\zeta_{m,n}." display="block"><mrow><mrow><msub><mi href="./1.1#p2.t1.r2">z</mi><mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msub><mo>=</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">m</mi><mo>=</mo><mn>0</mn></mrow><mi>p</mi></munderover><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>0</mn></mrow><mi>q</mi></munderover><msub><mi href="./1.9#Px26.p2">ζ</mi><mrow><mi href="./1.1#p2.t1.r5">m</mi><mo>,</mo><mi href="./1.1#p2.t1.r5">n</mi></mrow></msub></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E66.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>,
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./1.1#p2.t1.r5">m</mi></math>: nonnegative integer</a>,
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m181.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./1.9#Px26.p2" title="Double Sequences and Series ‣ §1.9(vii) Inversion of Limits ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m141.png" altimg-height="23px" altimg-valign="-8px" altimg-width="35px" alttext="\zeta_{p,q}" display="inline"><msub><mi href="./1.9#Px26.p2">ζ</mi><mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msub></math>: sum</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">If the limit exists, then the double series is <em class="ltx_emph ltx_font_italic">convergent</em>; otherwise it
is <em class="ltx_emph ltx_font_italic">divergent</em>. The double series is <em class="ltx_emph ltx_font_italic">absolutely convergent</em> if it is
convergent when <math class="ltx_Math" altimg="m140.png" altimg-height="23px" altimg-valign="-8px" altimg-width="43px" alttext="\zeta_{m,n}" display="inline"><msub><mi href="./1.9#Px26.p2">ζ</mi><mrow><mi href="./1.1#p2.t1.r5">m</mi><mo>,</mo><mi href="./1.1#p2.t1.r5">n</mi></mrow></msub></math> is replaced by <math class="ltx_Math" altimg="m233.png" altimg-height="24px" altimg-valign="-8px" altimg-width="54px" alttext="|\zeta_{m,n}|" display="inline"><mrow><mo stretchy="false">|</mo><msub><mi href="./1.9#Px26.p2">ζ</mi><mrow><mi href="./1.1#p2.t1.r5">m</mi><mo>,</mo><mi href="./1.1#p2.t1.r5">n</mi></mrow></msub><mo stretchy="false">|</mo></mrow></math>.
</p>
</div>
<div id="Px26.p3" class="ltx_para">
<p class="ltx_p">If a double series is absolutely convergent, then it is also convergent and its
sum is given by either of the repeated sums</p>
<table id="E67" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex24" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">1.9.67</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E67a.png" altimg-height="26px" altimg-valign="-8px" altimg-width="182px" alttext="\sum^{\infty}_{m=0}\left(\sum^{\infty}_{n=0}\zeta_{m,n}\right)," display="inline"><mrow><mrow><mstyle displaystyle="true"><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">m</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover></mstyle><mrow><mo>(</mo><mrow><mstyle displaystyle="true"><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover></mstyle><msub><mi href="./1.9#Px26.p2">ζ</mi><mrow><mi href="./1.1#p2.t1.r5">m</mi><mo>,</mo><mi href="./1.1#p2.t1.r5">n</mi></mrow></msub></mrow><mo>)</mo></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex25" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E67b.png" altimg-height="26px" altimg-valign="-8px" altimg-width="182px" alttext="\sum^{\infty}_{n=0}\left(\sum^{\infty}_{m=0}\zeta_{m,n}\right)." display="inline"><mrow><mrow><mstyle displaystyle="true"><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover></mstyle><mrow><mo>(</mo><mrow><mstyle displaystyle="true"><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">m</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover></mstyle><msub><mi href="./1.9#Px26.p2">ζ</mi><mrow><mi href="./1.1#p2.t1.r5">m</mi><mo>,</mo><mi href="./1.1#p2.t1.r5">n</mi></mrow></msub></mrow><mo>)</mo></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E67.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./1.1#p2.t1.r5">m</mi></math>: nonnegative integer</a>,
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m181.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./1.9#Px26.p2" title="Double Sequences and Series ‣ §1.9(vii) Inversion of Limits ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m141.png" altimg-height="23px" altimg-valign="-8px" altimg-width="35px" alttext="\zeta_{p,q}" display="inline"><msub><mi href="./1.9#Px26.p2">ζ</mi><mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msub></math>: sum</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px27" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Term-by-Term Integration</h3>
<div id="Px27.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px27.p1" class="ltx_para">
<p class="ltx_p">Suppose the series <math class="ltx_Math" altimg="m135.png" altimg-height="26px" altimg-valign="-8px" altimg-width="106px" alttext="\sum^{\infty}_{n=0}f_{n}(z)" display="inline"><mrow><msubsup><mo largeop="true" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></msubsup><mrow><msub><mi>f</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></math>, where <math class="ltx_Math" altimg="m170.png" altimg-height="23px" altimg-valign="-7px" altimg-width="51px" alttext="f_{n}(z)" display="inline"><mrow><msub><mi>f</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> is continuous,
converges uniformly on every <em class="ltx_emph ltx_font_italic">compact set</em>
of a domain <math class="ltx_Math" altimg="m71.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.9#Px27.p1">D</mi></math>, that is, every closed and bounded set in <math class="ltx_Math" altimg="m71.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.9#Px27.p1">D</mi></math>. Then</p>
<table id="E68" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.68</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E68.png" altimg-height="64px" altimg-valign="-27px" altimg-width="294px" alttext="\int_{C}\sum^{\infty}_{n=0}f_{n}(z)\mathrm{d}z=\sum^{\infty}_{n=0}\int_{C}f_{n%
}(z)\mathrm{d}z" display="block"><mrow><mrow><msub><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi href="./1.9#Px27.p1">C</mi></msub><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><mrow><msub><mi>f</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./1.1#p2.t1.r2">z</mi></mrow></mrow></mrow></mrow><mo>=</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><msub><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi href="./1.9#Px27.p1">C</mi></msub><mrow><mrow><msub><mi>f</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./1.1#p2.t1.r2">z</mi></mrow></mrow></mrow></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E68.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m118.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m96.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>,
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m181.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./1.9#Px27.p1" title="Term-by-Term Integration ‣ §1.9(vii) Inversion of Limits ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.9#Px27.p1">C</mi></math>: finite contour in <math class="ltx_Math" altimg="m71.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.9#Px27.p1">D</mi></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">for any finite contour <math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.9#Px27.p1">C</mi></math> in <math class="ltx_Math" altimg="m71.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.9#Px27.p1">D</mi></math>.</p>
</div>
</section>
<section id="Px28" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Dominated Convergence Theorem</h3>
<div id="Px28.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px28.p1" class="ltx_para">
<p class="ltx_p">Let <math class="ltx_Math" altimg="m50.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="(a,b)" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi>a</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi>b</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math> be a finite or infinite interval, and <math class="ltx_Math" altimg="m169.png" altimg-height="23px" altimg-valign="-7px" altimg-width="128px" alttext="f_{0}(t),f_{1}(t),\dots" display="inline"><mrow><mrow><msub><mi>f</mi><mn>0</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo><mrow><msub><mi>f</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo><mi mathvariant="normal">…</mi></mrow></math>
be real or complex continuous functions, <math class="ltx_Math" altimg="m194.png" altimg-height="23px" altimg-valign="-7px" altimg-width="79px" alttext="t\in(a,b)" display="inline"><mrow><mi>t</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi>a</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi>b</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></mrow></math>. Suppose
<math class="ltx_Math" altimg="m134.png" altimg-height="26px" altimg-valign="-8px" altimg-width="103px" alttext="\sum^{\infty}_{n=0}f_{n}(t)" display="inline"><mrow><msubsup><mo largeop="true" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></msubsup><mrow><msub><mi>f</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></math> converges uniformly in any compact interval in
<math class="ltx_Math" altimg="m50.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="(a,b)" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi>a</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi>b</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math>, and at least one of the following two conditions is satisfied:</p>
<table id="E69" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.69</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E69.png" altimg-height="64px" altimg-valign="-27px" altimg-width="200px" alttext="\int^{b}_{a}\sum^{\infty}_{n=0}|f_{n}(t)|\mathrm{d}t<\infty," display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi>a</mi><mi>b</mi></msubsup><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><mrow><mo stretchy="false">|</mo><mrow><msub><mi>f</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow><mo><</mo><mi mathvariant="normal">∞</mi></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E69.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m118.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m96.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a> and
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m181.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E70" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.70</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E70.png" altimg-height="64px" altimg-valign="-27px" altimg-width="200px" alttext="\sum^{\infty}_{n=0}\int^{b}_{a}|f_{n}(t)|\mathrm{d}t<\infty." display="block"><mrow><mrow><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi>a</mi><mi>b</mi></msubsup><mrow><mrow><mo stretchy="false">|</mo><mrow><msub><mi>f</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow><mo><</mo><mi mathvariant="normal">∞</mi></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E70.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m118.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m96.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a> and
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m181.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Then</p>
<table id="E71" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.71</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E71.png" altimg-height="64px" altimg-valign="-27px" altimg-width="295px" alttext="\int^{b}_{a}\sum^{\infty}_{n=0}f_{n}(t)\mathrm{d}t=\sum^{\infty}_{n=0}\int^{b}%
_{a}f_{n}(t)\mathrm{d}t." display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi>a</mi><mi>b</mi></msubsup><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><mrow><msub><mi>f</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow><mo>=</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi>a</mi><mi>b</mi></msubsup><mrow><mrow><msub><mi>f</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E71.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m118.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m96.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a> and
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m181.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
</section>
</section>
</div>
<div class="ltx_page_footer">
<span></div>
</div>
</body></text>
</html>
</page>
<page>
<!DOCTYPE html><html>
<head>
<title>DLMF: 1.13 Differential Equations</title>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<!--GOOGLE BOOTSTRAP--></head>
<text><body class="color_default textfont_default titlefont_default fontsize_default navbar_default">
<div class="ltx_page_navbar">
<div class="ltx_page_navlogo"></dd>
</dl>
</div>
</div>
<div id="SS1.p1" class="ltx_para">
<p class="ltx_p">A domain in the complex plane is <em class="ltx_emph ltx_font_italic">simply-connected</em>
if it has no “holes”; more precisely, if its complement in the extended plane
<math class="ltx_Math" altimg="m46.png" altimg-height="23px" altimg-valign="-7px" altimg-width="81px" alttext="\mathbb{C}\cup\{\infty\}" display="inline"><mrow><mi href="./front/introduction#Sx4.p1.t1.r1">ℂ</mi><mo href="./front/introduction#Sx4.p1.t1.r27">∪</mo><mrow><mo stretchy="false">{</mo><mi mathvariant="normal">∞</mi><mo stretchy="false">}</mo></mrow></mrow></math> is connected.</p>
</div>
<div id="SS1.p2" class="ltx_para">
<p class="ltx_p">The equation
</p>
<table id="E1" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.13.1</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E1.png" altimg-height="51px" altimg-valign="-18px" altimg-width="255px" alttext="\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+f(z)\frac{\mathrm{d}w}{\mathrm{d}z%
}+g(z)w=0," display="block"><mrow><mrow><mrow><mfrac><mrow><mpadded lspace="-1.7pt" width="-4.5pt"><msup><mo href="./1.4#E4">d</mo><mn>2</mn></msup></mpadded><mi href="./1.13#Px1.p1">w</mi></mrow><msup><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi href="./1.1#p2.t1.r2">z</mi></mrow><mn>2</mn></msup></mfrac><mo>+</mo><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi href="./1.13#Px1.p1">w</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi href="./1.1#p2.t1.r2">z</mi></mrow></mfrac></mrow><mo>+</mo><mrow><mrow><mi>g</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mi href="./1.13#Px1.p1">w</mi></mrow></mrow><mo>=</mo><mn>0</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#E4" title="(1.4.4) ‣ §1.4(iii) Derivatives ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="29px" altimg-valign="-9px" altimg-width="27px" alttext="\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: derivative of <math class="ltx_Math" altimg="m59.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a> and
<a href="./1.13#Px1.p1" title="Fundamental Pair ‣ §1.13(i) Existence of Solutions ‣ §1.13 Differential Equations ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m64.png" altimg-height="23px" altimg-valign="-7px" altimg-width="45px" alttext="w(z)" display="inline"><mrow><mi href="./1.13#Px1.p1">w</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: solution</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m71.png" altimg-height="18px" altimg-valign="-3px" altimg-width="56px" alttext="z\in D" display="inline"><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mi href="./1.13#SS1.p2">D</mi></mrow></math>, a simply-connected domain, and <math class="ltx_Math" altimg="m58.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(z)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math>, <math class="ltx_Math" altimg="m61.png" altimg-height="23px" altimg-valign="-7px" altimg-width="40px" alttext="g(z)" display="inline"><mrow><mi>g</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> are analytic in
<math class="ltx_Math" altimg="m22.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.13#SS1.p2">D</mi></math>, has an infinite number of analytic solutions in <math class="ltx_Math" altimg="m22.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.13#SS1.p2">D</mi></math>. A solution becomes
unique, for example, when <math class="ltx_Math" altimg="m65.png" altimg-height="13px" altimg-valign="-2px" altimg-width="19px" alttext="w" display="inline"><mi href="./1.13#Px1.p1">w</mi></math> and <math class="ltx_Math" altimg="m39.png" altimg-height="23px" altimg-valign="-7px" altimg-width="66px" alttext="\ifrac{\mathrm{d}w}{\mathrm{d}z}" display="inline"><mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi href="./1.13#Px1.p1">w</mi></mrow><mo href="./1.4#E4">/</mo><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi href="./1.1#p2.t1.r2">z</mi></mrow></mrow></math> are prescribed at a point
in <math class="ltx_Math" altimg="m22.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.13#SS1.p2">D</mi></math>.</p>
</div>
<section id="Px1" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Fundamental Pair</h3>
<div id="Px1.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px1.p1" class="ltx_para">
<p class="ltx_p">Two solutions <math class="ltx_Math" altimg="m67.png" altimg-height="23px" altimg-valign="-7px" altimg-width="53px" alttext="w_{1}(z)" display="inline"><mrow><msub><mi href="./1.13#Px1.p1">w</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> and <math class="ltx_Math" altimg="m68.png" altimg-height="23px" altimg-valign="-7px" altimg-width="53px" alttext="w_{2}(z)" display="inline"><mrow><msub><mi href="./1.13#Px1.p1">w</mi><mn>2</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> are called a <em class="ltx_emph ltx_font_italic">fundamental pair</em>
if any other solution <math class="ltx_Math" altimg="m64.png" altimg-height="23px" altimg-valign="-7px" altimg-width="45px" alttext="w(z)" display="inline"><mrow><mi href="./1.13#Px1.p1">w</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> is expressible as
</p>
<table id="E2" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.13.2</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E2.png" altimg-height="25px" altimg-valign="-7px" altimg-width="232px" alttext="w(z)=Aw_{1}(z)+Bw_{2}(z)," display="block"><mrow><mrow><mrow><mi href="./1.13#Px1.p1">w</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mi href="./1.13#Px1.p1">A</mi><mo></mo><mrow><msub><mi href="./1.13#Px1.p1">w</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mi href="./1.13#Px1.p1">B</mi><mo></mo><mrow><msub><mi href="./1.13#Px1.p1">w</mi><mn>2</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>,
<a href="./1.13#Px1.p1" title="Fundamental Pair ‣ §1.13(i) Existence of Solutions ‣ §1.13 Differential Equations ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m64.png" altimg-height="23px" altimg-valign="-7px" altimg-width="45px" alttext="w(z)" display="inline"><mrow><mi href="./1.13#Px1.p1">w</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: solution</a>,
<a href="./1.13#Px1.p1" title="Fundamental Pair ‣ §1.13(i) Existence of Solutions ‣ §1.13 Differential Equations ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="A" display="inline"><mi href="./1.13#Px1.p1">A</mi></math>: constant</a> and
<a href="./1.13#Px1.p1" title="Fundamental Pair ‣ §1.13(i) Existence of Solutions ‣ §1.13 Differential Equations ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m21.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="B" display="inline"><mi href="./1.13#Px1.p1">B</mi></math>: constant</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m20.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="A" display="inline"><mi href="./1.13#Px1.p1">A</mi></math> and <math class="ltx_Math" altimg="m21.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="B" display="inline"><mi href="./1.13#Px1.p1">B</mi></math> are constants. A fundamental pair can be obtained, for
example, by taking any <math class="ltx_Math" altimg="m72.png" altimg-height="21px" altimg-valign="-5px" altimg-width="64px" alttext="z_{0}\in D" display="inline"><mrow><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mi href="./1.13#SS1.p2">D</mi></mrow></math> and requiring that</p>
<table id="E3" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex1" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="4" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">1.13.3</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m15.png" altimg-height="25px" altimg-valign="-7px" altimg-width="63px" alttext="\displaystyle w_{1}(z_{0})" display="inline"><mrow><msub><mi href="./1.13#Px1.p1">w</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m4.png" altimg-height="22px" altimg-valign="-6px" altimg-width="43px" alttext="\displaystyle=1," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mn>1</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex2" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m16.png" altimg-height="26px" altimg-valign="-7px" altimg-width="63px" alttext="\displaystyle w_{1}^{\prime}(z_{0})" display="inline"><mrow><msubsup><mi href="./1.13#Px1.p1">w</mi><mn>1</mn><mo>′</mo></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m3.png" altimg-height="22px" altimg-valign="-6px" altimg-width="43px" alttext="\displaystyle=0," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mn>0</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex3" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m17.png" altimg-height="25px" altimg-valign="-7px" altimg-width="63px" alttext="\displaystyle w_{2}(z_{0})" display="inline"><mrow><msub><mi href="./1.13#Px1.p1">w</mi><mn>2</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m3.png" altimg-height="22px" altimg-valign="-6px" altimg-width="43px" alttext="\displaystyle=0," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mn>0</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex4" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m18.png" altimg-height="26px" altimg-valign="-7px" altimg-width="63px" alttext="\displaystyle w_{2}^{\prime}(z_{0})" display="inline"><mrow><msubsup><mi href="./1.13#Px1.p1">w</mi><mn>2</mn><mo>′</mo></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m5.png" altimg-height="18px" altimg-valign="-2px" altimg-width="43px" alttext="\displaystyle=1." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mn>1</mn></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a> and
<a href="./1.13#Px1.p1" title="Fundamental Pair ‣ §1.13(i) Existence of Solutions ‣ §1.13 Differential Equations ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m64.png" altimg-height="23px" altimg-valign="-7px" altimg-width="45px" alttext="w(z)" display="inline"><mrow><mi href="./1.13#Px1.p1">w</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: solution</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px2" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Wronskian</h3>
<div id="Px2.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px2.p1" class="ltx_para">
<p class="ltx_p">The <em class="ltx_emph ltx_font_italic">Wronskian</em> of <math class="ltx_Math" altimg="m67.png" altimg-height="23px" altimg-valign="-7px" altimg-width="53px" alttext="w_{1}(z)" display="inline"><mrow><msub><mi href="./1.13#Px1.p1">w</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> and <math class="ltx_Math" altimg="m68.png" altimg-height="23px" altimg-valign="-7px" altimg-width="53px" alttext="w_{2}(z)" display="inline"><mrow><msub><mi href="./1.13#Px1.p1">w</mi><mn>2</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> is defined by</p>
<table id="E4" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.13.4</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E4.png" altimg-height="26px" altimg-valign="-7px" altimg-width="409px" alttext="\mathscr{W}\left\{w_{1}(z),w_{2}(z)\right\}=w_{1}(z)w_{2}^{\prime}(z)-w_{2}(z)%
w_{1}^{\prime}(z)." display="block"><mrow><mrow><mrow><mi class="ltx_font_mathscript" href="./1.13#E4">𝒲</mi><mo></mo><mrow><mo>{</mo><mrow><mrow><msub><mi href="./1.13#Px1.p1">w</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo><mrow><msub><mi href="./1.13#Px1.p1">w</mi><mn>2</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>}</mo></mrow></mrow><mo>=</mo><mrow><mrow><mrow><msub><mi href="./1.13#Px1.p1">w</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><msubsup><mi href="./1.13#Px1.p1">w</mi><mn>2</mn><mo>′</mo></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>-</mo><mrow><mrow><msub><mi href="./1.13#Px1.p1">w</mi><mn>2</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><msubsup><mi href="./1.13#Px1.p1">w</mi><mn>1</mn><mo>′</mo></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E4.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m49.png" altimg-height="18px" altimg-valign="-2px" altimg-width="26px" alttext="\mathscr{W}" display="inline"><mi class="ltx_font_mathscript" href="./1.13#E4">𝒲</mi></math>: Wronskian</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a> and
<a href="./1.13#Px1.p1" title="Fundamental Pair ‣ §1.13(i) Existence of Solutions ‣ §1.13 Differential Equations ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m64.png" altimg-height="23px" altimg-valign="-7px" altimg-width="45px" alttext="w(z)" display="inline"><mrow><mi href="./1.13#Px1.p1">w</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: solution</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Then</p>
<table id="E5" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.13.5</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E5.png" altimg-height="30px" altimg-valign="-7px" altimg-width="284px" alttext="\mathscr{W}\left\{w_{1}(z),w_{2}(z)\right\}=ce^{-\int f(z)\mathrm{d}z}," display="block"><mrow><mrow><mrow><mi class="ltx_font_mathscript" href="./1.13#E4">𝒲</mi><mo></mo><mrow><mo>{</mo><mrow><mrow><msub><mi href="./1.13#Px1.p1">w</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo><mrow><msub><mi href="./1.13#Px1.p1">w</mi><mn>2</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>}</mo></mrow></mrow><mo>=</mo><mrow><mi>c</mi><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mstyle scriptlevel="-1"><mrow><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./1.1#p2.t1.r2">z</mi></mrow></mrow></mrow></mstyle></mrow></msup></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E5.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.13#E4" title="(1.13.4) ‣ Wronskian ‣ §1.13(i) Existence of Solutions ‣ §1.13 Differential Equations ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="18px" altimg-valign="-2px" altimg-width="26px" alttext="\mathscr{W}" display="inline"><mi class="ltx_font_mathscript" href="./1.13#E4">𝒲</mi></math>: Wronskian</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a> and
<a href="./1.13#Px1.p1" title="Fundamental Pair ‣ §1.13(i) Existence of Solutions ‣ §1.13 Differential Equations ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m64.png" altimg-height="23px" altimg-valign="-7px" altimg-width="45px" alttext="w(z)" display="inline"><mrow><mi href="./1.13#Px1.p1">w</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: solution</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m55.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="c" display="inline"><mi>c</mi></math> is independent of <math class="ltx_Math" altimg="m70.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>. If <math class="ltx_Math" altimg="m57.png" altimg-height="23px" altimg-valign="-7px" altimg-width="78px" alttext="f(z)=0" display="inline"><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mn>0</mn></mrow></math>, then the Wronskian is constant.</p>
</div>
<div id="Px2.p2" class="ltx_para">
<p class="ltx_p">The following three statements are equivalent: <math class="ltx_Math" altimg="m67.png" altimg-height="23px" altimg-valign="-7px" altimg-width="53px" alttext="w_{1}(z)" display="inline"><mrow><msub><mi href="./1.13#Px1.p1">w</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> and <math class="ltx_Math" altimg="m68.png" altimg-height="23px" altimg-valign="-7px" altimg-width="53px" alttext="w_{2}(z)" display="inline"><mrow><msub><mi href="./1.13#Px1.p1">w</mi><mn>2</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> comprise a
fundamental pair in <math class="ltx_Math" altimg="m22.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.13#SS1.p2">D</mi></math>; <math class="ltx_Math" altimg="m50.png" altimg-height="23px" altimg-valign="-7px" altimg-width="156px" alttext="\mathscr{W}\left\{w_{1}(z),w_{2}(z)\right\}" display="inline"><mrow><mi class="ltx_font_mathscript" href="./1.13#E4">𝒲</mi><mo></mo><mrow><mo>{</mo><mrow><mrow><msub><mi href="./1.13#Px1.p1">w</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo><mrow><msub><mi href="./1.13#Px1.p1">w</mi><mn>2</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>}</mo></mrow></mrow></math> does not vanish in <math class="ltx_Math" altimg="m22.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.13#SS1.p2">D</mi></math>;
<math class="ltx_Math" altimg="m67.png" altimg-height="23px" altimg-valign="-7px" altimg-width="53px" alttext="w_{1}(z)" display="inline"><mrow><msub><mi href="./1.13#Px1.p1">w</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> and <math class="ltx_Math" altimg="m68.png" altimg-height="23px" altimg-valign="-7px" altimg-width="53px" alttext="w_{2}(z)" display="inline"><mrow><msub><mi href="./1.13#Px1.p1">w</mi><mn>2</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> are <em class="ltx_emph ltx_font_italic">linearly independent</em>,
that is, the only constants <math class="ltx_Math" altimg="m20.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="A" display="inline"><mi href="./1.13#Px1.p1">A</mi></math> and <math class="ltx_Math" altimg="m21.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="B" display="inline"><mi href="./1.13#Px1.p1">B</mi></math> such that</p>
<table id="E6" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.13.6</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E6.png" altimg-height="25px" altimg-valign="-7px" altimg-width="201px" alttext="Aw_{1}(z)+Bw_{2}(z)=0," display="block"><mrow><mrow><mrow><mrow><mi href="./1.13#Px1.p1">A</mi><mo></mo><mrow><msub><mi href="./1.13#Px1.p1">w</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mi href="./1.13#Px1.p1">B</mi><mo></mo><mrow><msub><mi href="./1.13#Px1.p1">w</mi><mn>2</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow><mo>=</mo><mn>0</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m36.png" altimg-height="18px" altimg-valign="-3px" altimg-width="67px" alttext="\forall z\in D" display="inline"><mrow><mrow><mo href="./front/introduction#Sx4.p1.t1.r11">∀</mo><mi href="./1.1#p2.t1.r2">z</mi></mrow><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mi href="./1.13#SS1.p2">D</mi></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E6.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./front/introduction#Sx4.p1.t1.r9" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m42.png" altimg-height="15px" altimg-valign="-3px" altimg-width="18px" alttext="\in" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo></math>: element of</a>,
<a href="./front/introduction#Sx4.p1.t1.r11" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="\forall" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r11">∀</mo></math>: for every</a>,
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>,
<a href="./1.13#SS1.p2" title="§1.13(i) Existence of Solutions ‣ §1.13 Differential Equations ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.13#SS1.p2">D</mi></math>: simply-connected domain</a>,
<a href="./1.13#Px1.p1" title="Fundamental Pair ‣ §1.13(i) Existence of Solutions ‣ §1.13 Differential Equations ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m64.png" altimg-height="23px" altimg-valign="-7px" altimg-width="45px" alttext="w(z)" display="inline"><mrow><mi href="./1.13#Px1.p1">w</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: solution</a>,
<a href="./1.13#Px1.p1" title="Fundamental Pair ‣ §1.13(i) Existence of Solutions ‣ §1.13 Differential Equations ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="A" display="inline"><mi href="./1.13#Px1.p1">A</mi></math>: constant</a> and
<a href="./1.13#Px1.p1" title="Fundamental Pair ‣ §1.13(i) Existence of Solutions ‣ §1.13 Differential Equations ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m21.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="B" display="inline"><mi href="./1.13#Px1.p1">B</mi></math>: constant</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">are <math class="ltx_Math" altimg="m19.png" altimg-height="18px" altimg-valign="-2px" altimg-width="98px" alttext="A=B=0" display="inline"><mrow><mi href="./1.13#Px1.p1">A</mi><mo>=</mo><mi href="./1.13#Px1.p1">B</mi><mo>=</mo><mn>0</mn></mrow></math>.
</p>
</div>
</section>
</section>
<section id="ii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§1.13(ii) </span>Equations with a Parameter</h2>
<div id="SS2.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="SS2.p1" class="ltx_para">
<p class="ltx_p">Assume that in the equation
</p>
<table id="E7" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.13.7</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E7.png" altimg-height="51px" altimg-valign="-18px" altimg-width="296px" alttext="\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+f(u,z)\frac{\mathrm{d}w}{\mathrm{d%
}z}+g(u,z)w=0," display="block"><mrow><mrow><mrow><mfrac><mrow><mpadded lspace="-1.7pt" width="-4.5pt"><msup><mo href="./1.4#E4">d</mo><mn>2</mn></msup></mpadded><mi>w</mi></mrow><msup><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi href="./1.1#p2.t1.r2">z</mi></mrow><mn>2</mn></msup></mfrac><mo>+</mo><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>u</mi><mo>,</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi>w</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi href="./1.1#p2.t1.r2">z</mi></mrow></mfrac></mrow><mo>+</mo><mrow><mrow><mi>g</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>u</mi><mo>,</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mi>w</mi></mrow></mrow><mo>=</mo><mn>0</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E7.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#E4" title="(1.4.4) ‣ §1.4(iii) Derivatives ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="29px" altimg-valign="-9px" altimg-width="27px" alttext="\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: derivative of <math class="ltx_Math" altimg="m59.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p"><math class="ltx_Math" altimg="m63.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="u" display="inline"><mi>u</mi></math> and <math class="ltx_Math" altimg="m70.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math> belong to domains <math class="ltx_Math" altimg="m29.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="U" display="inline"><mi href="./1.13#SS2.p1">U</mi></math> and <math class="ltx_Math" altimg="m22.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.13#SS2.p1">D</mi></math> respectively, the coefficients
<math class="ltx_Math" altimg="m56.png" altimg-height="23px" altimg-valign="-7px" altimg-width="62px" alttext="f(u,z)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>u</mi><mo>,</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> and <math class="ltx_Math" altimg="m60.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="g(u,z)" display="inline"><mrow><mi>g</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>u</mi><mo>,</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> are continuous functions of both variables, and for each
fixed <math class="ltx_Math" altimg="m63.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="u" display="inline"><mi>u</mi></math> (fixed <math class="ltx_Math" altimg="m70.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>) the two functions are analytic in <math class="ltx_Math" altimg="m70.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math> (in <math class="ltx_Math" altimg="m63.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="u" display="inline"><mi>u</mi></math>). Suppose
also that at (a fixed) <math class="ltx_Math" altimg="m72.png" altimg-height="21px" altimg-valign="-5px" altimg-width="64px" alttext="z_{0}\in D" display="inline"><mrow><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mi href="./1.13#SS2.p1">D</mi></mrow></math>, <math class="ltx_Math" altimg="m65.png" altimg-height="13px" altimg-valign="-2px" altimg-width="19px" alttext="w" display="inline"><mi>w</mi></math> and <math class="ltx_Math" altimg="m40.png" altimg-height="23px" altimg-valign="-7px" altimg-width="67px" alttext="\ifrac{\partial w}{\partial z}" display="inline"><mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>w</mi></mrow><mo href="./1.5#E3">/</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.1#p2.t1.r2">z</mi></mrow></mrow></math> are analytic
functions of <math class="ltx_Math" altimg="m63.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="u" display="inline"><mi>u</mi></math>. Then at each <math class="ltx_Math" altimg="m71.png" altimg-height="18px" altimg-valign="-3px" altimg-width="56px" alttext="z\in D" display="inline"><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mi href="./1.13#SS2.p1">D</mi></mrow></math>, <math class="ltx_Math" altimg="m65.png" altimg-height="13px" altimg-valign="-2px" altimg-width="19px" alttext="w" display="inline"><mi>w</mi></math>, <math class="ltx_Math" altimg="m40.png" altimg-height="23px" altimg-valign="-7px" altimg-width="67px" alttext="\ifrac{\partial w}{\partial z}" display="inline"><mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>w</mi></mrow><mo href="./1.5#E3">/</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.1#p2.t1.r2">z</mi></mrow></mrow></math> and
<math class="ltx_Math" altimg="m41.png" altimg-height="28px" altimg-valign="-9px" altimg-width="87px" alttext="\ifrac{{\partial}^{2}w}{{\partial z}^{2}}" display="inline"><mrow><mrow><mpadded width="-1.7pt"><msup><mo href="./1.5#E3">∂</mo><mn>2</mn></msup></mpadded><mo href="./1.5#E3"></mo><mi>w</mi></mrow><mo href="./1.5#E3">/</mo><msup><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.1#p2.t1.r2">z</mi></mrow><mn>2</mn></msup></mrow></math> are analytic functions of <math class="ltx_Math" altimg="m63.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="u" display="inline"><mi>u</mi></math>.</p>
</div>
</section>
<section id="iii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§1.13(iii) </span>Inhomogeneous Equations</h2>
<div id="SS3.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="SS3.p1" class="ltx_para">
<p class="ltx_p">The <em class="ltx_emph ltx_font_italic">inhomogeneous</em>
(or <em class="ltx_emph ltx_font_italic">nonhomogeneous</em>)
equation</p>
<table id="E8" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.13.8</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E8.png" altimg-height="51px" altimg-valign="-18px" altimg-width="275px" alttext="\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+f(z)\frac{\mathrm{d}w}{\mathrm{d}z%
}+g(z)w=r(z)" display="block"><mrow><mrow><mfrac><mrow><mpadded lspace="-1.7pt" width="-4.5pt"><msup><mo href="./1.4#E4">d</mo><mn>2</mn></msup></mpadded><mi href="./1.13#SS1.p2">w</mi></mrow><msup><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi href="./1.1#p2.t1.r2">z</mi></mrow><mn>2</mn></msup></mfrac><mo>+</mo><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi href="./1.13#SS1.p2">w</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi href="./1.1#p2.t1.r2">z</mi></mrow></mfrac></mrow><mo>+</mo><mrow><mrow><mi>g</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mi href="./1.13#SS1.p2">w</mi></mrow></mrow><mo>=</mo><mrow><mi>r</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E8.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#E4" title="(1.4.4) ‣ §1.4(iii) Derivatives ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="29px" altimg-valign="-9px" altimg-width="27px" alttext="\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: derivative of <math class="ltx_Math" altimg="m59.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a> and
<a href="./1.13#SS1.p2" title="§1.13(i) Existence of Solutions ‣ §1.13 Differential Equations ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m64.png" altimg-height="23px" altimg-valign="-7px" altimg-width="45px" alttext="w(z)" display="inline"><mrow><mi href="./1.13#Px1.p1">w</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: solution</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">with <math class="ltx_Math" altimg="m58.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(z)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math>, <math class="ltx_Math" altimg="m61.png" altimg-height="23px" altimg-valign="-7px" altimg-width="40px" alttext="g(z)" display="inline"><mrow><mi>g</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math>, and <math class="ltx_Math" altimg="m62.png" altimg-height="23px" altimg-valign="-7px" altimg-width="39px" alttext="r(z)" display="inline"><mrow><mi>r</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> analytic in <math class="ltx_Math" altimg="m22.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.13#SS3.p1">D</mi></math> has infinitely many analytic
solutions in <math class="ltx_Math" altimg="m22.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.13#SS3.p1">D</mi></math>. If <math class="ltx_Math" altimg="m66.png" altimg-height="23px" altimg-valign="-7px" altimg-width="53px" alttext="w_{0}(z)" display="inline"><mrow><msub><mi href="./1.13#SS1.p2">w</mi><mn>0</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> is any one solution, and <math class="ltx_Math" altimg="m67.png" altimg-height="23px" altimg-valign="-7px" altimg-width="53px" alttext="w_{1}(z)" display="inline"><mrow><msub><mi href="./1.13#SS1.p2">w</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math>, <math class="ltx_Math" altimg="m68.png" altimg-height="23px" altimg-valign="-7px" altimg-width="53px" alttext="w_{2}(z)" display="inline"><mrow><msub><mi href="./1.13#SS1.p2">w</mi><mn>2</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> are a
fundamental pair of solutions of the corresponding homogeneous equation
() can be
expressed as</p>
<table id="E9" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.13.9</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E9.png" altimg-height="25px" altimg-valign="-7px" altimg-width="305px" alttext="w(z)=w_{0}(z)+Aw_{1}(z)+Bw_{2}(z)," display="block"><mrow><mrow><mrow><mi href="./1.13#SS1.p2">w</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mrow><msub><mi href="./1.13#SS1.p2">w</mi><mn>0</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>+</mo><mrow><mi href="./1.13#SS3.p1">A</mi><mo></mo><mrow><msub><mi href="./1.13#SS1.p2">w</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mi href="./1.13#SS3.p1">B</mi><mo></mo><mrow><msub><mi href="./1.13#SS1.p2">w</mi><mn>2</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E9.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>,
<a href="./1.13#SS1.p2" title="§1.13(i) Existence of Solutions ‣ §1.13 Differential Equations ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m64.png" altimg-height="23px" altimg-valign="-7px" altimg-width="45px" alttext="w(z)" display="inline"><mrow><mi href="./1.13#Px1.p1">w</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: solution</a>,
<a href="./1.13#SS3.p1" title="§1.13(iii) Inhomogeneous Equations ‣ §1.13 Differential Equations ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m21.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="B" display="inline"><mi href="./1.13#SS3.p1">B</mi></math>: constant</a> and
<a href="./1.13#SS3.p1" title="§1.13(iii) Inhomogeneous Equations ‣ §1.13 Differential Equations ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="A" display="inline"><mi href="./1.13#SS3.p1">A</mi></math>: constant</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m20.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="A" display="inline"><mi href="./1.13#SS3.p1">A</mi></math> and <math class="ltx_Math" altimg="m21.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="B" display="inline"><mi href="./1.13#SS3.p1">B</mi></math> are constants.</p>
</div>
<section id="Px3" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Variation of Parameters</h3>
<div id="Px3.info" class="ltx_metadata ltx_info">
)</p>
<table id="E10" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.13.10</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E10.png" altimg-height="53px" altimg-valign="-21px" altimg-width="617px" alttext="w_{0}(z)=w_{2}(z)\int\frac{w_{1}(z)r(z)}{\mathscr{W}\left\{w_{1}(z),w_{2}(z)%
\right\}}\mathrm{d}z-w_{1}(z)\int\frac{w_{2}(z)r(z)}{\mathscr{W}\left\{w_{1}(z%
),w_{2}(z)\right\}}\mathrm{d}z." display="block"><mrow><mrow><mrow><msub><mi href="./1.13#SS1.p2">w</mi><mn>0</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mrow><msub><mi href="./1.13#SS1.p2">w</mi><mn>2</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><mfrac><mrow><mrow><msub><mi href="./1.13#SS1.p2">w</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mi>r</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mrow><mi class="ltx_font_mathscript" href="./1.13#E4">𝒲</mi><mo></mo><mrow><mo>{</mo><mrow><mrow><msub><mi href="./1.13#SS1.p2">w</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo><mrow><msub><mi href="./1.13#SS1.p2">w</mi><mn>2</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>}</mo></mrow></mrow></mfrac><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./1.1#p2.t1.r2">z</mi></mrow></mrow></mrow></mrow><mo>-</mo><mrow><mrow><msub><mi href="./1.13#SS1.p2">w</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><mfrac><mrow><mrow><msub><mi href="./1.13#SS1.p2">w</mi><mn>2</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mi>r</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mrow><mi class="ltx_font_mathscript" href="./1.13#E4">𝒲</mi><mo></mo><mrow><mo>{</mo><mrow><mrow><msub><mi href="./1.13#SS1.p2">w</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo><mrow><msub><mi href="./1.13#SS1.p2">w</mi><mn>2</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>}</mo></mrow></mrow></mfrac><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./1.1#p2.t1.r2">z</mi></mrow></mrow></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E10.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.13#E4" title="(1.13.4) ‣ Wronskian ‣ §1.13(i) Existence of Solutions ‣ §1.13 Differential Equations ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="18px" altimg-valign="-2px" altimg-width="26px" alttext="\mathscr{W}" display="inline"><mi class="ltx_font_mathscript" href="./1.13#E4">𝒲</mi></math>: Wronskian</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a> and
<a href="./1.13#SS1.p2" title="§1.13(i) Existence of Solutions ‣ §1.13 Differential Equations ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m64.png" altimg-height="23px" altimg-valign="-7px" altimg-width="45px" alttext="w(z)" display="inline"><mrow><mi href="./1.13#Px1.p1">w</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: solution</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
</section>
<section id="iv" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§1.13(iv) </span>Change of Variables</h2>
<div id="SS4.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px4.p1" class="ltx_para">
<p class="ltx_p">The substitution <math class="ltx_Math" altimg="m51.png" altimg-height="23px" altimg-valign="-7px" altimg-width="71px" alttext="\xi=1/z" display="inline"><mrow><mi href="./1.13#Px4.p1">ξ</mi><mo>=</mo><mrow><mn>1</mn><mo>/</mo><mi href="./1.1#p2.t1.r2">z</mi></mrow></mrow></math> in () gives
</p>
<table id="E11" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.13.11</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E11.png" altimg-height="56px" altimg-valign="-22px" altimg-width="284px" alttext="\frac{{\mathrm{d}}^{2}W}{{\mathrm{d}\xi}^{2}}+F(\xi)\frac{\mathrm{d}W}{\mathrm%
{d}\xi}+G(\xi)W=0," display="block"><mrow><mrow><mrow><mfrac><mrow><mpadded lspace="-1.7pt" width="-4.5pt"><msup><mo href="./1.4#E4">d</mo><mn>2</mn></msup></mpadded><mi href="./1.13#Px5.p1">W</mi></mrow><msup><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi href="./1.13#Px4.p1">ξ</mi></mrow><mn>2</mn></msup></mfrac><mo>+</mo><mrow><mrow><mi href="./1.13#E12">F</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.13#Px4.p1">ξ</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi href="./1.13#Px5.p1">W</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi href="./1.13#Px4.p1">ξ</mi></mrow></mfrac></mrow><mo>+</mo><mrow><mrow><mi href="./1.13#E12">G</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.13#Px4.p1">ξ</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mi href="./1.13#Px5.p1">W</mi></mrow></mrow><mo>=</mo><mn>0</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E11.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#E4" title="(1.4.4) ‣ §1.4(iii) Derivatives ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="29px" altimg-valign="-9px" altimg-width="27px" alttext="\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: derivative of <math class="ltx_Math" altimg="m59.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.13#Px4.p1" title="Transformation of the Point at Infinity ‣ §1.13(iv) Change of Variables ‣ §1.13 Differential Equations ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="21px" altimg-valign="-6px" altimg-width="14px" alttext="\xi" display="inline"><mi href="./1.13#Px4.p1">ξ</mi></math>: change of variable</a>,
<a href="./1.13#E12" title="(1.13.12) ‣ Transformation of the Point at Infinity ‣ §1.13(iv) Change of Variables ‣ §1.13 Differential Equations ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m23.png" altimg-height="23px" altimg-valign="-7px" altimg-width="45px" alttext="F(\xi)" display="inline"><mrow><mi href="./1.13#E12">F</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.13#Px4.p1">ξ</mi><mo stretchy="false">)</mo></mrow></mrow></math></a>,
<a href="./1.13#E12" title="(1.13.12) ‣ Transformation of the Point at Infinity ‣ §1.13(iv) Change of Variables ‣ §1.13 Differential Equations ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m24.png" altimg-height="23px" altimg-valign="-7px" altimg-width="45px" alttext="G(\xi)" display="inline"><mrow><mi href="./1.13#E12">G</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.13#Px4.p1">ξ</mi><mo stretchy="false">)</mo></mrow></mrow></math></a> and
<a href="./1.13#Px5.p1" title="Elimination of First Derivative by Change of Dependent Variable ‣ §1.13(iv) Change of Variables ‣ §1.13 Differential Equations ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m32.png" altimg-height="23px" altimg-valign="-7px" altimg-width="52px" alttext="W(z)" display="inline"><mrow><mi href="./1.13#Px5.p1">W</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: change of variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where</p>
<table id="E12" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex5" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="3" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">1.13.12</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m11.png" altimg-height="25px" altimg-valign="-7px" altimg-width="53px" alttext="\displaystyle W(\xi)" display="inline"><mrow><mi href="./1.13#Px5.p1">W</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.13#Px4.p1">ξ</mi><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m8.png" altimg-height="53px" altimg-valign="-21px" altimg-width="98px" alttext="\displaystyle=w\left(\frac{1}{\xi}\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./1.13#SS1.p2">w</mi><mo></mo><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac><mn>1</mn><mi href="./1.13#Px4.p1">ξ</mi></mfrac></mstyle><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex6" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m9.png" altimg-height="25px" altimg-valign="-7px" altimg-width="47px" alttext="\displaystyle F(\xi)" display="inline"><mrow><mi href="./1.13#E12">F</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.13#Px4.p1">ξ</mi><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m7.png" altimg-height="53px" altimg-valign="-21px" altimg-width="158px" alttext="\displaystyle=\frac{2}{\xi}-\frac{1}{\xi^{2}}f\left(\frac{1}{\xi}\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><mfrac><mn>2</mn><mi href="./1.13#Px4.p1">ξ</mi></mfrac></mstyle><mo>-</mo><mrow><mstyle displaystyle="true"><mfrac><mn>1</mn><msup><mi href="./1.13#Px4.p1">ξ</mi><mn>2</mn></msup></mfrac></mstyle><mo></mo><mrow><mi>f</mi><mo></mo><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac><mn>1</mn><mi href="./1.13#Px4.p1">ξ</mi></mfrac></mstyle><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex7" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m10.png" altimg-height="25px" altimg-valign="-7px" altimg-width="47px" alttext="\displaystyle G(\xi)" display="inline"><mrow><mi href="./1.13#E12">G</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.13#Px4.p1">ξ</mi><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m6.png" altimg-height="53px" altimg-valign="-21px" altimg-width="117px" alttext="\displaystyle=\frac{1}{\xi^{4}}g\left(\frac{1}{\xi}\right)." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><mfrac><mn>1</mn><msup><mi href="./1.13#Px4.p1">ξ</mi><mn>4</mn></msup></mfrac></mstyle><mo></mo><mrow><mi>g</mi><mo></mo><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac><mn>1</mn><mi href="./1.13#Px4.p1">ξ</mi></mfrac></mstyle><mo>)</mo></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E12.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd>
<span class="ltx_text"><math class="ltx_Math" altimg="m23.png" altimg-height="23px" altimg-valign="-7px" altimg-width="45px" alttext="F(\xi)" display="inline"><mrow><mi href="./1.13#E12">F</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.13#Px4.p1">ξ</mi><mo stretchy="false">)</mo></mrow></mrow></math> (locally)</span> and
<span class="ltx_text"><math class="ltx_Math" altimg="m24.png" altimg-height="23px" altimg-valign="-7px" altimg-width="45px" alttext="G(\xi)" display="inline"><mrow><mi href="./1.13#E12">G</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.13#Px4.p1">ξ</mi><mo stretchy="false">)</mo></mrow></mrow></math> (locally)</span>
</dd>
<dt>Symbols:</dt>
<dd>
<a href="./1.13#SS1.p2" title="§1.13(i) Existence of Solutions ‣ §1.13 Differential Equations ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m64.png" altimg-height="23px" altimg-valign="-7px" altimg-width="45px" alttext="w(z)" display="inline"><mrow><mi href="./1.13#Px1.p1">w</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: solution</a>,
<a href="./1.13#Px4.p1" title="Transformation of the Point at Infinity ‣ §1.13(iv) Change of Variables ‣ §1.13 Differential Equations ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="21px" altimg-valign="-6px" altimg-width="14px" alttext="\xi" display="inline"><mi href="./1.13#Px4.p1">ξ</mi></math>: change of variable</a> and
<a href="./1.13#Px5.p1" title="Elimination of First Derivative by Change of Dependent Variable ‣ §1.13(iv) Change of Variables ‣ §1.13 Differential Equations ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m32.png" altimg-height="23px" altimg-valign="-7px" altimg-width="52px" alttext="W(z)" display="inline"><mrow><mi href="./1.13#Px5.p1">W</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: change of variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px5" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Elimination of First Derivative by Change of Dependent Variable</h3>
<div id="Px5.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px5.p1" class="ltx_para">
<p class="ltx_p">The substitution
</p>
<table id="E13" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.13.13</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E13.png" altimg-height="53px" altimg-valign="-21px" altimg-width="300px" alttext="w(z)=W(z)\exp\left(-\tfrac{1}{2}\int f(z)\mathrm{d}z\right)" display="block"><mrow><mrow><mi href="./1.13#SS1.p2">w</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mi href="./1.13#Px5.p1">W</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mrow><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./1.1#p2.t1.r2">z</mi></mrow></mrow></mrow></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E13.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E19" title="(4.2.19) ‣ §4.2(iii) The Exponential Function ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m35.png" altimg-height="16px" altimg-valign="-6px" altimg-width="48px" alttext="\exp\NVar{z}" display="inline"><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: exponential function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>,
<a href="./1.13#SS1.p2" title="§1.13(i) Existence of Solutions ‣ §1.13 Differential Equations ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m64.png" altimg-height="23px" altimg-valign="-7px" altimg-width="45px" alttext="w(z)" display="inline"><mrow><mi href="./1.13#Px1.p1">w</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: solution</a> and
<a href="./1.13#Px5.p1" title="Elimination of First Derivative by Change of Dependent Variable ‣ §1.13(iv) Change of Variables ‣ §1.13 Differential Equations ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m32.png" altimg-height="23px" altimg-valign="-7px" altimg-width="52px" alttext="W(z)" display="inline"><mrow><mi href="./1.13#Px5.p1">W</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: change of variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">in () gives</p>
<table id="E14" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.13.14</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E14.png" altimg-height="51px" altimg-valign="-18px" altimg-width="184px" alttext="\frac{{\mathrm{d}}^{2}W}{{\mathrm{d}z}^{2}}-H(z)W=0," display="block"><mrow><mrow><mrow><mfrac><mrow><mpadded lspace="-1.7pt" width="-4.5pt"><msup><mo href="./1.4#E4">d</mo><mn>2</mn></msup></mpadded><mi href="./1.13#Px5.p1">W</mi></mrow><msup><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi href="./1.1#p2.t1.r2">z</mi></mrow><mn>2</mn></msup></mfrac><mo>-</mo><mrow><mrow><mi href="./1.13#E15">H</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mi href="./1.13#Px5.p1">W</mi></mrow></mrow><mo>=</mo><mn>0</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E14.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#E4" title="(1.4.4) ‣ §1.4(iii) Derivatives ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="29px" altimg-valign="-9px" altimg-width="27px" alttext="\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: derivative of <math class="ltx_Math" altimg="m59.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>,
<a href="./1.13#Px5.p1" title="Elimination of First Derivative by Change of Dependent Variable ‣ §1.13(iv) Change of Variables ‣ §1.13 Differential Equations ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m32.png" altimg-height="23px" altimg-valign="-7px" altimg-width="52px" alttext="W(z)" display="inline"><mrow><mi href="./1.13#Px5.p1">W</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: change of variable</a> and
<a href="./1.13#E15" title="(1.13.15) ‣ Elimination of First Derivative by Change of Dependent Variable ‣ §1.13(iv) Change of Variables ‣ §1.13 Differential Equations ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m25.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="H(z)" display="inline"><mrow><mi href="./1.13#E15">H</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where</p>
<table id="E15" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.13.15</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E15.png" altimg-height="30px" altimg-valign="-9px" altimg-width="282px" alttext="H(z)=\tfrac{1}{4}f^{2}(z)+\tfrac{1}{2}f^{\prime}(z)-g(z)." display="block"><mrow><mrow><mrow><mi href="./1.13#E15">H</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>4</mn></mfrac></mstyle><mo></mo><mrow><msup><mi>f</mi><mn>2</mn></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mrow><msup><mi>f</mi><mo>′</mo></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow><mo>-</mo><mrow><mi>g</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E15.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text"><math class="ltx_Math" altimg="m25.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="H(z)" display="inline"><mrow><mi href="./1.13#E15">H</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> (locally)</span></dd>
<dt>Symbols:</dt>
<dd><a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a></dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px6" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Elimination of First Derivative by Change of Independent Variable</h3>
<div id="Px6.info" class="ltx_metadata ltx_info">
) substitute</p>
<table id="E16" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.13.16</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E16.png" altimg-height="53px" altimg-valign="-21px" altimg-width="260px" alttext="\eta=\int\exp\left(-\int f(z)\mathrm{d}z\right)\mathrm{d}z." display="block"><mrow><mrow><mi href="./1.13#E16">η</mi><mo>=</mo><mrow><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mrow><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./1.1#p2.t1.r2">z</mi></mrow></mrow></mrow></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./1.1#p2.t1.r2">z</mi></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E16.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text"><math class="ltx_Math" altimg="m34.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="\eta" display="inline"><mi href="./1.13#E16">η</mi></math>: change of variable (locally)</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E19" title="(4.2.19) ‣ §4.2(iii) The Exponential Function ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m35.png" altimg-height="16px" altimg-valign="-6px" altimg-width="48px" alttext="\exp\NVar{z}" display="inline"><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: exponential function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a> and
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Then</p>
<table id="E17" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.13.17</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E17.png" altimg-height="56px" altimg-valign="-22px" altimg-width="328px" alttext="\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}\eta}^{2}}+g(z)\exp\left(2\int f(z)\mathrm%
{d}z\right)w=0." display="block"><mrow><mrow><mrow><mfrac><mrow><mpadded lspace="-1.7pt" width="-4.5pt"><msup><mo href="./1.4#E4">d</mo><mn>2</mn></msup></mpadded><mi href="./1.13#SS1.p2">w</mi></mrow><msup><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi href="./1.13#E16">η</mi></mrow><mn>2</mn></msup></mfrac><mo>+</mo><mrow><mrow><mi>g</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><mrow><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./1.1#p2.t1.r2">z</mi></mrow></mrow></mrow></mrow><mo>)</mo></mrow></mrow><mo></mo><mi href="./1.13#SS1.p2">w</mi></mrow></mrow><mo>=</mo><mn>0</mn></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E17.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#E4" title="(1.4.4) ‣ §1.4(iii) Derivatives ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="29px" altimg-valign="-9px" altimg-width="27px" alttext="\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: derivative of <math class="ltx_Math" altimg="m59.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E19" title="(4.2.19) ‣ §4.2(iii) The Exponential Function ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m35.png" altimg-height="16px" altimg-valign="-6px" altimg-width="48px" alttext="\exp\NVar{z}" display="inline"><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: exponential function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>,
<a href="./1.13#SS1.p2" title="§1.13(i) Existence of Solutions ‣ §1.13 Differential Equations ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m64.png" altimg-height="23px" altimg-valign="-7px" altimg-width="45px" alttext="w(z)" display="inline"><mrow><mi href="./1.13#Px1.p1">w</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: solution</a> and
<a href="./1.13#E16" title="(1.13.16) ‣ Elimination of First Derivative by Change of Independent Variable ‣ §1.13(iv) Change of Variables ‣ §1.13 Differential Equations ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m34.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="\eta" display="inline"><mi href="./1.13#E16">η</mi></math>: change of variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px7" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Liouville Transformation</h3>
<div id="Px7.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px7.p1" class="ltx_para">
<p class="ltx_p">Let <math class="ltx_Math" altimg="m32.png" altimg-height="23px" altimg-valign="-7px" altimg-width="52px" alttext="W(z)" display="inline"><mrow><mi href="./1.13#Px5.p1">W</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> satisfy (), <math class="ltx_Math" altimg="m53.png" altimg-height="23px" altimg-valign="-7px" altimg-width="40px" alttext="\zeta(z)" display="inline"><mrow><mi href="./1.13#Px7.p1">ζ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> be any
thrice-differentiable function of <math class="ltx_Math" altimg="m70.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>, and</p>
<table id="E18" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.13.18</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E18.png" altimg-height="29px" altimg-valign="-7px" altimg-width="209px" alttext="U(z)=(\zeta^{\prime}(z))^{1/2}W(z)." display="block"><mrow><mrow><mrow><mi>U</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><msup><mi href="./1.13#Px7.p1">ζ</mi><mo>′</mo></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo></mo><mrow><mi href="./1.13#Px5.p1">W</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E18.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>,
<a href="./1.13#Px5.p1" title="Elimination of First Derivative by Change of Dependent Variable ‣ §1.13(iv) Change of Variables ‣ §1.13 Differential Equations ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m32.png" altimg-height="23px" altimg-valign="-7px" altimg-width="52px" alttext="W(z)" display="inline"><mrow><mi href="./1.13#Px5.p1">W</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: change of variable</a> and
<a href="./1.13#Px7.p1" title="Liouville Transformation ‣ §1.13(iv) Change of Variables ‣ §1.13 Differential Equations ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="23px" altimg-valign="-7px" altimg-width="40px" alttext="\zeta(z)" display="inline"><mrow><mi href="./1.13#Px7.p1">ζ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: thrice-differentiable function</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Then</p>
<table id="E19" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.13.19</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E19.png" altimg-height="56px" altimg-valign="-22px" altimg-width="266px" alttext="\frac{{\mathrm{d}}^{2}U}{{\mathrm{d}\zeta}^{2}}=\left(\dot{z}^{2}H(z)-\tfrac{1%
}{2}\left\{z,\zeta\right\}\right)U." display="block"><mrow><mrow><mfrac><mrow><mpadded lspace="-1.7pt" width="-4.5pt"><msup><mo href="./1.4#E4">d</mo><mn>2</mn></msup></mpadded><mi>U</mi></mrow><msup><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi href="./1.13#Px7.p1">ζ</mi></mrow><mn>2</mn></msup></mfrac><mo>=</mo><mrow><mrow><mo>(</mo><mrow><mrow><msup><mover accent="true"><mi href="./1.1#p2.t1.r2">z</mi><mo>˙</mo></mover><mn>2</mn></msup><mo></mo><mrow><mi href="./1.13#E15">H</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>-</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mrow><mo href="./1.13#E20">{</mo><mi href="./1.1#p2.t1.r2">z</mi><mo href="./1.13#E20">,</mo><mi href="./1.13#Px7.p1">ζ</mi><mo href="./1.13#E20">}</mo></mrow></mrow></mrow><mo>)</mo></mrow><mo></mo><mi>U</mi></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E19.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.13#E20" title="(1.13.20) ‣ Liouville Transformation ‣ §1.13(iv) Change of Variables ‣ §1.13 Differential Equations ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="23px" altimg-valign="-7px" altimg-width="53px" alttext="\left\{\NVar{z},\NVar{\zeta}\right\}" display="inline"><mrow><mo href="./1.13#E20">{</mo><mi class="ltx_nvar" href="./1.1#p2.t1.r2">z</mi><mo href="./1.13#E20">,</mo><mi class="ltx_nvar" href="./1.13#Px7.p1">ζ</mi><mo href="./1.13#E20">}</mo></mrow></math>: Schwarzian derivative</a>,
<a href="./1.4#E4" title="(1.4.4) ‣ §1.4(iii) Derivatives ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="29px" altimg-valign="-9px" altimg-width="27px" alttext="\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: derivative of <math class="ltx_Math" altimg="m59.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>,
<a href="./1.13#E15" title="(1.13.15) ‣ Elimination of First Derivative by Change of Dependent Variable ‣ §1.13(iv) Change of Variables ‣ §1.13 Differential Equations ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m25.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="H(z)" display="inline"><mrow><mi href="./1.13#E15">H</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math></a> and
<a href="./1.13#Px7.p1" title="Liouville Transformation ‣ §1.13(iv) Change of Variables ‣ §1.13 Differential Equations ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="23px" altimg-valign="-7px" altimg-width="40px" alttext="\zeta(z)" display="inline"><mrow><mi href="./1.13#Px7.p1">ζ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: thrice-differentiable function</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Here dots denote differentiations with respect to <math class="ltx_Math" altimg="m54.png" altimg-height="21px" altimg-valign="-6px" altimg-width="14px" alttext="\zeta" display="inline"><mi href="./1.13#Px7.p1">ζ</mi></math>, and
<math class="ltx_Math" altimg="m45.png" altimg-height="23px" altimg-valign="-7px" altimg-width="53px" alttext="\left\{z,\zeta\right\}" display="inline"><mrow><mo href="./1.13#E20">{</mo><mi href="./1.1#p2.t1.r2">z</mi><mo href="./1.13#E20">,</mo><mi href="./1.13#Px7.p1">ζ</mi><mo href="./1.13#E20">}</mo></mrow></math> is the <em class="ltx_emph ltx_font_italic">Schwarzian derivative</em>:
</p>
<table id="E20" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.13.20</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E20.png" altimg-height="58px" altimg-valign="-22px" altimg-width="396px" alttext="\left\{z,\zeta\right\}=-2\dot{z}^{\ifrac{1}{2}}\frac{{\mathrm{d}}^{2}}{{%
\mathrm{d}\zeta}^{2}}(\dot{z}^{-\ifrac{1}{2}})=\frac{\dddot{z}}{\dot{z}}-\frac%
{3}{2}\left(\frac{\ddot{z}}{\dot{z}}\right)^{2}." display="block"><mrow><mrow><mrow><mo href="./1.13#E20">{</mo><mi href="./1.1#p2.t1.r2">z</mi><mo href="./1.13#E20">,</mo><mi href="./1.13#Px7.p1">ζ</mi><mo href="./1.13#E20">}</mo></mrow><mo>=</mo><mrow><mo>-</mo><mrow><mn>2</mn><mo></mo><msup><mover accent="true"><mi href="./1.1#p2.t1.r2">z</mi><mo>˙</mo></mover><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo></mo><mrow><mfrac><mpadded lspace="-1.7pt" width="-4.5pt"><msup><mo href="./1.4#E4">d</mo><mn>2</mn></msup></mpadded><msup><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi href="./1.13#Px7.p1">ζ</mi></mrow><mn>2</mn></msup></mfrac><mo></mo><mrow><mo stretchy="false">(</mo><msup><mover accent="true"><mi href="./1.1#p2.t1.r2">z</mi><mo>˙</mo></mover><mrow><mo>-</mo><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></mrow></msup><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow><mo>=</mo><mrow><mfrac><mover accent="true"><mi href="./1.1#p2.t1.r2">z</mi><mo>˙˙˙</mo></mover><mover accent="true"><mi href="./1.1#p2.t1.r2">z</mi><mo>˙</mo></mover></mfrac><mo>-</mo><mrow><mfrac><mn>3</mn><mn>2</mn></mfrac><mo></mo><msup><mrow><mo>(</mo><mfrac><mover accent="true"><mi href="./1.1#p2.t1.r2">z</mi><mo>¨</mo></mover><mover accent="true"><mi href="./1.1#p2.t1.r2">z</mi><mo>˙</mo></mover></mfrac><mo>)</mo></mrow><mn>2</mn></msup></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E20.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m44.png" altimg-height="23px" altimg-valign="-7px" altimg-width="53px" alttext="\left\{\NVar{z},\NVar{\zeta}\right\}" display="inline"><mrow><mo href="./1.13#E20">{</mo><mi class="ltx_nvar" href="./1.1#p2.t1.r2">z</mi><mo href="./1.13#E20">,</mo><mi class="ltx_nvar" href="./1.13#Px7.p1">ζ</mi><mo href="./1.13#E20">}</mo></mrow></math>: Schwarzian derivative</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#E4" title="(1.4.4) ‣ §1.4(iii) Derivatives ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="29px" altimg-valign="-9px" altimg-width="27px" alttext="\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: derivative of <math class="ltx_Math" altimg="m59.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a> and
<a href="./1.13#Px7.p1" title="Liouville Transformation ‣ §1.13(iv) Change of Variables ‣ §1.13 Differential Equations ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="23px" altimg-valign="-7px" altimg-width="40px" alttext="\zeta(z)" display="inline"><mrow><mi href="./1.13#Px7.p1">ζ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: thrice-differentiable function</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px8" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Cayley’s Identity</h3>
<div id="Px8.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px8.p1" class="ltx_para">
<p class="ltx_p">For arbitrary <math class="ltx_Math" altimg="m52.png" altimg-height="21px" altimg-valign="-6px" altimg-width="14px" alttext="\xi" display="inline"><mi href="./1.13#Px4.p1">ξ</mi></math> and <math class="ltx_Math" altimg="m54.png" altimg-height="21px" altimg-valign="-6px" altimg-width="14px" alttext="\zeta" display="inline"><mi href="./1.13#Px7.p1">ζ</mi></math>,</p>
<table id="EGx1" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E21">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.13.21</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m14.png" altimg-height="25px" altimg-valign="-7px" altimg-width="55px" alttext="\displaystyle\left\{z,\zeta\right\}" display="inline"><mrow><mo href="./1.13#E20">{</mo><mi href="./1.1#p2.t1.r2">z</mi><mo href="./1.13#E20">,</mo><mi href="./1.13#Px7.p1">ζ</mi><mo href="./1.13#E20">}</mo></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m1.png" altimg-height="28px" altimg-valign="-7px" altimg-width="242px" alttext="\displaystyle=(\ifrac{\mathrm{d}\xi}{\mathrm{d}\zeta})^{2}\left\{z,\xi\right\}%
+\left\{\xi,\zeta\right\}." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi href="./1.13#Px4.p1">ξ</mi></mrow><mo href="./1.4#E4">/</mo><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi href="./1.13#Px7.p1">ζ</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo></mo><mrow><mo href="./1.13#E20">{</mo><mi href="./1.1#p2.t1.r2">z</mi><mo href="./1.13#E20">,</mo><mi href="./1.13#Px4.p1">ξ</mi><mo href="./1.13#E20">}</mo></mrow></mrow><mo>+</mo><mrow><mo href="./1.13#E20">{</mo><mi href="./1.13#Px4.p1">ξ</mi><mo href="./1.13#E20">,</mo><mi href="./1.13#Px7.p1">ζ</mi><mo href="./1.13#E20">}</mo></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E21.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.13#E20" title="(1.13.20) ‣ Liouville Transformation ‣ §1.13(iv) Change of Variables ‣ §1.13 Differential Equations ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="23px" altimg-valign="-7px" altimg-width="53px" alttext="\left\{\NVar{z},\NVar{\zeta}\right\}" display="inline"><mrow><mo href="./1.13#E20">{</mo><mi class="ltx_nvar" href="./1.1#p2.t1.r2">z</mi><mo href="./1.13#E20">,</mo><mi class="ltx_nvar" href="./1.13#Px7.p1">ζ</mi><mo href="./1.13#E20">}</mo></mrow></math>: Schwarzian derivative</a>,
<a href="./1.4#E4" title="(1.4.4) ‣ §1.4(iii) Derivatives ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="29px" altimg-valign="-9px" altimg-width="27px" alttext="\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: derivative of <math class="ltx_Math" altimg="m59.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>,
<a href="./1.13#Px4.p1" title="Transformation of the Point at Infinity ‣ §1.13(iv) Change of Variables ‣ §1.13 Differential Equations ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="21px" altimg-valign="-6px" altimg-width="14px" alttext="\xi" display="inline"><mi href="./1.13#Px4.p1">ξ</mi></math>: change of variable</a> and
<a href="./1.13#Px7.p1" title="Liouville Transformation ‣ §1.13(iv) Change of Variables ‣ §1.13 Differential Equations ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="23px" altimg-valign="-7px" altimg-width="40px" alttext="\zeta(z)" display="inline"><mrow><mi href="./1.13#Px7.p1">ζ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: thrice-differentiable function</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E22">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.13.22</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m14.png" altimg-height="25px" altimg-valign="-7px" altimg-width="55px" alttext="\displaystyle\left\{z,\zeta\right\}" display="inline"><mrow><mo href="./1.13#E20">{</mo><mi href="./1.1#p2.t1.r2">z</mi><mo href="./1.13#E20">,</mo><mi href="./1.13#Px7.p1">ζ</mi><mo href="./1.13#E20">}</mo></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m2.png" altimg-height="28px" altimg-valign="-7px" altimg-width="186px" alttext="\displaystyle=-(\ifrac{\mathrm{d}z}{\mathrm{d}\zeta})^{2}\left\{\zeta,z\right\}." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo>-</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi href="./1.1#p2.t1.r2">z</mi></mrow><mo href="./1.4#E4">/</mo><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi href="./1.13#Px7.p1">ζ</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo></mo><mrow><mo href="./1.13#E20">{</mo><mi href="./1.13#Px7.p1">ζ</mi><mo href="./1.13#E20">,</mo><mi href="./1.1#p2.t1.r2">z</mi><mo href="./1.13#E20">}</mo></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E22.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.13#E20" title="(1.13.20) ‣ Liouville Transformation ‣ §1.13(iv) Change of Variables ‣ §1.13 Differential Equations ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="23px" altimg-valign="-7px" altimg-width="53px" alttext="\left\{\NVar{z},\NVar{\zeta}\right\}" display="inline"><mrow><mo href="./1.13#E20">{</mo><mi class="ltx_nvar" href="./1.1#p2.t1.r2">z</mi><mo href="./1.13#E20">,</mo><mi class="ltx_nvar" href="./1.13#Px7.p1">ζ</mi><mo href="./1.13#E20">}</mo></mrow></math>: Schwarzian derivative</a>,
<a href="./1.4#E4" title="(1.4.4) ‣ §1.4(iii) Derivatives ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="29px" altimg-valign="-9px" altimg-width="27px" alttext="\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: derivative of <math class="ltx_Math" altimg="m59.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a> and
<a href="./1.13#Px7.p1" title="Liouville Transformation ‣ §1.13(iv) Change of Variables ‣ §1.13 Differential Equations ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="23px" altimg-valign="-7px" altimg-width="40px" alttext="\zeta(z)" display="inline"><mrow><mi href="./1.13#Px7.p1">ζ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: thrice-differentiable function</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
</div>
</section>
</section>
<section id="v" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§1.13(v) </span>Products of Solutions</h2>
<div id="SS5.info" class="ltx_metadata ltx_info">
) satisfies</p>
<table id="E23" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.13.23</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E23.png" altimg-height="51px" altimg-valign="-18px" altimg-width="498px" alttext="\frac{{\mathrm{d}}^{3}w}{{\mathrm{d}z}^{3}}+3f\frac{{\mathrm{d}}^{2}w}{{%
\mathrm{d}z}^{2}}+(2f^{2}+f^{\prime}+4g)\frac{\mathrm{d}w}{\mathrm{d}z}+(4fg+2%
g^{\prime})w=0." display="block"><mrow><mrow><mrow><mfrac><mrow><mpadded lspace="-1.7pt" width="-4.5pt"><msup><mo href="./1.4#E4">d</mo><mn>3</mn></msup></mpadded><mi href="./1.13#SS1.p2">w</mi></mrow><msup><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi href="./1.1#p2.t1.r2">z</mi></mrow><mn>3</mn></msup></mfrac><mo>+</mo><mrow><mn>3</mn><mo></mo><mi>f</mi><mo></mo><mfrac><mrow><mpadded lspace="-1.7pt" width="-4.5pt"><msup><mo href="./1.4#E4">d</mo><mn>2</mn></msup></mpadded><mi href="./1.13#SS1.p2">w</mi></mrow><msup><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi href="./1.1#p2.t1.r2">z</mi></mrow><mn>2</mn></msup></mfrac></mrow><mo>+</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>2</mn><mo></mo><msup><mi>f</mi><mn>2</mn></msup></mrow><mo>+</mo><msup><mi>f</mi><mo>′</mo></msup><mo>+</mo><mrow><mn>4</mn><mo></mo><mi>g</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi href="./1.13#SS1.p2">w</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi href="./1.1#p2.t1.r2">z</mi></mrow></mfrac></mrow><mo>+</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>4</mn><mo></mo><mi>f</mi><mo></mo><mi>g</mi></mrow><mo>+</mo><mrow><mn>2</mn><mo></mo><msup><mi>g</mi><mo>′</mo></msup></mrow></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi href="./1.13#SS1.p2">w</mi></mrow></mrow><mo>=</mo><mn>0</mn></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E23.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#E4" title="(1.4.4) ‣ §1.4(iii) Derivatives ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="29px" altimg-valign="-9px" altimg-width="27px" alttext="\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: derivative of <math class="ltx_Math" altimg="m59.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a> and
<a href="./1.13#SS1.p2" title="§1.13(i) Existence of Solutions ‣ §1.13 Differential Equations ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m64.png" altimg-height="23px" altimg-valign="-7px" altimg-width="45px" alttext="w(z)" display="inline"><mrow><mi href="./1.13#Px1.p1">w</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: solution</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS5.p2" class="ltx_para">
<p class="ltx_p">If <math class="ltx_Math" altimg="m28.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="U(z)" display="inline"><mrow><mi href="./1.13#SS5.p2">U</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> and <math class="ltx_Math" altimg="m30.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="V(z)" display="inline"><mrow><mi href="./1.13#SS5.p2">V</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> are respectively solutions of
</p>
<table id="E24" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex8" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">1.13.24</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m12.png" altimg-height="51px" altimg-valign="-18px" altimg-width="97px" alttext="\displaystyle\frac{{\mathrm{d}}^{2}U}{{\mathrm{d}z}^{2}}+IU" display="inline"><mrow><mstyle displaystyle="true"><mfrac><mrow><mpadded lspace="-1.7pt" width="-4.5pt"><msup><mo href="./1.4#E4">d</mo><mn>2</mn></msup></mpadded><mi href="./1.13#SS5.p2">U</mi></mrow><msup><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi href="./1.1#p2.t1.r2">z</mi></mrow><mn>2</mn></msup></mfrac></mstyle><mo>+</mo><mrow><mi href="./1.13#SS5.p2">I</mi><mo></mo><mi href="./1.13#SS5.p2">U</mi></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m3.png" altimg-height="22px" altimg-valign="-6px" altimg-width="43px" alttext="\displaystyle=0," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mn>0</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex9" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m13.png" altimg-height="51px" altimg-valign="-18px" altimg-width="100px" alttext="\displaystyle\frac{{\mathrm{d}}^{2}V}{{\mathrm{d}z}^{2}}+JV" display="inline"><mrow><mstyle displaystyle="true"><mfrac><mrow><mpadded lspace="-1.7pt" width="-4.5pt"><msup><mo href="./1.4#E4">d</mo><mn>2</mn></msup></mpadded><mi href="./1.13#SS5.p2">V</mi></mrow><msup><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi href="./1.1#p2.t1.r2">z</mi></mrow><mn>2</mn></msup></mfrac></mstyle><mo>+</mo><mrow><mi href="./1.13#SS5.p2">J</mi><mo></mo><mi href="./1.13#SS5.p2">V</mi></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m3.png" altimg-height="22px" altimg-valign="-6px" altimg-width="43px" alttext="\displaystyle=0," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mn>0</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E24.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#E4" title="(1.4.4) ‣ §1.4(iii) Derivatives ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="29px" altimg-valign="-9px" altimg-width="27px" alttext="\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: derivative of <math class="ltx_Math" altimg="m59.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>,
<a href="./1.13#SS5.p2" title="§1.13(v) Products of Solutions ‣ §1.13 Differential Equations ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m28.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="U(z)" display="inline"><mrow><mi href="./1.13#SS5.p2">U</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: solution</a>,
<a href="./1.13#SS5.p2" title="§1.13(v) Products of Solutions ‣ §1.13 Differential Equations ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m30.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="V(z)" display="inline"><mrow><mi href="./1.13#SS5.p2">V</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: solution</a>,
<a href="./1.13#SS5.p2" title="§1.13(v) Products of Solutions ‣ §1.13 Differential Equations ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m26.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="I" display="inline"><mi href="./1.13#SS5.p2">I</mi></math>: coefficient</a> and
<a href="./1.13#SS5.p2" title="§1.13(v) Products of Solutions ‣ §1.13 Differential Equations ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m27.png" altimg-height="18px" altimg-valign="-2px" altimg-width="17px" alttext="J" display="inline"><mi href="./1.13#SS5.p2">J</mi></math>: coefficient</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">then <math class="ltx_Math" altimg="m33.png" altimg-height="18px" altimg-valign="-2px" altimg-width="84px" alttext="W=UV" display="inline"><mrow><mi href="./1.13#SS5.p2">W</mi><mo>=</mo><mrow><mi href="./1.13#SS5.p2">U</mi><mo></mo><mi href="./1.13#SS5.p2">V</mi></mrow></mrow></math> is a solution of
</p>
<table id="E25" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.13.25</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E25.png" altimg-height="53px" altimg-valign="-21px" altimg-width="480px" alttext="\frac{\mathrm{d}}{\mathrm{d}z}\left(\frac{W^{\prime\prime\prime}+2(I+J)W^{%
\prime}+(I^{\prime}+J^{\prime})W}{I-J}\right)=-(I-J)W." display="block"><mrow><mrow><mrow><mfrac><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi href="./1.1#p2.t1.r2">z</mi></mrow></mfrac><mo></mo><mrow><mo>(</mo><mfrac><mrow><msup><mi href="./1.13#SS5.p2">W</mi><mo>′′′</mo></msup><mo>+</mo><mrow><mn>2</mn><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./1.13#SS5.p2">I</mi><mo>+</mo><mi href="./1.13#SS5.p2">J</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msup><mi href="./1.13#SS5.p2">W</mi><mo>′</mo></msup></mrow><mo>+</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><msup><mi href="./1.13#SS5.p2">I</mi><mo>′</mo></msup><mo>+</mo><msup><mi href="./1.13#SS5.p2">J</mi><mo>′</mo></msup></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi href="./1.13#SS5.p2">W</mi></mrow></mrow><mrow><mi href="./1.13#SS5.p2">I</mi><mo>-</mo><mi href="./1.13#SS5.p2">J</mi></mrow></mfrac><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mo>-</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./1.13#SS5.p2">I</mi><mo>-</mo><mi href="./1.13#SS5.p2">J</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi href="./1.13#SS5.p2">W</mi></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E25.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#E4" title="(1.4.4) ‣ §1.4(iii) Derivatives ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="29px" altimg-valign="-9px" altimg-width="27px" alttext="\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: derivative of <math class="ltx_Math" altimg="m59.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>,
<a href="./1.13#SS5.p2" title="§1.13(v) Products of Solutions ‣ §1.13 Differential Equations ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m26.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="I" display="inline"><mi href="./1.13#SS5.p2">I</mi></math>: coefficient</a>,
<a href="./1.13#SS5.p2" title="§1.13(v) Products of Solutions ‣ §1.13 Differential Equations ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m27.png" altimg-height="18px" altimg-valign="-2px" altimg-width="17px" alttext="J" display="inline"><mi href="./1.13#SS5.p2">J</mi></math>: coefficient</a> and
<a href="./1.13#SS5.p2" title="§1.13(v) Products of Solutions ‣ §1.13 Differential Equations ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m32.png" altimg-height="23px" altimg-valign="-7px" altimg-width="52px" alttext="W(z)" display="inline"><mrow><mi href="./1.13#SS5.p2">W</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: solution</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS5.p3" class="ltx_para">
<p class="ltx_p">For extensions of these results to linear homogeneous differential equations
of arbitrary order see <cite class="ltx_cite ltx_citemacro_citet">Spigler (</div>
</div>
</body></text>
</html>
</page>
<page>
<!DOCTYPE html><html>
<head>
<title>DLMF: 2.7 Differential Equations</title>
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<text><body class="color_default textfont_default titlefont_default fontsize_default navbar_default">
<div class="ltx_page_navbar">
<div class="ltx_page_navlogo"></dd>
</dl>
</div>
</div>
<div id="SS1.p1" class="ltx_para">
<p class="ltx_p">An <em class="ltx_emph ltx_font_italic">ordinary point</em> of the differential equation
</p>
<table id="E1" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">2.7.1</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E1.png" altimg-height="51px" altimg-valign="-18px" altimg-width="250px" alttext="\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+f(z)\frac{\mathrm{d}w}{\mathrm{d}z%
}+g(z)w=0" display="block"><mrow><mrow><mfrac><mrow><mpadded lspace="-1.7pt" width="-4.5pt"><msup><mo href="./1.4#E4">d</mo><mn>2</mn></msup></mpadded><mi href="./2.7#SS1.p1">w</mi></mrow><msup><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi>z</mi></mrow><mn>2</mn></msup></mfrac><mo>+</mo><mrow><mrow><mi href="./2.7#SS1.p1">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi href="./2.7#SS1.p1">w</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi>z</mi></mrow></mfrac></mrow><mo>+</mo><mrow><mrow><mi href="./2.7#SS1.p1">g</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mi href="./2.7#SS1.p1">w</mi></mrow></mrow><mo>=</mo><mn>0</mn></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#E4" title="(1.4.4) ‣ §1.4(iii) Derivatives ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m71.png" altimg-height="29px" altimg-valign="-9px" altimg-width="27px" alttext="\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: derivative of <math class="ltx_Math" altimg="m107.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m140.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./2.7#SS1.p1" title="§2.7(i) Regular Singularities: Fuchs–Frobenius Theory ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m126.png" altimg-height="13px" altimg-valign="-2px" altimg-width="19px" alttext="w" display="inline"><mi href="./2.7#SS1.p1">w</mi></math>: DE solution</a>,
<a href="./2.7#SS1.p1" title="§2.7(i) Regular Singularities: Fuchs–Frobenius Theory ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m104.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(z)" display="inline"><mrow><mi href="./2.7#SS1.p1">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: analytic function</a> and
<a href="./2.7#SS1.p1" title="§2.7(i) Regular Singularities: Fuchs–Frobenius Theory ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m112.png" altimg-height="23px" altimg-valign="-7px" altimg-width="40px" alttext="g(z)" display="inline"><mrow><mi href="./2.7#SS1.p1">g</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: analytic function</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">is one at which the coefficients <math class="ltx_Math" altimg="m104.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(z)" display="inline"><mrow><mi href="./2.7#SS1.p1">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math> and <math class="ltx_Math" altimg="m112.png" altimg-height="23px" altimg-valign="-7px" altimg-width="40px" alttext="g(z)" display="inline"><mrow><mi href="./2.7#SS1.p1">g</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math> are analytic. All solutions
are analytic at an ordinary point, and their Taylor-series expansions are found
by equating coefficients.
</p>
</div>
<div id="SS1.p2" class="ltx_para">
<p class="ltx_p">Other points <math class="ltx_Math" altimg="m147.png" altimg-height="16px" altimg-valign="-5px" altimg-width="23px" alttext="z_{0}" display="inline"><msub><mi>z</mi><mn>0</mn></msub></math> are <em class="ltx_emph ltx_font_italic">singularities</em> of the differential equation. If both
<math class="ltx_Math" altimg="m34.png" altimg-height="23px" altimg-valign="-7px" altimg-width="110px" alttext="(z-z_{0})f(z)" display="inline"><mrow><mrow><mo stretchy="false">(</mo><mrow><mi>z</mi><mo>-</mo><msub><mi>z</mi><mn>0</mn></msub></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mi href="./2.7#SS1.p1">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></math> and <math class="ltx_Math" altimg="m32.png" altimg-height="25px" altimg-valign="-7px" altimg-width="117px" alttext="(z-z_{0})^{2}g(z)" display="inline"><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mi>z</mi><mo>-</mo><msub><mi>z</mi><mn>0</mn></msub></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo></mo><mrow><mi href="./2.7#SS1.p1">g</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></math> are analytic at <math class="ltx_Math" altimg="m147.png" altimg-height="16px" altimg-valign="-5px" altimg-width="23px" alttext="z_{0}" display="inline"><msub><mi>z</mi><mn>0</mn></msub></math>, then <math class="ltx_Math" altimg="m147.png" altimg-height="16px" altimg-valign="-5px" altimg-width="23px" alttext="z_{0}" display="inline"><msub><mi>z</mi><mn>0</mn></msub></math> is a
<em class="ltx_emph ltx_font_italic">regular singularity</em> (or <em class="ltx_emph ltx_font_italic">singularity of the first kind</em>). All other
singularities are classified as <em class="ltx_emph ltx_font_italic">irregular</em>.</p>
</div>
<div id="SS1.p3" class="ltx_para">
<p class="ltx_p">In a punctured neighborhood <math class="ltx_Math" altimg="m80.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="\mathbf{N}" display="inline"><mi href="./2.7#SS1.p3" mathvariant="bold">N</mi></math> of a regular singularity <math class="ltx_Math" altimg="m147.png" altimg-height="16px" altimg-valign="-5px" altimg-width="23px" alttext="z_{0}" display="inline"><msub><mi>z</mi><mn>0</mn></msub></math>
</p>
<table id="E2" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex1" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">2.7.2</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m21.png" altimg-height="25px" altimg-valign="-7px" altimg-width="44px" alttext="\displaystyle f(z)" display="inline"><mrow><mi href="./2.7#SS1.p1">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m4.png" altimg-height="64px" altimg-valign="-27px" altimg-width="180px" alttext="\displaystyle=\sum_{s=0}^{\infty}f_{s}(z-z_{0})^{s-1}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi>s</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover></mstyle><mrow><msub><mi href="./2.7#SS1.p3">f</mi><mi>s</mi></msub><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi>z</mi><mo>-</mo><msub><mi>z</mi><mn>0</mn></msub></mrow><mo stretchy="false">)</mo></mrow><mrow><mi>s</mi><mo>-</mo><mn>1</mn></mrow></msup></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex2" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m22.png" altimg-height="25px" altimg-valign="-7px" altimg-width="42px" alttext="\displaystyle g(z)" display="inline"><mrow><mi href="./2.7#SS1.p1">g</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m5.png" altimg-height="64px" altimg-valign="-27px" altimg-width="180px" alttext="\displaystyle=\sum_{s=0}^{\infty}g_{s}(z-z_{0})^{s-2}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi>s</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover></mstyle><mrow><msub><mi href="./2.7#SS1.p3">g</mi><mi>s</mi></msub><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi>z</mi><mo>-</mo><msub><mi>z</mi><mn>0</mn></msub></mrow><mo stretchy="false">)</mo></mrow><mrow><mi>s</mi><mo>-</mo><mn>2</mn></mrow></msup></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./2.7#SS1.p1" title="§2.7(i) Regular Singularities: Fuchs–Frobenius Theory ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m104.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(z)" display="inline"><mrow><mi href="./2.7#SS1.p1">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: analytic function</a>,
<a href="./2.7#SS1.p1" title="§2.7(i) Regular Singularities: Fuchs–Frobenius Theory ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m112.png" altimg-height="23px" altimg-valign="-7px" altimg-width="40px" alttext="g(z)" display="inline"><mrow><mi href="./2.7#SS1.p1">g</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: analytic function</a>,
<a href="./2.7#SS1.p3" title="§2.7(i) Regular Singularities: Fuchs–Frobenius Theory ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m110.png" altimg-height="21px" altimg-valign="-6px" altimg-width="23px" alttext="f_{s}" display="inline"><msub><mi href="./2.7#SS1.p3">f</mi><mi>s</mi></msub></math>: coefficients</a> and
<a href="./2.7#SS1.p3" title="§2.7(i) Regular Singularities: Fuchs–Frobenius Theory ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m117.png" altimg-height="16px" altimg-valign="-6px" altimg-width="22px" alttext="g_{s}" display="inline"><msub><mi href="./2.7#SS1.p3">g</mi><mi>s</mi></msub></math>: coefficients</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">with at least one of the coefficients <math class="ltx_Math" altimg="m108.png" altimg-height="21px" altimg-valign="-6px" altimg-width="23px" alttext="f_{0}" display="inline"><msub><mi href="./2.7#SS1.p3">f</mi><mn>0</mn></msub></math>, <math class="ltx_Math" altimg="m115.png" altimg-height="16px" altimg-valign="-6px" altimg-width="23px" alttext="g_{0}" display="inline"><msub><mi href="./2.7#SS1.p3">g</mi><mn>0</mn></msub></math>, <math class="ltx_Math" altimg="m116.png" altimg-height="16px" altimg-valign="-6px" altimg-width="23px" alttext="g_{1}" display="inline"><msub><mi href="./2.7#SS1.p3">g</mi><mn>1</mn></msub></math> nonzero. Let
<math class="ltx_Math" altimg="m63.png" altimg-height="16px" altimg-valign="-5px" altimg-width="26px" alttext="\alpha_{1}" display="inline"><msub><mi>α</mi><mn>1</mn></msub></math>, <math class="ltx_Math" altimg="m64.png" altimg-height="16px" altimg-valign="-5px" altimg-width="26px" alttext="\alpha_{2}" display="inline"><msub><mi>α</mi><mn>2</mn></msub></math> denote the <em class="ltx_emph ltx_font_italic">indices</em> or <em class="ltx_emph ltx_font_italic">exponents</em>, that is,
the roots of the <em class="ltx_emph ltx_font_italic">indicial equation</em>
</p>
<table id="E3" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">2.7.3</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E3.png" altimg-height="25px" altimg-valign="-7px" altimg-width="293px" alttext="Q(\alpha)\equiv\alpha(\alpha-1)+f_{0}\alpha+g_{0}=0." display="block"><mrow><mrow><mrow><mi href="./2.7#SS1.p3">Q</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">)</mo></mrow></mrow><mo>≡</mo><mrow><mrow><mi>α</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>α</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>+</mo><mrow><msub><mi href="./2.7#SS1.p3">f</mi><mn>0</mn></msub><mo></mo><mi>α</mi></mrow><mo>+</mo><msub><mi href="./2.7#SS1.p3">g</mi><mn>0</mn></msub></mrow><mo>=</mo><mn>0</mn></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./2.7#SS1.p3" title="§2.7(i) Regular Singularities: Fuchs–Frobenius Theory ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m110.png" altimg-height="21px" altimg-valign="-6px" altimg-width="23px" alttext="f_{s}" display="inline"><msub><mi href="./2.7#SS1.p3">f</mi><mi>s</mi></msub></math>: coefficients</a>,
<a href="./2.7#SS1.p3" title="§2.7(i) Regular Singularities: Fuchs–Frobenius Theory ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m117.png" altimg-height="16px" altimg-valign="-6px" altimg-width="22px" alttext="g_{s}" display="inline"><msub><mi href="./2.7#SS1.p3">g</mi><mi>s</mi></msub></math>: coefficients</a> and
<a href="./2.7#SS1.p3" title="§2.7(i) Regular Singularities: Fuchs–Frobenius Theory ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="Q(\alpha)" display="inline"><mrow><mi href="./2.7#SS1.p3">Q</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">)</mo></mrow></mrow></math>: indicial function</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Provided that <math class="ltx_Math" altimg="m62.png" altimg-height="19px" altimg-valign="-5px" altimg-width="72px" alttext="\alpha_{1}-\alpha_{2}" display="inline"><mrow><msub><mi>α</mi><mn>1</mn></msub><mo>-</mo><msub><mi>α</mi><mn>2</mn></msub></mrow></math> is not zero or an integer, equation
() has independent solutions <math class="ltx_Math" altimg="m139.png" altimg-height="24px" altimg-valign="-8px" altimg-width="53px" alttext="w_{j}(z)" display="inline"><mrow><msub><mi href="./2.7#SS1.p3">w</mi><mi>j</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math>, <math class="ltx_Math" altimg="m118.png" altimg-height="20px" altimg-valign="-6px" altimg-width="69px" alttext="j=1,2" display="inline"><mrow><mi>j</mi><mo>=</mo><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></mrow></math>, such that
</p>
<table id="E4" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">2.7.4</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E4.png" altimg-height="64px" altimg-valign="-27px" altimg-width="317px" alttext="w_{j}(z)=(z-z_{0})^{\alpha_{j}}\sum_{s=0}^{\infty}a_{s,j}(z-z_{0})^{s}," display="block"><mrow><mrow><mrow><msub><mi href="./2.7#SS1.p3">w</mi><mi>j</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mi>z</mi><mo>-</mo><msub><mi>z</mi><mn>0</mn></msub></mrow><mo stretchy="false">)</mo></mrow><msub><mi>α</mi><mi>j</mi></msub></msup><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi>s</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><msub><mi href="./2.7#SS1.p3">a</mi><mrow><mi>s</mi><mo>,</mo><mi>j</mi></mrow></msub><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi>z</mi><mo>-</mo><msub><mi>z</mi><mn>0</mn></msub></mrow><mo stretchy="false">)</mo></mrow><mi>s</mi></msup></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m145.png" altimg-height="18px" altimg-valign="-3px" altimg-width="57px" alttext="z\in\mathbf{N}" display="inline"><mrow><mi>z</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mi href="./2.7#SS1.p3" mathvariant="bold">N</mi></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E4.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./front/introduction#Sx4.p1.t1.r9" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="15px" altimg-valign="-3px" altimg-width="18px" alttext="\in" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo></math>: element of</a>,
<a href="./2.7#SS1.p3" title="§2.7(i) Regular Singularities: Fuchs–Frobenius Theory ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m80.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="\mathbf{N}" display="inline"><mi href="./2.7#SS1.p3" mathvariant="bold">N</mi></math>: punctured neighborhood</a>,
<a href="./2.7#SS1.p3" title="§2.7(i) Regular Singularities: Fuchs–Frobenius Theory ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m139.png" altimg-height="24px" altimg-valign="-8px" altimg-width="53px" alttext="w_{j}(z)" display="inline"><mrow><msub><mi href="./2.7#SS1.p3">w</mi><mi>j</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: solutions</a> and
<a href="./2.7#SS1.p3" title="§2.7(i) Regular Singularities: Fuchs–Frobenius Theory ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m95.png" altimg-height="16px" altimg-valign="-5px" altimg-width="26px" alttext="a_{n}" display="inline"><msub><mi href="./2.7#SS1.p3">a</mi><mi>n</mi></msub></math>: coefficients</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">with <math class="ltx_Math" altimg="m92.png" altimg-height="22px" altimg-valign="-8px" altimg-width="72px" alttext="a_{0,j}=1" display="inline"><mrow><msub><mi href="./2.7#SS1.p3">a</mi><mrow><mn>0</mn><mo>,</mo><mi>j</mi></mrow></msub><mo>=</mo><mn>1</mn></mrow></math>, and</p>
<table id="E5" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">2.7.5</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E5.png" altimg-height="67px" altimg-valign="-27px" altimg-width="428px" alttext="Q(\alpha_{j}+s)a_{s,j}=-\sum_{r=0}^{s-1}\left((\alpha_{j}+r)f_{s-r}+g_{s-r}%
\right)a_{r,j}," display="block"><mrow><mrow><mrow><mrow><mi href="./2.7#SS1.p3">Q</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><msub><mi>α</mi><mi>j</mi></msub><mo>+</mo><mi>s</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo></mo><msub><mi href="./2.7#SS1.p3">a</mi><mrow><mi>s</mi><mo>,</mo><mi>j</mi></mrow></msub></mrow><mo>=</mo><mrow><mo>-</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi>r</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>s</mi><mo>-</mo><mn>1</mn></mrow></munderover><mrow><mrow><mo>(</mo><mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><msub><mi>α</mi><mi>j</mi></msub><mo>+</mo><mi>r</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msub><mi href="./2.7#SS1.p3">f</mi><mrow><mi>s</mi><mo>-</mo><mi>r</mi></mrow></msub></mrow><mo>+</mo><msub><mi href="./2.7#SS1.p3">g</mi><mrow><mi>s</mi><mo>-</mo><mi>r</mi></mrow></msub></mrow><mo>)</mo></mrow><mo></mo><msub><mi href="./2.7#SS1.p3">a</mi><mrow><mi>r</mi><mo>,</mo><mi>j</mi></mrow></msub></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E5.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./2.7#SS1.p3" title="§2.7(i) Regular Singularities: Fuchs–Frobenius Theory ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m110.png" altimg-height="21px" altimg-valign="-6px" altimg-width="23px" alttext="f_{s}" display="inline"><msub><mi href="./2.7#SS1.p3">f</mi><mi>s</mi></msub></math>: coefficients</a>,
<a href="./2.7#SS1.p3" title="§2.7(i) Regular Singularities: Fuchs–Frobenius Theory ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m117.png" altimg-height="16px" altimg-valign="-6px" altimg-width="22px" alttext="g_{s}" display="inline"><msub><mi href="./2.7#SS1.p3">g</mi><mi>s</mi></msub></math>: coefficients</a>,
<a href="./2.7#SS1.p3" title="§2.7(i) Regular Singularities: Fuchs–Frobenius Theory ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="Q(\alpha)" display="inline"><mrow><mi href="./2.7#SS1.p3">Q</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">)</mo></mrow></mrow></math>: indicial function</a> and
<a href="./2.7#SS1.p3" title="§2.7(i) Regular Singularities: Fuchs–Frobenius Theory ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m95.png" altimg-height="16px" altimg-valign="-5px" altimg-width="26px" alttext="a_{n}" display="inline"><msub><mi href="./2.7#SS1.p3">a</mi><mi>n</mi></msub></math>: coefficients</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">when <math class="ltx_Math" altimg="m122.png" altimg-height="20px" altimg-valign="-6px" altimg-width="120px" alttext="s=1,2,3,\dots" display="inline"><mrow><mi>s</mi><mo>=</mo><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>.</p>
</div>
<div id="SS1.p4" class="ltx_para">
<p class="ltx_p">If <math class="ltx_Math" altimg="m59.png" altimg-height="20px" altimg-valign="-6px" altimg-width="178px" alttext="\alpha_{1}-\alpha_{2}=0,1,2,\dots" display="inline"><mrow><mrow><msub><mi>α</mi><mn>1</mn></msub><mo>-</mo><msub><mi>α</mi><mn>2</mn></msub></mrow><mo>=</mo><mrow><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>, then () applies only
in the case <math class="ltx_Math" altimg="m119.png" altimg-height="20px" altimg-valign="-6px" altimg-width="50px" alttext="j=1" display="inline"><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow></math>. But there is an independent solution</p>
<table id="E6" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">2.7.6</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E6.png" altimg-height="78px" altimg-valign="-42px" altimg-width="520px" alttext="w_{2}(z)=(z-z_{0})^{\alpha_{2}}\sum_{\begin{subarray}{c}s=0\\
s\neq\alpha_{1}-\alpha_{2}\end{subarray}}^{\infty}b_{s}(z-z_{0})^{s}+cw_{1}(z)%
\ln\left(z-z_{0}\right)," display="block"><mrow><mrow><mrow><msub><mi href="./2.7#SS1.p3">w</mi><mn>2</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mi>z</mi><mo>-</mo><msub><mi>z</mi><mn>0</mn></msub></mrow><mo stretchy="false">)</mo></mrow><msub><mi>α</mi><mn>2</mn></msub></msup><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mtable class="ltx_align_c" rowspacing="0.0pt"><mtr><mtd><mrow><mi>s</mi><mo>=</mo><mn>0</mn></mrow></mtd></mtr><mtr><mtd><mrow><mi>s</mi><mo>≠</mo><mrow><msub><mi>α</mi><mn>1</mn></msub><mo>-</mo><msub><mi>α</mi><mn>2</mn></msub></mrow></mrow></mtd></mtr></mtable><mi mathvariant="normal">∞</mi></munderover><mrow><msub><mi href="./2.7#SS1.p4">b</mi><mi>s</mi></msub><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi>z</mi><mo>-</mo><msub><mi>z</mi><mn>0</mn></msub></mrow><mo stretchy="false">)</mo></mrow><mi>s</mi></msup></mrow></mrow></mrow><mo>+</mo><mrow><mi href="./2.7#SS1.p4">c</mi><mo></mo><mrow><msub><mi href="./2.7#SS1.p3">w</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><mo>(</mo><mrow><mi>z</mi><mo>-</mo><msub><mi>z</mi><mn>0</mn></msub></mrow><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m145.png" altimg-height="18px" altimg-valign="-3px" altimg-width="57px" alttext="z\in\mathbf{N}" display="inline"><mrow><mi>z</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mi href="./2.7#SS1.p3" mathvariant="bold">N</mi></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E6.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./front/introduction#Sx4.p1.t1.r9" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="15px" altimg-valign="-3px" altimg-width="18px" alttext="\in" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo></math>: element of</a>,
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m79.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a>,
<a href="./2.7#SS1.p4" title="§2.7(i) Regular Singularities: Fuchs–Frobenius Theory ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m100.png" altimg-height="21px" altimg-valign="-5px" altimg-width="21px" alttext="b_{s}" display="inline"><msub><mi href="./2.7#SS1.p4">b</mi><mi>s</mi></msub></math>: coefficients</a>,
<a href="./2.7#SS1.p4" title="§2.7(i) Regular Singularities: Fuchs–Frobenius Theory ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m102.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="c" display="inline"><mi href="./2.7#SS1.p4">c</mi></math>: constant</a>,
<a href="./2.7#SS1.p3" title="§2.7(i) Regular Singularities: Fuchs–Frobenius Theory ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m80.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="\mathbf{N}" display="inline"><mi href="./2.7#SS1.p3" mathvariant="bold">N</mi></math>: punctured neighborhood</a> and
<a href="./2.7#SS1.p3" title="§2.7(i) Regular Singularities: Fuchs–Frobenius Theory ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m139.png" altimg-height="24px" altimg-valign="-8px" altimg-width="53px" alttext="w_{j}(z)" display="inline"><mrow><msub><mi href="./2.7#SS1.p3">w</mi><mi>j</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: solutions</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">The coefficients <math class="ltx_Math" altimg="m100.png" altimg-height="21px" altimg-valign="-5px" altimg-width="21px" alttext="b_{s}" display="inline"><msub><mi href="./2.7#SS1.p4">b</mi><mi>s</mi></msub></math> and constant <math class="ltx_Math" altimg="m102.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="c" display="inline"><mi href="./2.7#SS1.p4">c</mi></math> are again determined by
equating coefficients in the differential equation, beginning with <math class="ltx_Math" altimg="m101.png" altimg-height="17px" altimg-valign="-2px" altimg-width="49px" alttext="c=1" display="inline"><mrow><mi href="./2.7#SS1.p4">c</mi><mo>=</mo><mn>1</mn></mrow></math> when
<math class="ltx_Math" altimg="m60.png" altimg-height="20px" altimg-valign="-5px" altimg-width="109px" alttext="\alpha_{1}-\alpha_{2}=0" display="inline"><mrow><mrow><msub><mi>α</mi><mn>1</mn></msub><mo>-</mo><msub><mi>α</mi><mn>2</mn></msub></mrow><mo>=</mo><mn>0</mn></mrow></math>, or with <math class="ltx_Math" altimg="m99.png" altimg-height="21px" altimg-valign="-5px" altimg-width="58px" alttext="b_{0}=1" display="inline"><mrow><msub><mi href="./2.7#SS1.p4">b</mi><mn>0</mn></msub><mo>=</mo><mn>1</mn></mrow></math> when <math class="ltx_Math" altimg="m61.png" altimg-height="20px" altimg-valign="-6px" altimg-width="178px" alttext="\alpha_{1}-\alpha_{2}=1,2,3,\dots" display="inline"><mrow><mrow><msub><mi>α</mi><mn>1</mn></msub><mo>-</mo><msub><mi>α</mi><mn>2</mn></msub></mrow><mo>=</mo><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>.</p>
</div>
<div id="SS1.p5" class="ltx_para">
<p class="ltx_p">The radii of convergence of the series ()
are not less than the distance of the next nearest singularity of the
differential equation from <math class="ltx_Math" altimg="m147.png" altimg-height="16px" altimg-valign="-5px" altimg-width="23px" alttext="z_{0}" display="inline"><msub><mi>z</mi><mn>0</mn></msub></math>.</p>
</div>
<div id="SS1.p6" class="ltx_para">
<p class="ltx_p">To include the point at infinity in the foregoing classification scheme, we
transform it into the origin by replacing <math class="ltx_Math" altimg="m144.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi>z</mi></math> in () with <math class="ltx_Math" altimg="m37.png" altimg-height="23px" altimg-valign="-7px" altimg-width="34px" alttext="1/z" display="inline"><mrow><mn>1</mn><mo>/</mo><mi>z</mi></mrow></math>;
see <cite class="ltx_cite ltx_citemacro_citet">Olver (</dd>
</dl>
</div>
</div>
<div id="SS2.p1" class="ltx_para">
<p class="ltx_p">If the singularities of <math class="ltx_Math" altimg="m104.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(z)" display="inline"><mrow><mi href="./2.7#SS1.p1">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math> and <math class="ltx_Math" altimg="m112.png" altimg-height="23px" altimg-valign="-7px" altimg-width="40px" alttext="g(z)" display="inline"><mrow><mi href="./2.7#SS1.p1">g</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math> at <math class="ltx_Math" altimg="m147.png" altimg-height="16px" altimg-valign="-5px" altimg-width="23px" alttext="z_{0}" display="inline"><msub><mi>z</mi><mn>0</mn></msub></math> are no worse than poles,
then <math class="ltx_Math" altimg="m147.png" altimg-height="16px" altimg-valign="-5px" altimg-width="23px" alttext="z_{0}" display="inline"><msub><mi>z</mi><mn>0</mn></msub></math> has <em class="ltx_emph ltx_font_italic">rank</em> <math class="ltx_Math" altimg="m66.png" altimg-height="19px" altimg-valign="-4px" altimg-width="47px" alttext="\ell-1" display="inline"><mrow><mi href="./2.7#SS2.p1" mathvariant="normal">ℓ</mi><mo>-</mo><mn>1</mn></mrow></math>, where <math class="ltx_Math" altimg="m67.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="\ell" display="inline"><mi href="./2.7#SS2.p1" mathvariant="normal">ℓ</mi></math> is the least integer such
that <math class="ltx_Math" altimg="m33.png" altimg-height="25px" altimg-valign="-7px" altimg-width="118px" alttext="(z-z_{0})^{\ell}f(z)" display="inline"><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mi>z</mi><mo>-</mo><msub><mi>z</mi><mn>0</mn></msub></mrow><mo stretchy="false">)</mo></mrow><mi href="./2.7#SS2.p1" mathvariant="normal">ℓ</mi></msup><mo></mo><mrow><mi href="./2.7#SS1.p1">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></math> and <math class="ltx_Math" altimg="m31.png" altimg-height="25px" altimg-valign="-7px" altimg-width="124px" alttext="(z-z_{0})^{2\ell}g(z)" display="inline"><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mi>z</mi><mo>-</mo><msub><mi>z</mi><mn>0</mn></msub></mrow><mo stretchy="false">)</mo></mrow><mrow><mn>2</mn><mo></mo><mi href="./2.7#SS2.p1" mathvariant="normal">ℓ</mi></mrow></msup><mo></mo><mrow><mi href="./2.7#SS1.p1">g</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></math> are analytic at
<math class="ltx_Math" altimg="m147.png" altimg-height="16px" altimg-valign="-5px" altimg-width="23px" alttext="z_{0}" display="inline"><msub><mi>z</mi><mn>0</mn></msub></math>. Thus a regular singularity has rank 0. The most common type of irregular
singularity for special functions has rank 1 and is located at infinity. Then</p>
<table id="E7" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex3" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">2.7.7</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m21.png" altimg-height="25px" altimg-valign="-7px" altimg-width="44px" alttext="\displaystyle f(z)" display="inline"><mrow><mi href="./2.7#SS1.p1">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m2.png" altimg-height="64px" altimg-valign="-27px" altimg-width="88px" alttext="\displaystyle=\sum_{s=0}^{\infty}\frac{f_{s}}{z^{s}}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi>s</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover></mstyle><mstyle displaystyle="true"><mfrac><msub><mi href="./2.7#SS1.p3">f</mi><mi>s</mi></msub><msup><mi>z</mi><mi>s</mi></msup></mfrac></mstyle></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex4" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m22.png" altimg-height="25px" altimg-valign="-7px" altimg-width="42px" alttext="\displaystyle g(z)" display="inline"><mrow><mi href="./2.7#SS1.p1">g</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m3.png" altimg-height="64px" altimg-valign="-27px" altimg-width="88px" alttext="\displaystyle=\sum_{s=0}^{\infty}\frac{g_{s}}{z^{s}}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi>s</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover></mstyle><mstyle displaystyle="true"><mfrac><msub><mi href="./2.7#SS1.p3">g</mi><mi>s</mi></msub><msup><mi>z</mi><mi>s</mi></msup></mfrac></mstyle></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E7.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./2.7#SS1.p1" title="§2.7(i) Regular Singularities: Fuchs–Frobenius Theory ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m104.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(z)" display="inline"><mrow><mi href="./2.7#SS1.p1">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: analytic function</a>,
<a href="./2.7#SS1.p1" title="§2.7(i) Regular Singularities: Fuchs–Frobenius Theory ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m112.png" altimg-height="23px" altimg-valign="-7px" altimg-width="40px" alttext="g(z)" display="inline"><mrow><mi href="./2.7#SS1.p1">g</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: analytic function</a>,
<a href="./2.7#SS1.p3" title="§2.7(i) Regular Singularities: Fuchs–Frobenius Theory ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m110.png" altimg-height="21px" altimg-valign="-6px" altimg-width="23px" alttext="f_{s}" display="inline"><msub><mi href="./2.7#SS1.p3">f</mi><mi>s</mi></msub></math>: coefficients</a> and
<a href="./2.7#SS1.p3" title="§2.7(i) Regular Singularities: Fuchs–Frobenius Theory ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m117.png" altimg-height="16px" altimg-valign="-6px" altimg-width="22px" alttext="g_{s}" display="inline"><msub><mi href="./2.7#SS1.p3">g</mi><mi>s</mi></msub></math>: coefficients</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">these series converging in an annulus <math class="ltx_Math" altimg="m149.png" altimg-height="23px" altimg-valign="-7px" altimg-width="63px" alttext="|z|>a" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mi>z</mi><mo stretchy="false">|</mo></mrow><mo>></mo><mi>a</mi></mrow></math>, with at least one of
<math class="ltx_Math" altimg="m108.png" altimg-height="21px" altimg-valign="-6px" altimg-width="23px" alttext="f_{0}" display="inline"><msub><mi href="./2.7#SS1.p3">f</mi><mn>0</mn></msub></math>, <math class="ltx_Math" altimg="m115.png" altimg-height="16px" altimg-valign="-6px" altimg-width="23px" alttext="g_{0}" display="inline"><msub><mi href="./2.7#SS1.p3">g</mi><mn>0</mn></msub></math>, <math class="ltx_Math" altimg="m116.png" altimg-height="16px" altimg-valign="-6px" altimg-width="23px" alttext="g_{1}" display="inline"><msub><mi href="./2.7#SS1.p3">g</mi><mn>1</mn></msub></math> nonzero.</p>
</div>
<div id="SS2.p2" class="ltx_para">
<p class="ltx_p">Formal solutions are
</p>
<table id="E8" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">2.7.8</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E8.png" altimg-height="64px" altimg-valign="-27px" altimg-width="150px" alttext="e^{\lambda_{j}z}z^{\mu_{j}}\sum_{s=0}^{\infty}\frac{a_{s,j}}{z^{s}}," display="block"><mrow><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><msub><mi href="./2.7#SS2.p2">λ</mi><mi>j</mi></msub><mo></mo><mi>z</mi></mrow></msup><mo></mo><msup><mi>z</mi><msub><mi href="./2.7#SS2.p2">μ</mi><mi>j</mi></msub></msup><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi>s</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mfrac><msub><mi href="./2.7#SS2.p2">a</mi><mrow><mi>s</mi><mo>,</mo><mi>j</mi></mrow></msub><msup><mi>z</mi><mi>s</mi></msup></mfrac></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m118.png" altimg-height="20px" altimg-valign="-6px" altimg-width="69px" alttext="j=1,2" display="inline"><mrow><mi>j</mi><mo>=</mo><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E8.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m86.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./2.7#SS2.p2" title="§2.7(ii) Irregular Singularities of Rank 1 ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m98.png" altimg-height="18px" altimg-valign="-8px" altimg-width="35px" alttext="a_{s,j}" display="inline"><msub><mi href="./2.7#SS2.p2">a</mi><mrow><mi>s</mi><mo>,</mo><mi>j</mi></mrow></msub></math>: coefficients</a>,
<a href="./2.7#SS2.p2" title="§2.7(ii) Irregular Singularities of Rank 1 ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="23px" altimg-valign="-8px" altimg-width="24px" alttext="\lambda_{j}" display="inline"><msub><mi href="./2.7#SS2.p2">λ</mi><mi>j</mi></msub></math>: roots</a> and
<a href="./2.7#SS2.p2" title="§2.7(ii) Irregular Singularities of Rank 1 ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="18px" altimg-valign="-8px" altimg-width="25px" alttext="\mu_{j}" display="inline"><msub><mi href="./2.7#SS2.p2">μ</mi><mi>j</mi></msub></math>: quantities</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m76.png" altimg-height="21px" altimg-valign="-5px" altimg-width="25px" alttext="\lambda_{1}" display="inline"><msub><mi href="./2.7#SS2.p2">λ</mi><mn>1</mn></msub></math>, <math class="ltx_Math" altimg="m77.png" altimg-height="21px" altimg-valign="-5px" altimg-width="25px" alttext="\lambda_{2}" display="inline"><msub><mi href="./2.7#SS2.p2">λ</mi><mn>2</mn></msub></math> are the roots of the <em class="ltx_emph ltx_font_italic">characteristic
equation</em>
</p>
<table id="E9" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">2.7.9</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E9.png" altimg-height="27px" altimg-valign="-6px" altimg-width="166px" alttext="\lambda^{2}+f_{0}\lambda+g_{0}=0," display="block"><mrow><mrow><mrow><msup><mi>λ</mi><mn>2</mn></msup><mo>+</mo><mrow><msub><mi href="./2.7#SS1.p3">f</mi><mn>0</mn></msub><mo></mo><mi>λ</mi></mrow><mo>+</mo><msub><mi href="./2.7#SS1.p3">g</mi><mn>0</mn></msub></mrow><mo>=</mo><mn>0</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E9.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./2.7#SS1.p3" title="§2.7(i) Regular Singularities: Fuchs–Frobenius Theory ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m110.png" altimg-height="21px" altimg-valign="-6px" altimg-width="23px" alttext="f_{s}" display="inline"><msub><mi href="./2.7#SS1.p3">f</mi><mi>s</mi></msub></math>: coefficients</a> and
<a href="./2.7#SS1.p3" title="§2.7(i) Regular Singularities: Fuchs–Frobenius Theory ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m117.png" altimg-height="16px" altimg-valign="-6px" altimg-width="22px" alttext="g_{s}" display="inline"><msub><mi href="./2.7#SS1.p3">g</mi><mi>s</mi></msub></math>: coefficients</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E10" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">2.7.10</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E10.png" altimg-height="26px" altimg-valign="-8px" altimg-width="270px" alttext="\mu_{j}=-(f_{1}\lambda_{j}+g_{1})/(f_{0}+2\lambda_{j})," display="block"><mrow><mrow><msub><mi href="./2.7#SS2.p2">μ</mi><mi>j</mi></msub><mo>=</mo><mrow><mo>-</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><msub><mi href="./2.7#SS1.p3">f</mi><mn>1</mn></msub><mo></mo><msub><mi href="./2.7#SS2.p2">λ</mi><mi>j</mi></msub></mrow><mo>+</mo><msub><mi href="./2.7#SS1.p3">g</mi><mn>1</mn></msub></mrow><mo stretchy="false">)</mo></mrow><mo>/</mo><mrow><mo stretchy="false">(</mo><mrow><msub><mi href="./2.7#SS1.p3">f</mi><mn>0</mn></msub><mo>+</mo><mrow><mn>2</mn><mo></mo><msub><mi href="./2.7#SS2.p2">λ</mi><mi>j</mi></msub></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E10.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./2.7#SS2.p2" title="§2.7(ii) Irregular Singularities of Rank 1 ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="23px" altimg-valign="-8px" altimg-width="24px" alttext="\lambda_{j}" display="inline"><msub><mi href="./2.7#SS2.p2">λ</mi><mi>j</mi></msub></math>: roots</a>,
<a href="./2.7#SS2.p2" title="§2.7(ii) Irregular Singularities of Rank 1 ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="18px" altimg-valign="-8px" altimg-width="25px" alttext="\mu_{j}" display="inline"><msub><mi href="./2.7#SS2.p2">μ</mi><mi>j</mi></msub></math>: quantities</a>,
<a href="./2.7#SS1.p3" title="§2.7(i) Regular Singularities: Fuchs–Frobenius Theory ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m110.png" altimg-height="21px" altimg-valign="-6px" altimg-width="23px" alttext="f_{s}" display="inline"><msub><mi href="./2.7#SS1.p3">f</mi><mi>s</mi></msub></math>: coefficients</a> and
<a href="./2.7#SS1.p3" title="§2.7(i) Regular Singularities: Fuchs–Frobenius Theory ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m117.png" altimg-height="16px" altimg-valign="-6px" altimg-width="22px" alttext="g_{s}" display="inline"><msub><mi href="./2.7#SS1.p3">g</mi><mi>s</mi></msub></math>: coefficients</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p"><math class="ltx_Math" altimg="m92.png" altimg-height="22px" altimg-valign="-8px" altimg-width="72px" alttext="a_{0,j}=1" display="inline"><mrow><msub><mi href="./2.7#SS2.p2">a</mi><mrow><mn>0</mn><mo>,</mo><mi>j</mi></mrow></msub><mo>=</mo><mn>1</mn></mrow></math>, and</p>
<table id="E11" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">2.7.11</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_Math ltx_eqn_cell ltx_align_center"><math class="ltx_Math" alttext="(f_{0}+2\lambda_{j})sa_{s,j}=(s-\mu_{j})(s-1-\mu_{j})a_{s-1,j}+\sum_{r=1}^{s}%
\left(\lambda_{j}f_{r+1}+g_{r+1}-(s-r-\mu_{j})f_{r}\right)a_{s-r,j}," display="block"><mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><msub><mi href="./2.7#SS1.p3">f</mi><mn>0</mn></msub><mo>+</mo><mrow><mn>2</mn><mo></mo><msub><mi href="./2.7#SS2.p2">λ</mi><mi>j</mi></msub></mrow></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi>s</mi><mo></mo><msub><mi href="./2.7#SS2.p2">a</mi><mrow><mi>s</mi><mo>,</mo><mi>j</mi></mrow></msub></mrow><mo>=</mo><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mrow><mrow><mo stretchy="false">(</mo><mrow><mi>s</mi><mo>-</mo><msub><mi href="./2.7#SS2.p2">μ</mi><mi>j</mi></msub></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>s</mi><mo>-</mo><mn>1</mn><mo>-</mo><msub><mi href="./2.7#SS2.p2">μ</mi><mi>j</mi></msub></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msub><mi href="./2.7#SS2.p2">a</mi><mrow><mrow><mi>s</mi><mo>-</mo><mn>1</mn></mrow><mo>,</mo><mi>j</mi></mrow></msub></mrow></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>+</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi>r</mi><mo>=</mo><mn>1</mn></mrow><mi>s</mi></munderover><mrow><mrow><mo>(</mo><mrow><mrow><mrow><msub><mi href="./2.7#SS2.p2">λ</mi><mi>j</mi></msub><mo></mo><msub><mi href="./2.7#SS1.p3">f</mi><mrow><mi>r</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><mo>+</mo><msub><mi href="./2.7#SS1.p3">g</mi><mrow><mi>r</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><mo>-</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mi>s</mi><mo>-</mo><mi>r</mi><mo>-</mo><msub><mi href="./2.7#SS2.p2">μ</mi><mi>j</mi></msub></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msub><mi href="./2.7#SS1.p3">f</mi><mi>r</mi></msub></mrow></mrow><mo>)</mo></mrow><mo></mo><msub><mi href="./2.7#SS2.p2">a</mi><mrow><mrow><mi>s</mi><mo>-</mo><mi>r</mi></mrow><mo>,</mo><mi>j</mi></mrow></msub></mrow></mrow><mo>,</mo></mtd></mtr></mtable></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E11.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./2.7#SS2.p2" title="§2.7(ii) Irregular Singularities of Rank 1 ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m98.png" altimg-height="18px" altimg-valign="-8px" altimg-width="35px" alttext="a_{s,j}" display="inline"><msub><mi href="./2.7#SS2.p2">a</mi><mrow><mi>s</mi><mo>,</mo><mi>j</mi></mrow></msub></math>: coefficients</a>,
<a href="./2.7#SS2.p2" title="§2.7(ii) Irregular Singularities of Rank 1 ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="23px" altimg-valign="-8px" altimg-width="24px" alttext="\lambda_{j}" display="inline"><msub><mi href="./2.7#SS2.p2">λ</mi><mi>j</mi></msub></math>: roots</a>,
<a href="./2.7#SS2.p2" title="§2.7(ii) Irregular Singularities of Rank 1 ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="18px" altimg-valign="-8px" altimg-width="25px" alttext="\mu_{j}" display="inline"><msub><mi href="./2.7#SS2.p2">μ</mi><mi>j</mi></msub></math>: quantities</a>,
<a href="./2.7#SS1.p3" title="§2.7(i) Regular Singularities: Fuchs–Frobenius Theory ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m110.png" altimg-height="21px" altimg-valign="-6px" altimg-width="23px" alttext="f_{s}" display="inline"><msub><mi href="./2.7#SS1.p3">f</mi><mi>s</mi></msub></math>: coefficients</a> and
<a href="./2.7#SS1.p3" title="§2.7(i) Regular Singularities: Fuchs–Frobenius Theory ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m117.png" altimg-height="16px" altimg-valign="-6px" altimg-width="22px" alttext="g_{s}" display="inline"><msub><mi href="./2.7#SS1.p3">g</mi><mi>s</mi></msub></math>: coefficients</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">when <math class="ltx_Math" altimg="m123.png" altimg-height="20px" altimg-valign="-6px" altimg-width="101px" alttext="s=1,2,\dots" display="inline"><mrow><mi>s</mi><mo>=</mo><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>. The construction fails iff <math class="ltx_Math" altimg="m75.png" altimg-height="21px" altimg-valign="-5px" altimg-width="72px" alttext="\lambda_{1}=\lambda_{2}" display="inline"><mrow><msub><mi href="./2.7#SS2.p2">λ</mi><mn>1</mn></msub><mo>=</mo><msub><mi href="./2.7#SS2.p2">λ</mi><mn>2</mn></msub></mrow></math>, that
is, when <math class="ltx_Math" altimg="m109.png" altimg-height="25px" altimg-valign="-7px" altimg-width="80px" alttext="f_{0}^{2}=4g_{0}" display="inline"><mrow><msubsup><mi href="./2.7#SS1.p3">f</mi><mn>0</mn><mn>2</mn></msubsup><mo>=</mo><mrow><mn>4</mn><mo></mo><msub><mi href="./2.7#SS1.p3">g</mi><mn>0</mn></msub></mrow></mrow></math>: this case is treated below.</p>
</div>
<div id="SS2.p3" class="ltx_para">
<p class="ltx_p">For large <math class="ltx_Math" altimg="m124.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="s" display="inline"><mi>s</mi></math>,</p>
<table id="EGx1" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E12">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">2.7.12</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m19.png" altimg-height="20px" altimg-valign="-8px" altimg-width="38px" alttext="\displaystyle a_{s,1}" display="inline"><msub><mi href="./2.7#SS2.p2">a</mi><mrow><mi>s</mi><mo>,</mo><mn>1</mn></mrow></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m17.png" altimg-height="67px" altimg-valign="-30px" altimg-width="451px" alttext="\displaystyle\sim\frac{\Lambda_{1}}{(\lambda_{1}-\lambda_{2})^{s}}\*\sum_{j=0}%
^{\infty}{a_{j,2}(\lambda_{1}-\lambda_{2})^{j}\Gamma\left(s+\mu_{2}-\mu_{1}-j%
\right)}," display="inline"><mrow><mrow><mi></mi><mo href="./2.1#SS3.p1">∼</mo><mrow><mstyle displaystyle="true"><mfrac><msub><mi href="./2.7#SS2.p3" mathvariant="normal">Λ</mi><mn>1</mn></msub><msup><mrow><mo stretchy="false">(</mo><mrow><msub><mi href="./2.7#SS2.p2">λ</mi><mn>1</mn></msub><mo>-</mo><msub><mi href="./2.7#SS2.p2">λ</mi><mn>2</mn></msub></mrow><mo stretchy="false">)</mo></mrow><mi>s</mi></msup></mfrac></mstyle><mo></mo><mrow><mstyle displaystyle="true"><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover></mstyle><mrow><msub><mi href="./2.7#SS2.p2">a</mi><mrow><mi>j</mi><mo>,</mo><mn>2</mn></mrow></msub><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><msub><mi href="./2.7#SS2.p2">λ</mi><mn>1</mn></msub><mo>-</mo><msub><mi href="./2.7#SS2.p2">λ</mi><mn>2</mn></msub></mrow><mo stretchy="false">)</mo></mrow><mi>j</mi></msup><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mi>s</mi><mo>+</mo><msub><mi href="./2.7#SS2.p2">μ</mi><mn>2</mn></msub></mrow><mo>-</mo><msub><mi href="./2.7#SS2.p2">μ</mi><mn>1</mn></msub><mo>-</mo><mi>j</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E12.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./2.1#SS3.p1" title="§2.1(iii) Asymptotic Expansions ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m90.png" altimg-height="12px" altimg-valign="-2px" altimg-width="20px" alttext="\sim" display="inline"><mo href="./2.1#SS3.p1">∼</mo></math>: Poincaré asymptotic expansion</a>,
<a href="./2.7#SS2.p2" title="§2.7(ii) Irregular Singularities of Rank 1 ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m98.png" altimg-height="18px" altimg-valign="-8px" altimg-width="35px" alttext="a_{s,j}" display="inline"><msub><mi href="./2.7#SS2.p2">a</mi><mrow><mi>s</mi><mo>,</mo><mi>j</mi></mrow></msub></math>: coefficients</a>,
<a href="./2.7#SS2.p2" title="§2.7(ii) Irregular Singularities of Rank 1 ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="23px" altimg-valign="-8px" altimg-width="24px" alttext="\lambda_{j}" display="inline"><msub><mi href="./2.7#SS2.p2">λ</mi><mi>j</mi></msub></math>: roots</a>,
<a href="./2.7#SS2.p2" title="§2.7(ii) Irregular Singularities of Rank 1 ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="18px" altimg-valign="-8px" altimg-width="25px" alttext="\mu_{j}" display="inline"><msub><mi href="./2.7#SS2.p2">μ</mi><mi>j</mi></msub></math>: quantities</a> and
<a href="./2.7#SS2.p3" title="§2.7(ii) Irregular Singularities of Rank 1 ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m55.png" altimg-height="23px" altimg-valign="-8px" altimg-width="27px" alttext="\Lambda_{j}" display="inline"><msub><mi href="./2.7#SS2.p3" mathvariant="normal">Λ</mi><mi>j</mi></msub></math>: constants</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E13">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">2.7.13</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m20.png" altimg-height="20px" altimg-valign="-8px" altimg-width="38px" alttext="\displaystyle a_{s,2}" display="inline"><msub><mi href="./2.7#SS2.p2">a</mi><mrow><mi>s</mi><mo>,</mo><mn>2</mn></mrow></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m18.png" altimg-height="67px" altimg-valign="-30px" altimg-width="451px" alttext="\displaystyle\sim\frac{\Lambda_{2}}{(\lambda_{2}-\lambda_{1})^{s}}\*\sum_{j=0}%
^{\infty}{a_{j,1}(\lambda_{2}-\lambda_{1})^{j}\Gamma\left(s+\mu_{1}-\mu_{2}-j%
\right)}," display="inline"><mrow><mrow><mi></mi><mo href="./2.1#SS3.p1">∼</mo><mrow><mstyle displaystyle="true"><mfrac><msub><mi href="./2.7#SS2.p3" mathvariant="normal">Λ</mi><mn>2</mn></msub><msup><mrow><mo stretchy="false">(</mo><mrow><msub><mi href="./2.7#SS2.p2">λ</mi><mn>2</mn></msub><mo>-</mo><msub><mi href="./2.7#SS2.p2">λ</mi><mn>1</mn></msub></mrow><mo stretchy="false">)</mo></mrow><mi>s</mi></msup></mfrac></mstyle><mo></mo><mrow><mstyle displaystyle="true"><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover></mstyle><mrow><msub><mi href="./2.7#SS2.p2">a</mi><mrow><mi>j</mi><mo>,</mo><mn>1</mn></mrow></msub><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><msub><mi href="./2.7#SS2.p2">λ</mi><mn>2</mn></msub><mo>-</mo><msub><mi href="./2.7#SS2.p2">λ</mi><mn>1</mn></msub></mrow><mo stretchy="false">)</mo></mrow><mi>j</mi></msup><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mi>s</mi><mo>+</mo><msub><mi href="./2.7#SS2.p2">μ</mi><mn>1</mn></msub></mrow><mo>-</mo><msub><mi href="./2.7#SS2.p2">μ</mi><mn>2</mn></msub><mo>-</mo><mi>j</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E13.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./2.1#SS3.p1" title="§2.1(iii) Asymptotic Expansions ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m90.png" altimg-height="12px" altimg-valign="-2px" altimg-width="20px" alttext="\sim" display="inline"><mo href="./2.1#SS3.p1">∼</mo></math>: Poincaré asymptotic expansion</a>,
<a href="./2.7#SS2.p2" title="§2.7(ii) Irregular Singularities of Rank 1 ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m98.png" altimg-height="18px" altimg-valign="-8px" altimg-width="35px" alttext="a_{s,j}" display="inline"><msub><mi href="./2.7#SS2.p2">a</mi><mrow><mi>s</mi><mo>,</mo><mi>j</mi></mrow></msub></math>: coefficients</a>,
<a href="./2.7#SS2.p2" title="§2.7(ii) Irregular Singularities of Rank 1 ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="23px" altimg-valign="-8px" altimg-width="24px" alttext="\lambda_{j}" display="inline"><msub><mi href="./2.7#SS2.p2">λ</mi><mi>j</mi></msub></math>: roots</a>,
<a href="./2.7#SS2.p2" title="§2.7(ii) Irregular Singularities of Rank 1 ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="18px" altimg-valign="-8px" altimg-width="25px" alttext="\mu_{j}" display="inline"><msub><mi href="./2.7#SS2.p2">μ</mi><mi>j</mi></msub></math>: quantities</a> and
<a href="./2.7#SS2.p3" title="§2.7(ii) Irregular Singularities of Rank 1 ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m55.png" altimg-height="23px" altimg-valign="-8px" altimg-width="27px" alttext="\Lambda_{j}" display="inline"><msub><mi href="./2.7#SS2.p3" mathvariant="normal">Λ</mi><mi>j</mi></msub></math>: constants</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m53.png" altimg-height="21px" altimg-valign="-5px" altimg-width="27px" alttext="\Lambda_{1}" display="inline"><msub><mi href="./2.7#SS2.p3" mathvariant="normal">Λ</mi><mn>1</mn></msub></math> and <math class="ltx_Math" altimg="m54.png" altimg-height="21px" altimg-valign="-5px" altimg-width="27px" alttext="\Lambda_{2}" display="inline"><msub><mi href="./2.7#SS2.p3" mathvariant="normal">Λ</mi><mn>2</mn></msub></math> are constants, and the <math class="ltx_Math" altimg="m47.png" altimg-height="18px" altimg-valign="-2px" altimg-width="17px" alttext="J" display="inline"><mi>J</mi></math>th remainder terms
in the sums are <math class="ltx_Math" altimg="m49.png" altimg-height="23px" altimg-valign="-7px" altimg-width="207px" alttext="O\left(\Gamma\left(s+\mu_{2}-\mu_{1}-J\right)\right)" display="inline"><mrow><mi href="./2.1#E3">O</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mi>s</mi><mo>+</mo><msub><mi href="./2.7#SS2.p2">μ</mi><mn>2</mn></msub></mrow><mo>-</mo><msub><mi href="./2.7#SS2.p2">μ</mi><mn>1</mn></msub><mo>-</mo><mi>J</mi></mrow><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow></math> and
<math class="ltx_Math" altimg="m48.png" altimg-height="23px" altimg-valign="-7px" altimg-width="207px" alttext="O\left(\Gamma\left(s+\mu_{1}-\mu_{2}-J\right)\right)" display="inline"><mrow><mi href="./2.1#E3">O</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mi>s</mi><mo>+</mo><msub><mi href="./2.7#SS2.p2">μ</mi><mn>1</mn></msub></mrow><mo>-</mo><msub><mi href="./2.7#SS2.p2">μ</mi><mn>2</mn></msub><mo>-</mo><mi>J</mi></mrow><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow></math>, respectively (<cite class="ltx_cite ltx_citemacro_citet">Olver () terminate (in which case the
corresponding <math class="ltx_Math" altimg="m55.png" altimg-height="23px" altimg-valign="-8px" altimg-width="27px" alttext="\Lambda_{j}" display="inline"><msub><mi href="./2.7#SS2.p3" mathvariant="normal">Λ</mi><mi>j</mi></msub></math> is zero) they diverge. However, there are unique and
linearly independent solutions <math class="ltx_Math" altimg="m139.png" altimg-height="24px" altimg-valign="-8px" altimg-width="53px" alttext="w_{j}(z)" display="inline"><mrow><msub><mi href="./2.7#SS1.p3">w</mi><mi>j</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math>, <math class="ltx_Math" altimg="m118.png" altimg-height="20px" altimg-valign="-6px" altimg-width="69px" alttext="j=1,2" display="inline"><mrow><mi>j</mi><mo>=</mo><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></mrow></math>, such that</p>
<table id="E14" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">2.7.14</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E14.png" altimg-height="64px" altimg-valign="-27px" altimg-width="312px" alttext="w_{j}(z)\sim e^{\lambda_{j}z}((\lambda_{2}-\lambda_{1})z)^{\mu_{j}}\sum_{s=0}^%
{\infty}\frac{a_{s,j}}{z^{s}}" display="block"><mrow><mrow><msub><mi href="./2.7#SS1.p3">w</mi><mi>j</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow><mo href="./2.1#SS3.p1">∼</mo><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><msub><mi href="./2.7#SS2.p2">λ</mi><mi>j</mi></msub><mo></mo><mi>z</mi></mrow></msup><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><msub><mi href="./2.7#SS2.p2">λ</mi><mn>2</mn></msub><mo>-</mo><msub><mi href="./2.7#SS2.p2">λ</mi><mn>1</mn></msub></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi>z</mi></mrow><mo stretchy="false">)</mo></mrow><msub><mi href="./2.7#SS2.p2">μ</mi><mi>j</mi></msub></msup><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi>s</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mfrac><msub><mi href="./2.7#SS2.p2">a</mi><mrow><mi>s</mi><mo>,</mo><mi>j</mi></mrow></msub><msup><mi>z</mi><mi>s</mi></msup></mfrac></mrow></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E14.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./2.1#SS3.p1" title="§2.1(iii) Asymptotic Expansions ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m90.png" altimg-height="12px" altimg-valign="-2px" altimg-width="20px" alttext="\sim" display="inline"><mo href="./2.1#SS3.p1">∼</mo></math>: Poincaré asymptotic expansion</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m86.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./2.7#SS2.p2" title="§2.7(ii) Irregular Singularities of Rank 1 ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m98.png" altimg-height="18px" altimg-valign="-8px" altimg-width="35px" alttext="a_{s,j}" display="inline"><msub><mi href="./2.7#SS2.p2">a</mi><mrow><mi>s</mi><mo>,</mo><mi>j</mi></mrow></msub></math>: coefficients</a>,
<a href="./2.7#SS2.p2" title="§2.7(ii) Irregular Singularities of Rank 1 ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="23px" altimg-valign="-8px" altimg-width="24px" alttext="\lambda_{j}" display="inline"><msub><mi href="./2.7#SS2.p2">λ</mi><mi>j</mi></msub></math>: roots</a>,
<a href="./2.7#SS2.p2" title="§2.7(ii) Irregular Singularities of Rank 1 ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="18px" altimg-valign="-8px" altimg-width="25px" alttext="\mu_{j}" display="inline"><msub><mi href="./2.7#SS2.p2">μ</mi><mi>j</mi></msub></math>: quantities</a> and
<a href="./2.7#SS1.p3" title="§2.7(i) Regular Singularities: Fuchs–Frobenius Theory ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m139.png" altimg-height="24px" altimg-valign="-8px" altimg-width="53px" alttext="w_{j}(z)" display="inline"><mrow><msub><mi href="./2.7#SS1.p3">w</mi><mi>j</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: solutions</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">as <math class="ltx_Math" altimg="m146.png" altimg-height="13px" altimg-valign="-2px" altimg-width="65px" alttext="z\to\infty" display="inline"><mrow><mi>z</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></math> in the sectors</p>
<table id="E15" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">2.7.15</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E15.png" altimg-height="29px" altimg-valign="-9px" altimg-width="329px" alttext="-\tfrac{3}{2}\pi+\delta\leq\operatorname{ph}\left((\lambda_{2}-\lambda_{1})z%
\right)\leq\tfrac{3}{2}\pi-\delta," display="block"><mrow><mrow><mrow><mrow><mo>-</mo><mrow><mstyle displaystyle="false"><mfrac><mn>3</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow><mo>+</mo><mi href="./2.7#SS2.p3">δ</mi></mrow><mo>≤</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><msub><mi href="./2.7#SS2.p2">λ</mi><mn>2</mn></msub><mo>-</mo><msub><mi href="./2.7#SS2.p2">λ</mi><mn>1</mn></msub></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi>z</mi></mrow><mo>)</mo></mrow></mrow><mo>≤</mo><mrow><mrow><mstyle displaystyle="false"><mfrac><mn>3</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo>-</mo><mi href="./2.7#SS2.p3">δ</mi></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m119.png" altimg-height="20px" altimg-valign="-6px" altimg-width="50px" alttext="j=1" display="inline"><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E15.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m89.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.9#E7" title="(1.9.7) ‣ Modulus and Phase ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m88.png" altimg-height="21px" altimg-valign="-6px" altimg-width="26px" alttext="\operatorname{ph}" display="inline"><mi href="./1.9#E7">ph</mi></math>: phase</a>,
<a href="./2.7#SS2.p2" title="§2.7(ii) Irregular Singularities of Rank 1 ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="23px" altimg-valign="-8px" altimg-width="24px" alttext="\lambda_{j}" display="inline"><msub><mi href="./2.7#SS2.p2">λ</mi><mi>j</mi></msub></math>: roots</a> and
<a href="./2.7#SS2.p3" title="§2.7(ii) Irregular Singularities of Rank 1 ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m65.png" altimg-height="18px" altimg-valign="-2px" altimg-width="14px" alttext="\delta" display="inline"><mi href="./2.7#SS2.p3">δ</mi></math>: arbitrary small positive constant</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E16" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">2.7.16</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E16.png" altimg-height="29px" altimg-valign="-9px" altimg-width="329px" alttext="-\tfrac{1}{2}\pi+\delta\leq\operatorname{ph}\left((\lambda_{2}-\lambda_{1})z%
\right)\leq\tfrac{5}{2}\pi-\delta," display="block"><mrow><mrow><mrow><mrow><mo>-</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow><mo>+</mo><mi href="./2.7#SS2.p3">δ</mi></mrow><mo>≤</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><msub><mi href="./2.7#SS2.p2">λ</mi><mn>2</mn></msub><mo>-</mo><msub><mi href="./2.7#SS2.p2">λ</mi><mn>1</mn></msub></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi>z</mi></mrow><mo>)</mo></mrow></mrow><mo>≤</mo><mrow><mrow><mstyle displaystyle="false"><mfrac><mn>5</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo>-</mo><mi href="./2.7#SS2.p3">δ</mi></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m120.png" altimg-height="20px" altimg-valign="-6px" altimg-width="50px" alttext="j=2" display="inline"><mrow><mi>j</mi><mo>=</mo><mn>2</mn></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E16.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m89.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.9#E7" title="(1.9.7) ‣ Modulus and Phase ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m88.png" altimg-height="21px" altimg-valign="-6px" altimg-width="26px" alttext="\operatorname{ph}" display="inline"><mi href="./1.9#E7">ph</mi></math>: phase</a>,
<a href="./2.7#SS2.p2" title="§2.7(ii) Irregular Singularities of Rank 1 ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="23px" altimg-valign="-8px" altimg-width="24px" alttext="\lambda_{j}" display="inline"><msub><mi href="./2.7#SS2.p2">λ</mi><mi>j</mi></msub></math>: roots</a> and
<a href="./2.7#SS2.p3" title="§2.7(ii) Irregular Singularities of Rank 1 ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m65.png" altimg-height="18px" altimg-valign="-2px" altimg-width="14px" alttext="\delta" display="inline"><mi href="./2.7#SS2.p3">δ</mi></math>: arbitrary small positive constant</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p"><math class="ltx_Math" altimg="m65.png" altimg-height="18px" altimg-valign="-2px" altimg-width="14px" alttext="\delta" display="inline"><mi href="./2.7#SS2.p3">δ</mi></math> being an arbitrary small positive constant.</p>
</div>
<div id="SS2.p4" class="ltx_para">
<p class="ltx_p">Although the expansions (), each solution <math class="ltx_Math" altimg="m139.png" altimg-height="24px" altimg-valign="-8px" altimg-width="53px" alttext="w_{j}(z)" display="inline"><mrow><msub><mi href="./2.7#SS1.p3">w</mi><mi>j</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math> can be
continued analytically into any other sector. Typical connection formulas are</p>
<table id="E17" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex5" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">2.7.17</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m26.png" altimg-height="25px" altimg-valign="-7px" altimg-width="55px" alttext="\displaystyle w_{1}(z)" display="inline"><mrow><msub><mi href="./2.7#SS1.p3">w</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m8.png" altimg-height="28px" altimg-valign="-7px" altimg-width="275px" alttext="\displaystyle=e^{2\pi i\mu_{1}}w_{1}(ze^{-2\pi i})+C_{1}w_{2}(z)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi><mo></mo><msub><mi href="./2.7#SS2.p2">μ</mi><mn>1</mn></msub></mrow></msup><mo></mo><mrow><msub><mi href="./2.7#SS1.p3">w</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>z</mi><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi></mrow></mrow></msup></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>+</mo><mrow><msub><mi href="./2.7#SS2.p4">C</mi><mn>1</mn></msub><mo></mo><mrow><msub><mi href="./2.7#SS1.p3">w</mi><mn>2</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex6" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m28.png" altimg-height="25px" altimg-valign="-7px" altimg-width="55px" alttext="\displaystyle w_{2}(z)" display="inline"><mrow><msub><mi href="./2.7#SS1.p3">w</mi><mn>2</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m6.png" altimg-height="28px" altimg-valign="-7px" altimg-width="275px" alttext="\displaystyle=e^{-2\pi i\mu_{2}}w_{2}(ze^{2\pi i})+C_{2}w_{1}(z)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi><mo></mo><msub><mi href="./2.7#SS2.p2">μ</mi><mn>2</mn></msub></mrow></mrow></msup><mo></mo><mrow><msub><mi href="./2.7#SS1.p3">w</mi><mn>2</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>z</mi><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi></mrow></msup></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>+</mo><mrow><msub><mi href="./2.7#SS2.p4">C</mi><mn>2</mn></msub><mo></mo><mrow><msub><mi href="./2.7#SS1.p3">w</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E17.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m89.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m86.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./2.7#SS2.p2" title="§2.7(ii) Irregular Singularities of Rank 1 ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="18px" altimg-valign="-8px" altimg-width="25px" alttext="\mu_{j}" display="inline"><msub><mi href="./2.7#SS2.p2">μ</mi><mi>j</mi></msub></math>: quantities</a>,
<a href="./2.7#SS2.p4" title="§2.7(ii) Irregular Singularities of Rank 1 ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m42.png" altimg-height="21px" altimg-valign="-5px" altimg-width="27px" alttext="C_{1}" display="inline"><msub><mi href="./2.7#SS2.p4">C</mi><mn>1</mn></msub></math>, <math class="ltx_Math" altimg="m43.png" altimg-height="21px" altimg-valign="-5px" altimg-width="27px" alttext="C_{2}" display="inline"><msub><mi href="./2.7#SS2.p4">C</mi><mn>2</mn></msub></math>: Stokes multipliers</a> and
<a href="./2.7#SS1.p3" title="§2.7(i) Regular Singularities: Fuchs–Frobenius Theory ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m139.png" altimg-height="24px" altimg-valign="-8px" altimg-width="53px" alttext="w_{j}(z)" display="inline"><mrow><msub><mi href="./2.7#SS1.p3">w</mi><mi>j</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: solutions</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">in which <math class="ltx_Math" altimg="m42.png" altimg-height="21px" altimg-valign="-5px" altimg-width="27px" alttext="C_{1}" display="inline"><msub><mi href="./2.7#SS2.p4">C</mi><mn>1</mn></msub></math>, <math class="ltx_Math" altimg="m43.png" altimg-height="21px" altimg-valign="-5px" altimg-width="27px" alttext="C_{2}" display="inline"><msub><mi href="./2.7#SS2.p4">C</mi><mn>2</mn></msub></math> are constants, the so-called <em class="ltx_emph ltx_font_italic">Stokes multipliers</em>.
In combination with () these formulas yield asymptotic
expansions for <math class="ltx_Math" altimg="m129.png" altimg-height="23px" altimg-valign="-7px" altimg-width="53px" alttext="w_{1}(z)" display="inline"><mrow><msub><mi href="./2.7#SS1.p3">w</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math> in <math class="ltx_Math" altimg="m70.png" altimg-height="27px" altimg-valign="-9px" altimg-width="307px" alttext="\frac{1}{2}\pi+\delta\leq\operatorname{ph}\left((\lambda_{2}-\lambda_{1})z%
\right)\leq\frac{5}{2}\pi-\delta" display="inline"><mrow><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo>+</mo><mi href="./2.7#SS2.p3">δ</mi></mrow><mo>≤</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><msub><mi href="./2.7#SS2.p2">λ</mi><mn>2</mn></msub><mo>-</mo><msub><mi href="./2.7#SS2.p2">λ</mi><mn>1</mn></msub></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi>z</mi></mrow><mo>)</mo></mrow></mrow><mo>≤</mo><mrow><mrow><mfrac><mn>5</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo>-</mo><mi href="./2.7#SS2.p3">δ</mi></mrow></mrow></math>, and <math class="ltx_Math" altimg="m132.png" altimg-height="23px" altimg-valign="-7px" altimg-width="53px" alttext="w_{2}(z)" display="inline"><mrow><msub><mi href="./2.7#SS1.p3">w</mi><mn>2</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math> in
<math class="ltx_Math" altimg="m35.png" altimg-height="27px" altimg-valign="-9px" altimg-width="323px" alttext="-\frac{3}{2}\pi+\delta\leq\operatorname{ph}\left((\lambda_{2}-\lambda_{1})z%
\right)\leq\frac{1}{2}\pi-\delta" display="inline"><mrow><mrow><mrow><mo>-</mo><mrow><mfrac><mn>3</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow><mo>+</mo><mi href="./2.7#SS2.p3">δ</mi></mrow><mo>≤</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><msub><mi href="./2.7#SS2.p2">λ</mi><mn>2</mn></msub><mo>-</mo><msub><mi href="./2.7#SS2.p2">λ</mi><mn>1</mn></msub></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi>z</mi></mrow><mo>)</mo></mrow></mrow><mo>≤</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo>-</mo><mi href="./2.7#SS2.p3">δ</mi></mrow></mrow></math>. Furthermore,</p>
<table id="E18" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex7" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">2.7.18</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m13.png" altimg-height="22px" altimg-valign="-5px" altimg-width="29px" alttext="\displaystyle\Lambda_{1}" display="inline"><msub><mi href="./2.7#SS2.p3" mathvariant="normal">Λ</mi><mn>1</mn></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m1.png" altimg-height="29px" altimg-valign="-7px" altimg-width="211px" alttext="\displaystyle=-ie^{(\mu_{2}-\mu_{1})\pi i}C_{1}/(2\pi)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo>-</mo><mrow><mrow><mi mathvariant="normal">i</mi><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mrow><mo stretchy="false">(</mo><mrow><msub><mi href="./2.7#SS2.p2">μ</mi><mn>2</mn></msub><mo>-</mo><msub><mi href="./2.7#SS2.p2">μ</mi><mn>1</mn></msub></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi></mrow></msup><mo></mo><msub><mi href="./2.7#SS2.p4">C</mi><mn>1</mn></msub></mrow><mo>/</mo><mrow><mo stretchy="false">(</mo><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex8" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m14.png" altimg-height="22px" altimg-valign="-5px" altimg-width="29px" alttext="\displaystyle\Lambda_{2}" display="inline"><msub><mi href="./2.7#SS2.p3" mathvariant="normal">Λ</mi><mn>2</mn></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m11.png" altimg-height="25px" altimg-valign="-7px" altimg-width="110px" alttext="\displaystyle=iC_{2}/(2\pi)." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mi mathvariant="normal">i</mi><mo></mo><msub><mi href="./2.7#SS2.p4">C</mi><mn>2</mn></msub></mrow><mo>/</mo><mrow><mo stretchy="false">(</mo><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E18.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m89.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m86.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./2.7#SS2.p2" title="§2.7(ii) Irregular Singularities of Rank 1 ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="18px" altimg-valign="-8px" altimg-width="25px" alttext="\mu_{j}" display="inline"><msub><mi href="./2.7#SS2.p2">μ</mi><mi>j</mi></msub></math>: quantities</a>,
<a href="./2.7#SS2.p3" title="§2.7(ii) Irregular Singularities of Rank 1 ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m55.png" altimg-height="23px" altimg-valign="-8px" altimg-width="27px" alttext="\Lambda_{j}" display="inline"><msub><mi href="./2.7#SS2.p3" mathvariant="normal">Λ</mi><mi>j</mi></msub></math>: constants</a> and
<a href="./2.7#SS2.p4" title="§2.7(ii) Irregular Singularities of Rank 1 ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m42.png" altimg-height="21px" altimg-valign="-5px" altimg-width="27px" alttext="C_{1}" display="inline"><msub><mi href="./2.7#SS2.p4">C</mi><mn>1</mn></msub></math>, <math class="ltx_Math" altimg="m43.png" altimg-height="21px" altimg-valign="-5px" altimg-width="27px" alttext="C_{2}" display="inline"><msub><mi href="./2.7#SS2.p4">C</mi><mn>2</mn></msub></math>: Stokes multipliers</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS2.p5" class="ltx_para">
<p class="ltx_p">Note that the coefficients in the expansions () for the “late” coefficients, that is, <math class="ltx_Math" altimg="m96.png" altimg-height="18px" altimg-valign="-8px" altimg-width="36px" alttext="a_{s,1}" display="inline"><msub><mi href="./2.7#SS2.p2">a</mi><mrow><mi>s</mi><mo>,</mo><mn>1</mn></mrow></msub></math>, <math class="ltx_Math" altimg="m97.png" altimg-height="18px" altimg-valign="-8px" altimg-width="36px" alttext="a_{s,2}" display="inline"><msub><mi href="./2.7#SS2.p2">a</mi><mrow><mi>s</mi><mo>,</mo><mn>2</mn></mrow></msub></math>
with <math class="ltx_Math" altimg="m124.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="s" display="inline"><mi>s</mi></math> large, are the “early” coefficients <math class="ltx_Math" altimg="m94.png" altimg-height="18px" altimg-valign="-8px" altimg-width="35px" alttext="a_{j,2}" display="inline"><msub><mi href="./2.7#SS2.p2">a</mi><mrow><mi>j</mi><mo>,</mo><mn>2</mn></mrow></msub></math>, <math class="ltx_Math" altimg="m93.png" altimg-height="18px" altimg-valign="-8px" altimg-width="35px" alttext="a_{j,1}" display="inline"><msub><mi href="./2.7#SS2.p2">a</mi><mrow><mi>j</mi><mo>,</mo><mn>1</mn></mrow></msub></math> with <math class="ltx_Math" altimg="m121.png" altimg-height="20px" altimg-valign="-6px" altimg-width="14px" alttext="j" display="inline"><mi>j</mi></math>
small. This phenomenon is an example of <em class="ltx_emph ltx_font_italic">resurgence</em>, a classification due
to <cite class="ltx_cite ltx_citemacro_citet">Écalle ( for
other examples.</p>
</div>
<div id="SS2.p6" class="ltx_para">
<p class="ltx_p">The exceptional case <math class="ltx_Math" altimg="m109.png" altimg-height="25px" altimg-valign="-7px" altimg-width="80px" alttext="f_{0}^{2}=4g_{0}" display="inline"><mrow><msubsup><mi href="./2.7#SS1.p3">f</mi><mn>0</mn><mn>2</mn></msubsup><mo>=</mo><mrow><mn>4</mn><mo></mo><msub><mi href="./2.7#SS1.p3">g</mi><mn>0</mn></msub></mrow></mrow></math> is handled by <em class="ltx_emph ltx_font_italic">Fabry’s transformation</em>:</p>
<table id="E19" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex9" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">2.7.19</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m24.png" altimg-height="14px" altimg-valign="-2px" altimg-width="21px" alttext="\displaystyle w" display="inline"><mi href="./2.7#SS1.p1">w</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m7.png" altimg-height="28px" altimg-valign="-6px" altimg-width="113px" alttext="\displaystyle=e^{-f_{0}z/2}W," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mrow><msub><mi href="./2.7#SS1.p3">f</mi><mn>0</mn></msub><mo></mo><mi>z</mi></mrow><mo>/</mo><mn>2</mn></mrow></mrow></msup><mo></mo><mi>W</mi></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex10" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m23.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="\displaystyle t" display="inline"><mi>t</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m12.png" altimg-height="24px" altimg-valign="-2px" altimg-width="68px" alttext="\displaystyle=z^{1/2}." display="inline"><mrow><mrow><mi></mi><mo>=</mo><msup><mi>z</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E19.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m86.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./2.7#SS1.p1" title="§2.7(i) Regular Singularities: Fuchs–Frobenius Theory ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m126.png" altimg-height="13px" altimg-valign="-2px" altimg-width="19px" alttext="w" display="inline"><mi href="./2.7#SS1.p1">w</mi></math>: DE solution</a> and
<a href="./2.7#SS1.p3" title="§2.7(i) Regular Singularities: Fuchs–Frobenius Theory ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m110.png" altimg-height="21px" altimg-valign="-6px" altimg-width="23px" alttext="f_{s}" display="inline"><msub><mi href="./2.7#SS1.p3">f</mi><mi>s</mi></msub></math>: coefficients</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">The transformed differential equation either has a regular singularity at
<math class="ltx_Math" altimg="m125.png" altimg-height="17px" altimg-valign="-2px" altimg-width="58px" alttext="t=\infty" display="inline"><mrow><mi>t</mi><mo>=</mo><mi mathvariant="normal">∞</mi></mrow></math>, or its characteristic equation has unequal roots.</p>
</div>
<div id="SS2.p7" class="ltx_para">
<p class="ltx_p">For error bounds for (</dd>
</dl>
</div>
</div>
<div id="Px1.p1" class="ltx_para">
<p class="ltx_p">In a finite or infinite interval <math class="ltx_Math" altimg="m30.png" altimg-height="23px" altimg-valign="-7px" altimg-width="68px" alttext="(a_{1},a_{2})" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><msub><mi href="./2.7#Px1.p1">a</mi><mn>1</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><msub><mi href="./2.7#Px1.p1">a</mi><mn>2</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math> let <math class="ltx_Math" altimg="m103.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="f(x)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math> be real, positive, and
twice-continuously differentiable, and <math class="ltx_Math" altimg="m111.png" altimg-height="23px" altimg-valign="-7px" altimg-width="41px" alttext="g(x)" display="inline"><mrow><mi>g</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math> be continuous. Then in
<math class="ltx_Math" altimg="m30.png" altimg-height="23px" altimg-valign="-7px" altimg-width="68px" alttext="(a_{1},a_{2})" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><msub><mi href="./2.7#Px1.p1">a</mi><mn>1</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><msub><mi href="./2.7#Px1.p1">a</mi><mn>2</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math> the differential equation</p>
<table id="E20" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">2.7.20</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E20.png" altimg-height="51px" altimg-valign="-18px" altimg-width="203px" alttext="\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}x}^{2}}=(f(x)+g(x))w" display="block"><mrow><mfrac><mrow><mpadded lspace="-1.7pt" width="-4.5pt"><msup><mo href="./1.4#E4">d</mo><mn>2</mn></msup></mpadded><mi href="./2.7#SS1.p1">w</mi></mrow><msup><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi>x</mi></mrow><mn>2</mn></msup></mfrac><mo>=</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow><mo>+</mo><mrow><mi>g</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi href="./2.7#SS1.p1">w</mi></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E20.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#E4" title="(1.4.4) ‣ §1.4(iii) Derivatives ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m71.png" altimg-height="29px" altimg-valign="-9px" altimg-width="27px" alttext="\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: derivative of <math class="ltx_Math" altimg="m107.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m140.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./2.7#SS1.p1" title="§2.7(i) Regular Singularities: Fuchs–Frobenius Theory ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m126.png" altimg-height="13px" altimg-valign="-2px" altimg-width="19px" alttext="w" display="inline"><mi href="./2.7#SS1.p1">w</mi></math>: DE solution</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">has twice-continuously differentiable solutions</p>
<table id="EGx2" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E21">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">2.7.21</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m25.png" altimg-height="25px" altimg-valign="-7px" altimg-width="56px" alttext="\displaystyle w_{1}(x)" display="inline"><mrow><msub><mi href="./2.7#SS1.p3">w</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m10.png" altimg-height="53px" altimg-valign="-21px" altimg-width="389px" alttext="\displaystyle=f^{-1/4}(x)\exp\left(\int f^{1/2}(x)\mathrm{d}x\right)\*\left(1+%
\epsilon_{1}(x)\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><msup><mi>f</mi><mrow><mo>-</mo><mrow><mn>1</mn><mo>/</mo><mn>4</mn></mrow></mrow></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mrow><mo>(</mo><mstyle displaystyle="true"><mrow><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><mrow><msup><mi>f</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>x</mi></mrow></mrow></mrow></mstyle><mo>)</mo></mrow></mrow><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mrow><msub><mi href="./2.7#Px1.p1">ϵ</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E21.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m140.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E19" title="(4.2.19) ‣ §4.2(iii) The Exponential Function ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="16px" altimg-valign="-6px" altimg-width="48px" alttext="\exp\NVar{z}" display="inline"><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: exponential function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./2.7#Px1.p1" title="Liouville–Green Approximation Theorem ‣ §2.7(iii) Liouville–Green (WKBJ) Approximation ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m68.png" altimg-height="24px" altimg-valign="-8px" altimg-width="48px" alttext="\epsilon_{j}(x)" display="inline"><mrow><msub><mi href="./2.7#Px1.p1">ϵ</mi><mi>j</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function</a> and
<a href="./2.7#SS1.p3" title="§2.7(i) Regular Singularities: Fuchs–Frobenius Theory ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m139.png" altimg-height="24px" altimg-valign="-8px" altimg-width="53px" alttext="w_{j}(z)" display="inline"><mrow><msub><mi href="./2.7#SS1.p3">w</mi><mi>j</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: solutions</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E22">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">2.7.22</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m27.png" altimg-height="25px" altimg-valign="-7px" altimg-width="56px" alttext="\displaystyle w_{2}(x)" display="inline"><mrow><msub><mi href="./2.7#SS1.p3">w</mi><mn>2</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m9.png" altimg-height="53px" altimg-valign="-21px" altimg-width="407px" alttext="\displaystyle=f^{-1/4}(x)\exp\left(-\int f^{1/2}(x)\mathrm{d}x\right)\*\left(1%
+\epsilon_{2}(x)\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><msup><mi>f</mi><mrow><mo>-</mo><mrow><mn>1</mn><mo>/</mo><mn>4</mn></mrow></mrow></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mstyle displaystyle="true"><mrow><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><mrow><msup><mi>f</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>x</mi></mrow></mrow></mrow></mstyle></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mrow><msub><mi href="./2.7#Px1.p1">ϵ</mi><mn>2</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E22.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m140.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E19" title="(4.2.19) ‣ §4.2(iii) The Exponential Function ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="16px" altimg-valign="-6px" altimg-width="48px" alttext="\exp\NVar{z}" display="inline"><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: exponential function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./2.7#Px1.p1" title="Liouville–Green Approximation Theorem ‣ §2.7(iii) Liouville–Green (WKBJ) Approximation ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m68.png" altimg-height="24px" altimg-valign="-8px" altimg-width="48px" alttext="\epsilon_{j}(x)" display="inline"><mrow><msub><mi href="./2.7#Px1.p1">ϵ</mi><mi>j</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function</a> and
<a href="./2.7#SS1.p3" title="§2.7(i) Regular Singularities: Fuchs–Frobenius Theory ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m139.png" altimg-height="24px" altimg-valign="-8px" altimg-width="53px" alttext="w_{j}(z)" display="inline"><mrow><msub><mi href="./2.7#SS1.p3">w</mi><mi>j</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: solutions</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
<p class="ltx_p">such that
</p>
<table id="E23" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">2.7.23</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E23.png" altimg-height="32px" altimg-valign="-10px" altimg-width="432px" alttext="|\epsilon_{j}(x)|,\;\;\tfrac{1}{2}f^{-1/2}(x)|\epsilon_{j}^{\prime}(x)|\leq%
\exp\left(\tfrac{1}{2}\mathcal{V}_{a_{j},x}\left(F\right)\right)-1," display="block"><mrow><mrow><mrow><mrow><mo stretchy="false">|</mo><mrow><msub><mi href="./2.7#Px1.p1">ϵ</mi><mi>j</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow><mo rspace="8.1pt">,</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mrow><msup><mi>f</mi><mrow><mo>-</mo><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></mrow></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo stretchy="false">|</mo><mrow><msubsup><mi href="./2.7#Px1.p1">ϵ</mi><mi>j</mi><mo>′</mo></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow></mrow></mrow><mo>≤</mo><mrow><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mrow><mo>(</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mrow><msub><mi class="ltx_font_mathcaligraphic" href="./1.4#E33">𝒱</mi><mrow><msub><mi href="./2.7#Px1.p1">a</mi><mi>j</mi></msub><mo>,</mo><mi>x</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./2.7#Px1.p1">F</mi><mo>)</mo></mrow></mrow></mrow><mo>)</mo></mrow></mrow><mo>-</mo><mn>1</mn></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m118.png" altimg-height="20px" altimg-valign="-6px" altimg-width="69px" alttext="j=1,2" display="inline"><mrow><mi>j</mi><mo>=</mo><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E23.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.2#E19" title="(4.2.19) ‣ §4.2(iii) The Exponential Function ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="16px" altimg-valign="-6px" altimg-width="48px" alttext="\exp\NVar{z}" display="inline"><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: exponential function</a>,
<a href="./1.4#E33" title="(1.4.33) ‣ Functions of Bounded Variation ‣ §1.4(v) Definite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="24px" altimg-valign="-8px" altimg-width="69px" alttext="\mathcal{V}_{\NVar{a,b}}\left(\NVar{f}\right)" display="inline"><mrow><msub><mi class="ltx_font_mathcaligraphic" href="./1.4#E33">𝒱</mi><mrow><mi class="ltx_nvar">a</mi><mo class="ltx_nvar">,</mo><mi class="ltx_nvar">b</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">f</mi><mo>)</mo></mrow></mrow></math>: total variation</a>,
<a href="./2.7#Px1.p1" title="Liouville–Green Approximation Theorem ‣ §2.7(iii) Liouville–Green (WKBJ) Approximation ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m30.png" altimg-height="23px" altimg-valign="-7px" altimg-width="68px" alttext="(a_{1},a_{2})" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><msub><mi href="./2.7#Px1.p1">a</mi><mn>1</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><msub><mi href="./2.7#Px1.p1">a</mi><mn>2</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math>: interval</a>,
<a href="./2.7#Px1.p1" title="Liouville–Green Approximation Theorem ‣ §2.7(iii) Liouville–Green (WKBJ) Approximation ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m68.png" altimg-height="24px" altimg-valign="-8px" altimg-width="48px" alttext="\epsilon_{j}(x)" display="inline"><mrow><msub><mi href="./2.7#Px1.p1">ϵ</mi><mi>j</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function</a> and
<a href="./2.7#Px1.p1" title="Liouville–Green Approximation Theorem ‣ §2.7(iii) Liouville–Green (WKBJ) Approximation ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m46.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="F" display="inline"><mi href="./2.7#Px1.p1">F</mi></math>: error-control function</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">provided that <math class="ltx_Math" altimg="m84.png" altimg-height="25px" altimg-valign="-9px" altimg-width="128px" alttext="\mathcal{V}_{a_{j},x}\left(F\right)<\infty" display="inline"><mrow><mrow><msub><mi class="ltx_font_mathcaligraphic" href="./1.4#E33">𝒱</mi><mrow><msub><mi href="./2.7#Px1.p1">a</mi><mi>j</mi></msub><mo>,</mo><mi>x</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./2.7#Px1.p1">F</mi><mo>)</mo></mrow></mrow><mo><</mo><mi mathvariant="normal">∞</mi></mrow></math>. Here <math class="ltx_Math" altimg="m45.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="F(x)" display="inline"><mrow><mi href="./2.7#Px1.p1">F</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math> is the
<em class="ltx_emph ltx_font_italic">error-control function</em>
</p>
<table id="E24" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">2.7.24</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E24.png" altimg-height="54px" altimg-valign="-21px" altimg-width="377px" alttext="F(x)=\int\left(\frac{1}{f^{1/4}}\frac{{\mathrm{d}}^{2}}{{\mathrm{d}x}^{2}}%
\left(\frac{1}{f^{1/4}}\right)-\frac{g}{f^{1/2}}\right)\mathrm{d}x," display="block"><mrow><mrow><mrow><mi href="./2.7#Px1.p1">F</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><mrow><mo>(</mo><mrow><mrow><mfrac><mn>1</mn><msup><mi>f</mi><mrow><mn>1</mn><mo>/</mo><mn>4</mn></mrow></msup></mfrac><mo></mo><mrow><mfrac><mpadded lspace="-1.7pt" width="-4.5pt"><msup><mo href="./1.4#E4">d</mo><mn>2</mn></msup></mpadded><msup><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi>x</mi></mrow><mn>2</mn></msup></mfrac><mo></mo><mrow><mo>(</mo><mfrac><mn>1</mn><msup><mi>f</mi><mrow><mn>1</mn><mo>/</mo><mn>4</mn></mrow></msup></mfrac><mo>)</mo></mrow></mrow></mrow><mo>-</mo><mfrac><mi>g</mi><msup><mi>f</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mfrac></mrow><mo>)</mo></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>x</mi></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E24.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#E4" title="(1.4.4) ‣ §1.4(iii) Derivatives ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m71.png" altimg-height="29px" altimg-valign="-9px" altimg-width="27px" alttext="\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: derivative of <math class="ltx_Math" altimg="m107.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m140.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m140.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a> and
<a href="./2.7#Px1.p1" title="Liouville–Green Approximation Theorem ‣ §2.7(iii) Liouville–Green (WKBJ) Approximation ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m46.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="F" display="inline"><mi href="./2.7#Px1.p1">F</mi></math>: error-control function</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and <math class="ltx_Math" altimg="m81.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="\mathcal{V}" display="inline"><mi class="ltx_font_mathcaligraphic" href="./1.4#E33">𝒱</mi></math> denotes the variational operator (§). Thus</p>
<table id="E25" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">2.7.25</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E25.png" altimg-height="57px" altimg-valign="-24px" altimg-width="499px" alttext="\mathcal{V}_{a_{j},x}\left(F\right)=\int_{a_{j}}^{x}\left|\left(\frac{1}{f^{1/%
4}(t)}\frac{{\mathrm{d}}^{2}}{{\mathrm{d}t}^{2}}\left(\frac{1}{f^{1/4}(t)}%
\right)-\frac{g(t)}{f^{1/2}(t)}\right)\mathrm{d}t\right|." display="block"><mrow><mrow><mrow><msub><mi class="ltx_font_mathcaligraphic" href="./1.4#E33">𝒱</mi><mrow><msub><mi href="./2.7#Px1.p1">a</mi><mi>j</mi></msub><mo>,</mo><mi>x</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./2.7#Px1.p1">F</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><msub><mi href="./2.7#Px1.p1">a</mi><mi>j</mi></msub><mi>x</mi></msubsup><mrow><mo>|</mo><mrow><mrow><mo>(</mo><mrow><mrow><mfrac><mn>1</mn><mrow><msup><mi>f</mi><mrow><mn>1</mn><mo>/</mo><mn>4</mn></mrow></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></mfrac><mo></mo><mrow><mfrac><mpadded lspace="-1.7pt" width="-4.5pt"><msup><mo href="./1.4#E4">d</mo><mn>2</mn></msup></mpadded><msup><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi>t</mi></mrow><mn>2</mn></msup></mfrac><mo></mo><mrow><mo>(</mo><mfrac><mn>1</mn><mrow><msup><mi>f</mi><mrow><mn>1</mn><mo>/</mo><mn>4</mn></mrow></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></mfrac><mo>)</mo></mrow></mrow></mrow><mo>-</mo><mfrac><mrow><mi>g</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mrow><msup><mi>f</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></mfrac></mrow><mo>)</mo></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow><mo>|</mo></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E25.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#E4" title="(1.4.4) ‣ §1.4(iii) Derivatives ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m71.png" altimg-height="29px" altimg-valign="-9px" altimg-width="27px" alttext="\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: derivative of <math class="ltx_Math" altimg="m107.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m140.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m140.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.4#E33" title="(1.4.33) ‣ Functions of Bounded Variation ‣ §1.4(v) Definite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="24px" altimg-valign="-8px" altimg-width="69px" alttext="\mathcal{V}_{\NVar{a,b}}\left(\NVar{f}\right)" display="inline"><mrow><msub><mi class="ltx_font_mathcaligraphic" href="./1.4#E33">𝒱</mi><mrow><mi class="ltx_nvar">a</mi><mo class="ltx_nvar">,</mo><mi class="ltx_nvar">b</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">f</mi><mo>)</mo></mrow></mrow></math>: total variation</a>,
<a href="./2.7#Px1.p1" title="Liouville–Green Approximation Theorem ‣ §2.7(iii) Liouville–Green (WKBJ) Approximation ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m30.png" altimg-height="23px" altimg-valign="-7px" altimg-width="68px" alttext="(a_{1},a_{2})" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><msub><mi href="./2.7#Px1.p1">a</mi><mn>1</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><msub><mi href="./2.7#Px1.p1">a</mi><mn>2</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math>: interval</a> and
<a href="./2.7#Px1.p1" title="Liouville–Green Approximation Theorem ‣ §2.7(iii) Liouville–Green (WKBJ) Approximation ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m46.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="F" display="inline"><mi href="./2.7#Px1.p1">F</mi></math>: error-control function</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="Px1.p2" class="ltx_para">
<p class="ltx_p">Assuming also <math class="ltx_Math" altimg="m83.png" altimg-height="24px" altimg-valign="-8px" altimg-width="136px" alttext="\mathcal{V}_{a_{1},a_{2}}\left(F\right)<\infty" display="inline"><mrow><mrow><msub><mi class="ltx_font_mathcaligraphic" href="./1.4#E33">𝒱</mi><mrow><msub><mi href="./2.7#Px1.p1">a</mi><mn>1</mn></msub><mo>,</mo><msub><mi href="./2.7#Px1.p1">a</mi><mn>2</mn></msub></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./2.7#Px1.p1">F</mi><mo>)</mo></mrow></mrow><mo><</mo><mi mathvariant="normal">∞</mi></mrow></math>, we have
</p>
<table id="E26" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">2.7.26</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E26.png" altimg-height="53px" altimg-valign="-21px" altimg-width="344px" alttext="w_{1}(x)\sim f^{-1/4}(x)\exp\left(\int f^{1/2}(x)\mathrm{d}x\right)," display="block"><mrow><mrow><mrow><msub><mi href="./2.7#SS1.p3">w</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow><mo href="./2.1#E1">∼</mo><mrow><mrow><msup><mi>f</mi><mrow><mo>-</mo><mrow><mn>1</mn><mo>/</mo><mn>4</mn></mrow></mrow></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mrow><mo>(</mo><mrow><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><mrow><msup><mi>f</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>x</mi></mrow></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m141.png" altimg-height="19px" altimg-valign="-5px" altimg-width="82px" alttext="x\to a_{1}+" display="inline"><mrow><mi>x</mi><mo>→</mo><mrow><msub><mi href="./2.7#Px1.p1">a</mi><mn>1</mn></msub><mo>+</mo></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E26.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./2.1#E1" title="(2.1.1) ‣ §2.1(i) Asymptotic and Order Symbols ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m90.png" altimg-height="12px" altimg-valign="-2px" altimg-width="20px" alttext="\sim" display="inline"><mo href="./2.1#E1">∼</mo></math>: asymptotic equality</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m140.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E19" title="(4.2.19) ‣ §4.2(iii) The Exponential Function ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="16px" altimg-valign="-6px" altimg-width="48px" alttext="\exp\NVar{z}" display="inline"><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: exponential function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./2.7#Px1.p1" title="Liouville–Green Approximation Theorem ‣ §2.7(iii) Liouville–Green (WKBJ) Approximation ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m30.png" altimg-height="23px" altimg-valign="-7px" altimg-width="68px" alttext="(a_{1},a_{2})" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><msub><mi href="./2.7#Px1.p1">a</mi><mn>1</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><msub><mi href="./2.7#Px1.p1">a</mi><mn>2</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math>: interval</a> and
<a href="./2.7#SS1.p3" title="§2.7(i) Regular Singularities: Fuchs–Frobenius Theory ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m139.png" altimg-height="24px" altimg-valign="-8px" altimg-width="53px" alttext="w_{j}(z)" display="inline"><mrow><msub><mi href="./2.7#SS1.p3">w</mi><mi>j</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: solutions</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E27" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">2.7.27</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E27.png" altimg-height="53px" altimg-valign="-21px" altimg-width="363px" alttext="w_{2}(x)\sim f^{-1/4}(x)\exp\left(-\int f^{1/2}(x)\mathrm{d}x\right)," display="block"><mrow><mrow><mrow><msub><mi href="./2.7#SS1.p3">w</mi><mn>2</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow><mo href="./2.1#E1">∼</mo><mrow><mrow><msup><mi>f</mi><mrow><mo>-</mo><mrow><mn>1</mn><mo>/</mo><mn>4</mn></mrow></mrow></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mrow><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><mrow><msup><mi>f</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>x</mi></mrow></mrow></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m142.png" altimg-height="19px" altimg-valign="-5px" altimg-width="82px" alttext="x\to a_{2}-" display="inline"><mrow><mi>x</mi><mo>→</mo><mrow><msub><mi href="./2.7#Px1.p1">a</mi><mn>2</mn></msub><mo>-</mo></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E27.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./2.1#E1" title="(2.1.1) ‣ §2.1(i) Asymptotic and Order Symbols ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m90.png" altimg-height="12px" altimg-valign="-2px" altimg-width="20px" alttext="\sim" display="inline"><mo href="./2.1#E1">∼</mo></math>: asymptotic equality</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m140.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E19" title="(4.2.19) ‣ §4.2(iii) The Exponential Function ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="16px" altimg-valign="-6px" altimg-width="48px" alttext="\exp\NVar{z}" display="inline"><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: exponential function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./2.7#Px1.p1" title="Liouville–Green Approximation Theorem ‣ §2.7(iii) Liouville–Green (WKBJ) Approximation ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m30.png" altimg-height="23px" altimg-valign="-7px" altimg-width="68px" alttext="(a_{1},a_{2})" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><msub><mi href="./2.7#Px1.p1">a</mi><mn>1</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><msub><mi href="./2.7#Px1.p1">a</mi><mn>2</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math>: interval</a> and
<a href="./2.7#SS1.p3" title="§2.7(i) Regular Singularities: Fuchs–Frobenius Theory ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m139.png" altimg-height="24px" altimg-valign="-8px" altimg-width="53px" alttext="w_{j}(z)" display="inline"><mrow><msub><mi href="./2.7#SS1.p3">w</mi><mi>j</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: solutions</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Suppose in addition <math class="ltx_Math" altimg="m148.png" altimg-height="27px" altimg-valign="-8px" altimg-width="122px" alttext="|\int f^{1/2}(x)\mathrm{d}x|" display="inline"><mrow><mo stretchy="false">|</mo><mrow><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><mrow><msup><mi>f</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>x</mi></mrow></mrow></mrow><mo stretchy="false">|</mo></mrow></math> is unbounded as <math class="ltx_Math" altimg="m141.png" altimg-height="19px" altimg-valign="-5px" altimg-width="82px" alttext="x\to a_{1}+" display="inline"><mrow><mi>x</mi><mo>→</mo><mrow><msub><mi href="./2.7#Px1.p1">a</mi><mn>1</mn></msub><mo>+</mo></mrow></mrow></math>
and <math class="ltx_Math" altimg="m142.png" altimg-height="19px" altimg-valign="-5px" altimg-width="82px" alttext="x\to a_{2}-" display="inline"><mrow><mi>x</mi><mo>→</mo><mrow><msub><mi href="./2.7#Px1.p1">a</mi><mn>2</mn></msub><mo>-</mo></mrow></mrow></math>. Then there are solutions <math class="ltx_Math" altimg="m133.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="w_{3}(x)" display="inline"><mrow><msub><mi href="./2.7#SS1.p3">w</mi><mn>3</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math>, <math class="ltx_Math" altimg="m136.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="w_{4}(x)" display="inline"><mrow><msub><mi href="./2.7#SS1.p3">w</mi><mn>4</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math>, such that
</p>
<table id="E28" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">2.7.28</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E28.png" altimg-height="53px" altimg-valign="-21px" altimg-width="344px" alttext="w_{3}(x)\sim f^{-1/4}(x)\exp\left(\int f^{1/2}(x)\mathrm{d}x\right)," display="block"><mrow><mrow><mrow><msub><mi href="./2.7#SS1.p3">w</mi><mn>3</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow><mo href="./2.1#E1">∼</mo><mrow><mrow><msup><mi>f</mi><mrow><mo>-</mo><mrow><mn>1</mn><mo>/</mo><mn>4</mn></mrow></mrow></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mrow><mo>(</mo><mrow><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><mrow><msup><mi>f</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>x</mi></mrow></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m142.png" altimg-height="19px" altimg-valign="-5px" altimg-width="82px" alttext="x\to a_{2}-" display="inline"><mrow><mi>x</mi><mo>→</mo><mrow><msub><mi href="./2.7#Px1.p1">a</mi><mn>2</mn></msub><mo>-</mo></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E28.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./2.1#E1" title="(2.1.1) ‣ §2.1(i) Asymptotic and Order Symbols ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m90.png" altimg-height="12px" altimg-valign="-2px" altimg-width="20px" alttext="\sim" display="inline"><mo href="./2.1#E1">∼</mo></math>: asymptotic equality</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m140.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E19" title="(4.2.19) ‣ §4.2(iii) The Exponential Function ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="16px" altimg-valign="-6px" altimg-width="48px" alttext="\exp\NVar{z}" display="inline"><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: exponential function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./2.7#Px1.p1" title="Liouville–Green Approximation Theorem ‣ §2.7(iii) Liouville–Green (WKBJ) Approximation ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m30.png" altimg-height="23px" altimg-valign="-7px" altimg-width="68px" alttext="(a_{1},a_{2})" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><msub><mi href="./2.7#Px1.p1">a</mi><mn>1</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><msub><mi href="./2.7#Px1.p1">a</mi><mn>2</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math>: interval</a> and
<a href="./2.7#SS1.p3" title="§2.7(i) Regular Singularities: Fuchs–Frobenius Theory ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m139.png" altimg-height="24px" altimg-valign="-8px" altimg-width="53px" alttext="w_{j}(z)" display="inline"><mrow><msub><mi href="./2.7#SS1.p3">w</mi><mi>j</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: solutions</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E29" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">2.7.29</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E29.png" altimg-height="53px" altimg-valign="-21px" altimg-width="363px" alttext="w_{4}(x)\sim f^{-1/4}(x)\exp\left(-\int f^{1/2}(x)\mathrm{d}x\right)," display="block"><mrow><mrow><mrow><msub><mi href="./2.7#SS1.p3">w</mi><mn>4</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow><mo href="./2.1#E1">∼</mo><mrow><mrow><msup><mi>f</mi><mrow><mo>-</mo><mrow><mn>1</mn><mo>/</mo><mn>4</mn></mrow></mrow></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mrow><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><mrow><msup><mi>f</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>x</mi></mrow></mrow></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m141.png" altimg-height="19px" altimg-valign="-5px" altimg-width="82px" alttext="x\to a_{1}+" display="inline"><mrow><mi>x</mi><mo>→</mo><mrow><msub><mi href="./2.7#Px1.p1">a</mi><mn>1</mn></msub><mo>+</mo></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E29.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./2.1#E1" title="(2.1.1) ‣ §2.1(i) Asymptotic and Order Symbols ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m90.png" altimg-height="12px" altimg-valign="-2px" altimg-width="20px" alttext="\sim" display="inline"><mo href="./2.1#E1">∼</mo></math>: asymptotic equality</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m140.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E19" title="(4.2.19) ‣ §4.2(iii) The Exponential Function ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="16px" altimg-valign="-6px" altimg-width="48px" alttext="\exp\NVar{z}" display="inline"><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: exponential function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./2.7#Px1.p1" title="Liouville–Green Approximation Theorem ‣ §2.7(iii) Liouville–Green (WKBJ) Approximation ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m30.png" altimg-height="23px" altimg-valign="-7px" altimg-width="68px" alttext="(a_{1},a_{2})" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><msub><mi href="./2.7#Px1.p1">a</mi><mn>1</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><msub><mi href="./2.7#Px1.p1">a</mi><mn>2</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math>: interval</a> and
<a href="./2.7#SS1.p3" title="§2.7(i) Regular Singularities: Fuchs–Frobenius Theory ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m139.png" altimg-height="24px" altimg-valign="-8px" altimg-width="53px" alttext="w_{j}(z)" display="inline"><mrow><msub><mi href="./2.7#SS1.p3">w</mi><mi>j</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: solutions</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">The solutions with the properties (). In fact, since</p>
<table id="E30" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">2.7.30</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E30.png" altimg-height="25px" altimg-valign="-7px" altimg-width="163px" alttext="w_{1}(x)/w_{4}(x)\to 0," display="block"><mrow><mrow><mrow><mrow><msub><mi href="./2.7#SS1.p3">w</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow><mo>/</mo><mrow><msub><mi href="./2.7#SS1.p3">w</mi><mn>4</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>→</mo><mn>0</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m141.png" altimg-height="19px" altimg-valign="-5px" altimg-width="82px" alttext="x\to a_{1}+" display="inline"><mrow><mi>x</mi><mo>→</mo><mrow><msub><mi href="./2.7#Px1.p1">a</mi><mn>1</mn></msub><mo>+</mo></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E30.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./2.7#Px1.p1" title="Liouville–Green Approximation Theorem ‣ §2.7(iii) Liouville–Green (WKBJ) Approximation ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m30.png" altimg-height="23px" altimg-valign="-7px" altimg-width="68px" alttext="(a_{1},a_{2})" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><msub><mi href="./2.7#Px1.p1">a</mi><mn>1</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><msub><mi href="./2.7#Px1.p1">a</mi><mn>2</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math>: interval</a> and
<a href="./2.7#SS1.p3" title="§2.7(i) Regular Singularities: Fuchs–Frobenius Theory ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m139.png" altimg-height="24px" altimg-valign="-8px" altimg-width="53px" alttext="w_{j}(z)" display="inline"><mrow><msub><mi href="./2.7#SS1.p3">w</mi><mi>j</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: solutions</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p"><math class="ltx_Math" altimg="m127.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="w_{1}(x)" display="inline"><mrow><msub><mi href="./2.7#SS1.p3">w</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math> is a <em class="ltx_emph ltx_font_italic">recessive</em> (or <em class="ltx_emph ltx_font_italic">subdominant</em>) solution as
<math class="ltx_Math" altimg="m141.png" altimg-height="19px" altimg-valign="-5px" altimg-width="82px" alttext="x\to a_{1}+" display="inline"><mrow><mi>x</mi><mo>→</mo><mrow><msub><mi href="./2.7#Px1.p1">a</mi><mn>1</mn></msub><mo>+</mo></mrow></mrow></math>, and <math class="ltx_Math" altimg="m136.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="w_{4}(x)" display="inline"><mrow><msub><mi href="./2.7#SS1.p3">w</mi><mn>4</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math> is a <em class="ltx_emph ltx_font_italic">dominant</em> solution as <math class="ltx_Math" altimg="m141.png" altimg-height="19px" altimg-valign="-5px" altimg-width="82px" alttext="x\to a_{1}+" display="inline"><mrow><mi>x</mi><mo>→</mo><mrow><msub><mi href="./2.7#Px1.p1">a</mi><mn>1</mn></msub><mo>+</mo></mrow></mrow></math>.
Similarly for <math class="ltx_Math" altimg="m130.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="w_{2}(x)" display="inline"><mrow><msub><mi href="./2.7#SS1.p3">w</mi><mn>2</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math> and <math class="ltx_Math" altimg="m133.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="w_{3}(x)" display="inline"><mrow><msub><mi href="./2.7#SS1.p3">w</mi><mn>3</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math> as <math class="ltx_Math" altimg="m142.png" altimg-height="19px" altimg-valign="-5px" altimg-width="82px" alttext="x\to a_{2}-" display="inline"><mrow><mi>x</mi><mo>→</mo><mrow><msub><mi href="./2.7#Px1.p1">a</mi><mn>2</mn></msub><mo>-</mo></mrow></mrow></math>.</p>
</div>
</section>
<section id="Px2" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Example</h3>
<div id="Px2.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px2.p1" class="ltx_para">
<table id="E31" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">2.7.31</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E31.png" altimg-height="51px" altimg-valign="-18px" altimg-width="175px" alttext="\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}x}^{2}}=(x+\ln x)w," display="block"><mrow><mrow><mfrac><mrow><mpadded lspace="-1.7pt" width="-4.5pt"><msup><mo href="./1.4#E4">d</mo><mn>2</mn></msup></mpadded><mi href="./2.7#SS1.p1">w</mi></mrow><msup><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi>x</mi></mrow><mn>2</mn></msup></mfrac><mo>=</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mi>x</mi><mo>+</mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi>x</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi href="./2.7#SS1.p1">w</mi></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m36.png" altimg-height="17px" altimg-valign="-3px" altimg-width="99px" alttext="0<x<\infty" display="inline"><mrow><mn>0</mn><mo><</mo><mi>x</mi><mo><</mo><mi mathvariant="normal">∞</mi></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E31.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#E4" title="(1.4.4) ‣ §1.4(iii) Derivatives ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m71.png" altimg-height="29px" altimg-valign="-9px" altimg-width="27px" alttext="\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: derivative of <math class="ltx_Math" altimg="m107.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m140.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m79.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a> and
<a href="./2.7#SS1.p1" title="§2.7(i) Regular Singularities: Fuchs–Frobenius Theory ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m126.png" altimg-height="13px" altimg-valign="-2px" altimg-width="19px" alttext="w" display="inline"><mi href="./2.7#SS1.p1">w</mi></math>: DE solution</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">We cannot take <math class="ltx_Math" altimg="m106.png" altimg-height="21px" altimg-valign="-6px" altimg-width="54px" alttext="f=x" display="inline"><mrow><mi>f</mi><mo>=</mo><mi>x</mi></mrow></math> and <math class="ltx_Math" altimg="m114.png" altimg-height="21px" altimg-valign="-6px" altimg-width="72px" alttext="g=\ln x" display="inline"><mrow><mi>g</mi><mo>=</mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi>x</mi></mrow></mrow></math> because <math class="ltx_Math" altimg="m73.png" altimg-height="27px" altimg-valign="-8px" altimg-width="103px" alttext="\int gf^{-1/2}\mathrm{d}x" display="inline"><mrow><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><mi>g</mi><mo></mo><msup><mi>f</mi><mrow><mo>-</mo><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></mrow></msup><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>x</mi></mrow></mrow></mrow></math>
would diverge as <math class="ltx_Math" altimg="m143.png" altimg-height="17px" altimg-valign="-4px" altimg-width="82px" alttext="x\to+\infty" display="inline"><mrow><mi>x</mi><mo>→</mo><mrow><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></mrow></math>. Instead set <math class="ltx_Math" altimg="m105.png" altimg-height="21px" altimg-valign="-6px" altimg-width="110px" alttext="f=x+\ln x" display="inline"><mrow><mi>f</mi><mo>=</mo><mrow><mi>x</mi><mo>+</mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi>x</mi></mrow></mrow></mrow></math>, <math class="ltx_Math" altimg="m113.png" altimg-height="20px" altimg-valign="-6px" altimg-width="51px" alttext="g=0" display="inline"><mrow><mi>g</mi><mo>=</mo><mn>0</mn></mrow></math>. By
approximating</p>
<table id="E32" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">2.7.32</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E32.png" altimg-height="41px" altimg-valign="-15px" altimg-width="407px" alttext="f^{1/2}=x^{1/2}+\tfrac{1}{2}x^{-1/2}\ln x+O\left(x^{-3/2}(\ln x)^{2}\right)," display="block"><mrow><mrow><msup><mi>f</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>=</mo><mrow><msup><mi>x</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>+</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><msup><mi>x</mi><mrow><mo>-</mo><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></mrow></msup><mo></mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi>x</mi></mrow></mrow><mo>+</mo><mrow><mi href="./2.1#E3">O</mi><mo></mo><mrow><mo>(</mo><mrow><msup><mi>x</mi><mrow><mo>-</mo><mrow><mn>3</mn><mo>/</mo><mn>2</mn></mrow></mrow></msup><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi>x</mi></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E32.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./2.1#E3" title="(2.1.3) ‣ §2.1(i) Asymptotic and Order Symbols ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="23px" altimg-valign="-7px" altimg-width="50px" alttext="O\left(\NVar{x}\right)" display="inline"><mrow><mi href="./2.1#E3">O</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>)</mo></mrow></mrow></math>: order not exceeding</a> and
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m79.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">we arrive at</p>
<table id="EGx3" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E33">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">2.7.33</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m27.png" altimg-height="25px" altimg-valign="-7px" altimg-width="56px" alttext="\displaystyle w_{2}(x)" display="inline"><mrow><msub><mi href="./2.7#SS1.p3">w</mi><mn>2</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m16.png" altimg-height="41px" altimg-valign="-15px" altimg-width="313px" alttext="\displaystyle\sim x^{-(1/4)-\sqrt{x}}\exp\left(2x^{1/2}-\tfrac{2}{3}x^{3/2}%
\right)," display="inline"><mrow><mrow><mi></mi><mo href="./2.1#E1">∼</mo><mrow><msup><mi>x</mi><mrow><mrow><mo>-</mo><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>/</mo><mn>4</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>-</mo><msqrt><mi>x</mi></msqrt></mrow></msup><mo></mo><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mn>2</mn><mo></mo><msup><mi>x</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow><mo>-</mo><mrow><mfrac><mn>2</mn><mn>3</mn></mfrac><mo></mo><msup><mi>x</mi><mrow><mn>3</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E33.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./2.1#E1" title="(2.1.1) ‣ §2.1(i) Asymptotic and Order Symbols ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m90.png" altimg-height="12px" altimg-valign="-2px" altimg-width="20px" alttext="\sim" display="inline"><mo href="./2.1#E1">∼</mo></math>: asymptotic equality</a>,
<a href="./4.2#E19" title="(4.2.19) ‣ §4.2(iii) The Exponential Function ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="16px" altimg-valign="-6px" altimg-width="48px" alttext="\exp\NVar{z}" display="inline"><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: exponential function</a> and
<a href="./2.7#SS1.p3" title="§2.7(i) Regular Singularities: Fuchs–Frobenius Theory ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m139.png" altimg-height="24px" altimg-valign="-8px" altimg-width="53px" alttext="w_{j}(z)" display="inline"><mrow><msub><mi href="./2.7#SS1.p3">w</mi><mi>j</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: solutions</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E34">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">2.7.34</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m29.png" altimg-height="25px" altimg-valign="-7px" altimg-width="56px" alttext="\displaystyle w_{3}(x)" display="inline"><mrow><msub><mi href="./2.7#SS1.p3">w</mi><mn>3</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m15.png" altimg-height="41px" altimg-valign="-15px" altimg-width="312px" alttext="\displaystyle\sim x^{-(1/4)+\sqrt{x}}\exp\left(\tfrac{2}{3}x^{3/2}-2x^{1/2}%
\right)," display="inline"><mrow><mrow><mi></mi><mo href="./2.1#E1">∼</mo><mrow><msup><mi>x</mi><mrow><mrow><mo>-</mo><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>/</mo><mn>4</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>+</mo><msqrt><mi>x</mi></msqrt></mrow></msup><mo></mo><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mfrac><mn>2</mn><mn>3</mn></mfrac><mo></mo><msup><mi>x</mi><mrow><mn>3</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow><mo>-</mo><mrow><mn>2</mn><mo></mo><msup><mi>x</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E34.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./2.1#E1" title="(2.1.1) ‣ §2.1(i) Asymptotic and Order Symbols ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m90.png" altimg-height="12px" altimg-valign="-2px" altimg-width="20px" alttext="\sim" display="inline"><mo href="./2.1#E1">∼</mo></math>: asymptotic equality</a>,
<a href="./4.2#E19" title="(4.2.19) ‣ §4.2(iii) The Exponential Function ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="16px" altimg-valign="-6px" altimg-width="48px" alttext="\exp\NVar{z}" display="inline"><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: exponential function</a> and
<a href="./2.7#SS1.p3" title="§2.7(i) Regular Singularities: Fuchs–Frobenius Theory ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m139.png" altimg-height="24px" altimg-valign="-8px" altimg-width="53px" alttext="w_{j}(z)" display="inline"><mrow><msub><mi href="./2.7#SS1.p3">w</mi><mi>j</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: solutions</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
<p class="ltx_p">as <math class="ltx_Math" altimg="m143.png" altimg-height="17px" altimg-valign="-4px" altimg-width="82px" alttext="x\to+\infty" display="inline"><mrow><mi>x</mi><mo>→</mo><mrow><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></mrow></math>, <math class="ltx_Math" altimg="m130.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="w_{2}(x)" display="inline"><mrow><msub><mi href="./2.7#SS1.p3">w</mi><mn>2</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math> being recessive and <math class="ltx_Math" altimg="m133.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="w_{3}(x)" display="inline"><mrow><msub><mi href="./2.7#SS1.p3">w</mi><mn>3</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math> dominant.</p>
</div>
<div id="Px2.p2" class="ltx_para">
<p class="ltx_p">For other examples, and also the corresponding results when <math class="ltx_Math" altimg="m103.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="f(x)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math> is negative, see
<cite class="ltx_cite ltx_citemacro_citet">Olver (</dd>
</dl>
</div>
</div>
<div id="SS4.p1" class="ltx_para">
<p class="ltx_p">One pair of independent solutions of the equation</p>
<table id="E35" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">2.7.35</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E35.png" altimg-height="30px" altimg-valign="-9px" altimg-width="129px" alttext="\ifrac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}=w" display="block"><mrow><mrow><mrow><mpadded lspace="-1.7pt" width="-4.5pt"><msup><mo href="./1.4#E4">d</mo><mn>2</mn></msup></mpadded><mi href="./2.7#SS1.p1">w</mi></mrow><mo href="./1.4#E4">/</mo><msup><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi>z</mi></mrow><mn>2</mn></msup></mrow><mo>=</mo><mi href="./2.7#SS1.p1">w</mi></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E35.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#E4" title="(1.4.4) ‣ §1.4(iii) Derivatives ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m71.png" altimg-height="29px" altimg-valign="-9px" altimg-width="27px" alttext="\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: derivative of <math class="ltx_Math" altimg="m107.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m140.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./2.7#SS1.p1" title="§2.7(i) Regular Singularities: Fuchs–Frobenius Theory ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m126.png" altimg-height="13px" altimg-valign="-2px" altimg-width="19px" alttext="w" display="inline"><mi href="./2.7#SS1.p1">w</mi></math>: DE solution</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">is <math class="ltx_Math" altimg="m128.png" altimg-height="23px" altimg-valign="-7px" altimg-width="98px" alttext="w_{1}(z)=e^{z}" display="inline"><mrow><mrow><msub><mi href="./2.7#SS1.p3">w</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mi>z</mi></msup></mrow></math>, <math class="ltx_Math" altimg="m131.png" altimg-height="24px" altimg-valign="-7px" altimg-width="111px" alttext="w_{2}(z)=e^{-z}" display="inline"><mrow><mrow><msub><mi href="./2.7#SS1.p3">w</mi><mn>2</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mi>z</mi></mrow></msup></mrow></math>. Another is <math class="ltx_Math" altimg="m134.png" altimg-height="23px" altimg-valign="-7px" altimg-width="131px" alttext="w_{3}(z)=\cosh z" display="inline"><mrow><mrow><msub><mi href="./2.7#SS1.p3">w</mi><mn>3</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mi href="./4.28#E2">cosh</mi><mo></mo><mi>z</mi></mrow></mrow></math>,
<math class="ltx_Math" altimg="m137.png" altimg-height="23px" altimg-valign="-7px" altimg-width="129px" alttext="w_{4}(z)=\sinh z" display="inline"><mrow><mrow><msub><mi href="./2.7#SS1.p3">w</mi><mn>4</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mi href="./4.28#E1">sinh</mi><mo></mo><mi>z</mi></mrow></mrow></math>. In theory either pair may be used to construct any other
solution</p>
<table id="E36" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">2.7.36</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E36.png" altimg-height="25px" altimg-valign="-7px" altimg-width="232px" alttext="w(z)=Aw_{1}(z)+Bw_{2}(z)," display="block"><mrow><mrow><mrow><mi href="./2.7#SS1.p1">w</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mi href="./2.7#SS4.p1">A</mi><mo></mo><mrow><msub><mi href="./2.7#SS1.p3">w</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mi href="./2.7#SS4.p1">B</mi><mo></mo><mrow><msub><mi href="./2.7#SS1.p3">w</mi><mn>2</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E36.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./2.7#SS1.p1" title="§2.7(i) Regular Singularities: Fuchs–Frobenius Theory ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m126.png" altimg-height="13px" altimg-valign="-2px" altimg-width="19px" alttext="w" display="inline"><mi href="./2.7#SS1.p1">w</mi></math>: DE solution</a>,
<a href="./2.7#SS4.p1" title="§2.7(iv) Numerically Satisfactory Solutions ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="A" display="inline"><mi href="./2.7#SS4.p1">A</mi></math>: constant</a>,
<a href="./2.7#SS4.p1" title="§2.7(iv) Numerically Satisfactory Solutions ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m40.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="B" display="inline"><mi href="./2.7#SS4.p1">B</mi></math>: constant</a> and
<a href="./2.7#SS1.p3" title="§2.7(i) Regular Singularities: Fuchs–Frobenius Theory ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m139.png" altimg-height="24px" altimg-valign="-8px" altimg-width="53px" alttext="w_{j}(z)" display="inline"><mrow><msub><mi href="./2.7#SS1.p3">w</mi><mi>j</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: solutions</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">or</p>
<table id="E37" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">2.7.37</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E37.png" altimg-height="25px" altimg-valign="-7px" altimg-width="233px" alttext="w(z)=Cw_{3}(z)+Dw_{4}(z)," display="block"><mrow><mrow><mrow><mi href="./2.7#SS1.p1">w</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mi href="./2.7#SS4.p1">C</mi><mo></mo><mrow><msub><mi href="./2.7#SS1.p3">w</mi><mn>3</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mi href="./2.7#SS4.p1">D</mi><mo></mo><mrow><msub><mi href="./2.7#SS1.p3">w</mi><mn>4</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E37.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./2.7#SS1.p1" title="§2.7(i) Regular Singularities: Fuchs–Frobenius Theory ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m126.png" altimg-height="13px" altimg-valign="-2px" altimg-width="19px" alttext="w" display="inline"><mi href="./2.7#SS1.p1">w</mi></math>: DE solution</a>,
<a href="./2.7#SS4.p1" title="§2.7(iv) Numerically Satisfactory Solutions ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m41.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./2.7#SS4.p1">C</mi></math>: constant</a>,
<a href="./2.7#SS4.p1" title="§2.7(iv) Numerically Satisfactory Solutions ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./2.7#SS4.p1">D</mi></math>: constant</a> and
<a href="./2.7#SS1.p3" title="§2.7(i) Regular Singularities: Fuchs–Frobenius Theory ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m139.png" altimg-height="24px" altimg-valign="-8px" altimg-width="53px" alttext="w_{j}(z)" display="inline"><mrow><msub><mi href="./2.7#SS1.p3">w</mi><mi>j</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: solutions</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m38.png" altimg-height="21px" altimg-valign="-6px" altimg-width="94px" alttext="A,B,C,D" display="inline"><mrow><mi href="./2.7#SS4.p1">A</mi><mo>,</mo><mi href="./2.7#SS4.p1">B</mi><mo>,</mo><mi href="./2.7#SS4.p1">C</mi><mo>,</mo><mi href="./2.7#SS4.p1">D</mi></mrow></math> are constants. From the numerical standpoint, however, the pair
<math class="ltx_Math" altimg="m135.png" altimg-height="23px" altimg-valign="-7px" altimg-width="53px" alttext="w_{3}(z)" display="inline"><mrow><msub><mi href="./2.7#SS1.p3">w</mi><mn>3</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math> and <math class="ltx_Math" altimg="m138.png" altimg-height="23px" altimg-valign="-7px" altimg-width="53px" alttext="w_{4}(z)" display="inline"><mrow><msub><mi href="./2.7#SS1.p3">w</mi><mn>4</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math> has the drawback that severe numerical cancellation can
occur with certain combinations of <math class="ltx_Math" altimg="m41.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./2.7#SS4.p1">C</mi></math> and <math class="ltx_Math" altimg="m44.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./2.7#SS4.p1">D</mi></math>, for example if <math class="ltx_Math" altimg="m41.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./2.7#SS4.p1">C</mi></math> and <math class="ltx_Math" altimg="m44.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./2.7#SS4.p1">D</mi></math> are
equal, or nearly equal, and <math class="ltx_Math" altimg="m144.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi>z</mi></math>, or <math class="ltx_Math" altimg="m56.png" altimg-height="18px" altimg-valign="-2px" altimg-width="29px" alttext="\Re z" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>z</mi></mrow></math>, is large and negative. This
kind of cancellation cannot take place with <math class="ltx_Math" altimg="m129.png" altimg-height="23px" altimg-valign="-7px" altimg-width="53px" alttext="w_{1}(z)" display="inline"><mrow><msub><mi href="./2.7#SS1.p3">w</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math> and <math class="ltx_Math" altimg="m132.png" altimg-height="23px" altimg-valign="-7px" altimg-width="53px" alttext="w_{2}(z)" display="inline"><mrow><msub><mi href="./2.7#SS1.p3">w</mi><mn>2</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math>, and for this
reason, and following <cite class="ltx_cite ltx_citemacro_citet">Miller ()</cite>, we call <math class="ltx_Math" altimg="m129.png" altimg-height="23px" altimg-valign="-7px" altimg-width="53px" alttext="w_{1}(z)" display="inline"><mrow><msub><mi href="./2.7#SS1.p3">w</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math> and <math class="ltx_Math" altimg="m132.png" altimg-height="23px" altimg-valign="-7px" altimg-width="53px" alttext="w_{2}(z)" display="inline"><mrow><msub><mi href="./2.7#SS1.p3">w</mi><mn>2</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math> a
<em class="ltx_emph ltx_font_italic">numerically satisfactory pair</em>
of solutions.
</p>
</div>
<div id="SS4.p2" class="ltx_para">
<p class="ltx_p">The solutions <math class="ltx_Math" altimg="m129.png" altimg-height="23px" altimg-valign="-7px" altimg-width="53px" alttext="w_{1}(z)" display="inline"><mrow><msub><mi href="./2.7#SS1.p3">w</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math> and <math class="ltx_Math" altimg="m132.png" altimg-height="23px" altimg-valign="-7px" altimg-width="53px" alttext="w_{2}(z)" display="inline"><mrow><msub><mi href="./2.7#SS1.p3">w</mi><mn>2</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math> are respectively recessive and dominant as
<math class="ltx_Math" altimg="m58.png" altimg-height="19px" altimg-valign="-4px" altimg-width="95px" alttext="\Re z\to-\infty" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>z</mi></mrow><mo>→</mo><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow></mrow></math>, and <em class="ltx_emph ltx_font_italic">vice versa</em> as
<math class="ltx_Math" altimg="m57.png" altimg-height="19px" altimg-valign="-4px" altimg-width="95px" alttext="\Re z\to+\infty" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>z</mi></mrow><mo>→</mo><mrow><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></mrow></math>. This is characteristic of numerically satisfactory
pairs. In a neighborhood, or sectorial neighborhood of a singularity, one
member has to be recessive. In consequence, if a differential equation has more
than one singularity in the extended plane, then usually more than two standard
solutions need to be chosen in order to have numerically satisfactory
representations everywhere.</p>
</div>
<div id="SS4.p3" class="ltx_para">
<p class="ltx_p">In oscillatory intervals, and again following <cite class="ltx_cite ltx_citemacro_citet">Miller ()</cite>, we call
a pair of solutions numerically satisfactory if asymptotically they have the
same amplitude and are <math class="ltx_Math" altimg="m91.png" altimg-height="27px" altimg-valign="-9px" altimg-width="29px" alttext="\tfrac{1}{2}\pi" display="inline"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow></math> out of phase.</p>
</div>
</section>
</section>
</div>
<div class="ltx_page_footer">
<span></div>
</div>
</body></text>
</html>
</page>
<page>
<!DOCTYPE html><html>
<head>
<title>DLMF: 2.5 Mellin Transform Methods</title>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<!--GOOGLE BOOTSTRAP--></head>
<text><body class="color_default textfont_default titlefont_default fontsize_default navbar_default">
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<div class="ltx_page_navlogo"></dd>
</dl>
</div>
</div>
<div id="SS1.p1" class="ltx_para">
<p class="ltx_p">Let <math class="ltx_Math" altimg="m127.png" altimg-height="23px" altimg-valign="-7px" altimg-width="39px" alttext="f(t)" display="inline"><mrow><mi href="./2.5#SS1.p1">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math> be a <em class="ltx_emph ltx_font_italic">locally integrable</em> function on <math class="ltx_Math" altimg="m14.png" altimg-height="23px" altimg-valign="-7px" altimg-width="59px" alttext="(0,\infty)" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mn>0</mn><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi mathvariant="normal">∞</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math>, that is,
<math class="ltx_Math" altimg="m67.png" altimg-height="34px" altimg-valign="-12px" altimg-width="86px" alttext="\int_{\rho}^{T}f(t)\mathrm{d}t" display="inline"><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi>ρ</mi><mi>T</mi></msubsup><mrow><mrow><mi href="./2.5#SS1.p1">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></math> exists for all <math class="ltx_Math" altimg="m102.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="\rho" display="inline"><mi>ρ</mi></math> and <math class="ltx_Math" altimg="m40.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="T" display="inline"><mi>T</mi></math> satisfying
<math class="ltx_Math" altimg="m24.png" altimg-height="21px" altimg-valign="-6px" altimg-width="139px" alttext="0<\rho<T<\infty" display="inline"><mrow><mn>0</mn><mo><</mo><mi>ρ</mi><mo><</mo><mi>T</mi><mo><</mo><mi mathvariant="normal">∞</mi></mrow></math>. The <em class="ltx_emph ltx_font_italic">Mellin transform</em> of <math class="ltx_Math" altimg="m127.png" altimg-height="23px" altimg-valign="-7px" altimg-width="39px" alttext="f(t)" display="inline"><mrow><mi href="./2.5#SS1.p1">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math> is defined by
</p>
<table id="E1" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">2.5.1</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E1.png" altimg-height="53px" altimg-valign="-20px" altimg-width="233px" alttext="\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(z\right)=\int_{0}^{\infty}t^{z-1}f%
(t)\mathrm{d}t," display="block"><mrow><mrow><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><mi href="./2.5#SS1.p1">f</mi></mpadded><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><mrow><msup><mi>t</mi><mrow><mi>z</mi><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mrow><mi href="./2.5#SS1.p1">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.14#E32" title="(1.14.32) ‣ §1.14(iv) Mellin Transform ‣ §1.14 Integral Transforms ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="23px" altimg-valign="-7px" altimg-width="88px" alttext="\mathscr{M}\left(\NVar{f}\right)\left(\NVar{s}\right)" display="inline"><mrow><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">f</mi><mo>)</mo></mrow><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">s</mi><mo>)</mo></mrow></mrow></math>: Mellin transform</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m161.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a> and
<a href="./2.5#SS1.p1" title="§2.5(i) Introduction ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m128.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="f(x)" display="inline"><mrow><mi href="./2.5#SS1.p1">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math>: locally integrable function</a>
</dd>
<dt>Notational Change (effective with 1.0.15):</dt>
<dd>
The notation for the Mellin transform was changed to <math class="ltx_Math" altimg="m82.png" altimg-height="23px" altimg-valign="-7px" altimg-width="70px" alttext="\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(z\right)" display="inline"><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><mi href="./2.5#SS1.p1">f</mi></mpadded><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></math> from <math class="ltx_Math" altimg="m77.png" altimg-height="23px" altimg-valign="-7px" altimg-width="75px" alttext="\mathscr{M}(f;z)" display="inline"><mrow><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./2.5#SS1.p1">f</mi><mo>;</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math>.
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">when this integral converges. The domain of analyticity of
<math class="ltx_Math" altimg="m82.png" altimg-height="23px" altimg-valign="-7px" altimg-width="70px" alttext="\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(z\right)" display="inline"><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><mi href="./2.5#SS1.p1">f</mi></mpadded><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></math> is usually an infinite strip <math class="ltx_Math" altimg="m107.png" altimg-height="18px" altimg-valign="-3px" altimg-width="101px" alttext="a<\Re z<b" display="inline"><mrow><mi href="./2.5#SS1.p1">a</mi><mo><</mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>z</mi></mrow><mo><</mo><mi href="./2.5#SS1.p1">b</mi></mrow></math>
parallel to the imaginary axis. The inversion formula is given by
</p>
<table id="E2" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">2.5.2</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E2.png" altimg-height="57px" altimg-valign="-22px" altimg-width="294px" alttext="f(t)=\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}t^{-z}\mathscr{M}\mskip-3.0mu%
f\mskip 3.0mu \left(z\right)\mathrm{d}z," display="block"><mrow><mrow><mrow><mi href="./2.5#SS1.p1">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mfrac><mn>1</mn><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi></mrow></mfrac><mo></mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><mi href="./2.5#SS1.p1">c</mi><mo>-</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi mathvariant="normal">∞</mi></mrow></mrow><mrow><mi href="./2.5#SS1.p1">c</mi><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi mathvariant="normal">∞</mi></mrow></mrow></msubsup><mrow><msup><mi>t</mi><mrow><mo>-</mo><mi>z</mi></mrow></msup><mo></mo><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><mi href="./2.5#SS1.p1">f</mi></mpadded><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>z</mi></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.14#E32" title="(1.14.32) ‣ §1.14(iv) Mellin Transform ‣ §1.14 Integral Transforms ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="23px" altimg-valign="-7px" altimg-width="88px" alttext="\mathscr{M}\left(\NVar{f}\right)\left(\NVar{s}\right)" display="inline"><mrow><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">f</mi><mo>)</mo></mrow><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">s</mi><mo>)</mo></mrow></mrow></math>: Mellin transform</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m98.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m161.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./2.5#SS1.p1" title="§2.5(i) Introduction ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m128.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="f(x)" display="inline"><mrow><mi href="./2.5#SS1.p1">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math>: locally integrable function</a> and
<a href="./2.5#SS1.p1" title="§2.5(i) Introduction ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m119.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="c" display="inline"><mi href="./2.5#SS1.p1">c</mi></math>: point</a>
</dd>
<dt>Notational Change (effective with 1.0.15):</dt>
<dd>
The notation for the Mellin transform was changed to <math class="ltx_Math" altimg="m82.png" altimg-height="23px" altimg-valign="-7px" altimg-width="70px" alttext="\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(z\right)" display="inline"><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><mi href="./2.5#SS1.p1">f</mi></mpadded><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></math> from <math class="ltx_Math" altimg="m77.png" altimg-height="23px" altimg-valign="-7px" altimg-width="75px" alttext="\mathscr{M}(f;z)" display="inline"><mrow><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./2.5#SS1.p1">f</mi><mo>;</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math>.
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">with <math class="ltx_Math" altimg="m108.png" altimg-height="18px" altimg-valign="-3px" altimg-width="85px" alttext="a<c<b" display="inline"><mrow><mi href="./2.5#SS1.p1">a</mi><mo><</mo><mi href="./2.5#SS1.p1">c</mi><mo><</mo><mi href="./2.5#SS1.p1">b</mi></mrow></math>.</p>
</div>
<div id="SS1.p2" class="ltx_para">
<p class="ltx_p">One of the two convolution integrals associated with the Mellin transform is
of the form
</p>
<table id="E3" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">2.5.3</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E3.png" altimg-height="53px" altimg-valign="-20px" altimg-width="217px" alttext="I(x)=\int_{0}^{\infty}f(t)\,h(xt)\mathrm{d}t," display="block"><mrow><mrow><mrow><mi href="./2.5#SS1.p2">I</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><mrow><mrow><mi href="./2.5#SS1.p1">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo rspace="4.2pt" stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mi href="./2.5#SS1.p2">h</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>x</mi><mo></mo><mi>t</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m160.png" altimg-height="17px" altimg-valign="-3px" altimg-width="52px" alttext="x>0" display="inline"><mrow><mi>x</mi><mo>></mo><mn>0</mn></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m161.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./2.5#SS1.p1" title="§2.5(i) Introduction ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m128.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="f(x)" display="inline"><mrow><mi href="./2.5#SS1.p1">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math>: locally integrable function</a>,
<a href="./2.5#SS1.p2" title="§2.5(i) Introduction ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m34.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="I(x)" display="inline"><mrow><mi href="./2.5#SS1.p2">I</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math>: convolution integral</a> and
<a href="./2.5#SS1.p2" title="§2.5(i) Introduction ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m136.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="h(x)" display="inline"><mrow><mi href="./2.5#SS1.p2">h</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and</p>
<table id="E4" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">2.5.4</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E4.png" altimg-height="25px" altimg-valign="-7px" altimg-width="274px" alttext="\mathscr{M}\mskip-3.0mu I\mskip 3.0mu \left(z\right)=\mathscr{M}\mskip-3.0mu f%
\mskip 3.0mu \left(1-z\right)\mathscr{M}\mskip-3.0mu h\mskip 3.0mu \left(z%
\right)." display="block"><mrow><mrow><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><mi href="./2.5#SS1.p2">I</mi></mpadded><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mrow><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><mi href="./2.5#SS1.p1">f</mi></mpadded><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mi>z</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo></mo><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><mi href="./2.5#SS1.p2">h</mi></mpadded><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E4.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.14#E32" title="(1.14.32) ‣ §1.14(iv) Mellin Transform ‣ §1.14 Integral Transforms ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="23px" altimg-valign="-7px" altimg-width="88px" alttext="\mathscr{M}\left(\NVar{f}\right)\left(\NVar{s}\right)" display="inline"><mrow><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">f</mi><mo>)</mo></mrow><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">s</mi><mo>)</mo></mrow></mrow></math>: Mellin transform</a>,
<a href="./2.5#SS1.p1" title="§2.5(i) Introduction ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m128.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="f(x)" display="inline"><mrow><mi href="./2.5#SS1.p1">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math>: locally integrable function</a>,
<a href="./2.5#SS1.p2" title="§2.5(i) Introduction ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m34.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="I(x)" display="inline"><mrow><mi href="./2.5#SS1.p2">I</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math>: convolution integral</a> and
<a href="./2.5#SS1.p2" title="§2.5(i) Introduction ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m136.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="h(x)" display="inline"><mrow><mi href="./2.5#SS1.p2">h</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function</a>
</dd>
<dt>Notational Change (effective with 1.0.15):</dt>
<dd>
The notation for the Mellin transform was changed to <math class="ltx_Math" altimg="m82.png" altimg-height="23px" altimg-valign="-7px" altimg-width="70px" alttext="\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(z\right)" display="inline"><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><mi href="./2.5#SS1.p1">f</mi></mpadded><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></math> from <math class="ltx_Math" altimg="m77.png" altimg-height="23px" altimg-valign="-7px" altimg-width="75px" alttext="\mathscr{M}(f;z)" display="inline"><mrow><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./2.5#SS1.p1">f</mi><mo>;</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math>.
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">If <math class="ltx_Math" altimg="m80.png" altimg-height="23px" altimg-valign="-7px" altimg-width="104px" alttext="\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(1-z\right)" display="inline"><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><mi href="./2.5#SS1.p1">f</mi></mpadded><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mi>z</mi></mrow><mo>)</mo></mrow></mrow></mrow></math> and <math class="ltx_Math" altimg="m91.png" altimg-height="23px" altimg-valign="-7px" altimg-width="69px" alttext="\mathscr{M}\mskip-3.0mu h\mskip 3.0mu \left(z\right)" display="inline"><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><mi href="./2.5#SS1.p2">h</mi></mpadded><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></math> have a common strip of
analyticity <math class="ltx_Math" altimg="m107.png" altimg-height="18px" altimg-valign="-3px" altimg-width="101px" alttext="a<\Re z<b" display="inline"><mrow><mi href="./2.5#SS1.p1">a</mi><mo><</mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>z</mi></mrow><mo><</mo><mi href="./2.5#SS1.p1">b</mi></mrow></math>, then</p>
<table id="E5" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">2.5.5</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E5.png" altimg-height="57px" altimg-valign="-22px" altimg-width="403px" alttext="I(x)=\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}x^{-z}\mathscr{M}\mskip-3.0mu%
f\mskip 3.0mu \left(1-z\right)\mathscr{M}\mskip-3.0mu h\mskip 3.0mu \left(z%
\right)\mathrm{d}z," display="block"><mrow><mrow><mrow><mi href="./2.5#SS1.p2">I</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mfrac><mn>1</mn><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi></mrow></mfrac><mo></mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><mi href="./2.5#SS1.p1">c</mi><mo>-</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi mathvariant="normal">∞</mi></mrow></mrow><mrow><mi href="./2.5#SS1.p1">c</mi><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi mathvariant="normal">∞</mi></mrow></mrow></msubsup><mrow><msup><mi>x</mi><mrow><mo>-</mo><mi>z</mi></mrow></msup><mo></mo><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><mi href="./2.5#SS1.p1">f</mi></mpadded><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mi>z</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo></mo><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><mi href="./2.5#SS1.p2">h</mi></mpadded><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>z</mi></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E5.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.14#E32" title="(1.14.32) ‣ §1.14(iv) Mellin Transform ‣ §1.14 Integral Transforms ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="23px" altimg-valign="-7px" altimg-width="88px" alttext="\mathscr{M}\left(\NVar{f}\right)\left(\NVar{s}\right)" display="inline"><mrow><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">f</mi><mo>)</mo></mrow><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">s</mi><mo>)</mo></mrow></mrow></math>: Mellin transform</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m98.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m161.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./2.5#SS1.p1" title="§2.5(i) Introduction ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m128.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="f(x)" display="inline"><mrow><mi href="./2.5#SS1.p1">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math>: locally integrable function</a>,
<a href="./2.5#SS1.p1" title="§2.5(i) Introduction ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m119.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="c" display="inline"><mi href="./2.5#SS1.p1">c</mi></math>: point</a>,
<a href="./2.5#SS1.p2" title="§2.5(i) Introduction ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m34.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="I(x)" display="inline"><mrow><mi href="./2.5#SS1.p2">I</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math>: convolution integral</a> and
<a href="./2.5#SS1.p2" title="§2.5(i) Introduction ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m136.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="h(x)" display="inline"><mrow><mi href="./2.5#SS1.p2">h</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function</a>
</dd>
<dt>Notational Change (effective with 1.0.15):</dt>
<dd>
The notation for the Mellin transform was changed to <math class="ltx_Math" altimg="m82.png" altimg-height="23px" altimg-valign="-7px" altimg-width="70px" alttext="\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(z\right)" display="inline"><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><mi href="./2.5#SS1.p1">f</mi></mpadded><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></math> from <math class="ltx_Math" altimg="m77.png" altimg-height="23px" altimg-valign="-7px" altimg-width="75px" alttext="\mathscr{M}(f;z)" display="inline"><mrow><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./2.5#SS1.p1">f</mi><mo>;</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math>.
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m108.png" altimg-height="18px" altimg-valign="-3px" altimg-width="85px" alttext="a<c<b" display="inline"><mrow><mi href="./2.5#SS1.p1">a</mi><mo><</mo><mi href="./2.5#SS1.p1">c</mi><mo><</mo><mi href="./2.5#SS1.p1">b</mi></mrow></math>. When <math class="ltx_Math" altimg="m159.png" altimg-height="17px" altimg-valign="-2px" altimg-width="52px" alttext="x=1" display="inline"><mrow><mi>x</mi><mo>=</mo><mn>1</mn></mrow></math>, this identity is a Parseval-type formula;
compare §.
</p>
</div>
<div id="SS1.p3" class="ltx_para">
<p class="ltx_p">If <math class="ltx_Math" altimg="m80.png" altimg-height="23px" altimg-valign="-7px" altimg-width="104px" alttext="\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(1-z\right)" display="inline"><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><mi href="./2.5#SS1.p1">f</mi></mpadded><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mi>z</mi></mrow><mo>)</mo></mrow></mrow></mrow></math> and <math class="ltx_Math" altimg="m91.png" altimg-height="23px" altimg-valign="-7px" altimg-width="69px" alttext="\mathscr{M}\mskip-3.0mu h\mskip 3.0mu \left(z\right)" display="inline"><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><mi href="./2.5#SS1.p2">h</mi></mpadded><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></math> can be continued
analytically to meromorphic functions in a left half-plane, and if the contour
<math class="ltx_Math" altimg="m45.png" altimg-height="18px" altimg-valign="-2px" altimg-width="64px" alttext="\Re z=c" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>z</mi></mrow><mo>=</mo><mi href="./2.5#SS1.p1">c</mi></mrow></math> can be translated to <math class="ltx_Math" altimg="m46.png" altimg-height="18px" altimg-valign="-2px" altimg-width="66px" alttext="\Re z=d" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>z</mi></mrow><mo>=</mo><mi href="./2.5#SS1.p3">d</mi></mrow></math> with <math class="ltx_Math" altimg="m122.png" altimg-height="18px" altimg-valign="-3px" altimg-width="50px" alttext="d<c" display="inline"><mrow><mi href="./2.5#SS1.p3">d</mi><mo><</mo><mi href="./2.5#SS1.p1">c</mi></mrow></math>, then</p>
<table id="E6" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">2.5.6</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E6.png" altimg-height="53px" altimg-valign="-29px" altimg-width="452px" alttext="I(x)=\sum\limits_{d<\Re z<c}\Residue\left[x^{-z}\mathscr{M}\mskip-3.0mu f%
\mskip 3.0mu \left(1-z\right)\mathscr{M}\mskip-3.0mu h\mskip 3.0mu \left(z%
\right)\right]+E(x)," display="block"><mrow><mrow><mrow><mi href="./2.5#SS1.p2">I</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mrow><munder><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./2.5#SS1.p3">d</mi><mo><</mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>z</mi></mrow><mo><</mo><mi href="./2.5#SS1.p1">c</mi></mrow></munder><mrow><mo href="./1.10#SS3.p5">res</mo><mo></mo><mrow><mo>[</mo><mrow><msup><mi>x</mi><mrow><mo>-</mo><mi>z</mi></mrow></msup><mo></mo><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><mi href="./2.5#SS1.p1">f</mi></mpadded><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mi>z</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo></mo><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><mi href="./2.5#SS1.p2">h</mi></mpadded><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>]</mo></mrow></mrow></mrow><mo>+</mo><mrow><mi href="./2.5#SS1.p3">E</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E6.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.14#E32" title="(1.14.32) ‣ §1.14(iv) Mellin Transform ‣ §1.14 Integral Transforms ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="23px" altimg-valign="-7px" altimg-width="88px" alttext="\mathscr{M}\left(\NVar{f}\right)\left(\NVar{s}\right)" display="inline"><mrow><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">f</mi><mo>)</mo></mrow><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">s</mi><mo>)</mo></mrow></mrow></math>: Mellin transform</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m55.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./1.10#SS3.p5" title="§1.10(iii) Laurent Series ‣ §1.10 Functions of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m57.png" altimg-height="13px" altimg-valign="-2px" altimg-width="29px" alttext="\Residue" display="inline"><mo href="./1.10#SS3.p5">res</mo></math>: residue</a>,
<a href="./2.5#SS1.p1" title="§2.5(i) Introduction ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m128.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="f(x)" display="inline"><mrow><mi href="./2.5#SS1.p1">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math>: locally integrable function</a>,
<a href="./2.5#SS1.p1" title="§2.5(i) Introduction ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m119.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="c" display="inline"><mi href="./2.5#SS1.p1">c</mi></math>: point</a>,
<a href="./2.5#SS1.p2" title="§2.5(i) Introduction ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m34.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="I(x)" display="inline"><mrow><mi href="./2.5#SS1.p2">I</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math>: convolution integral</a>,
<a href="./2.5#SS1.p2" title="§2.5(i) Introduction ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m136.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="h(x)" display="inline"><mrow><mi href="./2.5#SS1.p2">h</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function</a>,
<a href="./2.5#SS1.p3" title="§2.5(i) Introduction ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m124.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="d" display="inline"><mi href="./2.5#SS1.p3">d</mi></math>: point</a> and
<a href="./2.5#SS1.p3" title="§2.5(i) Introduction ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m30.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="E(x)" display="inline"><mrow><mi href="./2.5#SS1.p3">E</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function</a>
</dd>
<dt>Notational Change (effective with 1.0.15):</dt>
<dd>
The notation for the Mellin transform was changed to <math class="ltx_Math" altimg="m82.png" altimg-height="23px" altimg-valign="-7px" altimg-width="70px" alttext="\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(z\right)" display="inline"><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><mi href="./2.5#SS1.p1">f</mi></mpadded><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></math> from <math class="ltx_Math" altimg="m77.png" altimg-height="23px" altimg-valign="-7px" altimg-width="75px" alttext="\mathscr{M}(f;z)" display="inline"><mrow><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./2.5#SS1.p1">f</mi><mo>;</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math>.
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where
</p>
<table id="E7" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">2.5.7</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E7.png" altimg-height="57px" altimg-valign="-22px" altimg-width="410px" alttext="E(x)=\frac{1}{2\pi i}\int_{d-i\infty}^{d+i\infty}x^{-z}\mathscr{M}\mskip-3.0mu%
f\mskip 3.0mu \left(1-z\right)\mathscr{M}\mskip-3.0mu h\mskip 3.0mu \left(z%
\right)\mathrm{d}z." display="block"><mrow><mrow><mrow><mi href="./2.5#SS1.p3">E</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mfrac><mn>1</mn><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi></mrow></mfrac><mo></mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><mi href="./2.5#SS1.p3">d</mi><mo>-</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi mathvariant="normal">∞</mi></mrow></mrow><mrow><mi href="./2.5#SS1.p3">d</mi><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi mathvariant="normal">∞</mi></mrow></mrow></msubsup><mrow><msup><mi>x</mi><mrow><mo>-</mo><mi>z</mi></mrow></msup><mo></mo><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><mi href="./2.5#SS1.p1">f</mi></mpadded><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mi>z</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo></mo><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><mi href="./2.5#SS1.p2">h</mi></mpadded><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>z</mi></mrow></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E7.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.14#E32" title="(1.14.32) ‣ §1.14(iv) Mellin Transform ‣ §1.14 Integral Transforms ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="23px" altimg-valign="-7px" altimg-width="88px" alttext="\mathscr{M}\left(\NVar{f}\right)\left(\NVar{s}\right)" display="inline"><mrow><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">f</mi><mo>)</mo></mrow><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">s</mi><mo>)</mo></mrow></mrow></math>: Mellin transform</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m98.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m161.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./2.5#SS1.p1" title="§2.5(i) Introduction ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m128.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="f(x)" display="inline"><mrow><mi href="./2.5#SS1.p1">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math>: locally integrable function</a>,
<a href="./2.5#SS1.p2" title="§2.5(i) Introduction ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m136.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="h(x)" display="inline"><mrow><mi href="./2.5#SS1.p2">h</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function</a>,
<a href="./2.5#SS1.p3" title="§2.5(i) Introduction ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m124.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="d" display="inline"><mi href="./2.5#SS1.p3">d</mi></math>: point</a> and
<a href="./2.5#SS1.p3" title="§2.5(i) Introduction ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m30.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="E(x)" display="inline"><mrow><mi href="./2.5#SS1.p3">E</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function</a>
</dd>
<dt>Notational Change (effective with 1.0.15):</dt>
<dd>
The notation for the Mellin transform was changed to <math class="ltx_Math" altimg="m82.png" altimg-height="23px" altimg-valign="-7px" altimg-width="70px" alttext="\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(z\right)" display="inline"><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><mi href="./2.5#SS1.p1">f</mi></mpadded><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></math> from <math class="ltx_Math" altimg="m77.png" altimg-height="23px" altimg-valign="-7px" altimg-width="75px" alttext="\mathscr{M}(f;z)" display="inline"><mrow><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./2.5#SS1.p1">f</mi><mo>;</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math>.
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">The sum in () is taken over all poles of
<math class="ltx_Math" altimg="m165.png" altimg-height="24px" altimg-valign="-7px" altimg-width="206px" alttext="x^{-z}\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(1-z\right)\mathscr{M}\mskip-%
3.0mu h\mskip 3.0mu \left(z\right)" display="inline"><mrow><msup><mi>x</mi><mrow><mo>-</mo><mi>z</mi></mrow></msup><mo></mo><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><mi href="./2.5#SS1.p1">f</mi></mpadded><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mi>z</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo></mo><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><mi href="./2.5#SS1.p2">h</mi></mpadded><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></mrow></math> in the strip
<math class="ltx_Math" altimg="m121.png" altimg-height="18px" altimg-valign="-3px" altimg-width="101px" alttext="d<\Re z<c" display="inline"><mrow><mi href="./2.5#SS1.p3">d</mi><mo><</mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>z</mi></mrow><mo><</mo><mi href="./2.5#SS1.p1">c</mi></mrow></math>, and it provides the asymptotic expansion of <math class="ltx_Math" altimg="m34.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="I(x)" display="inline"><mrow><mi href="./2.5#SS1.p2">I</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math> for
small values of <math class="ltx_Math" altimg="m161.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math>. Similarly, if <math class="ltx_Math" altimg="m80.png" altimg-height="23px" altimg-valign="-7px" altimg-width="104px" alttext="\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(1-z\right)" display="inline"><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><mi href="./2.5#SS1.p1">f</mi></mpadded><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mi>z</mi></mrow><mo>)</mo></mrow></mrow></mrow></math> and
<math class="ltx_Math" altimg="m91.png" altimg-height="23px" altimg-valign="-7px" altimg-width="69px" alttext="\mathscr{M}\mskip-3.0mu h\mskip 3.0mu \left(z\right)" display="inline"><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><mi href="./2.5#SS1.p2">h</mi></mpadded><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></math> can be continued analytically to meromorphic functions in
a right half-plane, and if the vertical line of integration can be translated
to the right, then we obtain an asymptotic expansion for <math class="ltx_Math" altimg="m34.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="I(x)" display="inline"><mrow><mi href="./2.5#SS1.p2">I</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math> for large
values of <math class="ltx_Math" altimg="m161.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math>.
</p>
</div>
<section id="Px1" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Example</h3>
<div id="Px1.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px1.p1" class="ltx_para">
<table id="E8" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">2.5.8</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E8.png" altimg-height="53px" altimg-valign="-20px" altimg-width="198px" alttext="I(x)=\int_{0}^{\infty}\frac{{J_{\nu}^{2}}\left(xt\right)}{1+t}\mathrm{d}t," display="block"><mrow><mrow><mrow><mi href="./2.5#SS1.p2">I</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><mrow><mfrac><mrow><msubsup><mi href="./10.2#E2">J</mi><mi>ν</mi><mn>2</mn></msubsup><mo></mo><mrow><mo>(</mo><mrow><mi>x</mi><mo></mo><mi>t</mi></mrow><mo>)</mo></mrow></mrow><mrow><mn>1</mn><mo>+</mo><mi>t</mi></mrow></mfrac><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m97.png" altimg-height="27px" altimg-valign="-9px" altimg-width="70px" alttext="\nu>-\tfrac{1}{2}" display="inline"><mrow><mi>ν</mi><mo>></mo><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E8.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m36.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m161.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a> and
<a href="./2.5#SS1.p2" title="§2.5(i) Introduction ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m34.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="I(x)" display="inline"><mrow><mi href="./2.5#SS1.p2">I</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math>: convolution integral</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m37.png" altimg-height="21px" altimg-valign="-5px" altimg-width="25px" alttext="J_{\nu}" display="inline"><msub><mi href="./10.2#E2">J</mi><mi>ν</mi></msub></math> denotes the Bessel function (§), and
<math class="ltx_Math" altimg="m161.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math> is a large positive parameter. Let <math class="ltx_Math" altimg="m134.png" altimg-height="25px" altimg-valign="-7px" altimg-width="113px" alttext="h(t)={J_{\nu}^{2}}\left(t\right)" display="inline"><mrow><mrow><mi href="./2.5#Px1.p1">h</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><msubsup><mi href="./10.2#E2">J</mi><mi>ν</mi><mn>2</mn></msubsup><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></mrow></math> and
<math class="ltx_Math" altimg="m126.png" altimg-height="23px" altimg-valign="-7px" altimg-width="142px" alttext="f(t)=1/(1+t)" display="inline"><mrow><mrow><mi href="./2.5#Px1.p1">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mn>1</mn><mo>/</mo><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>+</mo><mi>t</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></math>. Then from Table , p. 403)</cite></p>
<table id="E9" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">2.5.9</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E9.png" altimg-height="46px" altimg-valign="-21px" altimg-width="208px" alttext="\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(1-z\right)=\frac{\pi}{\sin\left(%
\pi z\right)}," display="block"><mrow><mrow><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><mi href="./2.5#Px1.p1">f</mi></mpadded><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mi>z</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mfrac><mi href="./3.12#E1">π</mi><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi>z</mi></mrow><mo>)</mo></mrow></mrow></mfrac></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m23.png" altimg-height="18px" altimg-valign="-3px" altimg-width="102px" alttext="0<\Re z<1" display="inline"><mrow><mn>0</mn><mo><</mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>z</mi></mrow><mo><</mo><mn>1</mn></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E9.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.14#E32" title="(1.14.32) ‣ §1.14(iv) Mellin Transform ‣ §1.14 Integral Transforms ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="23px" altimg-valign="-7px" altimg-width="88px" alttext="\mathscr{M}\left(\NVar{f}\right)\left(\NVar{s}\right)" display="inline"><mrow><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">f</mi><mo>)</mo></mrow><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">s</mi><mo>)</mo></mrow></mrow></math>: Mellin transform</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m98.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m55.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m104.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a> and
<a href="./2.5#Px1.p1" title="Example ‣ §2.5(i) Introduction ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m126.png" altimg-height="23px" altimg-valign="-7px" altimg-width="142px" alttext="f(t)=1/(1+t)" display="inline"><mrow><mrow><mi href="./2.5#Px1.p1">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mn>1</mn><mo>/</mo><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>+</mo><mi>t</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></math>: function</a>
</dd>
<dt>Notational Change (effective with 1.0.15):</dt>
<dd>
The notation for the Mellin transform was changed to <math class="ltx_Math" altimg="m82.png" altimg-height="23px" altimg-valign="-7px" altimg-width="70px" alttext="\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(z\right)" display="inline"><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><mi href="./2.5#Px1.p1">f</mi></mpadded><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></math> from <math class="ltx_Math" altimg="m77.png" altimg-height="23px" altimg-valign="-7px" altimg-width="75px" alttext="\mathscr{M}(f;z)" display="inline"><mrow><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./2.5#Px1.p1">f</mi><mo>;</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math>.
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E10" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">2.5.10</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E10.png" altimg-height="59px" altimg-valign="-24px" altimg-width="453px" alttext="\mathscr{M}\mskip-3.0mu h\mskip 3.0mu \left(z\right)=\frac{2^{z-1}\Gamma\left(%
\nu+\frac{1}{2}z\right)}{{\Gamma^{2}}\left(1-\frac{1}{2}z\right)\Gamma\left(1+%
\nu-\frac{1}{2}z\right)\Gamma\left(z\right)}\frac{\pi}{\sin\left(\pi z\right)}," display="block"><mrow><mrow><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><mi href="./2.5#Px1.p1">h</mi></mpadded><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mrow><mfrac><mrow><msup><mn>2</mn><mrow><mi>z</mi><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi>ν</mi><mo>+</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi>z</mi></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mrow><mrow><msup><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mn>2</mn></msup><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi>z</mi></mrow></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mn>1</mn><mo>+</mo><mi>ν</mi></mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi>z</mi></mrow></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></mfrac><mo></mo><mfrac><mi href="./3.12#E1">π</mi><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi>z</mi></mrow><mo>)</mo></mrow></mrow></mfrac></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m16.png" altimg-height="19px" altimg-valign="-4px" altimg-width="128px" alttext="-2\nu<\Re z<1" display="inline"><mrow><mrow><mo>-</mo><mrow><mn>2</mn><mo></mo><mi>ν</mi></mrow></mrow><mo><</mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>z</mi></mrow><mo><</mo><mn>1</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E10.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m41.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./1.14#E32" title="(1.14.32) ‣ §1.14(iv) Mellin Transform ‣ §1.14 Integral Transforms ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="23px" altimg-valign="-7px" altimg-width="88px" alttext="\mathscr{M}\left(\NVar{f}\right)\left(\NVar{s}\right)" display="inline"><mrow><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">f</mi><mo>)</mo></mrow><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">s</mi><mo>)</mo></mrow></mrow></math>: Mellin transform</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m98.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m55.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m104.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./2.5#Px1.p1" title="Example ‣ §2.5(i) Introduction ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m134.png" altimg-height="25px" altimg-valign="-7px" altimg-width="113px" alttext="h(t)={J_{\nu}^{2}}\left(t\right)" display="inline"><mrow><mrow><mi href="./2.5#Px1.p1">h</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><msubsup><mi href="./10.2#E2">J</mi><mi>ν</mi><mn>2</mn></msubsup><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></mrow></math>: function</a> and
<a href="./2.5#Px1.p1" title="Example ‣ §2.5(i) Introduction ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m126.png" altimg-height="23px" altimg-valign="-7px" altimg-width="142px" alttext="f(t)=1/(1+t)" display="inline"><mrow><mrow><mi href="./2.5#Px1.p1">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mn>1</mn><mo>/</mo><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>+</mo><mi>t</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></math>: function</a>
</dd>
<dt>Notational Change (effective with 1.0.15):</dt>
<dd>
The notation for the Mellin transform was changed to <math class="ltx_Math" altimg="m82.png" altimg-height="23px" altimg-valign="-7px" altimg-width="70px" alttext="\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(z\right)" display="inline"><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><mi href="./2.5#Px1.p1">f</mi></mpadded><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></math> from <math class="ltx_Math" altimg="m77.png" altimg-height="23px" altimg-valign="-7px" altimg-width="75px" alttext="\mathscr{M}(f;z)" display="inline"><mrow><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./2.5#Px1.p1">f</mi><mo>;</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math>.
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">In the half-plane <math class="ltx_Math" altimg="m49.png" altimg-height="23px" altimg-valign="-7px" altimg-width="163px" alttext="\Re z>\max(0,-2\nu)" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>z</mi></mrow><mo>></mo><mrow><mi>max</mi><mo></mo><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mrow><mo>-</mo><mrow><mn>2</mn><mo></mo><mi>ν</mi></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></math>, the product
<math class="ltx_Math" altimg="m81.png" altimg-height="23px" altimg-valign="-7px" altimg-width="173px" alttext="\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(1-z\right)\mathscr{M}\mskip-3.0mu %
h\mskip 3.0mu \left(z\right)" display="inline"><mrow><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><mi href="./2.5#Px1.p1">f</mi></mpadded><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mi>z</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo></mo><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><mi href="./2.5#Px1.p1">h</mi></mpadded><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></mrow></math> has a pole of order two at each
positive integer, and</p>
<table id="E11" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">2.5.11</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E11.png" altimg-height="36px" altimg-valign="-16px" altimg-width="444px" alttext="\Residue_{z=n}\left[x^{-z}\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(1-z%
\right)\mathscr{M}\mskip-3.0mu h\mskip 3.0mu \left(z\right)\right]=(a_{n}\ln x%
+b_{n})x^{-n}," display="block"><mrow><mrow><mrow><msub><mo href="./1.10#SS3.p5">res</mo><mrow><mi>z</mi><mo>=</mo><mi>n</mi></mrow></msub><mo></mo><mrow><mo>[</mo><mrow><msup><mi>x</mi><mrow><mo>-</mo><mi>z</mi></mrow></msup><mo></mo><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><mi href="./2.5#Px1.p1">f</mi></mpadded><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mi>z</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo></mo><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><mi href="./2.5#Px1.p1">h</mi></mpadded><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>]</mo></mrow></mrow><mo>=</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><msub><mi href="./2.5#EGx1">a</mi><mi>n</mi></msub><mo></mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi>x</mi></mrow></mrow><mo>+</mo><msub><mi href="./2.5#EGx1">b</mi><mi>n</mi></msub></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msup><mi>x</mi><mrow><mo>-</mo><mi>n</mi></mrow></msup></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E11.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.14#E32" title="(1.14.32) ‣ §1.14(iv) Mellin Transform ‣ §1.14 Integral Transforms ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="23px" altimg-valign="-7px" altimg-width="88px" alttext="\mathscr{M}\left(\NVar{f}\right)\left(\NVar{s}\right)" display="inline"><mrow><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">f</mi><mo>)</mo></mrow><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">s</mi><mo>)</mo></mrow></mrow></math>: Mellin transform</a>,
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m71.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a>,
<a href="./1.10#SS3.p5" title="§1.10(iii) Laurent Series ‣ §1.10 Functions of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m57.png" altimg-height="13px" altimg-valign="-2px" altimg-width="29px" alttext="\Residue" display="inline"><mo href="./1.10#SS3.p5">res</mo></math>: residue</a>,
<a href="./2.5#Px1.p1" title="Example ‣ §2.5(i) Introduction ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m134.png" altimg-height="25px" altimg-valign="-7px" altimg-width="113px" alttext="h(t)={J_{\nu}^{2}}\left(t\right)" display="inline"><mrow><mrow><mi href="./2.5#Px1.p1">h</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><msubsup><mi href="./10.2#E2">J</mi><mi>ν</mi><mn>2</mn></msubsup><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></mrow></math>: function</a>,
<a href="./2.5#EGx1" title="Example ‣ §2.5(i) Introduction ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m110.png" altimg-height="16px" altimg-valign="-5px" altimg-width="26px" alttext="a_{n}" display="inline"><msub><mi href="./2.5#EGx1">a</mi><mi>n</mi></msub></math>: coefficients</a>,
<a href="./2.5#EGx1" title="Example ‣ §2.5(i) Introduction ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m115.png" altimg-height="21px" altimg-valign="-5px" altimg-width="24px" alttext="b_{n}" display="inline"><msub><mi href="./2.5#EGx1">b</mi><mi>n</mi></msub></math>: coefficients</a> and
<a href="./2.5#Px1.p1" title="Example ‣ §2.5(i) Introduction ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m126.png" altimg-height="23px" altimg-valign="-7px" altimg-width="142px" alttext="f(t)=1/(1+t)" display="inline"><mrow><mrow><mi href="./2.5#Px1.p1">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mn>1</mn><mo>/</mo><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>+</mo><mi>t</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></math>: function</a>
</dd>
<dt>Notational Change (effective with 1.0.15):</dt>
<dd>
The notation for the Mellin transform was changed to <math class="ltx_Math" altimg="m82.png" altimg-height="23px" altimg-valign="-7px" altimg-width="70px" alttext="\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(z\right)" display="inline"><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><mi href="./2.5#Px1.p1">f</mi></mpadded><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></math> from <math class="ltx_Math" altimg="m77.png" altimg-height="23px" altimg-valign="-7px" altimg-width="75px" alttext="\mathscr{M}(f;z)" display="inline"><mrow><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./2.5#Px1.p1">f</mi><mo>;</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math>.
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where</p>
<table id="EGx1" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E12">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">2.5.12</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m7.png" altimg-height="17px" altimg-valign="-5px" altimg-width="27px" alttext="\displaystyle a_{n}" display="inline"><msub><mi href="./2.5#EGx1">a</mi><mi>n</mi></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m4.png" altimg-height="59px" altimg-valign="-24px" altimg-width="318px" alttext="\displaystyle=\frac{2^{n-1}\Gamma\left(\nu+\tfrac{1}{2}n\right)}{{\Gamma^{2}}%
\left(1-\tfrac{1}{2}n\right)\Gamma\left(1+\nu-\tfrac{1}{2}n\right)\Gamma\left(%
n\right)}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mstyle displaystyle="true"><mfrac><mrow><msup><mn>2</mn><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi>ν</mi><mo>+</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi>n</mi></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mrow><mrow><msup><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mn>2</mn></msup><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi>n</mi></mrow></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mn>1</mn><mo>+</mo><mi>ν</mi></mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi>n</mi></mrow></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></mrow></mfrac></mstyle></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E12.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m41.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a> and
<a href="./2.5#EGx1" title="Example ‣ §2.5(i) Introduction ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m110.png" altimg-height="16px" altimg-valign="-5px" altimg-width="26px" alttext="a_{n}" display="inline"><msub><mi href="./2.5#EGx1">a</mi><mi>n</mi></msub></math>: coefficients</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E13">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">2.5.13</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m8.png" altimg-height="22px" altimg-valign="-5px" altimg-width="25px" alttext="\displaystyle b_{n}" display="inline"><msub><mi href="./2.5#EGx1">b</mi><mi>n</mi></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m1.png" altimg-height="29px" altimg-valign="-9px" altimg-width="611px" alttext="\displaystyle=-a_{n}\left(\ln 2+\tfrac{1}{2}\psi\left(\nu+\tfrac{1}{2}n\right)%
+\psi\left(1-\tfrac{1}{2}n\right)+\tfrac{1}{2}\psi\left(1+\nu-\tfrac{1}{2}n%
\right)-\psi\left(n\right)\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo>-</mo><mrow><msub><mi href="./2.5#EGx1">a</mi><mi>n</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mrow><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mn>2</mn></mrow><mo>+</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mrow><mi>ν</mi><mo>+</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi>n</mi></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi>n</mi></mrow></mrow><mo>)</mo></mrow></mrow><mo>+</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mn>1</mn><mo>+</mo><mi>ν</mi></mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi>n</mi></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>-</mo><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E13.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E2" title="(5.2.2) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m100.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="\psi\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: psi (or digamma) function</a>,
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m71.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a>,
<a href="./2.5#EGx1" title="Example ‣ §2.5(i) Introduction ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m110.png" altimg-height="16px" altimg-valign="-5px" altimg-width="26px" alttext="a_{n}" display="inline"><msub><mi href="./2.5#EGx1">a</mi><mi>n</mi></msub></math>: coefficients</a> and
<a href="./2.5#EGx1" title="Example ‣ §2.5(i) Introduction ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m115.png" altimg-height="21px" altimg-valign="-5px" altimg-width="24px" alttext="b_{n}" display="inline"><msub><mi href="./2.5#EGx1">b</mi><mi>n</mi></msub></math>: coefficients</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
<p class="ltx_p">and <math class="ltx_Math" altimg="m99.png" altimg-height="21px" altimg-valign="-6px" altimg-width="18px" alttext="\psi" display="inline"><mi href="./5.2#E2">ψ</mi></math> is the logarithmic derivative of the gamma function
(§) with <math class="ltx_Math" altimg="m96.png" altimg-height="23px" altimg-valign="-7px" altimg-width="184px" alttext="\max(0,-2\nu)<c<1" display="inline"><mrow><mrow><mi>max</mi><mo></mo><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mrow><mo>-</mo><mrow><mn>2</mn><mo></mo><mi>ν</mi></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow><mo><</mo><mi href="./2.5#SS1.p1">c</mi><mo><</mo><mn>1</mn></mrow></math>, and then
translate the integration contour to the right. This is allowable in view of
the asymptotic formula</p>
<table id="E14" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">2.5.14</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E14.png" altimg-height="29px" altimg-valign="-7px" altimg-width="418px" alttext="|\Gamma\left(x+iy\right)|=\sqrt{2\pi}e^{-\pi|y|/2}|y|^{x-(1/2)}\left(1+o\left(%
1\right)\right)," display="block"><mrow><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi>y</mi></mrow></mrow><mo>)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow><mo>=</mo><mrow><msqrt><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi></mrow></msqrt><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mrow><mi href="./3.12#E1">π</mi><mo></mo><mrow><mo stretchy="false">|</mo><mi>y</mi><mo stretchy="false">|</mo></mrow></mrow><mo>/</mo><mn>2</mn></mrow></mrow></msup><mo></mo><msup><mrow><mo stretchy="false">|</mo><mi>y</mi><mo stretchy="false">|</mo></mrow><mrow><mi>x</mi><mo>-</mo><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow><mo stretchy="false">)</mo></mrow></mrow></msup><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mrow><mi href="./2.1#E2">o</mi><mo></mo><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E14.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m41.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m98.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m73.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a> and
<a href="./2.1#E2" title="(2.1.2) ‣ §2.1(i) Asymptotic and Order Symbols ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m142.png" altimg-height="23px" altimg-valign="-7px" altimg-width="44px" alttext="o\left(\NVar{x}\right)" display="inline"><mrow><mi href="./2.1#E2">o</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>)</mo></mrow></mrow></math>: order less than</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">as <math class="ltx_Math" altimg="m166.png" altimg-height="19px" altimg-valign="-6px" altimg-width="81px" alttext="y\to\pm\infty" display="inline"><mrow><mi>y</mi><mo>→</mo><mrow><mo>±</mo><mi mathvariant="normal">∞</mi></mrow></mrow></math>, uniformly for bounded <math class="ltx_Math" altimg="m168.png" altimg-height="23px" altimg-valign="-7px" altimg-width="27px" alttext="|x|" display="inline"><mrow><mo stretchy="false">|</mo><mi>x</mi><mo stretchy="false">|</mo></mrow></math>; see (), with
<math class="ltx_Math" altimg="m123.png" altimg-height="19px" altimg-valign="-4px" altimg-width="130px" alttext="d=2n+1-\epsilon" display="inline"><mrow><mi href="./2.5#SS1.p3">d</mi><mo>=</mo><mrow><mrow><mrow><mn>2</mn><mo></mo><mi>n</mi></mrow><mo>+</mo><mn>1</mn></mrow><mo>-</mo><mi href="./2.5#Px1.p2">ϵ</mi></mrow></mrow></math> <math class="ltx_Math" altimg="m15.png" altimg-height="23px" altimg-valign="-7px" altimg-width="101px" alttext="(0<\epsilon<1)" display="inline"><mrow><mo stretchy="false">(</mo><mrow><mn>0</mn><mo><</mo><mi href="./2.5#Px1.p2">ϵ</mi><mo><</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></math>, we obtain
</p>
<table id="E15" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">2.5.15</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E15.png" altimg-height="67px" altimg-valign="-27px" altimg-width="413px" alttext="I(x)=-\sum_{s=0}^{2n}(a_{s}\ln x+b_{s})x^{-s}+O\left(x^{-2n-1+\epsilon}\right)," display="block"><mrow><mrow><mrow><mi href="./2.5#SS1.p2">I</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mo>-</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi>s</mi><mo>=</mo><mn>0</mn></mrow><mrow><mn>2</mn><mo></mo><mi>n</mi></mrow></munderover><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><msub><mi href="./2.5#EGx1">a</mi><mi>s</mi></msub><mo></mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi>x</mi></mrow></mrow><mo>+</mo><msub><mi href="./2.5#EGx1">b</mi><mi>s</mi></msub></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msup><mi>x</mi><mrow><mo>-</mo><mi>s</mi></mrow></msup></mrow></mrow></mrow><mo>+</mo><mrow><mi href="./2.1#E3">O</mi><mo></mo><mrow><mo>(</mo><msup><mi>x</mi><mrow><mrow><mrow><mo>-</mo><mrow><mn>2</mn><mo></mo><mi>n</mi></mrow></mrow><mo>-</mo><mn>1</mn></mrow><mo>+</mo><mi href="./2.5#Px1.p2">ϵ</mi></mrow></msup><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m141.png" altimg-height="20px" altimg-valign="-6px" altimg-width="123px" alttext="n=0,1,2,\dots" display="inline"><mrow><mi>n</mi><mo>=</mo><mrow><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E15.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./2.1#E3" title="(2.1.3) ‣ §2.1(i) Asymptotic and Order Symbols ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="23px" altimg-valign="-7px" altimg-width="50px" alttext="O\left(\NVar{x}\right)" display="inline"><mrow><mi href="./2.1#E3">O</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>)</mo></mrow></mrow></math>: order not exceeding</a>,
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m71.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a>,
<a href="./2.5#EGx1" title="Example ‣ §2.5(i) Introduction ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m110.png" altimg-height="16px" altimg-valign="-5px" altimg-width="26px" alttext="a_{n}" display="inline"><msub><mi href="./2.5#EGx1">a</mi><mi>n</mi></msub></math>: coefficients</a>,
<a href="./2.5#EGx1" title="Example ‣ §2.5(i) Introduction ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m115.png" altimg-height="21px" altimg-valign="-5px" altimg-width="24px" alttext="b_{n}" display="inline"><msub><mi href="./2.5#EGx1">b</mi><mi>n</mi></msub></math>: coefficients</a>,
<a href="./2.5#Px1.p2" title="Example ‣ §2.5(i) Introduction ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m62.png" altimg-height="13px" altimg-valign="-2px" altimg-width="12px" alttext="\epsilon" display="inline"><mi href="./2.5#Px1.p2">ϵ</mi></math>: parameter</a> and
<a href="./2.5#SS1.p2" title="§2.5(i) Introduction ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m34.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="I(x)" display="inline"><mrow><mi href="./2.5#SS1.p2">I</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math>: convolution integral</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">From (), it is seen that
<math class="ltx_Math" altimg="m111.png" altimg-height="21px" altimg-valign="-5px" altimg-width="104px" alttext="a_{s}=b_{s}=0" display="inline"><mrow><msub><mi href="./2.5#EGx1">a</mi><mi>s</mi></msub><mo>=</mo><msub><mi href="./2.5#EGx1">b</mi><mi>s</mi></msub><mo>=</mo><mn>0</mn></mrow></math> when <math class="ltx_Math" altimg="m153.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="s" display="inline"><mi>s</mi></math> is even. Hence</p>
<table id="E16" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">2.5.16</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E16.png" altimg-height="67px" altimg-valign="-27px" altimg-width="424px" alttext="I(x)=\sum_{s=0}^{n-1}(c_{s}\ln x+d_{s})x^{-2s-1}+O\left(x^{-2n-1+\epsilon}%
\right)," display="block"><mrow><mrow><mrow><mi href="./2.5#SS1.p2">I</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi>s</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></munderover><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><msub><mi href="./2.5#SS1.p1">c</mi><mi>s</mi></msub><mo></mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi>x</mi></mrow></mrow><mo>+</mo><msub><mi href="./2.5#SS1.p3">d</mi><mi>s</mi></msub></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msup><mi>x</mi><mrow><mrow><mo>-</mo><mrow><mn>2</mn><mo></mo><mi>s</mi></mrow></mrow><mo>-</mo><mn>1</mn></mrow></msup></mrow></mrow><mo>+</mo><mrow><mi href="./2.1#E3">O</mi><mo></mo><mrow><mo>(</mo><msup><mi>x</mi><mrow><mrow><mrow><mo>-</mo><mrow><mn>2</mn><mo></mo><mi>n</mi></mrow></mrow><mo>-</mo><mn>1</mn></mrow><mo>+</mo><mi href="./2.5#Px1.p2">ϵ</mi></mrow></msup><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E16.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./2.1#E3" title="(2.1.3) ‣ §2.1(i) Asymptotic and Order Symbols ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="23px" altimg-valign="-7px" altimg-width="50px" alttext="O\left(\NVar{x}\right)" display="inline"><mrow><mi href="./2.1#E3">O</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>)</mo></mrow></mrow></math>: order not exceeding</a>,
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m71.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a>,
<a href="./2.5#Px1.p2" title="Example ‣ §2.5(i) Introduction ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m62.png" altimg-height="13px" altimg-valign="-2px" altimg-width="12px" alttext="\epsilon" display="inline"><mi href="./2.5#Px1.p2">ϵ</mi></math>: parameter</a>,
<a href="./2.5#SS1.p1" title="§2.5(i) Introduction ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m119.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="c" display="inline"><mi href="./2.5#SS1.p1">c</mi></math>: point</a>,
<a href="./2.5#SS1.p2" title="§2.5(i) Introduction ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m34.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="I(x)" display="inline"><mrow><mi href="./2.5#SS1.p2">I</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math>: convolution integral</a> and
<a href="./2.5#SS1.p3" title="§2.5(i) Introduction ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m124.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="d" display="inline"><mi href="./2.5#SS1.p3">d</mi></math>: point</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m120.png" altimg-height="20px" altimg-valign="-7px" altimg-width="111px" alttext="c_{s}=-a_{2s+1}" display="inline"><mrow><msub><mi href="./2.5#SS1.p1">c</mi><mi>s</mi></msub><mo>=</mo><mrow><mo>-</mo><msub><mi href="./2.5#EGx1">a</mi><mrow><mrow><mn>2</mn><mo></mo><mi>s</mi></mrow><mo>+</mo><mn>1</mn></mrow></msub></mrow></mrow></math>, <math class="ltx_Math" altimg="m125.png" altimg-height="22px" altimg-valign="-7px" altimg-width="110px" alttext="d_{s}=-b_{2s+1}" display="inline"><mrow><msub><mi href="./2.5#SS1.p3">d</mi><mi>s</mi></msub><mo>=</mo><mrow><mo>-</mo><msub><mi href="./2.5#EGx1">b</mi><mrow><mrow><mn>2</mn><mo></mo><mi>s</mi></mrow><mo>+</mo><mn>1</mn></mrow></msub></mrow></mrow></math>.</p>
</div>
</section>
</section>
<section id="ii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§2.5(ii) </span>Extensions</h2>
<div id="SS2.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="SS2.p1" class="ltx_para">
<p class="ltx_p">Let <math class="ltx_Math" altimg="m127.png" altimg-height="23px" altimg-valign="-7px" altimg-width="39px" alttext="f(t)" display="inline"><mrow><mi href="./2.5#SS2.p1">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math> and <math class="ltx_Math" altimg="m135.png" altimg-height="23px" altimg-valign="-7px" altimg-width="38px" alttext="h(t)" display="inline"><mrow><mi href="./2.5#SS2.p1">h</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math> be locally integrable on <math class="ltx_Math" altimg="m14.png" altimg-height="23px" altimg-valign="-7px" altimg-width="59px" alttext="(0,\infty)" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mn>0</mn><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi mathvariant="normal">∞</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math> and
</p>
<table id="E17" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">2.5.17</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E17.png" altimg-height="64px" altimg-valign="-27px" altimg-width="150px" alttext="f(t)\sim\sum_{s=0}^{\infty}a_{s}t^{\alpha_{s}}," display="block"><mrow><mrow><mrow><mi href="./2.5#SS2.p1">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo href="./2.1#SS3.p1">∼</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi>s</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><msub><mi href="./2.5#SS2.p1">a</mi><mi>s</mi></msub><mo></mo><msup><mi>t</mi><msub><mi>α</mi><mi>s</mi></msub></msup></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m156.png" altimg-height="18px" altimg-valign="-4px" altimg-width="68px" alttext="t\to 0+" display="inline"><mrow><mi>t</mi><mo>→</mo><mrow><mn>0</mn><mo>+</mo></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E17.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./2.1#SS3.p1" title="§2.1(iii) Asymptotic Expansions ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="12px" altimg-valign="-2px" altimg-width="20px" alttext="\sim" display="inline"><mo href="./2.1#SS3.p1">∼</mo></math>: Poincaré asymptotic expansion</a>,
<a href="./2.5#SS2.p1" title="§2.5(ii) Extensions ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m128.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="f(x)" display="inline"><mrow><mi href="./2.5#SS2.p1">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math>: locally integrable function</a> and
<a href="./2.5#SS2.p1" title="§2.5(ii) Extensions ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m112.png" altimg-height="16px" altimg-valign="-5px" altimg-width="23px" alttext="a_{s}" display="inline"><msub><mi href="./2.5#SS2.p1">a</mi><mi>s</mi></msub></math>: coefficients</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m51.png" altimg-height="21px" altimg-valign="-5px" altimg-width="108px" alttext="\Re\alpha_{s}>\Re\alpha_{s^{\prime}}" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><msub><mi>α</mi><mi>s</mi></msub></mrow><mo>></mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><msub><mi>α</mi><msup><mi>s</mi><mo>′</mo></msup></msub></mrow></mrow></math> for <math class="ltx_Math" altimg="m152.png" altimg-height="19px" altimg-valign="-3px" altimg-width="55px" alttext="s>s^{\prime}" display="inline"><mrow><mi>s</mi><mo>></mo><msup><mi>s</mi><mo>′</mo></msup></mrow></math>, and
<math class="ltx_Math" altimg="m52.png" altimg-height="21px" altimg-valign="-5px" altimg-width="106px" alttext="\Re\alpha_{s}\to+\infty" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><msub><mi>α</mi><mi>s</mi></msub></mrow><mo>→</mo><mrow><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></mrow></math> as <math class="ltx_Math" altimg="m154.png" altimg-height="13px" altimg-valign="-2px" altimg-width="65px" alttext="s\to\infty" display="inline"><mrow><mi>s</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></math>. Also, let
</p>
<table id="E18" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">2.5.18</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E18.png" altimg-height="64px" altimg-valign="-27px" altimg-width="246px" alttext="h(t)\sim\exp\left(i\kappa t^{p}\right)\sum_{s=0}^{\infty}b_{s}t^{-\beta_{s}}," display="block"><mrow><mrow><mrow><mi href="./2.5#SS2.p1">h</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo href="./2.1#SS3.p1">∼</mo><mrow><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mrow><mo>(</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./2.5#SS2.p1">κ</mi><mo></mo><msup><mi>t</mi><mi href="./2.5#SS2.p1">p</mi></msup></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi>s</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><msub><mi href="./2.5#SS1.p1">b</mi><mi>s</mi></msub><mo></mo><msup><mi>t</mi><mrow><mo>-</mo><msub><mi>β</mi><mi>s</mi></msub></mrow></msup></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m157.png" altimg-height="18px" altimg-valign="-4px" altimg-width="78px" alttext="t\to+\infty" display="inline"><mrow><mi>t</mi><mo>→</mo><mrow><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E18.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./2.1#SS3.p1" title="§2.1(iii) Asymptotic Expansions ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="12px" altimg-valign="-2px" altimg-width="20px" alttext="\sim" display="inline"><mo href="./2.1#SS3.p1">∼</mo></math>: Poincaré asymptotic expansion</a>,
<a href="./4.2#E19" title="(4.2.19) ‣ §4.2(iii) The Exponential Function ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="16px" altimg-valign="-6px" altimg-width="48px" alttext="\exp\NVar{z}" display="inline"><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: exponential function</a>,
<a href="./2.5#SS2.p1" title="§2.5(ii) Extensions ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m136.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="h(x)" display="inline"><mrow><mi href="./2.5#SS2.p1">h</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math>: locally integrable function</a>,
<a href="./2.5#SS2.p1" title="§2.5(ii) Extensions ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\kappa" display="inline"><mi href="./2.5#SS2.p1">κ</mi></math>: real</a>,
<a href="./2.5#SS2.p1" title="§2.5(ii) Extensions ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m144.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="p" display="inline"><mi href="./2.5#SS2.p1">p</mi></math>: positive</a> and
<a href="./2.5#SS1.p1" title="§2.5(i) Introduction ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m114.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="b" display="inline"><mi href="./2.5#SS1.p1">b</mi></math>: right endpoint</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\kappa" display="inline"><mi href="./2.5#SS2.p1">κ</mi></math> is real, <math class="ltx_Math" altimg="m143.png" altimg-height="20px" altimg-valign="-6px" altimg-width="51px" alttext="p>0" display="inline"><mrow><mi href="./2.5#SS2.p1">p</mi><mo>></mo><mn>0</mn></mrow></math>,
<math class="ltx_Math" altimg="m53.png" altimg-height="21px" altimg-valign="-6px" altimg-width="105px" alttext="\Re\beta_{s}>\Re\beta_{s^{\prime}}" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><msub><mi>β</mi><mi>s</mi></msub></mrow><mo>></mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><msub><mi>β</mi><msup><mi>s</mi><mo>′</mo></msup></msub></mrow></mrow></math> for <math class="ltx_Math" altimg="m152.png" altimg-height="19px" altimg-valign="-3px" altimg-width="55px" alttext="s>s^{\prime}" display="inline"><mrow><mi>s</mi><mo>></mo><msup><mi>s</mi><mo>′</mo></msup></mrow></math>, and
<math class="ltx_Math" altimg="m54.png" altimg-height="21px" altimg-valign="-6px" altimg-width="105px" alttext="\Re\beta_{s}\to+\infty" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><msub><mi>β</mi><mi>s</mi></msub></mrow><mo>→</mo><mrow><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></mrow></math> as
<math class="ltx_Math" altimg="m154.png" altimg-height="13px" altimg-valign="-2px" altimg-width="65px" alttext="s\to\infty" display="inline"><mrow><mi>s</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></math>. To ensure that the integral () converges we
assume that</p>
<table id="E19" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">2.5.19</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E19.png" altimg-height="30px" altimg-valign="-9px" altimg-width="141px" alttext="f(t)=O\left(t^{-b}\right)," display="block"><mrow><mrow><mrow><mi href="./2.5#SS2.p1">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mi href="./2.1#E3">O</mi><mo></mo><mrow><mo>(</mo><msup><mi>t</mi><mrow><mo>-</mo><mi href="./2.5#SS1.p1">b</mi></mrow></msup><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m157.png" altimg-height="18px" altimg-valign="-4px" altimg-width="78px" alttext="t\to+\infty" display="inline"><mrow><mi>t</mi><mo>→</mo><mrow><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E19.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./2.1#E3" title="(2.1.3) ‣ §2.1(i) Asymptotic and Order Symbols ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="23px" altimg-valign="-7px" altimg-width="50px" alttext="O\left(\NVar{x}\right)" display="inline"><mrow><mi href="./2.1#E3">O</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>)</mo></mrow></mrow></math>: order not exceeding</a>,
<a href="./2.5#SS2.p1" title="§2.5(ii) Extensions ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m128.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="f(x)" display="inline"><mrow><mi href="./2.5#SS2.p1">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math>: locally integrable function</a> and
<a href="./2.5#SS1.p1" title="§2.5(i) Introduction ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m114.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="b" display="inline"><mi href="./2.5#SS1.p1">b</mi></math>: right endpoint</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">with <math class="ltx_Math" altimg="m113.png" altimg-height="21px" altimg-valign="-6px" altimg-width="108px" alttext="b+\Re\beta_{0}>1" display="inline"><mrow><mrow><mi href="./2.5#SS1.p1">b</mi><mo>+</mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><msub><mi>β</mi><mn>0</mn></msub></mrow></mrow><mo>></mo><mn>1</mn></mrow></math>, and</p>
<table id="E20" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">2.5.20</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E20.png" altimg-height="25px" altimg-valign="-7px" altimg-width="126px" alttext="h(t)=O\left(t^{c}\right)," display="block"><mrow><mrow><mrow><mi href="./2.5#SS2.p1">h</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mi href="./2.1#E3">O</mi><mo></mo><mrow><mo>(</mo><msup><mi>t</mi><mi href="./2.5#SS1.p1">c</mi></msup><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m156.png" altimg-height="18px" altimg-valign="-4px" altimg-width="68px" alttext="t\to 0+" display="inline"><mrow><mi>t</mi><mo>→</mo><mrow><mn>0</mn><mo>+</mo></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E20.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./2.1#E3" title="(2.1.3) ‣ §2.1(i) Asymptotic and Order Symbols ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="23px" altimg-valign="-7px" altimg-width="50px" alttext="O\left(\NVar{x}\right)" display="inline"><mrow><mi href="./2.1#E3">O</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>)</mo></mrow></mrow></math>: order not exceeding</a>,
<a href="./2.5#SS2.p1" title="§2.5(ii) Extensions ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m136.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="h(x)" display="inline"><mrow><mi href="./2.5#SS2.p1">h</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math>: locally integrable function</a> and
<a href="./2.5#SS1.p1" title="§2.5(i) Introduction ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m119.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="c" display="inline"><mi href="./2.5#SS1.p1">c</mi></math>: point</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">with <math class="ltx_Math" altimg="m116.png" altimg-height="21px" altimg-valign="-5px" altimg-width="125px" alttext="c+\Re\alpha_{0}>-1" display="inline"><mrow><mrow><mi href="./2.5#SS1.p1">c</mi><mo>+</mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><msub><mi>α</mi><mn>0</mn></msub></mrow></mrow><mo>></mo><mrow><mo>-</mo><mn>1</mn></mrow></mrow></math>. To apply the Mellin transform method
outlined in §, we require the transforms
<math class="ltx_Math" altimg="m80.png" altimg-height="23px" altimg-valign="-7px" altimg-width="104px" alttext="\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(1-z\right)" display="inline"><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><mi href="./2.5#SS2.p1">f</mi></mpadded><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mi>z</mi></mrow><mo>)</mo></mrow></mrow></mrow></math> and <math class="ltx_Math" altimg="m91.png" altimg-height="23px" altimg-valign="-7px" altimg-width="69px" alttext="\mathscr{M}\mskip-3.0mu h\mskip 3.0mu \left(z\right)" display="inline"><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><mi href="./2.5#SS2.p1">h</mi></mpadded><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></math> to have a common strip of
analyticity. This, in turn, requires <math class="ltx_Math" altimg="m18.png" altimg-height="21px" altimg-valign="-5px" altimg-width="91px" alttext="-b<\Re\alpha_{0}" display="inline"><mrow><mrow><mo>-</mo><mi href="./2.5#SS1.p1">b</mi></mrow><mo><</mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><msub><mi>α</mi><mn>0</mn></msub></mrow></mrow></math>, <math class="ltx_Math" altimg="m21.png" altimg-height="21px" altimg-valign="-6px" altimg-width="90px" alttext="-c<\Re\beta_{0}" display="inline"><mrow><mrow><mo>-</mo><mi href="./2.5#SS1.p1">c</mi></mrow><mo><</mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><msub><mi>β</mi><mn>0</mn></msub></mrow></mrow></math>, and either <math class="ltx_Math" altimg="m20.png" altimg-height="21px" altimg-valign="-5px" altimg-width="125px" alttext="-c<\Re\alpha_{0}+1" display="inline"><mrow><mrow><mo>-</mo><mi href="./2.5#SS1.p1">c</mi></mrow><mo><</mo><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><msub><mi>α</mi><mn>0</mn></msub></mrow><mo>+</mo><mn>1</mn></mrow></mrow></math> or <math class="ltx_Math" altimg="m26.png" altimg-height="21px" altimg-valign="-6px" altimg-width="108px" alttext="1-b<\Re\beta_{0}" display="inline"><mrow><mrow><mn>1</mn><mo>-</mo><mi href="./2.5#SS1.p1">b</mi></mrow><mo><</mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><msub><mi>β</mi><mn>0</mn></msub></mrow></mrow></math>. Following <cite class="ltx_cite ltx_citemacro_citet">Handelsman and Lew ()</cite>
we now give an extension of this method in which none of these conditions is
required.</p>
</div>
<div id="SS2.p2" class="ltx_para">
<p class="ltx_p">First, we introduce the truncated functions <math class="ltx_Math" altimg="m130.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="f_{1}(t)" display="inline"><mrow><msub><mi href="./2.5#EGx2">f</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math> and <math class="ltx_Math" altimg="m131.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="f_{2}(t)" display="inline"><mrow><msub><mi href="./2.5#EGx2">f</mi><mn>2</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math> defined by</p>
<table id="EGx2" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E21">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">2.5.21</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m9.png" altimg-height="25px" altimg-valign="-7px" altimg-width="47px" alttext="\displaystyle f_{1}(t)" display="inline"><mrow><msub><mi href="./2.5#EGx2">f</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m2.png" altimg-height="65px" altimg-valign="-27px" altimg-width="201px" alttext="\displaystyle=\begin{cases}f(t),&0<t\leq 1,\\
0,&1<t<\infty,\end{cases}" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mo>{</mo><mtable columnspacing="5pt" rowspacing="0pt"><mtr><mtd columnalign="left"><mrow><mrow><mi href="./2.5#SS2.p1">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mrow><mn>0</mn><mo><</mo><mi>t</mi><mo>≤</mo><mn>1</mn></mrow><mo>,</mo></mrow></mtd></mtr><mtr><mtd columnalign="left"><mrow><mn>0</mn><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mrow><mn>1</mn><mo><</mo><mi>t</mi><mo><</mo><mi mathvariant="normal">∞</mi></mrow><mo>,</mo></mrow></mtd></mtr></mtable></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E21.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./2.5#SS2.p1" title="§2.5(ii) Extensions ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m128.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="f(x)" display="inline"><mrow><mi href="./2.5#SS2.p1">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math>: locally integrable function</a> and
<a href="./2.5#EGx2" title="§2.5(ii) Extensions ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m132.png" altimg-height="24px" altimg-valign="-8px" altimg-width="45px" alttext="f_{j}(t)" display="inline"><mrow><msub><mi href="./2.5#EGx2">f</mi><mi>j</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math>: truncated functions</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E22">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">2.5.22</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m10.png" altimg-height="25px" altimg-valign="-7px" altimg-width="47px" alttext="\displaystyle f_{2}(t)" display="inline"><mrow><msub><mi href="./2.5#EGx2">f</mi><mn>2</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m5.png" altimg-height="25px" altimg-valign="-7px" altimg-width="133px" alttext="\displaystyle=f(t)-f_{1}(t)." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mi href="./2.5#SS2.p1">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo>-</mo><mrow><msub><mi href="./2.5#EGx2">f</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E22.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./2.5#SS2.p1" title="§2.5(ii) Extensions ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m128.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="f(x)" display="inline"><mrow><mi href="./2.5#SS2.p1">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math>: locally integrable function</a> and
<a href="./2.5#EGx2" title="§2.5(ii) Extensions ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m132.png" altimg-height="24px" altimg-valign="-8px" altimg-width="45px" alttext="f_{j}(t)" display="inline"><mrow><msub><mi href="./2.5#EGx2">f</mi><mi>j</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math>: truncated functions</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
<p class="ltx_p">Similarly,</p>
<table id="EGx3" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E23">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">2.5.23</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m11.png" altimg-height="25px" altimg-valign="-7px" altimg-width="49px" alttext="\displaystyle h_{1}(t)" display="inline"><mrow><msub><mi href="./2.5#SS2.p1">h</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m3.png" altimg-height="65px" altimg-valign="-27px" altimg-width="201px" alttext="\displaystyle=\begin{cases}h(t),&0<t\leq 1,\\
0,&1<t<\infty,\end{cases}" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mo>{</mo><mtable columnspacing="5pt" rowspacing="0pt"><mtr><mtd columnalign="left"><mrow><mrow><mi href="./2.5#SS2.p1">h</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mrow><mn>0</mn><mo><</mo><mi>t</mi><mo>≤</mo><mn>1</mn></mrow><mo>,</mo></mrow></mtd></mtr><mtr><mtd columnalign="left"><mrow><mn>0</mn><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mrow><mn>1</mn><mo><</mo><mi>t</mi><mo><</mo><mi mathvariant="normal">∞</mi></mrow><mo>,</mo></mrow></mtd></mtr></mtable></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E23.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./2.5#SS2.p1" title="§2.5(ii) Extensions ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m136.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="h(x)" display="inline"><mrow><mi href="./2.5#SS2.p1">h</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math>: locally integrable function</a></dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E24">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">2.5.24</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m12.png" altimg-height="25px" altimg-valign="-7px" altimg-width="49px" alttext="\displaystyle h_{2}(t)" display="inline"><mrow><msub><mi href="./2.5#SS2.p1">h</mi><mn>2</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m6.png" altimg-height="25px" altimg-valign="-7px" altimg-width="134px" alttext="\displaystyle=h(t)-h_{1}(t)." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mi href="./2.5#SS2.p1">h</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo>-</mo><mrow><msub><mi href="./2.5#SS2.p1">h</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E24.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./2.5#SS2.p1" title="§2.5(ii) Extensions ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m136.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="h(x)" display="inline"><mrow><mi href="./2.5#SS2.p1">h</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math>: locally integrable function</a></dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
<p class="ltx_p">With these definitions and the conditions (.
</p>
</div>
<figure id="T1" class="ltx_table">
<figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_table">Table 2.5.1: </span>Domains of convergence for Mellin transforms.
</figcaption>
<table id="T1.t1" class="ltx_tabular ltx_centering ltx_guessed_headers ltx_align_middle">
<thead class="ltx_thead">
<tr id="T1.t1.r1" class="ltx_tr">
<th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;">Transform</th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;">Domain of Convergence</th>
</tr>
</thead>
<tbody class="ltx_tbody">
<tr id="T1.t1.r2" class="ltx_tr">
<td class="ltx_td ltx_align_center ltx_border_t" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m85.png" altimg-height="23px" altimg-valign="-7px" altimg-width="77px" alttext="\mathscr{M}\mskip-3.0mu f_{1}\mskip 3.0mu \left(z\right)" display="inline"><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><msub><mi href="./2.5#EGx2">f</mi><mn>1</mn></msub></mpadded><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_center ltx_border_t" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m47.png" altimg-height="21px" altimg-valign="-5px" altimg-width="107px" alttext="\Re z>-\Re\alpha_{0}" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>z</mi></mrow><mo>></mo><mrow><mo>-</mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><msub><mi>α</mi><mn>0</mn></msub></mrow></mrow></mrow></math></td>
</tr>
<tr id="T1.t1.r3" class="ltx_tr">
<td class="ltx_td ltx_align_center" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m88.png" altimg-height="23px" altimg-valign="-7px" altimg-width="77px" alttext="\mathscr{M}\mskip-3.0mu f_{2}\mskip 3.0mu \left(z\right)" display="inline"><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><msub><mi href="./2.5#EGx2">f</mi><mn>2</mn></msub></mpadded><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_center" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m44.png" altimg-height="18px" altimg-valign="-3px" altimg-width="64px" alttext="\Re z<b" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>z</mi></mrow><mo><</mo><mi href="./2.5#SS1.p1">b</mi></mrow></math></td>
</tr>
<tr id="T1.t1.r4" class="ltx_tr">
<td class="ltx_td ltx_align_center" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m92.png" altimg-height="23px" altimg-valign="-7px" altimg-width="78px" alttext="\mathscr{M}\mskip-3.0mu h_{1}\mskip 3.0mu \left(z\right)" display="inline"><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><msub><mi href="./2.5#SS2.p1">h</mi><mn>1</mn></msub></mpadded><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_center" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m48.png" altimg-height="19px" altimg-valign="-4px" altimg-width="80px" alttext="\Re z>-c" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>z</mi></mrow><mo>></mo><mrow><mo>-</mo><mi href="./2.5#SS1.p1">c</mi></mrow></mrow></math></td>
</tr>
<tr id="T1.t1.r5" class="ltx_tr">
<td class="ltx_td ltx_align_center ltx_border_b" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m93.png" altimg-height="23px" altimg-valign="-7px" altimg-width="78px" alttext="\mathscr{M}\mskip-3.0mu h_{2}\mskip 3.0mu \left(z\right)" display="inline"><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><msub><mi href="./2.5#SS2.p1">h</mi><mn>2</mn></msub></mpadded><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_center ltx_border_b" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m42.png" altimg-height="21px" altimg-valign="-6px" altimg-width="90px" alttext="\Re z<\Re\beta_{0}" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>z</mi></mrow><mo><</mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><msub><mi>β</mi><mn>0</mn></msub></mrow></mrow></math></td>
</tr>
</tbody>
</table>
<div id="T1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.14#E32" title="(1.14.32) ‣ §1.14(iv) Mellin Transform ‣ §1.14 Integral Transforms ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="23px" altimg-valign="-7px" altimg-width="88px" alttext="\mathscr{M}\left(\NVar{f}\right)\left(\NVar{s}\right)" display="inline"><mrow><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">f</mi><mo>)</mo></mrow><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">s</mi><mo>)</mo></mrow></mrow></math>: Mellin transform</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m55.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./2.5#SS2.p1" title="§2.5(ii) Extensions ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m128.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="f(x)" display="inline"><mrow><mi href="./2.5#SS2.p1">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math>: locally integrable function</a>,
<a href="./2.5#SS2.p1" title="§2.5(ii) Extensions ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m136.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="h(x)" display="inline"><mrow><mi href="./2.5#SS2.p1">h</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math>: locally integrable function</a>,
<a href="./2.5#EGx2" title="§2.5(ii) Extensions ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m132.png" altimg-height="24px" altimg-valign="-8px" altimg-width="45px" alttext="f_{j}(t)" display="inline"><mrow><msub><mi href="./2.5#EGx2">f</mi><mi>j</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math>: truncated functions</a>,
<a href="./2.5#SS1.p1" title="§2.5(i) Introduction ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m114.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="b" display="inline"><mi href="./2.5#SS1.p1">b</mi></math>: right endpoint</a> and
<a href="./2.5#SS1.p1" title="§2.5(i) Introduction ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m119.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="c" display="inline"><mi href="./2.5#SS1.p1">c</mi></math>: point</a>
</dd>
<dt>Notational Change (effective with 1.0.15):</dt>
<dd>
The notation for the Mellin transform was changed to <math class="ltx_Math" altimg="m82.png" altimg-height="23px" altimg-valign="-7px" altimg-width="70px" alttext="\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(z\right)" display="inline"><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><mi href="./2.5#SS2.p1">f</mi></mpadded><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></math> from <math class="ltx_Math" altimg="m77.png" altimg-height="23px" altimg-valign="-7px" altimg-width="75px" alttext="\mathscr{M}(f;z)" display="inline"><mrow><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./2.5#SS2.p1">f</mi><mo>;</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math>.
</dd>
</dl>
</div>
</div>
</figure>
<div id="SS2.p3" class="ltx_para">
<p class="ltx_p">Furthermore, <math class="ltx_Math" altimg="m85.png" altimg-height="23px" altimg-valign="-7px" altimg-width="77px" alttext="\mathscr{M}\mskip-3.0mu f_{1}\mskip 3.0mu \left(z\right)" display="inline"><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><msub><mi href="./2.5#EGx2">f</mi><mn>1</mn></msub></mpadded><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></math> can be continued analytically to a
meromorphic function on the entire <math class="ltx_Math" altimg="m167.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi>z</mi></math>-plane, whose singularities are simple
poles at <math class="ltx_Math" altimg="m17.png" altimg-height="19px" altimg-valign="-5px" altimg-width="41px" alttext="-\alpha_{s}" display="inline"><mrow><mo>-</mo><msub><mi>α</mi><mi>s</mi></msub></mrow></math>, <math class="ltx_Math" altimg="m151.png" altimg-height="20px" altimg-valign="-6px" altimg-width="120px" alttext="s=0,1,2,\dots" display="inline"><mrow><mi>s</mi><mo>=</mo><mrow><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>, with principal part</p>
<table id="E25" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">2.5.25</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E25.png" altimg-height="25px" altimg-valign="-7px" altimg-width="118px" alttext="a_{s}/\left(z+\alpha_{s}\right)." display="block"><mrow><mrow><msub><mi href="./2.5#SS2.p1">a</mi><mi>s</mi></msub><mo>/</mo><mrow><mo>(</mo><mrow><mi>z</mi><mo>+</mo><msub><mi>α</mi><mi>s</mi></msub></mrow><mo>)</mo></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E25.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./2.5#SS2.p1" title="§2.5(ii) Extensions ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m112.png" altimg-height="16px" altimg-valign="-5px" altimg-width="23px" alttext="a_{s}" display="inline"><msub><mi href="./2.5#SS2.p1">a</mi><mi>s</mi></msub></math>: coefficients</a></dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS2.p4" class="ltx_para">
<p class="ltx_p">By Table , <math class="ltx_Math" altimg="m88.png" altimg-height="23px" altimg-valign="-7px" altimg-width="77px" alttext="\mathscr{M}\mskip-3.0mu f_{2}\mskip 3.0mu \left(z\right)" display="inline"><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><msub><mi href="./2.5#EGx2">f</mi><mn>2</mn></msub></mpadded><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></math> is an analytic function in
the half-plane <math class="ltx_Math" altimg="m44.png" altimg-height="18px" altimg-valign="-3px" altimg-width="64px" alttext="\Re z<b" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>z</mi></mrow><mo><</mo><mi href="./2.5#SS1.p1">b</mi></mrow></math>. Hence we can extend the definition of the
Mellin transform of <math class="ltx_Math" altimg="m129.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi href="./2.5#SS2.p1">f</mi></math> by setting</p>
<table id="E26" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">2.5.26</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E26.png" altimg-height="25px" altimg-valign="-7px" altimg-width="267px" alttext="\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(z\right)=\mathscr{M}\mskip-3.0mu f%
_{1}\mskip 3.0mu \left(z\right)+\mathscr{M}\mskip-3.0mu f_{2}\mskip 3.0mu %
\left(z\right)" display="block"><mrow><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><mi href="./2.5#SS2.p1">f</mi></mpadded><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mrow><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><msub><mi href="./2.5#EGx2">f</mi><mn>1</mn></msub></mpadded><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><msub><mi href="./2.5#EGx2">f</mi><mn>2</mn></msub></mpadded><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E26.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.14#E32" title="(1.14.32) ‣ §1.14(iv) Mellin Transform ‣ §1.14 Integral Transforms ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="23px" altimg-valign="-7px" altimg-width="88px" alttext="\mathscr{M}\left(\NVar{f}\right)\left(\NVar{s}\right)" display="inline"><mrow><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">f</mi><mo>)</mo></mrow><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">s</mi><mo>)</mo></mrow></mrow></math>: Mellin transform</a>,
<a href="./2.5#SS2.p1" title="§2.5(ii) Extensions ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m128.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="f(x)" display="inline"><mrow><mi href="./2.5#SS2.p1">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math>: locally integrable function</a> and
<a href="./2.5#EGx2" title="§2.5(ii) Extensions ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m132.png" altimg-height="24px" altimg-valign="-8px" altimg-width="45px" alttext="f_{j}(t)" display="inline"><mrow><msub><mi href="./2.5#EGx2">f</mi><mi>j</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math>: truncated functions</a>
</dd>
<dt>Notational Change (effective with 1.0.15):</dt>
<dd>
The notation for the Mellin transform was changed to <math class="ltx_Math" altimg="m82.png" altimg-height="23px" altimg-valign="-7px" altimg-width="70px" alttext="\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(z\right)" display="inline"><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><mi href="./2.5#SS2.p1">f</mi></mpadded><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></math> from <math class="ltx_Math" altimg="m77.png" altimg-height="23px" altimg-valign="-7px" altimg-width="75px" alttext="\mathscr{M}(f;z)" display="inline"><mrow><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./2.5#SS2.p1">f</mi><mo>;</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math>.
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">for <math class="ltx_Math" altimg="m44.png" altimg-height="18px" altimg-valign="-3px" altimg-width="64px" alttext="\Re z<b" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>z</mi></mrow><mo><</mo><mi href="./2.5#SS1.p1">b</mi></mrow></math>. The extended transform <math class="ltx_Math" altimg="m82.png" altimg-height="23px" altimg-valign="-7px" altimg-width="70px" alttext="\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(z\right)" display="inline"><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><mi href="./2.5#SS2.p1">f</mi></mpadded><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></math> has the
same properties as <math class="ltx_Math" altimg="m85.png" altimg-height="23px" altimg-valign="-7px" altimg-width="77px" alttext="\mathscr{M}\mskip-3.0mu f_{1}\mskip 3.0mu \left(z\right)" display="inline"><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><msub><mi href="./2.5#EGx2">f</mi><mn>1</mn></msub></mpadded><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></math> in the half-plane <math class="ltx_Math" altimg="m44.png" altimg-height="18px" altimg-valign="-3px" altimg-width="64px" alttext="\Re z<b" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>z</mi></mrow><mo><</mo><mi href="./2.5#SS1.p1">b</mi></mrow></math>.</p>
</div>
<div id="SS2.p5" class="ltx_para">
<p class="ltx_p">Similarly, if <math class="ltx_Math" altimg="m68.png" altimg-height="17px" altimg-valign="-2px" altimg-width="52px" alttext="\kappa=0" display="inline"><mrow><mi href="./2.5#SS2.p1">κ</mi><mo>=</mo><mn>0</mn></mrow></math> in (), then <math class="ltx_Math" altimg="m93.png" altimg-height="23px" altimg-valign="-7px" altimg-width="78px" alttext="\mathscr{M}\mskip-3.0mu h_{2}\mskip 3.0mu \left(z\right)" display="inline"><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><msub><mi href="./2.5#SS2.p1">h</mi><mn>2</mn></msub></mpadded><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></math>
can be continued analytically to a meromorphic function on the entire <math class="ltx_Math" altimg="m167.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi>z</mi></math>-plane
with simple poles at <math class="ltx_Math" altimg="m59.png" altimg-height="21px" altimg-valign="-6px" altimg-width="24px" alttext="\beta_{s}" display="inline"><msub><mi>β</mi><mi>s</mi></msub></math>, <math class="ltx_Math" altimg="m151.png" altimg-height="20px" altimg-valign="-6px" altimg-width="120px" alttext="s=0,1,2,\dots" display="inline"><mrow><mi>s</mi><mo>=</mo><mrow><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>, with principal part
</p>
<table id="E27" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">2.5.27</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E27.png" altimg-height="25px" altimg-valign="-7px" altimg-width="131px" alttext="-b_{s}/\left(z-\beta_{s}\right)." display="block"><mrow><mrow><mo>-</mo><mrow><msub><mi href="./2.5#SS1.p1">b</mi><mi>s</mi></msub><mo>/</mo><mrow><mo>(</mo><mrow><mi>z</mi><mo>-</mo><msub><mi>β</mi><mi>s</mi></msub></mrow><mo>)</mo></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E27.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./2.5#SS1.p1" title="§2.5(i) Introduction ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m114.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="b" display="inline"><mi href="./2.5#SS1.p1">b</mi></math>: right endpoint</a></dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Alternatively, if <math class="ltx_Math" altimg="m70.png" altimg-height="21px" altimg-valign="-6px" altimg-width="52px" alttext="\kappa\neq 0" display="inline"><mrow><mi href="./2.5#SS2.p1">κ</mi><mo>≠</mo><mn>0</mn></mrow></math> in (), then
<math class="ltx_Math" altimg="m93.png" altimg-height="23px" altimg-valign="-7px" altimg-width="78px" alttext="\mathscr{M}\mskip-3.0mu h_{2}\mskip 3.0mu \left(z\right)" display="inline"><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><msub><mi href="./2.5#SS2.p1">h</mi><mn>2</mn></msub></mpadded><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></math> can be continued analytically to an entire function.</p>
</div>
<div id="SS2.p6" class="ltx_para">
<p class="ltx_p">Since <math class="ltx_Math" altimg="m92.png" altimg-height="23px" altimg-valign="-7px" altimg-width="78px" alttext="\mathscr{M}\mskip-3.0mu h_{1}\mskip 3.0mu \left(z\right)" display="inline"><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><msub><mi href="./2.5#SS2.p1">h</mi><mn>1</mn></msub></mpadded><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></math> is analytic for <math class="ltx_Math" altimg="m48.png" altimg-height="19px" altimg-valign="-4px" altimg-width="80px" alttext="\Re z>-c" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>z</mi></mrow><mo>></mo><mrow><mo>-</mo><mi href="./2.5#SS1.p1">c</mi></mrow></mrow></math> by
Table , the analytically-continued <math class="ltx_Math" altimg="m93.png" altimg-height="23px" altimg-valign="-7px" altimg-width="78px" alttext="\mathscr{M}\mskip-3.0mu h_{2}\mskip 3.0mu \left(z\right)" display="inline"><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><msub><mi href="./2.5#SS2.p1">h</mi><mn>2</mn></msub></mpadded><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></math>
allows us to extend the Mellin transform of <math class="ltx_Math" altimg="m137.png" altimg-height="18px" altimg-valign="-2px" altimg-width="16px" alttext="h" display="inline"><mi href="./2.5#SS2.p1">h</mi></math> via</p>
<table id="E28" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">2.5.28</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E28.png" altimg-height="25px" altimg-valign="-7px" altimg-width="270px" alttext="\mathscr{M}\mskip-3.0mu h\mskip 3.0mu \left(z\right)=\mathscr{M}\mskip-3.0mu h%
_{1}\mskip 3.0mu \left(z\right)+\mathscr{M}\mskip-3.0mu h_{2}\mskip 3.0mu %
\left(z\right)" display="block"><mrow><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><mi href="./2.5#SS2.p1">h</mi></mpadded><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mrow><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><msub><mi href="./2.5#SS2.p1">h</mi><mn>1</mn></msub></mpadded><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><msub><mi href="./2.5#SS2.p1">h</mi><mn>2</mn></msub></mpadded><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E28.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.14#E32" title="(1.14.32) ‣ §1.14(iv) Mellin Transform ‣ §1.14 Integral Transforms ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="23px" altimg-valign="-7px" altimg-width="88px" alttext="\mathscr{M}\left(\NVar{f}\right)\left(\NVar{s}\right)" display="inline"><mrow><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">f</mi><mo>)</mo></mrow><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">s</mi><mo>)</mo></mrow></mrow></math>: Mellin transform</a>,
<a href="./2.5#SS2.p1" title="§2.5(ii) Extensions ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m128.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="f(x)" display="inline"><mrow><mi href="./2.5#SS2.p1">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math>: locally integrable function</a> and
<a href="./2.5#SS2.p1" title="§2.5(ii) Extensions ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m136.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="h(x)" display="inline"><mrow><mi href="./2.5#SS2.p1">h</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math>: locally integrable function</a>
</dd>
<dt>Notational Change (effective with 1.0.15):</dt>
<dd>
The notation for the Mellin transform was changed to <math class="ltx_Math" altimg="m82.png" altimg-height="23px" altimg-valign="-7px" altimg-width="70px" alttext="\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(z\right)" display="inline"><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><mi href="./2.5#SS2.p1">f</mi></mpadded><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></math> from <math class="ltx_Math" altimg="m77.png" altimg-height="23px" altimg-valign="-7px" altimg-width="75px" alttext="\mathscr{M}(f;z)" display="inline"><mrow><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./2.5#SS2.p1">f</mi><mo>;</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math>.
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">in the same half-plane. From (), it
follows that both <math class="ltx_Math" altimg="m80.png" altimg-height="23px" altimg-valign="-7px" altimg-width="104px" alttext="\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(1-z\right)" display="inline"><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><mi href="./2.5#SS2.p1">f</mi></mpadded><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mi>z</mi></mrow><mo>)</mo></mrow></mrow></mrow></math> and <math class="ltx_Math" altimg="m91.png" altimg-height="23px" altimg-valign="-7px" altimg-width="69px" alttext="\mathscr{M}\mskip-3.0mu h\mskip 3.0mu \left(z\right)" display="inline"><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><mi href="./2.5#SS2.p1">h</mi></mpadded><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></math> are defined
in the half-plane <math class="ltx_Math" altimg="m50.png" altimg-height="23px" altimg-valign="-7px" altimg-width="184px" alttext="\Re z>\max(1-b,-c)" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>z</mi></mrow><mo>></mo><mrow><mi>max</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><mi href="./2.5#SS1.p1">b</mi></mrow><mo>,</mo><mrow><mo>-</mo><mi href="./2.5#SS1.p1">c</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></math>.</p>
</div>
<div id="SS2.p7" class="ltx_para">
<p class="ltx_p">We are now ready to derive the asymptotic expansion of the integral <math class="ltx_Math" altimg="m34.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="I(x)" display="inline"><mrow><mi href="./2.5#SS1.p2">I</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math> in
() as <math class="ltx_Math" altimg="m164.png" altimg-height="13px" altimg-valign="-2px" altimg-width="67px" alttext="x\to\infty" display="inline"><mrow><mi>x</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></math>. First we note that
</p>
<table id="E29" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">2.5.29</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E29.png" altimg-height="70px" altimg-valign="-31px" altimg-width="172px" alttext="I(x)=\sum\limits_{j,k=1}^{2}I_{jk}(x)," display="block"><mrow><mrow><mrow><mi href="./2.5#SS1.p2">I</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mrow><mi>j</mi><mo>,</mo><mi>k</mi></mrow><mo>=</mo><mn>1</mn></mrow><mn>2</mn></munderover><mrow><msub><mi href="./2.5#SS1.p2">I</mi><mrow><mi>j</mi><mo></mo><mi>k</mi></mrow></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E29.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./2.5#SS1.p2" title="§2.5(i) Introduction ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m34.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="I(x)" display="inline"><mrow><mi href="./2.5#SS1.p2">I</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math>: convolution integral</a></dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where</p>
<table id="E30" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">2.5.30</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E30.png" altimg-height="53px" altimg-valign="-20px" altimg-width="246px" alttext="I_{jk}(x)=\int_{0}^{\infty}f_{j}(t)h_{k}(xt)\mathrm{d}t." display="block"><mrow><mrow><mrow><msub><mi href="./2.5#SS1.p2">I</mi><mrow><mi>j</mi><mo></mo><mi>k</mi></mrow></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><mrow><mrow><msub><mi href="./2.5#EGx2">f</mi><mi>j</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./2.5#SS2.p1">h</mi><mi>k</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>x</mi><mo></mo><mi>t</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E30.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m161.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./2.5#SS2.p1" title="§2.5(ii) Extensions ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m136.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="h(x)" display="inline"><mrow><mi href="./2.5#SS2.p1">h</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math>: locally integrable function</a>,
<a href="./2.5#EGx2" title="§2.5(ii) Extensions ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m132.png" altimg-height="24px" altimg-valign="-8px" altimg-width="45px" alttext="f_{j}(t)" display="inline"><mrow><msub><mi href="./2.5#EGx2">f</mi><mi>j</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math>: truncated functions</a> and
<a href="./2.5#SS1.p2" title="§2.5(i) Introduction ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m34.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="I(x)" display="inline"><mrow><mi href="./2.5#SS1.p2">I</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math>: convolution integral</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">By direct computation</p>
<table id="E31" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">2.5.31</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E31.png" altimg-height="25px" altimg-valign="-7px" altimg-width="101px" alttext="I_{21}(x)=0," display="block"><mrow><mrow><mrow><msub><mi href="./2.5#SS1.p2">I</mi><mn>21</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mn>0</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint">for <math class="ltx_Math" altimg="m162.png" altimg-height="19px" altimg-valign="-5px" altimg-width="52px" alttext="x\geq 1" display="inline"><mrow><mi>x</mi><mo>≥</mo><mn>1</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E31.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./2.5#SS1.p2" title="§2.5(i) Introduction ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m34.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="I(x)" display="inline"><mrow><mi href="./2.5#SS1.p2">I</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math>: convolution integral</a></dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Next from Table we observe that the integrals for the
transform pair <math class="ltx_Math" altimg="m89.png" altimg-height="24px" altimg-valign="-8px" altimg-width="110px" alttext="\mathscr{M}\mskip-3.0mu f_{j}\mskip 3.0mu \left(1-z\right)" display="inline"><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><msub><mi href="./2.5#EGx2">f</mi><mi>j</mi></msub></mpadded><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mi>z</mi></mrow><mo>)</mo></mrow></mrow></mrow></math> and <math class="ltx_Math" altimg="m94.png" altimg-height="23px" altimg-valign="-7px" altimg-width="79px" alttext="\mathscr{M}\mskip-3.0mu h_{k}\mskip 3.0mu \left(z\right)" display="inline"><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><msub><mi href="./2.5#SS2.p1">h</mi><mi>k</mi></msub></mpadded><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></math> are
absolutely convergent in the domain <math class="ltx_Math" altimg="m29.png" altimg-height="23px" altimg-valign="-8px" altimg-width="38px" alttext="D_{jk}" display="inline"><msub><mi href="./2.5#SS2.p7">D</mi><mrow><mi>j</mi><mo></mo><mi>k</mi></mrow></msub></math> specified in Table
).
</p>
</div>
<figure id="T2" class="ltx_table">
<figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_table">Table 2.5.2: </span>Domains of analyticity for Mellin transforms.
</figcaption>
<table id="T2.t1" class="ltx_tabular ltx_centering ltx_guessed_headers ltx_align_middle">
<thead class="ltx_thead">
<tr id="T2.t1.r1" class="ltx_tr">
<th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" style="padding:2.083333333333333px 3.0pt;">Transform Pair</th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" style="padding:2.083333333333333px 3.0pt;">Domain <math class="ltx_Math" altimg="m29.png" altimg-height="23px" altimg-valign="-8px" altimg-width="38px" alttext="D_{jk}" display="inline"><msub><mi href="./2.5#SS2.p7">D</mi><mrow><mi>j</mi><mo></mo><mi>k</mi></mrow></msub></math>
</th>
</tr>
</thead>
<tbody class="ltx_tbody">
<tr id="T2.t1.r2" class="ltx_tr">
<td class="ltx_td ltx_align_center ltx_border_t" style="padding:2.083333333333333px 3.0pt;"><math class="ltx_Math" altimg="m83.png" altimg-height="23px" altimg-valign="-7px" altimg-width="203px" alttext="\mathscr{M}\mskip-3.0mu f_{1}\mskip 3.0mu \left(1-z\right),\;\mathscr{M}\mskip%
-3.0mu h_{1}\mskip 3.0mu \left(z\right)" display="inline"><mrow><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><msub><mi href="./2.5#EGx2">f</mi><mn>1</mn></msub></mpadded><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mi>z</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo rspace="5.3pt">,</mo><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><msub><mi href="./2.5#SS2.p1">h</mi><mn>1</mn></msub></mpadded><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_center ltx_border_t" style="padding:2.083333333333333px 3.0pt;"><math class="ltx_Math" altimg="m19.png" altimg-height="21px" altimg-valign="-5px" altimg-width="176px" alttext="-c<\Re z<1+\Re\alpha_{0}" display="inline"><mrow><mrow><mo>-</mo><mi href="./2.5#SS1.p1">c</mi></mrow><mo><</mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>z</mi></mrow><mo><</mo><mrow><mn>1</mn><mo>+</mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><msub><mi>α</mi><mn>0</mn></msub></mrow></mrow></mrow></math></td>
</tr>
<tr id="T2.t1.r3" class="ltx_tr">
<td class="ltx_td ltx_align_center" style="padding:2.083333333333333px 3.0pt;"><math class="ltx_Math" altimg="m84.png" altimg-height="23px" altimg-valign="-7px" altimg-width="203px" alttext="\mathscr{M}\mskip-3.0mu f_{1}\mskip 3.0mu \left(1-z\right),\;\mathscr{M}\mskip%
-3.0mu h_{2}\mskip 3.0mu \left(z\right)" display="inline"><mrow><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><msub><mi href="./2.5#EGx2">f</mi><mn>1</mn></msub></mpadded><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mi>z</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo rspace="5.3pt">,</mo><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><msub><mi href="./2.5#SS2.p1">h</mi><mn>2</mn></msub></mpadded><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_center" style="padding:2.083333333333333px 3.0pt;"><math class="ltx_Math" altimg="m43.png" altimg-height="23px" altimg-valign="-7px" altimg-width="218px" alttext="\Re z<\min(1+\Re\alpha_{0},\Re\beta_{0})" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>z</mi></mrow><mo><</mo><mrow><mi>min</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>+</mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><msub><mi>α</mi><mn>0</mn></msub></mrow></mrow><mo>,</mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><msub><mi>β</mi><mn>0</mn></msub></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></math></td>
</tr>
<tr id="T2.t1.r4" class="ltx_tr">
<td class="ltx_td ltx_align_center" style="padding:2.083333333333333px 3.0pt;"><math class="ltx_Math" altimg="m86.png" altimg-height="23px" altimg-valign="-7px" altimg-width="203px" alttext="\mathscr{M}\mskip-3.0mu f_{2}\mskip 3.0mu \left(1-z\right),\;\mathscr{M}\mskip%
-3.0mu h_{1}\mskip 3.0mu \left(z\right)" display="inline"><mrow><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><msub><mi href="./2.5#EGx2">f</mi><mn>2</mn></msub></mpadded><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mi>z</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo rspace="5.3pt">,</mo><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><msub><mi href="./2.5#SS2.p1">h</mi><mn>1</mn></msub></mpadded><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_center" style="padding:2.083333333333333px 3.0pt;"><math class="ltx_Math" altimg="m95.png" altimg-height="23px" altimg-valign="-7px" altimg-width="184px" alttext="\max(-c,1-b)<\Re z" display="inline"><mrow><mrow><mi>max</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mi href="./2.5#SS1.p1">c</mi></mrow><mo>,</mo><mrow><mn>1</mn><mo>-</mo><mi href="./2.5#SS1.p1">b</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo><</mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>z</mi></mrow></mrow></math></td>
</tr>
<tr id="T2.t1.r5" class="ltx_tr">
<td class="ltx_td ltx_align_center ltx_border_b" style="padding:2.083333333333333px 3.0pt;"><math class="ltx_Math" altimg="m87.png" altimg-height="23px" altimg-valign="-7px" altimg-width="203px" alttext="\mathscr{M}\mskip-3.0mu f_{2}\mskip 3.0mu \left(1-z\right),\;\mathscr{M}\mskip%
-3.0mu h_{2}\mskip 3.0mu \left(z\right)" display="inline"><mrow><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><msub><mi href="./2.5#EGx2">f</mi><mn>2</mn></msub></mpadded><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mi>z</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo rspace="5.3pt">,</mo><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><msub><mi href="./2.5#SS2.p1">h</mi><mn>2</mn></msub></mpadded><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_center ltx_border_b" style="padding:2.083333333333333px 3.0pt;"><math class="ltx_Math" altimg="m25.png" altimg-height="21px" altimg-valign="-6px" altimg-width="159px" alttext="1-b<\Re z<\Re\beta_{0}" display="inline"><mrow><mrow><mn>1</mn><mo>-</mo><mi href="./2.5#SS1.p1">b</mi></mrow><mo><</mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>z</mi></mrow><mo><</mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><msub><mi>β</mi><mn>0</mn></msub></mrow></mrow></math></td>
</tr>
</tbody>
</table>
<div id="T2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.14#E32" title="(1.14.32) ‣ §1.14(iv) Mellin Transform ‣ §1.14 Integral Transforms ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="23px" altimg-valign="-7px" altimg-width="88px" alttext="\mathscr{M}\left(\NVar{f}\right)\left(\NVar{s}\right)" display="inline"><mrow><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">f</mi><mo>)</mo></mrow><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">s</mi><mo>)</mo></mrow></mrow></math>: Mellin transform</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m55.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./2.5#SS2.p1" title="§2.5(ii) Extensions ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m128.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="f(x)" display="inline"><mrow><mi href="./2.5#SS2.p1">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math>: locally integrable function</a>,
<a href="./2.5#SS2.p1" title="§2.5(ii) Extensions ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m136.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="h(x)" display="inline"><mrow><mi href="./2.5#SS2.p1">h</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math>: locally integrable function</a>,
<a href="./2.5#EGx2" title="§2.5(ii) Extensions ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m132.png" altimg-height="24px" altimg-valign="-8px" altimg-width="45px" alttext="f_{j}(t)" display="inline"><mrow><msub><mi href="./2.5#EGx2">f</mi><mi>j</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math>: truncated functions</a>,
<a href="./2.5#SS2.p7" title="§2.5(ii) Extensions ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="23px" altimg-valign="-8px" altimg-width="38px" alttext="D_{jk}" display="inline"><msub><mi href="./2.5#SS2.p7">D</mi><mrow><mi>j</mi><mo></mo><mi>k</mi></mrow></msub></math>: domain</a>,
<a href="./2.5#SS1.p1" title="§2.5(i) Introduction ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m114.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="b" display="inline"><mi href="./2.5#SS1.p1">b</mi></math>: right endpoint</a> and
<a href="./2.5#SS1.p1" title="§2.5(i) Introduction ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m119.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="c" display="inline"><mi href="./2.5#SS1.p1">c</mi></math>: point</a>
</dd>
<dt>Notational Change (effective with 1.0.15):</dt>
<dd>
The notation for the Mellin transform was changed to <math class="ltx_Math" altimg="m82.png" altimg-height="23px" altimg-valign="-7px" altimg-width="70px" alttext="\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(z\right)" display="inline"><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><mi href="./2.5#SS2.p1">f</mi></mpadded><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></math> from <math class="ltx_Math" altimg="m77.png" altimg-height="23px" altimg-valign="-7px" altimg-width="75px" alttext="\mathscr{M}(f;z)" display="inline"><mrow><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./2.5#SS2.p1">f</mi><mo>;</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math>.
</dd>
</dl>
</div>
</div>
</figure>
<div id="SS2.p8" class="ltx_para">
<p class="ltx_p">For simplicity, write
</p>
<table id="E32" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">2.5.32</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E32.png" altimg-height="26px" altimg-valign="-8px" altimg-width="284px" alttext="G_{jk}(z)=\mathscr{M}\mskip-3.0mu f_{j}\mskip 3.0mu \left(1-z\right)\mathscr{M%
}\mskip-3.0mu h_{k}\mskip 3.0mu \left(z\right)." display="block"><mrow><mrow><mrow><msub><mi href="./2.5#SS2.p8">G</mi><mrow><mi>j</mi><mo></mo><mi>k</mi></mrow></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><msub><mi href="./2.5#EGx2">f</mi><mi>j</mi></msub></mpadded><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mi>z</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo></mo><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><msub><mi href="./2.5#SS2.p1">h</mi><mi>k</mi></msub></mpadded><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E32.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.14#E32" title="(1.14.32) ‣ §1.14(iv) Mellin Transform ‣ §1.14 Integral Transforms ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="23px" altimg-valign="-7px" altimg-width="88px" alttext="\mathscr{M}\left(\NVar{f}\right)\left(\NVar{s}\right)" display="inline"><mrow><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">f</mi><mo>)</mo></mrow><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">s</mi><mo>)</mo></mrow></mrow></math>: Mellin transform</a>,
<a href="./2.5#SS2.p1" title="§2.5(ii) Extensions ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m128.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="f(x)" display="inline"><mrow><mi href="./2.5#SS2.p1">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math>: locally integrable function</a>,
<a href="./2.5#SS2.p1" title="§2.5(ii) Extensions ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m136.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="h(x)" display="inline"><mrow><mi href="./2.5#SS2.p1">h</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math>: locally integrable function</a>,
<a href="./2.5#EGx2" title="§2.5(ii) Extensions ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m132.png" altimg-height="24px" altimg-valign="-8px" altimg-width="45px" alttext="f_{j}(t)" display="inline"><mrow><msub><mi href="./2.5#EGx2">f</mi><mi>j</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math>: truncated functions</a> and
<a href="./2.5#SS2.p8" title="§2.5(ii) Extensions ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m33.png" altimg-height="24px" altimg-valign="-8px" altimg-width="63px" alttext="G_{jk}(z)" display="inline"><mrow><msub><mi href="./2.5#SS2.p8">G</mi><mrow><mi>j</mi><mo></mo><mi>k</mi></mrow></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function</a>
</dd>
<dt>Notational Change (effective with 1.0.15):</dt>
<dd>
The notation for the Mellin transform was changed to <math class="ltx_Math" altimg="m82.png" altimg-height="23px" altimg-valign="-7px" altimg-width="70px" alttext="\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(z\right)" display="inline"><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><mi href="./2.5#SS2.p1">f</mi></mpadded><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></math> from <math class="ltx_Math" altimg="m77.png" altimg-height="23px" altimg-valign="-7px" altimg-width="75px" alttext="\mathscr{M}(f;z)" display="inline"><mrow><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./2.5#SS2.p1">f</mi><mo>;</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math>.
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">From Table , we see that each <math class="ltx_Math" altimg="m33.png" altimg-height="24px" altimg-valign="-8px" altimg-width="63px" alttext="G_{jk}(z)" display="inline"><mrow><msub><mi href="./2.5#SS2.p8">G</mi><mrow><mi>j</mi><mo></mo><mi>k</mi></mrow></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math> is analytic in the
domain <math class="ltx_Math" altimg="m29.png" altimg-height="23px" altimg-valign="-8px" altimg-width="38px" alttext="D_{jk}" display="inline"><msub><mi href="./2.5#SS2.p7">D</mi><mrow><mi>j</mi><mo></mo><mi>k</mi></mrow></msub></math>. Furthermore, each <math class="ltx_Math" altimg="m33.png" altimg-height="24px" altimg-valign="-8px" altimg-width="63px" alttext="G_{jk}(z)" display="inline"><mrow><msub><mi href="./2.5#SS2.p8">G</mi><mrow><mi>j</mi><mo></mo><mi>k</mi></mrow></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math> has an analytic or meromorphic
extension to a half-plane containing <math class="ltx_Math" altimg="m29.png" altimg-height="23px" altimg-valign="-8px" altimg-width="38px" alttext="D_{jk}" display="inline"><msub><mi href="./2.5#SS2.p7">D</mi><mrow><mi>j</mi><mo></mo><mi>k</mi></mrow></msub></math>. Now suppose that there is a
real number <math class="ltx_Math" altimg="m148.png" altimg-height="18px" altimg-valign="-8px" altimg-width="32px" alttext="p_{jk}" display="inline"><msub><mi href="./2.5#SS2.p8">p</mi><mrow><mi>j</mi><mo></mo><mi>k</mi></mrow></msub></math> in <math class="ltx_Math" altimg="m29.png" altimg-height="23px" altimg-valign="-8px" altimg-width="38px" alttext="D_{jk}" display="inline"><msub><mi href="./2.5#SS2.p7">D</mi><mrow><mi>j</mi><mo></mo><mi>k</mi></mrow></msub></math> such that the Parseval formula
() applies and</p>
<table id="E33" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">2.5.33</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E33.png" altimg-height="59px" altimg-valign="-24px" altimg-width="322px" alttext="I_{jk}(x)=\frac{1}{2\pi i}\int_{p_{jk}-i\infty}^{p_{jk}+i\infty}x^{-z}G_{jk}(z%
)\mathrm{d}z." display="block"><mrow><mrow><mrow><msub><mi href="./2.5#SS1.p2">I</mi><mrow><mi>j</mi><mo></mo><mi>k</mi></mrow></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mfrac><mn>1</mn><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi></mrow></mfrac><mo></mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><msub><mi href="./2.5#SS2.p8">p</mi><mrow><mi>j</mi><mo></mo><mi>k</mi></mrow></msub><mo>-</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi mathvariant="normal">∞</mi></mrow></mrow><mrow><msub><mi href="./2.5#SS2.p8">p</mi><mrow><mi>j</mi><mo></mo><mi>k</mi></mrow></msub><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi mathvariant="normal">∞</mi></mrow></mrow></msubsup><mrow><msup><mi>x</mi><mrow><mo>-</mo><mi>z</mi></mrow></msup><mo></mo><mrow><msub><mi href="./2.5#SS2.p8">G</mi><mrow><mi>j</mi><mo></mo><mi>k</mi></mrow></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>z</mi></mrow></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E33.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m98.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m161.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./2.5#SS2.p8" title="§2.5(ii) Extensions ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m33.png" altimg-height="24px" altimg-valign="-8px" altimg-width="63px" alttext="G_{jk}(z)" display="inline"><mrow><msub><mi href="./2.5#SS2.p8">G</mi><mrow><mi>j</mi><mo></mo><mi>k</mi></mrow></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function</a>,
<a href="./2.5#SS2.p8" title="§2.5(ii) Extensions ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m148.png" altimg-height="18px" altimg-valign="-8px" altimg-width="32px" alttext="p_{jk}" display="inline"><msub><mi href="./2.5#SS2.p8">p</mi><mrow><mi>j</mi><mo></mo><mi>k</mi></mrow></msub></math>: real number</a> and
<a href="./2.5#SS1.p2" title="§2.5(i) Introduction ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m34.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="I(x)" display="inline"><mrow><mi href="./2.5#SS1.p2">I</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math>: convolution integral</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">If, in addition, there exists a number <math class="ltx_Math" altimg="m150.png" altimg-height="20px" altimg-valign="-8px" altimg-width="84px" alttext="q_{jk}>p_{jk}" display="inline"><mrow><msub><mi href="./2.5#SS2.p8">q</mi><mrow><mi>j</mi><mo></mo><mi>k</mi></mrow></msub><mo>></mo><msub><mi href="./2.5#SS2.p8">p</mi><mrow><mi>j</mi><mo></mo><mi>k</mi></mrow></msub></mrow></math> such that
</p>
<table id="E34" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">2.5.34</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E34.png" altimg-height="43px" altimg-valign="-25px" altimg-width="253px" alttext="\sup_{p_{jk}\leq x\leq q_{jk}}\left|G_{jk}(x+iy)\right|\to 0," display="block"><mrow><mrow><mrow><munder><mo href="./front/introduction#Sx4.p1.t1.r25" movablelimits="false">sup</mo><mrow><msub><mi href="./2.5#SS2.p8">p</mi><mrow><mi>j</mi><mo></mo><mi>k</mi></mrow></msub><mo>≤</mo><mi>x</mi><mo>≤</mo><msub><mi href="./2.5#SS2.p8">q</mi><mrow><mi>j</mi><mo></mo><mi>k</mi></mrow></msub></mrow></munder><mo></mo><mrow><mo>|</mo><mrow><msub><mi href="./2.5#SS2.p8">G</mi><mrow><mi>j</mi><mo></mo><mi>k</mi></mrow></msub><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>x</mi><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi>y</mi></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>|</mo></mrow></mrow><mo>→</mo><mn>0</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m166.png" altimg-height="19px" altimg-valign="-6px" altimg-width="81px" alttext="y\to\pm\infty" display="inline"><mrow><mi>y</mi><mo>→</mo><mrow><mo>±</mo><mi mathvariant="normal">∞</mi></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E34.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./front/introduction#Sx4.p1.t1.r25" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m105.png" altimg-height="16px" altimg-valign="-6px" altimg-width="34px" alttext="\sup" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r25">sup</mo></math>: least upper bound (supremum)</a>,
<a href="./2.5#SS2.p8" title="§2.5(ii) Extensions ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m33.png" altimg-height="24px" altimg-valign="-8px" altimg-width="63px" alttext="G_{jk}(z)" display="inline"><mrow><msub><mi href="./2.5#SS2.p8">G</mi><mrow><mi>j</mi><mo></mo><mi>k</mi></mrow></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function</a>,
<a href="./2.5#SS2.p8" title="§2.5(ii) Extensions ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m148.png" altimg-height="18px" altimg-valign="-8px" altimg-width="32px" alttext="p_{jk}" display="inline"><msub><mi href="./2.5#SS2.p8">p</mi><mrow><mi>j</mi><mo></mo><mi>k</mi></mrow></msub></math>: real number</a> and
<a href="./2.5#SS2.p8" title="§2.5(ii) Extensions ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m150.png" altimg-height="20px" altimg-valign="-8px" altimg-width="84px" alttext="q_{jk}>p_{jk}" display="inline"><mrow><msub><mi href="./2.5#SS2.p8">q</mi><mrow><mi>j</mi><mo></mo><mi>k</mi></mrow></msub><mo>></mo><msub><mi href="./2.5#SS2.p8">p</mi><mrow><mi>j</mi><mo></mo><mi>k</mi></mrow></msub></mrow></math>: real number</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">then</p>
<table id="E35" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">2.5.35</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E35.png" altimg-height="56px" altimg-valign="-32px" altimg-width="419px" alttext="I_{jk}(x)=\sum_{p_{jk}<\Re z<q_{jk}}\Residue\left[-x^{-z}G_{jk}(z)\right]+E_{%
jk}(x)," display="block"><mrow><mrow><mrow><msub><mi href="./2.5#SS1.p2">I</mi><mrow><mi>j</mi><mo></mo><mi>k</mi></mrow></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mrow><munder><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><msub><mi href="./2.5#SS2.p8">p</mi><mrow><mi>j</mi><mo></mo><mi>k</mi></mrow></msub><mo><</mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>z</mi></mrow><mo><</mo><msub><mi href="./2.5#SS2.p8">q</mi><mrow><mi>j</mi><mo></mo><mi>k</mi></mrow></msub></mrow></munder><mrow><mo href="./1.10#SS3.p5">res</mo><mo></mo><mrow><mo>[</mo><mrow><mo>-</mo><mrow><msup><mi>x</mi><mrow><mo>-</mo><mi>z</mi></mrow></msup><mo></mo><mrow><msub><mi href="./2.5#SS2.p8">G</mi><mrow><mi>j</mi><mo></mo><mi>k</mi></mrow></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow><mo>]</mo></mrow></mrow></mrow><mo>+</mo><mrow><msub><mi href="./2.5#SS2.p8">E</mi><mrow><mi>j</mi><mo></mo><mi>k</mi></mrow></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E35.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m55.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./1.10#SS3.p5" title="§1.10(iii) Laurent Series ‣ §1.10 Functions of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m57.png" altimg-height="13px" altimg-valign="-2px" altimg-width="29px" alttext="\Residue" display="inline"><mo href="./1.10#SS3.p5">res</mo></math>: residue</a>,
<a href="./2.5#SS2.p8" title="§2.5(ii) Extensions ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m33.png" altimg-height="24px" altimg-valign="-8px" altimg-width="63px" alttext="G_{jk}(z)" display="inline"><mrow><msub><mi href="./2.5#SS2.p8">G</mi><mrow><mi>j</mi><mo></mo><mi>k</mi></mrow></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function</a>,
<a href="./2.5#SS2.p8" title="§2.5(ii) Extensions ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m148.png" altimg-height="18px" altimg-valign="-8px" altimg-width="32px" alttext="p_{jk}" display="inline"><msub><mi href="./2.5#SS2.p8">p</mi><mrow><mi>j</mi><mo></mo><mi>k</mi></mrow></msub></math>: real number</a>,
<a href="./2.5#SS2.p8" title="§2.5(ii) Extensions ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m150.png" altimg-height="20px" altimg-valign="-8px" altimg-width="84px" alttext="q_{jk}>p_{jk}" display="inline"><mrow><msub><mi href="./2.5#SS2.p8">q</mi><mrow><mi>j</mi><mo></mo><mi>k</mi></mrow></msub><mo>></mo><msub><mi href="./2.5#SS2.p8">p</mi><mrow><mi>j</mi><mo></mo><mi>k</mi></mrow></msub></mrow></math>: real number</a>,
<a href="./2.5#SS2.p8" title="§2.5(ii) Extensions ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m32.png" altimg-height="24px" altimg-valign="-8px" altimg-width="63px" alttext="E_{jk}(x)" display="inline"><mrow><msub><mi href="./2.5#SS2.p8">E</mi><mrow><mi>j</mi><mo></mo><mi>k</mi></mrow></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function</a> and
<a href="./2.5#SS1.p2" title="§2.5(i) Introduction ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m34.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="I(x)" display="inline"><mrow><mi href="./2.5#SS1.p2">I</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math>: convolution integral</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where
</p>
<table id="E36" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">2.5.36</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E36.png" altimg-height="59px" altimg-valign="-24px" altimg-width="427px" alttext="E_{jk}(x)=\frac{1}{2\pi i}\int_{q_{jk}-i\infty}^{q_{jk}+i\infty}x^{-z}G_{jk}(z%
)\mathrm{d}z=o\left(x^{-q_{jk}}\right)" display="block"><mrow><mrow><msub><mi href="./2.5#SS2.p8">E</mi><mrow><mi>j</mi><mo></mo><mi>k</mi></mrow></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mfrac><mn>1</mn><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi></mrow></mfrac><mo></mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><msub><mi href="./2.5#SS2.p8">q</mi><mrow><mi>j</mi><mo></mo><mi>k</mi></mrow></msub><mo>-</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi mathvariant="normal">∞</mi></mrow></mrow><mrow><msub><mi href="./2.5#SS2.p8">q</mi><mrow><mi>j</mi><mo></mo><mi>k</mi></mrow></msub><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi mathvariant="normal">∞</mi></mrow></mrow></msubsup><mrow><msup><mi>x</mi><mrow><mo>-</mo><mi>z</mi></mrow></msup><mo></mo><mrow><msub><mi href="./2.5#SS2.p8">G</mi><mrow><mi>j</mi><mo></mo><mi>k</mi></mrow></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>z</mi></mrow></mrow></mrow></mrow><mo>=</mo><mrow><mi href="./2.1#E2">o</mi><mo></mo><mrow><mo>(</mo><msup><mi>x</mi><mrow><mo>-</mo><msub><mi href="./2.5#SS2.p8">q</mi><mrow><mi>j</mi><mo></mo><mi>k</mi></mrow></msub></mrow></msup><mo>)</mo></mrow></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E36.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m98.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m161.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./2.1#E2" title="(2.1.2) ‣ §2.1(i) Asymptotic and Order Symbols ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m142.png" altimg-height="23px" altimg-valign="-7px" altimg-width="44px" alttext="o\left(\NVar{x}\right)" display="inline"><mrow><mi href="./2.1#E2">o</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>)</mo></mrow></mrow></math>: order less than</a>,
<a href="./2.5#SS2.p8" title="§2.5(ii) Extensions ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m33.png" altimg-height="24px" altimg-valign="-8px" altimg-width="63px" alttext="G_{jk}(z)" display="inline"><mrow><msub><mi href="./2.5#SS2.p8">G</mi><mrow><mi>j</mi><mo></mo><mi>k</mi></mrow></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function</a>,
<a href="./2.5#SS2.p8" title="§2.5(ii) Extensions ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m150.png" altimg-height="20px" altimg-valign="-8px" altimg-width="84px" alttext="q_{jk}>p_{jk}" display="inline"><mrow><msub><mi href="./2.5#SS2.p8">q</mi><mrow><mi>j</mi><mo></mo><mi>k</mi></mrow></msub><mo>></mo><msub><mi href="./2.5#SS2.p8">p</mi><mrow><mi>j</mi><mo></mo><mi>k</mi></mrow></msub></mrow></math>: real number</a> and
<a href="./2.5#SS2.p8" title="§2.5(ii) Extensions ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m32.png" altimg-height="24px" altimg-valign="-8px" altimg-width="63px" alttext="E_{jk}(x)" display="inline"><mrow><msub><mi href="./2.5#SS2.p8">E</mi><mrow><mi>j</mi><mo></mo><mi>k</mi></mrow></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">as <math class="ltx_Math" altimg="m163.png" altimg-height="17px" altimg-valign="-4px" altimg-width="82px" alttext="x\to+\infty" display="inline"><mrow><mi>x</mi><mo>→</mo><mrow><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></mrow></math>. (The last order estimate follows from the Riemann–Lebesgue
lemma, §.) The asymptotic expansion of <math class="ltx_Math" altimg="m34.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="I(x)" display="inline"><mrow><mi href="./2.5#SS1.p2">I</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math> is then
obtained from (</dd>
</dl>
</div>
</div>
<div id="SS3.p1" class="ltx_para">
<p class="ltx_p">Let <math class="ltx_Math" altimg="m135.png" altimg-height="23px" altimg-valign="-7px" altimg-width="38px" alttext="h(t)" display="inline"><mrow><mi href="./2.5#SS2.p1">h</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math> satisfy () with <math class="ltx_Math" altimg="m118.png" altimg-height="18px" altimg-valign="-4px" altimg-width="65px" alttext="c>-1" display="inline"><mrow><mi href="./2.5#SS1.p1">c</mi><mo>></mo><mrow><mo>-</mo><mn>1</mn></mrow></mrow></math>,
and consider the Laplace transform</p>
<table id="E37" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">2.5.37</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E37.png" altimg-height="53px" altimg-valign="-20px" altimg-width="229px" alttext="\mathscr{L}\mskip-3.0mu h\mskip 3.0mu \left(\zeta\right)=\int_{0}^{\infty}h(t)%
e^{-\zeta t}\mathrm{d}t." display="block"><mrow><mrow><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E17">ℒ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><mi href="./2.5#SS2.p1">h</mi></mpadded><mo></mo><mrow><mo>(</mo><mi>ζ</mi><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><mrow><mrow><mi href="./2.5#SS2.p1">h</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mi>ζ</mi><mo></mo><mi>t</mi></mrow></mrow></msup><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E37.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.14#E17" title="(1.14.17) ‣ §1.14(iii) Laplace Transform ‣ §1.14 Integral Transforms ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m75.png" altimg-height="23px" altimg-valign="-7px" altimg-width="84px" alttext="\mathscr{L}\left(\NVar{f}\right)\left(\NVar{s}\right)" display="inline"><mrow><mi class="ltx_font_mathscript" href="./1.14#E17">ℒ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">f</mi><mo>)</mo></mrow><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">s</mi><mo>)</mo></mrow></mrow></math>: Laplace transform</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m161.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m73.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a> and
<a href="./2.5#SS2.p1" title="§2.5(ii) Extensions ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m136.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="h(x)" display="inline"><mrow><mi href="./2.5#SS2.p1">h</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math>: locally integrable function</a>
</dd>
<dt>Notational Change (effective with 1.0.15):</dt>
<dd>
The notation for the Laplace transform was changed to <math class="ltx_Math" altimg="m76.png" altimg-height="23px" altimg-valign="-7px" altimg-width="66px" alttext="\mathscr{L}\mskip-3.0mu f\mskip 3.0mu \left(z\right)" display="inline"><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E17">ℒ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><mi>f</mi></mpadded><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></math> from <math class="ltx_Math" altimg="m74.png" altimg-height="23px" altimg-valign="-7px" altimg-width="72px" alttext="\mathscr{L}(f;z)" display="inline"><mrow><mi class="ltx_font_mathscript" href="./1.14#E17">ℒ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>f</mi><mo>;</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math>.
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Put <math class="ltx_Math" altimg="m158.png" altimg-height="23px" altimg-valign="-7px" altimg-width="72px" alttext="x=1/\zeta" display="inline"><mrow><mi>x</mi><mo>=</mo><mrow><mn>1</mn><mo>/</mo><mi>ζ</mi></mrow></mrow></math> and break the integration range at <math class="ltx_Math" altimg="m155.png" altimg-height="17px" altimg-valign="-2px" altimg-width="48px" alttext="t=1" display="inline"><mrow><mi>t</mi><mo>=</mo><mn>1</mn></mrow></math>, as in
(). Then</p>
<table id="E38" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">2.5.38</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E38.png" altimg-height="25px" altimg-valign="-7px" altimg-width="224px" alttext="\zeta\mathscr{L}\mskip-3.0mu h\mskip 3.0mu \left(\zeta\right)=I_{1}(x)+I_{2}(x)," display="block"><mrow><mrow><mrow><mi>ζ</mi><mo></mo><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E17">ℒ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><mi href="./2.5#SS2.p1">h</mi></mpadded><mo></mo><mrow><mo>(</mo><mi>ζ</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>=</mo><mrow><mrow><msub><mi href="./2.5#SS3.p1">I</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow><mo>+</mo><mrow><msub><mi href="./2.5#SS3.p1">I</mi><mn>2</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E38.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.14#E17" title="(1.14.17) ‣ §1.14(iii) Laplace Transform ‣ §1.14 Integral Transforms ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m75.png" altimg-height="23px" altimg-valign="-7px" altimg-width="84px" alttext="\mathscr{L}\left(\NVar{f}\right)\left(\NVar{s}\right)" display="inline"><mrow><mi class="ltx_font_mathscript" href="./1.14#E17">ℒ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">f</mi><mo>)</mo></mrow><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">s</mi><mo>)</mo></mrow></mrow></math>: Laplace transform</a>,
<a href="./2.5#SS2.p1" title="§2.5(ii) Extensions ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m136.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="h(x)" display="inline"><mrow><mi href="./2.5#SS2.p1">h</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math>: locally integrable function</a> and
<a href="./2.5#SS3.p1" title="§2.5(iii) Laplace Transforms with Small Parameters ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m35.png" altimg-height="24px" altimg-valign="-8px" altimg-width="48px" alttext="I_{j}(x)" display="inline"><mrow><msub><mi href="./2.5#SS3.p1">I</mi><mi>j</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math>: integral</a>
</dd>
<dt>Notational Change (effective with 1.0.15):</dt>
<dd>
The notation for the Laplace transform was changed to <math class="ltx_Math" altimg="m76.png" altimg-height="23px" altimg-valign="-7px" altimg-width="66px" alttext="\mathscr{L}\mskip-3.0mu f\mskip 3.0mu \left(z\right)" display="inline"><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E17">ℒ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><mi>f</mi></mpadded><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></math> from <math class="ltx_Math" altimg="m74.png" altimg-height="23px" altimg-valign="-7px" altimg-width="72px" alttext="\mathscr{L}(f;z)" display="inline"><mrow><mi class="ltx_font_mathscript" href="./1.14#E17">ℒ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>f</mi><mo>;</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math>.
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where
</p>
<table id="E39" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">2.5.39</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E39.png" altimg-height="53px" altimg-valign="-20px" altimg-width="223px" alttext="I_{j}(x)=\int_{0}^{\infty}e^{-t}h_{j}(xt)\mathrm{d}t," display="block"><mrow><mrow><mrow><msub><mi href="./2.5#SS3.p1">I</mi><mi>j</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mi>t</mi></mrow></msup><mo></mo><mrow><msub><mi href="./2.5#SS2.p1">h</mi><mi>j</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>x</mi><mo></mo><mi>t</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m138.png" altimg-height="20px" altimg-valign="-6px" altimg-width="69px" alttext="j=1,2" display="inline"><mrow><mi>j</mi><mo>=</mo><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E39.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m161.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m73.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./2.5#SS2.p1" title="§2.5(ii) Extensions ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m136.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="h(x)" display="inline"><mrow><mi href="./2.5#SS2.p1">h</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math>: locally integrable function</a> and
<a href="./2.5#SS3.p1" title="§2.5(iii) Laplace Transforms with Small Parameters ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m35.png" altimg-height="24px" altimg-valign="-8px" altimg-width="48px" alttext="I_{j}(x)" display="inline"><mrow><msub><mi href="./2.5#SS3.p1">I</mi><mi>j</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math>: integral</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Since <math class="ltx_Math" altimg="m79.png" altimg-height="24px" altimg-valign="-7px" altimg-width="155px" alttext="\mathscr{M}\mskip-3.0mu e^{-t}\mskip 3.0mu \left(z\right)=\Gamma\left(z\right)" display="inline"><mrow><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mi>t</mi></mrow></msup></mpadded><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></math>, by the Parseval formula
(), there are real numbers <math class="ltx_Math" altimg="m145.png" altimg-height="16px" altimg-valign="-6px" altimg-width="23px" alttext="p_{1}" display="inline"><msub><mi href="./2.5#SS3.p1">p</mi><mn>1</mn></msub></math> and <math class="ltx_Math" altimg="m147.png" altimg-height="16px" altimg-valign="-6px" altimg-width="23px" alttext="p_{2}" display="inline"><msub><mi href="./2.5#SS3.p1">p</mi><mn>2</mn></msub></math> such that
<math class="ltx_Math" altimg="m22.png" altimg-height="20px" altimg-valign="-6px" altimg-width="110px" alttext="-c<p_{1}<1" display="inline"><mrow><mrow><mo>-</mo><mi href="./2.5#SS1.p1">c</mi></mrow><mo><</mo><msub><mi href="./2.5#SS3.p1">p</mi><mn>1</mn></msub><mo><</mo><mn>1</mn></mrow></math>, <math class="ltx_Math" altimg="m146.png" altimg-height="23px" altimg-valign="-7px" altimg-width="152px" alttext="p_{2}<\min(1,\Re\beta_{0})" display="inline"><mrow><msub><mi href="./2.5#SS3.p1">p</mi><mn>2</mn></msub><mo><</mo><mrow><mi>min</mi><mo></mo><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><msub><mi>β</mi><mn>0</mn></msub></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></math>, and</p>
<table id="E40" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">2.5.40</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E40.png" altimg-height="59px" altimg-valign="-24px" altimg-width="403px" alttext="I_{j}(x)=\frac{1}{2\pi i}\int_{p_{j}-i\infty}^{p_{j}+i\infty}x^{-z}\Gamma\left%
(1-z\right)\mathscr{M}\mskip-3.0mu h_{j}\mskip 3.0mu \left(z\right)\mathrm{d}z," display="block"><mrow><mrow><mrow><msub><mi href="./2.5#SS3.p1">I</mi><mi>j</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mfrac><mn>1</mn><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi></mrow></mfrac><mo></mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><msub><mi href="./2.5#SS3.p1">p</mi><mi>j</mi></msub><mo>-</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi mathvariant="normal">∞</mi></mrow></mrow><mrow><msub><mi href="./2.5#SS3.p1">p</mi><mi>j</mi></msub><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi mathvariant="normal">∞</mi></mrow></mrow></msubsup><mrow><msup><mi>x</mi><mrow><mo>-</mo><mi>z</mi></mrow></msup><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mi>z</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><msub><mi href="./2.5#SS2.p1">h</mi><mi>j</mi></msub></mpadded><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>z</mi></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m138.png" altimg-height="20px" altimg-valign="-6px" altimg-width="69px" alttext="j=1,2" display="inline"><mrow><mi>j</mi><mo>=</mo><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E40.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m41.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./1.14#E32" title="(1.14.32) ‣ §1.14(iv) Mellin Transform ‣ §1.14 Integral Transforms ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="23px" altimg-valign="-7px" altimg-width="88px" alttext="\mathscr{M}\left(\NVar{f}\right)\left(\NVar{s}\right)" display="inline"><mrow><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">f</mi><mo>)</mo></mrow><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">s</mi><mo>)</mo></mrow></mrow></math>: Mellin transform</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m98.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m161.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./2.5#SS2.p1" title="§2.5(ii) Extensions ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m136.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="h(x)" display="inline"><mrow><mi href="./2.5#SS2.p1">h</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math>: locally integrable function</a>,
<a href="./2.5#SS3.p1" title="§2.5(iii) Laplace Transforms with Small Parameters ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m35.png" altimg-height="24px" altimg-valign="-8px" altimg-width="48px" alttext="I_{j}(x)" display="inline"><mrow><msub><mi href="./2.5#SS3.p1">I</mi><mi>j</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math>: integral</a> and
<a href="./2.5#SS3.p1" title="§2.5(iii) Laplace Transforms with Small Parameters ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m149.png" altimg-height="18px" altimg-valign="-8px" altimg-width="23px" alttext="p_{j}" display="inline"><msub><mi href="./2.5#SS3.p1">p</mi><mi>j</mi></msub></math>: real numbers</a>
</dd>
<dt>Notational Change (effective with 1.0.15):</dt>
<dd>
The notation for the Mellin transform was changed to <math class="ltx_Math" altimg="m82.png" altimg-height="23px" altimg-valign="-7px" altimg-width="70px" alttext="\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(z\right)" display="inline"><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><mi>f</mi></mpadded><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></math> from <math class="ltx_Math" altimg="m77.png" altimg-height="23px" altimg-valign="-7px" altimg-width="75px" alttext="\mathscr{M}(f;z)" display="inline"><mrow><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>f</mi><mo>;</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math>.
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Since <math class="ltx_Math" altimg="m91.png" altimg-height="23px" altimg-valign="-7px" altimg-width="69px" alttext="\mathscr{M}\mskip-3.0mu h\mskip 3.0mu \left(z\right)" display="inline"><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><mi href="./2.5#SS2.p1">h</mi></mpadded><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></math> is analytic for <math class="ltx_Math" altimg="m48.png" altimg-height="19px" altimg-valign="-4px" altimg-width="80px" alttext="\Re z>-c" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>z</mi></mrow><mo>></mo><mrow><mo>-</mo><mi href="./2.5#SS1.p1">c</mi></mrow></mrow></math>, by
(),</p>
<table id="E41" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">2.5.41</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E41.png" altimg-height="59px" altimg-valign="-23px" altimg-width="531px" alttext="I_{1}(x)=\mathscr{M}\mskip-3.0mu h_{1}\mskip 3.0mu \left(1\right)x^{-1}+\frac{%
1}{2\pi i}\int_{\rho-i\infty}^{\rho+i\infty}x^{-z}\Gamma\left(1-z\right)%
\mathscr{M}\mskip-3.0mu h_{1}\mskip 3.0mu \left(z\right)\mathrm{d}z," display="block"><mrow><mrow><mrow><msub><mi href="./2.5#SS3.p1">I</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><msub><mi href="./2.5#SS2.p1">h</mi><mn>1</mn></msub></mpadded><mo></mo><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></mrow><mo></mo><msup><mi>x</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></mrow><mo>+</mo><mrow><mfrac><mn>1</mn><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi></mrow></mfrac><mo></mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><mi href="./2.5#SS3.p1">ρ</mi><mo>-</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi mathvariant="normal">∞</mi></mrow></mrow><mrow><mi href="./2.5#SS3.p1">ρ</mi><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi mathvariant="normal">∞</mi></mrow></mrow></msubsup><mrow><msup><mi>x</mi><mrow><mo>-</mo><mi>z</mi></mrow></msup><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mi>z</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><msub><mi href="./2.5#SS2.p1">h</mi><mn>1</mn></msub></mpadded><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>z</mi></mrow></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E41.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m41.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./1.14#E32" title="(1.14.32) ‣ §1.14(iv) Mellin Transform ‣ §1.14 Integral Transforms ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="23px" altimg-valign="-7px" altimg-width="88px" alttext="\mathscr{M}\left(\NVar{f}\right)\left(\NVar{s}\right)" display="inline"><mrow><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">f</mi><mo>)</mo></mrow><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">s</mi><mo>)</mo></mrow></mrow></math>: Mellin transform</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m98.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m161.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./2.5#SS2.p1" title="§2.5(ii) Extensions ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m136.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="h(x)" display="inline"><mrow><mi href="./2.5#SS2.p1">h</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math>: locally integrable function</a>,
<a href="./2.5#SS3.p1" title="§2.5(iii) Laplace Transforms with Small Parameters ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m35.png" altimg-height="24px" altimg-valign="-8px" altimg-width="48px" alttext="I_{j}(x)" display="inline"><mrow><msub><mi href="./2.5#SS3.p1">I</mi><mi>j</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math>: integral</a> and
<a href="./2.5#SS3.p1" title="§2.5(iii) Laplace Transforms with Small Parameters ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m102.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="\rho" display="inline"><mi href="./2.5#SS3.p1">ρ</mi></math>: parameter</a>
</dd>
<dt>Notational Change (effective with 1.0.15):</dt>
<dd>
The notation for the Mellin transform was changed to <math class="ltx_Math" altimg="m82.png" altimg-height="23px" altimg-valign="-7px" altimg-width="70px" alttext="\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(z\right)" display="inline"><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><mi>f</mi></mpadded><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></math> from <math class="ltx_Math" altimg="m77.png" altimg-height="23px" altimg-valign="-7px" altimg-width="75px" alttext="\mathscr{M}(f;z)" display="inline"><mrow><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>f</mi><mo>;</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math>.
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">for any <math class="ltx_Math" altimg="m102.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="\rho" display="inline"><mi href="./2.5#SS3.p1">ρ</mi></math> satisfying <math class="ltx_Math" altimg="m28.png" altimg-height="20px" altimg-valign="-6px" altimg-width="88px" alttext="1<\rho<2" display="inline"><mrow><mn>1</mn><mo><</mo><mi href="./2.5#SS3.p1">ρ</mi><mo><</mo><mn>2</mn></mrow></math>. Similarly, since
<math class="ltx_Math" altimg="m93.png" altimg-height="23px" altimg-valign="-7px" altimg-width="78px" alttext="\mathscr{M}\mskip-3.0mu h_{2}\mskip 3.0mu \left(z\right)" display="inline"><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><msub><mi href="./2.5#SS2.p1">h</mi><mn>2</mn></msub></mpadded><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></math> can be continued analytically to a meromorphic function
(when <math class="ltx_Math" altimg="m68.png" altimg-height="17px" altimg-valign="-2px" altimg-width="52px" alttext="\kappa=0" display="inline"><mrow><mi href="./2.5#SS2.p1">κ</mi><mo>=</mo><mn>0</mn></mrow></math>) or to an entire function (when <math class="ltx_Math" altimg="m70.png" altimg-height="21px" altimg-valign="-6px" altimg-width="52px" alttext="\kappa\neq 0" display="inline"><mrow><mi href="./2.5#SS2.p1">κ</mi><mo>≠</mo><mn>0</mn></mrow></math>), we can
choose <math class="ltx_Math" altimg="m102.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="\rho" display="inline"><mi href="./2.5#SS3.p1">ρ</mi></math> so that <math class="ltx_Math" altimg="m93.png" altimg-height="23px" altimg-valign="-7px" altimg-width="78px" alttext="\mathscr{M}\mskip-3.0mu h_{2}\mskip 3.0mu \left(z\right)" display="inline"><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><msub><mi href="./2.5#SS2.p1">h</mi><mn>2</mn></msub></mpadded><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></math> has no poles in
<math class="ltx_Math" altimg="m27.png" altimg-height="21px" altimg-valign="-6px" altimg-width="139px" alttext="1<\Re z\leq\rho<2" display="inline"><mrow><mn>1</mn><mo><</mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>z</mi></mrow><mo>≤</mo><mi href="./2.5#SS3.p1">ρ</mi><mo><</mo><mn>2</mn></mrow></math>. Thus</p>
<table id="E42" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">2.5.42</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_Math ltx_eqn_cell ltx_align_center"><math class="ltx_Math" alttext="I_{2}(x)=\sum_{\Re\beta_{0}\leq\Re z\leq 1}\Residue\left[-x^{-z}\Gamma\left(1-%
z\right)\mathscr{M}\mskip-3.0mu h_{2}\mskip 3.0mu \left(z\right)\right]+\frac{%
1}{2\pi i}\int_{\rho-i\infty}^{\rho+i\infty}x^{-z}\Gamma\left(1-z\right)%
\mathscr{M}\mskip-3.0mu h_{2}\mskip 3.0mu \left(z\right)\mathrm{d}z." display="block"><mrow><mrow><msub><mi href="./2.5#SS3.p1">I</mi><mn>2</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mrow><munder><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><msub><mi>β</mi><mn>0</mn></msub></mrow><mo>≤</mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>z</mi></mrow><mo>≤</mo><mn>1</mn></mrow></munder><mrow><mo href="./1.10#SS3.p5">res</mo><mo></mo><mrow><mo>[</mo><mrow><mo>-</mo><mrow><msup><mi>x</mi><mrow><mo>-</mo><mi>z</mi></mrow></msup><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mi>z</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><msub><mi href="./2.5#SS2.p1">h</mi><mn>2</mn></msub></mpadded><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>]</mo></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>+</mo><mrow><mfrac><mn>1</mn><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi></mrow></mfrac><mo></mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><mi href="./2.5#SS3.p1">ρ</mi><mo>-</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi mathvariant="normal">∞</mi></mrow></mrow><mrow><mi href="./2.5#SS3.p1">ρ</mi><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi mathvariant="normal">∞</mi></mrow></mrow></msubsup><mrow><msup><mi>x</mi><mrow><mo>-</mo><mi>z</mi></mrow></msup><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mi>z</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><msub><mi href="./2.5#SS2.p1">h</mi><mn>2</mn></msub></mpadded><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>z</mi></mrow></mrow></mrow></mrow><mo>.</mo></mtd></mtr></mtable></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E42.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m41.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./1.14#E32" title="(1.14.32) ‣ §1.14(iv) Mellin Transform ‣ §1.14 Integral Transforms ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="23px" altimg-valign="-7px" altimg-width="88px" alttext="\mathscr{M}\left(\NVar{f}\right)\left(\NVar{s}\right)" display="inline"><mrow><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">f</mi><mo>)</mo></mrow><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">s</mi><mo>)</mo></mrow></mrow></math>: Mellin transform</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m98.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m161.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m55.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./1.10#SS3.p5" title="§1.10(iii) Laurent Series ‣ §1.10 Functions of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m57.png" altimg-height="13px" altimg-valign="-2px" altimg-width="29px" alttext="\Residue" display="inline"><mo href="./1.10#SS3.p5">res</mo></math>: residue</a>,
<a href="./2.5#SS2.p1" title="§2.5(ii) Extensions ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m136.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="h(x)" display="inline"><mrow><mi href="./2.5#SS2.p1">h</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math>: locally integrable function</a>,
<a href="./2.5#SS3.p1" title="§2.5(iii) Laplace Transforms with Small Parameters ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m35.png" altimg-height="24px" altimg-valign="-8px" altimg-width="48px" alttext="I_{j}(x)" display="inline"><mrow><msub><mi href="./2.5#SS3.p1">I</mi><mi>j</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math>: integral</a> and
<a href="./2.5#SS3.p1" title="§2.5(iii) Laplace Transforms with Small Parameters ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m102.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="\rho" display="inline"><mi href="./2.5#SS3.p1">ρ</mi></math>: parameter</a>
</dd>
<dt>Notational Change (effective with 1.0.15):</dt>
<dd>
The notation for the Mellin transform was changed to <math class="ltx_Math" altimg="m82.png" altimg-height="23px" altimg-valign="-7px" altimg-width="70px" alttext="\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(z\right)" display="inline"><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><mi>f</mi></mpadded><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></math> from <math class="ltx_Math" altimg="m77.png" altimg-height="23px" altimg-valign="-7px" altimg-width="75px" alttext="\mathscr{M}(f;z)" display="inline"><mrow><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>f</mi><mo>;</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math>.
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS3.p2" class="ltx_para">
<p class="ltx_p">On substituting (), we obtain</p>
<table id="E43" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">2.5.43</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_Math ltx_eqn_cell ltx_align_center"><math class="ltx_Math" alttext="\mathscr{L}\mskip-3.0mu h\mskip 3.0mu \left(\zeta\right)=\mathscr{M}\mskip-3.0%
mu h_{1}\mskip 3.0mu \left(1\right)+\sum_{\Re\beta_{0}\leq\Re z\leq 1}\Residue%
\left[-\zeta^{z-1}\Gamma\left(1-z\right)\mathscr{M}\mskip-3.0mu h_{2}\mskip 3.%
0mu \left(z\right)\right]+\sum\limits_{1<\Re z<l}\Residue\left[-\zeta^{z-1}%
\Gamma\left(1-z\right)\mathscr{M}\mskip-3.0mu h\mskip 3.0mu \left(z\right)%
\right]+\frac{1}{2\pi i}\int_{l-\delta-i\infty}^{l-\delta+i\infty}\zeta^{z-1}%
\Gamma\left(1-z\right)\mathscr{M}\mskip-3.0mu h\mskip 3.0mu \left(z\right)%
\mathrm{d}z," display="block"><mrow><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E17">ℒ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><mi href="./2.5#SS2.p1">h</mi></mpadded><mo></mo><mrow><mo>(</mo><mi>ζ</mi><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><msub><mi href="./2.5#SS2.p1">h</mi><mn>1</mn></msub></mpadded><mo></mo><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></mrow><mo>+</mo><mrow><munder><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><msub><mi>β</mi><mn>0</mn></msub></mrow><mo>≤</mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>z</mi></mrow><mo>≤</mo><mn>1</mn></mrow></munder><mrow><mo href="./1.10#SS3.p5">res</mo><mo></mo><mrow><mo>[</mo><mrow><mo>-</mo><mrow><msup><mi>ζ</mi><mrow><mi>z</mi><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mi>z</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><msub><mi href="./2.5#SS2.p1">h</mi><mn>2</mn></msub></mpadded><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>]</mo></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>+</mo><mrow><munder><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mn>1</mn><mo><</mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>z</mi></mrow><mo><</mo><mi>l</mi></mrow></munder><mrow><mo href="./1.10#SS3.p5">res</mo><mo></mo><mrow><mo>[</mo><mrow><mo>-</mo><mrow><msup><mi>ζ</mi><mrow><mi>z</mi><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mi>z</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><mi href="./2.5#SS2.p1">h</mi></mpadded><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>]</mo></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>+</mo><mrow><mfrac><mn>1</mn><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi></mrow></mfrac><mo></mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><mi>l</mi><mo>-</mo><mi href="./2.5#SS3.p2">δ</mi><mo>-</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi mathvariant="normal">∞</mi></mrow></mrow><mrow><mrow><mi>l</mi><mo>-</mo><mi href="./2.5#SS3.p2">δ</mi></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi mathvariant="normal">∞</mi></mrow></mrow></msubsup><mrow><msup><mi>ζ</mi><mrow><mi>z</mi><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mi>z</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><mi href="./2.5#SS2.p1">h</mi></mpadded><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>z</mi></mrow></mrow></mrow></mrow><mo>,</mo></mtd></mtr></mtable></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E43.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m41.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./1.14#E17" title="(1.14.17) ‣ §1.14(iii) Laplace Transform ‣ §1.14 Integral Transforms ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m75.png" altimg-height="23px" altimg-valign="-7px" altimg-width="84px" alttext="\mathscr{L}\left(\NVar{f}\right)\left(\NVar{s}\right)" display="inline"><mrow><mi class="ltx_font_mathscript" href="./1.14#E17">ℒ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">f</mi><mo>)</mo></mrow><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">s</mi><mo>)</mo></mrow></mrow></math>: Laplace transform</a>,
<a href="./1.14#E32" title="(1.14.32) ‣ §1.14(iv) Mellin Transform ‣ §1.14 Integral Transforms ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="23px" altimg-valign="-7px" altimg-width="88px" alttext="\mathscr{M}\left(\NVar{f}\right)\left(\NVar{s}\right)" display="inline"><mrow><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">f</mi><mo>)</mo></mrow><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">s</mi><mo>)</mo></mrow></mrow></math>: Mellin transform</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m98.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m161.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m55.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./1.10#SS3.p5" title="§1.10(iii) Laurent Series ‣ §1.10 Functions of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m57.png" altimg-height="13px" altimg-valign="-2px" altimg-width="29px" alttext="\Residue" display="inline"><mo href="./1.10#SS3.p5">res</mo></math>: residue</a>,
<a href="./2.5#SS2.p1" title="§2.5(ii) Extensions ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m136.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="h(x)" display="inline"><mrow><mi href="./2.5#SS2.p1">h</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math>: locally integrable function</a> and
<a href="./2.5#SS3.p2" title="§2.5(iii) Laplace Transforms with Small Parameters ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m61.png" altimg-height="18px" altimg-valign="-2px" altimg-width="14px" alttext="\delta" display="inline"><mi href="./2.5#SS3.p2">δ</mi></math>: arbitrary small positive constant</a>
</dd>
<dt>Notational Change (effective with 1.0.15):</dt>
<dd>
The notation for the Laplace transform was changed to <math class="ltx_Math" altimg="m76.png" altimg-height="23px" altimg-valign="-7px" altimg-width="66px" alttext="\mathscr{L}\mskip-3.0mu f\mskip 3.0mu \left(z\right)" display="inline"><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E17">ℒ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><mi>f</mi></mpadded><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></math> from <math class="ltx_Math" altimg="m74.png" altimg-height="23px" altimg-valign="-7px" altimg-width="72px" alttext="\mathscr{L}(f;z)" display="inline"><mrow><mi class="ltx_font_mathscript" href="./1.14#E17">ℒ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>f</mi><mo>;</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math>.
In addition, the notation for the Mellin transform was changed to <math class="ltx_Math" altimg="m82.png" altimg-height="23px" altimg-valign="-7px" altimg-width="70px" alttext="\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(z\right)" display="inline"><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><mi>f</mi></mpadded><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></math>
from <math class="ltx_Math" altimg="m77.png" altimg-height="23px" altimg-valign="-7px" altimg-width="75px" alttext="\mathscr{M}(f;z)" display="inline"><mrow><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>f</mi><mo>;</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math>.
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m139.png" altimg-height="18px" altimg-valign="-2px" altimg-width="11px" alttext="l" display="inline"><mi>l</mi></math> (<math class="ltx_Math" altimg="m65.png" altimg-height="19px" altimg-valign="-5px" altimg-width="35px" alttext="\geq 2" display="inline"><mrow><mi></mi><mo>≥</mo><mn>2</mn></mrow></math>) is an arbitrary integer and <math class="ltx_Math" altimg="m61.png" altimg-height="18px" altimg-valign="-2px" altimg-width="14px" alttext="\delta" display="inline"><mi href="./2.5#SS3.p2">δ</mi></math> is an arbitrary small
positive constant.
The last term is clearly <math class="ltx_Math" altimg="m39.png" altimg-height="27px" altimg-valign="-9px" altimg-width="99px" alttext="O\left(\zeta^{l-\delta-1}\right)" display="inline"><mrow><mi href="./2.1#E3">O</mi><mo></mo><mrow><mo>(</mo><msup><mi>ζ</mi><mrow><mi>l</mi><mo>-</mo><mi href="./2.5#SS3.p2">δ</mi><mo>-</mo><mn>1</mn></mrow></msup><mo>)</mo></mrow></mrow></math> as <math class="ltx_Math" altimg="m106.png" altimg-height="21px" altimg-valign="-6px" altimg-width="71px" alttext="\zeta\to 0+" display="inline"><mrow><mi>ζ</mi><mo>→</mo><mrow><mn>0</mn><mo>+</mo></mrow></mrow></math>.
</p>
</div>
<div id="SS3.p3" class="ltx_para">
<p class="ltx_p">If <math class="ltx_Math" altimg="m68.png" altimg-height="17px" altimg-valign="-2px" altimg-width="52px" alttext="\kappa=0" display="inline"><mrow><mi href="./2.5#SS2.p1">κ</mi><mo>=</mo><mn>0</mn></mrow></math> in () and <math class="ltx_Math" altimg="m118.png" altimg-height="18px" altimg-valign="-4px" altimg-width="65px" alttext="c>-1" display="inline"><mrow><mi href="./2.5#SS1.p1">c</mi><mo>></mo><mrow><mo>-</mo><mn>1</mn></mrow></mrow></math> in () gives the following useful result:</p>
<table id="E44" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">2.5.44</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E44.png" altimg-height="64px" altimg-valign="-27px" altimg-width="513px" alttext="\mathscr{L}\mskip-3.0mu h\mskip 3.0mu \left(\zeta\right)\sim\sum_{n=0}^{\infty%
}b_{n}\Gamma\left(1-\beta_{n}\right)\zeta^{\beta_{n}-1}+\sum\limits_{n=0}^{%
\infty}\frac{(-\zeta)^{n}}{n!}\mathscr{M}\mskip-3.0mu h\mskip 3.0mu \left(n+1%
\right)," display="block"><mrow><mrow><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E17">ℒ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><mi href="./2.5#SS2.p1">h</mi></mpadded><mo></mo><mrow><mo>(</mo><mi>ζ</mi><mo>)</mo></mrow></mrow></mrow><mo href="./2.1#SS3.p1">∼</mo><mrow><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><msub><mi href="./2.5#SS1.p1">b</mi><mi>n</mi></msub><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><msub><mi>β</mi><mi>n</mi></msub></mrow><mo>)</mo></mrow></mrow><mo></mo><msup><mi>ζ</mi><mrow><msub><mi>β</mi><mi>n</mi></msub><mo>-</mo><mn>1</mn></mrow></msup></mrow></mrow><mo>+</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><mfrac><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mi>ζ</mi></mrow><mo stretchy="false">)</mo></mrow><mi>n</mi></msup><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mfrac><mo></mo><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><mi href="./2.5#SS2.p1">h</mi></mpadded><mo></mo><mrow><mo>(</mo><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m106.png" altimg-height="21px" altimg-valign="-6px" altimg-width="71px" alttext="\zeta\to 0+" display="inline"><mrow><mi>ζ</mi><mo>→</mo><mrow><mn>0</mn><mo>+</mo></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E44.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m41.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./1.14#E17" title="(1.14.17) ‣ §1.14(iii) Laplace Transform ‣ §1.14 Integral Transforms ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m75.png" altimg-height="23px" altimg-valign="-7px" altimg-width="84px" alttext="\mathscr{L}\left(\NVar{f}\right)\left(\NVar{s}\right)" display="inline"><mrow><mi class="ltx_font_mathscript" href="./1.14#E17">ℒ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">f</mi><mo>)</mo></mrow><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">s</mi><mo>)</mo></mrow></mrow></math>: Laplace transform</a>,
<a href="./1.14#E32" title="(1.14.32) ‣ §1.14(iv) Mellin Transform ‣ §1.14 Integral Transforms ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="23px" altimg-valign="-7px" altimg-width="88px" alttext="\mathscr{M}\left(\NVar{f}\right)\left(\NVar{s}\right)" display="inline"><mrow><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">f</mi><mo>)</mo></mrow><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">s</mi><mo>)</mo></mrow></mrow></math>: Mellin transform</a>,
<a href="./2.1#SS3.p1" title="§2.1(iii) Asymptotic Expansions ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="12px" altimg-valign="-2px" altimg-width="20px" alttext="\sim" display="inline"><mo href="./2.1#SS3.p1">∼</mo></math>: Poincaré asymptotic expansion</a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m13.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m140.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./2.5#SS2.p1" title="§2.5(ii) Extensions ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m136.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="h(x)" display="inline"><mrow><mi href="./2.5#SS2.p1">h</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math>: locally integrable function</a> and
<a href="./2.5#SS1.p1" title="§2.5(i) Introduction ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m114.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="b" display="inline"><mi href="./2.5#SS1.p1">b</mi></math>: right endpoint</a>
</dd>
<dt>Notational Change (effective with 1.0.15):</dt>
<dd>
The notation for the Laplace transform was changed to <math class="ltx_Math" altimg="m76.png" altimg-height="23px" altimg-valign="-7px" altimg-width="66px" alttext="\mathscr{L}\mskip-3.0mu f\mskip 3.0mu \left(z\right)" display="inline"><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E17">ℒ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><mi>f</mi></mpadded><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></math> from <math class="ltx_Math" altimg="m74.png" altimg-height="23px" altimg-valign="-7px" altimg-width="72px" alttext="\mathscr{L}(f;z)" display="inline"><mrow><mi class="ltx_font_mathscript" href="./1.14#E17">ℒ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>f</mi><mo>;</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math>.
In addition, the notation for the Mellin transform was changed to <math class="ltx_Math" altimg="m82.png" altimg-height="23px" altimg-valign="-7px" altimg-width="70px" alttext="\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(z\right)" display="inline"><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><mi>f</mi></mpadded><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></math> from
<math class="ltx_Math" altimg="m77.png" altimg-height="23px" altimg-valign="-7px" altimg-width="75px" alttext="\mathscr{M}(f;z)" display="inline"><mrow><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>f</mi><mo>;</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math>.
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<section id="Px2" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Example</h3>
<div id="Px2.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px2.p1" class="ltx_para">
<table id="E45" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">2.5.45</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E45.png" altimg-height="54px" altimg-valign="-20px" altimg-width="205px" alttext="\mathscr{L}\mskip-3.0mu h\mskip 3.0mu \left(\zeta\right)=\int_{0}^{\infty}%
\frac{e^{-\zeta t}}{1+t}\mathrm{d}t," display="block"><mrow><mrow><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E17">ℒ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><mi href="./2.5#SS2.p1">h</mi></mpadded><mo></mo><mrow><mo>(</mo><mi>ζ</mi><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><mrow><mfrac><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mi>ζ</mi><mo></mo><mi>t</mi></mrow></mrow></msup><mrow><mn>1</mn><mo>+</mo><mi>t</mi></mrow></mfrac><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m56.png" altimg-height="21px" altimg-valign="-6px" altimg-width="65px" alttext="\Re\zeta>0" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>ζ</mi></mrow><mo>></mo><mn>0</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E45.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.14#E17" title="(1.14.17) ‣ §1.14(iii) Laplace Transform ‣ §1.14 Integral Transforms ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m75.png" altimg-height="23px" altimg-valign="-7px" altimg-width="84px" alttext="\mathscr{L}\left(\NVar{f}\right)\left(\NVar{s}\right)" display="inline"><mrow><mi class="ltx_font_mathscript" href="./1.14#E17">ℒ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">f</mi><mo>)</mo></mrow><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">s</mi><mo>)</mo></mrow></mrow></math>: Laplace transform</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m161.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m73.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m55.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a> and
<a href="./2.5#SS2.p1" title="§2.5(ii) Extensions ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m136.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="h(x)" display="inline"><mrow><mi href="./2.5#SS2.p1">h</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math>: locally integrable function</a>
</dd>
<dt>Notational Change (effective with 1.0.15):</dt>
<dd>
The notation for the Laplace transform was changed to <math class="ltx_Math" altimg="m76.png" altimg-height="23px" altimg-valign="-7px" altimg-width="66px" alttext="\mathscr{L}\mskip-3.0mu f\mskip 3.0mu \left(z\right)" display="inline"><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E17">ℒ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><mi>f</mi></mpadded><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></math> from <math class="ltx_Math" altimg="m74.png" altimg-height="23px" altimg-valign="-7px" altimg-width="72px" alttext="\mathscr{L}(f;z)" display="inline"><mrow><mi class="ltx_font_mathscript" href="./1.14#E17">ℒ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>f</mi><mo>;</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math>.
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">With <math class="ltx_Math" altimg="m133.png" altimg-height="23px" altimg-valign="-7px" altimg-width="142px" alttext="h(t)=1/(1+t)" display="inline"><mrow><mrow><mi href="./2.5#SS2.p1">h</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mn>1</mn><mo>/</mo><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>+</mo><mi>t</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></math>, we have <math class="ltx_Math" altimg="m90.png" altimg-height="23px" altimg-valign="-7px" altimg-width="178px" alttext="\mathscr{M}\mskip-3.0mu h\mskip 3.0mu \left(z\right)=\pi\csc\left(\pi z\right)" display="inline"><mrow><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><mi href="./2.5#SS2.p1">h</mi></mpadded><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mrow><mi href="./3.12#E1">π</mi><mo></mo><mrow><mi href="./4.14#E5">csc</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi>z</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow></math> for
<math class="ltx_Math" altimg="m23.png" altimg-height="18px" altimg-valign="-3px" altimg-width="102px" alttext="0<\Re z<1" display="inline"><mrow><mn>0</mn><mo><</mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>z</mi></mrow><mo><</mo><mn>1</mn></mrow></math>. In the notation of (), <math class="ltx_Math" altimg="m68.png" altimg-height="17px" altimg-valign="-2px" altimg-width="52px" alttext="\kappa=0" display="inline"><mrow><mi href="./2.5#SS2.p1">κ</mi><mo>=</mo><mn>0</mn></mrow></math>, <math class="ltx_Math" altimg="m58.png" altimg-height="21px" altimg-valign="-6px" altimg-width="94px" alttext="\beta_{s}=s+1" display="inline"><mrow><msub><mi>β</mi><mi>s</mi></msub><mo>=</mo><mrow><mi>s</mi><mo>+</mo><mn>1</mn></mrow></mrow></math>, and <math class="ltx_Math" altimg="m117.png" altimg-height="17px" altimg-valign="-2px" altimg-width="49px" alttext="c=0" display="inline"><mrow><mi href="./2.5#SS1.p1">c</mi><mo>=</mo><mn>0</mn></mrow></math>.
Straightforward calculation gives</p>
<table id="E46" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">2.5.46</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E46.png" altimg-height="55px" altimg-valign="-21px" altimg-width="515px" alttext="\Residue_{z=k}\left[-\zeta^{z-1}\Gamma\left(1-z\right)\pi\csc\left(\pi z\right%
)\right]=\left(-\ln\zeta+\psi\left(k\right)\right)\dfrac{\zeta^{k-1}}{(k-1)!}," display="block"><mrow><mrow><mrow><msub><mo href="./1.10#SS3.p5">res</mo><mrow><mi>z</mi><mo>=</mo><mi>k</mi></mrow></msub><mo></mo><mrow><mo>[</mo><mrow><mo>-</mo><mrow><msup><mi>ζ</mi><mrow><mi>z</mi><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mi>z</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mrow><mi href="./4.14#E5">csc</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi>z</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>]</mo></mrow></mrow><mo>=</mo><mrow><mrow><mo>(</mo><mrow><mrow><mo>-</mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi>ζ</mi></mrow></mrow><mo>+</mo><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></mrow></mrow><mo>)</mo></mrow><mo></mo><mfrac><msup><mi>ζ</mi><mrow><mi>k</mi><mo>-</mo><mn>1</mn></mrow></msup><mrow><mrow><mo stretchy="false">(</mo><mrow><mi>k</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mfrac></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E46.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m41.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m98.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.14#E5" title="(4.14.5) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="13px" altimg-valign="-2px" altimg-width="43px" alttext="\csc\NVar{z}" display="inline"><mrow><mi href="./4.14#E5">csc</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosecant function</a>,
<a href="./5.2#E2" title="(5.2.2) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m100.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="\psi\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: psi (or digamma) function</a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m13.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m140.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m71.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a> and
<a href="./1.10#SS3.p5" title="§1.10(iii) Laurent Series ‣ §1.10 Functions of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m57.png" altimg-height="13px" altimg-valign="-2px" altimg-width="29px" alttext="\Residue" display="inline"><mo href="./1.10#SS3.p5">res</mo></math>: residue</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m101.png" altimg-height="24px" altimg-valign="-7px" altimg-width="175px" alttext="\psi\left(z\right)=\Gamma'\left(z\right)/\Gamma\left(z\right)" display="inline"><mrow><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mrow><msup><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo>′</mo></msup><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow><mo>/</mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></mrow></math>. From
()</p>
<table id="E47" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">2.5.47</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E47.png" altimg-height="38px" altimg-valign="-16px" altimg-width="501px" alttext="\Residue_{z=1}\left[-\zeta^{z-1}\Gamma\left(1-z\right)\mathscr{M}\mskip-3.0mu %
h_{2}\mskip 3.0mu \left(z\right)\right]=\left(-\ln\zeta-\gamma\right)-\mathscr%
{M}\mskip-3.0mu h_{1}\mskip 3.0mu \left(1\right)," display="block"><mrow><mrow><mrow><msub><mo href="./1.10#SS3.p5">res</mo><mrow><mi>z</mi><mo>=</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>[</mo><mrow><mo>-</mo><mrow><msup><mi>ζ</mi><mrow><mi>z</mi><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mi>z</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><msub><mi href="./2.5#SS2.p1">h</mi><mn>2</mn></msub></mpadded><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>]</mo></mrow></mrow><mo>=</mo><mrow><mrow><mo>(</mo><mrow><mrow><mo>-</mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi>ζ</mi></mrow></mrow><mo>-</mo><mi href="./5.2#E3">γ</mi></mrow><mo>)</mo></mrow><mo>-</mo><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><msub><mi href="./2.5#SS2.p1">h</mi><mn>1</mn></msub></mpadded><mo></mo><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E47.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m41.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./5.2#E3" title="(5.2.3) ‣ §5.2(ii) Euler’s Constant ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m64.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\gamma" display="inline"><mi href="./5.2#E3">γ</mi></math>: Euler’s constant</a>,
<a href="./1.14#E32" title="(1.14.32) ‣ §1.14(iv) Mellin Transform ‣ §1.14 Integral Transforms ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="23px" altimg-valign="-7px" altimg-width="88px" alttext="\mathscr{M}\left(\NVar{f}\right)\left(\NVar{s}\right)" display="inline"><mrow><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">f</mi><mo>)</mo></mrow><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">s</mi><mo>)</mo></mrow></mrow></math>: Mellin transform</a>,
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m71.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a>,
<a href="./1.10#SS3.p5" title="§1.10(iii) Laurent Series ‣ §1.10 Functions of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m57.png" altimg-height="13px" altimg-valign="-2px" altimg-width="29px" alttext="\Residue" display="inline"><mo href="./1.10#SS3.p5">res</mo></math>: residue</a> and
<a href="./2.5#SS2.p1" title="§2.5(ii) Extensions ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m136.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="h(x)" display="inline"><mrow><mi href="./2.5#SS2.p1">h</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math>: locally integrable function</a>
</dd>
<dt>Notational Change (effective with 1.0.15):</dt>
<dd>
The notation for the Mellin transform was changed to <math class="ltx_Math" altimg="m82.png" altimg-height="23px" altimg-valign="-7px" altimg-width="70px" alttext="\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(z\right)" display="inline"><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><mi>f</mi></mpadded><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></math> from <math class="ltx_Math" altimg="m77.png" altimg-height="23px" altimg-valign="-7px" altimg-width="75px" alttext="\mathscr{M}(f;z)" display="inline"><mrow><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>f</mi><mo>;</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math>.
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m64.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\gamma" display="inline"><mi href="./5.2#E3">γ</mi></math> is Euler’s constant (§) yields</p>
<table id="E48" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">2.5.48</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E48.png" altimg-height="64px" altimg-valign="-28px" altimg-width="387px" alttext="\mathscr{L}\mskip-3.0mu h\mskip 3.0mu \left(\zeta\right)\sim(-\ln\zeta)\sum_{k%
=0}^{\infty}\frac{\zeta^{k}}{k!}+\sum_{k=0}^{\infty}\psi\left(k+1\right)\frac{%
\zeta^{k}}{k!}," display="block"><mrow><mrow><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E17">ℒ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><mi href="./2.5#SS2.p1">h</mi></mpadded><mo></mo><mrow><mo>(</mo><mi>ζ</mi><mo>)</mo></mrow></mrow></mrow><mo href="./2.1#SS3.p1">∼</mo><mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi>ζ</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mfrac><msup><mi>ζ</mi><mi>k</mi></msup><mrow><mi>k</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mfrac></mrow></mrow><mo>+</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mo></mo><mfrac><msup><mi>ζ</mi><mi>k</mi></msup><mrow><mi>k</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mfrac></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m106.png" altimg-height="21px" altimg-valign="-6px" altimg-width="71px" alttext="\zeta\to 0+" display="inline"><mrow><mi>ζ</mi><mo>→</mo><mrow><mn>0</mn><mo>+</mo></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E48.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.14#E17" title="(1.14.17) ‣ §1.14(iii) Laplace Transform ‣ §1.14 Integral Transforms ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m75.png" altimg-height="23px" altimg-valign="-7px" altimg-width="84px" alttext="\mathscr{L}\left(\NVar{f}\right)\left(\NVar{s}\right)" display="inline"><mrow><mi class="ltx_font_mathscript" href="./1.14#E17">ℒ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">f</mi><mo>)</mo></mrow><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">s</mi><mo>)</mo></mrow></mrow></math>: Laplace transform</a>,
<a href="./2.1#SS3.p1" title="§2.1(iii) Asymptotic Expansions ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="12px" altimg-valign="-2px" altimg-width="20px" alttext="\sim" display="inline"><mo href="./2.1#SS3.p1">∼</mo></math>: Poincaré asymptotic expansion</a>,
<a href="./5.2#E2" title="(5.2.2) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m100.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="\psi\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: psi (or digamma) function</a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m13.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m140.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m71.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a> and
<a href="./2.5#SS2.p1" title="§2.5(ii) Extensions ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m136.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="h(x)" display="inline"><mrow><mi href="./2.5#SS2.p1">h</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math>: locally integrable function</a>
</dd>
<dt>Notational Change (effective with 1.0.15):</dt>
<dd>
The notation for the Laplace transform was changed to <math class="ltx_Math" altimg="m76.png" altimg-height="23px" altimg-valign="-7px" altimg-width="66px" alttext="\mathscr{L}\mskip-3.0mu f\mskip 3.0mu \left(z\right)" display="inline"><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E17">ℒ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><mi>f</mi></mpadded><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></math> from <math class="ltx_Math" altimg="m74.png" altimg-height="23px" altimg-valign="-7px" altimg-width="72px" alttext="\mathscr{L}(f;z)" display="inline"><mrow><mi class="ltx_font_mathscript" href="./1.14#E17">ℒ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>f</mi><mo>;</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math>.
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">To verify () we may use
</p>
<table id="E49" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">2.5.49</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E49.png" altimg-height="28px" altimg-valign="-7px" altimg-width="174px" alttext="\mathscr{L}\mskip-3.0mu h\mskip 3.0mu \left(\zeta\right)=e^{\zeta}E_{1}\left(%
\zeta\right);" display="block"><mrow><mrow><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E17">ℒ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><mi href="./2.5#SS2.p1">h</mi></mpadded><mo></mo><mrow><mo>(</mo><mi>ζ</mi><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mi>ζ</mi></msup><mo></mo><mrow><msub><mi href="./6.2#E1">E</mi><mn>1</mn></msub><mo></mo><mrow><mo>(</mo><mi>ζ</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>;</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E49.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.14#E17" title="(1.14.17) ‣ §1.14(iii) Laplace Transform ‣ §1.14 Integral Transforms ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m75.png" altimg-height="23px" altimg-valign="-7px" altimg-width="84px" alttext="\mathscr{L}\left(\NVar{f}\right)\left(\NVar{s}\right)" display="inline"><mrow><mi class="ltx_font_mathscript" href="./1.14#E17">ℒ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">f</mi><mo>)</mo></mrow><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">s</mi><mo>)</mo></mrow></mrow></math>: Laplace transform</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m73.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./6.2#E1" title="(6.2.1) ‣ §6.2(i) Exponential and Logarithmic Integrals ‣ §6.2 Definitions and Interrelations ‣ Properties ‣ Chapter 6 Exponential, Logarithmic, Sine, and Cosine Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m31.png" altimg-height="23px" altimg-valign="-7px" altimg-width="57px" alttext="E_{1}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./6.2#E1">E</mi><mn>1</mn></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./6.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: exponential integral</a> and
<a href="./2.5#SS2.p1" title="§2.5(ii) Extensions ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m136.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="h(x)" display="inline"><mrow><mi href="./2.5#SS2.p1">h</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math>: locally integrable function</a>
</dd>
<dt>Notational Change (effective with 1.0.15):</dt>
<dd>
The notation for the Laplace transform was changed to <math class="ltx_Math" altimg="m76.png" altimg-height="23px" altimg-valign="-7px" altimg-width="66px" alttext="\mathscr{L}\mskip-3.0mu f\mskip 3.0mu \left(z\right)" display="inline"><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E17">ℒ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><mi>f</mi></mpadded><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></math> from <math class="ltx_Math" altimg="m74.png" altimg-height="23px" altimg-valign="-7px" altimg-width="72px" alttext="\mathscr{L}(f;z)" display="inline"><mrow><mi class="ltx_font_mathscript" href="./1.14#E17">ℒ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>f</mi><mo>;</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math>.
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">compare ().</p>
</div>
<div id="Px2.p2" class="ltx_para">
<p class="ltx_p">For examples in which the integral defining the Mellin transform
<math class="ltx_Math" altimg="m91.png" altimg-height="23px" altimg-valign="-7px" altimg-width="69px" alttext="\mathscr{M}\mskip-3.0mu h\mskip 3.0mu \left(z\right)" display="inline"><mrow><mpadded width="-1.7pt"><mi class="ltx_font_mathscript" href="./1.14#E32">ℳ</mi></mpadded><mo></mo><mrow><mpadded width="+1.7pt"><mi href="./2.5#SS2.p1">h</mi></mpadded><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></math> does not exist for any value of <math class="ltx_Math" altimg="m167.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi>z</mi></math>, see
<cite class="ltx_cite ltx_citemacro_citet">Wong (</div>
</div>
</body></text>
</html>
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<page>
<!DOCTYPE html><html>
<head>
<title>DLMF: 1.15 Summability Methods</title>
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</dl>
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<div id="SS1.p1" class="ltx_para">
<table id="E1" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.15.1</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E1.png" altimg-height="64px" altimg-valign="-28px" altimg-width="111px" alttext="s_{n}=\sum_{k=0}^{n}a_{k}." display="block"><mrow><mrow><msub><mi>s</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo>=</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r4">k</mi><mo>=</mo><mn>0</mn></mrow><mi href="./1.1#p2.t1.r5">n</mi></munderover><msub><mi>a</mi><mi href="./1.1#p2.t1.r4">k</mi></msub></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.1#p2.t1.r4" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./1.1#p2.t1.r4">k</mi></math>: integer</a> and
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<section id="Px1" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Abel Summability</h3>
<div id="Px1.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px1.p1" class="ltx_para">
<table id="E2" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.15.2</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E2.png" altimg-height="64px" altimg-valign="-27px" altimg-width="154px" alttext="\sum^{\infty}_{n=0}a_{n}=s\;\;\;\textit{(A)}," display="block"><mrow><mrow><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></mrow><mo>=</mo><mrow><mpadded width="+8.3pt"><mi>s</mi></mpadded><mo></mo><mtext mathvariant="italic">(A)</mtext></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a></dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">if</p>
<table id="E3" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.15.3</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E3.png" altimg-height="64px" altimg-valign="-27px" altimg-width="182px" alttext="\lim_{x\to 1-}\sum^{\infty}_{n=0}a_{n}x^{n}=s." display="block"><mrow><mrow><mrow><munder><mo movablelimits="false">lim</mo><mrow><mi>x</mi><mo>→</mo><mrow><mn>1</mn><mo>-</mo></mrow></mrow></munder><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><msup><mi>x</mi><mi href="./1.1#p2.t1.r5">n</mi></msup></mrow></mrow></mrow><mo>=</mo><mi>s</mi></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a></dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px2" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Cesàro Summability</h3>
<div id="Px2.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px2.p1" class="ltx_para">
<table id="E4" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.15.4</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E4.png" altimg-height="64px" altimg-valign="-27px" altimg-width="170px" alttext="\sum^{\infty}_{n=0}a_{n}=s\;\;\;\textit{(C,1)}," display="block"><mrow><mrow><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></mrow><mo>=</mo><mrow><mpadded width="+8.3pt"><mi>s</mi></mpadded><mo></mo><mtext mathvariant="italic">(C,1)</mtext></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E4.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a></dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">if</p>
<table id="E5" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.15.5</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E5.png" altimg-height="46px" altimg-valign="-17px" altimg-width="255px" alttext="\lim_{n\to\infty}\frac{s_{0}+s_{1}+\dots+s_{n}}{n+1}=s." display="block"><mrow><mrow><mrow><munder><mo movablelimits="false">lim</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><mo></mo><mfrac><mrow><msub><mi>s</mi><mn>0</mn></msub><mo>+</mo><msub><mi>s</mi><mn>1</mn></msub><mo>+</mo><mi mathvariant="normal">…</mi><mo>+</mo><msub><mi>s</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></mrow><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mrow><mo>=</mo><mi>s</mi></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E5.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a></dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px3" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">General Cesàro Summability</h3>
<div id="Px3.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px3.p1" class="ltx_para">
<p class="ltx_p">For <math class="ltx_Math" altimg="m44.png" altimg-height="18px" altimg-valign="-4px" altimg-width="69px" alttext="\alpha>-1" display="inline"><mrow><mi>α</mi><mo>></mo><mrow><mo>-</mo><mn>1</mn></mrow></mrow></math>,
</p>
<table id="E6" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.15.6</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E6.png" altimg-height="64px" altimg-valign="-27px" altimg-width="173px" alttext="\sum^{\infty}_{n=0}a_{n}=s\;\;\;\textit{(C,$\alpha$)}," display="block"><mrow><mrow><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></mrow><mo>=</mo><mrow><mpadded width="+8.3pt"><mi>s</mi></mpadded><mo></mo><mrow><mtext mathvariant="italic">(C,</mtext><mi>α</mi><mtext mathvariant="italic">)</mtext></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E6.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a></dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">if</p>
<table id="E7" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.15.7</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E7.png" altimg-height="64px" altimg-valign="-28px" altimg-width="335px" alttext="\lim_{n\to\infty}\frac{n!}{(\alpha+1)_{n}}\sum^{n}_{k=0}\frac{(\alpha+1)_{k}}{%
k!}a_{n-k}=s." display="block"><mrow><mrow><mrow><munder><mo movablelimits="false">lim</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><mo></mo><mrow><mfrac><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow><msub><mrow><mo stretchy="false">(</mo><mrow><mi>α</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./1.1#p2.t1.r5">n</mi></msub></mfrac><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r4">k</mi><mo>=</mo><mn>0</mn></mrow><mi href="./1.1#p2.t1.r5">n</mi></munderover><mrow><mfrac><msub><mrow><mo stretchy="false">(</mo><mrow><mi>α</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./1.1#p2.t1.r4">k</mi></msub><mrow><mi href="./1.1#p2.t1.r4">k</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mfrac><mo></mo><msub><mi>a</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mi href="./1.1#p2.t1.r4">k</mi></mrow></msub></mrow></mrow></mrow></mrow><mo>=</mo><mi>s</mi></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E7.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m8.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m75.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./1.1#p2.t1.r4" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./1.1#p2.t1.r4">k</mi></math>: integer</a> and
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px4" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Borel Summability</h3>
<div id="Px4.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px4.p1" class="ltx_para">
<table id="E8" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.15.8</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E8.png" altimg-height="64px" altimg-valign="-27px" altimg-width="154px" alttext="\sum^{\infty}_{n=0}a_{n}=s\;\;\;\textit{(B)}," display="block"><mrow><mrow><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></mrow><mo>=</mo><mrow><mpadded width="+8.3pt"><mi>s</mi></mpadded><mo></mo><mtext mathvariant="italic">(B)</mtext></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E8.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a></dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">if</p>
<table id="E9" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.15.9</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E9.png" altimg-height="64px" altimg-valign="-27px" altimg-width="206px" alttext="\lim_{t\to\infty}e^{-t}\sum^{\infty}_{n=0}\frac{s_{n}}{n!}t^{n}=s." display="block"><mrow><mrow><mrow><munder><mo movablelimits="false">lim</mo><mrow><mi>t</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><mo></mo><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mi>t</mi></mrow></msup><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><mfrac><msub><mi>s</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mfrac><mo></mo><msup><mi>t</mi><mi href="./1.1#p2.t1.r5">n</mi></msup></mrow></mrow></mrow></mrow><mo>=</mo><mi>s</mi></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E9.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m55.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m8.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m75.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a> and
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
</section>
<section id="ii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§1.15(ii) </span>Regularity</h2>
<div id="SS2.info" class="ltx_metadata ltx_info">
are regular. For example if</p>
<table id="E10" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.15.10</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E10.png" altimg-height="64px" altimg-valign="-27px" altimg-width="107px" alttext="\sum^{\infty}_{n=0}a_{n}=s," display="block"><mrow><mrow><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></mrow><mo>=</mo><mi>s</mi></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E10.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a></dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">then</p>
<table id="E11" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.15.11</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E11.png" altimg-height="64px" altimg-valign="-27px" altimg-width="154px" alttext="\sum^{\infty}_{n=0}a_{n}=s\;\;\;\textit{(A)}." display="block"><mrow><mrow><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></mrow><mo>=</mo><mrow><mpadded width="+8.3pt"><mi>s</mi></mpadded><mo></mo><mtext mathvariant="italic">(A)</mtext></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E11.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a></dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="iii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§1.15(iii) </span>Summability of Fourier Series</h2>
<div id="SS3.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px5.p1" class="ltx_para">
<table id="E12" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.15.12</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E12.png" altimg-height="65px" altimg-valign="-29px" altimg-width="385px" alttext="P(r,\theta)=\frac{1-r^{2}}{1-2r\cos\theta+r^{2}}=\sum^{\infty}_{n=-\infty}r^{|%
n|}e^{in\theta}," display="block"><mrow><mrow><mrow><mi href="./1.15#E12">P</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>r</mi><mo>,</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mfrac><mrow><mn>1</mn><mo>-</mo><msup><mi>r</mi><mn>2</mn></msup></mrow><mrow><mrow><mn>1</mn><mo>-</mo><mrow><mn>2</mn><mo></mo><mi>r</mi><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi>θ</mi></mrow></mrow></mrow><mo>+</mo><msup><mi>r</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><msup><mi>r</mi><mrow><mo stretchy="false">|</mo><mi href="./1.1#p2.t1.r5">n</mi><mo stretchy="false">|</mo></mrow></msup><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./1.1#p2.t1.r5">n</mi><mo></mo><mi>θ</mi></mrow></msup></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m17.png" altimg-height="19px" altimg-valign="-5px" altimg-width="87px" alttext="0\leq r<1" display="inline"><mrow><mn>0</mn><mo>≤</mo><mi>r</mi><mo><</mo><mn>1</mn></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E12.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text"><math class="ltx_Math" altimg="m32.png" altimg-height="23px" altimg-valign="-7px" altimg-width="63px" alttext="P(r,\theta)" display="inline"><mrow><mi href="./1.15#E12">P</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>r</mi><mo>,</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow></mrow></math>: Poisson kernel (locally)</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m55.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a> and
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E13" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.15.13</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E13.png" altimg-height="56px" altimg-valign="-20px" altimg-width="199px" alttext="\frac{1}{2\pi}\int^{2\pi}_{0}P(r,\theta)\mathrm{d}\theta=1." display="block"><mrow><mrow><mrow><mfrac><mn>1</mn><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi></mrow></mfrac><mo></mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi></mrow></msubsup><mrow><mrow><mi href="./1.15#E12">P</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>r</mi><mo>,</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>θ</mi></mrow></mrow></mrow></mrow><mo>=</mo><mn>1</mn></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E13.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m57.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m84.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a> and
<a href="./1.15#E12" title="(1.15.12) ‣ Poisson Kernel ‣ §1.15(iii) Summability of Fourier Series ‣ §1.15 Summability Methods ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m32.png" altimg-height="23px" altimg-valign="-7px" altimg-width="63px" alttext="P(r,\theta)" display="inline"><mrow><mi href="./1.15#E12">P</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>r</mi><mo>,</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow></mrow></math>: Poisson kernel</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">As <math class="ltx_Math" altimg="m80.png" altimg-height="18px" altimg-valign="-4px" altimg-width="70px" alttext="r\to 1-" display="inline"><mrow><mi>r</mi><mo>→</mo><mrow><mn>1</mn><mo>-</mo></mrow></mrow></math></p>
<table id="E14" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.15.14</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E14.png" altimg-height="25px" altimg-valign="-7px" altimg-width="111px" alttext="P(r,\theta)\to 0," display="block"><mrow><mrow><mrow><mi href="./1.15#E12">P</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>r</mi><mo>,</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow></mrow><mo>→</mo><mn>0</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E14.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./1.15#E12" title="(1.15.12) ‣ Poisson Kernel ‣ §1.15(iii) Summability of Fourier Series ‣ §1.15 Summability Methods ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m32.png" altimg-height="23px" altimg-valign="-7px" altimg-width="63px" alttext="P(r,\theta)" display="inline"><mrow><mi href="./1.15#E12">P</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>r</mi><mo>,</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow></mrow></math>: Poisson kernel</a></dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">uniformly for <math class="ltx_Math" altimg="m63.png" altimg-height="23px" altimg-valign="-7px" altimg-width="123px" alttext="\theta\in[\delta,2\pi-\delta]" display="inline"><mrow><mi>θ</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">[</mo><mi>δ</mi><mo href="./front/introduction#Sx4.p1.t1.r29">,</mo><mrow><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo>-</mo><mi>δ</mi></mrow><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">]</mo></mrow></mrow></math>. (Here and elsewhere in this
subsection <math class="ltx_Math" altimg="m49.png" altimg-height="18px" altimg-valign="-2px" altimg-width="14px" alttext="\delta" display="inline"><mi>δ</mi></math> is a constant such that <math class="ltx_Math" altimg="m15.png" altimg-height="18px" altimg-valign="-3px" altimg-width="89px" alttext="0<\delta<\pi" display="inline"><mrow><mn>0</mn><mo><</mo><mi>δ</mi><mo><</mo><mi href="./3.12#E1">π</mi></mrow></math>.)</p>
</div>
</section>
<section id="Px6" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Fejér Kernel</h3>
<div id="Px6.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px6.p1" class="ltx_para">
<p class="ltx_p">For <math class="ltx_Math" altimg="m76.png" altimg-height="20px" altimg-valign="-6px" altimg-width="123px" alttext="n=0,1,2,\dots" display="inline"><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mrow><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>,</p>
<table id="E15" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.15.15</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E15.png" altimg-height="69px" altimg-valign="-27px" altimg-width="318px" alttext="K_{n}(\theta)=\frac{1}{n+1}\left(\frac{\sin\left(\tfrac{1}{2}(n+1)\theta\right%
)}{\sin\left(\tfrac{1}{2}\theta\right)}\right)^{2}," display="block"><mrow><mrow><mrow><msub><mi href="./1.15#E15">K</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mfrac><mn>1</mn><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mo></mo><msup><mrow><mo>(</mo><mfrac><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi>θ</mi></mrow><mo>)</mo></mrow></mrow><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi>θ</mi></mrow><mo>)</mo></mrow></mrow></mfrac><mo>)</mo></mrow><mn>2</mn></msup></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E15.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text"><math class="ltx_Math" altimg="m30.png" altimg-height="23px" altimg-valign="-7px" altimg-width="57px" alttext="K_{n}(\theta)" display="inline"><mrow><msub><mi href="./1.15#E15">K</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow></mrow></math>: Fejér kernel (locally)</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m61.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a> and
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E16" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.15.16</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E16.png" altimg-height="56px" altimg-valign="-20px" altimg-width="194px" alttext="\frac{1}{2\pi}\int^{2\pi}_{0}K_{n}(\theta)\mathrm{d}\theta=1." display="block"><mrow><mrow><mrow><mfrac><mn>1</mn><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi></mrow></mfrac><mo></mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi></mrow></msubsup><mrow><mrow><msub><mi href="./1.15#E15">K</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>θ</mi></mrow></mrow></mrow></mrow><mo>=</mo><mn>1</mn></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E16.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m57.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m84.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./1.15#E15" title="(1.15.15) ‣ Fejér Kernel ‣ §1.15(iii) Summability of Fourier Series ‣ §1.15 Summability Methods ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m30.png" altimg-height="23px" altimg-valign="-7px" altimg-width="57px" alttext="K_{n}(\theta)" display="inline"><mrow><msub><mi href="./1.15#E15">K</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow></mrow></math>: Fejér kernel</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">As <math class="ltx_Math" altimg="m79.png" altimg-height="13px" altimg-valign="-2px" altimg-width="67px" alttext="n\to\infty" display="inline"><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></math></p>
<table id="E17" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.15.17</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E17.png" altimg-height="25px" altimg-valign="-7px" altimg-width="106px" alttext="K_{n}(\theta)\to 0," display="block"><mrow><mrow><mrow><msub><mi href="./1.15#E15">K</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow></mrow><mo>→</mo><mn>0</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E17.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./1.15#E15" title="(1.15.15) ‣ Fejér Kernel ‣ §1.15(iii) Summability of Fourier Series ‣ §1.15 Summability Methods ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m30.png" altimg-height="23px" altimg-valign="-7px" altimg-width="57px" alttext="K_{n}(\theta)" display="inline"><mrow><msub><mi href="./1.15#E15">K</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow></mrow></math>: Fejér kernel</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">uniformly for <math class="ltx_Math" altimg="m63.png" altimg-height="23px" altimg-valign="-7px" altimg-width="123px" alttext="\theta\in[\delta,2\pi-\delta]" display="inline"><mrow><mi>θ</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">[</mo><mi>δ</mi><mo href="./front/introduction#Sx4.p1.t1.r29">,</mo><mrow><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo>-</mo><mi>δ</mi></mrow><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">]</mo></mrow></mrow></math>.</p>
</div>
</section>
<section id="Px7" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Abel Means</h3>
<div id="Px7.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px7.p1" class="ltx_para">
<table id="E18" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.15.18</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E18.png" altimg-height="65px" altimg-valign="-29px" altimg-width="256px" alttext="A(r,\theta)=\sum^{\infty}_{n=-\infty}r^{|n|}F(n)e^{in\theta}," display="block"><mrow><mrow><mrow><mi href="./1.15#E18">A</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>r</mi><mo>,</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><msup><mi>r</mi><mrow><mo stretchy="false">|</mo><mi href="./1.1#p2.t1.r5">n</mi><mo stretchy="false">|</mo></mrow></msup><mo></mo><mrow><mi href="./1.15#E19">F</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r5">n</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./1.1#p2.t1.r5">n</mi><mo></mo><mi>θ</mi></mrow></msup></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E18.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text"><math class="ltx_Math" altimg="m18.png" altimg-height="23px" altimg-valign="-7px" altimg-width="62px" alttext="A(r,\theta)" display="inline"><mrow><mi href="./1.15#E18">A</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>r</mi><mo>,</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow></mrow></math>: Abel mean (locally)</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m55.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./1.15#E19" title="(1.15.19) ‣ Abel Means ‣ §1.15(iii) Summability of Fourier Series ‣ §1.15 Summability Methods ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m23.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="F(n)" display="inline"><mrow><mi href="./1.15#E19">F</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r5">n</mi><mo stretchy="false">)</mo></mrow></mrow></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where</p>
<table id="E19" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.15.19</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E19.png" altimg-height="56px" altimg-valign="-20px" altimg-width="250px" alttext="F(n)=\frac{1}{2\pi}\int^{2\pi}_{0}f(t)e^{-int}\mathrm{d}t." display="block"><mrow><mrow><mrow><mi href="./1.15#E19">F</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r5">n</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mfrac><mn>1</mn><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi></mrow></mfrac><mo></mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi></mrow></msubsup><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./1.1#p2.t1.r5">n</mi><mo></mo><mi>t</mi></mrow></mrow></msup><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E19.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text"><math class="ltx_Math" altimg="m23.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="F(n)" display="inline"><mrow><mi href="./1.15#E19">F</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r5">n</mi><mo stretchy="false">)</mo></mrow></mrow></math> (locally)</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m57.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m84.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m55.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a> and
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p"><math class="ltx_Math" altimg="m18.png" altimg-height="23px" altimg-valign="-7px" altimg-width="62px" alttext="A(r,\theta)" display="inline"><mrow><mi href="./1.15#E18">A</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>r</mi><mo>,</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow></mrow></math> is a harmonic function in polar coordinates
(()), and</p>
<table id="E20" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.15.20</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E20.png" altimg-height="56px" altimg-valign="-20px" altimg-width="311px" alttext="A(r,\theta)=\frac{1}{2\pi}\int^{2\pi}_{0}P(r,\theta-t)f(t)\mathrm{d}t." display="block"><mrow><mrow><mrow><mi href="./1.15#E18">A</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>r</mi><mo>,</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mfrac><mn>1</mn><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi></mrow></mfrac><mo></mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi></mrow></msubsup><mrow><mrow><mi href="./1.15#E12">P</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>r</mi><mo>,</mo><mrow><mi>θ</mi><mo>-</mo><mi>t</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E20.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m57.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m84.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.15#E12" title="(1.15.12) ‣ Poisson Kernel ‣ §1.15(iii) Summability of Fourier Series ‣ §1.15 Summability Methods ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m32.png" altimg-height="23px" altimg-valign="-7px" altimg-width="63px" alttext="P(r,\theta)" display="inline"><mrow><mi href="./1.15#E12">P</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>r</mi><mo>,</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow></mrow></math>: Poisson kernel</a> and
<a href="./1.15#E18" title="(1.15.18) ‣ Abel Means ‣ §1.15(iii) Summability of Fourier Series ‣ §1.15 Summability Methods ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m18.png" altimg-height="23px" altimg-valign="-7px" altimg-width="62px" alttext="A(r,\theta)" display="inline"><mrow><mi href="./1.15#E18">A</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>r</mi><mo>,</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow></mrow></math>: Abel mean</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px8" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Cesàro (or <span class="ltx_text ltx_font_italic">(C,1)</span>) Means</h3>
<div id="Px8.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px8.p1" class="ltx_para">
<p class="ltx_p">Let</p>
<table id="E21" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.15.21</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E21.png" altimg-height="49px" altimg-valign="-17px" altimg-width="320px" alttext="\sigma_{n}(\theta)=\frac{s_{0}(\theta)+s_{1}(\theta)+\dots+s_{n}(\theta)}{n+1}," display="block"><mrow><mrow><mrow><msub><mi href="./1.15#E21">σ</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mfrac><mrow><mrow><msub><mi>s</mi><mn>0</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow></mrow><mo>+</mo><mrow><msub><mi>s</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow></mrow><mo>+</mo><mi mathvariant="normal">…</mi><mo>+</mo><mrow><msub><mi>s</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E21.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text"><math class="ltx_Math" altimg="m60.png" altimg-height="23px" altimg-valign="-7px" altimg-width="52px" alttext="\sigma_{n}(\theta)" display="inline"><mrow><msub><mi href="./1.15#E21">σ</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow></mrow></math>: Cesàro mean (locally)</span></dd>
<dt>Symbols:</dt>
<dd><a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a></dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p"><math class="ltx_Math" altimg="m76.png" altimg-height="20px" altimg-valign="-6px" altimg-width="123px" alttext="n=0,1,2,\dots" display="inline"><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mrow><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>, where</p>
<table id="E22" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.15.22</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E22.png" altimg-height="66px" altimg-valign="-30px" altimg-width="205px" alttext="s_{n}(\theta)=\sum^{n}_{k=-n}F(k)e^{ik\theta}." display="block"><mrow><mrow><mrow><msub><mi>s</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r4">k</mi><mo>=</mo><mrow><mo>-</mo><mi href="./1.1#p2.t1.r5">n</mi></mrow></mrow><mi href="./1.1#p2.t1.r5">n</mi></munderover><mrow><mrow><mi href="./1.15#E19">F</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r4">k</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./1.1#p2.t1.r4">k</mi><mo></mo><mi>θ</mi></mrow></msup></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E22.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m55.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./1.1#p2.t1.r4" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./1.1#p2.t1.r4">k</mi></math>: integer</a>,
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./1.15#E19" title="(1.15.19) ‣ Abel Means ‣ §1.15(iii) Summability of Fourier Series ‣ §1.15 Summability Methods ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m23.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="F(n)" display="inline"><mrow><mi href="./1.15#E19">F</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r5">n</mi><mo stretchy="false">)</mo></mrow></mrow></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Then</p>
<table id="E23" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.15.23</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E23.png" altimg-height="56px" altimg-valign="-20px" altimg-width="295px" alttext="\sigma_{n}(\theta)=\frac{1}{2\pi}\int^{2\pi}_{0}K_{n}(\theta-t)f(t)\mathrm{d}t." display="block"><mrow><mrow><mrow><msub><mi href="./1.15#E21">σ</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mfrac><mn>1</mn><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi></mrow></mfrac><mo></mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi></mrow></msubsup><mrow><mrow><msub><mi href="./1.15#E15">K</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>θ</mi><mo>-</mo><mi>t</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E23.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m57.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m84.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a>,
<a href="./1.15#E15" title="(1.15.15) ‣ Fejér Kernel ‣ §1.15(iii) Summability of Fourier Series ‣ §1.15 Summability Methods ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m30.png" altimg-height="23px" altimg-valign="-7px" altimg-width="57px" alttext="K_{n}(\theta)" display="inline"><mrow><msub><mi href="./1.15#E15">K</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow></mrow></math>: Fejér kernel</a> and
<a href="./1.15#E21" title="(1.15.21) ‣ Cesàro (or (C,1)) Means ‣ §1.15(iii) Summability of Fourier Series ‣ §1.15 Summability Methods ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="23px" altimg-valign="-7px" altimg-width="52px" alttext="\sigma_{n}(\theta)" display="inline"><mrow><msub><mi href="./1.15#E21">σ</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow></mrow></math>: Cesàro mean</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px9" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Convergence</h3>
<div id="Px9.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px9.p1" class="ltx_para">
<p class="ltx_p">If <math class="ltx_Math" altimg="m68.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(\theta)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow></mrow></math> is periodic and integrable on <math class="ltx_Math" altimg="m35.png" altimg-height="23px" altimg-valign="-7px" altimg-width="56px" alttext="[0,2\pi]" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">[</mo><mn>0</mn><mo href="./front/introduction#Sx4.p1.t1.r29">,</mo><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">]</mo></mrow></math>, then as <math class="ltx_Math" altimg="m79.png" altimg-height="13px" altimg-valign="-2px" altimg-width="67px" alttext="n\to\infty" display="inline"><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></math>
the Abel means <math class="ltx_Math" altimg="m18.png" altimg-height="23px" altimg-valign="-7px" altimg-width="62px" alttext="A(r,\theta)" display="inline"><mrow><mi href="./1.15#E18">A</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>r</mi><mo>,</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow></mrow></math> and the <span class="ltx_text ltx_font_italic">(C,1)</span> means <math class="ltx_Math" altimg="m60.png" altimg-height="23px" altimg-valign="-7px" altimg-width="52px" alttext="\sigma_{n}(\theta)" display="inline"><mrow><msub><mi href="./1.15#E21">σ</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow></mrow></math> converge
to
</p>
<table id="E24" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.15.24</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E24.png" altimg-height="29px" altimg-valign="-9px" altimg-width="164px" alttext="\tfrac{1}{2}(f(\theta+)+f(\theta-))" display="block"><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>θ</mi><mo>+</mo></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>+</mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>θ</mi><mo>-</mo></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E24.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">at every point <math class="ltx_Math" altimg="m62.png" altimg-height="18px" altimg-valign="-2px" altimg-width="14px" alttext="\theta" display="inline"><mi>θ</mi></math> where both limits exist. If <math class="ltx_Math" altimg="m68.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(\theta)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow></mrow></math> is also
continuous, then the convergence is uniform for all <math class="ltx_Math" altimg="m62.png" altimg-height="18px" altimg-valign="-2px" altimg-width="14px" alttext="\theta" display="inline"><mi>θ</mi></math>.</p>
</div>
<div id="Px9.p2" class="ltx_para">
<p class="ltx_p">For real-valued <math class="ltx_Math" altimg="m68.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(\theta)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow></mrow></math>, if</p>
<table id="E25" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.15.25</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E25.png" altimg-height="65px" altimg-valign="-29px" altimg-width="141px" alttext="\sum^{\infty}_{n=-\infty}F(n)e^{in\theta}" display="block"><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><mrow><mi href="./1.15#E19">F</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r5">n</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./1.1#p2.t1.r5">n</mi><mo></mo><mi>θ</mi></mrow></msup></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E25.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m55.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./1.15#E19" title="(1.15.19) ‣ Abel Means ‣ §1.15(iii) Summability of Fourier Series ‣ §1.15 Summability Methods ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m23.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="F(n)" display="inline"><mrow><mi href="./1.15#E19">F</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r5">n</mi><mo stretchy="false">)</mo></mrow></mrow></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">is the Fourier series of <math class="ltx_Math" altimg="m68.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(\theta)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow></mrow></math>, then the series</p>
<table id="E26" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.15.26</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E26.png" altimg-height="64px" altimg-valign="-27px" altimg-width="199px" alttext="F(0)+2\sum^{\infty}_{n=1}F(n)e^{in\theta}" display="block"><mrow><mrow><mi href="./1.15#E19">F</mi><mo></mo><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></mrow><mo>+</mo><mrow><mn>2</mn><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><mrow><mi href="./1.15#E19">F</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r5">n</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./1.1#p2.t1.r5">n</mi><mo></mo><mi>θ</mi></mrow></msup></mrow></mrow></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E26.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m55.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./1.15#E19" title="(1.15.19) ‣ Abel Means ‣ §1.15(iii) Summability of Fourier Series ‣ §1.15 Summability Methods ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m23.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="F(n)" display="inline"><mrow><mi href="./1.15#E19">F</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r5">n</mi><mo stretchy="false">)</mo></mrow></mrow></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">can be extended to the interior of the unit circle as an analytic function</p>
<table id="E27" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.15.27</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E27.png" altimg-height="64px" altimg-valign="-27px" altimg-width="538px" alttext="G(z)=G(x+iy)=u(x,y)+iv(x,y)=F(0)+2\sum^{\infty}_{n=1}F(n)z^{n}." display="block"><mrow><mrow><mrow><mi>G</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mi>G</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>x</mi><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi>y</mi></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mi href="./1.15#Px9.p2">u</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi>v</mi><mo></mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi>x</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi>y</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></mrow></mrow><mo>=</mo><mrow><mrow><mi href="./1.15#E19">F</mi><mo></mo><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></mrow><mo>+</mo><mrow><mn>2</mn><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><mrow><mi href="./1.15#E19">F</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r5">n</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><msup><mi href="./1.1#p2.t1.r2">z</mi><mi href="./1.1#p2.t1.r5">n</mi></msup></mrow></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E27.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./front/introduction#Sx4.p1.t1.r28" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m10.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="(\NVar{a},\NVar{b})" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi class="ltx_nvar">a</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi class="ltx_nvar">b</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math>: open interval</a>,
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m88.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>,
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a>,
<a href="./1.15#E19" title="(1.15.19) ‣ Abel Means ‣ §1.15(iii) Summability of Fourier Series ‣ §1.15 Summability Methods ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m23.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="F(n)" display="inline"><mrow><mi href="./1.15#E19">F</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r5">n</mi><mo stretchy="false">)</mo></mrow></mrow></math></a> and
<a href="./1.15#Px9.p2" title="Convergence ‣ §1.15(iii) Summability of Fourier Series ‣ §1.15 Summability Methods ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="23px" altimg-valign="-7px" altimg-width="62px" alttext="u(x,y)" display="inline"><mrow><mi href="./1.15#Px9.p2">u</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Here <math class="ltx_Math" altimg="m81.png" altimg-height="23px" altimg-valign="-7px" altimg-width="146px" alttext="u(x,y)=A(r,\theta)" display="inline"><mrow><mrow><mi href="./1.15#Px9.p2">u</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mi href="./1.15#E18">A</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>r</mi><mo>,</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></math> is the <em class="ltx_emph ltx_font_italic">Abel</em> (or <em class="ltx_emph ltx_font_italic">Poisson</em>) <em class="ltx_emph ltx_font_italic">sum</em>
of <math class="ltx_Math" altimg="m68.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(\theta)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow></mrow></math>, and <math class="ltx_Math" altimg="m83.png" altimg-height="23px" altimg-valign="-7px" altimg-width="61px" alttext="v(x,y)" display="inline"><mrow><mi>v</mi><mo></mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi>x</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi>y</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></mrow></math> has the series representation</p>
<table id="E28" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.15.28</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E28.png" altimg-height="65px" altimg-valign="-29px" altimg-width="267px" alttext="-\sum^{\infty}_{n=-\infty}i(\operatorname{sign}n)F(n)r^{|n|}e^{in\theta};" display="block"><mrow><mrow><mo>-</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><mi mathvariant="normal">i</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./front/introduction#Sx4.p2.t1.r18">sign</mi><mo></mo><mi href="./1.1#p2.t1.r5">n</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mi href="./1.15#E19">F</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r5">n</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><msup><mi>r</mi><mrow><mo stretchy="false">|</mo><mi href="./1.1#p2.t1.r5">n</mi><mo stretchy="false">|</mo></mrow></msup><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./1.1#p2.t1.r5">n</mi><mo></mo><mi>θ</mi></mrow></msup></mrow></mrow></mrow><mo>;</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E28.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m55.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./front/introduction#Sx4.p2.t1.r18" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="21px" altimg-valign="-6px" altimg-width="53px" alttext="\operatorname{sign}\NVar{x}" display="inline"><mrow><mi href="./front/introduction#Sx4.p2.t1.r18">sign</mi><mo></mo><mi class="ltx_nvar">x</mi></mrow></math>: sign of <math class="ltx_Math" altimg="m84.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./1.15#E19" title="(1.15.19) ‣ Abel Means ‣ §1.15(iii) Summability of Fourier Series ‣ §1.15 Summability Methods ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m23.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="F(n)" display="inline"><mrow><mi href="./1.15#E19">F</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r5">n</mi><mo stretchy="false">)</mo></mrow></mrow></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">compare §</dd>
</dl>
</div>
</div>
<div id="Px10.p1" class="ltx_para">
<p class="ltx_p"><math class="ltx_Math" altimg="m52.png" altimg-height="29px" altimg-valign="-11px" altimg-width="99px" alttext="\int^{\infty}_{-\infty}f(t)\mathrm{d}t" display="inline"><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mi mathvariant="normal">∞</mi></msubsup><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></math> is <em class="ltx_emph ltx_font_italic">Abel summable</em> to <math class="ltx_Math" altimg="m31.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="L" display="inline"><mi>L</mi></math>, or</p>
<table id="E29" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.15.29</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E29.png" altimg-height="54px" altimg-valign="-22px" altimg-width="196px" alttext="\int^{\infty}_{-\infty}f(t)\mathrm{d}t=L\;\;\;\textit{(A)}," display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mi mathvariant="normal">∞</mi></msubsup><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow><mo>=</mo><mrow><mpadded width="+8.3pt"><mi>L</mi></mpadded><mo></mo><mtext mathvariant="italic">(A)</mtext></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E29.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m84.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">when</p>
<table id="E30" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.15.30</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E30.png" altimg-height="54px" altimg-valign="-22px" altimg-width="244px" alttext="\lim_{\epsilon\to 0+}\int^{\infty}_{-\infty}e^{-\epsilon|t|}f(t)\mathrm{d}t=L." display="block"><mrow><mrow><mrow><munder><mo movablelimits="false">lim</mo><mrow><mi>ϵ</mi><mo>→</mo><mrow><mn>0</mn><mo>+</mo></mrow></mrow></munder><mo></mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mi mathvariant="normal">∞</mi></msubsup><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mi>ϵ</mi><mo></mo><mrow><mo stretchy="false">|</mo><mi>t</mi><mo stretchy="false">|</mo></mrow></mrow></mrow></msup><mo></mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow><mo>=</mo><mi>L</mi></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E30.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m84.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m55.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a> and
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px11" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Cesàro Summability</h3>
<div id="Px11.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px11.p1" class="ltx_para">
<p class="ltx_p"><math class="ltx_Math" altimg="m52.png" altimg-height="29px" altimg-valign="-11px" altimg-width="99px" alttext="\int^{\infty}_{-\infty}f(t)\mathrm{d}t" display="inline"><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mi mathvariant="normal">∞</mi></msubsup><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></math> is <em class="ltx_emph ltx_font_italic">(C,1) summable</em> to <math class="ltx_Math" altimg="m31.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="L" display="inline"><mi>L</mi></math>, or</p>
<table id="E31" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.15.31</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E31.png" altimg-height="54px" altimg-valign="-22px" altimg-width="212px" alttext="\int^{\infty}_{-\infty}f(t)\mathrm{d}t=L\;\;\;\textit{(C,1)}," display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mi mathvariant="normal">∞</mi></msubsup><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow><mo>=</mo><mrow><mpadded width="+8.3pt"><mi>L</mi></mpadded><mo></mo><mtext mathvariant="italic">(C,1)</mtext></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E31.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m84.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">when</p>
<table id="E32" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.15.32</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E32.png" altimg-height="57px" altimg-valign="-22px" altimg-width="286px" alttext="\lim_{R\to\infty}\int^{R}_{-R}\left(1-\frac{|t|}{R}\right)f(t)\mathrm{d}t=L." display="block"><mrow><mrow><mrow><munder><mo movablelimits="false">lim</mo><mrow><mi>R</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><mo></mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><mo>-</mo><mi>R</mi></mrow><mi>R</mi></msubsup><mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mfrac><mrow><mo stretchy="false">|</mo><mi>t</mi><mo stretchy="false">|</mo></mrow><mi>R</mi></mfrac></mrow><mo>)</mo></mrow><mo></mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow><mo>=</mo><mi>L</mi></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E32.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m84.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="Px11.p2" class="ltx_para">
<p class="ltx_p">If <math class="ltx_Math" altimg="m52.png" altimg-height="29px" altimg-valign="-11px" altimg-width="99px" alttext="\int^{\infty}_{-\infty}f(t)\mathrm{d}t" display="inline"><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mi mathvariant="normal">∞</mi></msubsup><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></math> converges and equals <math class="ltx_Math" altimg="m31.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="L" display="inline"><mi>L</mi></math>, then the
integral is Abel and Cesàro summable to <math class="ltx_Math" altimg="m31.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="L" display="inline"><mi>L</mi></math>.
</p>
</div>
</section>
</section>
<section id="v" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§1.15(v) </span>Summability of Fourier Integrals</h2>
<div id="SS5.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px12.p1" class="ltx_para">
<table id="E33" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.15.33</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E33.png" altimg-height="49px" altimg-valign="-20px" altimg-width="169px" alttext="P(x,y)=\frac{2y}{x^{2}+y^{2}}," display="block"><mrow><mrow><mrow><mi href="./1.15#E33">P</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mfrac><mrow><mn>2</mn><mo></mo><mi>y</mi></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup></mrow></mfrac></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m86.png" altimg-height="20px" altimg-valign="-6px" altimg-width="51px" alttext="y>0" display="inline"><mrow><mi>y</mi><mo>></mo><mn>0</mn></mrow></math>, <math class="ltx_Math" altimg="m13.png" altimg-height="17px" altimg-valign="-4px" altimg-width="124px" alttext="-\infty<x<\infty" display="inline"><mrow><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mo><</mo><mi>x</mi><mo><</mo><mi mathvariant="normal">∞</mi></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E33.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text"><math class="ltx_Math" altimg="m33.png" altimg-height="23px" altimg-valign="-7px" altimg-width="66px" alttext="P(x,y)" display="inline"><mrow><mi href="./1.15#E33">P</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></math>: Poisson kernel (locally)</span></dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E34" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.15.34</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E34.png" altimg-height="54px" altimg-valign="-22px" altimg-width="206px" alttext="\frac{1}{2\pi}\int^{\infty}_{-\infty}P(x,y)\mathrm{d}x=1." display="block"><mrow><mrow><mrow><mfrac><mn>1</mn><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi></mrow></mfrac><mo></mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mi mathvariant="normal">∞</mi></msubsup><mrow><mrow><mi href="./1.15#E33">P</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>x</mi></mrow></mrow></mrow></mrow><mo>=</mo><mn>1</mn></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E34.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m57.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m84.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a> and
<a href="./1.15#E33" title="(1.15.33) ‣ Poisson Kernel ‣ §1.15(v) Summability of Fourier Integrals ‣ §1.15 Summability Methods ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m33.png" altimg-height="23px" altimg-valign="-7px" altimg-width="66px" alttext="P(x,y)" display="inline"><mrow><mi href="./1.15#E33">P</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></math>: Poisson kernel</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">For each <math class="ltx_Math" altimg="m48.png" altimg-height="18px" altimg-valign="-3px" altimg-width="50px" alttext="\delta>0" display="inline"><mrow><mi>δ</mi><mo>></mo><mn>0</mn></mrow></math>,</p>
<table id="E35" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.15.35</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E35.png" altimg-height="54px" altimg-valign="-24px" altimg-width="195px" alttext="\int_{|x|\geq\delta}P(x,y)\mathrm{d}x\to 0," display="block"><mrow><mrow><mrow><msub><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><mrow><mo stretchy="false">|</mo><mi>x</mi><mo stretchy="false">|</mo></mrow><mo>≥</mo><mi>δ</mi></mrow></msub><mrow><mrow><mi href="./1.15#E33">P</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>x</mi></mrow></mrow></mrow><mo>→</mo><mn>0</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint">as <math class="ltx_Math" altimg="m87.png" altimg-height="20px" altimg-valign="-6px" altimg-width="56px" alttext="y\to 0" display="inline"><mrow><mi>y</mi><mo>→</mo><mn>0</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E35.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m84.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a> and
<a href="./1.15#E33" title="(1.15.33) ‣ Poisson Kernel ‣ §1.15(v) Summability of Fourier Integrals ‣ §1.15 Summability Methods ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m33.png" altimg-height="23px" altimg-valign="-7px" altimg-width="66px" alttext="P(x,y)" display="inline"><mrow><mi href="./1.15#E33">P</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></math>: Poisson kernel</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Let
</p>
<table id="E36" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.15.36</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E36.png" altimg-height="54px" altimg-valign="-22px" altimg-width="333px" alttext="h(x,y)=\frac{1}{\sqrt{2\pi}}\int^{\infty}_{-\infty}e^{-y|t|}e^{-ixt}F(t)%
\mathrm{d}t," display="block"><mrow><mrow><mrow><mi href="./1.15#Px12.p1">h</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mfrac><mn>1</mn><msqrt><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi></mrow></msqrt></mfrac><mo></mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mi mathvariant="normal">∞</mi></msubsup><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mi>y</mi><mo></mo><mrow><mo stretchy="false">|</mo><mi>t</mi><mo stretchy="false">|</mo></mrow></mrow></mrow></msup><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi>x</mi><mo></mo><mi>t</mi></mrow></mrow></msup><mo></mo><mrow><mi href="./1.14#SS1.p2">F</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E36.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m57.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m84.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m55.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.14#SS1.p2" title="§1.14(i) Fourier Transform ‣ §1.14 Integral Transforms ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m25.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="F(x)" display="inline"><mrow><mi href="./1.14#SS1.p2">F</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math>: Fourier transform of <math class="ltx_Math" altimg="m69.png" altimg-height="23px" altimg-valign="-7px" altimg-width="39px" alttext="f(t)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math></a> and
<a href="./1.15#Px12.p1" title="Poisson Kernel ‣ §1.15(v) Summability of Fourier Integrals ‣ §1.15 Summability Methods ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="23px" altimg-valign="-7px" altimg-width="62px" alttext="h(x,y)" display="inline"><mrow><mi href="./1.15#Px12.p1">h</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m24.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="F(t)" display="inline"><mrow><mi href="./1.14#SS1.p2">F</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math> is the Fourier transform of <math class="ltx_Math" altimg="m70.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="f(x)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math> (§). Then</p>
<table id="E37" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.15.37</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E37.png" altimg-height="54px" altimg-valign="-22px" altimg-width="310px" alttext="h(x,y)=\frac{1}{2\pi}\int^{\infty}_{-\infty}f(t)P(x-t,y)\mathrm{d}t" display="block"><mrow><mrow><mi href="./1.15#Px12.p1">h</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mfrac><mn>1</mn><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi></mrow></mfrac><mo></mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mi mathvariant="normal">∞</mi></msubsup><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mi href="./1.15#E33">P</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>x</mi><mo>-</mo><mi>t</mi></mrow><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E37.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m57.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m84.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.15#E33" title="(1.15.33) ‣ Poisson Kernel ‣ §1.15(v) Summability of Fourier Integrals ‣ §1.15 Summability Methods ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m33.png" altimg-height="23px" altimg-valign="-7px" altimg-width="66px" alttext="P(x,y)" display="inline"><mrow><mi href="./1.15#E33">P</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></math>: Poisson kernel</a> and
<a href="./1.15#Px12.p1" title="Poisson Kernel ‣ §1.15(v) Summability of Fourier Integrals ‣ §1.15 Summability Methods ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="23px" altimg-valign="-7px" altimg-width="62px" alttext="h(x,y)" display="inline"><mrow><mi href="./1.15#Px12.p1">h</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">is the <em class="ltx_emph ltx_font_italic">Poisson integral</em> of <math class="ltx_Math" altimg="m69.png" altimg-height="23px" altimg-valign="-7px" altimg-width="39px" alttext="f(t)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math>.
</p>
</div>
<div id="Px12.p2" class="ltx_para">
<p class="ltx_p">If <math class="ltx_Math" altimg="m70.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="f(x)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math> is integrable on <math class="ltx_Math" altimg="m9.png" altimg-height="23px" altimg-valign="-7px" altimg-width="84px" alttext="(-\infty,\infty)" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi mathvariant="normal">∞</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math>, then</p>
<table id="E38" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.15.38</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E38.png" altimg-height="54px" altimg-valign="-22px" altimg-width="299px" alttext="\lim_{y\to 0+}\int^{\infty}_{-\infty}|h(x,y)-f(x)|\mathrm{d}x=0." display="block"><mrow><mrow><mrow><munder><mo movablelimits="false">lim</mo><mrow><mi>y</mi><mo>→</mo><mrow><mn>0</mn><mo>+</mo></mrow></mrow></munder><mo></mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mi mathvariant="normal">∞</mi></msubsup><mrow><mrow><mo stretchy="false">|</mo><mrow><mrow><mi href="./1.15#Px12.p1">h</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow><mo>-</mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo stretchy="false">|</mo></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>x</mi></mrow></mrow></mrow></mrow><mo>=</mo><mn>0</mn></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E38.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m84.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a> and
<a href="./1.15#Px12.p1" title="Poisson Kernel ‣ §1.15(v) Summability of Fourier Integrals ‣ §1.15 Summability Methods ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="23px" altimg-valign="-7px" altimg-width="62px" alttext="h(x,y)" display="inline"><mrow><mi href="./1.15#Px12.p1">h</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="Px12.p3" class="ltx_para">
<p class="ltx_p">Suppose now <math class="ltx_Math" altimg="m70.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="f(x)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math> is real-valued and integrable on <math class="ltx_Math" altimg="m9.png" altimg-height="23px" altimg-valign="-7px" altimg-width="84px" alttext="(-\infty,\infty)" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi mathvariant="normal">∞</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math>. Let
</p>
<table id="E39" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.15.39</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E39.png" altimg-height="54px" altimg-valign="-22px" altimg-width="409px" alttext="\Phi(z)=\Phi(x+iy)=\frac{i}{\pi}\int^{\infty}_{-\infty}f(t)\frac{1}{(x-t)+iy}%
\mathrm{d}t," display="block"><mrow><mrow><mrow><mi href="./1.15#Px12.p3" mathvariant="normal">Φ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mi href="./1.15#Px12.p3" mathvariant="normal">Φ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>x</mi><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi>y</mi></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mfrac><mi mathvariant="normal">i</mi><mi href="./3.12#E1">π</mi></mfrac><mo></mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mi mathvariant="normal">∞</mi></msubsup><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mfrac><mn>1</mn><mrow><mrow><mo stretchy="false">(</mo><mrow><mi>x</mi><mo>-</mo><mi>t</mi></mrow><mo stretchy="false">)</mo></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi>y</mi></mrow></mrow></mfrac><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E39.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m57.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m84.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m88.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a> and
<a href="./1.15#Px12.p3" title="Poisson Kernel ‣ §1.15(v) Summability of Fourier Integrals ‣ §1.15 Summability Methods ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="23px" altimg-valign="-7px" altimg-width="44px" alttext="\Phi(z)" display="inline"><mrow><mi href="./1.15#Px12.p3" mathvariant="normal">Φ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m86.png" altimg-height="20px" altimg-valign="-6px" altimg-width="51px" alttext="y>0" display="inline"><mrow><mi>y</mi><mo>></mo><mn>0</mn></mrow></math> and <math class="ltx_Math" altimg="m13.png" altimg-height="17px" altimg-valign="-4px" altimg-width="124px" alttext="-\infty<x<\infty" display="inline"><mrow><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mo><</mo><mi>x</mi><mo><</mo><mi mathvariant="normal">∞</mi></mrow></math>. Then <math class="ltx_Math" altimg="m39.png" altimg-height="23px" altimg-valign="-7px" altimg-width="44px" alttext="\Phi(z)" display="inline"><mrow><mi href="./1.15#Px12.p3" mathvariant="normal">Φ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> is an analytic
function in the upper half-plane and its real part is the Poisson integral
<math class="ltx_Math" altimg="m72.png" altimg-height="23px" altimg-valign="-7px" altimg-width="62px" alttext="h(x,y)" display="inline"><mrow><mi href="./1.15#Px12.p1">h</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></math>; compare (). The imaginary part</p>
<table id="E40" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.15.40</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E40.png" altimg-height="54px" altimg-valign="-22px" altimg-width="363px" alttext="\Im\Phi(x+iy)=\frac{1}{\pi}\int^{\infty}_{-\infty}f(t)\frac{x-t}{(x-t)^{2}+y^{%
2}}\mathrm{d}t" display="block"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℑ</mi><mo></mo><mrow><mi href="./1.15#Px12.p3" mathvariant="normal">Φ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>x</mi><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi>y</mi></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>=</mo><mrow><mfrac><mn>1</mn><mi href="./3.12#E1">π</mi></mfrac><mo></mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mi mathvariant="normal">∞</mi></msubsup><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mfrac><mrow><mi>x</mi><mo>-</mo><mi>t</mi></mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mi>x</mi><mo>-</mo><mi>t</mi></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup></mrow></mfrac><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E40.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m57.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m84.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Im" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℑ</mi><mo></mo></mrow></math>: imaginary part</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a> and
<a href="./1.15#Px12.p3" title="Poisson Kernel ‣ §1.15(v) Summability of Fourier Integrals ‣ §1.15 Summability Methods ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="23px" altimg-valign="-7px" altimg-width="44px" alttext="\Phi(z)" display="inline"><mrow><mi href="./1.15#Px12.p3" mathvariant="normal">Φ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">is the <em class="ltx_emph ltx_font_italic">conjugate Poisson integral</em>
of <math class="ltx_Math" altimg="m70.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="f(x)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math>. Moreover, <math class="ltx_Math" altimg="m53.png" altimg-height="24px" altimg-valign="-8px" altimg-width="178px" alttext="\lim_{y\to 0+}\Im\Phi(x+iy)" display="inline"><mrow><msub><mo>lim</mo><mrow><mi>y</mi><mo>→</mo><mrow><mn>0</mn><mo>+</mo></mrow></mrow></msub><mo></mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℑ</mi><mo></mo><mrow><mi href="./1.15#Px12.p3" mathvariant="normal">Φ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>x</mi><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi>y</mi></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow></math> is the Hilbert
transform of <math class="ltx_Math" altimg="m70.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="f(x)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math> (§</dd>
</dl>
</div>
</div>
<div id="Px13.p1" class="ltx_para">
<table id="E41" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.15.41</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E41.png" altimg-height="48px" altimg-valign="-16px" altimg-width="234px" alttext="K_{R}(s)=\frac{1}{\pi R}\frac{1-\cos\left(Rs\right)}{s^{2}}," display="block"><mrow><mrow><mrow><msub><mi href="./1.15#E41">K</mi><mi>R</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mfrac><mn>1</mn><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi>R</mi></mrow></mfrac><mo></mo><mfrac><mrow><mn>1</mn><mo>-</mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mi>R</mi><mo></mo><mi>s</mi></mrow><mo>)</mo></mrow></mrow></mrow><msup><mi>s</mi><mn>2</mn></msup></mfrac></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E41.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text"><math class="ltx_Math" altimg="m29.png" altimg-height="23px" altimg-valign="-7px" altimg-width="59px" alttext="K_{R}(s)" display="inline"><mrow><msub><mi href="./1.15#E41">K</mi><mi>R</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mrow></mrow></math>: Fejér kernel (locally)</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m57.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a> and
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E42" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.15.42</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E42.png" altimg-height="54px" altimg-valign="-22px" altimg-width="167px" alttext="\int^{\infty}_{-\infty}K_{R}(s)\mathrm{d}s=1." display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mi mathvariant="normal">∞</mi></msubsup><mrow><mrow><msub><mi href="./1.15#E41">K</mi><mi>R</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>s</mi></mrow></mrow></mrow><mo>=</mo><mn>1</mn></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E42.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m84.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a> and
<a href="./1.15#E41" title="(1.15.41) ‣ Fejér Kernel ‣ §1.15(v) Summability of Fourier Integrals ‣ §1.15 Summability Methods ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="23px" altimg-valign="-7px" altimg-width="59px" alttext="K_{R}(s)" display="inline"><mrow><msub><mi href="./1.15#E41">K</mi><mi>R</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mrow></mrow></math>: Fejér kernel</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">For each <math class="ltx_Math" altimg="m48.png" altimg-height="18px" altimg-valign="-3px" altimg-width="50px" alttext="\delta>0" display="inline"><mrow><mi>δ</mi><mo>></mo><mn>0</mn></mrow></math>,</p>
<table id="E43" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.15.43</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E43.png" altimg-height="54px" altimg-valign="-24px" altimg-width="185px" alttext="\int_{|s|\geq\delta}K_{R}(s)\mathrm{d}s\to 0," display="block"><mrow><mrow><mrow><msub><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><mrow><mo stretchy="false">|</mo><mi>s</mi><mo stretchy="false">|</mo></mrow><mo>≥</mo><mi>δ</mi></mrow></msub><mrow><mrow><msub><mi href="./1.15#E41">K</mi><mi>R</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>s</mi></mrow></mrow></mrow><mo>→</mo><mn>0</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint">as <math class="ltx_Math" altimg="m34.png" altimg-height="18px" altimg-valign="-2px" altimg-width="71px" alttext="R\to\infty" display="inline"><mrow><mi>R</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E43.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m84.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a> and
<a href="./1.15#E41" title="(1.15.41) ‣ Fejér Kernel ‣ §1.15(v) Summability of Fourier Integrals ‣ §1.15 Summability Methods ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="23px" altimg-valign="-7px" altimg-width="59px" alttext="K_{R}(s)" display="inline"><mrow><msub><mi href="./1.15#E41">K</mi><mi>R</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mrow></mrow></math>: Fejér kernel</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="Px13.p2" class="ltx_para">
<p class="ltx_p">Let</p>
<table id="EGx1" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E44">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.15.44</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m5.png" altimg-height="25px" altimg-valign="-7px" altimg-width="56px" alttext="\displaystyle\sigma_{R}(\theta)" display="inline"><mrow><msub><mi href="./1.15#EGx1">σ</mi><mi>R</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m1.png" altimg-height="57px" altimg-valign="-22px" altimg-width="308px" alttext="\displaystyle=\frac{1}{\sqrt{2\pi}}\int^{R}_{-R}\left(1-\frac{|t|}{R}\right)e^%
{-i\theta t}F(t)\mathrm{d}t," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><mfrac><mn>1</mn><msqrt><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi></mrow></msqrt></mfrac></mstyle><mo></mo><mrow><mstyle displaystyle="true"><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><mo>-</mo><mi>R</mi></mrow><mi>R</mi></msubsup></mstyle><mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mstyle displaystyle="true"><mfrac><mrow><mo stretchy="false">|</mo><mi>t</mi><mo stretchy="false">|</mo></mrow><mi>R</mi></mfrac></mstyle></mrow><mo>)</mo></mrow><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi>θ</mi><mo></mo><mi>t</mi></mrow></mrow></msup><mo></mo><mrow><mi href="./1.14#SS1.p2">F</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E44.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m57.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m84.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m55.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.14#SS1.p2" title="§1.14(i) Fourier Transform ‣ §1.14 Integral Transforms ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m25.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="F(x)" display="inline"><mrow><mi href="./1.14#SS1.p2">F</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math>: Fourier transform of <math class="ltx_Math" altimg="m69.png" altimg-height="23px" altimg-valign="-7px" altimg-width="39px" alttext="f(t)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math></a> and
<a href="./1.15#EGx1" title="Fejér Kernel ‣ §1.15(v) Summability of Fourier Integrals ‣ §1.15 Summability Methods ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="\sigma_{R}(\theta)" display="inline"><mrow><msub><mi href="./1.15#EGx1">σ</mi><mi>R</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tr class="ltx_eqn_row ltx_align_baseline"><td class="ltx_eqn_cell ltx_align_left" style="white-space:normal;" colspan="5">then</td></tr>
<tbody id="E45">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.15.45</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m5.png" altimg-height="25px" altimg-valign="-7px" altimg-width="56px" alttext="\displaystyle\sigma_{R}(\theta)" display="inline"><mrow><msub><mi href="./1.15#EGx1">σ</mi><mi>R</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m2.png" altimg-height="54px" altimg-valign="-22px" altimg-width="216px" alttext="\displaystyle=\int^{\infty}_{-\infty}f(t)K_{R}(\theta-t)\mathrm{d}t." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mi mathvariant="normal">∞</mi></msubsup></mstyle><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./1.15#E41">K</mi><mi>R</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>θ</mi><mo>-</mo><mi>t</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E45.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m84.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.15#E41" title="(1.15.41) ‣ Fejér Kernel ‣ §1.15(v) Summability of Fourier Integrals ‣ §1.15 Summability Methods ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="23px" altimg-valign="-7px" altimg-width="59px" alttext="K_{R}(s)" display="inline"><mrow><msub><mi href="./1.15#E41">K</mi><mi>R</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mrow></mrow></math>: Fejér kernel</a> and
<a href="./1.15#EGx1" title="Fejér Kernel ‣ §1.15(v) Summability of Fourier Integrals ‣ §1.15 Summability Methods ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="\sigma_{R}(\theta)" display="inline"><mrow><msub><mi href="./1.15#EGx1">σ</mi><mi>R</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
</div>
<div id="Px13.p3" class="ltx_para">
<p class="ltx_p">If <math class="ltx_Math" altimg="m68.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(\theta)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow></mrow></math> is integrable on <math class="ltx_Math" altimg="m9.png" altimg-height="23px" altimg-valign="-7px" altimg-width="84px" alttext="(-\infty,\infty)" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi mathvariant="normal">∞</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math>, then</p>
<table id="E46" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.15.46</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E46.png" altimg-height="54px" altimg-valign="-22px" altimg-width="287px" alttext="\lim_{R\to\infty}\int^{\infty}_{-\infty}|\sigma_{R}(\theta)-f(\theta)|\mathrm{%
d}\theta=0." display="block"><mrow><mrow><mrow><munder><mo movablelimits="false">lim</mo><mrow><mi>R</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><mo></mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mi mathvariant="normal">∞</mi></msubsup><mrow><mrow><mo stretchy="false">|</mo><mrow><mrow><msub><mi href="./1.15#EGx1">σ</mi><mi>R</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow></mrow><mo>-</mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo stretchy="false">|</mo></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>θ</mi></mrow></mrow></mrow></mrow><mo>=</mo><mn>0</mn></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E46.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m84.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a> and
<a href="./1.15#EGx1" title="Fejér Kernel ‣ §1.15(v) Summability of Fourier Integrals ‣ §1.15 Summability Methods ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="\sigma_{R}(\theta)" display="inline"><mrow><msub><mi href="./1.15#EGx1">σ</mi><mi>R</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
</section>
<section id="vi" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§1.15(vi) </span>Fractional Integrals</h2>
<div id="SS6.info" class="ltx_metadata ltx_info">
</dd>
<dt>Modification (effective with 1.0.15):</dt>
<dd>
The word <em class="ltx_emph ltx_font_italic">operator</em> was removed from the definition of a fractional integral,
<p><span class="ltx_font_italic">Suggested 2017-04-22 by Tom Koornwinder</span></p>
</dd>
<dt>Modification (effective with 1.0.14):</dt>
<dd>
The <em class="ltx_emph ltx_font_italic">fractional integral operator of order</em> <math class="ltx_Math" altimg="m45.png" altimg-height="13px" altimg-valign="-2px" altimg-width="17px" alttext="\alpha" display="inline"><mi>α</mi></math> was more precisely identified as
the <em class="ltx_emph ltx_font_italic">Riemann-Liouville fractional integral operator of order</em> <math class="ltx_Math" altimg="m45.png" altimg-height="13px" altimg-valign="-2px" altimg-width="17px" alttext="\alpha" display="inline"><mi>α</mi></math>.
Also, a paragraph with a generalization of (), a
general property, and references to examples in the DLMF has been added at
the end of this subsection.
</dd>
<dt>Errata (effective with 1.0.1):</dt>
<dd>
The formulas in this subsection are only valid for <math class="ltx_Math" altimg="m85.png" altimg-height="19px" altimg-valign="-5px" altimg-width="52px" alttext="x\geq 0" display="inline"><mrow><mi>x</mi><mo>≥</mo><mn>0</mn></mrow></math>.
No conditions on <math class="ltx_Math" altimg="m84.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math> were given originally.
<p><span class="ltx_font_italic">Reported 2010-10-18 by Andreas Kurt Richter</span></p>
</dd>
</dl>
</div>
</div>
<div id="SS6.p1" class="ltx_para">
<p class="ltx_p">For <math class="ltx_Math" altimg="m40.png" altimg-height="18px" altimg-valign="-3px" altimg-width="68px" alttext="\Re\alpha>0" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>α</mi></mrow><mo>></mo><mn>0</mn></mrow></math> and <math class="ltx_Math" altimg="m85.png" altimg-height="19px" altimg-valign="-5px" altimg-width="52px" alttext="x\geq 0" display="inline"><mrow><mi>x</mi><mo>≥</mo><mn>0</mn></mrow></math>, the Riemann-Liouville <em class="ltx_emph ltx_font_italic">fractional integral of order</em>
<math class="ltx_Math" altimg="m45.png" altimg-height="13px" altimg-valign="-2px" altimg-width="17px" alttext="\alpha" display="inline"><mi>α</mi></math> is defined by</p>
<table id="E47" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.15.47</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E47.png" altimg-height="52px" altimg-valign="-21px" altimg-width="324px" alttext="I^{\alpha}f(x)=\frac{1}{\Gamma\left(\alpha\right)}\int^{x}_{0}(x-t)^{\alpha-1}%
f(t)\mathrm{d}t." display="block"><mrow><mrow><mrow><mrow><msup><mo href="./1.15#E47" mathvariant="italic">I</mo><mi>α</mi></msup><mo></mo><mi>f</mi></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mfrac><mn>1</mn><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow></mrow></mfrac><mo></mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi>x</mi></msubsup><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mi>x</mi><mo>-</mo><mi>t</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mi>α</mi><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E47.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text"><math class="ltx_Math" altimg="m27.png" altimg-height="18px" altimg-valign="-2px" altimg-width="26px" alttext="I^{\alpha}" display="inline"><msup><mo href="./1.15#E47" mathvariant="italic">I</mo><mi>α</mi></msup></math>: fractional integral (locally)</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m36.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m84.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">For <math class="ltx_Math" altimg="m37.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="\Gamma\left(\alpha\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow></mrow></math> see §) in the case when <math class="ltx_Math" altimg="m45.png" altimg-height="13px" altimg-valign="-2px" altimg-width="17px" alttext="\alpha" display="inline"><mi>α</mi></math> is a positive integer.</p>
<table id="E48" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.15.48</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E48.png" altimg-height="27px" altimg-valign="-6px" altimg-width="125px" alttext="I^{\alpha}I^{\beta}=I^{\alpha+\beta}," display="block"><mrow><mrow><mrow><msup><mo href="./1.15#E47" mathvariant="italic">I</mo><mi>α</mi></msup><mo></mo><msup><mo href="./1.15#E47" mathvariant="italic">I</mo><mi>β</mi></msup></mrow><mo>=</mo><msup><mo href="./1.15#E47" mathvariant="italic">I</mo><mrow><mi>α</mi><mo>+</mo><mi>β</mi></mrow></msup></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m40.png" altimg-height="18px" altimg-valign="-3px" altimg-width="68px" alttext="\Re\alpha>0" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>α</mi></mrow><mo>></mo><mn>0</mn></mrow></math>, <math class="ltx_Math" altimg="m41.png" altimg-height="21px" altimg-valign="-6px" altimg-width="68px" alttext="\Re\beta>0" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>β</mi></mrow><mo>></mo><mn>0</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E48.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m42.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a> and
<a href="./1.15#E47" title="(1.15.47) ‣ §1.15(vi) Fractional Integrals ‣ §1.15 Summability Methods ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m27.png" altimg-height="18px" altimg-valign="-2px" altimg-width="26px" alttext="I^{\alpha}" display="inline"><msup><mo href="./1.15#E47" mathvariant="italic">I</mo><mi>α</mi></msup></math>: fractional integral</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">For extensions of ()</cite>.</p>
</div>
<div id="SS6.p2" class="ltx_para">
<p class="ltx_p">If</p>
<table id="E49" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.15.49</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E49.png" altimg-height="64px" altimg-valign="-28px" altimg-width="151px" alttext="f(x)=\sum^{\infty}_{k=0}a_{k}x^{k}," display="block"><mrow><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r4">k</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><msub><mi>a</mi><mi href="./1.1#p2.t1.r4">k</mi></msub><mo></mo><msup><mi>x</mi><mi href="./1.1#p2.t1.r4">k</mi></msup></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E49.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./1.1#p2.t1.r4" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./1.1#p2.t1.r4">k</mi></math>: integer</a></dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">then</p>
<table id="E50" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.15.50</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E50.png" altimg-height="64px" altimg-valign="-28px" altimg-width="314px" alttext="I^{\alpha}f(x)=\sum^{\infty}_{k=0}\frac{k!}{\Gamma\left(k+\alpha+1\right)}a_{k%
}x^{k+\alpha}." display="block"><mrow><mrow><mrow><mrow><msup><mo href="./1.15#E47" mathvariant="italic">I</mo><mi>α</mi></msup><mo></mo><mi>f</mi></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r4">k</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><mfrac><mrow><mi href="./1.1#p2.t1.r4">k</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./1.1#p2.t1.r4">k</mi><mo>+</mo><mi>α</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mfrac><mo></mo><msub><mi>a</mi><mi href="./1.1#p2.t1.r4">k</mi></msub><mo></mo><msup><mi>x</mi><mrow><mi href="./1.1#p2.t1.r4">k</mi><mo>+</mo><mi>α</mi></mrow></msup></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E50.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m36.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m8.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m75.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./1.1#p2.t1.r4" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./1.1#p2.t1.r4">k</mi></math>: integer</a> and
<a href="./1.15#E47" title="(1.15.47) ‣ §1.15(vi) Fractional Integrals ‣ §1.15 Summability Methods ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m27.png" altimg-height="18px" altimg-valign="-2px" altimg-width="26px" alttext="I^{\alpha}" display="inline"><msup><mo href="./1.15#E47" mathvariant="italic">I</mo><mi>α</mi></msup></math>: fractional integral</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS6.p3" class="ltx_para">
<p class="ltx_p">The lower limit <math class="ltx_Math" altimg="m16.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math> of the integral in () can be replaced by any constant <math class="ltx_Math" altimg="m66.png" altimg-height="19px" altimg-valign="-5px" altimg-width="53px" alttext="a\leq x" display="inline"><mrow><mi>a</mi><mo>≤</mo><mi>x</mi></mrow></math> . Also, we can
replace the lower and upper limits of the integral by <math class="ltx_Math" altimg="m84.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math> and <math class="ltx_Math" altimg="m65.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi>a</mi></math>, respectively. In that case we must
also replace <math class="ltx_Math" altimg="m12.png" altimg-height="23px" altimg-valign="-7px" altimg-width="63px" alttext="(x-t)" display="inline"><mrow><mo stretchy="false">(</mo><mrow><mi>x</mi><mo>-</mo><mi>t</mi></mrow><mo stretchy="false">)</mo></mrow></math> in the integrand by <math class="ltx_Math" altimg="m11.png" altimg-height="23px" altimg-valign="-7px" altimg-width="63px" alttext="(t-x)" display="inline"><mrow><mo stretchy="false">(</mo><mrow><mi>t</mi><mo>-</mo><mi>x</mi></mrow><mo stretchy="false">)</mo></mrow></math> and we can even set <math class="ltx_Math" altimg="m64.png" altimg-height="13px" altimg-valign="-2px" altimg-width="61px" alttext="a=\infty" display="inline"><mrow><mi>a</mi><mo>=</mo><mi mathvariant="normal">∞</mi></mrow></math>.
See (</dd>
<dt>Clarification (effective with 1.0.14):</dt>
<dd>
The first sentence was clarified by adding “the fractional derivative of order <math class="ltx_Math" altimg="m45.png" altimg-height="13px" altimg-valign="-2px" altimg-width="17px" alttext="\alpha" display="inline"><mi>α</mi></math> is defined by”
ahead of ().
</dd>
<dt>Errata (effective with 1.0.1):</dt>
<dd>
The formulas in this subsection are only valid for <math class="ltx_Math" altimg="m85.png" altimg-height="19px" altimg-valign="-5px" altimg-width="52px" alttext="x\geq 0" display="inline"><mrow><mi>x</mi><mo>≥</mo><mn>0</mn></mrow></math>.
No conditions on <math class="ltx_Math" altimg="m84.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math> were given originally.
<p><span class="ltx_font_italic">Reported 2010-10-18 by Andreas Kurt Richter</span></p>
</dd>
</dl>
</div>
</div>
<div id="SS7.p1" class="ltx_para">
<p class="ltx_p">For <math class="ltx_Math" altimg="m14.png" altimg-height="18px" altimg-valign="-3px" altimg-width="107px" alttext="0<\Re\alpha<n" display="inline"><mrow><mn>0</mn><mo><</mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>α</mi></mrow><mo><</mo><mi href="./1.1#p2.t1.r5">n</mi></mrow></math>, <math class="ltx_Math" altimg="m78.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math> an integer, and <math class="ltx_Math" altimg="m85.png" altimg-height="19px" altimg-valign="-5px" altimg-width="52px" alttext="x\geq 0" display="inline"><mrow><mi>x</mi><mo>≥</mo><mn>0</mn></mrow></math>,
the fractional derivative of order <math class="ltx_Math" altimg="m45.png" altimg-height="13px" altimg-valign="-2px" altimg-width="17px" alttext="\alpha" display="inline"><mi>α</mi></math> is defined by
</p>
<table id="E51" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.15.51</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E51.png" altimg-height="47px" altimg-valign="-16px" altimg-width="226px" alttext="D^{\alpha}f(x)=\frac{{\mathrm{d}}^{n}}{{\mathrm{d}x}^{n}}I^{n-\alpha}f(x)," display="block"><mrow><mrow><mrow><mrow><msup><mo href="./1.15#E51" mathvariant="italic">D</mo><mi>α</mi></msup><mo></mo><mi>f</mi></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mfrac><mpadded lspace="-1.7pt" width="-4.5pt"><msup><mo href="./1.4#E4">d</mo><mi href="./1.1#p2.t1.r5">n</mi></msup></mpadded><msup><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi>x</mi></mrow><mi href="./1.4#E4">n</mi></msup></mfrac><mo></mo><mrow><mrow><msup><mo href="./1.15#E47" mathvariant="italic">I</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mi>α</mi></mrow></msup><mo></mo><mi>f</mi></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E51.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text"><math class="ltx_Math" altimg="m22.png" altimg-height="18px" altimg-valign="-2px" altimg-width="33px" alttext="D^{\alpha}" display="inline"><msup><mo href="./1.15#E51" mathvariant="italic">D</mo><mi>α</mi></msup></math>: fractional derivative (locally)</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#E4" title="(1.4.4) ‣ §1.4(iii) Derivatives ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="29px" altimg-valign="-9px" altimg-width="27px" alttext="\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: derivative of <math class="ltx_Math" altimg="m71.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m84.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./1.15#E47" title="(1.15.47) ‣ §1.15(vi) Fractional Integrals ‣ §1.15 Summability Methods ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m27.png" altimg-height="18px" altimg-valign="-2px" altimg-width="26px" alttext="I^{\alpha}" display="inline"><msup><mo href="./1.15#E47" mathvariant="italic">I</mo><mi>α</mi></msup></math>: fractional integral</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and satisfies the property</p>
<table id="E52" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.15.52</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E52.png" altimg-height="27px" altimg-valign="-6px" altimg-width="180px" alttext="D^{k}I^{\alpha}=D^{n}I^{\alpha+n-k}," display="block"><mrow><mrow><mrow><msup><mo href="./1.15#E51" mathvariant="italic">D</mo><mi href="./1.1#p2.t1.r4">k</mi></msup><mo></mo><msup><mo href="./1.15#E47" mathvariant="italic">I</mo><mi>α</mi></msup></mrow><mo>=</mo><mrow><msup><mo href="./1.15#E51" mathvariant="italic">D</mo><mi href="./1.1#p2.t1.r5">n</mi></msup><mo></mo><msup><mo href="./1.15#E47" mathvariant="italic">I</mo><mrow><mrow><mi>α</mi><mo>+</mo><mi href="./1.1#p2.t1.r5">n</mi></mrow><mo>-</mo><mi href="./1.1#p2.t1.r4">k</mi></mrow></msup></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m73.png" altimg-height="21px" altimg-valign="-6px" altimg-width="127px" alttext="k=1,2,\dots,n" display="inline"><mrow><mi href="./1.1#p2.t1.r4">k</mi><mo>=</mo><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><mi href="./1.1#p2.t1.r5">n</mi></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E52.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.1#p2.t1.r4" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./1.1#p2.t1.r4">k</mi></math>: integer</a>,
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a>,
<a href="./1.15#E47" title="(1.15.47) ‣ §1.15(vi) Fractional Integrals ‣ §1.15 Summability Methods ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m27.png" altimg-height="18px" altimg-valign="-2px" altimg-width="26px" alttext="I^{\alpha}" display="inline"><msup><mo href="./1.15#E47" mathvariant="italic">I</mo><mi>α</mi></msup></math>: fractional integral</a> and
<a href="./1.15#E51" title="(1.15.51) ‣ §1.15(vii) Fractional Derivatives ‣ §1.15 Summability Methods ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="18px" altimg-valign="-2px" altimg-width="33px" alttext="D^{\alpha}" display="inline"><msup><mo href="./1.15#E51" mathvariant="italic">D</mo><mi>α</mi></msup></math>: fractional derivative</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">When none of <math class="ltx_Math" altimg="m45.png" altimg-height="13px" altimg-valign="-2px" altimg-width="17px" alttext="\alpha" display="inline"><mi>α</mi></math>, <math class="ltx_Math" altimg="m46.png" altimg-height="21px" altimg-valign="-6px" altimg-width="17px" alttext="\beta" display="inline"><mi>β</mi></math>, and <math class="ltx_Math" altimg="m43.png" altimg-height="21px" altimg-valign="-6px" altimg-width="54px" alttext="\alpha+\beta" display="inline"><mrow><mi>α</mi><mo>+</mo><mi>β</mi></mrow></math> is an integer
</p>
<table id="E53" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.15.53</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E53.png" altimg-height="23px" altimg-valign="-2px" altimg-width="145px" alttext="D^{\alpha}D^{\beta}=D^{\alpha+\beta}." display="block"><mrow><mrow><mrow><msup><mo href="./1.15#E51" mathvariant="italic">D</mo><mi>α</mi></msup><mo></mo><msup><mo href="./1.15#E51" mathvariant="italic">D</mo><mi>β</mi></msup></mrow><mo>=</mo><msup><mo href="./1.15#E51" mathvariant="italic">D</mo><mrow><mi>α</mi><mo>+</mo><mi>β</mi></mrow></msup></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E53.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./1.15#E51" title="(1.15.51) ‣ §1.15(vii) Fractional Derivatives ‣ §1.15 Summability Methods ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="18px" altimg-valign="-2px" altimg-width="33px" alttext="D^{\alpha}" display="inline"><msup><mo href="./1.15#E51" mathvariant="italic">D</mo><mi>α</mi></msup></math>: fractional derivative</a></dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Note that <math class="ltx_Math" altimg="m20.png" altimg-height="25px" altimg-valign="-6px" altimg-width="132px" alttext="D^{1/2}D\not=D^{3/2}" display="inline"><mrow><mrow><msup><mo href="./1.15#E51" mathvariant="italic">D</mo><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo></mo><mo href="./1.15#E51" mathvariant="italic">D</mo></mrow><mo>≠</mo><msup><mo href="./1.15#E51" mathvariant="italic">D</mo><mrow><mn>3</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></math>. See also <cite class="ltx_cite ltx_citemacro_citet">Love (</dd>
</dl>
</div>
</div>
<div id="SS8.p1" class="ltx_para">
<p class="ltx_p">If</p>
<table id="E54" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex1" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="3" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">1.15.54</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m6.png" altimg-height="64px" altimg-valign="-27px" altimg-width="65px" alttext="\displaystyle\sum^{\infty}_{n=0}a_{n}" display="inline"><mrow><mstyle displaystyle="true"><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover></mstyle><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m3.png" altimg-height="25px" altimg-valign="-7px" altimg-width="90px" alttext="\displaystyle=s\;\;\;\textit{(A)}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mpadded width="+8.3pt"><mi>s</mi></mpadded><mo></mo><mtext mathvariant="italic">(A)</mtext></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex2" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m7.png" altimg-height="17px" altimg-valign="-5px" altimg-width="27px" alttext="\displaystyle a_{n}" display="inline"><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell">
<math class="ltx_Math" altimg="m4.png" altimg-height="46px" altimg-valign="-16px" altimg-width="66px" alttext="\displaystyle>-\frac{K}{n}" display="inline"><mrow><mi></mi><mo>></mo><mrow><mo>-</mo><mstyle displaystyle="true"><mfrac><mi>K</mi><mi href="./1.1#p2.t1.r5">n</mi></mfrac></mstyle></mrow></mrow></math>,</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m77.png" altimg-height="17px" altimg-valign="-3px" altimg-width="53px" alttext="n>0" display="inline"><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>></mo><mn>0</mn></mrow></math>, <math class="ltx_Math" altimg="m28.png" altimg-height="18px" altimg-valign="-3px" altimg-width="59px" alttext="K>0" display="inline"><mrow><mi>K</mi><mo>></mo><mn>0</mn></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E54.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a></dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">then</p>
<table id="E55" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.15.55</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E55.png" altimg-height="64px" altimg-valign="-27px" altimg-width="107px" alttext="\sum^{\infty}_{n=0}a_{n}=s." display="block"><mrow><mrow><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></mrow><mo>=</mo><mi>s</mi></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E55.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a></dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS8.p2" class="ltx_para">
<p class="ltx_p">If</p>
<table id="E56" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.15.56</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E56.png" altimg-height="64px" altimg-valign="-27px" altimg-width="243px" alttext="\lim_{x\to 1-}(1-x)\sum^{\infty}_{n=0}a_{n}x^{n}=s," display="block"><mrow><mrow><mrow><munder><mo movablelimits="false">lim</mo><mrow><mi>x</mi><mo>→</mo><mrow><mn>1</mn><mo>-</mo></mrow></mrow></munder><mo></mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><mi>x</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><msup><mi>x</mi><mi href="./1.1#p2.t1.r5">n</mi></msup></mrow></mrow></mrow></mrow><mo>=</mo><mi>s</mi></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E56.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a></dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and either <math class="ltx_Math" altimg="m89.png" altimg-height="23px" altimg-valign="-7px" altimg-width="82px" alttext="|a_{n}|\leq K" display="inline"><mrow><mrow><mo stretchy="false">|</mo><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo stretchy="false">|</mo></mrow><mo>≤</mo><mi>K</mi></mrow></math> or <math class="ltx_Math" altimg="m67.png" altimg-height="20px" altimg-valign="-5px" altimg-width="62px" alttext="a_{n}\geq 0" display="inline"><mrow><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo>≥</mo><mn>0</mn></mrow></math>, then</p>
<table id="E57" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.15.57</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E57.png" altimg-height="46px" altimg-valign="-17px" altimg-width="258px" alttext="\lim_{n\to\infty}\frac{a_{0}+a_{1}+\dots+a_{n}}{n+1}=s." display="block"><mrow><mrow><mrow><munder><mo movablelimits="false">lim</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><mo></mo><mfrac><mrow><msub><mi>a</mi><mn>0</mn></msub><mo>+</mo><msub><mi>a</mi><mn>1</mn></msub><mo>+</mo><mi mathvariant="normal">…</mi><mo>+</mo><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></mrow><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mrow><mo>=</mo><mi>s</mi></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E57.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a></dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
</section>
</div>
<div class="ltx_page_footer">
<span></div>
</div>
</body></text>
</html>
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<head>
<title>DLMF: 24.2 Definitions and Generating Functions</title>
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<div class="ltx_page_navbar">
<div class="ltx_page_navlogo"><div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd>
<span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m234.png" altimg-height="21px" altimg-valign="-5px" altimg-width="30px" alttext="B_{\NVar{n}}" display="inline"><msub><mi href="./24.2#i">B</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub></math>: Bernoulli numbers</span> and
<span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m235.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="B_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./24.2#i">B</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></math>: Bernoulli polynomials</span>
</dd>
<dt>Keywords:</dt>
<dd></dd>
</dl>
</div>
</div>
<div id="SS1.p1" class="ltx_para">
<table id="EGx1" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E1">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">24.2.1</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m26.png" altimg-height="47px" altimg-valign="-17px" altimg-width="61px" alttext="\displaystyle\frac{t}{e^{t}-1}" display="inline"><mstyle displaystyle="true"><mfrac><mi href="./24.1#p2.t1.r2">t</mi><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mi href="./24.1#p2.t1.r2">t</mi></msup><mo>-</mo><mn>1</mn></mrow></mfrac></mstyle></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m11.png" altimg-height="64px" altimg-valign="-27px" altimg-width="115px" alttext="\displaystyle=\sum_{n=0}^{\infty}B_{n}\frac{t^{n}}{n!}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./24.1#p2.t1.r1">n</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover></mstyle><mrow><msub><mi href="./24.2#i">B</mi><mi href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mstyle displaystyle="true"><mfrac><msup><mi href="./24.1#p2.t1.r2">t</mi><mi href="./24.1#p2.t1.r1">n</mi></msup><mrow><mi href="./24.1#p2.t1.r1">n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mfrac></mstyle></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m330.png" altimg-height="23px" altimg-valign="-7px" altimg-width="71px" alttext="|t|<2\pi" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mi href="./24.1#p2.t1.r2">t</mi><mo stretchy="false">|</mo></mrow><mo><</mo><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./24.2#i" title="§24.2(i) Bernoulli Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m234.png" altimg-height="21px" altimg-valign="-5px" altimg-width="30px" alttext="B_{\NVar{n}}" display="inline"><msub><mi href="./24.2#i">B</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub></math>: Bernoulli numbers</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m251.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m250.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m31.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m315.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m317.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./24.1#p2.t1.r1">n</mi></math>: integer</a> and
<a href="./24.1#p2.t1.r2" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m318.png" altimg-height="17px" altimg-valign="-2px" altimg-width="11px" alttext="t" display="inline"><mi href="./24.1#p2.t1.r2">t</mi></math>: real or complex</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tr id="E2" class="ltx_eqn_row"><td class="ltx_eqn_cell"></td></tr>
<tr id="Ex1" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="3" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">24.2.2</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m17.png" altimg-height="24px" altimg-valign="-7px" altimg-width="60px" alttext="\displaystyle B_{2n+1}" display="inline"><msub><mi href="./24.2#i">B</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./24.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mn>1</mn></mrow></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell">
<math class="ltx_Math" altimg="m4.png" altimg-height="18px" altimg-valign="-2px" altimg-width="37px" alttext="\displaystyle=0" display="inline"><mrow><mi></mi><mo>=</mo><mn>0</mn></mrow></math>,</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex2" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m1.png" altimg-height="28px" altimg-valign="-7px" altimg-width="112px" alttext="\displaystyle(-1)^{n+1}B_{2n}" display="inline"><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mrow><mi href="./24.1#p2.t1.r1">n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo></mo><msub><mi href="./24.2#i">B</mi><mrow><mn>2</mn><mo></mo><mi href="./24.1#p2.t1.r1">n</mi></mrow></msub></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell">
<math class="ltx_Math" altimg="m16.png" altimg-height="19px" altimg-valign="-3px" altimg-width="37px" alttext="\displaystyle>0" display="inline"><mrow><mi></mi><mo>></mo><mn>0</mn></mrow></math>,</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m316.png" altimg-height="20px" altimg-valign="-6px" altimg-width="104px" alttext="n=1,2,\dots" display="inline"><mrow><mi href="./24.1#p2.t1.r1">n</mi><mo>=</mo><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./24.2#i" title="§24.2(i) Bernoulli Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m234.png" altimg-height="21px" altimg-valign="-5px" altimg-width="30px" alttext="B_{\NVar{n}}" display="inline"><msub><mi href="./24.2#i">B</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub></math>: Bernoulli numbers</a> and
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m317.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./24.1#p2.t1.r1">n</mi></math>: integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
<tbody id="E3">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">24.2.3</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m25.png" altimg-height="50px" altimg-valign="-17px" altimg-width="61px" alttext="\displaystyle\frac{te^{xt}}{e^{t}-1}" display="inline"><mstyle displaystyle="true"><mfrac><mrow><mi href="./24.1#p2.t1.r2">t</mi><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mi href="./24.1#p2.t1.r2">x</mi><mo></mo><mi href="./24.1#p2.t1.r2">t</mi></mrow></msup></mrow><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mi href="./24.1#p2.t1.r2">t</mi></msup><mo>-</mo><mn>1</mn></mrow></mfrac></mstyle></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m12.png" altimg-height="64px" altimg-valign="-27px" altimg-width="148px" alttext="\displaystyle=\sum_{n=0}^{\infty}B_{n}\left(x\right)\frac{t^{n}}{n!}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./24.1#p2.t1.r1">n</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover></mstyle><mrow><mrow><msub><mi href="./24.2#i">B</mi><mi href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow><mo></mo><mstyle displaystyle="true"><mfrac><msup><mi href="./24.1#p2.t1.r2">t</mi><mi href="./24.1#p2.t1.r1">n</mi></msup><mrow><mi href="./24.1#p2.t1.r1">n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mfrac></mstyle></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m330.png" altimg-height="23px" altimg-valign="-7px" altimg-width="71px" alttext="|t|<2\pi" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mi href="./24.1#p2.t1.r2">t</mi><mo stretchy="false">|</mo></mrow><mo><</mo><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./24.2#i" title="§24.2(i) Bernoulli Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m235.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="B_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./24.2#i">B</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></math>: Bernoulli polynomials</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m251.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m250.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m31.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m315.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m317.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./24.1#p2.t1.r1">n</mi></math>: integer</a>,
<a href="./24.1#p2.t1.r2" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m318.png" altimg-height="17px" altimg-valign="-2px" altimg-width="11px" alttext="t" display="inline"><mi href="./24.1#p2.t1.r2">t</mi></math>: real or complex</a> and
<a href="./24.1#p2.t1.r2" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m320.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./24.1#p2.t1.r2">x</mi></math>: real or complex</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">23.1.1</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E4">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">24.2.4</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m18.png" altimg-height="22px" altimg-valign="-5px" altimg-width="32px" alttext="\displaystyle B_{n}" display="inline"><msub><mi href="./24.2#i">B</mi><mi href="./24.1#p2.t1.r1">n</mi></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m6.png" altimg-height="25px" altimg-valign="-7px" altimg-width="91px" alttext="\displaystyle=B_{n}\left(0\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msub><mi href="./24.2#i">B</mi><mi href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E4.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./24.2#i" title="§24.2(i) Bernoulli Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m234.png" altimg-height="21px" altimg-valign="-5px" altimg-width="30px" alttext="B_{\NVar{n}}" display="inline"><msub><mi href="./24.2#i">B</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub></math>: Bernoulli numbers</a>,
<a href="./24.2#i" title="§24.2(i) Bernoulli Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m235.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="B_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./24.2#i">B</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></math>: Bernoulli polynomials</a> and
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m317.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./24.1#p2.t1.r1">n</mi></math>: integer</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">23.1.2</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E5">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">24.2.5</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m19.png" altimg-height="25px" altimg-valign="-7px" altimg-width="62px" alttext="\displaystyle B_{n}\left(x\right)" display="inline"><mrow><msub><mi href="./24.2#i">B</mi><mi href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m9.png" altimg-height="64px" altimg-valign="-28px" altimg-width="175px" alttext="\displaystyle=\sum_{k=0}^{n}{n\choose k}B_{k}x^{n-k}." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./24.1#p2.t1.r1">k</mi><mo>=</mo><mn>0</mn></mrow><mi href="./24.1#p2.t1.r1">n</mi></munderover></mstyle><mrow><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac linethickness="0pt"><mi href="./24.1#p2.t1.r1">n</mi><mi href="./24.1#p2.t1.r1">k</mi></mfrac></mstyle><mo>)</mo></mrow><mo></mo><msub><mi href="./24.2#i">B</mi><mi href="./24.1#p2.t1.r1">k</mi></msub><mo></mo><msup><mi href="./24.1#p2.t1.r2">x</mi><mrow><mi href="./24.1#p2.t1.r1">n</mi><mo>-</mo><mi href="./24.1#p2.t1.r1">k</mi></mrow></msup></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E5.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./24.2#i" title="§24.2(i) Bernoulli Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m234.png" altimg-height="21px" altimg-valign="-5px" altimg-width="30px" alttext="B_{\NVar{n}}" display="inline"><msub><mi href="./24.2#i">B</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub></math>: Bernoulli numbers</a>,
<a href="./24.2#i" title="§24.2(i) Bernoulli Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m235.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="B_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./24.2#i">B</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></math>: Bernoulli polynomials</a>,
<a href="./1.2#i" title="§1.2(i) Binomial Coefficients ‣ §1.2 Elementary Algebra ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m247.png" altimg-height="27px" altimg-valign="-9px" altimg-width="37px" alttext="\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}" display="inline"><mrow><mo href="./1.2#i">(</mo><mfrac linethickness="0.0pt"><mi class="ltx_nvar" href="./1.1#p2.t1.r5">m</mi><mi class="ltx_nvar" href="./1.1#p2.t1.r5">n</mi></mfrac><mo href="./1.2#i">)</mo></mrow></math>: binomial coefficient</a>,
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m314.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./24.1#p2.t1.r1">k</mi></math>: integer</a>,
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m317.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./24.1#p2.t1.r1">n</mi></math>: integer</a> and
<a href="./24.1#p2.t1.r2" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m320.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./24.1#p2.t1.r2">x</mi></math>: real or complex</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
</div>
<div id="SS1.p2" class="ltx_para">
<p class="ltx_p">See also §§<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd>
<span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m241.png" altimg-height="21px" altimg-valign="-5px" altimg-width="30px" alttext="E_{\NVar{n}}" display="inline"><msub><mi href="./24.2#ii">E</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub></math>: Euler numbers</span> and
<span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m242.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="E_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./24.2#ii">E</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></math>: Euler polynomials</span>
</dd>
<dt>Keywords:</dt>
<dd>
</dd>
</dl>
</div>
</div>
<div id="SS2.p1" class="ltx_para">
<table id="EGx2" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E6">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">24.2.6</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m23.png" altimg-height="50px" altimg-valign="-17px" altimg-width="69px" alttext="\displaystyle\frac{2e^{t}}{e^{2t}+1}" display="inline"><mstyle displaystyle="true"><mfrac><mrow><mn>2</mn><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mi href="./24.1#p2.t1.r2">t</mi></msup></mrow><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mn>2</mn><mo></mo><mi href="./24.1#p2.t1.r2">t</mi></mrow></msup><mo>+</mo><mn>1</mn></mrow></mfrac></mstyle></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m13.png" altimg-height="64px" altimg-valign="-27px" altimg-width="114px" alttext="\displaystyle=\sum_{n=0}^{\infty}E_{n}\frac{t^{n}}{n!}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./24.1#p2.t1.r1">n</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover></mstyle><mrow><msub><mi href="./24.2#ii">E</mi><mi href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mstyle displaystyle="true"><mfrac><msup><mi href="./24.1#p2.t1.r2">t</mi><mi href="./24.1#p2.t1.r1">n</mi></msup><mrow><mi href="./24.1#p2.t1.r1">n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mfrac></mstyle></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m332.png" altimg-height="27px" altimg-valign="-9px" altimg-width="74px" alttext="|t|<\tfrac{1}{2}\pi" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mi href="./24.1#p2.t1.r2">t</mi><mo stretchy="false">|</mo></mrow><mo><</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E6.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./24.2#ii" title="§24.2(ii) Euler Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m241.png" altimg-height="21px" altimg-valign="-5px" altimg-width="30px" alttext="E_{\NVar{n}}" display="inline"><msub><mi href="./24.2#ii">E</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub></math>: Euler numbers</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m251.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m250.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m31.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m315.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m317.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./24.1#p2.t1.r1">n</mi></math>: integer</a> and
<a href="./24.1#p2.t1.r2" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m318.png" altimg-height="17px" altimg-valign="-2px" altimg-width="11px" alttext="t" display="inline"><mi href="./24.1#p2.t1.r2">t</mi></math>: real or complex</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tr id="E7" class="ltx_eqn_row"><td class="ltx_eqn_cell"></td></tr>
<tr id="Ex3" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">24.2.7</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m20.png" altimg-height="24px" altimg-valign="-7px" altimg-width="60px" alttext="\displaystyle E_{2n+1}" display="inline"><msub><mi href="./24.2#ii">E</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./24.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mn>1</mn></mrow></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell">
<math class="ltx_Math" altimg="m4.png" altimg-height="18px" altimg-valign="-2px" altimg-width="37px" alttext="\displaystyle=0" display="inline"><mrow><mi></mi><mo>=</mo><mn>0</mn></mrow></math>,</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex4" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m2.png" altimg-height="25px" altimg-valign="-7px" altimg-width="91px" alttext="\displaystyle(-1)^{n}E_{2n}" display="inline"><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./24.1#p2.t1.r1">n</mi></msup><mo></mo><msub><mi href="./24.2#ii">E</mi><mrow><mn>2</mn><mo></mo><mi href="./24.1#p2.t1.r1">n</mi></mrow></msub></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell">
<math class="ltx_Math" altimg="m16.png" altimg-height="19px" altimg-valign="-3px" altimg-width="37px" alttext="\displaystyle>0" display="inline"><mrow><mi></mi><mo>></mo><mn>0</mn></mrow></math>.</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E7.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./24.2#ii" title="§24.2(ii) Euler Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m241.png" altimg-height="21px" altimg-valign="-5px" altimg-width="30px" alttext="E_{\NVar{n}}" display="inline"><msub><mi href="./24.2#ii">E</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub></math>: Euler numbers</a> and
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m317.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./24.1#p2.t1.r1">n</mi></math>: integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
<tbody id="E8">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">24.2.8</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m24.png" altimg-height="50px" altimg-valign="-17px" altimg-width="61px" alttext="\displaystyle\frac{2e^{xt}}{e^{t}+1}" display="inline"><mstyle displaystyle="true"><mfrac><mrow><mn>2</mn><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mi href="./24.1#p2.t1.r2">x</mi><mo></mo><mi href="./24.1#p2.t1.r2">t</mi></mrow></msup></mrow><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mi href="./24.1#p2.t1.r2">t</mi></msup><mo>+</mo><mn>1</mn></mrow></mfrac></mstyle></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m14.png" altimg-height="64px" altimg-valign="-27px" altimg-width="148px" alttext="\displaystyle=\sum_{n=0}^{\infty}E_{n}\left(x\right)\frac{t^{n}}{n!}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./24.1#p2.t1.r1">n</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover></mstyle><mrow><mrow><msub><mi href="./24.2#ii">E</mi><mi href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow><mo></mo><mstyle displaystyle="true"><mfrac><msup><mi href="./24.1#p2.t1.r2">t</mi><mi href="./24.1#p2.t1.r1">n</mi></msup><mrow><mi href="./24.1#p2.t1.r1">n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mfrac></mstyle></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m331.png" altimg-height="23px" altimg-valign="-7px" altimg-width="61px" alttext="|t|<\pi" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mi href="./24.1#p2.t1.r2">t</mi><mo stretchy="false">|</mo></mrow><mo><</mo><mi href="./3.12#E1">π</mi></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E8.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./24.2#ii" title="§24.2(ii) Euler Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m242.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="E_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./24.2#ii">E</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></math>: Euler polynomials</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m251.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m250.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m31.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m315.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m317.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./24.1#p2.t1.r1">n</mi></math>: integer</a>,
<a href="./24.1#p2.t1.r2" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m318.png" altimg-height="17px" altimg-valign="-2px" altimg-width="11px" alttext="t" display="inline"><mi href="./24.1#p2.t1.r2">t</mi></math>: real or complex</a> and
<a href="./24.1#p2.t1.r2" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m320.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./24.1#p2.t1.r2">x</mi></math>: real or complex</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">23.1.1</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E9">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">24.2.9</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m21.png" altimg-height="22px" altimg-valign="-5px" altimg-width="32px" alttext="\displaystyle E_{n}" display="inline"><msub><mi href="./24.2#ii">E</mi><mi href="./24.1#p2.t1.r1">n</mi></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m5.png" altimg-height="29px" altimg-valign="-9px" altimg-width="199px" alttext="\displaystyle=2^{n}E_{n}\left(\tfrac{1}{2}\right)=\text{integer}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msup><mn>2</mn><mi href="./24.1#p2.t1.r1">n</mi></msup><mo></mo><mrow><msub><mi href="./24.2#ii">E</mi><mi href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mtext>integer</mtext></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E9.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./24.2#ii" title="§24.2(ii) Euler Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m241.png" altimg-height="21px" altimg-valign="-5px" altimg-width="30px" alttext="E_{\NVar{n}}" display="inline"><msub><mi href="./24.2#ii">E</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub></math>: Euler numbers</a>,
<a href="./24.2#ii" title="§24.2(ii) Euler Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m242.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="E_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./24.2#ii">E</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></math>: Euler polynomials</a> and
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m317.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./24.1#p2.t1.r1">n</mi></math>: integer</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">23.1.2</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E10">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">24.2.10</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m22.png" altimg-height="25px" altimg-valign="-7px" altimg-width="62px" alttext="\displaystyle E_{n}\left(x\right)" display="inline"><mrow><msub><mi href="./24.2#ii">E</mi><mi href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m10.png" altimg-height="64px" altimg-valign="-28px" altimg-width="232px" alttext="\displaystyle=\sum_{k=0}^{n}{n\choose k}\frac{E_{k}}{2^{k}}(x-\tfrac{1}{2})^{n%
-k}." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./24.1#p2.t1.r1">k</mi><mo>=</mo><mn>0</mn></mrow><mi href="./24.1#p2.t1.r1">n</mi></munderover></mstyle><mrow><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac linethickness="0pt"><mi href="./24.1#p2.t1.r1">n</mi><mi href="./24.1#p2.t1.r1">k</mi></mfrac></mstyle><mo>)</mo></mrow><mo></mo><mstyle displaystyle="true"><mfrac><msub><mi href="./24.2#ii">E</mi><mi href="./24.1#p2.t1.r1">k</mi></msub><msup><mn>2</mn><mi href="./24.1#p2.t1.r1">k</mi></msup></mfrac></mstyle><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./24.1#p2.t1.r2">x</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo stretchy="false">)</mo></mrow><mrow><mi href="./24.1#p2.t1.r1">n</mi><mo>-</mo><mi href="./24.1#p2.t1.r1">k</mi></mrow></msup></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E10.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./24.2#ii" title="§24.2(ii) Euler Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m241.png" altimg-height="21px" altimg-valign="-5px" altimg-width="30px" alttext="E_{\NVar{n}}" display="inline"><msub><mi href="./24.2#ii">E</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub></math>: Euler numbers</a>,
<a href="./24.2#ii" title="§24.2(ii) Euler Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m242.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="E_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./24.2#ii">E</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></math>: Euler polynomials</a>,
<a href="./1.2#i" title="§1.2(i) Binomial Coefficients ‣ §1.2 Elementary Algebra ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m247.png" altimg-height="27px" altimg-valign="-9px" altimg-width="37px" alttext="\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}" display="inline"><mrow><mo href="./1.2#i">(</mo><mfrac linethickness="0.0pt"><mi class="ltx_nvar" href="./1.1#p2.t1.r5">m</mi><mi class="ltx_nvar" href="./1.1#p2.t1.r5">n</mi></mfrac><mo href="./1.2#i">)</mo></mrow></math>: binomial coefficient</a>,
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m314.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./24.1#p2.t1.r1">k</mi></math>: integer</a>,
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m317.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./24.1#p2.t1.r1">n</mi></math>: integer</a> and
<a href="./24.1#p2.t1.r2" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m320.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./24.1#p2.t1.r2">x</mi></math>: real or complex</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">23.1.7</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
</div>
<div id="SS2.p2" class="ltx_para">
<p class="ltx_p">See also (<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd>
<span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m310.png" altimg-height="28px" altimg-valign="-7px" altimg-width="60px" alttext="\widetilde{B}_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mover accent="true"><mi href="./24.2#iii">B</mi><mo href="./24.2#iii">~</mo></mover><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></math>: periodic Bernoulli functions</span> and
<span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m311.png" altimg-height="28px" altimg-valign="-7px" altimg-width="60px" alttext="\widetilde{E}_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mover accent="true"><mi href="./24.2#iii">E</mi><mo href="./24.2#iii">~</mo></mover><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></math>: periodic Euler functions</span>
</dd>
<dt>Keywords:</dt>
<dd>
</dd>
</dl>
</div>
</div>
<div id="SS3.p1" class="ltx_para">
<table id="E11" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex5" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="3" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">24.2.11</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m28.png" altimg-height="30px" altimg-valign="-7px" altimg-width="62px" alttext="\displaystyle\widetilde{B}_{n}\left(x\right)" display="inline"><mrow><msub><mover accent="true"><mi href="./24.2#iii">B</mi><mo href="./24.2#iii">~</mo></mover><mi href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell">
<math class="ltx_Math" altimg="m7.png" altimg-height="25px" altimg-valign="-7px" altimg-width="83px" alttext="\displaystyle=B_{n}\left(x\right)" display="inline"><mrow><mi></mi><mo>=</mo><mrow><msub><mi href="./24.2#i">B</mi><mi href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></mrow></math>,</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex6" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m30.png" altimg-height="30px" altimg-valign="-7px" altimg-width="62px" alttext="\displaystyle\widetilde{E}_{n}\left(x\right)" display="inline"><mrow><msub><mover accent="true"><mi href="./24.2#iii">E</mi><mo href="./24.2#iii">~</mo></mover><mi href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell">
<math class="ltx_Math" altimg="m8.png" altimg-height="25px" altimg-valign="-7px" altimg-width="83px" alttext="\displaystyle=E_{n}\left(x\right)" display="inline"><mrow><mi></mi><mo>=</mo><mrow><msub><mi href="./24.2#ii">E</mi><mi href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></mrow></math>,</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m128.png" altimg-height="19px" altimg-valign="-5px" altimg-width="89px" alttext="0\leq x<1" display="inline"><mrow><mn>0</mn><mo>≤</mo><mi href="./24.1#p2.t1.r2">x</mi><mo><</mo><mn>1</mn></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E11.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./24.2#i" title="§24.2(i) Bernoulli Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m235.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="B_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./24.2#i">B</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></math>: Bernoulli polynomials</a>,
<a href="./24.2#ii" title="§24.2(ii) Euler Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m242.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="E_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./24.2#ii">E</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></math>: Euler polynomials</a>,
<a href="./24.2#iii" title="§24.2(iii) Periodic Bernoulli and Euler Functions ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m310.png" altimg-height="28px" altimg-valign="-7px" altimg-width="60px" alttext="\widetilde{B}_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mover accent="true"><mi href="./24.2#iii">B</mi><mo href="./24.2#iii">~</mo></mover><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></math>: periodic Bernoulli functions</a>,
<a href="./24.2#iii" title="§24.2(iii) Periodic Bernoulli and Euler Functions ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m311.png" altimg-height="28px" altimg-valign="-7px" altimg-width="60px" alttext="\widetilde{E}_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mover accent="true"><mi href="./24.2#iii">E</mi><mo href="./24.2#iii">~</mo></mover><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></math>: periodic Euler functions</a>,
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m317.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./24.1#p2.t1.r1">n</mi></math>: integer</a> and
<a href="./24.1#p2.t1.r2" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m320.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./24.1#p2.t1.r2">x</mi></math>: real or complex</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E12" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex7" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="3" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">24.2.12</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m27.png" altimg-height="30px" altimg-valign="-7px" altimg-width="97px" alttext="\displaystyle\widetilde{B}_{n}\left(x+1\right)" display="inline"><mrow><msub><mover accent="true"><mi href="./24.2#iii">B</mi><mo href="./24.2#iii">~</mo></mover><mi href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi href="./24.1#p2.t1.r2">x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m15.png" altimg-height="30px" altimg-valign="-7px" altimg-width="92px" alttext="\displaystyle=\widetilde{B}_{n}\left(x\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msub><mover accent="true"><mi href="./24.2#iii">B</mi><mo href="./24.2#iii">~</mo></mover><mi href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex8" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m29.png" altimg-height="30px" altimg-valign="-7px" altimg-width="96px" alttext="\displaystyle\widetilde{E}_{n}\left(x+1\right)" display="inline"><mrow><msub><mover accent="true"><mi href="./24.2#iii">E</mi><mo href="./24.2#iii">~</mo></mover><mi href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi href="./24.1#p2.t1.r2">x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell">
<math class="ltx_Math" altimg="m3.png" altimg-height="30px" altimg-valign="-7px" altimg-width="98px" alttext="\displaystyle=-\widetilde{E}_{n}\left(x\right)" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mo>-</mo><mrow><msub><mover accent="true"><mi href="./24.2#iii">E</mi><mo href="./24.2#iii">~</mo></mover><mi href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></mrow></mrow></math>,</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m321.png" altimg-height="18px" altimg-valign="-3px" altimg-width="54px" alttext="x\in\mathbb{R}" display="inline"><mrow><mi href="./24.1#p2.t1.r2">x</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mi href="./front/introduction#Sx4.p2.t1.r14">ℝ</mi></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E12.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./front/introduction#Sx4.p1.t1.r9" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m248.png" altimg-height="15px" altimg-valign="-3px" altimg-width="18px" alttext="\in" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo></math>: element of</a>,
<a href="./24.2#iii" title="§24.2(iii) Periodic Bernoulli and Euler Functions ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m310.png" altimg-height="28px" altimg-valign="-7px" altimg-width="60px" alttext="\widetilde{B}_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mover accent="true"><mi href="./24.2#iii">B</mi><mo href="./24.2#iii">~</mo></mover><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></math>: periodic Bernoulli functions</a>,
<a href="./24.2#iii" title="§24.2(iii) Periodic Bernoulli and Euler Functions ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m311.png" altimg-height="28px" altimg-valign="-7px" altimg-width="60px" alttext="\widetilde{E}_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mover accent="true"><mi href="./24.2#iii">E</mi><mo href="./24.2#iii">~</mo></mover><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></math>: periodic Euler functions</a>,
<a href="./front/introduction#Sx4.p2.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m249.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\mathbb{R}" display="inline"><mi href="./front/introduction#Sx4.p2.t1.r14">ℝ</mi></math>: real line</a>,
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m317.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./24.1#p2.t1.r1">n</mi></math>: integer</a> and
<a href="./24.1#p2.t1.r2" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m320.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./24.1#p2.t1.r2">x</mi></math>: real or complex</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div class="ltx_pagination ltx_role_newpage"></div>
</section>
<section id="iv" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§24.2(iv) </span>Tables</h2>
<div id="SS4.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<figure id="SS4.tab1" class="ltx_table">
<table style="width:100%;">
<tr>
<td class="ltx_subtable">
<figure id="T1" class="ltx_table">
<figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_table">Table 24.2.1: </span>Bernoulli and Euler numbers.
</figcaption>
<table id="T1.t1" class="ltx_tabular ltx_centering ltx_guessed_headers ltx_align_middle">
<thead class="ltx_thead">
<tr id="T1.t1.r1" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_border_tt" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m317.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./24.1#p2.t1.r1">n</mi></math></th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_border_tt" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m237.png" altimg-height="21px" altimg-valign="-5px" altimg-width="30px" alttext="B_{n}" display="inline"><msub><mi href="./24.2#i">B</mi><mi href="./24.1#p2.t1.r1">n</mi></msub></math></th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_border_tt" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m243.png" altimg-height="21px" altimg-valign="-5px" altimg-width="30px" alttext="E_{n}" display="inline"><msub><mi href="./24.2#ii">E</mi><mi href="./24.1#p2.t1.r1">n</mi></msub></math></th>
</tr>
</thead>
<tbody class="ltx_tbody">
<tr id="T1.t1.r2" class="ltx_tr">
<td class="ltx_td ltx_align_right ltx_border_t" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_t" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m157.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="1" display="inline"><mn>1</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_t" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m157.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="1" display="inline"><mn>1</mn></math></td>
</tr>
<tr id="T1.t1.r3" class="ltx_tr">
<td class="ltx_td ltx_align_right" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m157.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="1" display="inline"><mn>1</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m101.png" altimg-height="27px" altimg-valign="-9px" altimg-width="33px" alttext="-\tfrac{1}{2}" display="inline"><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math></td>
<td class="ltx_td ltx_align_right" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
</tr>
<tr id="T1.t1.r4" class="ltx_tr">
<td class="ltx_td ltx_align_right" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m175.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="2" display="inline"><mn>2</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m262.png" altimg-height="27px" altimg-valign="-9px" altimg-width="17px" alttext="\tfrac{1}{6}" display="inline"><mfrac><mn>1</mn><mn>6</mn></mfrac></math></td>
<td class="ltx_td ltx_align_right" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m45.png" altimg-height="18px" altimg-valign="-4px" altimg-width="30px" alttext="-1" display="inline"><mrow><mo>-</mo><mn>1</mn></mrow></math></td>
</tr>
<tr id="T1.t1.r5" class="ltx_tr">
<td class="ltx_td ltx_align_right" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m197.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="4" display="inline"><mn>4</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m102.png" altimg-height="27px" altimg-valign="-9px" altimg-width="40px" alttext="-\tfrac{1}{30}" display="inline"><mrow><mo>-</mo><mfrac><mn>1</mn><mn>30</mn></mfrac></mrow></math></td>
<td class="ltx_td ltx_align_right" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m208.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="5" display="inline"><mn>5</mn></math></td>
</tr>
<tr id="T1.t1.r6" class="ltx_tr">
<td class="ltx_td ltx_align_right" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m217.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="6" display="inline"><mn>6</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m260.png" altimg-height="27px" altimg-valign="-9px" altimg-width="25px" alttext="\tfrac{1}{42}" display="inline"><mfrac><mn>1</mn><mn>42</mn></mfrac></math></td>
<td class="ltx_td ltx_align_right" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m79.png" altimg-height="18px" altimg-valign="-4px" altimg-width="40px" alttext="-61" display="inline"><mrow><mo>-</mo><mn>61</mn></mrow></math></td>
</tr>
<tr id="T1.t1.r7" class="ltx_tr">
<td class="ltx_td ltx_align_right" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m230.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="8" display="inline"><mn>8</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m102.png" altimg-height="27px" altimg-valign="-9px" altimg-width="40px" alttext="-\tfrac{1}{30}" display="inline"><mrow><mo>-</mo><mfrac><mn>1</mn><mn>30</mn></mfrac></mrow></math></td>
<td class="ltx_td ltx_align_right" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m138.png" altimg-height="17px" altimg-valign="-2px" altimg-width="44px" alttext="1385" display="inline"><mn>1385</mn></math></td>
</tr>
<tr id="T1.t1.r8" class="ltx_tr">
<td class="ltx_td ltx_align_right" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m133.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="10" display="inline"><mn>10</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m275.png" altimg-height="27px" altimg-valign="-9px" altimg-width="25px" alttext="\tfrac{5}{66}" display="inline"><mfrac><mn>5</mn><mn>66</mn></mfrac></math></td>
<td class="ltx_td ltx_align_right" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m75.png" altimg-height="18px" altimg-valign="-4px" altimg-width="70px" alttext="-50521" display="inline"><mrow><mo>-</mo><mn>50521</mn></mrow></math></td>
</tr>
<tr id="T1.t1.r9" class="ltx_tr">
<td class="ltx_td ltx_align_right" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m136.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="12" display="inline"><mn>12</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m119.png" altimg-height="27px" altimg-valign="-9px" altimg-width="56px" alttext="-\tfrac{691}{2730}" display="inline"><mrow><mo>-</mo><mfrac><mn>691</mn><mn>2730</mn></mfrac></mrow></math></td>
<td class="ltx_td ltx_align_right" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m170.png" altimg-height="17px" altimg-valign="-2px" altimg-width="80px" alttext="27\;02765" display="inline"><mn>27 02765</mn></math></td>
</tr>
<tr id="T1.t1.r10" class="ltx_tr">
<td class="ltx_td ltx_align_right" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m142.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="14" display="inline"><mn>14</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m282.png" altimg-height="27px" altimg-valign="-9px" altimg-width="17px" alttext="\tfrac{7}{6}" display="inline"><mfrac><mn>7</mn><mn>6</mn></mfrac></math></td>
<td class="ltx_td ltx_align_right" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m43.png" altimg-height="18px" altimg-valign="-4px" altimg-width="115px" alttext="-1993\;60981" display="inline"><mrow><mo>-</mo><mn>1993 60981</mn></mrow></math></td>
</tr>
<tr id="T1.t1.r11" class="ltx_tr">
<td class="ltx_td ltx_align_right ltx_border_b" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m148.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="16" display="inline"><mn>16</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_b" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m110.png" altimg-height="27px" altimg-valign="-9px" altimg-width="56px" alttext="-\tfrac{3617}{510}" display="inline"><mrow><mo>-</mo><mfrac><mn>3617</mn><mn>510</mn></mfrac></mrow></math></td>
<td class="ltx_td ltx_align_right ltx_border_b" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m156.png" altimg-height="17px" altimg-valign="-2px" altimg-width="125px" alttext="1\;93915\;12145" display="inline"><mn>1 93915 12145</mn></math></td>
</tr>
</tbody>
</table>
<div id="T1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./24.2#i" title="§24.2(i) Bernoulli Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m234.png" altimg-height="21px" altimg-valign="-5px" altimg-width="30px" alttext="B_{\NVar{n}}" display="inline"><msub><mi href="./24.2#i">B</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub></math>: Bernoulli numbers</a>,
<a href="./24.2#ii" title="§24.2(ii) Euler Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m241.png" altimg-height="21px" altimg-valign="-5px" altimg-width="30px" alttext="E_{\NVar{n}}" display="inline"><msub><mi href="./24.2#ii">E</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub></math>: Euler numbers</a> and
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m317.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./24.1#p2.t1.r1">n</mi></math>: integer</a>
</dd>
<dt>Keywords:</dt>
<dd>
</dd>
</dl>
</div>
</div>
</figure>
</td>
<td class="ltx_subtable">
<figure id="T2" class="ltx_table">
<figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_table">Table 24.2.2: </span>Bernoulli and Euler polynomials.
</figcaption>
<table id="T2.t1" class="ltx_tabular ltx_centering ltx_guessed_headers ltx_align_middle">
<thead class="ltx_thead">
<tr id="T2.t1.r1" class="ltx_tr">
<th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_th_row ltx_border_tt" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m317.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./24.1#p2.t1.r1">n</mi></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m239.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="B_{n}\left(x\right)" display="inline"><mrow><msub><mi href="./24.2#i">B</mi><mi href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m245.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="E_{n}\left(x\right)" display="inline"><mrow><msub><mi href="./24.2#ii">E</mi><mi href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></math></th>
</tr>
</thead>
<tbody class="ltx_tbody">
<tr id="T2.t1.r2" class="ltx_tr">
<th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_t" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></th>
<td class="ltx_td ltx_align_center ltx_border_t" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m157.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="1" display="inline"><mn>1</mn></math></td>
<td class="ltx_td ltx_align_center ltx_border_t" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m157.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="1" display="inline"><mn>1</mn></math></td>
</tr>
<tr id="T2.t1.r3" class="ltx_tr">
<th class="ltx_td ltx_align_center ltx_th ltx_th_row" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m157.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="1" display="inline"><mn>1</mn></math></th>
<td class="ltx_td ltx_align_center" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m319.png" altimg-height="27px" altimg-valign="-9px" altimg-width="53px" alttext="x-\frac{1}{2}" display="inline"><mrow><mi href="./24.1#p2.t1.r2">x</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math></td>
<td class="ltx_td ltx_align_center" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m319.png" altimg-height="27px" altimg-valign="-9px" altimg-width="53px" alttext="x-\frac{1}{2}" display="inline"><mrow><mi href="./24.1#p2.t1.r2">x</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math></td>
</tr>
<tr id="T2.t1.r4" class="ltx_tr">
<th class="ltx_td ltx_align_center ltx_th ltx_th_row" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m175.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="2" display="inline"><mn>2</mn></math></th>
<td class="ltx_td ltx_align_center" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m322.png" altimg-height="27px" altimg-valign="-9px" altimg-width="97px" alttext="x^{2}-x+\frac{1}{6}" display="inline"><mrow><mrow><msup><mi href="./24.1#p2.t1.r2">x</mi><mn>2</mn></msup><mo>-</mo><mi href="./24.1#p2.t1.r2">x</mi></mrow><mo>+</mo><mfrac><mn>1</mn><mn>6</mn></mfrac></mrow></math></td>
<td class="ltx_td ltx_align_center" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m323.png" altimg-height="21px" altimg-valign="-4px" altimg-width="60px" alttext="x^{2}-x" display="inline"><mrow><msup><mi href="./24.1#p2.t1.r2">x</mi><mn>2</mn></msup><mo>-</mo><mi href="./24.1#p2.t1.r2">x</mi></mrow></math></td>
</tr>
<tr id="T2.t1.r5" class="ltx_tr">
<th class="ltx_td ltx_align_center ltx_th ltx_th_row" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m185.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="3" display="inline"><mn>3</mn></math></th>
<td class="ltx_td ltx_align_center" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m324.png" altimg-height="27px" altimg-valign="-9px" altimg-width="130px" alttext="x^{3}-\frac{3}{2}x^{2}+\frac{1}{2}x" display="inline"><mrow><mrow><msup><mi href="./24.1#p2.t1.r2">x</mi><mn>3</mn></msup><mo>-</mo><mrow><mfrac><mn>3</mn><mn>2</mn></mfrac><mo></mo><msup><mi href="./24.1#p2.t1.r2">x</mi><mn>2</mn></msup></mrow></mrow><mo>+</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./24.1#p2.t1.r2">x</mi></mrow></mrow></math></td>
<td class="ltx_td ltx_align_center" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m325.png" altimg-height="27px" altimg-valign="-9px" altimg-width="119px" alttext="x^{3}-\frac{3}{2}x^{2}+\frac{1}{4}" display="inline"><mrow><mrow><msup><mi href="./24.1#p2.t1.r2">x</mi><mn>3</mn></msup><mo>-</mo><mrow><mfrac><mn>3</mn><mn>2</mn></mfrac><mo></mo><msup><mi href="./24.1#p2.t1.r2">x</mi><mn>2</mn></msup></mrow></mrow><mo>+</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></math></td>
</tr>
<tr id="T2.t1.r6" class="ltx_tr">
<th class="ltx_td ltx_align_center ltx_th ltx_th_row" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m197.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="4" display="inline"><mn>4</mn></math></th>
<td class="ltx_td ltx_align_center" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m327.png" altimg-height="27px" altimg-valign="-9px" altimg-width="169px" alttext="x^{4}-2x^{3}+x^{2}-\frac{1}{30}" display="inline"><mrow><mrow><mrow><msup><mi href="./24.1#p2.t1.r2">x</mi><mn>4</mn></msup><mo>-</mo><mrow><mn>2</mn><mo></mo><msup><mi href="./24.1#p2.t1.r2">x</mi><mn>3</mn></msup></mrow></mrow><mo>+</mo><msup><mi href="./24.1#p2.t1.r2">x</mi><mn>2</mn></msup></mrow><mo>-</mo><mfrac><mn>1</mn><mn>30</mn></mfrac></mrow></math></td>
<td class="ltx_td ltx_align_center" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m326.png" altimg-height="21px" altimg-valign="-4px" altimg-width="115px" alttext="x^{4}-2x^{3}+x" display="inline"><mrow><mrow><msup><mi href="./24.1#p2.t1.r2">x</mi><mn>4</mn></msup><mo>-</mo><mrow><mn>2</mn><mo></mo><msup><mi href="./24.1#p2.t1.r2">x</mi><mn>3</mn></msup></mrow></mrow><mo>+</mo><mi href="./24.1#p2.t1.r2">x</mi></mrow></math></td>
</tr>
<tr id="T2.t1.r7" class="ltx_tr">
<th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_b" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m208.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="5" display="inline"><mn>5</mn></math></th>
<td class="ltx_td ltx_align_center ltx_border_b" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m329.png" altimg-height="27px" altimg-valign="-9px" altimg-width="188px" alttext="x^{5}-\frac{5}{2}x^{4}+\frac{5}{3}x^{3}-\frac{1}{6}x" display="inline"><mrow><mrow><mrow><msup><mi href="./24.1#p2.t1.r2">x</mi><mn>5</mn></msup><mo>-</mo><mrow><mfrac><mn>5</mn><mn>2</mn></mfrac><mo></mo><msup><mi href="./24.1#p2.t1.r2">x</mi><mn>4</mn></msup></mrow></mrow><mo>+</mo><mrow><mfrac><mn>5</mn><mn>3</mn></mfrac><mo></mo><msup><mi href="./24.1#p2.t1.r2">x</mi><mn>3</mn></msup></mrow></mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>6</mn></mfrac><mo></mo><mi href="./24.1#p2.t1.r2">x</mi></mrow></mrow></math></td>
<td class="ltx_td ltx_align_center ltx_border_b" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m328.png" altimg-height="27px" altimg-valign="-9px" altimg-width="177px" alttext="x^{5}-\frac{5}{2}x^{4}+\frac{5}{2}x^{2}-\frac{1}{2}" display="inline"><mrow><mrow><mrow><msup><mi href="./24.1#p2.t1.r2">x</mi><mn>5</mn></msup><mo>-</mo><mrow><mfrac><mn>5</mn><mn>2</mn></mfrac><mo></mo><msup><mi href="./24.1#p2.t1.r2">x</mi><mn>4</mn></msup></mrow></mrow><mo>+</mo><mrow><mfrac><mn>5</mn><mn>2</mn></mfrac><mo></mo><msup><mi href="./24.1#p2.t1.r2">x</mi><mn>2</mn></msup></mrow></mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math></td>
</tr>
</tbody>
</table>
<div id="T2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./24.2#i" title="§24.2(i) Bernoulli Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m235.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="B_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./24.2#i">B</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></math>: Bernoulli polynomials</a>,
<a href="./24.2#ii" title="§24.2(ii) Euler Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m242.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="E_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./24.2#ii">E</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></math>: Euler polynomials</a>,
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m317.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./24.1#p2.t1.r1">n</mi></math>: integer</a> and
<a href="./24.1#p2.t1.r2" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m320.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./24.1#p2.t1.r2">x</mi></math>: real or complex</a>
</dd>
<dt>Keywords:</dt>
<dd>
</dd>
</dl>
</div>
</div>
</figure>
</td>
</tr>
</table>
</figure>
<div id="SS4.p1" class="ltx_para">
</div>
<figure id="T3" class="ltx_table">
<figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_table">Table 24.2.3: </span>Bernoulli numbers <math class="ltx_Math" altimg="m236.png" altimg-height="23px" altimg-valign="-7px" altimg-width="100px" alttext="B_{n}=N/D" display="inline"><mrow><msub><mi href="./24.2#i">B</mi><mi href="./24.1#p2.t1.r1">n</mi></msub><mo>=</mo><mrow><mi>N</mi><mo>/</mo><mi>D</mi></mrow></mrow></math>.
</figcaption>
<table id="T3.t1" class="ltx_tabular ltx_centering ltx_guessed_headers ltx_align_middle">
<thead class="ltx_thead">
<tr id="T3.t1.r1" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_border_tt"><math class="ltx_Math" altimg="m317.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./24.1#p2.t1.r1">n</mi></math></th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_border_tt"><math class="ltx_Math" altimg="m246.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="N" display="inline"><mi>N</mi></math></th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_border_tt"><math class="ltx_Math" altimg="m240.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi>D</mi></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" colspan="2"><math class="ltx_Math" altimg="m237.png" altimg-height="21px" altimg-valign="-5px" altimg-width="30px" alttext="B_{n}" display="inline"><msub><mi href="./24.2#i">B</mi><mi href="./24.1#p2.t1.r1">n</mi></msub></math></th>
</tr>
</thead>
<tbody class="ltx_tbody">
<tr id="T3.t1.r2" class="ltx_tr">
<td class="ltx_td ltx_align_right ltx_border_t"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_t"><math class="ltx_Math" altimg="m157.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="1" display="inline"><mn>1</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_t"><math class="ltx_Math" altimg="m157.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="1" display="inline"><mn>1</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_t"><math class="ltx_Math" altimg="m129.png" altimg-height="17px" altimg-valign="-2px" altimg-width="115px" alttext="1.00000\;0000" display="inline"><mn>1.00000 0000</mn></math></td>
<td class="ltx_td ltx_align_left ltx_border_t"></td>
</tr>
<tr id="T3.t1.r3" class="ltx_tr">
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m157.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="1" display="inline"><mn>1</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m45.png" altimg-height="18px" altimg-valign="-4px" altimg-width="30px" alttext="-1" display="inline"><mrow><mo>-</mo><mn>1</mn></mrow></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m175.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="2" display="inline"><mn>2</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m72.png" altimg-height="18px" altimg-valign="-4px" altimg-width="130px" alttext="-5.00000\;0000" display="inline"><mrow><mo>-</mo><mn>5.00000 0000</mn></mrow></math></td>
<td class="ltx_td ltx_align_left"><math class="ltx_Math" altimg="m286.png" altimg-height="21px" altimg-valign="-4px" altimg-width="61px" alttext="\times 10^{-1}" display="inline"><mn>×10⁻¹</mn></math></td>
</tr>
<tr id="T3.t1.r4" class="ltx_tr">
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m175.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="2" display="inline"><mn>2</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m157.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="1" display="inline"><mn>1</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m217.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="6" display="inline"><mn>6</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m132.png" altimg-height="17px" altimg-valign="-2px" altimg-width="115px" alttext="1.66666\;6667" display="inline"><mn>1.66666 6667</mn></math></td>
<td class="ltx_td ltx_align_left"><math class="ltx_Math" altimg="m286.png" altimg-height="21px" altimg-valign="-4px" altimg-width="61px" alttext="\times 10^{-1}" display="inline"><mn>×10⁻¹</mn></math></td>
</tr>
<tr id="T3.t1.r5" class="ltx_tr">
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m197.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="4" display="inline"><mn>4</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m45.png" altimg-height="18px" altimg-valign="-4px" altimg-width="30px" alttext="-1" display="inline"><mrow><mo>-</mo><mn>1</mn></mrow></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m177.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="30" display="inline"><mn>30</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m61.png" altimg-height="18px" altimg-valign="-4px" altimg-width="130px" alttext="-3.33333\;3333" display="inline"><mrow><mo>-</mo><mn>3.33333 3333</mn></mrow></math></td>
<td class="ltx_td ltx_align_left"><math class="ltx_Math" altimg="m287.png" altimg-height="21px" altimg-valign="-4px" altimg-width="61px" alttext="\times 10^{-2}" display="inline"><mn>×10⁻²</mn></math></td>
</tr>
<tr id="T3.t1.r6" class="ltx_tr">
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m217.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="6" display="inline"><mn>6</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m157.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="1" display="inline"><mn>1</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m190.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="42" display="inline"><mn>42</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m159.png" altimg-height="17px" altimg-valign="-2px" altimg-width="115px" alttext="2.38095\;2381" display="inline"><mn>2.38095 2381</mn></math></td>
<td class="ltx_td ltx_align_left"><math class="ltx_Math" altimg="m287.png" altimg-height="21px" altimg-valign="-4px" altimg-width="61px" alttext="\times 10^{-2}" display="inline"><mn>×10⁻²</mn></math></td>
</tr>
<tr id="T3.t1.r7" class="ltx_tr">
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m230.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="8" display="inline"><mn>8</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m45.png" altimg-height="18px" altimg-valign="-4px" altimg-width="30px" alttext="-1" display="inline"><mrow><mo>-</mo><mn>1</mn></mrow></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m177.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="30" display="inline"><mn>30</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m61.png" altimg-height="18px" altimg-valign="-4px" altimg-width="130px" alttext="-3.33333\;3333" display="inline"><mrow><mo>-</mo><mn>3.33333 3333</mn></mrow></math></td>
<td class="ltx_td ltx_align_left"><math class="ltx_Math" altimg="m287.png" altimg-height="21px" altimg-valign="-4px" altimg-width="61px" alttext="\times 10^{-2}" display="inline"><mn>×10⁻²</mn></math></td>
</tr>
<tr id="T3.t1.r8" class="ltx_tr">
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m133.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="10" display="inline"><mn>10</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m208.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="5" display="inline"><mn>5</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m215.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="66" display="inline"><mn>66</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m219.png" altimg-height="17px" altimg-valign="-2px" altimg-width="115px" alttext="7.57575\;7576" display="inline"><mn>7.57575 7576</mn></math></td>
<td class="ltx_td ltx_align_left"><math class="ltx_Math" altimg="m287.png" altimg-height="21px" altimg-valign="-4px" altimg-width="61px" alttext="\times 10^{-2}" display="inline"><mn>×10⁻²</mn></math></td>
</tr>
<tr id="T3.t1.r9" class="ltx_tr">
<td class="ltx_td ltx_align_right ltx_border_T"><math class="ltx_Math" altimg="m136.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="12" display="inline"><mn>12</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_T"><math class="ltx_Math" altimg="m82.png" altimg-height="18px" altimg-valign="-4px" altimg-width="50px" alttext="-691" display="inline"><mrow><mo>-</mo><mn>691</mn></mrow></math></td>
<td class="ltx_td ltx_align_right ltx_border_T"><math class="ltx_Math" altimg="m169.png" altimg-height="17px" altimg-valign="-2px" altimg-width="44px" alttext="2730" display="inline"><mn>2730</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_T"><math class="ltx_Math" altimg="m47.png" altimg-height="18px" altimg-valign="-4px" altimg-width="130px" alttext="-2.53113\;5531" display="inline"><mrow><mo>-</mo><mn>2.53113 5531</mn></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_border_T"><math class="ltx_Math" altimg="m286.png" altimg-height="21px" altimg-valign="-4px" altimg-width="61px" alttext="\times 10^{-1}" display="inline"><mn>×10⁻¹</mn></math></td>
</tr>
<tr id="T3.t1.r10" class="ltx_tr">
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m142.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="14" display="inline"><mn>14</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m223.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="7" display="inline"><mn>7</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m217.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="6" display="inline"><mn>6</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m130.png" altimg-height="17px" altimg-valign="-2px" altimg-width="115px" alttext="1.16666\;6667" display="inline"><mn>1.16666 6667</mn></math></td>
<td class="ltx_td ltx_align_left"></td>
</tr>
<tr id="T3.t1.r11" class="ltx_tr">
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m148.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="16" display="inline"><mn>16</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m64.png" altimg-height="18px" altimg-valign="-4px" altimg-width="60px" alttext="-3617" display="inline"><mrow><mo>-</mo><mn>3617</mn></mrow></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m200.png" altimg-height="17px" altimg-valign="-2px" altimg-width="34px" alttext="510" display="inline"><mn>510</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m85.png" altimg-height="18px" altimg-valign="-4px" altimg-width="130px" alttext="-7.09215\;6863" display="inline"><mrow><mo>-</mo><mn>7.09215 6863</mn></mrow></math></td>
<td class="ltx_td ltx_align_left"></td>
</tr>
<tr id="T3.t1.r12" class="ltx_tr">
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m152.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="18" display="inline"><mn>18</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m191.png" altimg-height="17px" altimg-valign="-2px" altimg-width="54px" alttext="43867" display="inline"><mn>43867</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m221.png" altimg-height="17px" altimg-valign="-2px" altimg-width="34px" alttext="798" display="inline"><mn>798</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m198.png" altimg-height="17px" altimg-valign="-2px" altimg-width="115px" alttext="5.49711\;7794" display="inline"><mn>5.49711 7794</mn></math></td>
<td class="ltx_td ltx_align_left"><math class="ltx_Math" altimg="m295.png" altimg-height="21px" altimg-valign="-4px" altimg-width="49px" alttext="\times 10^{1}" display="inline"><mn>×10¹</mn></math></td>
</tr>
<tr id="T3.t1.r13" class="ltx_tr">
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m162.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="20" display="inline"><mn>20</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m44.png" altimg-height="18px" altimg-valign="-4px" altimg-width="85px" alttext="-1\;74611" display="inline"><mrow><mo>-</mo><mn>1 74611</mn></mrow></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m179.png" altimg-height="17px" altimg-valign="-2px" altimg-width="34px" alttext="330" display="inline"><mn>330</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m74.png" altimg-height="18px" altimg-valign="-4px" altimg-width="130px" alttext="-5.29124\;2424" display="inline"><mrow><mo>-</mo><mn>5.29124 2424</mn></mrow></math></td>
<td class="ltx_td ltx_align_left"><math class="ltx_Math" altimg="m301.png" altimg-height="21px" altimg-valign="-4px" altimg-width="49px" alttext="\times 10^{2}" display="inline"><mn>×10²</mn></math></td>
</tr>
<tr id="T3.t1.r14" class="ltx_tr">
<td class="ltx_td ltx_align_right ltx_border_T"><math class="ltx_Math" altimg="m164.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="22" display="inline"><mn>22</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_T"><math class="ltx_Math" altimg="m229.png" altimg-height="17px" altimg-valign="-2px" altimg-width="70px" alttext="8\;54513" display="inline"><mn>8 54513</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_T"><math class="ltx_Math" altimg="m139.png" altimg-height="17px" altimg-valign="-2px" altimg-width="34px" alttext="138" display="inline"><mn>138</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_T"><math class="ltx_Math" altimg="m210.png" altimg-height="17px" altimg-valign="-2px" altimg-width="115px" alttext="6.19212\;3188" display="inline"><mn>6.19212 3188</mn></math></td>
<td class="ltx_td ltx_align_left ltx_border_T"><math class="ltx_Math" altimg="m305.png" altimg-height="21px" altimg-valign="-4px" altimg-width="49px" alttext="\times 10^{3}" display="inline"><mn>×10³</mn></math></td>
</tr>
<tr id="T3.t1.r15" class="ltx_tr">
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m165.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="24" display="inline"><mn>24</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-4px" altimg-width="115px" alttext="-2363\;64091" display="inline"><mrow><mo>-</mo><mn>2363 64091</mn></mrow></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m169.png" altimg-height="17px" altimg-valign="-2px" altimg-width="44px" alttext="2730" display="inline"><mn>2730</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m89.png" altimg-height="18px" altimg-valign="-4px" altimg-width="130px" alttext="-8.65802\;5311" display="inline"><mrow><mo>-</mo><mn>8.65802 5311</mn></mrow></math></td>
<td class="ltx_td ltx_align_left"><math class="ltx_Math" altimg="m306.png" altimg-height="21px" altimg-valign="-4px" altimg-width="49px" alttext="\times 10^{4}" display="inline"><mn>×10⁴</mn></math></td>
</tr>
<tr id="T3.t1.r16" class="ltx_tr">
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m168.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="26" display="inline"><mn>26</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m226.png" altimg-height="17px" altimg-valign="-2px" altimg-width="80px" alttext="85\;53103" display="inline"><mn>85 53103</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m217.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="6" display="inline"><mn>6</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m131.png" altimg-height="17px" altimg-valign="-2px" altimg-width="115px" alttext="1.42551\;7167" display="inline"><mn>1.42551 7167</mn></math></td>
<td class="ltx_td ltx_align_left"><math class="ltx_Math" altimg="m307.png" altimg-height="21px" altimg-valign="-4px" altimg-width="49px" alttext="\times 10^{6}" display="inline"><mn>×10⁶</mn></math></td>
</tr>
<tr id="T3.t1.r17" class="ltx_tr">
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m172.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="28" display="inline"><mn>28</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m58.png" altimg-height="18px" altimg-valign="-4px" altimg-width="140px" alttext="-2\;37494\;61029" display="inline"><mrow><mo>-</mo><mn>2 37494 61029</mn></mrow></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m228.png" altimg-height="17px" altimg-valign="-2px" altimg-width="34px" alttext="870" display="inline"><mn>870</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m48.png" altimg-height="18px" altimg-valign="-4px" altimg-width="130px" alttext="-2.72982\;3107" display="inline"><mrow><mo>-</mo><mn>2.72982 3107</mn></mrow></math></td>
<td class="ltx_td ltx_align_left"><math class="ltx_Math" altimg="m308.png" altimg-height="21px" altimg-valign="-4px" altimg-width="49px" alttext="\times 10^{7}" display="inline"><mn>×10⁷</mn></math></td>
</tr>
<tr id="T3.t1.r18" class="ltx_tr">
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m177.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="30" display="inline"><mn>30</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m227.png" altimg-height="17px" altimg-valign="-2px" altimg-width="145px" alttext="861\;58412\;76005" display="inline"><mn>861 58412 76005</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m141.png" altimg-height="17px" altimg-valign="-2px" altimg-width="54px" alttext="14322" display="inline"><mn>14322</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m209.png" altimg-height="17px" altimg-valign="-2px" altimg-width="115px" alttext="6.01580\;8739" display="inline"><mn>6.01580 8739</mn></math></td>
<td class="ltx_td ltx_align_left"><math class="ltx_Math" altimg="m309.png" altimg-height="21px" altimg-valign="-4px" altimg-width="49px" alttext="\times 10^{8}" display="inline"><mn>×10⁸</mn></math></td>
</tr>
<tr id="T3.t1.r19" class="ltx_tr">
<td class="ltx_td ltx_align_right ltx_border_T"><math class="ltx_Math" altimg="m178.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="32" display="inline"><mn>32</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_T"><math class="ltx_Math" altimg="m86.png" altimg-height="18px" altimg-valign="-4px" altimg-width="160px" alttext="-770\;93210\;41217" display="inline"><mrow><mo>-</mo><mn>770 93210 41217</mn></mrow></math></td>
<td class="ltx_td ltx_align_right ltx_border_T"><math class="ltx_Math" altimg="m200.png" altimg-height="17px" altimg-valign="-2px" altimg-width="34px" alttext="510" display="inline"><mn>510</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_T"><math class="ltx_Math" altimg="m34.png" altimg-height="18px" altimg-valign="-4px" altimg-width="130px" alttext="-1.51163\;1577" display="inline"><mrow><mo>-</mo><mn>1.51163 1577</mn></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_border_T"><math class="ltx_Math" altimg="m288.png" altimg-height="21px" altimg-valign="-4px" altimg-width="57px" alttext="\times 10^{10}" display="inline"><mn>×10¹⁰</mn></math></td>
</tr>
<tr id="T3.t1.r20" class="ltx_tr">
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m180.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="34" display="inline"><mn>34</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m167.png" altimg-height="17px" altimg-valign="-2px" altimg-width="145px" alttext="257\;76878\;58367" display="inline"><mn>257 76878 58367</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m217.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="6" display="inline"><mn>6</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m186.png" altimg-height="17px" altimg-valign="-2px" altimg-width="115px" alttext="4.29614\;6431" display="inline"><mn>4.29614 6431</mn></math></td>
<td class="ltx_td ltx_align_left"><math class="ltx_Math" altimg="m289.png" altimg-height="21px" altimg-valign="-4px" altimg-width="57px" alttext="\times 10^{11}" display="inline"><mn>×10¹¹</mn></math></td>
</tr>
<tr id="T3.t1.r21" class="ltx_tr">
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m182.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="36" display="inline"><mn>36</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m55.png" altimg-height="18px" altimg-valign="-4px" altimg-width="236px" alttext="-26315\;27155\;30534\;77373" display="inline"><mrow><mo>-</mo><mn>26315 27155 30534 77373</mn></mrow></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m153.png" altimg-height="17px" altimg-valign="-2px" altimg-width="80px" alttext="19\;19190" display="inline"><mn>19 19190</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m33.png" altimg-height="18px" altimg-valign="-4px" altimg-width="130px" alttext="-1.37116\;5521" display="inline"><mrow><mo>-</mo><mn>1.37116 5521</mn></mrow></math></td>
<td class="ltx_td ltx_align_left"><math class="ltx_Math" altimg="m290.png" altimg-height="21px" altimg-valign="-4px" altimg-width="57px" alttext="\times 10^{13}" display="inline"><mn>×10¹³</mn></math></td>
</tr>
<tr id="T3.t1.r22" class="ltx_tr">
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m184.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="38" display="inline"><mn>38</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m174.png" altimg-height="17px" altimg-valign="-2px" altimg-width="180px" alttext="2\;92999\;39138\;41559" display="inline"><mn>2 92999 39138 41559</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m217.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="6" display="inline"><mn>6</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m187.png" altimg-height="17px" altimg-valign="-2px" altimg-width="115px" alttext="4.88332\;3190" display="inline"><mn>4.88332 3190</mn></math></td>
<td class="ltx_td ltx_align_left"><math class="ltx_Math" altimg="m291.png" altimg-height="21px" altimg-valign="-4px" altimg-width="57px" alttext="\times 10^{14}" display="inline"><mn>×10¹⁴</mn></math></td>
</tr>
<tr id="T3.t1.r23" class="ltx_tr">
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m188.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="40" display="inline"><mn>40</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-4px" altimg-width="251px" alttext="-2\;61082\;71849\;64491\;22051" display="inline"><mrow><mo>-</mo><mn>2 61082 71849 64491 22051</mn></mrow></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m137.png" altimg-height="17px" altimg-valign="-2px" altimg-width="54px" alttext="13530" display="inline"><mn>13530</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m35.png" altimg-height="18px" altimg-valign="-4px" altimg-width="130px" alttext="-1.92965\;7934" display="inline"><mrow><mo>-</mo><mn>1.92965 7934</mn></mrow></math></td>
<td class="ltx_td ltx_align_left"><math class="ltx_Math" altimg="m292.png" altimg-height="21px" altimg-valign="-4px" altimg-width="57px" alttext="\times 10^{16}" display="inline"><mn>×10¹⁶</mn></math></td>
</tr>
<tr id="T3.t1.r24" class="ltx_tr">
<td class="ltx_td ltx_align_right ltx_border_T"><math class="ltx_Math" altimg="m190.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="42" display="inline"><mn>42</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_T"><math class="ltx_Math" altimg="m145.png" altimg-height="17px" altimg-valign="-2px" altimg-width="246px" alttext="15\;20097\;64391\;80708\;02691" display="inline"><mn>15 20097 64391 80708 02691</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_T"><math class="ltx_Math" altimg="m151.png" altimg-height="17px" altimg-valign="-2px" altimg-width="44px" alttext="1806" display="inline"><mn>1806</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_T"><math class="ltx_Math" altimg="m224.png" altimg-height="17px" altimg-valign="-2px" altimg-width="115px" alttext="8.41693\;0476" display="inline"><mn>8.41693 0476</mn></math></td>
<td class="ltx_td ltx_align_left ltx_border_T"><math class="ltx_Math" altimg="m293.png" altimg-height="21px" altimg-valign="-4px" altimg-width="57px" alttext="\times 10^{17}" display="inline"><mn>×10¹⁷</mn></math></td>
</tr>
<tr id="T3.t1.r25" class="ltx_tr">
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m192.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="44" display="inline"><mn>44</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m56.png" altimg-height="18px" altimg-valign="-4px" altimg-width="271px" alttext="-278\;33269\;57930\;10242\;35023" display="inline"><mrow><mo>-</mo><mn>278 33269 57930 10242 35023</mn></mrow></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m216.png" altimg-height="17px" altimg-valign="-2px" altimg-width="34px" alttext="690" display="inline"><mn>690</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m68.png" altimg-height="18px" altimg-valign="-4px" altimg-width="130px" alttext="-4.03380\;7185" display="inline"><mrow><mo>-</mo><mn>4.03380 7185</mn></mrow></math></td>
<td class="ltx_td ltx_align_left"><math class="ltx_Math" altimg="m294.png" altimg-height="21px" altimg-valign="-4px" altimg-width="57px" alttext="\times 10^{19}" display="inline"><mn>×10¹⁹</mn></math></td>
</tr>
<tr id="T3.t1.r26" class="ltx_tr">
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m194.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="46" display="inline"><mn>46</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m207.png" altimg-height="17px" altimg-valign="-2px" altimg-width="266px" alttext="5964\;51111\;59391\;21632\;77961" display="inline"><mn>5964 51111 59391 21632 77961</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m171.png" altimg-height="17px" altimg-valign="-2px" altimg-width="34px" alttext="282" display="inline"><mn>282</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m158.png" altimg-height="17px" altimg-valign="-2px" altimg-width="115px" alttext="2.11507\;4864" display="inline"><mn>2.11507 4864</mn></math></td>
<td class="ltx_td ltx_align_left"><math class="ltx_Math" altimg="m296.png" altimg-height="21px" altimg-valign="-4px" altimg-width="57px" alttext="\times 10^{21}" display="inline"><mn>×10²¹</mn></math></td>
</tr>
<tr id="T3.t1.r27" class="ltx_tr">
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m195.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="48" display="inline"><mn>48</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m76.png" altimg-height="18px" altimg-valign="-4px" altimg-width="326px" alttext="-560\;94033\;68997\;81768\;62491\;27547" display="inline"><mrow><mo>-</mo><mn>560 94033 68997 81768 62491 27547</mn></mrow></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m193.png" altimg-height="17px" altimg-valign="-2px" altimg-width="54px" alttext="46410" display="inline"><mn>46410</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m32.png" altimg-height="18px" altimg-valign="-4px" altimg-width="130px" alttext="-1.20866\;2652" display="inline"><mrow><mo>-</mo><mn>1.20866 2652</mn></mrow></math></td>
<td class="ltx_td ltx_align_left"><math class="ltx_Math" altimg="m297.png" altimg-height="21px" altimg-valign="-4px" altimg-width="57px" alttext="\times 10^{23}" display="inline"><mn>×10²³</mn></math></td>
</tr>
<tr id="T3.t1.r28" class="ltx_tr">
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m199.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="50" display="inline"><mn>50</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m196.png" altimg-height="17px" altimg-valign="-2px" altimg-width="301px" alttext="49\;50572\;05241\;07964\;82124\;77525" display="inline"><mn>49 50572 05241 07964 82124 77525</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m215.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="66" display="inline"><mn>66</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m218.png" altimg-height="17px" altimg-valign="-2px" altimg-width="115px" alttext="7.50086\;6746" display="inline"><mn>7.50086 6746</mn></math></td>
<td class="ltx_td ltx_align_left"><math class="ltx_Math" altimg="m298.png" altimg-height="21px" altimg-valign="-4px" altimg-width="57px" alttext="\times 10^{24}" display="inline"><mn>×10²⁴</mn></math></td>
</tr>
<tr id="T3.t1.r29" class="ltx_tr">
<td class="ltx_td ltx_align_right ltx_border_T"><math class="ltx_Math" altimg="m201.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="52" display="inline"><mn>52</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_T"><math class="ltx_Math" altimg="m90.png" altimg-height="18px" altimg-valign="-4px" altimg-width="346px" alttext="-80116\;57181\;35489\;95734\;79249\;91853" display="inline"><mrow><mo>-</mo><mn>80116 57181 35489 95734 79249 91853</mn></mrow></math></td>
<td class="ltx_td ltx_align_right ltx_border_T"><math class="ltx_Math" altimg="m144.png" altimg-height="17px" altimg-valign="-2px" altimg-width="44px" alttext="1590" display="inline"><mn>1590</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_T"><math class="ltx_Math" altimg="m73.png" altimg-height="18px" altimg-valign="-4px" altimg-width="130px" alttext="-5.03877\;8101" display="inline"><mrow><mo>-</mo><mn>5.03877 8101</mn></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_border_T"><math class="ltx_Math" altimg="m299.png" altimg-height="21px" altimg-valign="-4px" altimg-width="57px" alttext="\times 10^{26}" display="inline"><mn>×10²⁶</mn></math></td>
</tr>
<tr id="T3.t1.r30" class="ltx_tr">
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m202.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="54" display="inline"><mn>54</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m173.png" altimg-height="17px" altimg-valign="-2px" altimg-width="356px" alttext="29\;14996\;36348\;84862\;42141\;81238\;12691" display="inline"><mn>29 14996 36348 84862 42141 81238 12691</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m221.png" altimg-height="17px" altimg-valign="-2px" altimg-width="34px" alttext="798" display="inline"><mn>798</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m176.png" altimg-height="17px" altimg-valign="-2px" altimg-width="115px" alttext="3.65287\;7648" display="inline"><mn>3.65287 7648</mn></math></td>
<td class="ltx_td ltx_align_left"><math class="ltx_Math" altimg="m300.png" altimg-height="21px" altimg-valign="-4px" altimg-width="57px" alttext="\times 10^{28}" display="inline"><mn>×10²⁸</mn></math></td>
</tr>
<tr id="T3.t1.r31" class="ltx_tr">
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m205.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="56" display="inline"><mn>56</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m54.png" altimg-height="18px" altimg-valign="-4px" altimg-width="392px" alttext="-2479\;39292\;93132\;26753\;68541\;57396\;63229" display="inline"><mrow><mo>-</mo><mn>2479 39292 93132 26753 68541 57396 63229</mn></mrow></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m228.png" altimg-height="17px" altimg-valign="-2px" altimg-width="34px" alttext="870" display="inline"><mn>870</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m49.png" altimg-height="18px" altimg-valign="-4px" altimg-width="130px" alttext="-2.84987\;6930" display="inline"><mrow><mo>-</mo><mn>2.84987 6930</mn></mrow></math></td>
<td class="ltx_td ltx_align_left"><math class="ltx_Math" altimg="m302.png" altimg-height="21px" altimg-valign="-4px" altimg-width="57px" alttext="\times 10^{30}" display="inline"><mn>×10³⁰</mn></math></td>
</tr>
<tr id="T3.t1.r32" class="ltx_tr">
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m206.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="58" display="inline"><mn>58</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m225.png" altimg-height="17px" altimg-valign="-2px" altimg-width="386px" alttext="84483\;61334\;88800\;41862\;04677\;59940\;36021" display="inline"><mn>84483 61334 88800 41862 04677 59940 36021</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m181.png" altimg-height="17px" altimg-valign="-2px" altimg-width="34px" alttext="354" display="inline"><mn>354</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m160.png" altimg-height="17px" altimg-valign="-2px" altimg-width="115px" alttext="2.38654\;2750" display="inline"><mn>2.38654 2750</mn></math></td>
<td class="ltx_td ltx_align_left"><math class="ltx_Math" altimg="m303.png" altimg-height="21px" altimg-valign="-4px" altimg-width="57px" alttext="\times 10^{32}" display="inline"><mn>×10³²</mn></math></td>
</tr>
<tr id="T3.t1.r33" class="ltx_tr">
<td class="ltx_td ltx_align_right ltx_border_b"><math class="ltx_Math" altimg="m212.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="60" display="inline"><mn>60</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_b"><math class="ltx_Math" altimg="m39.png" altimg-height="18px" altimg-valign="-4px" altimg-width="492px" alttext="-121\;52331\;40483\;75557\;20403\;04994\;07982\;02460\;41491" display="inline"><mrow><mo>-</mo><mn>121 52331 40483 75557 20403 04994 07982 02460 41491</mn></mrow></math></td>
<td class="ltx_td ltx_align_right ltx_border_b"><math class="ltx_Math" altimg="m204.png" altimg-height="17px" altimg-valign="-2px" altimg-width="90px" alttext="567\;86730" display="inline"><mn>567 86730</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_b"><math class="ltx_Math" altimg="m46.png" altimg-height="18px" altimg-valign="-4px" altimg-width="130px" alttext="-2.13999\;4926" display="inline"><mrow><mo>-</mo><mn>2.13999 4926</mn></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_border_b"><math class="ltx_Math" altimg="m304.png" altimg-height="21px" altimg-valign="-4px" altimg-width="57px" alttext="\times 10^{34}" display="inline"><mn>×10³⁴</mn></math></td>
</tr>
</tbody>
</table>
<div id="T3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./24.2#i" title="§24.2(i) Bernoulli Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m234.png" altimg-height="21px" altimg-valign="-5px" altimg-width="30px" alttext="B_{\NVar{n}}" display="inline"><msub><mi href="./24.2#i">B</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub></math>: Bernoulli numbers</a> and
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m317.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./24.1#p2.t1.r1">n</mi></math>: integer</a>
</dd>
</dl>
</div>
</div>
</figure>
<figure id="T4" class="ltx_table">
<figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_table">Table 24.2.4: </span>Euler numbers <math class="ltx_Math" altimg="m243.png" altimg-height="21px" altimg-valign="-5px" altimg-width="30px" alttext="E_{n}" display="inline"><msub><mi href="./24.2#ii">E</mi><mi href="./24.1#p2.t1.r1">n</mi></msub></math>.
</figcaption>
<table id="T4.t1" class="ltx_tabular ltx_centering ltx_guessed_headers ltx_align_middle">
<thead class="ltx_thead">
<tr id="T4.t1.r1" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_border_tt"><math class="ltx_Math" altimg="m317.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./24.1#p2.t1.r1">n</mi></math></th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_border_tt"><math class="ltx_Math" altimg="m243.png" altimg-height="21px" altimg-valign="-5px" altimg-width="30px" alttext="E_{n}" display="inline"><msub><mi href="./24.2#ii">E</mi><mi href="./24.1#p2.t1.r1">n</mi></msub></math></th>
</tr>
</thead>
<tbody class="ltx_tbody">
<tr id="T4.t1.r2" class="ltx_tr">
<td class="ltx_td ltx_align_right ltx_border_t"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_t"><math class="ltx_Math" altimg="m157.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="1" display="inline"><mn>1</mn></math></td>
</tr>
<tr id="T4.t1.r3" class="ltx_tr">
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m175.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="2" display="inline"><mn>2</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m45.png" altimg-height="18px" altimg-valign="-4px" altimg-width="30px" alttext="-1" display="inline"><mrow><mo>-</mo><mn>1</mn></mrow></math></td>
</tr>
<tr id="T4.t1.r4" class="ltx_tr">
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m197.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="4" display="inline"><mn>4</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m208.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="5" display="inline"><mn>5</mn></math></td>
</tr>
<tr id="T4.t1.r5" class="ltx_tr">
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m217.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="6" display="inline"><mn>6</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m79.png" altimg-height="18px" altimg-valign="-4px" altimg-width="40px" alttext="-61" display="inline"><mrow><mo>-</mo><mn>61</mn></mrow></math></td>
</tr>
<tr id="T4.t1.r6" class="ltx_tr">
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m230.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="8" display="inline"><mn>8</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m138.png" altimg-height="17px" altimg-valign="-2px" altimg-width="44px" alttext="1385" display="inline"><mn>1385</mn></math></td>
</tr>
<tr id="T4.t1.r7" class="ltx_tr">
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m133.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="10" display="inline"><mn>10</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m75.png" altimg-height="18px" altimg-valign="-4px" altimg-width="70px" alttext="-50521" display="inline"><mrow><mo>-</mo><mn>50521</mn></mrow></math></td>
</tr>
<tr id="T4.t1.r8" class="ltx_tr">
<td class="ltx_td ltx_align_right ltx_border_T"><math class="ltx_Math" altimg="m136.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="12" display="inline"><mn>12</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_T"><math class="ltx_Math" altimg="m170.png" altimg-height="17px" altimg-valign="-2px" altimg-width="80px" alttext="27\;02765" display="inline"><mn>27 02765</mn></math></td>
</tr>
<tr id="T4.t1.r9" class="ltx_tr">
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m142.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="14" display="inline"><mn>14</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m43.png" altimg-height="18px" altimg-valign="-4px" altimg-width="115px" alttext="-1993\;60981" display="inline"><mrow><mo>-</mo><mn>1993 60981</mn></mrow></math></td>
</tr>
<tr id="T4.t1.r10" class="ltx_tr">
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m148.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="16" display="inline"><mn>16</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m156.png" altimg-height="17px" altimg-valign="-2px" altimg-width="125px" alttext="1\;93915\;12145" display="inline"><mn>1 93915 12145</mn></math></td>
</tr>
<tr id="T4.t1.r11" class="ltx_tr">
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m152.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="18" display="inline"><mn>18</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m53.png" altimg-height="18px" altimg-valign="-4px" altimg-width="160px" alttext="-240\;48796\;75441" display="inline"><mrow><mo>-</mo><mn>240 48796 75441</mn></mrow></math></td>
</tr>
<tr id="T4.t1.r12" class="ltx_tr">
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m162.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="20" display="inline"><mn>20</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m183.png" altimg-height="17px" altimg-valign="-2px" altimg-width="165px" alttext="37037\;11882\;37525" display="inline"><mn>37037 11882 37525</mn></math></td>
</tr>
<tr id="T4.t1.r13" class="ltx_tr">
<td class="ltx_td ltx_align_right ltx_border_T"><math class="ltx_Math" altimg="m164.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="22" display="inline"><mn>22</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_T"><math class="ltx_Math" altimg="m83.png" altimg-height="18px" altimg-valign="-4px" altimg-width="206px" alttext="-69\;34887\;43931\;37901" display="inline"><mrow><mo>-</mo><mn>69 34887 43931 37901</mn></mrow></math></td>
</tr>
<tr id="T4.t1.r14" class="ltx_tr">
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m165.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="24" display="inline"><mn>24</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m143.png" altimg-height="17px" altimg-valign="-2px" altimg-width="220px" alttext="15514\;53416\;35570\;86905" display="inline"><mn>15514 53416 35570 86905</mn></math></td>
</tr>
<tr id="T4.t1.r15" class="ltx_tr">
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m168.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="26" display="inline"><mn>26</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m69.png" altimg-height="18px" altimg-valign="-4px" altimg-width="261px" alttext="-40\;87072\;50929\;31238\;92361" display="inline"><mrow><mo>-</mo><mn>40 87072 50929 31238 92361</mn></mrow></math></td>
</tr>
<tr id="T4.t1.r16" class="ltx_tr">
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m172.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="28" display="inline"><mn>28</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m135.png" altimg-height="17px" altimg-valign="-2px" altimg-width="275px" alttext="12522\;59641\;40362\;98654\;68285" display="inline"><mn>12522 59641 40362 98654 68285</mn></math></td>
</tr>
<tr id="T4.t1.r17" class="ltx_tr">
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m177.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="30" display="inline"><mn>30</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m70.png" altimg-height="18px" altimg-valign="-4px" altimg-width="316px" alttext="-44\;15438\;93249\;02310\;45536\;82821" display="inline"><mrow><mo>-</mo><mn>44 15438 93249 02310 45536 82821</mn></mrow></math></td>
</tr>
<tr id="T4.t1.r18" class="ltx_tr">
<td class="ltx_td ltx_align_right ltx_border_T"><math class="ltx_Math" altimg="m178.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="32" display="inline"><mn>32</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_T"><math class="ltx_Math" altimg="m149.png" altimg-height="17px" altimg-valign="-2px" altimg-width="331px" alttext="17751\;93915\;79539\;28943\;66647\;89665" display="inline"><mn>17751 93915 79539 28943 66647 89665</mn></math></td>
</tr>
<tr id="T4.t1.r19" class="ltx_tr">
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m180.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="34" display="inline"><mn>34</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m91.png" altimg-height="18px" altimg-valign="-4px" altimg-width="372px" alttext="-80\;72329\;92358\;87898\;06216\;82474\;53281" display="inline"><mrow><mo>-</mo><mn>80 72329 92358 87898 06216 82474 53281</mn></mrow></math></td>
</tr>
<tr id="T4.t1.r20" class="ltx_tr">
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m182.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="36" display="inline"><mn>36</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m189.png" altimg-height="17px" altimg-valign="-2px" altimg-width="386px" alttext="41222\;06033\;95177\;02122\;34707\;96712\;59045" display="inline"><mn>41222 06033 95177 02122 34707 96712 59045</mn></math></td>
</tr>
<tr id="T4.t1.r21" class="ltx_tr">
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m184.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="38" display="inline"><mn>38</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m51.png" altimg-height="18px" altimg-valign="-4px" altimg-width="437px" alttext="-234\;89580\;52704\;31082\;52017\;82857\;61989\;47741" display="inline"><mrow><mo>-</mo><mn>234 89580 52704 31082 52017 82857 61989 47741</mn></mrow></math></td>
</tr>
<tr id="T4.t1.r22" class="ltx_tr">
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m188.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="40" display="inline"><mn>40</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m154.png" altimg-height="17px" altimg-valign="-2px" altimg-width="457px" alttext="1\;48511\;50718\;11498\;00178\;77156\;78140\;58266\;84425" display="inline"><mn>1 48511 50718 11498 00178 77156 78140 58266 84425</mn></math></td>
</tr>
<tr id="T4.t1.r23" class="ltx_tr">
<td class="ltx_td ltx_align_right ltx_border_T"><math class="ltx_Math" altimg="m190.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="42" display="inline"><mn>42</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_T"><math class="ltx_Math" altimg="m37.png" altimg-height="18px" altimg-valign="-4px" altimg-width="502px" alttext="-1036\;46227\;33519\;61211\;93979\;57304\;74518\;59763\;10201" display="inline"><mrow><mo>-</mo><mn>1036 46227 33519 61211 93979 57304 74518 59763 10201</mn></mrow></math></td>
</tr>
<tr id="T4.t1.r24" class="ltx_tr">
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m192.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="44" display="inline"><mn>44</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m222.png" altimg-height="17px" altimg-valign="-2px" altimg-width="512px" alttext="7\;94757\;94225\;97592\;70360\;80405\;10088\;07061\;95192\;73805" display="inline"><mn>7 94757 94225 97592 70360 80405 10088 07061 95192 73805</mn></math></td>
</tr>
<tr id="T4.t1.r25" class="ltx_tr">
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m194.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="46" display="inline"><mn>46</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m81.png" altimg-height="18px" altimg-valign="-4px" altimg-width="558px" alttext="-6667\;53751\;66855\;44977\;43502\;84747\;73748\;19752\;41076\;84661" display="inline"><mrow><mo>-</mo><mn>6667 53751 66855 44977 43502 84747 73748 19752 41076 84661</mn></mrow></math></td>
</tr>
<tr id="T4.t1.r26" class="ltx_tr">
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m195.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="48" display="inline"><mn>48</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m211.png" altimg-height="17px" altimg-valign="-2px" altimg-width="578px" alttext="60\;96278\;64556\;85421\;58691\;68574\;28768\;43153\;97653\;90444\;35185" display="inline"><mn>60 96278 64556 85421 58691 68574 28768 43153 97653 90444 35185</mn></math></td>
</tr>
<tr id="T4.t1.r27" class="ltx_tr">
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m199.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="50" display="inline"><mn>50</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m78.png" altimg-height="18px" altimg-valign="-4px" altimg-width="623px" alttext="-60532\;85248\;18862\;18963\;14383\;78511\;16490\;88103\;49822\;51468\;15121" display="inline"><mrow><mo>-</mo><mn>60532 85248 18862 18963 14383 78511 16490 88103 49822 51468 15121</mn></mrow></math></td>
</tr>
<tr id="T4.t1.r28" class="ltx_tr">
<td class="ltx_td ltx_align_right ltx_border_T"><math class="ltx_Math" altimg="m201.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="52" display="inline"><mn>52</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_T"><math class="ltx_Math" altimg="m214.png" altimg-height="17px" altimg-valign="-2px" altimg-width="643px" alttext="650\;61624\;86684\;60884\;77158\;70634\;08082\;29834\;83644\;23676\;53855\;76565" display="inline"><mn>650 61624 86684 60884 77158 70634 08082 29834 83644 23676 53855 76565</mn></math></td>
</tr>
<tr id="T4.t1.r29" class="ltx_tr">
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m202.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="54" display="inline"><mn>54</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m87.png" altimg-height="18px" altimg-valign="-4px" altimg-width="694px" alttext="-7\;54665\;99390\;08739\;09806\;14325\;65889\;73674\;42122\;40024\;71169\;9858%
6\;45581" display="inline"><mrow><mo>-</mo><mn>7 54665 99390 08739 09806 14325 65889 73674 42122 40024 71169 98586 45581</mn></mrow></math></td>
</tr>
<tr id="T4.t1.r30" class="ltx_tr">
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m205.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="56" display="inline"><mn>56</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m232.png" altimg-height="17px" altimg-valign="-2px" altimg-width="708px" alttext="9420\;32189\;64202\;41204\;20228\;62376\;90583\;22720\;93888\;52599\;64600\;93%
949\;05945" display="inline"><mn>9420 32189 64202 41204 20228 62376 90583 22720 93888 52599 64600 93949 05945</mn></math></td>
</tr>
<tr id="T4.t1.r31" class="ltx_tr">
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m206.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="58" display="inline"><mn>58</mn></math></td>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m40.png" altimg-height="18px" altimg-valign="-4px" altimg-width="769px" alttext="-126\;22019\;25180\;62187\;19903\;40923\;72874\;89255\;48234\;10611\;91825\;59%
406\;99649\;20041" display="inline"><mrow><mo>-</mo><mn>126 22019 25180 62187 19903 40923 72874 89255 48234 10611 91825 59406 99649 20041</mn></mrow></math></td>
</tr>
<tr id="T4.t1.r32" class="ltx_tr">
<td class="ltx_td ltx_align_right ltx_border_b"><math class="ltx_Math" altimg="m212.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="60" display="inline"><mn>60</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_b"><math class="ltx_Math" altimg="m155.png" altimg-height="17px" altimg-valign="-2px" altimg-width="789px" alttext="1\;81089\;11496\;57923\;04965\;45807\;74165\;21586\;88733\;48734\;92363\;14106%
\;00809\;54542\;31325" display="inline"><mn>1 81089 11496 57923 04965 45807 74165 21586 88733 48734 92363 14106 00809 54542 31325</mn></math></td>
</tr>
</tbody>
</table>
<div id="T4.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./24.2#ii" title="§24.2(ii) Euler Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m241.png" altimg-height="21px" altimg-valign="-5px" altimg-width="30px" alttext="E_{\NVar{n}}" display="inline"><msub><mi href="./24.2#ii">E</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub></math>: Euler numbers</a> and
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m317.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./24.1#p2.t1.r1">n</mi></math>: integer</a>
</dd>
</dl>
</div>
</div>
</figure>
<figure id="T5" class="ltx_table">
<figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_table">Table 24.2.5: </span>Coefficients <math class="ltx_Math" altimg="m312.png" altimg-height="23px" altimg-valign="-8px" altimg-width="37px" alttext="b_{n,k}" display="inline"><msub><mi>b</mi><mrow><mi href="./24.1#p2.t1.r1">n</mi><mo>,</mo><mi href="./24.1#p2.t1.r1">k</mi></mrow></msub></math> of the Bernoulli polynomials
<math class="ltx_Math" altimg="m238.png" altimg-height="26px" altimg-valign="-8px" altimg-width="195px" alttext="B_{n}\left(x\right)=\sum_{k=0}^{n}b_{n,k}x^{k}" display="inline"><mrow><mrow><msub><mi href="./24.2#i">B</mi><mi href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msubsup><mo largeop="true" symmetric="true">∑</mo><mrow><mi href="./24.1#p2.t1.r1">k</mi><mo>=</mo><mn>0</mn></mrow><mi href="./24.1#p2.t1.r1">n</mi></msubsup><mrow><msub><mi>b</mi><mrow><mi href="./24.1#p2.t1.r1">n</mi><mo>,</mo><mi href="./24.1#p2.t1.r1">k</mi></mrow></msub><mo></mo><msup><mi href="./24.1#p2.t1.r2">x</mi><mi href="./24.1#p2.t1.r1">k</mi></msup></mrow></mrow></mrow></math>.
</figcaption>
<table id="T5.t1" class="ltx_tabular ltx_centering ltx_guessed_headers ltx_align_middle">
<thead class="ltx_thead">
<tr id="T5.t1.r1" class="ltx_tr">
<th class="ltx_td ltx_th ltx_th_column ltx_th_row ltx_border_tt" style="padding:2.083333333333333px 4.2pt;"></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" style="padding:2.083333333333333px 4.2pt;" colspan="16"><math class="ltx_Math" altimg="m314.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./24.1#p2.t1.r1">k</mi></math></th>
</tr>
<tr id="T5.t1.r2" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_th_row ltx_border_r" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m317.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./24.1#p2.t1.r1">n</mi></math></th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_border_t" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_border_t" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m157.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="1" display="inline"><mn>1</mn></math></th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_border_t" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m175.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="2" display="inline"><mn>2</mn></math></th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_border_r ltx_border_t" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m185.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="3" display="inline"><mn>3</mn></math></th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_border_t" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m197.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="4" display="inline"><mn>4</mn></math></th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_border_t" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m208.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="5" display="inline"><mn>5</mn></math></th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_border_t" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m217.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="6" display="inline"><mn>6</mn></math></th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_border_r ltx_border_t" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m223.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="7" display="inline"><mn>7</mn></math></th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_border_t" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m230.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="8" display="inline"><mn>8</mn></math></th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_border_t" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m233.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="9" display="inline"><mn>9</mn></math></th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_border_t" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m133.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="10" display="inline"><mn>10</mn></math></th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_border_r ltx_border_t" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m134.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="11" display="inline"><mn>11</mn></math></th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_border_t" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m136.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="12" display="inline"><mn>12</mn></math></th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_border_t" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m140.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="13" display="inline"><mn>13</mn></math></th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_border_t" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m142.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="14" display="inline"><mn>14</mn></math></th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_border_r ltx_border_t" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m146.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="15" display="inline"><mn>15</mn></math></th>
</tr>
</thead>
<tbody class="ltx_tbody">
<tr id="T5.t1.r3" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_r ltx_border_t" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></th>
<td class="ltx_td ltx_align_right ltx_border_t" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m157.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="1" display="inline"><mn>1</mn></math></td>
<td class="ltx_td ltx_border_t" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td ltx_border_t" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td ltx_border_r ltx_border_t" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td ltx_border_t" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td ltx_border_t" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td ltx_border_t" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td ltx_border_r ltx_border_t" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td ltx_border_t" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td ltx_border_t" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td ltx_border_t" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td ltx_border_r ltx_border_t" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td ltx_border_t" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td ltx_border_t" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td ltx_border_t" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td ltx_border_r ltx_border_t" style="padding:2.083333333333333px 4.2pt;"></td>
</tr>
<tr id="T5.t1.r4" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_r" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m157.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="1" display="inline"><mn>1</mn></math></th>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m101.png" altimg-height="27px" altimg-valign="-9px" altimg-width="33px" alttext="-\tfrac{1}{2}" display="inline"><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m157.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="1" display="inline"><mn>1</mn></math></td>
<td class="ltx_td" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td ltx_border_r" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td ltx_border_r" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td ltx_border_r" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td ltx_border_r" style="padding:2.083333333333333px 4.2pt;"></td>
</tr>
<tr id="T5.t1.r5" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_r" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m175.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="2" display="inline"><mn>2</mn></math></th>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m262.png" altimg-height="27px" altimg-valign="-9px" altimg-width="17px" alttext="\tfrac{1}{6}" display="inline"><mfrac><mn>1</mn><mn>6</mn></mfrac></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m45.png" altimg-height="18px" altimg-valign="-4px" altimg-width="30px" alttext="-1" display="inline"><mrow><mo>-</mo><mn>1</mn></mrow></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m157.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="1" display="inline"><mn>1</mn></math></td>
<td class="ltx_td ltx_border_r" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td ltx_border_r" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td ltx_border_r" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td ltx_border_r" style="padding:2.083333333333333px 4.2pt;"></td>
</tr>
<tr id="T5.t1.r6" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_r" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m185.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="3" display="inline"><mn>3</mn></math></th>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m259.png" altimg-height="27px" altimg-valign="-9px" altimg-width="17px" alttext="\tfrac{1}{2}" display="inline"><mfrac><mn>1</mn><mn>2</mn></mfrac></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m112.png" altimg-height="27px" altimg-valign="-9px" altimg-width="33px" alttext="-\tfrac{3}{2}" display="inline"><mrow><mo>-</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow></math></td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m157.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="1" display="inline"><mn>1</mn></math></td>
<td class="ltx_td" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td ltx_border_r" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td ltx_border_r" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td ltx_border_r" style="padding:2.083333333333333px 4.2pt;"></td>
</tr>
<tr id="T5.t1.r7" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_T ltx_border_r" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m197.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="4" display="inline"><mn>4</mn></math></th>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m102.png" altimg-height="27px" altimg-valign="-9px" altimg-width="40px" alttext="-\tfrac{1}{30}" display="inline"><mrow><mo>-</mo><mfrac><mn>1</mn><mn>30</mn></mfrac></mrow></math></td>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m157.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="1" display="inline"><mn>1</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_T ltx_border_r" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m60.png" altimg-height="18px" altimg-valign="-4px" altimg-width="30px" alttext="-2" display="inline"><mrow><mo>-</mo><mn>2</mn></mrow></math></td>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m157.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="1" display="inline"><mn>1</mn></math></td>
<td class="ltx_td ltx_border_T" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td ltx_border_T" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td ltx_border_T ltx_border_r" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td ltx_border_T" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td ltx_border_T" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td ltx_border_T" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td ltx_border_T ltx_border_r" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td ltx_border_T" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td ltx_border_T" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td ltx_border_T" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td ltx_border_T ltx_border_r" style="padding:2.083333333333333px 4.2pt;"></td>
</tr>
<tr id="T5.t1.r8" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_r" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m208.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="5" display="inline"><mn>5</mn></math></th>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m103.png" altimg-height="27px" altimg-valign="-9px" altimg-width="33px" alttext="-\tfrac{1}{6}" display="inline"><mrow><mo>-</mo><mfrac><mn>1</mn><mn>6</mn></mfrac></mrow></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m274.png" altimg-height="27px" altimg-valign="-9px" altimg-width="17px" alttext="\tfrac{5}{3}" display="inline"><mfrac><mn>5</mn><mn>3</mn></mfrac></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m117.png" altimg-height="27px" altimg-valign="-9px" altimg-width="33px" alttext="-\tfrac{5}{2}" display="inline"><mrow><mo>-</mo><mfrac><mn>5</mn><mn>2</mn></mfrac></mrow></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m157.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="1" display="inline"><mn>1</mn></math></td>
<td class="ltx_td" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td ltx_border_r" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td ltx_border_r" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td ltx_border_r" style="padding:2.083333333333333px 4.2pt;"></td>
</tr>
<tr id="T5.t1.r9" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_r" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m217.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="6" display="inline"><mn>6</mn></math></th>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m260.png" altimg-height="27px" altimg-valign="-9px" altimg-width="25px" alttext="\tfrac{1}{42}" display="inline"><mfrac><mn>1</mn><mn>42</mn></mfrac></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m101.png" altimg-height="27px" altimg-valign="-9px" altimg-width="33px" alttext="-\tfrac{1}{2}" display="inline"><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math></td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m273.png" altimg-height="27px" altimg-valign="-9px" altimg-width="17px" alttext="\tfrac{5}{2}" display="inline"><mfrac><mn>5</mn><mn>2</mn></mfrac></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m67.png" altimg-height="18px" altimg-valign="-4px" altimg-width="30px" alttext="-3" display="inline"><mrow><mo>-</mo><mn>3</mn></mrow></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m157.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="1" display="inline"><mn>1</mn></math></td>
<td class="ltx_td ltx_border_r" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td ltx_border_r" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td ltx_border_r" style="padding:2.083333333333333px 4.2pt;"></td>
</tr>
<tr id="T5.t1.r10" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_r" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m223.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="7" display="inline"><mn>7</mn></math></th>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m262.png" altimg-height="27px" altimg-valign="-9px" altimg-width="17px" alttext="\tfrac{1}{6}" display="inline"><mfrac><mn>1</mn><mn>6</mn></mfrac></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m124.png" altimg-height="27px" altimg-valign="-9px" altimg-width="33px" alttext="-\tfrac{7}{6}" display="inline"><mrow><mo>-</mo><mfrac><mn>7</mn><mn>6</mn></mfrac></mrow></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m281.png" altimg-height="27px" altimg-valign="-9px" altimg-width="17px" alttext="\tfrac{7}{2}" display="inline"><mfrac><mn>7</mn><mn>2</mn></mfrac></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m122.png" altimg-height="27px" altimg-valign="-9px" altimg-width="33px" alttext="-\tfrac{7}{2}" display="inline"><mrow><mo>-</mo><mfrac><mn>7</mn><mn>2</mn></mfrac></mrow></math></td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m157.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="1" display="inline"><mn>1</mn></math></td>
<td class="ltx_td" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td ltx_border_r" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td ltx_border_r" style="padding:2.083333333333333px 4.2pt;"></td>
</tr>
<tr id="T5.t1.r11" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_T ltx_border_r" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m230.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="8" display="inline"><mn>8</mn></math></th>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m102.png" altimg-height="27px" altimg-valign="-9px" altimg-width="40px" alttext="-\tfrac{1}{30}" display="inline"><mrow><mo>-</mo><mfrac><mn>1</mn><mn>30</mn></mfrac></mrow></math></td>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m266.png" altimg-height="27px" altimg-valign="-9px" altimg-width="17px" alttext="\tfrac{2}{3}" display="inline"><mfrac><mn>2</mn><mn>3</mn></mfrac></math></td>
<td class="ltx_td ltx_align_right ltx_border_T ltx_border_r" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m123.png" altimg-height="27px" altimg-valign="-9px" altimg-width="33px" alttext="-\tfrac{7}{3}" display="inline"><mrow><mo>-</mo><mfrac><mn>7</mn><mn>3</mn></mfrac></mrow></math></td>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m254.png" altimg-height="27px" altimg-valign="-9px" altimg-width="25px" alttext="\tfrac{14}{3}" display="inline"><mfrac><mn>14</mn><mn>3</mn></mfrac></math></td>
<td class="ltx_td ltx_align_right ltx_border_T ltx_border_r" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m71.png" altimg-height="18px" altimg-valign="-4px" altimg-width="30px" alttext="-4" display="inline"><mrow><mo>-</mo><mn>4</mn></mrow></math></td>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m157.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="1" display="inline"><mn>1</mn></math></td>
<td class="ltx_td ltx_border_T" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td ltx_border_T" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td ltx_border_T ltx_border_r" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td ltx_border_T" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td ltx_border_T" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td ltx_border_T" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td ltx_border_T ltx_border_r" style="padding:2.083333333333333px 4.2pt;"></td>
</tr>
<tr id="T5.t1.r12" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_r" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m233.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="9" display="inline"><mn>9</mn></math></th>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m111.png" altimg-height="27px" altimg-valign="-9px" altimg-width="40px" alttext="-\tfrac{3}{10}" display="inline"><mrow><mo>-</mo><mfrac><mn>3</mn><mn>10</mn></mfrac></mrow></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m175.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="2" display="inline"><mn>2</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m105.png" altimg-height="27px" altimg-valign="-9px" altimg-width="40px" alttext="-\tfrac{21}{5}" display="inline"><mrow><mo>-</mo><mfrac><mn>21</mn><mn>5</mn></mfrac></mrow></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m217.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="6" display="inline"><mn>6</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m126.png" altimg-height="27px" altimg-valign="-9px" altimg-width="33px" alttext="-\tfrac{9}{2}" display="inline"><mrow><mo>-</mo><mfrac><mn>9</mn><mn>2</mn></mfrac></mrow></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m157.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="1" display="inline"><mn>1</mn></math></td>
<td class="ltx_td" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td ltx_border_r" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td ltx_border_r" style="padding:2.083333333333333px 4.2pt;"></td>
</tr>
<tr id="T5.t1.r13" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_r" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m133.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="10" display="inline"><mn>10</mn></math></th>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m275.png" altimg-height="27px" altimg-valign="-9px" altimg-width="25px" alttext="\tfrac{5}{66}" display="inline"><mfrac><mn>5</mn><mn>66</mn></mfrac></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m112.png" altimg-height="27px" altimg-valign="-9px" altimg-width="33px" alttext="-\tfrac{3}{2}" display="inline"><mrow><mo>-</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow></math></td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m208.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="5" display="inline"><mn>5</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m88.png" altimg-height="18px" altimg-valign="-4px" altimg-width="30px" alttext="-7" display="inline"><mrow><mo>-</mo><mn>7</mn></mrow></math></td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m256.png" altimg-height="27px" altimg-valign="-9px" altimg-width="25px" alttext="\tfrac{15}{2}" display="inline"><mfrac><mn>15</mn><mn>2</mn></mfrac></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m77.png" altimg-height="18px" altimg-valign="-4px" altimg-width="30px" alttext="-5" display="inline"><mrow><mo>-</mo><mn>5</mn></mrow></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m157.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="1" display="inline"><mn>1</mn></math></td>
<td class="ltx_td ltx_border_r" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td ltx_border_r" style="padding:2.083333333333333px 4.2pt;"></td>
</tr>
<tr id="T5.t1.r14" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_r" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m134.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="11" display="inline"><mn>11</mn></math></th>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m276.png" altimg-height="27px" altimg-valign="-9px" altimg-width="17px" alttext="\tfrac{5}{6}" display="inline"><mfrac><mn>5</mn><mn>6</mn></mfrac></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m94.png" altimg-height="27px" altimg-valign="-9px" altimg-width="40px" alttext="-\tfrac{11}{2}" display="inline"><mrow><mo>-</mo><mfrac><mn>11</mn><mn>2</mn></mfrac></mrow></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m134.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="11" display="inline"><mn>11</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m38.png" altimg-height="18px" altimg-valign="-4px" altimg-width="40px" alttext="-11" display="inline"><mrow><mo>-</mo><mn>11</mn></mrow></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m272.png" altimg-height="27px" altimg-valign="-9px" altimg-width="25px" alttext="\tfrac{55}{6}" display="inline"><mfrac><mn>55</mn><mn>6</mn></mfrac></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m94.png" altimg-height="27px" altimg-valign="-9px" altimg-width="40px" alttext="-\tfrac{11}{2}" display="inline"><mrow><mo>-</mo><mfrac><mn>11</mn><mn>2</mn></mfrac></mrow></math></td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m157.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="1" display="inline"><mn>1</mn></math></td>
<td class="ltx_td" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td ltx_border_r" style="padding:2.083333333333333px 4.2pt;"></td>
</tr>
<tr id="T5.t1.r15" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_T ltx_border_r" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m136.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="12" display="inline"><mn>12</mn></math></th>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m119.png" altimg-height="27px" altimg-valign="-9px" altimg-width="56px" alttext="-\tfrac{691}{2730}" display="inline"><mrow><mo>-</mo><mfrac><mn>691</mn><mn>2730</mn></mfrac></mrow></math></td>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m208.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="5" display="inline"><mn>5</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_T ltx_border_r" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m109.png" altimg-height="27px" altimg-valign="-9px" altimg-width="40px" alttext="-\tfrac{33}{2}" display="inline"><mrow><mo>-</mo><mfrac><mn>33</mn><mn>2</mn></mfrac></mrow></math></td>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m164.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="22" display="inline"><mn>22</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_T ltx_border_r" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m109.png" altimg-height="27px" altimg-valign="-9px" altimg-width="40px" alttext="-\tfrac{33}{2}" display="inline"><mrow><mo>-</mo><mfrac><mn>33</mn><mn>2</mn></mfrac></mrow></math></td>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m134.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="11" display="inline"><mn>11</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_T ltx_border_r" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m84.png" altimg-height="18px" altimg-valign="-4px" altimg-width="30px" alttext="-6" display="inline"><mrow><mo>-</mo><mn>6</mn></mrow></math></td>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m157.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="1" display="inline"><mn>1</mn></math></td>
<td class="ltx_td ltx_border_T" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td ltx_border_T" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td ltx_border_T ltx_border_r" style="padding:2.083333333333333px 4.2pt;"></td>
</tr>
<tr id="T5.t1.r16" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_r" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m140.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="13" display="inline"><mn>13</mn></math></th>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m118.png" altimg-height="27px" altimg-valign="-9px" altimg-width="48px" alttext="-\tfrac{691}{210}" display="inline"><mrow><mo>-</mo><mfrac><mn>691</mn><mn>210</mn></mfrac></mrow></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m277.png" altimg-height="27px" altimg-valign="-9px" altimg-width="25px" alttext="\tfrac{65}{3}" display="inline"><mfrac><mn>65</mn><mn>3</mn></mfrac></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m113.png" altimg-height="27px" altimg-valign="-9px" altimg-width="48px" alttext="-\tfrac{429}{10}" display="inline"><mrow><mo>-</mo><mfrac><mn>429</mn><mn>10</mn></mfrac></mrow></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m265.png" altimg-height="27px" altimg-valign="-9px" altimg-width="33px" alttext="\tfrac{286}{7}" display="inline"><mfrac><mn>286</mn><mn>7</mn></mfrac></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m97.png" altimg-height="27px" altimg-valign="-9px" altimg-width="48px" alttext="-\tfrac{143}{6}" display="inline"><mrow><mo>-</mo><mfrac><mn>143</mn><mn>6</mn></mfrac></mrow></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m140.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="13" display="inline"><mn>13</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m96.png" altimg-height="27px" altimg-valign="-9px" altimg-width="40px" alttext="-\tfrac{13}{2}" display="inline"><mrow><mo>-</mo><mfrac><mn>13</mn><mn>2</mn></mfrac></mrow></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m157.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="1" display="inline"><mn>1</mn></math></td>
<td class="ltx_td" style="padding:2.083333333333333px 4.2pt;"></td>
<td class="ltx_td ltx_border_r" style="padding:2.083333333333333px 4.2pt;"></td>
</tr>
<tr id="T5.t1.r17" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_r" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m142.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="14" display="inline"><mn>14</mn></math></th>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m282.png" altimg-height="27px" altimg-valign="-9px" altimg-width="17px" alttext="\tfrac{7}{6}" display="inline"><mfrac><mn>7</mn><mn>6</mn></mfrac></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m120.png" altimg-height="27px" altimg-valign="-9px" altimg-width="48px" alttext="-\tfrac{691}{30}" display="inline"><mrow><mo>-</mo><mfrac><mn>691</mn><mn>30</mn></mfrac></mrow></math></td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m271.png" altimg-height="27px" altimg-valign="-9px" altimg-width="33px" alttext="\tfrac{455}{6}" display="inline"><mfrac><mn>455</mn><mn>6</mn></mfrac></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m92.png" altimg-height="27px" altimg-valign="-9px" altimg-width="56px" alttext="-\tfrac{1001}{10}" display="inline"><mrow><mo>-</mo><mfrac><mn>1001</mn><mn>10</mn></mfrac></mrow></math></td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m253.png" altimg-height="27px" altimg-valign="-9px" altimg-width="33px" alttext="\tfrac{143}{2}" display="inline"><mfrac><mn>143</mn><mn>2</mn></mfrac></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m93.png" altimg-height="27px" altimg-valign="-9px" altimg-width="56px" alttext="-\tfrac{1001}{30}" display="inline"><mrow><mo>-</mo><mfrac><mn>1001</mn><mn>30</mn></mfrac></mrow></math></td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m283.png" altimg-height="27px" altimg-valign="-9px" altimg-width="25px" alttext="\tfrac{91}{6}" display="inline"><mfrac><mn>91</mn><mn>6</mn></mfrac></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m88.png" altimg-height="18px" altimg-valign="-4px" altimg-width="30px" alttext="-7" display="inline"><mrow><mo>-</mo><mn>7</mn></mrow></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m157.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="1" display="inline"><mn>1</mn></math></td>
<td class="ltx_td ltx_border_r" style="padding:2.083333333333333px 4.2pt;"></td>
</tr>
<tr id="T5.t1.r18" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_b ltx_border_r" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m146.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="15" display="inline"><mn>15</mn></math></th>
<td class="ltx_td ltx_align_right ltx_border_b" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_b" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m267.png" altimg-height="27px" altimg-valign="-9px" altimg-width="25px" alttext="\tfrac{35}{2}" display="inline"><mfrac><mn>35</mn><mn>2</mn></mfrac></math></td>
<td class="ltx_td ltx_align_right ltx_border_b" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_b ltx_border_r" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m121.png" altimg-height="27px" altimg-valign="-9px" altimg-width="48px" alttext="-\tfrac{691}{6}" display="inline"><mrow><mo>-</mo><mfrac><mn>691</mn><mn>6</mn></mfrac></mrow></math></td>
<td class="ltx_td ltx_align_right ltx_border_b" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_b" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m269.png" altimg-height="27px" altimg-valign="-9px" altimg-width="33px" alttext="\tfrac{455}{2}" display="inline"><mfrac><mn>455</mn><mn>2</mn></mfrac></math></td>
<td class="ltx_td ltx_align_right ltx_border_b" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_b ltx_border_r" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m114.png" altimg-height="27px" altimg-valign="-9px" altimg-width="48px" alttext="-\tfrac{429}{2}" display="inline"><mrow><mo>-</mo><mfrac><mn>429</mn><mn>2</mn></mfrac></mrow></math></td>
<td class="ltx_td ltx_align_right ltx_border_b" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_b" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m279.png" altimg-height="27px" altimg-valign="-9px" altimg-width="33px" alttext="\tfrac{715}{6}" display="inline"><mfrac><mn>715</mn><mn>6</mn></mfrac></math></td>
<td class="ltx_td ltx_align_right ltx_border_b" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_b ltx_border_r" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m125.png" altimg-height="27px" altimg-valign="-9px" altimg-width="40px" alttext="-\tfrac{91}{2}" display="inline"><mrow><mo>-</mo><mfrac><mn>91</mn><mn>2</mn></mfrac></mrow></math></td>
<td class="ltx_td ltx_align_right ltx_border_b" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_b" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m267.png" altimg-height="27px" altimg-valign="-9px" altimg-width="25px" alttext="\tfrac{35}{2}" display="inline"><mfrac><mn>35</mn><mn>2</mn></mfrac></math></td>
<td class="ltx_td ltx_align_right ltx_border_b" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m99.png" altimg-height="27px" altimg-valign="-9px" altimg-width="40px" alttext="-\tfrac{15}{2}" display="inline"><mrow><mo>-</mo><mfrac><mn>15</mn><mn>2</mn></mfrac></mrow></math></td>
<td class="ltx_td ltx_align_right ltx_border_b ltx_border_r" style="padding:2.083333333333333px 4.2pt;"><math class="ltx_Math" altimg="m157.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="1" display="inline"><mn>1</mn></math></td>
</tr>
</tbody>
</table>
<div id="T5.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./24.2#i" title="§24.2(i) Bernoulli Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m235.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="B_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./24.2#i">B</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></math>: Bernoulli polynomials</a>,
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m314.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./24.1#p2.t1.r1">k</mi></math>: integer</a>,
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m317.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./24.1#p2.t1.r1">n</mi></math>: integer</a> and
<a href="./24.1#p2.t1.r2" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m320.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./24.1#p2.t1.r2">x</mi></math>: real or complex</a>
</dd>
</dl>
</div>
</div>
</figure>
<figure id="T6" class="ltx_table">
<figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_table">Table 24.2.6: </span>Coefficients <math class="ltx_Math" altimg="m313.png" altimg-height="18px" altimg-valign="-8px" altimg-width="38px" alttext="e_{n,k}" display="inline"><msub><mi>e</mi><mrow><mi href="./24.1#p2.t1.r1">n</mi><mo>,</mo><mi href="./24.1#p2.t1.r1">k</mi></mrow></msub></math> of the Euler polynomials
<math class="ltx_Math" altimg="m244.png" altimg-height="26px" altimg-valign="-8px" altimg-width="196px" alttext="E_{n}\left(x\right)=\sum_{k=0}^{n}e_{n,k}x^{k}" display="inline"><mrow><mrow><msub><mi href="./24.2#ii">E</mi><mi href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msubsup><mo largeop="true" symmetric="true">∑</mo><mrow><mi href="./24.1#p2.t1.r1">k</mi><mo>=</mo><mn>0</mn></mrow><mi href="./24.1#p2.t1.r1">n</mi></msubsup><mrow><msub><mi>e</mi><mrow><mi href="./24.1#p2.t1.r1">n</mi><mo>,</mo><mi href="./24.1#p2.t1.r1">k</mi></mrow></msub><mo></mo><msup><mi href="./24.1#p2.t1.r2">x</mi><mi href="./24.1#p2.t1.r1">k</mi></msup></mrow></mrow></mrow></math>.
</figcaption>
<table id="T6.t1" class="ltx_tabular ltx_centering ltx_guessed_headers ltx_align_middle">
<thead class="ltx_thead">
<tr id="T6.t1.r1" class="ltx_tr">
<th class="ltx_td ltx_th ltx_th_column ltx_th_row ltx_border_tt" style="padding:2.083333333333333px 3.6pt;"></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" style="padding:2.083333333333333px 3.6pt;" colspan="16"><math class="ltx_Math" altimg="m314.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./24.1#p2.t1.r1">k</mi></math></th>
</tr>
<tr id="T6.t1.r2" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_th_row ltx_border_r" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m317.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./24.1#p2.t1.r1">n</mi></math></th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_border_t" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_border_t" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m157.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="1" display="inline"><mn>1</mn></math></th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_border_t" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m175.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="2" display="inline"><mn>2</mn></math></th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_border_r ltx_border_t" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m185.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="3" display="inline"><mn>3</mn></math></th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_border_t" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m197.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="4" display="inline"><mn>4</mn></math></th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_border_t" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m208.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="5" display="inline"><mn>5</mn></math></th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_border_t" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m217.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="6" display="inline"><mn>6</mn></math></th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_border_r ltx_border_t" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m223.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="7" display="inline"><mn>7</mn></math></th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_border_t" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m230.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="8" display="inline"><mn>8</mn></math></th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_border_t" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m233.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="9" display="inline"><mn>9</mn></math></th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_border_t" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m133.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="10" display="inline"><mn>10</mn></math></th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_border_r ltx_border_t" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m134.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="11" display="inline"><mn>11</mn></math></th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_border_t" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m136.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="12" display="inline"><mn>12</mn></math></th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_border_t" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m140.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="13" display="inline"><mn>13</mn></math></th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_border_t" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m142.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="14" display="inline"><mn>14</mn></math></th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_border_r ltx_border_t" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m146.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="15" display="inline"><mn>15</mn></math></th>
</tr>
</thead>
<tbody class="ltx_tbody">
<tr id="T6.t1.r3" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_r ltx_border_t" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></th>
<td class="ltx_td ltx_align_right ltx_border_t" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m157.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="1" display="inline"><mn>1</mn></math></td>
<td class="ltx_td ltx_border_t" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td ltx_border_t" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td ltx_border_r ltx_border_t" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td ltx_border_t" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td ltx_border_t" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td ltx_border_t" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td ltx_border_r ltx_border_t" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td ltx_border_t" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td ltx_border_t" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td ltx_border_t" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td ltx_border_r ltx_border_t" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td ltx_border_t" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td ltx_border_t" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td ltx_border_t" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td ltx_border_r ltx_border_t" style="padding:2.083333333333333px 3.6pt;"></td>
</tr>
<tr id="T6.t1.r4" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_r" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m157.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="1" display="inline"><mn>1</mn></math></th>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m101.png" altimg-height="27px" altimg-valign="-9px" altimg-width="33px" alttext="-\tfrac{1}{2}" display="inline"><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m157.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="1" display="inline"><mn>1</mn></math></td>
<td class="ltx_td" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td ltx_border_r" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td ltx_border_r" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td ltx_border_r" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td ltx_border_r" style="padding:2.083333333333333px 3.6pt;"></td>
</tr>
<tr id="T6.t1.r5" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_r" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m175.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="2" display="inline"><mn>2</mn></math></th>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m45.png" altimg-height="18px" altimg-valign="-4px" altimg-width="30px" alttext="-1" display="inline"><mrow><mo>-</mo><mn>1</mn></mrow></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m157.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="1" display="inline"><mn>1</mn></math></td>
<td class="ltx_td ltx_border_r" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td ltx_border_r" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td ltx_border_r" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td ltx_border_r" style="padding:2.083333333333333px 3.6pt;"></td>
</tr>
<tr id="T6.t1.r6" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_r" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m185.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="3" display="inline"><mn>3</mn></math></th>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m261.png" altimg-height="27px" altimg-valign="-9px" altimg-width="17px" alttext="\tfrac{1}{4}" display="inline"><mfrac><mn>1</mn><mn>4</mn></mfrac></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m112.png" altimg-height="27px" altimg-valign="-9px" altimg-width="33px" alttext="-\tfrac{3}{2}" display="inline"><mrow><mo>-</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow></math></td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m157.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="1" display="inline"><mn>1</mn></math></td>
<td class="ltx_td" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td ltx_border_r" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td ltx_border_r" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td ltx_border_r" style="padding:2.083333333333333px 3.6pt;"></td>
</tr>
<tr id="T6.t1.r7" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_T ltx_border_r" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m197.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="4" display="inline"><mn>4</mn></math></th>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m157.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="1" display="inline"><mn>1</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding:2.083333333333333px 3.6pt;">0</td>
<td class="ltx_td ltx_align_right ltx_border_T ltx_border_r" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m60.png" altimg-height="18px" altimg-valign="-4px" altimg-width="30px" alttext="-2" display="inline"><mrow><mo>-</mo><mn>2</mn></mrow></math></td>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m157.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="1" display="inline"><mn>1</mn></math></td>
<td class="ltx_td ltx_border_T" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td ltx_border_T" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td ltx_border_T ltx_border_r" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td ltx_border_T" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td ltx_border_T" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td ltx_border_T" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td ltx_border_T ltx_border_r" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td ltx_border_T" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td ltx_border_T" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td ltx_border_T" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td ltx_border_T ltx_border_r" style="padding:2.083333333333333px 3.6pt;"></td>
</tr>
<tr id="T6.t1.r8" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_r" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m208.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="5" display="inline"><mn>5</mn></math></th>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m101.png" altimg-height="27px" altimg-valign="-9px" altimg-width="33px" alttext="-\tfrac{1}{2}" display="inline"><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m273.png" altimg-height="27px" altimg-valign="-9px" altimg-width="17px" alttext="\tfrac{5}{2}" display="inline"><mfrac><mn>5</mn><mn>2</mn></mfrac></math></td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m117.png" altimg-height="27px" altimg-valign="-9px" altimg-width="33px" alttext="-\tfrac{5}{2}" display="inline"><mrow><mo>-</mo><mfrac><mn>5</mn><mn>2</mn></mfrac></mrow></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m157.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="1" display="inline"><mn>1</mn></math></td>
<td class="ltx_td" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td ltx_border_r" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td ltx_border_r" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td ltx_border_r" style="padding:2.083333333333333px 3.6pt;"></td>
</tr>
<tr id="T6.t1.r9" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_r" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m217.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="6" display="inline"><mn>6</mn></math></th>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m67.png" altimg-height="18px" altimg-valign="-4px" altimg-width="30px" alttext="-3" display="inline"><mrow><mo>-</mo><mn>3</mn></mrow></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m208.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="5" display="inline"><mn>5</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m67.png" altimg-height="18px" altimg-valign="-4px" altimg-width="30px" alttext="-3" display="inline"><mrow><mo>-</mo><mn>3</mn></mrow></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m157.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="1" display="inline"><mn>1</mn></math></td>
<td class="ltx_td ltx_border_r" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td ltx_border_r" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td ltx_border_r" style="padding:2.083333333333333px 3.6pt;"></td>
</tr>
<tr id="T6.t1.r10" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_r" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m223.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="7" display="inline"><mn>7</mn></math></th>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m258.png" altimg-height="27px" altimg-valign="-9px" altimg-width="25px" alttext="\tfrac{17}{8}" display="inline"><mfrac><mn>17</mn><mn>8</mn></mfrac></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m104.png" altimg-height="27px" altimg-valign="-9px" altimg-width="40px" alttext="-\tfrac{21}{2}" display="inline"><mrow><mo>-</mo><mfrac><mn>21</mn><mn>2</mn></mfrac></mrow></math></td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m268.png" altimg-height="27px" altimg-valign="-9px" altimg-width="25px" alttext="\tfrac{35}{4}" display="inline"><mfrac><mn>35</mn><mn>4</mn></mfrac></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m122.png" altimg-height="27px" altimg-valign="-9px" altimg-width="33px" alttext="-\tfrac{7}{2}" display="inline"><mrow><mo>-</mo><mfrac><mn>7</mn><mn>2</mn></mfrac></mrow></math></td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m157.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="1" display="inline"><mn>1</mn></math></td>
<td class="ltx_td" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td ltx_border_r" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td ltx_border_r" style="padding:2.083333333333333px 3.6pt;"></td>
</tr>
<tr id="T6.t1.r11" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_T ltx_border_r" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m230.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="8" display="inline"><mn>8</mn></math></th>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m150.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="17" display="inline"><mn>17</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_T ltx_border_r" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m57.png" altimg-height="18px" altimg-valign="-4px" altimg-width="40px" alttext="-28" display="inline"><mrow><mo>-</mo><mn>28</mn></mrow></math></td>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m142.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="14" display="inline"><mn>14</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_T ltx_border_r" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m71.png" altimg-height="18px" altimg-valign="-4px" altimg-width="30px" alttext="-4" display="inline"><mrow><mo>-</mo><mn>4</mn></mrow></math></td>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m157.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="1" display="inline"><mn>1</mn></math></td>
<td class="ltx_td ltx_border_T" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td ltx_border_T" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td ltx_border_T ltx_border_r" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td ltx_border_T" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td ltx_border_T" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td ltx_border_T" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td ltx_border_T ltx_border_r" style="padding:2.083333333333333px 3.6pt;"></td>
</tr>
<tr id="T6.t1.r12" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_r" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m233.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="9" display="inline"><mn>9</mn></math></th>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m108.png" altimg-height="27px" altimg-valign="-9px" altimg-width="40px" alttext="-\tfrac{31}{2}" display="inline"><mrow><mo>-</mo><mfrac><mn>31</mn><mn>2</mn></mfrac></mrow></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m255.png" altimg-height="27px" altimg-valign="-9px" altimg-width="33px" alttext="\tfrac{153}{2}" display="inline"><mfrac><mn>153</mn><mn>2</mn></mfrac></math></td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m80.png" altimg-height="18px" altimg-valign="-4px" altimg-width="40px" alttext="-63" display="inline"><mrow><mo>-</mo><mn>63</mn></mrow></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m163.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="21" display="inline"><mn>21</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m126.png" altimg-height="27px" altimg-valign="-9px" altimg-width="33px" alttext="-\tfrac{9}{2}" display="inline"><mrow><mo>-</mo><mfrac><mn>9</mn><mn>2</mn></mfrac></mrow></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m157.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="1" display="inline"><mn>1</mn></math></td>
<td class="ltx_td" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td ltx_border_r" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td ltx_border_r" style="padding:2.083333333333333px 3.6pt;"></td>
</tr>
<tr id="T6.t1.r13" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_r" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m133.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="10" display="inline"><mn>10</mn></math></th>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m42.png" altimg-height="18px" altimg-valign="-4px" altimg-width="50px" alttext="-155" display="inline"><mrow><mo>-</mo><mn>155</mn></mrow></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m166.png" altimg-height="17px" altimg-valign="-2px" altimg-width="34px" alttext="255" display="inline"><mn>255</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m41.png" altimg-height="18px" altimg-valign="-4px" altimg-width="50px" alttext="-126" display="inline"><mrow><mo>-</mo><mn>126</mn></mrow></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m177.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="30" display="inline"><mn>30</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m77.png" altimg-height="18px" altimg-valign="-4px" altimg-width="30px" alttext="-5" display="inline"><mrow><mo>-</mo><mn>5</mn></mrow></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m157.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="1" display="inline"><mn>1</mn></math></td>
<td class="ltx_td ltx_border_r" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td ltx_border_r" style="padding:2.083333333333333px 3.6pt;"></td>
</tr>
<tr id="T6.t1.r14" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_r" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m134.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="11" display="inline"><mn>11</mn></math></th>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m278.png" altimg-height="27px" altimg-valign="-9px" altimg-width="33px" alttext="\tfrac{691}{4}" display="inline"><mfrac><mn>691</mn><mn>4</mn></mfrac></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m100.png" altimg-height="27px" altimg-valign="-9px" altimg-width="56px" alttext="-\tfrac{1705}{2}" display="inline"><mrow><mo>-</mo><mfrac><mn>1705</mn><mn>2</mn></mfrac></mrow></math></td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m264.png" altimg-height="27px" altimg-valign="-9px" altimg-width="41px" alttext="\tfrac{2805}{4}" display="inline"><mfrac><mn>2805</mn><mn>4</mn></mfrac></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m50.png" altimg-height="18px" altimg-valign="-4px" altimg-width="50px" alttext="-231" display="inline"><mrow><mo>-</mo><mn>231</mn></mrow></math></td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m257.png" altimg-height="27px" altimg-valign="-9px" altimg-width="33px" alttext="\tfrac{165}{4}" display="inline"><mfrac><mn>165</mn><mn>4</mn></mfrac></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m94.png" altimg-height="27px" altimg-valign="-9px" altimg-width="40px" alttext="-\tfrac{11}{2}" display="inline"><mrow><mo>-</mo><mfrac><mn>11</mn><mn>2</mn></mfrac></mrow></math></td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m157.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="1" display="inline"><mn>1</mn></math></td>
<td class="ltx_td" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td ltx_border_r" style="padding:2.083333333333333px 3.6pt;"></td>
</tr>
<tr id="T6.t1.r15" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_T ltx_border_r" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m136.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="12" display="inline"><mn>12</mn></math></th>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m161.png" altimg-height="17px" altimg-valign="-2px" altimg-width="44px" alttext="2073" display="inline"><mn>2073</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_T ltx_border_r" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-4px" altimg-width="60px" alttext="-3410" display="inline"><mrow><mo>-</mo><mn>3410</mn></mrow></math></td>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m147.png" altimg-height="17px" altimg-valign="-2px" altimg-width="44px" alttext="1683" display="inline"><mn>1683</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_T ltx_border_r" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m66.png" altimg-height="18px" altimg-valign="-4px" altimg-width="50px" alttext="-396" display="inline"><mrow><mo>-</mo><mn>396</mn></mrow></math></td>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m203.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="55" display="inline"><mn>55</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_T ltx_border_r" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m84.png" altimg-height="18px" altimg-valign="-4px" altimg-width="30px" alttext="-6" display="inline"><mrow><mo>-</mo><mn>6</mn></mrow></math></td>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m157.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="1" display="inline"><mn>1</mn></math></td>
<td class="ltx_td ltx_border_T" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td ltx_border_T" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td ltx_border_T ltx_border_r" style="padding:2.083333333333333px 3.6pt;"></td>
</tr>
<tr id="T6.t1.r16" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_r" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m140.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="13" display="inline"><mn>13</mn></math></th>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m115.png" altimg-height="27px" altimg-valign="-9px" altimg-width="56px" alttext="-\tfrac{5461}{2}" display="inline"><mrow><mo>-</mo><mfrac><mn>5461</mn><mn>2</mn></mfrac></mrow></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m263.png" altimg-height="27px" altimg-valign="-9px" altimg-width="49px" alttext="\tfrac{26949}{2}" display="inline"><mfrac><mn>26949</mn><mn>2</mn></mfrac></math></td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m106.png" altimg-height="27px" altimg-valign="-9px" altimg-width="64px" alttext="-\tfrac{22165}{2}" display="inline"><mrow><mo>-</mo><mfrac><mn>22165</mn><mn>2</mn></mfrac></mrow></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m280.png" altimg-height="27px" altimg-valign="-9px" altimg-width="41px" alttext="\tfrac{7293}{2}" display="inline"><mfrac><mn>7293</mn><mn>2</mn></mfrac></math></td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m95.png" altimg-height="27px" altimg-valign="-9px" altimg-width="56px" alttext="-\tfrac{1287}{2}" display="inline"><mrow><mo>-</mo><mfrac><mn>1287</mn><mn>2</mn></mfrac></mrow></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m253.png" altimg-height="27px" altimg-valign="-9px" altimg-width="33px" alttext="\tfrac{143}{2}" display="inline"><mfrac><mn>143</mn><mn>2</mn></mfrac></math></td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m96.png" altimg-height="27px" altimg-valign="-9px" altimg-width="40px" alttext="-\tfrac{13}{2}" display="inline"><mrow><mo>-</mo><mfrac><mn>13</mn><mn>2</mn></mfrac></mrow></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m157.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="1" display="inline"><mn>1</mn></math></td>
<td class="ltx_td" style="padding:2.083333333333333px 3.6pt;"></td>
<td class="ltx_td ltx_border_r" style="padding:2.083333333333333px 3.6pt;"></td>
</tr>
<tr id="T6.t1.r17" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_r" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m142.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="14" display="inline"><mn>14</mn></math></th>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m65.png" altimg-height="18px" altimg-valign="-4px" altimg-width="70px" alttext="-38227" display="inline"><mrow><mo>-</mo><mn>38227</mn></mrow></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m213.png" altimg-height="17px" altimg-valign="-2px" altimg-width="54px" alttext="62881" display="inline"><mn>62881</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m62.png" altimg-height="18px" altimg-valign="-4px" altimg-width="70px" alttext="-31031" display="inline"><mrow><mo>-</mo><mn>31031</mn></mrow></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m220.png" altimg-height="17px" altimg-valign="-2px" altimg-width="44px" alttext="7293" display="inline"><mn>7293</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m36.png" altimg-height="18px" altimg-valign="-4px" altimg-width="60px" alttext="-1001" display="inline"><mrow><mo>-</mo><mn>1001</mn></mrow></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m231.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="91" display="inline"><mn>91</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m88.png" altimg-height="18px" altimg-valign="-4px" altimg-width="30px" alttext="-7" display="inline"><mrow><mo>-</mo><mn>7</mn></mrow></math></td>
<td class="ltx_td ltx_align_right" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m157.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="1" display="inline"><mn>1</mn></math></td>
<td class="ltx_td ltx_border_r" style="padding:2.083333333333333px 3.6pt;"></td>
</tr>
<tr id="T6.t1.r18" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_b ltx_border_r" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m146.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="15" display="inline"><mn>15</mn></math></th>
<td class="ltx_td ltx_align_right ltx_border_b" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m284.png" altimg-height="27px" altimg-valign="-9px" altimg-width="57px" alttext="\tfrac{929569}{16}" display="inline"><mfrac><mn>929569</mn><mn>16</mn></mfrac></math></td>
<td class="ltx_td ltx_align_right ltx_border_b" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_b" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m116.png" altimg-height="27px" altimg-valign="-9px" altimg-width="72px" alttext="-\tfrac{573405}{2}" display="inline"><mrow><mo>-</mo><mfrac><mn>573405</mn><mn>2</mn></mfrac></mrow></math></td>
<td class="ltx_td ltx_align_right ltx_border_b ltx_border_r" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_b" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m285.png" altimg-height="27px" altimg-valign="-9px" altimg-width="57px" alttext="\tfrac{943215}{4}" display="inline"><mfrac><mn>943215</mn><mn>4</mn></mfrac></math></td>
<td class="ltx_td ltx_align_right ltx_border_b" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_b" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m98.png" altimg-height="27px" altimg-valign="-9px" altimg-width="72px" alttext="-\tfrac{155155}{2}" display="inline"><mrow><mo>-</mo><mfrac><mn>155155</mn><mn>2</mn></mfrac></mrow></math></td>
<td class="ltx_td ltx_align_right ltx_border_b ltx_border_r" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_b" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m252.png" altimg-height="27px" altimg-valign="-9px" altimg-width="57px" alttext="\tfrac{109395}{8}" display="inline"><mfrac><mn>109395</mn><mn>8</mn></mfrac></math></td>
<td class="ltx_td ltx_align_right ltx_border_b" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_b" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m107.png" altimg-height="27px" altimg-valign="-9px" altimg-width="56px" alttext="-\tfrac{3003}{2}" display="inline"><mrow><mo>-</mo><mfrac><mn>3003</mn><mn>2</mn></mfrac></mrow></math></td>
<td class="ltx_td ltx_align_right ltx_border_b ltx_border_r" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_b" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m270.png" altimg-height="27px" altimg-valign="-9px" altimg-width="33px" alttext="\tfrac{455}{4}" display="inline"><mfrac><mn>455</mn><mn>4</mn></mfrac></math></td>
<td class="ltx_td ltx_align_right ltx_border_b" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m127.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_b" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m99.png" altimg-height="27px" altimg-valign="-9px" altimg-width="40px" alttext="-\tfrac{15}{2}" display="inline"><mrow><mo>-</mo><mfrac><mn>15</mn><mn>2</mn></mfrac></mrow></math></td>
<td class="ltx_td ltx_align_right ltx_border_b ltx_border_r" style="padding:2.083333333333333px 3.6pt;"><math class="ltx_Math" altimg="m157.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="1" display="inline"><mn>1</mn></math></td>
</tr>
</tbody>
</table>
<div id="T6.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./24.2#ii" title="§24.2(ii) Euler Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m242.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="E_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./24.2#ii">E</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></math>: Euler polynomials</a>,
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m314.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./24.1#p2.t1.r1">k</mi></math>: integer</a>,
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m317.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./24.1#p2.t1.r1">n</mi></math>: integer</a> and
<a href="./24.1#p2.t1.r2" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m320.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./24.1#p2.t1.r2">x</mi></math>: real or complex</a>
</dd>
</dl>
</div>
</div>
</figure>
</section>
</section>
</div>
<div class="ltx_page_footer">
<span></div>
</div>
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<page>
<!DOCTYPE html><html>
<head>
<title>DLMF: 24.4 Basic Properties</title>
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<div class="ltx_page_navlogo"></dd>
</dl>
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<div id="SS1.p1" class="ltx_para">
<table id="EGx1" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E1">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">24.4.1</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m34.png" altimg-height="25px" altimg-valign="-7px" altimg-width="177px" alttext="\displaystyle B_{n}\left(x+1\right)-B_{n}\left(x\right)" display="inline"><mrow><mrow><msub><mi href="./24.2#i">B</mi><mi href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi href="./24.1#p2.t1.r2">x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mo>-</mo><mrow><msub><mi href="./24.2#i">B</mi><mi href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m28.png" altimg-height="27px" altimg-valign="-6px" altimg-width="87px" alttext="\displaystyle=nx^{n-1}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./24.1#p2.t1.r1">n</mi><mo></mo><msup><mi href="./24.1#p2.t1.r2">x</mi><mrow><mi href="./24.1#p2.t1.r1">n</mi><mo>-</mo><mn>1</mn></mrow></msup></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./24.2#i" title="§24.2(i) Bernoulli Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m55.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="B_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./24.2#i">B</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></math>: Bernoulli polynomials</a>,
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./24.1#p2.t1.r1">n</mi></math>: integer</a> and
<a href="./24.1#p2.t1.r2" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m76.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./24.1#p2.t1.r2">x</mi></math>: real or complex</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">23.1.6</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E2">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">24.4.2</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m41.png" altimg-height="25px" altimg-valign="-7px" altimg-width="176px" alttext="\displaystyle E_{n}\left(x+1\right)+E_{n}\left(x\right)" display="inline"><mrow><mrow><msub><mi href="./24.2#ii">E</mi><mi href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi href="./24.1#p2.t1.r2">x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mo>+</mo><mrow><msub><mi href="./24.2#ii">E</mi><mi href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m12.png" altimg-height="20px" altimg-valign="-2px" altimg-width="65px" alttext="\displaystyle=2x^{n}." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mn>2</mn><mo></mo><msup><mi href="./24.1#p2.t1.r2">x</mi><mi href="./24.1#p2.t1.r1">n</mi></msup></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./24.2#ii" title="§24.2(ii) Euler Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m57.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="E_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./24.2#ii">E</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></math>: Euler polynomials</a>,
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./24.1#p2.t1.r1">n</mi></math>: integer</a> and
<a href="./24.1#p2.t1.r2" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m76.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./24.1#p2.t1.r2">x</mi></math>: real or complex</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">23.1.6</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
</div>
</section>
<section id="ii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§24.4(ii) </span>Symmetry</h2>
<div id="SS2.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="SS2.p1" class="ltx_para">
<table id="EGx2" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E3">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">24.4.3</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m32.png" altimg-height="25px" altimg-valign="-7px" altimg-width="97px" alttext="\displaystyle B_{n}\left(1-x\right)" display="inline"><mrow><msub><mi href="./24.2#i">B</mi><mi href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mi href="./24.1#p2.t1.r2">x</mi></mrow><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m3.png" altimg-height="25px" altimg-valign="-7px" altimg-width="144px" alttext="\displaystyle=(-1)^{n}B_{n}\left(x\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./24.1#p2.t1.r1">n</mi></msup><mo></mo><mrow><msub><mi href="./24.2#i">B</mi><mi href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./24.2#i" title="§24.2(i) Bernoulli Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m55.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="B_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./24.2#i">B</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></math>: Bernoulli polynomials</a>,
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./24.1#p2.t1.r1">n</mi></math>: integer</a> and
<a href="./24.1#p2.t1.r2" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m76.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./24.1#p2.t1.r2">x</mi></math>: real or complex</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">23.1.8</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E4">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">24.4.4</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m39.png" altimg-height="25px" altimg-valign="-7px" altimg-width="96px" alttext="\displaystyle E_{n}\left(1-x\right)" display="inline"><mrow><msub><mi href="./24.2#ii">E</mi><mi href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mi href="./24.1#p2.t1.r2">x</mi></mrow><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m4.png" altimg-height="25px" altimg-valign="-7px" altimg-width="143px" alttext="\displaystyle=(-1)^{n}E_{n}\left(x\right)." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./24.1#p2.t1.r1">n</mi></msup><mo></mo><mrow><msub><mi href="./24.2#ii">E</mi><mi href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E4.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./24.2#ii" title="§24.2(ii) Euler Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m57.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="E_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./24.2#ii">E</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></math>: Euler polynomials</a>,
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./24.1#p2.t1.r1">n</mi></math>: integer</a> and
<a href="./24.1#p2.t1.r2" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m76.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./24.1#p2.t1.r2">x</mi></math>: real or complex</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">23.1.8</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E5">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">24.4.5</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m2.png" altimg-height="25px" altimg-valign="-7px" altimg-width="130px" alttext="\displaystyle(-1)^{n}B_{n}\left(-x\right)" display="inline"><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./24.1#p2.t1.r1">n</mi></msup><mo></mo><mrow><msub><mi href="./24.2#i">B</mi><mi href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mi href="./24.1#p2.t1.r2">x</mi></mrow><mo>)</mo></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m13.png" altimg-height="28px" altimg-valign="-7px" altimg-width="168px" alttext="\displaystyle=B_{n}\left(x\right)+nx^{n-1}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><msub><mi href="./24.2#i">B</mi><mi href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow><mo>+</mo><mrow><mi href="./24.1#p2.t1.r1">n</mi><mo></mo><msup><mi href="./24.1#p2.t1.r2">x</mi><mrow><mi href="./24.1#p2.t1.r1">n</mi><mo>-</mo><mn>1</mn></mrow></msup></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E5.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./24.2#i" title="§24.2(i) Bernoulli Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m55.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="B_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./24.2#i">B</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></math>: Bernoulli polynomials</a>,
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./24.1#p2.t1.r1">n</mi></math>: integer</a> and
<a href="./24.1#p2.t1.r2" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m76.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./24.1#p2.t1.r2">x</mi></math>: real or complex</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">23.1.9</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E6">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">24.4.6</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m1.png" altimg-height="28px" altimg-valign="-7px" altimg-width="149px" alttext="\displaystyle(-1)^{n+1}E_{n}\left(-x\right)" display="inline"><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mrow><mi href="./24.1#p2.t1.r1">n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo></mo><mrow><msub><mi href="./24.2#ii">E</mi><mi href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mi href="./24.1#p2.t1.r2">x</mi></mrow><mo>)</mo></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m14.png" altimg-height="25px" altimg-valign="-7px" altimg-width="145px" alttext="\displaystyle=E_{n}\left(x\right)-2x^{n}." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><msub><mi href="./24.2#ii">E</mi><mi href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow><mo>-</mo><mrow><mn>2</mn><mo></mo><msup><mi href="./24.1#p2.t1.r2">x</mi><mi href="./24.1#p2.t1.r1">n</mi></msup></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E6.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./24.2#ii" title="§24.2(ii) Euler Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m57.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="E_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./24.2#ii">E</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></math>: Euler polynomials</a>,
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./24.1#p2.t1.r1">n</mi></math>: integer</a> and
<a href="./24.1#p2.t1.r2" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m76.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./24.1#p2.t1.r2">x</mi></math>: real or complex</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">23.1.9</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
</div>
</section>
<section id="iii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§24.4(iii) </span>Sums of Powers</h2>
<div id="SS3.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="SS3.p1" class="ltx_para">
<table id="EGx3" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E7">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">24.4.7</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m51.png" altimg-height="64px" altimg-valign="-28px" altimg-width="65px" alttext="\displaystyle\sum_{k=1}^{m}k^{n}" display="inline"><mrow><mstyle displaystyle="true"><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./24.1#p2.t1.r1">k</mi><mo>=</mo><mn>1</mn></mrow><mi href="./24.1#p2.t1.r1">m</mi></munderover></mstyle><msup><mi href="./24.1#p2.t1.r1">k</mi><mi href="./24.1#p2.t1.r1">n</mi></msup></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m22.png" altimg-height="49px" altimg-valign="-17px" altimg-width="225px" alttext="\displaystyle=\frac{B_{n+1}\left(m+1\right)-B_{n+1}}{n+1}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mstyle displaystyle="true"><mfrac><mrow><mrow><msub><mi href="./24.2#i">B</mi><mrow><mi href="./24.1#p2.t1.r1">n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mrow><mi href="./24.1#p2.t1.r1">m</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mo>-</mo><msub><mi href="./24.2#i">B</mi><mrow><mi href="./24.1#p2.t1.r1">n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><mrow><mi href="./24.1#p2.t1.r1">n</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mstyle></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E7.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./24.2#i" title="§24.2(i) Bernoulli Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="21px" altimg-valign="-5px" altimg-width="30px" alttext="B_{\NVar{n}}" display="inline"><msub><mi href="./24.2#i">B</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub></math>: Bernoulli numbers</a>,
<a href="./24.2#i" title="§24.2(i) Bernoulli Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m55.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="B_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./24.2#i">B</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></math>: Bernoulli polynomials</a>,
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./24.1#p2.t1.r1">k</mi></math>: integer</a>,
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./24.1#p2.t1.r1">m</mi></math>: integer</a> and
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./24.1#p2.t1.r1">n</mi></math>: integer</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">23.1.4</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E8">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">24.4.8</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m50.png" altimg-height="64px" altimg-valign="-28px" altimg-width="139px" alttext="\displaystyle\sum_{k=1}^{m}(-1)^{m-k}k^{n}" display="inline"><mrow><mstyle displaystyle="true"><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./24.1#p2.t1.r1">k</mi><mo>=</mo><mn>1</mn></mrow><mi href="./24.1#p2.t1.r1">m</mi></munderover></mstyle><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mrow><mi href="./24.1#p2.t1.r1">m</mi><mo>-</mo><mi href="./24.1#p2.t1.r1">k</mi></mrow></msup><mo></mo><msup><mi href="./24.1#p2.t1.r1">k</mi><mi href="./24.1#p2.t1.r1">n</mi></msup></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m23.png" altimg-height="48px" altimg-valign="-16px" altimg-width="268px" alttext="\displaystyle=\frac{E_{n}\left(m+1\right)+(-1)^{m}E_{n}\left(0\right)}{2}." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mstyle displaystyle="true"><mfrac><mrow><mrow><msub><mi href="./24.2#ii">E</mi><mi href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi href="./24.1#p2.t1.r1">m</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mo>+</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./24.1#p2.t1.r1">m</mi></msup><mo></mo><mrow><msub><mi href="./24.2#ii">E</mi><mi href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow></mrow></mrow></mrow><mn>2</mn></mfrac></mstyle></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E8.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./24.2#ii" title="§24.2(ii) Euler Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m57.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="E_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./24.2#ii">E</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></math>: Euler polynomials</a>,
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./24.1#p2.t1.r1">k</mi></math>: integer</a>,
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./24.1#p2.t1.r1">m</mi></math>: integer</a> and
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./24.1#p2.t1.r1">n</mi></math>: integer</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">23.1.4</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
<table id="EGx4" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E9">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">24.4.9</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m49.png" altimg-height="67px" altimg-valign="-28px" altimg-width="123px" alttext="\displaystyle\sum_{k=0}^{m-1}(a+dk)^{n}" display="inline"><mrow><mstyle displaystyle="true"><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./24.1#p2.t1.r1">k</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi href="./24.1#p2.t1.r1">m</mi><mo>-</mo><mn>1</mn></mrow></munderover></mstyle><msup><mrow><mo stretchy="false">(</mo><mrow><mi>a</mi><mo>+</mo><mrow><mi>d</mi><mo></mo><mi href="./24.1#p2.t1.r1">k</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mi href="./24.1#p2.t1.r1">n</mi></msup></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m30.png" altimg-height="48px" altimg-valign="-17px" altimg-width="354px" alttext="\displaystyle={\frac{d^{n}}{n+1}\left(B_{n+1}\left(m+\frac{a}{d}\right)-B_{n+1%
}\left(\frac{a}{d}\right)\right)}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><mfrac><msup><mi>d</mi><mi href="./24.1#p2.t1.r1">n</mi></msup><mrow><mi href="./24.1#p2.t1.r1">n</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mstyle><mo></mo><mrow><mo>(</mo><mrow><mrow><msub><mi href="./24.2#i">B</mi><mrow><mi href="./24.1#p2.t1.r1">n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mrow><mi href="./24.1#p2.t1.r1">m</mi><mo>+</mo><mstyle displaystyle="true"><mfrac><mi>a</mi><mi>d</mi></mfrac></mstyle></mrow><mo>)</mo></mrow></mrow><mo>-</mo><mrow><msub><mi href="./24.2#i">B</mi><mrow><mi href="./24.1#p2.t1.r1">n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac><mi>a</mi><mi>d</mi></mfrac></mstyle><mo>)</mo></mrow></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E9.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./24.2#i" title="§24.2(i) Bernoulli Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m55.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="B_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./24.2#i">B</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></math>: Bernoulli polynomials</a>,
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./24.1#p2.t1.r1">k</mi></math>: integer</a>,
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./24.1#p2.t1.r1">m</mi></math>: integer</a> and
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./24.1#p2.t1.r1">n</mi></math>: integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E10">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">24.4.10</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m48.png" altimg-height="67px" altimg-valign="-28px" altimg-width="174px" alttext="\displaystyle\sum_{k=0}^{m-1}(-1)^{k}(a+dk)^{n}" display="inline"><mrow><mstyle displaystyle="true"><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./24.1#p2.t1.r1">k</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi href="./24.1#p2.t1.r1">m</mi><mo>-</mo><mn>1</mn></mrow></munderover></mstyle><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./24.1#p2.t1.r1">k</mi></msup><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi>a</mi><mo>+</mo><mrow><mi>d</mi><mo></mo><mi href="./24.1#p2.t1.r1">k</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mi href="./24.1#p2.t1.r1">n</mi></msup></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m29.png" altimg-height="47px" altimg-valign="-16px" altimg-width="364px" alttext="\displaystyle={\frac{d^{n}}{2}\left((-1)^{m-1}E_{n}\left(m+\frac{a}{d}\right)+%
E_{n}\left(\frac{a}{d}\right)\right)}." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><mfrac><msup><mi>d</mi><mi href="./24.1#p2.t1.r1">n</mi></msup><mn>2</mn></mfrac></mstyle><mo></mo><mrow><mo>(</mo><mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mrow><mi href="./24.1#p2.t1.r1">m</mi><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mrow><msub><mi href="./24.2#ii">E</mi><mi href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi href="./24.1#p2.t1.r1">m</mi><mo>+</mo><mstyle displaystyle="true"><mfrac><mi>a</mi><mi>d</mi></mfrac></mstyle></mrow><mo>)</mo></mrow></mrow></mrow><mo>+</mo><mrow><msub><mi href="./24.2#ii">E</mi><mi href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac><mi>a</mi><mi>d</mi></mfrac></mstyle><mo>)</mo></mrow></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E10.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./24.2#ii" title="§24.2(ii) Euler Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m57.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="E_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./24.2#ii">E</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></math>: Euler polynomials</a>,
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./24.1#p2.t1.r1">k</mi></math>: integer</a>,
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./24.1#p2.t1.r1">m</mi></math>: integer</a> and
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./24.1#p2.t1.r1">n</mi></math>: integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E11">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">24.4.11</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m47.png" altimg-height="80px" altimg-valign="-44px" altimg-width="92px" alttext="\displaystyle\sum_{\begin{subarray}{c}k=1\\
\left(k,m\right)=1\end{subarray}}^{m}k^{n}" display="inline"><mrow><mstyle displaystyle="true"><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mtable class="ltx_align_c" rowspacing="0.0pt"><mtr><mtd><mrow><mi href="./24.1#p2.t1.r1">k</mi><mo>=</mo><mn>1</mn></mrow></mtd></mtr><mtr><mtd><mrow><mrow><mo href="./27.1#p2.t1.r3">(</mo><mrow><mi href="./24.1#p2.t1.r1">k</mi><mo>,</mo><mi href="./24.1#p2.t1.r1">m</mi></mrow><mo href="./27.1#p2.t1.r3">)</mo></mrow><mo>=</mo><mn>1</mn></mrow></mtd></mtr></mtable><mi href="./24.1#p2.t1.r1">m</mi></munderover></mstyle><msup><mi href="./24.1#p2.t1.r1">k</mi><mi href="./24.1#p2.t1.r1">n</mi></msup></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m17.png" altimg-height="77px" altimg-valign="-33px" altimg-width="450px" alttext="\displaystyle=\frac{1}{n+1}\sum_{j=1}^{n+1}{n+1\choose j}\*\left(\prod_{p%
\mathbin{|}m}(1-p^{n-j})B_{n+1-j}\right)m^{j}." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><mfrac><mn>1</mn><mrow><mi href="./24.1#p2.t1.r1">n</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mstyle><mo></mo><mrow><mstyle displaystyle="true"><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./24.1#p2.t1.r1">j</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi href="./24.1#p2.t1.r1">n</mi><mo>+</mo><mn>1</mn></mrow></munderover></mstyle><mrow><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac linethickness="0pt"><mrow><mi href="./24.1#p2.t1.r1">n</mi><mo>+</mo><mn>1</mn></mrow><mi href="./24.1#p2.t1.r1">j</mi></mfrac></mstyle><mo>)</mo></mrow><mo></mo><mrow><mo>(</mo><mrow><mstyle displaystyle="true"><munder><mo largeop="true" movablelimits="false" symmetric="true">∏</mo><mrow><mi href="./24.1#p2.t1.r3">p</mi><mo stretchy="false">|</mo><mi href="./24.1#p2.t1.r1">m</mi></mrow></munder></mstyle><mrow><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><msup><mi href="./24.1#p2.t1.r3">p</mi><mrow><mi href="./24.1#p2.t1.r1">n</mi><mo>-</mo><mi href="./24.1#p2.t1.r1">j</mi></mrow></msup></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msub><mi href="./24.2#i">B</mi><mrow><mrow><mi href="./24.1#p2.t1.r1">n</mi><mo>+</mo><mn>1</mn></mrow><mo>-</mo><mi href="./24.1#p2.t1.r1">j</mi></mrow></msub></mrow></mrow><mo>)</mo></mrow><mo></mo><msup><mi href="./24.1#p2.t1.r1">m</mi><mi href="./24.1#p2.t1.r1">j</mi></msup></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E11.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./24.2#i" title="§24.2(i) Bernoulli Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="21px" altimg-valign="-5px" altimg-width="30px" alttext="B_{\NVar{n}}" display="inline"><msub><mi href="./24.2#i">B</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub></math>: Bernoulli numbers</a>,
<a href="./1.2#i" title="§1.2(i) Binomial Coefficients ‣ §1.2 Elementary Algebra ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="27px" altimg-valign="-9px" altimg-width="37px" alttext="\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}" display="inline"><mrow><mo href="./1.2#i">(</mo><mfrac linethickness="0.0pt"><mi class="ltx_nvar" href="./1.1#p2.t1.r5">m</mi><mi class="ltx_nvar" href="./1.1#p2.t1.r5">n</mi></mfrac><mo href="./1.2#i">)</mo></mrow></math>: binomial coefficient</a>,
<a href="./27.1#p2.t1.r3" title="§27.1 Special Notation ‣ Notation ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m61.png" altimg-height="23px" altimg-valign="-7px" altimg-width="58px" alttext="\left(\NVar{m},\NVar{n}\right)" display="inline"><mrow><mo href="./27.1#p2.t1.r3">(</mo><mrow><mi class="ltx_nvar" href="./27.1#p2.t1.r1">m</mi><mo>,</mo><mi class="ltx_nvar" href="./27.1#p2.t1.r1">n</mi></mrow><mo href="./27.1#p2.t1.r3">)</mo></mrow></math>: greatest common divisor (gcd)</a>,
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m65.png" altimg-height="20px" altimg-valign="-6px" altimg-width="14px" alttext="j" display="inline"><mi href="./24.1#p2.t1.r1">j</mi></math>: integer</a>,
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./24.1#p2.t1.r1">k</mi></math>: integer</a>,
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./24.1#p2.t1.r1">m</mi></math>: integer</a>,
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./24.1#p2.t1.r1">n</mi></math>: integer</a> and
<a href="./24.1#p2.t1.r3" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m75.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="p" display="inline"><mi href="./24.1#p2.t1.r3">p</mi></math>: prime</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
</div>
</section>
<section id="iv" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§24.4(iv) </span>Finite Expansions</h2>
<div id="SS4.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="SS4.p1" class="ltx_para">
<table id="EGx5" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E12">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">24.4.12</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m35.png" altimg-height="25px" altimg-valign="-7px" altimg-width="98px" alttext="\displaystyle B_{n}\left(x+h\right)" display="inline"><mrow><msub><mi href="./24.2#i">B</mi><mi href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi href="./24.1#p2.t1.r2">x</mi><mo>+</mo><mi>h</mi></mrow><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m24.png" altimg-height="64px" altimg-valign="-28px" altimg-width="208px" alttext="\displaystyle=\sum_{k=0}^{n}{n\choose k}B_{k}\left(x\right)h^{n-k}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./24.1#p2.t1.r1">k</mi><mo>=</mo><mn>0</mn></mrow><mi href="./24.1#p2.t1.r1">n</mi></munderover></mstyle><mrow><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac linethickness="0pt"><mi href="./24.1#p2.t1.r1">n</mi><mi href="./24.1#p2.t1.r1">k</mi></mfrac></mstyle><mo>)</mo></mrow><mo></mo><mrow><msub><mi href="./24.2#i">B</mi><mi href="./24.1#p2.t1.r1">k</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow><mo></mo><msup><mi>h</mi><mrow><mi href="./24.1#p2.t1.r1">n</mi><mo>-</mo><mi href="./24.1#p2.t1.r1">k</mi></mrow></msup></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E12.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./24.2#i" title="§24.2(i) Bernoulli Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m55.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="B_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./24.2#i">B</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></math>: Bernoulli polynomials</a>,
<a href="./1.2#i" title="§1.2(i) Binomial Coefficients ‣ §1.2 Elementary Algebra ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="27px" altimg-valign="-9px" altimg-width="37px" alttext="\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}" display="inline"><mrow><mo href="./1.2#i">(</mo><mfrac linethickness="0.0pt"><mi class="ltx_nvar" href="./1.1#p2.t1.r5">m</mi><mi class="ltx_nvar" href="./1.1#p2.t1.r5">n</mi></mfrac><mo href="./1.2#i">)</mo></mrow></math>: binomial coefficient</a>,
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./24.1#p2.t1.r1">k</mi></math>: integer</a>,
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./24.1#p2.t1.r1">n</mi></math>: integer</a> and
<a href="./24.1#p2.t1.r2" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m76.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./24.1#p2.t1.r2">x</mi></math>: real or complex</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">23.1.7</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E13">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">24.4.13</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m42.png" altimg-height="25px" altimg-valign="-7px" altimg-width="98px" alttext="\displaystyle E_{n}\left(x+h\right)" display="inline"><mrow><msub><mi href="./24.2#ii">E</mi><mi href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi href="./24.1#p2.t1.r2">x</mi><mo>+</mo><mi>h</mi></mrow><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m25.png" altimg-height="64px" altimg-valign="-28px" altimg-width="208px" alttext="\displaystyle=\sum_{k=0}^{n}{n\choose k}E_{k}\left(x\right)h^{n-k}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./24.1#p2.t1.r1">k</mi><mo>=</mo><mn>0</mn></mrow><mi href="./24.1#p2.t1.r1">n</mi></munderover></mstyle><mrow><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac linethickness="0pt"><mi href="./24.1#p2.t1.r1">n</mi><mi href="./24.1#p2.t1.r1">k</mi></mfrac></mstyle><mo>)</mo></mrow><mo></mo><mrow><msub><mi href="./24.2#ii">E</mi><mi href="./24.1#p2.t1.r1">k</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow><mo></mo><msup><mi>h</mi><mrow><mi href="./24.1#p2.t1.r1">n</mi><mo>-</mo><mi href="./24.1#p2.t1.r1">k</mi></mrow></msup></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E13.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./24.2#ii" title="§24.2(ii) Euler Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m57.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="E_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./24.2#ii">E</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></math>: Euler polynomials</a>,
<a href="./1.2#i" title="§1.2(i) Binomial Coefficients ‣ §1.2 Elementary Algebra ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="27px" altimg-valign="-9px" altimg-width="37px" alttext="\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}" display="inline"><mrow><mo href="./1.2#i">(</mo><mfrac linethickness="0.0pt"><mi class="ltx_nvar" href="./1.1#p2.t1.r5">m</mi><mi class="ltx_nvar" href="./1.1#p2.t1.r5">n</mi></mfrac><mo href="./1.2#i">)</mo></mrow></math>: binomial coefficient</a>,
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./24.1#p2.t1.r1">k</mi></math>: integer</a>,
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./24.1#p2.t1.r1">n</mi></math>: integer</a> and
<a href="./24.1#p2.t1.r2" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m76.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./24.1#p2.t1.r2">x</mi></math>: real or complex</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">23.1.7</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E14">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">24.4.14</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m38.png" altimg-height="25px" altimg-valign="-7px" altimg-width="82px" alttext="\displaystyle E_{n-1}\left(x\right)" display="inline"><mrow><msub><mi href="./24.2#ii">E</mi><mrow><mi href="./24.1#p2.t1.r1">n</mi><mo>-</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m21.png" altimg-height="64px" altimg-valign="-28px" altimg-width="264px" alttext="\displaystyle=\frac{2}{n}\sum_{k=0}^{n}{n\choose k}(1-2^{k})B_{k}x^{n-k}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><mfrac><mn>2</mn><mi href="./24.1#p2.t1.r1">n</mi></mfrac></mstyle><mo></mo><mrow><mstyle displaystyle="true"><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./24.1#p2.t1.r1">k</mi><mo>=</mo><mn>0</mn></mrow><mi href="./24.1#p2.t1.r1">n</mi></munderover></mstyle><mrow><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac linethickness="0pt"><mi href="./24.1#p2.t1.r1">n</mi><mi href="./24.1#p2.t1.r1">k</mi></mfrac></mstyle><mo>)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><msup><mn>2</mn><mi href="./24.1#p2.t1.r1">k</mi></msup></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msub><mi href="./24.2#i">B</mi><mi href="./24.1#p2.t1.r1">k</mi></msub><mo></mo><msup><mi href="./24.1#p2.t1.r2">x</mi><mrow><mi href="./24.1#p2.t1.r1">n</mi><mo>-</mo><mi href="./24.1#p2.t1.r1">k</mi></mrow></msup></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E14.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./24.2#i" title="§24.2(i) Bernoulli Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="21px" altimg-valign="-5px" altimg-width="30px" alttext="B_{\NVar{n}}" display="inline"><msub><mi href="./24.2#i">B</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub></math>: Bernoulli numbers</a>,
<a href="./24.2#ii" title="§24.2(ii) Euler Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m57.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="E_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./24.2#ii">E</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></math>: Euler polynomials</a>,
<a href="./1.2#i" title="§1.2(i) Binomial Coefficients ‣ §1.2 Elementary Algebra ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="27px" altimg-valign="-9px" altimg-width="37px" alttext="\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}" display="inline"><mrow><mo href="./1.2#i">(</mo><mfrac linethickness="0.0pt"><mi class="ltx_nvar" href="./1.1#p2.t1.r5">m</mi><mi class="ltx_nvar" href="./1.1#p2.t1.r5">n</mi></mfrac><mo href="./1.2#i">)</mo></mrow></math>: binomial coefficient</a>,
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./24.1#p2.t1.r1">k</mi></math>: integer</a>,
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./24.1#p2.t1.r1">n</mi></math>: integer</a> and
<a href="./24.1#p2.t1.r2" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m76.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./24.1#p2.t1.r2">x</mi></math>: real or complex</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
<table id="EGx6" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E15">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">24.4.15</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m31.png" altimg-height="22px" altimg-valign="-5px" altimg-width="40px" alttext="\displaystyle B_{2n}" display="inline"><msub><mi href="./24.2#i">B</mi><mrow><mn>2</mn><mo></mo><mi href="./24.1#p2.t1.r1">n</mi></mrow></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m19.png" altimg-height="67px" altimg-valign="-28px" altimg-width="300px" alttext="\displaystyle=\frac{2n}{2^{2n}(2^{2n}-1)}\sum_{k=0}^{n-1}{2n-1\choose 2k}E_{2k}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><mfrac><mrow><mn>2</mn><mo></mo><mi href="./24.1#p2.t1.r1">n</mi></mrow><mrow><msup><mn>2</mn><mrow><mn>2</mn><mo></mo><mi href="./24.1#p2.t1.r1">n</mi></mrow></msup><mo></mo><mrow><mo stretchy="false">(</mo><mrow><msup><mn>2</mn><mrow><mn>2</mn><mo></mo><mi href="./24.1#p2.t1.r1">n</mi></mrow></msup><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow></mfrac></mstyle><mo></mo><mrow><mstyle displaystyle="true"><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./24.1#p2.t1.r1">k</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi href="./24.1#p2.t1.r1">n</mi><mo>-</mo><mn>1</mn></mrow></munderover></mstyle><mrow><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac linethickness="0pt"><mrow><mrow><mn>2</mn><mo></mo><mi href="./24.1#p2.t1.r1">n</mi></mrow><mo>-</mo><mn>1</mn></mrow><mrow><mn>2</mn><mo></mo><mi href="./24.1#p2.t1.r1">k</mi></mrow></mfrac></mstyle><mo>)</mo></mrow><mo></mo><msub><mi href="./24.2#ii">E</mi><mrow><mn>2</mn><mo></mo><mi href="./24.1#p2.t1.r1">k</mi></mrow></msub></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E15.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./24.2#i" title="§24.2(i) Bernoulli Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="21px" altimg-valign="-5px" altimg-width="30px" alttext="B_{\NVar{n}}" display="inline"><msub><mi href="./24.2#i">B</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub></math>: Bernoulli numbers</a>,
<a href="./24.2#ii" title="§24.2(ii) Euler Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="21px" altimg-valign="-5px" altimg-width="30px" alttext="E_{\NVar{n}}" display="inline"><msub><mi href="./24.2#ii">E</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub></math>: Euler numbers</a>,
<a href="./1.2#i" title="§1.2(i) Binomial Coefficients ‣ §1.2 Elementary Algebra ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="27px" altimg-valign="-9px" altimg-width="37px" alttext="\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}" display="inline"><mrow><mo href="./1.2#i">(</mo><mfrac linethickness="0.0pt"><mi class="ltx_nvar" href="./1.1#p2.t1.r5">m</mi><mi class="ltx_nvar" href="./1.1#p2.t1.r5">n</mi></mfrac><mo href="./1.2#i">)</mo></mrow></math>: binomial coefficient</a>,
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./24.1#p2.t1.r1">k</mi></math>: integer</a> and
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./24.1#p2.t1.r1">n</mi></math>: integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E16">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">24.4.16</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m37.png" altimg-height="22px" altimg-valign="-5px" altimg-width="40px" alttext="\displaystyle E_{2n}" display="inline"><msub><mi href="./24.2#ii">E</mi><mrow><mn>2</mn><mo></mo><mi href="./24.1#p2.t1.r1">n</mi></mrow></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m16.png" altimg-height="64px" altimg-valign="-28px" altimg-width="398px" alttext="\displaystyle=\frac{1}{2n+1}-\sum_{k=1}^{n}{2n\choose 2k-1}\frac{2^{2k}(2^{2k-%
1}-1)B_{2k}}{k}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><mfrac><mn>1</mn><mrow><mrow><mn>2</mn><mo></mo><mi href="./24.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mn>1</mn></mrow></mfrac></mstyle><mo>-</mo><mrow><mstyle displaystyle="true"><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./24.1#p2.t1.r1">k</mi><mo>=</mo><mn>1</mn></mrow><mi href="./24.1#p2.t1.r1">n</mi></munderover></mstyle><mrow><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac linethickness="0pt"><mrow><mn>2</mn><mo></mo><mi href="./24.1#p2.t1.r1">n</mi></mrow><mrow><mrow><mn>2</mn><mo></mo><mi href="./24.1#p2.t1.r1">k</mi></mrow><mo>-</mo><mn>1</mn></mrow></mfrac></mstyle><mo>)</mo></mrow><mo></mo><mstyle displaystyle="true"><mfrac><mrow><msup><mn>2</mn><mrow><mn>2</mn><mo></mo><mi href="./24.1#p2.t1.r1">k</mi></mrow></msup><mo></mo><mrow><mo stretchy="false">(</mo><mrow><msup><mn>2</mn><mrow><mrow><mn>2</mn><mo></mo><mi href="./24.1#p2.t1.r1">k</mi></mrow><mo>-</mo><mn>1</mn></mrow></msup><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msub><mi href="./24.2#i">B</mi><mrow><mn>2</mn><mo></mo><mi href="./24.1#p2.t1.r1">k</mi></mrow></msub></mrow><mi href="./24.1#p2.t1.r1">k</mi></mfrac></mstyle></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E16.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./24.2#i" title="§24.2(i) Bernoulli Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="21px" altimg-valign="-5px" altimg-width="30px" alttext="B_{\NVar{n}}" display="inline"><msub><mi href="./24.2#i">B</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub></math>: Bernoulli numbers</a>,
<a href="./24.2#ii" title="§24.2(ii) Euler Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="21px" altimg-valign="-5px" altimg-width="30px" alttext="E_{\NVar{n}}" display="inline"><msub><mi href="./24.2#ii">E</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub></math>: Euler numbers</a>,
<a href="./1.2#i" title="§1.2(i) Binomial Coefficients ‣ §1.2 Elementary Algebra ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="27px" altimg-valign="-9px" altimg-width="37px" alttext="\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}" display="inline"><mrow><mo href="./1.2#i">(</mo><mfrac linethickness="0.0pt"><mi class="ltx_nvar" href="./1.1#p2.t1.r5">m</mi><mi class="ltx_nvar" href="./1.1#p2.t1.r5">n</mi></mfrac><mo href="./1.2#i">)</mo></mrow></math>: binomial coefficient</a>,
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./24.1#p2.t1.r1">k</mi></math>: integer</a> and
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./24.1#p2.t1.r1">n</mi></math>: integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E17">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">24.4.17</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m37.png" altimg-height="22px" altimg-valign="-5px" altimg-width="40px" alttext="\displaystyle E_{2n}" display="inline"><msub><mi href="./24.2#ii">E</mi><mrow><mn>2</mn><mo></mo><mi href="./24.1#p2.t1.r1">n</mi></mrow></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m8.png" altimg-height="64px" altimg-valign="-28px" altimg-width="327px" alttext="\displaystyle=1-\sum_{k=1}^{n}{2n\choose 2k-1}\frac{2^{2k}(2^{2k}-1)B_{2k}}{2k}." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mn>1</mn><mo>-</mo><mrow><mstyle displaystyle="true"><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./24.1#p2.t1.r1">k</mi><mo>=</mo><mn>1</mn></mrow><mi href="./24.1#p2.t1.r1">n</mi></munderover></mstyle><mrow><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac linethickness="0pt"><mrow><mn>2</mn><mo></mo><mi href="./24.1#p2.t1.r1">n</mi></mrow><mrow><mrow><mn>2</mn><mo></mo><mi href="./24.1#p2.t1.r1">k</mi></mrow><mo>-</mo><mn>1</mn></mrow></mfrac></mstyle><mo>)</mo></mrow><mo></mo><mstyle displaystyle="true"><mfrac><mrow><msup><mn>2</mn><mrow><mn>2</mn><mo></mo><mi href="./24.1#p2.t1.r1">k</mi></mrow></msup><mo></mo><mrow><mo stretchy="false">(</mo><mrow><msup><mn>2</mn><mrow><mn>2</mn><mo></mo><mi href="./24.1#p2.t1.r1">k</mi></mrow></msup><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msub><mi href="./24.2#i">B</mi><mrow><mn>2</mn><mo></mo><mi href="./24.1#p2.t1.r1">k</mi></mrow></msub></mrow><mrow><mn>2</mn><mo></mo><mi href="./24.1#p2.t1.r1">k</mi></mrow></mfrac></mstyle></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E17.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./24.2#i" title="§24.2(i) Bernoulli Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="21px" altimg-valign="-5px" altimg-width="30px" alttext="B_{\NVar{n}}" display="inline"><msub><mi href="./24.2#i">B</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub></math>: Bernoulli numbers</a>,
<a href="./24.2#ii" title="§24.2(ii) Euler Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="21px" altimg-valign="-5px" altimg-width="30px" alttext="E_{\NVar{n}}" display="inline"><msub><mi href="./24.2#ii">E</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub></math>: Euler numbers</a>,
<a href="./1.2#i" title="§1.2(i) Binomial Coefficients ‣ §1.2 Elementary Algebra ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="27px" altimg-valign="-9px" altimg-width="37px" alttext="\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}" display="inline"><mrow><mo href="./1.2#i">(</mo><mfrac linethickness="0.0pt"><mi class="ltx_nvar" href="./1.1#p2.t1.r5">m</mi><mi class="ltx_nvar" href="./1.1#p2.t1.r5">n</mi></mfrac><mo href="./1.2#i">)</mo></mrow></math>: binomial coefficient</a>,
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./24.1#p2.t1.r1">k</mi></math>: integer</a> and
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./24.1#p2.t1.r1">n</mi></math>: integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
</div>
</section>
<section id="v" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§24.4(v) </span>Multiplication Formulas</h2>
<div id="SS5.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px1.p1" class="ltx_para">
<table id="E18" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">24.4.18</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E18.png" altimg-height="67px" altimg-valign="-28px" altimg-width="322px" alttext="B_{n}\left(mx\right)=m^{n-1}\sum_{k=0}^{m-1}B_{n}\left(x+\frac{k}{m}\right)." display="block"><mrow><mrow><mrow><msub><mi href="./24.2#i">B</mi><mi href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi href="./24.1#p2.t1.r1">m</mi><mo></mo><mi href="./24.1#p2.t1.r2">x</mi></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msup><mi href="./24.1#p2.t1.r1">m</mi><mrow><mi href="./24.1#p2.t1.r1">n</mi><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./24.1#p2.t1.r1">k</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi href="./24.1#p2.t1.r1">m</mi><mo>-</mo><mn>1</mn></mrow></munderover><mrow><msub><mi href="./24.2#i">B</mi><mi href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi href="./24.1#p2.t1.r2">x</mi><mo>+</mo><mfrac><mi href="./24.1#p2.t1.r1">k</mi><mi href="./24.1#p2.t1.r1">m</mi></mfrac></mrow><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E18.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./24.2#i" title="§24.2(i) Bernoulli Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m55.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="B_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./24.2#i">B</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></math>: Bernoulli polynomials</a>,
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./24.1#p2.t1.r1">k</mi></math>: integer</a>,
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./24.1#p2.t1.r1">m</mi></math>: integer</a>,
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./24.1#p2.t1.r1">n</mi></math>: integer</a> and
<a href="./24.1#p2.t1.r2" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m76.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./24.1#p2.t1.r2">x</mi></math>: real or complex</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">23.1.10</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="Px1.p2" class="ltx_para">
<p class="ltx_p">Next,
</p>
<table id="E19" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">24.4.19</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E19.png" altimg-height="67px" altimg-valign="-28px" altimg-width="407px" alttext="E_{n}\left(mx\right)=-\frac{2m^{n}}{n+1}\sum_{k=0}^{m-1}(-1)^{k}B_{n+1}\left(x%
+\frac{k}{m}\right)," display="block"><mrow><mrow><mrow><msub><mi href="./24.2#ii">E</mi><mi href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi href="./24.1#p2.t1.r1">m</mi><mo></mo><mi href="./24.1#p2.t1.r2">x</mi></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mo>-</mo><mrow><mfrac><mrow><mn>2</mn><mo></mo><msup><mi href="./24.1#p2.t1.r1">m</mi><mi href="./24.1#p2.t1.r1">n</mi></msup></mrow><mrow><mi href="./24.1#p2.t1.r1">n</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./24.1#p2.t1.r1">k</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi href="./24.1#p2.t1.r1">m</mi><mo>-</mo><mn>1</mn></mrow></munderover><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./24.1#p2.t1.r1">k</mi></msup><mo></mo><mrow><msub><mi href="./24.2#i">B</mi><mrow><mi href="./24.1#p2.t1.r1">n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mrow><mi href="./24.1#p2.t1.r2">x</mi><mo>+</mo><mfrac><mi href="./24.1#p2.t1.r1">k</mi><mi href="./24.1#p2.t1.r1">m</mi></mfrac></mrow><mo>)</mo></mrow></mrow></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m69.png" altimg-height="20px" altimg-valign="-6px" altimg-width="128px" alttext="m=2,4,6,\dots" display="inline"><mrow><mi href="./24.1#p2.t1.r1">m</mi><mo>=</mo><mrow><mn>2</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>6</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E19.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./24.2#i" title="§24.2(i) Bernoulli Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m55.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="B_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./24.2#i">B</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></math>: Bernoulli polynomials</a>,
<a href="./24.2#ii" title="§24.2(ii) Euler Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m57.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="E_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./24.2#ii">E</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></math>: Euler polynomials</a>,
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./24.1#p2.t1.r1">k</mi></math>: integer</a>,
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./24.1#p2.t1.r1">m</mi></math>: integer</a>,
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./24.1#p2.t1.r1">n</mi></math>: integer</a> and
<a href="./24.1#p2.t1.r2" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m76.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./24.1#p2.t1.r2">x</mi></math>: real or complex</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">23.1.10</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E20" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">24.4.20</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E20.png" altimg-height="67px" altimg-valign="-28px" altimg-width="348px" alttext="E_{n}\left(mx\right)=m^{n}\sum_{k=0}^{m-1}(-1)^{k}E_{n}\left(x+\frac{k}{m}%
\right)," display="block"><mrow><mrow><mrow><msub><mi href="./24.2#ii">E</mi><mi href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi href="./24.1#p2.t1.r1">m</mi><mo></mo><mi href="./24.1#p2.t1.r2">x</mi></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msup><mi href="./24.1#p2.t1.r1">m</mi><mi href="./24.1#p2.t1.r1">n</mi></msup><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./24.1#p2.t1.r1">k</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi href="./24.1#p2.t1.r1">m</mi><mo>-</mo><mn>1</mn></mrow></munderover><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./24.1#p2.t1.r1">k</mi></msup><mo></mo><mrow><msub><mi href="./24.2#ii">E</mi><mi href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi href="./24.1#p2.t1.r2">x</mi><mo>+</mo><mfrac><mi href="./24.1#p2.t1.r1">k</mi><mi href="./24.1#p2.t1.r1">m</mi></mfrac></mrow><mo>)</mo></mrow></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m67.png" altimg-height="20px" altimg-valign="-6px" altimg-width="128px" alttext="m=1,3,5,\dots" display="inline"><mrow><mi href="./24.1#p2.t1.r1">m</mi><mo>=</mo><mrow><mn>1</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>5</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E20.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./24.2#ii" title="§24.2(ii) Euler Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m57.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="E_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./24.2#ii">E</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></math>: Euler polynomials</a>,
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./24.1#p2.t1.r1">k</mi></math>: integer</a>,
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./24.1#p2.t1.r1">m</mi></math>: integer</a>,
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./24.1#p2.t1.r1">n</mi></math>: integer</a> and
<a href="./24.1#p2.t1.r2" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m76.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./24.1#p2.t1.r2">x</mi></math>: real or complex</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">23.1.10</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="EGx7" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E21">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">24.4.21</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m36.png" altimg-height="25px" altimg-valign="-7px" altimg-width="62px" alttext="\displaystyle B_{n}\left(x\right)" display="inline"><mrow><msub><mi href="./24.2#i">B</mi><mi href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m11.png" altimg-height="30px" altimg-valign="-9px" altimg-width="303px" alttext="\displaystyle=2^{n-1}\left(B_{n}\left(\tfrac{1}{2}x\right)+B_{n}\left(\tfrac{1%
}{2}x+\tfrac{1}{2}\right)\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msup><mn>2</mn><mrow><mi href="./24.1#p2.t1.r1">n</mi><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mrow><mo>(</mo><mrow><mrow><msub><mi href="./24.2#i">B</mi><mi href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./24.1#p2.t1.r2">x</mi></mrow><mo>)</mo></mrow></mrow><mo>+</mo><mrow><msub><mi href="./24.2#i">B</mi><mi href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./24.1#p2.t1.r2">x</mi></mrow><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E21.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./24.2#i" title="§24.2(i) Bernoulli Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m55.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="B_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./24.2#i">B</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></math>: Bernoulli polynomials</a>,
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./24.1#p2.t1.r1">n</mi></math>: integer</a> and
<a href="./24.1#p2.t1.r2" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m76.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./24.1#p2.t1.r2">x</mi></math>: real or complex</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E22">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">24.4.22</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m38.png" altimg-height="25px" altimg-valign="-7px" altimg-width="82px" alttext="\displaystyle E_{n-1}\left(x\right)" display="inline"><mrow><msub><mi href="./24.2#ii">E</mi><mrow><mi href="./24.1#p2.t1.r1">n</mi><mo>-</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m20.png" altimg-height="46px" altimg-valign="-16px" altimg-width="247px" alttext="\displaystyle=\frac{2}{n}\left(B_{n}\left(x\right)-2^{n}B_{n}\left(\tfrac{1}{2%
}x\right)\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><mfrac><mn>2</mn><mi href="./24.1#p2.t1.r1">n</mi></mfrac></mstyle><mo></mo><mrow><mo>(</mo><mrow><mrow><msub><mi href="./24.2#i">B</mi><mi href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow><mo>-</mo><mrow><msup><mn>2</mn><mi href="./24.1#p2.t1.r1">n</mi></msup><mo></mo><mrow><msub><mi href="./24.2#i">B</mi><mi href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./24.1#p2.t1.r2">x</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E22.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./24.2#i" title="§24.2(i) Bernoulli Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m55.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="B_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./24.2#i">B</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></math>: Bernoulli polynomials</a>,
<a href="./24.2#ii" title="§24.2(ii) Euler Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m57.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="E_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./24.2#ii">E</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></math>: Euler polynomials</a>,
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./24.1#p2.t1.r1">n</mi></math>: integer</a> and
<a href="./24.1#p2.t1.r2" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m76.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./24.1#p2.t1.r2">x</mi></math>: real or complex</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">23.1.27</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E23">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">24.4.23</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m38.png" altimg-height="25px" altimg-valign="-7px" altimg-width="82px" alttext="\displaystyle E_{n-1}\left(x\right)" display="inline"><mrow><msub><mi href="./24.2#ii">E</mi><mrow><mi href="./24.1#p2.t1.r1">n</mi><mo>-</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m18.png" altimg-height="46px" altimg-valign="-16px" altimg-width="288px" alttext="\displaystyle=\frac{2^{n}}{n}\left(B_{n}\left(\tfrac{1}{2}x+\tfrac{1}{2}\right%
)-B_{n}\left(\tfrac{1}{2}x\right)\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><mfrac><msup><mn>2</mn><mi href="./24.1#p2.t1.r1">n</mi></msup><mi href="./24.1#p2.t1.r1">n</mi></mfrac></mstyle><mo></mo><mrow><mo>(</mo><mrow><mrow><msub><mi href="./24.2#i">B</mi><mi href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./24.1#p2.t1.r2">x</mi></mrow><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mo>-</mo><mrow><msub><mi href="./24.2#i">B</mi><mi href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./24.1#p2.t1.r2">x</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E23.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./24.2#i" title="§24.2(i) Bernoulli Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m55.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="B_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./24.2#i">B</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></math>: Bernoulli polynomials</a>,
<a href="./24.2#ii" title="§24.2(ii) Euler Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m57.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="E_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./24.2#ii">E</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></math>: Euler polynomials</a>,
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./24.1#p2.t1.r1">n</mi></math>: integer</a> and
<a href="./24.1#p2.t1.r2" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m76.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./24.1#p2.t1.r2">x</mi></math>: real or complex</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">23.1.27</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
<table id="E24" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">24.4.24</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E24.png" altimg-height="70px" altimg-valign="-30px" altimg-width="706px" alttext="B_{n}\left(mx\right)=m^{n}B_{n}\left(x\right)+n\sum_{k=1}^{n}\sum_{j=0}^{k-1}(%
-1)^{j}{n\choose k}\*\left(\sum_{r=1}^{m-1}\frac{e^{2\pi i(k-j)r/m}}{(1-e^{2%
\pi ir/m})^{n}}\right)(j+mx)^{n-1}," display="block"><mrow><mrow><mrow><msub><mi href="./24.2#i">B</mi><mi href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi href="./24.1#p2.t1.r1">m</mi><mo></mo><mi href="./24.1#p2.t1.r2">x</mi></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mrow><msup><mi href="./24.1#p2.t1.r1">m</mi><mi href="./24.1#p2.t1.r1">n</mi></msup><mo></mo><mrow><msub><mi href="./24.2#i">B</mi><mi href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mi href="./24.1#p2.t1.r1">n</mi><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./24.1#p2.t1.r1">k</mi><mo>=</mo><mn>1</mn></mrow><mi href="./24.1#p2.t1.r1">n</mi></munderover><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./24.1#p2.t1.r1">j</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi href="./24.1#p2.t1.r1">k</mi><mo>-</mo><mn>1</mn></mrow></munderover><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./24.1#p2.t1.r1">j</mi></msup><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mi href="./24.1#p2.t1.r1">n</mi><mi href="./24.1#p2.t1.r1">k</mi></mfrac><mo>)</mo></mrow><mo></mo><mrow><mo>(</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi>r</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi href="./24.1#p2.t1.r1">m</mi><mo>-</mo><mn>1</mn></mrow></munderover><mfrac><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./24.1#p2.t1.r1">k</mi><mo>-</mo><mi href="./24.1#p2.t1.r1">j</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi>r</mi></mrow><mo>/</mo><mi href="./24.1#p2.t1.r1">m</mi></mrow></msup><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi><mo></mo><mi>r</mi></mrow><mo>/</mo><mi href="./24.1#p2.t1.r1">m</mi></mrow></msup></mrow><mo stretchy="false">)</mo></mrow><mi href="./24.1#p2.t1.r1">n</mi></msup></mfrac></mrow><mo>)</mo></mrow><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./24.1#p2.t1.r1">j</mi><mo>+</mo><mrow><mi href="./24.1#p2.t1.r1">m</mi><mo></mo><mi href="./24.1#p2.t1.r2">x</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mrow><mi href="./24.1#p2.t1.r1">n</mi><mo>-</mo><mn>1</mn></mrow></msup></mrow></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m72.png" altimg-height="20px" altimg-valign="-6px" altimg-width="104px" alttext="n=1,2,\dots" display="inline"><mrow><mi href="./24.1#p2.t1.r1">n</mi><mo>=</mo><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>, <math class="ltx_Math" altimg="m68.png" altimg-height="20px" altimg-valign="-6px" altimg-width="109px" alttext="m=2,3,\dots" display="inline"><mrow><mi href="./24.1#p2.t1.r1">m</mi><mo>=</mo><mrow><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E24.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./24.2#i" title="§24.2(i) Bernoulli Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m55.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="B_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./24.2#i">B</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></math>: Bernoulli polynomials</a>,
<a href="./1.2#i" title="§1.2(i) Binomial Coefficients ‣ §1.2 Elementary Algebra ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="27px" altimg-valign="-9px" altimg-width="37px" alttext="\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}" display="inline"><mrow><mo href="./1.2#i">(</mo><mfrac linethickness="0.0pt"><mi class="ltx_nvar" href="./1.1#p2.t1.r5">m</mi><mi class="ltx_nvar" href="./1.1#p2.t1.r5">n</mi></mfrac><mo href="./1.2#i">)</mo></mrow></math>: binomial coefficient</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m62.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m65.png" altimg-height="20px" altimg-valign="-6px" altimg-width="14px" alttext="j" display="inline"><mi href="./24.1#p2.t1.r1">j</mi></math>: integer</a>,
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./24.1#p2.t1.r1">k</mi></math>: integer</a>,
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./24.1#p2.t1.r1">m</mi></math>: integer</a>,
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./24.1#p2.t1.r1">n</mi></math>: integer</a> and
<a href="./24.1#p2.t1.r2" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m76.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./24.1#p2.t1.r2">x</mi></math>: real or complex</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
</section>
<section id="vi" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§24.4(vi) </span>Special Values</h2>
<div id="SS6.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="SS6.p1" class="ltx_para">
<table id="E25" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">24.4.25</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E25.png" altimg-height="25px" altimg-valign="-7px" altimg-width="252px" alttext="B_{n}\left(0\right)=(-1)^{n}B_{n}\left(1\right)=B_{n}." display="block"><mrow><mrow><mrow><msub><mi href="./24.2#i">B</mi><mi href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./24.1#p2.t1.r1">n</mi></msup><mo></mo><mrow><msub><mi href="./24.2#i">B</mi><mi href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></mrow><mo>=</mo><msub><mi href="./24.2#i">B</mi><mi href="./24.1#p2.t1.r1">n</mi></msub></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E25.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./24.2#i" title="§24.2(i) Bernoulli Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="21px" altimg-valign="-5px" altimg-width="30px" alttext="B_{\NVar{n}}" display="inline"><msub><mi href="./24.2#i">B</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub></math>: Bernoulli numbers</a>,
<a href="./24.2#i" title="§24.2(i) Bernoulli Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m55.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="B_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./24.2#i">B</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></math>: Bernoulli polynomials</a> and
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./24.1#p2.t1.r1">n</mi></math>: integer</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">23.1.20</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E26" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">24.4.26</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E26.png" altimg-height="47px" altimg-valign="-17px" altimg-width="392px" alttext="E_{n}\left(0\right)=-E_{n}\left(1\right)=-\frac{2}{n+1}(2^{n+1}-1)B_{n+1}," display="block"><mrow><mrow><mrow><msub><mi href="./24.2#ii">E</mi><mi href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mo>-</mo><mrow><msub><mi href="./24.2#ii">E</mi><mi href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mrow><mo>-</mo><mrow><mfrac><mn>2</mn><mrow><mi href="./24.1#p2.t1.r1">n</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mo></mo><mrow><mo stretchy="false">(</mo><mrow><msup><mn>2</mn><mrow><mi href="./24.1#p2.t1.r1">n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msub><mi href="./24.2#i">B</mi><mrow><mi href="./24.1#p2.t1.r1">n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m73.png" altimg-height="17px" altimg-valign="-3px" altimg-width="53px" alttext="n>0" display="inline"><mrow><mi href="./24.1#p2.t1.r1">n</mi><mo>></mo><mn>0</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E26.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./24.2#i" title="§24.2(i) Bernoulli Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="21px" altimg-valign="-5px" altimg-width="30px" alttext="B_{\NVar{n}}" display="inline"><msub><mi href="./24.2#i">B</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub></math>: Bernoulli numbers</a>,
<a href="./24.2#ii" title="§24.2(ii) Euler Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m57.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="E_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./24.2#ii">E</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></math>: Euler polynomials</a> and
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./24.1#p2.t1.r1">n</mi></math>: integer</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">23.1.20</span></span>
</dd>
<dt>Errata (effective with 1.0.5):</dt>
<dd>
This equation is true only for <math class="ltx_Math" altimg="m73.png" altimg-height="17px" altimg-valign="-3px" altimg-width="53px" alttext="n>0" display="inline"><mrow><mi href="./24.1#p2.t1.r1">n</mi><mo>></mo><mn>0</mn></mrow></math>. Previously, <math class="ltx_Math" altimg="m71.png" altimg-height="17px" altimg-valign="-2px" altimg-width="53px" alttext="n=0" display="inline"><mrow><mi href="./24.1#p2.t1.r1">n</mi><mo>=</mo><mn>0</mn></mrow></math> was also allowed.
<p><span class="ltx_font_italic">Reported 2012-05-14 by Vladimir Yurovsky</span></p>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="EGx8" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E27">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">24.4.27</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m33.png" altimg-height="29px" altimg-valign="-9px" altimg-width="66px" alttext="\displaystyle B_{n}\left(\tfrac{1}{2}\right)" display="inline"><mrow><msub><mi href="./24.2#i">B</mi><mi href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m7.png" altimg-height="28px" altimg-valign="-7px" altimg-width="165px" alttext="\displaystyle=-(1-2^{1-n})B_{n}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo>-</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><msup><mn>2</mn><mrow><mn>1</mn><mo>-</mo><mi href="./24.1#p2.t1.r1">n</mi></mrow></msup></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msub><mi href="./24.2#i">B</mi><mi href="./24.1#p2.t1.r1">n</mi></msub></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E27.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./24.2#i" title="§24.2(i) Bernoulli Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="21px" altimg-valign="-5px" altimg-width="30px" alttext="B_{\NVar{n}}" display="inline"><msub><mi href="./24.2#i">B</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub></math>: Bernoulli numbers</a>,
<a href="./24.2#i" title="§24.2(i) Bernoulli Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m55.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="B_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./24.2#i">B</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></math>: Bernoulli polynomials</a> and
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./24.1#p2.t1.r1">n</mi></math>: integer</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">23.1.21</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E28">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">24.4.28</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m40.png" altimg-height="29px" altimg-valign="-9px" altimg-width="66px" alttext="\displaystyle E_{n}\left(\tfrac{1}{2}\right)" display="inline"><mrow><msub><mi href="./24.2#ii">E</mi><mi href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m10.png" altimg-height="25px" altimg-valign="-5px" altimg-width="91px" alttext="\displaystyle=2^{-n}E_{n}." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msup><mn>2</mn><mrow><mo>-</mo><mi href="./24.1#p2.t1.r1">n</mi></mrow></msup><mo></mo><msub><mi href="./24.2#ii">E</mi><mi href="./24.1#p2.t1.r1">n</mi></msub></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E28.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./24.2#ii" title="§24.2(ii) Euler Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="21px" altimg-valign="-5px" altimg-width="30px" alttext="E_{\NVar{n}}" display="inline"><msub><mi href="./24.2#ii">E</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub></math>: Euler numbers</a>,
<a href="./24.2#ii" title="§24.2(ii) Euler Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m57.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="E_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./24.2#ii">E</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></math>: Euler polynomials</a> and
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./24.1#p2.t1.r1">n</mi></math>: integer</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">23.1.21</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
<table id="E29" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">24.4.29</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E29.png" altimg-height="30px" altimg-valign="-9px" altimg-width="362px" alttext="B_{2n}\left(\tfrac{1}{3}\right)=B_{2n}\left(\tfrac{2}{3}\right)=-\tfrac{1}{2}(%
1-3^{1-2n})B_{2n}." display="block"><mrow><mrow><mrow><msub><mi href="./24.2#i">B</mi><mrow><mn>2</mn><mo></mo><mi href="./24.1#p2.t1.r1">n</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>3</mn></mfrac></mstyle><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msub><mi href="./24.2#i">B</mi><mrow><mn>2</mn><mo></mo><mi href="./24.1#p2.t1.r1">n</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mn>2</mn><mn>3</mn></mfrac></mstyle><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mo>-</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><msup><mn>3</mn><mrow><mn>1</mn><mo>-</mo><mrow><mn>2</mn><mo></mo><mi href="./24.1#p2.t1.r1">n</mi></mrow></mrow></msup></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msub><mi href="./24.2#i">B</mi><mrow><mn>2</mn><mo></mo><mi href="./24.1#p2.t1.r1">n</mi></mrow></msub></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E29.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./24.2#i" title="§24.2(i) Bernoulli Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="21px" altimg-valign="-5px" altimg-width="30px" alttext="B_{\NVar{n}}" display="inline"><msub><mi href="./24.2#i">B</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub></math>: Bernoulli numbers</a>,
<a href="./24.2#i" title="§24.2(i) Bernoulli Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m55.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="B_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./24.2#i">B</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></math>: Bernoulli polynomials</a> and
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./24.1#p2.t1.r1">n</mi></math>: integer</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">23.1.23</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E30" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">24.4.30</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E30.png" altimg-height="50px" altimg-valign="-16px" altimg-width="488px" alttext="E_{2n-1}\left(\tfrac{1}{3}\right)=-E_{2n-1}\left(\tfrac{2}{3}\right)=-\frac{(1%
-3^{1-2n})(2^{2n}-1)}{2n}B_{2n}," display="block"><mrow><mrow><mrow><msub><mi href="./24.2#ii">E</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./24.1#p2.t1.r1">n</mi></mrow><mo>-</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>3</mn></mfrac></mstyle><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mo>-</mo><mrow><msub><mi href="./24.2#ii">E</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./24.1#p2.t1.r1">n</mi></mrow><mo>-</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mn>2</mn><mn>3</mn></mfrac></mstyle><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mrow><mo>-</mo><mrow><mfrac><mrow><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><msup><mn>3</mn><mrow><mn>1</mn><mo>-</mo><mrow><mn>2</mn><mo></mo><mi href="./24.1#p2.t1.r1">n</mi></mrow></mrow></msup></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><msup><mn>2</mn><mrow><mn>2</mn><mo></mo><mi href="./24.1#p2.t1.r1">n</mi></mrow></msup><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mrow><mn>2</mn><mo></mo><mi href="./24.1#p2.t1.r1">n</mi></mrow></mfrac><mo></mo><msub><mi href="./24.2#i">B</mi><mrow><mn>2</mn><mo></mo><mi href="./24.1#p2.t1.r1">n</mi></mrow></msub></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m72.png" altimg-height="20px" altimg-valign="-6px" altimg-width="104px" alttext="n=1,2,\dots" display="inline"><mrow><mi href="./24.1#p2.t1.r1">n</mi><mo>=</mo><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E30.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./24.2#i" title="§24.2(i) Bernoulli Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="21px" altimg-valign="-5px" altimg-width="30px" alttext="B_{\NVar{n}}" display="inline"><msub><mi href="./24.2#i">B</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub></math>: Bernoulli numbers</a>,
<a href="./24.2#ii" title="§24.2(ii) Euler Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m57.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="E_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./24.2#ii">E</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></math>: Euler polynomials</a> and
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./24.1#p2.t1.r1">n</mi></math>: integer</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">23.1.22</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E31" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">24.4.31</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E31.png" altimg-height="50px" altimg-valign="-16px" altimg-width="455px" alttext="B_{n}\left(\tfrac{1}{4}\right)=(-1)^{n}B_{n}\left(\tfrac{3}{4}\right)=-\frac{1%
-2^{1-n}}{2^{n}}B_{n}-\frac{n}{4^{n}}E_{n-1}," display="block"><mrow><mrow><mrow><msub><mi href="./24.2#i">B</mi><mi href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>4</mn></mfrac></mstyle><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./24.1#p2.t1.r1">n</mi></msup><mo></mo><mrow><msub><mi href="./24.2#i">B</mi><mi href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mn>3</mn><mn>4</mn></mfrac></mstyle><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mrow><mrow><mo>-</mo><mrow><mfrac><mrow><mn>1</mn><mo>-</mo><msup><mn>2</mn><mrow><mn>1</mn><mo>-</mo><mi href="./24.1#p2.t1.r1">n</mi></mrow></msup></mrow><msup><mn>2</mn><mi href="./24.1#p2.t1.r1">n</mi></msup></mfrac><mo></mo><msub><mi href="./24.2#i">B</mi><mi href="./24.1#p2.t1.r1">n</mi></msub></mrow></mrow><mo>-</mo><mrow><mfrac><mi href="./24.1#p2.t1.r1">n</mi><msup><mn>4</mn><mi href="./24.1#p2.t1.r1">n</mi></msup></mfrac><mo></mo><msub><mi href="./24.2#ii">E</mi><mrow><mi href="./24.1#p2.t1.r1">n</mi><mo>-</mo><mn>1</mn></mrow></msub></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m72.png" altimg-height="20px" altimg-valign="-6px" altimg-width="104px" alttext="n=1,2,\dots" display="inline"><mrow><mi href="./24.1#p2.t1.r1">n</mi><mo>=</mo><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E31.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./24.2#i" title="§24.2(i) Bernoulli Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="21px" altimg-valign="-5px" altimg-width="30px" alttext="B_{\NVar{n}}" display="inline"><msub><mi href="./24.2#i">B</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub></math>: Bernoulli numbers</a>,
<a href="./24.2#i" title="§24.2(i) Bernoulli Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m55.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="B_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./24.2#i">B</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></math>: Bernoulli polynomials</a>,
<a href="./24.2#ii" title="§24.2(ii) Euler Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="21px" altimg-valign="-5px" altimg-width="30px" alttext="E_{\NVar{n}}" display="inline"><msub><mi href="./24.2#ii">E</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub></math>: Euler numbers</a> and
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./24.1#p2.t1.r1">n</mi></math>: integer</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">23.1.22</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E32" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">24.4.32</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E32.png" altimg-height="30px" altimg-valign="-9px" altimg-width="446px" alttext="B_{2n}\left(\tfrac{1}{6}\right)=B_{2n}\left(\tfrac{5}{6}\right)=\tfrac{1}{2}(1%
-2^{1-2n})(1-3^{1-2n})B_{2n}," display="block"><mrow><mrow><mrow><msub><mi href="./24.2#i">B</mi><mrow><mn>2</mn><mo></mo><mi href="./24.1#p2.t1.r1">n</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>6</mn></mfrac></mstyle><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msub><mi href="./24.2#i">B</mi><mrow><mn>2</mn><mo></mo><mi href="./24.1#p2.t1.r1">n</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mn>5</mn><mn>6</mn></mfrac></mstyle><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><msup><mn>2</mn><mrow><mn>1</mn><mo>-</mo><mrow><mn>2</mn><mo></mo><mi href="./24.1#p2.t1.r1">n</mi></mrow></mrow></msup></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><msup><mn>3</mn><mrow><mn>1</mn><mo>-</mo><mrow><mn>2</mn><mo></mo><mi href="./24.1#p2.t1.r1">n</mi></mrow></mrow></msup></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msub><mi href="./24.2#i">B</mi><mrow><mn>2</mn><mo></mo><mi href="./24.1#p2.t1.r1">n</mi></mrow></msub></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E32.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./24.2#i" title="§24.2(i) Bernoulli Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="21px" altimg-valign="-5px" altimg-width="30px" alttext="B_{\NVar{n}}" display="inline"><msub><mi href="./24.2#i">B</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub></math>: Bernoulli numbers</a>,
<a href="./24.2#i" title="§24.2(i) Bernoulli Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m55.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="B_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./24.2#i">B</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></math>: Bernoulli polynomials</a> and
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./24.1#p2.t1.r1">n</mi></math>: integer</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">23.1.24</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E33" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">24.4.33</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E33.png" altimg-height="50px" altimg-valign="-16px" altimg-width="314px" alttext="E_{2n}\left(\tfrac{1}{6}\right)=E_{2n}\left(\tfrac{5}{6}\right)=\frac{1+3^{-2n%
}}{2^{2n+1}}E_{2n}." display="block"><mrow><mrow><mrow><msub><mi href="./24.2#ii">E</mi><mrow><mn>2</mn><mo></mo><mi href="./24.1#p2.t1.r1">n</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>6</mn></mfrac></mstyle><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msub><mi href="./24.2#ii">E</mi><mrow><mn>2</mn><mo></mo><mi href="./24.1#p2.t1.r1">n</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mn>5</mn><mn>6</mn></mfrac></mstyle><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mfrac><mrow><mn>1</mn><mo>+</mo><msup><mn>3</mn><mrow><mo>-</mo><mrow><mn>2</mn><mo></mo><mi href="./24.1#p2.t1.r1">n</mi></mrow></mrow></msup></mrow><msup><mn>2</mn><mrow><mrow><mn>2</mn><mo></mo><mi href="./24.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mn>1</mn></mrow></msup></mfrac><mo></mo><msub><mi href="./24.2#ii">E</mi><mrow><mn>2</mn><mo></mo><mi href="./24.1#p2.t1.r1">n</mi></mrow></msub></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E33.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./24.2#ii" title="§24.2(ii) Euler Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="21px" altimg-valign="-5px" altimg-width="30px" alttext="E_{\NVar{n}}" display="inline"><msub><mi href="./24.2#ii">E</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub></math>: Euler numbers</a>,
<a href="./24.2#ii" title="§24.2(ii) Euler Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m57.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="E_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./24.2#ii">E</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></math>: Euler polynomials</a> and
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./24.1#p2.t1.r1">n</mi></math>: integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="vii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§24.4(vii) </span>Derivatives</h2>
<div id="SS7.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="SS7.p1" class="ltx_para">
<table id="EGx9" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E34">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">24.4.34</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m45.png" altimg-height="47px" altimg-valign="-16px" altimg-width="89px" alttext="\displaystyle\frac{\mathrm{d}}{\mathrm{d}x}B_{n}\left(x\right)" display="inline"><mrow><mstyle displaystyle="true"><mfrac><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi href="./24.1#p2.t1.r2">x</mi></mrow></mfrac></mstyle><mo></mo><mrow><msub><mi href="./24.2#i">B</mi><mi href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m26.png" altimg-height="25px" altimg-valign="-7px" altimg-width="124px" alttext="\displaystyle=nB_{n-1}\left(x\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./24.1#p2.t1.r1">n</mi><mo></mo><mrow><msub><mi href="./24.2#i">B</mi><mrow><mi href="./24.1#p2.t1.r1">n</mi><mo>-</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m72.png" altimg-height="20px" altimg-valign="-6px" altimg-width="104px" alttext="n=1,2,\dots" display="inline"><mrow><mi href="./24.1#p2.t1.r1">n</mi><mo>=</mo><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E34.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./24.2#i" title="§24.2(i) Bernoulli Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m55.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="B_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./24.2#i">B</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></math>: Bernoulli polynomials</a>,
<a href="./1.4#E4" title="(1.4.4) ‣ §1.4(iii) Derivatives ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="29px" altimg-valign="-9px" altimg-width="27px" alttext="\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: derivative of <math class="ltx_Math" altimg="m64.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m76.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./24.1#p2.t1.r1">n</mi></math>: integer</a> and
<a href="./24.1#p2.t1.r2" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m76.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./24.1#p2.t1.r2">x</mi></math>: real or complex</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">23.1.5</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E35">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">24.4.35</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m46.png" altimg-height="47px" altimg-valign="-16px" altimg-width="89px" alttext="\displaystyle\frac{\mathrm{d}}{\mathrm{d}x}E_{n}\left(x\right)" display="inline"><mrow><mstyle displaystyle="true"><mfrac><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi href="./24.1#p2.t1.r2">x</mi></mrow></mfrac></mstyle><mo></mo><mrow><msub><mi href="./24.2#ii">E</mi><mi href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m27.png" altimg-height="25px" altimg-valign="-7px" altimg-width="124px" alttext="\displaystyle=nE_{n-1}\left(x\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./24.1#p2.t1.r1">n</mi><mo></mo><mrow><msub><mi href="./24.2#ii">E</mi><mrow><mi href="./24.1#p2.t1.r1">n</mi><mo>-</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m72.png" altimg-height="20px" altimg-valign="-6px" altimg-width="104px" alttext="n=1,2,\dots" display="inline"><mrow><mi href="./24.1#p2.t1.r1">n</mi><mo>=</mo><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E35.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./24.2#ii" title="§24.2(ii) Euler Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m57.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="E_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./24.2#ii">E</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></math>: Euler polynomials</a>,
<a href="./1.4#E4" title="(1.4.4) ‣ §1.4(iii) Derivatives ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="29px" altimg-valign="-9px" altimg-width="27px" alttext="\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: derivative of <math class="ltx_Math" altimg="m64.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m76.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./24.1#p2.t1.r1">n</mi></math>: integer</a> and
<a href="./24.1#p2.t1.r2" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m76.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./24.1#p2.t1.r2">x</mi></math>: real or complex</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">23.1.5</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
</div>
</section>
<section id="viii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§24.4(viii) </span>Symbolic Operations</h2>
<div id="SS8.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="SS8.p1" class="ltx_para">
<p class="ltx_p">Let <math class="ltx_Math" altimg="m58.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="P(x)" display="inline"><mrow><mi>P</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./24.1#p2.t1.r2">x</mi><mo stretchy="false">)</mo></mrow></mrow></math> denote any polynomial in <math class="ltx_Math" altimg="m76.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./24.1#p2.t1.r2">x</mi></math>, and after expanding set
<math class="ltx_Math" altimg="m52.png" altimg-height="23px" altimg-valign="-7px" altimg-width="156px" alttext="(B(x))^{n}=B_{n}\left(x\right)" display="inline"><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mi>B</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./24.1#p2.t1.r2">x</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow><mi href="./24.1#p2.t1.r1">n</mi></msup><mo>=</mo><mrow><msub><mi href="./24.2#i">B</mi><mi href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></mrow></math> and <math class="ltx_Math" altimg="m53.png" altimg-height="23px" altimg-valign="-7px" altimg-width="156px" alttext="(E(x))^{n}=E_{n}\left(x\right)" display="inline"><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mi>E</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./24.1#p2.t1.r2">x</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow><mi href="./24.1#p2.t1.r1">n</mi></msup><mo>=</mo><mrow><msub><mi href="./24.2#ii">E</mi><mi href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></mrow></math>. Then</p>
<table id="EGx10" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E36">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">24.4.36</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m43.png" altimg-height="25px" altimg-valign="-7px" altimg-width="213px" alttext="\displaystyle P(B(x)+1)-P(B(x))" display="inline"><mrow><mrow><mi>P</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mi>B</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./24.1#p2.t1.r2">x</mi><mo stretchy="false">)</mo></mrow></mrow><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>-</mo><mrow><mi>P</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>B</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./24.1#p2.t1.r2">x</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m15.png" altimg-height="26px" altimg-valign="-7px" altimg-width="81px" alttext="\displaystyle=P^{\prime}(x)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msup><mi>P</mi><mo>′</mo></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./24.1#p2.t1.r2">x</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E36.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./24.1#p2.t1.r2" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m76.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./24.1#p2.t1.r2">x</mi></math>: real or complex</a></dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">23.1.25</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E37">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">24.4.37</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m35.png" altimg-height="25px" altimg-valign="-7px" altimg-width="98px" alttext="\displaystyle B_{n}\left(x+h\right)" display="inline"><mrow><msub><mi href="./24.2#i">B</mi><mi href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi href="./24.1#p2.t1.r2">x</mi><mo>+</mo><mi>h</mi></mrow><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m5.png" altimg-height="25px" altimg-valign="-7px" altimg-width="138px" alttext="\displaystyle=(B(x)+h)^{n}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mrow><mrow><mi>B</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./24.1#p2.t1.r2">x</mi><mo stretchy="false">)</mo></mrow></mrow><mo>+</mo><mi>h</mi></mrow><mo stretchy="false">)</mo></mrow><mi href="./24.1#p2.t1.r1">n</mi></msup></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E37.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./24.2#i" title="§24.2(i) Bernoulli Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m55.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="B_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./24.2#i">B</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></math>: Bernoulli polynomials</a>,
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./24.1#p2.t1.r1">n</mi></math>: integer</a> and
<a href="./24.1#p2.t1.r2" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m76.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./24.1#p2.t1.r2">x</mi></math>: real or complex</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">23.1.26</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E38">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">24.4.38</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m44.png" altimg-height="25px" altimg-valign="-7px" altimg-width="212px" alttext="\displaystyle P(E(x)+1)+P(E(x))" display="inline"><mrow><mrow><mi>P</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mi>E</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./24.1#p2.t1.r2">x</mi><mo stretchy="false">)</mo></mrow></mrow><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>+</mo><mrow><mi>P</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>E</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./24.1#p2.t1.r2">x</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m9.png" altimg-height="25px" altimg-valign="-7px" altimg-width="85px" alttext="\displaystyle=2P(x)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mn>2</mn><mo></mo><mrow><mi>P</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./24.1#p2.t1.r2">x</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E38.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./24.1#p2.t1.r2" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m76.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./24.1#p2.t1.r2">x</mi></math>: real or complex</a></dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">23.1.25</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E39">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">24.4.39</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m42.png" altimg-height="25px" altimg-valign="-7px" altimg-width="98px" alttext="\displaystyle E_{n}\left(x+h\right)" display="inline"><mrow><msub><mi href="./24.2#ii">E</mi><mi href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi href="./24.1#p2.t1.r2">x</mi><mo>+</mo><mi>h</mi></mrow><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m6.png" altimg-height="25px" altimg-valign="-7px" altimg-width="138px" alttext="\displaystyle=(E(x)+h)^{n}." display="inline"><mrow><mrow><mi></mi><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mrow><mrow><mi>E</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./24.1#p2.t1.r2">x</mi><mo stretchy="false">)</mo></mrow></mrow><mo>+</mo><mi>h</mi></mrow><mo stretchy="false">)</mo></mrow><mi href="./24.1#p2.t1.r1">n</mi></msup></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E39.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./24.2#ii" title="§24.2(ii) Euler Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m57.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="E_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./24.2#ii">E</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></math>: Euler polynomials</a>,
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./24.1#p2.t1.r1">n</mi></math>: integer</a> and
<a href="./24.1#p2.t1.r2" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m76.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./24.1#p2.t1.r2">x</mi></math>: real or complex</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">23.1.26</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
<p class="ltx_p">For these results and also connections with the umbral calculus see
<cite class="ltx_cite ltx_citemacro_citet">Gessel (</div>
</div>
</body></text>
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<title>DLMF: 24.13 Integrals</title>
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<div id="SS1.p1" class="ltx_para">
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<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">24.13.1</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m6.png" altimg-height="51px" altimg-valign="-19px" altimg-width="103px" alttext="\displaystyle\int B_{n}\left(t\right)\mathrm{d}t" display="inline"><mstyle displaystyle="true"><mrow><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><mrow><msub><mi href="./24.2#i">B</mi><mi href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./24.1#p2.t1.r2">t</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./24.1#p2.t1.r2">t</mi></mrow></mrow></mrow></mstyle></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m2.png" altimg-height="49px" altimg-valign="-17px" altimg-width="185px" alttext="\displaystyle=\frac{B_{n+1}\left(t\right)}{n+1}+\text{const.}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><mfrac><mrow><msub><mi href="./24.2#i">B</mi><mrow><mi href="./24.1#p2.t1.r1">n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./24.1#p2.t1.r2">t</mi><mo>)</mo></mrow></mrow><mrow><mi href="./24.1#p2.t1.r1">n</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mstyle><mo>+</mo><mtext>const.</mtext></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./24.2#i" title="§24.2(i) Bernoulli Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m13.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="B_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./24.2#i">B</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></math>: Bernoulli polynomials</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m26.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m16.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m24.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./24.1#p2.t1.r1">n</mi></math>: integer</a> and
<a href="./24.1#p2.t1.r2" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m25.png" altimg-height="17px" altimg-valign="-2px" altimg-width="11px" alttext="t" display="inline"><mi href="./24.1#p2.t1.r2">t</mi></math>: real or complex</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">23.1.11</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E2">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">24.13.2</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m10.png" altimg-height="56px" altimg-valign="-20px" altimg-width="133px" alttext="\displaystyle\int_{x}^{x+1}B_{n}\left(t\right)\mathrm{d}t" display="inline"><mrow><mstyle displaystyle="true"><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi href="./24.1#p2.t1.r2">x</mi><mrow><mi href="./24.1#p2.t1.r2">x</mi><mo>+</mo><mn>1</mn></mrow></msubsup></mstyle><mrow><mrow><msub><mi href="./24.2#i">B</mi><mi href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./24.1#p2.t1.r2">t</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./24.1#p2.t1.r2">t</mi></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m5.png" altimg-height="24px" altimg-valign="-6px" altimg-width="55px" alttext="\displaystyle=x^{n}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><msup><mi href="./24.1#p2.t1.r2">x</mi><mi href="./24.1#p2.t1.r1">n</mi></msup></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m23.png" altimg-height="20px" altimg-valign="-6px" altimg-width="104px" alttext="n=1,2,\dots" display="inline"><mrow><mi href="./24.1#p2.t1.r1">n</mi><mo>=</mo><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./24.2#i" title="§24.2(i) Bernoulli Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m13.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="B_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./24.2#i">B</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></math>: Bernoulli polynomials</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m26.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m16.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m24.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./24.1#p2.t1.r1">n</mi></math>: integer</a>,
<a href="./24.1#p2.t1.r2" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m25.png" altimg-height="17px" altimg-valign="-2px" altimg-width="11px" alttext="t" display="inline"><mi href="./24.1#p2.t1.r2">t</mi></math>: real or complex</a> and
<a href="./24.1#p2.t1.r2" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m26.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./24.1#p2.t1.r2">x</mi></math>: real or complex</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E3">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">24.13.3</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m9.png" altimg-height="56px" altimg-valign="-20px" altimg-width="162px" alttext="\displaystyle\int_{x}^{x+(1/2)}B_{n}\left(t\right)\mathrm{d}t" display="inline"><mrow><mstyle displaystyle="true"><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi href="./24.1#p2.t1.r2">x</mi><mrow><mi href="./24.1#p2.t1.r2">x</mi><mo>+</mo><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow><mo stretchy="false">)</mo></mrow></mrow></msubsup></mstyle><mrow><mrow><msub><mi href="./24.2#i">B</mi><mi href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./24.1#p2.t1.r2">t</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./24.1#p2.t1.r2">t</mi></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m3.png" altimg-height="48px" altimg-valign="-16px" altimg-width="103px" alttext="\displaystyle=\frac{E_{n}\left(2x\right)}{2^{n+1}}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mstyle displaystyle="true"><mfrac><mrow><msub><mi href="./24.2#ii">E</mi><mi href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><mi href="./24.1#p2.t1.r2">x</mi></mrow><mo>)</mo></mrow></mrow><msup><mn>2</mn><mrow><mi href="./24.1#p2.t1.r1">n</mi><mo>+</mo><mn>1</mn></mrow></msup></mfrac></mstyle></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./24.2#i" title="§24.2(i) Bernoulli Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m13.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="B_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./24.2#i">B</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></math>: Bernoulli polynomials</a>,
<a href="./24.2#ii" title="§24.2(ii) Euler Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m15.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="E_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./24.2#ii">E</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></math>: Euler polynomials</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m26.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m16.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m24.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./24.1#p2.t1.r1">n</mi></math>: integer</a>,
<a href="./24.1#p2.t1.r2" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m25.png" altimg-height="17px" altimg-valign="-2px" altimg-width="11px" alttext="t" display="inline"><mi href="./24.1#p2.t1.r2">t</mi></math>: real or complex</a> and
<a href="./24.1#p2.t1.r2" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m26.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./24.1#p2.t1.r2">x</mi></math>: real or complex</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E4">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">24.13.4</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m7.png" altimg-height="56px" altimg-valign="-20px" altimg-width="128px" alttext="\displaystyle\int_{0}^{1/2}B_{n}\left(t\right)\mathrm{d}t" display="inline"><mrow><mstyle displaystyle="true"><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msubsup></mstyle><mrow><mrow><msub><mi href="./24.2#i">B</mi><mi href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./24.1#p2.t1.r2">t</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./24.1#p2.t1.r2">t</mi></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m1.png" altimg-height="51px" altimg-valign="-17px" altimg-width="164px" alttext="\displaystyle=\frac{1-2^{n+1}}{2^{n}}\frac{B_{n+1}}{n+1}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><mfrac><mrow><mn>1</mn><mo>-</mo><msup><mn>2</mn><mrow><mi href="./24.1#p2.t1.r1">n</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><msup><mn>2</mn><mi href="./24.1#p2.t1.r1">n</mi></msup></mfrac></mstyle><mo></mo><mstyle displaystyle="true"><mfrac><msub><mi href="./24.2#i">B</mi><mrow><mi href="./24.1#p2.t1.r1">n</mi><mo>+</mo><mn>1</mn></mrow></msub><mrow><mi href="./24.1#p2.t1.r1">n</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mstyle></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E4.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./24.2#i" title="§24.2(i) Bernoulli Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m12.png" altimg-height="21px" altimg-valign="-5px" altimg-width="30px" alttext="B_{\NVar{n}}" display="inline"><msub><mi href="./24.2#i">B</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub></math>: Bernoulli numbers</a>,
<a href="./24.2#i" title="§24.2(i) Bernoulli Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m13.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="B_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./24.2#i">B</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></math>: Bernoulli polynomials</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m26.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m16.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m24.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./24.1#p2.t1.r1">n</mi></math>: integer</a> and
<a href="./24.1#p2.t1.r2" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m25.png" altimg-height="17px" altimg-valign="-2px" altimg-width="11px" alttext="t" display="inline"><mi href="./24.1#p2.t1.r2">t</mi></math>: real or complex</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E5">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">24.13.5</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m8.png" altimg-height="61px" altimg-valign="-24px" altimg-width="128px" alttext="\displaystyle\int_{1/4}^{3/4}B_{n}\left(t\right)\mathrm{d}t" display="inline"><mrow><mstyle displaystyle="true"><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><mn>1</mn><mo>/</mo><mn>4</mn></mrow><mrow><mn>3</mn><mo>/</mo><mn>4</mn></mrow></msubsup></mstyle><mrow><mrow><msub><mi href="./24.2#i">B</mi><mi href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./24.1#p2.t1.r2">t</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./24.1#p2.t1.r2">t</mi></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m4.png" altimg-height="46px" altimg-valign="-16px" altimg-width="86px" alttext="\displaystyle=\frac{E_{n}}{2^{2n+1}}." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mstyle displaystyle="true"><mfrac><msub><mi href="./24.2#ii">E</mi><mi href="./24.1#p2.t1.r1">n</mi></msub><msup><mn>2</mn><mrow><mrow><mn>2</mn><mo></mo><mi href="./24.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mn>1</mn></mrow></msup></mfrac></mstyle></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E5.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./24.2#i" title="§24.2(i) Bernoulli Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m13.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="B_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./24.2#i">B</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></math>: Bernoulli polynomials</a>,
<a href="./24.2#ii" title="§24.2(ii) Euler Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m14.png" altimg-height="21px" altimg-valign="-5px" altimg-width="30px" alttext="E_{\NVar{n}}" display="inline"><msub><mi href="./24.2#ii">E</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub></math>: Euler numbers</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m26.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m16.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m24.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./24.1#p2.t1.r1">n</mi></math>: integer</a> and
<a href="./24.1#p2.t1.r2" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m25.png" altimg-height="17px" altimg-valign="-2px" altimg-width="11px" alttext="t" display="inline"><mi href="./24.1#p2.t1.r2">t</mi></math>: real or complex</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
</div>
<div id="SS1.p2" class="ltx_para">
<p class="ltx_p">For <math class="ltx_Math" altimg="m20.png" altimg-height="20px" altimg-valign="-6px" altimg-width="130px" alttext="m,n=1,2,\ldots" display="inline"><mrow><mrow><mrow><mi href="./24.1#p2.t1.r1">m</mi><mo>,</mo><mi href="./24.1#p2.t1.r1">n</mi></mrow><mo>=</mo><mn>1</mn></mrow><mo>,</mo><mrow><mn>2</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>,</p>
<table id="E6" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">24.13.6</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E6.png" altimg-height="55px" altimg-valign="-21px" altimg-width="373px" alttext="\int_{0}^{1}B_{n}\left(t\right)B_{m}\left(t\right)\mathrm{d}t=\frac{(-1)^{n-1}%
m!n!}{(m+n)!}B_{m+n}." display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mn>1</mn></msubsup><mrow><mrow><msub><mi href="./24.2#i">B</mi><mi href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./24.1#p2.t1.r2">t</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./24.2#i">B</mi><mi href="./24.1#p2.t1.r1">m</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./24.1#p2.t1.r2">t</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./24.1#p2.t1.r2">t</mi></mrow></mrow></mrow><mo>=</mo><mrow><mfrac><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mrow><mi href="./24.1#p2.t1.r1">n</mi><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mrow><mi href="./24.1#p2.t1.r1">m</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow><mo></mo><mrow><mi href="./24.1#p2.t1.r1">n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./24.1#p2.t1.r1">m</mi><mo>+</mo><mi href="./24.1#p2.t1.r1">n</mi></mrow><mo stretchy="false">)</mo></mrow><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mfrac><mo></mo><msub><mi href="./24.2#i">B</mi><mrow><mi href="./24.1#p2.t1.r1">m</mi><mo>+</mo><mi href="./24.1#p2.t1.r1">n</mi></mrow></msub></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E6.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./24.2#i" title="§24.2(i) Bernoulli Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m12.png" altimg-height="21px" altimg-valign="-5px" altimg-width="30px" alttext="B_{\NVar{n}}" display="inline"><msub><mi href="./24.2#i">B</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub></math>: Bernoulli numbers</a>,
<a href="./24.2#i" title="§24.2(i) Bernoulli Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m13.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="B_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./24.2#i">B</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></math>: Bernoulli polynomials</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m26.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m11.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m22.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m16.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m21.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./24.1#p2.t1.r1">m</mi></math>: integer</a>,
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m24.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./24.1#p2.t1.r1">n</mi></math>: integer</a> and
<a href="./24.1#p2.t1.r2" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m25.png" altimg-height="17px" altimg-valign="-2px" altimg-width="11px" alttext="t" display="inline"><mi href="./24.1#p2.t1.r2">t</mi></math>: real or complex</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">23.1.12</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS1.p3" class="ltx_para">
<p class="ltx_p">For integrals of the form <math class="ltx_Math" altimg="m18.png" altimg-height="28px" altimg-valign="-9px" altimg-width="164px" alttext="\int_{0}^{x}B_{n}\left(t\right)B_{m}\left(t\right)\mathrm{d}t" display="inline"><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi href="./24.1#p2.t1.r2">x</mi></msubsup><mrow><mrow><msub><mi href="./24.2#i">B</mi><mi href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./24.1#p2.t1.r2">t</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./24.2#i">B</mi><mi href="./24.1#p2.t1.r1">m</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./24.1#p2.t1.r2">t</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./24.1#p2.t1.r2">t</mi></mrow></mrow></mrow></math> and
<math class="ltx_Math" altimg="m17.png" altimg-height="28px" altimg-valign="-9px" altimg-width="218px" alttext="\int_{0}^{x}B_{n}\left(t\right)B_{m}\left(t\right)B_{k}\left(t\right)\mathrm{d}t" display="inline"><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi href="./24.1#p2.t1.r2">x</mi></msubsup><mrow><mrow><msub><mi href="./24.2#i">B</mi><mi href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./24.1#p2.t1.r2">t</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./24.2#i">B</mi><mi href="./24.1#p2.t1.r1">m</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./24.1#p2.t1.r2">t</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./24.2#i">B</mi><mi href="./24.1#p2.t1.r1">k</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./24.1#p2.t1.r2">t</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./24.1#p2.t1.r2">t</mi></mrow></mrow></mrow></math>
see <cite class="ltx_cite ltx_citemacro_citet">Agoh and Dilcher (</dd>
</dl>
</div>
</div>
<div id="SS2.p1" class="ltx_para">
<table id="E7" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">24.13.7</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E7.png" altimg-height="51px" altimg-valign="-19px" altimg-width="286px" alttext="\int E_{n}\left(t\right)\mathrm{d}t=\frac{E_{n+1}\left(t\right)}{n+1}+\text{%
const.}," display="block"><mrow><mrow><mrow><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><mrow><msub><mi href="./24.2#ii">E</mi><mi href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./24.1#p2.t1.r2">t</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./24.1#p2.t1.r2">t</mi></mrow></mrow></mrow><mo>=</mo><mrow><mfrac><mrow><msub><mi href="./24.2#ii">E</mi><mrow><mi href="./24.1#p2.t1.r1">n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./24.1#p2.t1.r2">t</mi><mo>)</mo></mrow></mrow><mrow><mi href="./24.1#p2.t1.r1">n</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mo>+</mo><mtext>const.</mtext></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E7.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./24.2#ii" title="§24.2(ii) Euler Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m15.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="E_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./24.2#ii">E</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></math>: Euler polynomials</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m26.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m16.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m24.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./24.1#p2.t1.r1">n</mi></math>: integer</a> and
<a href="./24.1#p2.t1.r2" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m25.png" altimg-height="17px" altimg-valign="-2px" altimg-width="11px" alttext="t" display="inline"><mi href="./24.1#p2.t1.r2">t</mi></math>: real or complex</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">23.1.11</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E8" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">24.13.8</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E8.png" altimg-height="55px" altimg-valign="-21px" altimg-width="449px" alttext="\int_{0}^{1}E_{n}\left(t\right)\mathrm{d}t=-2\frac{E_{n+1}\left(0\right)}{n+1}%
=\frac{4(2^{n+2}-1)}{(n+1)(n+2)}B_{n+2}," display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mn>1</mn></msubsup><mrow><mrow><msub><mi href="./24.2#ii">E</mi><mi href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./24.1#p2.t1.r2">t</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./24.1#p2.t1.r2">t</mi></mrow></mrow></mrow><mo>=</mo><mrow><mo>-</mo><mrow><mn>2</mn><mo></mo><mfrac><mrow><msub><mi href="./24.2#ii">E</mi><mrow><mi href="./24.1#p2.t1.r1">n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow></mrow><mrow><mi href="./24.1#p2.t1.r1">n</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mrow></mrow><mo>=</mo><mrow><mfrac><mrow><mn>4</mn><mo></mo><mrow><mo stretchy="false">(</mo><mrow><msup><mn>2</mn><mrow><mi href="./24.1#p2.t1.r1">n</mi><mo>+</mo><mn>2</mn></mrow></msup><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./24.1#p2.t1.r1">n</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./24.1#p2.t1.r1">n</mi><mo>+</mo><mn>2</mn></mrow><mo stretchy="false">)</mo></mrow></mrow></mfrac><mo></mo><msub><mi href="./24.2#i">B</mi><mrow><mi href="./24.1#p2.t1.r1">n</mi><mo>+</mo><mn>2</mn></mrow></msub></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E8.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./24.2#i" title="§24.2(i) Bernoulli Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m12.png" altimg-height="21px" altimg-valign="-5px" altimg-width="30px" alttext="B_{\NVar{n}}" display="inline"><msub><mi href="./24.2#i">B</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub></math>: Bernoulli numbers</a>,
<a href="./24.2#ii" title="§24.2(ii) Euler Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m15.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="E_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./24.2#ii">E</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></math>: Euler polynomials</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m26.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m16.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m24.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./24.1#p2.t1.r1">n</mi></math>: integer</a> and
<a href="./24.1#p2.t1.r2" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m25.png" altimg-height="17px" altimg-valign="-2px" altimg-width="11px" alttext="t" display="inline"><mi href="./24.1#p2.t1.r2">t</mi></math>: real or complex</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E9" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">24.13.9</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E9.png" altimg-height="58px" altimg-valign="-21px" altimg-width="464px" alttext="\int_{0}^{1/2}E_{2n}\left(t\right)\mathrm{d}t=-\frac{E_{2n+1}\left(0\right)}{2%
n+1}=\frac{2(2^{2n+2}-1)B_{2n+2}}{(2n+1)(2n+2)}," display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msubsup><mrow><mrow><msub><mi href="./24.2#ii">E</mi><mrow><mn>2</mn><mo></mo><mi href="./24.1#p2.t1.r1">n</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./24.1#p2.t1.r2">t</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./24.1#p2.t1.r2">t</mi></mrow></mrow></mrow><mo>=</mo><mrow><mo>-</mo><mfrac><mrow><msub><mi href="./24.2#ii">E</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./24.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow></mrow><mrow><mrow><mn>2</mn><mo></mo><mi href="./24.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mn>1</mn></mrow></mfrac></mrow><mo>=</mo><mfrac><mrow><mn>2</mn><mo></mo><mrow><mo stretchy="false">(</mo><mrow><msup><mn>2</mn><mrow><mrow><mn>2</mn><mo></mo><mi href="./24.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mn>2</mn></mrow></msup><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msub><mi href="./24.2#i">B</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./24.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mn>2</mn></mrow></msub></mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>2</mn><mo></mo><mi href="./24.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>2</mn><mo></mo><mi href="./24.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mn>2</mn></mrow><mo stretchy="false">)</mo></mrow></mrow></mfrac></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E9.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./24.2#i" title="§24.2(i) Bernoulli Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m12.png" altimg-height="21px" altimg-valign="-5px" altimg-width="30px" alttext="B_{\NVar{n}}" display="inline"><msub><mi href="./24.2#i">B</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub></math>: Bernoulli numbers</a>,
<a href="./24.2#ii" title="§24.2(ii) Euler Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m15.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="E_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./24.2#ii">E</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></math>: Euler polynomials</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m26.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m16.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m24.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./24.1#p2.t1.r1">n</mi></math>: integer</a> and
<a href="./24.1#p2.t1.r2" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m25.png" altimg-height="17px" altimg-valign="-2px" altimg-width="11px" alttext="t" display="inline"><mi href="./24.1#p2.t1.r2">t</mi></math>: real or complex</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E10" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">24.13.10</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E10.png" altimg-height="56px" altimg-valign="-20px" altimg-width="254px" alttext="\int_{0}^{1/2}E_{2n-1}\left(t\right)\mathrm{d}t=\frac{E_{2n}}{n2^{2n+1}}," display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msubsup><mrow><mrow><msub><mi href="./24.2#ii">E</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./24.1#p2.t1.r1">n</mi></mrow><mo>-</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./24.1#p2.t1.r2">t</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./24.1#p2.t1.r2">t</mi></mrow></mrow></mrow><mo>=</mo><mfrac><msub><mi href="./24.2#ii">E</mi><mrow><mn>2</mn><mo></mo><mi href="./24.1#p2.t1.r1">n</mi></mrow></msub><mrow><mi href="./24.1#p2.t1.r1">n</mi><mo></mo><msup><mn>2</mn><mrow><mrow><mn>2</mn><mo></mo><mi href="./24.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mn>1</mn></mrow></msup></mrow></mfrac></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m23.png" altimg-height="20px" altimg-valign="-6px" altimg-width="104px" alttext="n=1,2,\dots" display="inline"><mrow><mi href="./24.1#p2.t1.r1">n</mi><mo>=</mo><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E10.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./24.2#ii" title="§24.2(ii) Euler Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m14.png" altimg-height="21px" altimg-valign="-5px" altimg-width="30px" alttext="E_{\NVar{n}}" display="inline"><msub><mi href="./24.2#ii">E</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub></math>: Euler numbers</a>,
<a href="./24.2#ii" title="§24.2(ii) Euler Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m15.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="E_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./24.2#ii">E</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></math>: Euler polynomials</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m26.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m16.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m24.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./24.1#p2.t1.r1">n</mi></math>: integer</a> and
<a href="./24.1#p2.t1.r2" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m25.png" altimg-height="17px" altimg-valign="-2px" altimg-width="11px" alttext="t" display="inline"><mi href="./24.1#p2.t1.r2">t</mi></math>: real or complex</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS2.p2" class="ltx_para">
<p class="ltx_p">For <math class="ltx_Math" altimg="m20.png" altimg-height="20px" altimg-valign="-6px" altimg-width="130px" alttext="m,n=1,2,\ldots" display="inline"><mrow><mrow><mrow><mi href="./24.1#p2.t1.r1">m</mi><mo>,</mo><mi href="./24.1#p2.t1.r1">n</mi></mrow><mo>=</mo><mn>1</mn></mrow><mo>,</mo><mrow><mn>2</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>,</p>
<table id="E11" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">24.13.11</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E11.png" altimg-height="55px" altimg-valign="-21px" altimg-width="499px" alttext="\int_{0}^{1}E_{n}\left(t\right)E_{m}\left(t\right)\mathrm{d}t=(-1)^{n}4\frac{(%
2^{m+n+2}-1)m!n!}{(m+n+2)!}B_{m+n+2}." display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mn>1</mn></msubsup><mrow><mrow><msub><mi href="./24.2#ii">E</mi><mi href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./24.1#p2.t1.r2">t</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./24.2#ii">E</mi><mi href="./24.1#p2.t1.r1">m</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./24.1#p2.t1.r2">t</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./24.1#p2.t1.r2">t</mi></mrow></mrow></mrow><mo>=</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./24.1#p2.t1.r1">n</mi></msup><mo></mo><mn>4</mn><mo></mo><mfrac><mrow><mrow><mo stretchy="false">(</mo><mrow><msup><mn>2</mn><mrow><mi href="./24.1#p2.t1.r1">m</mi><mo>+</mo><mi href="./24.1#p2.t1.r1">n</mi><mo>+</mo><mn>2</mn></mrow></msup><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mi href="./24.1#p2.t1.r1">m</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow><mo></mo><mrow><mi href="./24.1#p2.t1.r1">n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./24.1#p2.t1.r1">m</mi><mo>+</mo><mi href="./24.1#p2.t1.r1">n</mi><mo>+</mo><mn>2</mn></mrow><mo stretchy="false">)</mo></mrow><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mfrac><mo></mo><msub><mi href="./24.2#i">B</mi><mrow><mi href="./24.1#p2.t1.r1">m</mi><mo>+</mo><mi href="./24.1#p2.t1.r1">n</mi><mo>+</mo><mn>2</mn></mrow></msub></mrow></mrow><mo>.</mo></mrow></math></td>
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</tr>
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<div id="E11.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./24.2#i" title="§24.2(i) Bernoulli Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m12.png" altimg-height="21px" altimg-valign="-5px" altimg-width="30px" alttext="B_{\NVar{n}}" display="inline"><msub><mi href="./24.2#i">B</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub></math>: Bernoulli numbers</a>,
<a href="./24.2#ii" title="§24.2(ii) Euler Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m15.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="E_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./24.2#ii">E</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./24.1#p2.t1.r2">x</mi><mo>)</mo></mrow></mrow></math>: Euler polynomials</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m26.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m11.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m22.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m16.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m21.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./24.1#p2.t1.r1">m</mi></math>: integer</a>,
<a href="./24.1#p2.t1.r1" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m24.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./24.1#p2.t1.r1">n</mi></math>: integer</a> and
<a href="./24.1#p2.t1.r2" title="§24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m25.png" altimg-height="17px" altimg-valign="-2px" altimg-width="11px" alttext="t" display="inline"><mi href="./24.1#p2.t1.r2">t</mi></math>: real or complex</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">23.1.12</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="iii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§24.13(iii) </span>Compendia</h2>
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</body></text>
</html>
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<title>DLMF: 10.18 Modulus and Phase Functions</title>
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<div id="SS1.p1" class="ltx_para">
<p class="ltx_p">For <math class="ltx_Math" altimg="m46.png" altimg-height="19px" altimg-valign="-5px" altimg-width="52px" alttext="\nu\geq 0" display="inline"><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>≥</mo><mn>0</mn></mrow></math> and <math class="ltx_Math" altimg="m62.png" altimg-height="17px" altimg-valign="-3px" altimg-width="52px" alttext="x>0" display="inline"><mrow><mi href="./10.1#p2.t1.r3">x</mi><mo>></mo><mn>0</mn></mrow></math></p>
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<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.18.1</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m16.png" altimg-height="29px" altimg-valign="-7px" altimg-width="120px" alttext="\displaystyle M_{\nu}\left(x\right)e^{i\!\theta_{\nu}\left(x\right)}" display="inline"><mrow><mrow><msub><mi href="./10.18#E1">M</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mpadded width="-1.7pt"><mi mathvariant="normal">i</mi></mpadded><mo></mo><mrow><msub><mi href="./10.18#E3">θ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mrow></msup></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m12.png" altimg-height="29px" altimg-valign="-7px" altimg-width="106px" alttext="\displaystyle={H^{(1)}_{\nu}}\left(x\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msubsup><mi href="./10.2#E5">H</mi><mi href="./10.1#p2.t1.r5">ν</mi><mrow><mo href="./10.2#E5" stretchy="false">(</mo><mn>1</mn><mo href="./10.2#E5" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
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<div id="E1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m35.png" altimg-height="23px" altimg-valign="-7px" altimg-width="64px" alttext="M_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.18#E1">M</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: modulus of Bessel functions</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E5" title="(10.2.5) ‣ Bessel Functions of the Third Kind (Hankel Functions) ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="29px" altimg-valign="-7px" altimg-width="73px" alttext="{H^{(1)}_{\NVar{\nu}}}\left(\NVar{z}\right)" display="inline"><mrow><msubsup><mi href="./10.2#E5">H</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi><mrow><mo href="./10.2#E5" stretchy="false">(</mo><mn>1</mn><mo href="./10.2#E5" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the third kind (or Hankel function)</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m42.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./10.18#E3" title="(10.18.3) ‣ §10.18(i) Definitions ‣ §10.18 Modulus and Phase Functions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="\theta_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.18#E3">θ</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: phase of Bessel functions</a>,
<a href="./10.1#p2.t1.r3" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./10.1#p2.t1.r3">x</mi></math>: real variable</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m45.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
</dl>
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<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.18.2</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m18.png" altimg-height="29px" altimg-valign="-7px" altimg-width="119px" alttext="\displaystyle N_{\nu}\left(x\right)e^{i\!\phi_{\nu}\left(x\right)}" display="inline"><mrow><mrow><msub><mi href="./10.18#E2">N</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mpadded width="-1.7pt"><mi mathvariant="normal">i</mi></mpadded><mo></mo><mrow><msub><mi href="./10.18#E3">ϕ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mrow></msup></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m11.png" altimg-height="32px" altimg-valign="-7px" altimg-width="111px" alttext="\displaystyle={H^{(1)}_{\nu}}'\left(x\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mmultiscripts><mi href="./10.2#E5">H</mi><mi href="./10.1#p2.t1.r5">ν</mi><mrow><mo href="./10.2#E5" stretchy="false">(</mo><mn>1</mn><mo href="./10.2#E5" stretchy="false">)</mo></mrow><none></none><mo>′</mo></mmultiscripts><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
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<div id="E2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m37.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="N_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.18#E2">N</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: modulus of derivatives of Bessel functions</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E5" title="(10.2.5) ‣ Bessel Functions of the Third Kind (Hankel Functions) ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="29px" altimg-valign="-7px" altimg-width="73px" alttext="{H^{(1)}_{\NVar{\nu}}}\left(\NVar{z}\right)" display="inline"><mrow><msubsup><mi href="./10.2#E5">H</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi><mrow><mo href="./10.2#E5" stretchy="false">(</mo><mn>1</mn><mo href="./10.2#E5" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the third kind (or Hankel function)</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m42.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./10.18#E3" title="(10.18.3) ‣ §10.18(i) Definitions ‣ §10.18 Modulus and Phase Functions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="23px" altimg-valign="-7px" altimg-width="56px" alttext="\phi_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.18#E3">ϕ</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: phase of derivatives of Bessel functions</a>,
<a href="./10.1#p2.t1.r3" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./10.1#p2.t1.r3">x</mi></math>: real variable</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m45.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m36.png" altimg-height="23px" altimg-valign="-7px" altimg-width="64px" alttext="M_{\nu}\left(x\right)" display="inline"><mrow><msub><mi href="./10.18#E1">M</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math> <math class="ltx_Math" altimg="m31.png" altimg-height="23px" altimg-valign="-7px" altimg-width="51px" alttext="(>0)" display="inline"><mrow><mo stretchy="false">(</mo><mrow><mi></mi><mo>></mo><mn>0</mn></mrow><mo stretchy="false">)</mo></mrow></math>, <math class="ltx_Math" altimg="m38.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="N_{\nu}\left(x\right)" display="inline"><mrow><msub><mi href="./10.18#E2">N</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math> <math class="ltx_Math" altimg="m31.png" altimg-height="23px" altimg-valign="-7px" altimg-width="51px" alttext="(>0)" display="inline"><mrow><mo stretchy="false">(</mo><mrow><mi></mi><mo>></mo><mn>0</mn></mrow><mo stretchy="false">)</mo></mrow></math>, <math class="ltx_Math" altimg="m55.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="\theta_{\nu}\left(x\right)" display="inline"><mrow><msub><mi href="./10.18#E3">θ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>,
and <math class="ltx_Math" altimg="m49.png" altimg-height="23px" altimg-valign="-7px" altimg-width="56px" alttext="\phi_{\nu}\left(x\right)" display="inline"><mrow><msub><mi href="./10.18#E3">ϕ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math> are continuous real functions of <math class="ltx_Math" altimg="m45.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math> and <math class="ltx_Math" altimg="m63.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./10.1#p2.t1.r3">x</mi></math>, with
the branches of <math class="ltx_Math" altimg="m55.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="\theta_{\nu}\left(x\right)" display="inline"><mrow><msub><mi href="./10.18#E3">θ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math> and <math class="ltx_Math" altimg="m49.png" altimg-height="23px" altimg-valign="-7px" altimg-width="56px" alttext="\phi_{\nu}\left(x\right)" display="inline"><mrow><msub><mi href="./10.18#E3">ϕ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math> fixed by</p>
<table id="E3" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex1" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="3" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">10.18.3</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m24.png" altimg-height="25px" altimg-valign="-7px" altimg-width="56px" alttext="\displaystyle\theta_{\nu}\left(x\right)" display="inline"><mrow><msub><mi href="./10.18#E3">θ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m25.png" altimg-height="29px" altimg-valign="-9px" altimg-width="77px" alttext="\displaystyle\to-\tfrac{1}{2}\pi," display="inline"><mrow><mrow><mi></mi><mo>→</mo><mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex2" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m21.png" altimg-height="25px" altimg-valign="-7px" altimg-width="58px" alttext="\displaystyle\phi_{\nu}\left(x\right)" display="inline"><mrow><msub><mi href="./10.18#E3">ϕ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell">
<math class="ltx_Math" altimg="m26.png" altimg-height="29px" altimg-valign="-9px" altimg-width="56px" alttext="\displaystyle\to\tfrac{1}{2}\pi" display="inline"><mrow><mi></mi><mo>→</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow></math>,</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m64.png" altimg-height="18px" altimg-valign="-4px" altimg-width="72px" alttext="x\to 0+" display="inline"><mrow><mi href="./10.1#p2.t1.r3">x</mi><mo>→</mo><mrow><mn>0</mn><mo>+</mo></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd>
<span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m48.png" altimg-height="23px" altimg-valign="-7px" altimg-width="56px" alttext="\phi_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.18#E3">ϕ</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: phase of derivatives of Bessel functions</span> and
<span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m54.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="\theta_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.18#E3">θ</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: phase of Bessel functions</span>
</dd>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./10.1#p2.t1.r3" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./10.1#p2.t1.r3">x</mi></math>: real variable</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m45.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="ii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§10.18(ii) </span>Basic Properties</h2>
<div id="SS2.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="SS2.p1" class="ltx_para">
<table id="E4" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex3" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">10.18.4</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m14.png" altimg-height="25px" altimg-valign="-7px" altimg-width="57px" alttext="\displaystyle J_{\nu}\left(x\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m1.png" altimg-height="25px" altimg-valign="-7px" altimg-width="178px" alttext="\displaystyle=M_{\nu}\left(x\right)\cos\theta_{\nu}\left(x\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><msub><mi href="./10.18#E1">M</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><msub><mi href="./10.18#E3">θ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex4" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m20.png" altimg-height="25px" altimg-valign="-7px" altimg-width="58px" alttext="\displaystyle Y_{\nu}\left(x\right)" display="inline"><mrow><msub><mi href="./10.2#E3">Y</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m2.png" altimg-height="25px" altimg-valign="-7px" altimg-width="176px" alttext="\displaystyle=M_{\nu}\left(x\right)\sin\theta_{\nu}\left(x\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><msub><mi href="./10.18#E1">M</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><msub><mi href="./10.18#E3">θ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E4.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m34.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./10.2#E3" title="(10.2.3) ‣ Bessel Function of the Second Kind (Weber’s Function) ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m40.png" altimg-height="23px" altimg-valign="-7px" altimg-width="55px" alttext="Y_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E3">Y</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the second kind</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m41.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./10.18#E1" title="(10.18.1) ‣ §10.18(i) Definitions ‣ §10.18 Modulus and Phase Functions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m35.png" altimg-height="23px" altimg-valign="-7px" altimg-width="64px" alttext="M_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.18#E1">M</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: modulus of Bessel functions</a>,
<a href="./10.18#E3" title="(10.18.3) ‣ §10.18(i) Definitions ‣ §10.18 Modulus and Phase Functions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="\theta_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.18#E3">θ</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: phase of Bessel functions</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./10.1#p2.t1.r3" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./10.1#p2.t1.r3">x</mi></math>: real variable</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m45.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">9.2.19</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E5" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex5" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">10.18.5</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m13.png" altimg-height="26px" altimg-valign="-7px" altimg-width="57px" alttext="\displaystyle J_{\nu}'\left(x\right)" display="inline"><mrow><msubsup><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′</mo></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m3.png" altimg-height="25px" altimg-valign="-7px" altimg-width="177px" alttext="\displaystyle=N_{\nu}\left(x\right)\cos\phi_{\nu}\left(x\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><msub><mi href="./10.18#E2">N</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><msub><mi href="./10.18#E3">ϕ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex6" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m19.png" altimg-height="26px" altimg-valign="-7px" altimg-width="58px" alttext="\displaystyle Y_{\nu}'\left(x\right)" display="inline"><mrow><msubsup><mi href="./10.2#E3">Y</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′</mo></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m4.png" altimg-height="25px" altimg-valign="-7px" altimg-width="175px" alttext="\displaystyle=N_{\nu}\left(x\right)\sin\phi_{\nu}\left(x\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><msub><mi href="./10.18#E2">N</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><msub><mi href="./10.18#E3">ϕ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E5.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m34.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./10.2#E3" title="(10.2.3) ‣ Bessel Function of the Second Kind (Weber’s Function) ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m40.png" altimg-height="23px" altimg-valign="-7px" altimg-width="55px" alttext="Y_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E3">Y</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the second kind</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m41.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./10.18#E2" title="(10.18.2) ‣ §10.18(i) Definitions ‣ §10.18 Modulus and Phase Functions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="N_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.18#E2">N</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: modulus of derivatives of Bessel functions</a>,
<a href="./10.18#E3" title="(10.18.3) ‣ §10.18(i) Definitions ‣ §10.18 Modulus and Phase Functions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="23px" altimg-valign="-7px" altimg-width="56px" alttext="\phi_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.18#E3">ϕ</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: phase of derivatives of Bessel functions</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./10.1#p2.t1.r3" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./10.1#p2.t1.r3">x</mi></math>: real variable</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m45.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">9.2.20</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E6" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex7" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">10.18.6</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m15.png" altimg-height="25px" altimg-valign="-7px" altimg-width="65px" alttext="\displaystyle M_{\nu}\left(x\right)" display="inline"><mrow><msub><mi href="./10.18#E1">M</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m8.png" altimg-height="37px" altimg-valign="-9px" altimg-width="196px" alttext="\displaystyle=\left({J_{\nu}^{2}}\left(x\right)+{Y_{\nu}^{2}}\left(x\right)%
\right)^{\frac{1}{2}}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><msup><mrow><mo>(</mo><mrow><mrow><msubsup><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi><mn>2</mn></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow><mo>+</mo><mrow><msubsup><mi href="./10.2#E3">Y</mi><mi href="./10.1#p2.t1.r5">ν</mi><mn>2</mn></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mrow><mo>)</mo></mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></msup></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex8" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m17.png" altimg-height="25px" altimg-valign="-7px" altimg-width="62px" alttext="\displaystyle N_{\nu}\left(x\right)" display="inline"><mrow><msub><mi href="./10.18#E2">N</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m7.png" altimg-height="37px" altimg-valign="-9px" altimg-width="205px" alttext="\displaystyle=\left({J_{\nu}'^{2}}\left(x\right)+{Y_{\nu}'^{2}}\left(x\right)%
\right)^{\frac{1}{2}}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><msup><mrow><mo>(</mo><mrow><mrow><mmultiscripts><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′</mo><none></none><mn>2</mn></mmultiscripts><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow><mo>+</mo><mrow><mmultiscripts><mi href="./10.2#E3">Y</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′</mo><none></none><mn>2</mn></mmultiscripts><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mrow><mo>)</mo></mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></msup></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E6.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m34.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./10.2#E3" title="(10.2.3) ‣ Bessel Function of the Second Kind (Weber’s Function) ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m40.png" altimg-height="23px" altimg-valign="-7px" altimg-width="55px" alttext="Y_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E3">Y</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the second kind</a>,
<a href="./10.18#E1" title="(10.18.1) ‣ §10.18(i) Definitions ‣ §10.18 Modulus and Phase Functions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m35.png" altimg-height="23px" altimg-valign="-7px" altimg-width="64px" alttext="M_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.18#E1">M</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: modulus of Bessel functions</a>,
<a href="./10.18#E2" title="(10.18.2) ‣ §10.18(i) Definitions ‣ §10.18 Modulus and Phase Functions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="N_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.18#E2">N</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: modulus of derivatives of Bessel functions</a>,
<a href="./10.1#p2.t1.r3" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./10.1#p2.t1.r3">x</mi></math>: real variable</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m45.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">9.2.17, 9.2.18</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E7" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex9" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">10.18.7</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m24.png" altimg-height="25px" altimg-valign="-7px" altimg-width="56px" alttext="\displaystyle\theta_{\nu}\left(x\right)" display="inline"><mrow><msub><mi href="./10.18#E3">θ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m10.png" altimg-height="25px" altimg-valign="-7px" altimg-width="231px" alttext="\displaystyle=\operatorname{Arctan}\left(Y_{\nu}\left(x\right)/J_{\nu}\left(x%
\right)\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./4.23#E3">Arctan</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><msub><mi href="./10.2#E3">Y</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow><mo>/</mo><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex10" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m21.png" altimg-height="25px" altimg-valign="-7px" altimg-width="58px" alttext="\displaystyle\phi_{\nu}\left(x\right)" display="inline"><mrow><msub><mi href="./10.18#E3">ϕ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m9.png" altimg-height="26px" altimg-valign="-7px" altimg-width="231px" alttext="\displaystyle=\operatorname{Arctan}\left(Y_{\nu}'\left(x\right)/J_{\nu}'\left(%
x\right)\right)." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./4.23#E3">Arctan</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><msubsup><mi href="./10.2#E3">Y</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′</mo></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow><mo>/</mo><mrow><msubsup><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′</mo></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E7.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m34.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./10.2#E3" title="(10.2.3) ‣ Bessel Function of the Second Kind (Weber’s Function) ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m40.png" altimg-height="23px" altimg-valign="-7px" altimg-width="55px" alttext="Y_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E3">Y</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the second kind</a>,
<a href="./4.23#E3" title="(4.23.3) ‣ §4.23(i) General Definitions ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="18px" altimg-valign="-2px" altimg-width="78px" alttext="\operatorname{Arctan}\NVar{z}" display="inline"><mrow><mi href="./4.23#E3">Arctan</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: general arctangent function</a>,
<a href="./10.18#E3" title="(10.18.3) ‣ §10.18(i) Definitions ‣ §10.18 Modulus and Phase Functions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="23px" altimg-valign="-7px" altimg-width="56px" alttext="\phi_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.18#E3">ϕ</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: phase of derivatives of Bessel functions</a>,
<a href="./10.18#E3" title="(10.18.3) ‣ §10.18(i) Definitions ‣ §10.18 Modulus and Phase Functions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="\theta_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.18#E3">θ</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: phase of Bessel functions</a>,
<a href="./10.1#p2.t1.r3" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./10.1#p2.t1.r3">x</mi></math>: real variable</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m45.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">9.2.17, 9.2.18</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E8" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex11" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">10.18.8</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m28.png" altimg-height="28px" altimg-valign="-7px" altimg-width="119px" alttext="\displaystyle{M_{\nu}^{2}}\left(x\right)\theta_{\nu}'\left(x\right)" display="inline"><mrow><mrow><msubsup><mi href="./10.18#E1">M</mi><mi href="./10.1#p2.t1.r5">ν</mi><mn>2</mn></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><msubsup><mi href="./10.18#E3">θ</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′</mo></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m6.png" altimg-height="46px" altimg-valign="-16px" altimg-width="61px" alttext="\displaystyle=\frac{2}{\pi x}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mstyle displaystyle="true"><mfrac><mn>2</mn><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi href="./10.1#p2.t1.r3">x</mi></mrow></mfrac></mstyle></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex12" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m29.png" altimg-height="28px" altimg-valign="-7px" altimg-width="119px" alttext="\displaystyle{N_{\nu}^{2}}\left(x\right)\phi_{\nu}'\left(x\right)" display="inline"><mrow><mrow><msubsup><mi href="./10.18#E2">N</mi><mi href="./10.1#p2.t1.r5">ν</mi><mn>2</mn></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><msubsup><mi href="./10.18#E3">ϕ</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′</mo></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m5.png" altimg-height="50px" altimg-valign="-16px" altimg-width="128px" alttext="\displaystyle=\frac{2(x^{2}-\nu^{2})}{\pi x^{3}}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mstyle displaystyle="true"><mfrac><mrow><mn>2</mn><mo></mo><mrow><mo stretchy="false">(</mo><mrow><msup><mi href="./10.1#p2.t1.r3">x</mi><mn>2</mn></msup><mo>-</mo><msup><mi href="./10.1#p2.t1.r5">ν</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow></mrow><mrow><mi href="./3.12#E1">π</mi><mo></mo><msup><mi href="./10.1#p2.t1.r3">x</mi><mn>3</mn></msup></mrow></mfrac></mstyle></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E8.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./10.18#E1" title="(10.18.1) ‣ §10.18(i) Definitions ‣ §10.18 Modulus and Phase Functions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m35.png" altimg-height="23px" altimg-valign="-7px" altimg-width="64px" alttext="M_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.18#E1">M</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: modulus of Bessel functions</a>,
<a href="./10.18#E2" title="(10.18.2) ‣ §10.18(i) Definitions ‣ §10.18 Modulus and Phase Functions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="N_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.18#E2">N</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: modulus of derivatives of Bessel functions</a>,
<a href="./10.18#E3" title="(10.18.3) ‣ §10.18(i) Definitions ‣ §10.18 Modulus and Phase Functions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="23px" altimg-valign="-7px" altimg-width="56px" alttext="\phi_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.18#E3">ϕ</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: phase of derivatives of Bessel functions</a>,
<a href="./10.18#E3" title="(10.18.3) ‣ §10.18(i) Definitions ‣ §10.18 Modulus and Phase Functions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="\theta_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.18#E3">θ</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: phase of Bessel functions</a>,
<a href="./10.1#p2.t1.r3" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./10.1#p2.t1.r3">x</mi></math>: real variable</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m45.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">9.2.21</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E9" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.18.9</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E9.png" altimg-height="51px" altimg-valign="-21px" altimg-width="531px" alttext="{N_{\nu}^{2}}\left(x\right)={M_{\nu}'^{2}}\left(x\right)+{M_{\nu}^{2}}\left(x%
\right){\theta_{\nu}'^{2}}\left(x\right)={M_{\nu}'^{2}}\left(x\right)+\frac{4}%
{(\pi xM_{\nu}\left(x\right))^{2}}," display="block"><mrow><mrow><mrow><msubsup><mi href="./10.18#E2">N</mi><mi href="./10.1#p2.t1.r5">ν</mi><mn>2</mn></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mmultiscripts><mi href="./10.18#E1">M</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′</mo><none></none><mn>2</mn></mmultiscripts><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow><mo>+</mo><mrow><mrow><msubsup><mi href="./10.18#E1">M</mi><mi href="./10.1#p2.t1.r5">ν</mi><mn>2</mn></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mmultiscripts><mi href="./10.18#E3">θ</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′</mo><none></none><mn>2</mn></mmultiscripts><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>=</mo><mrow><mrow><mmultiscripts><mi href="./10.18#E1">M</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′</mo><none></none><mn>2</mn></mmultiscripts><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow><mo>+</mo><mfrac><mn>4</mn><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi href="./10.1#p2.t1.r3">x</mi><mo></mo><mrow><msub><mi href="./10.18#E1">M</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mfrac></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E9.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./10.18#E1" title="(10.18.1) ‣ §10.18(i) Definitions ‣ §10.18 Modulus and Phase Functions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m35.png" altimg-height="23px" altimg-valign="-7px" altimg-width="64px" alttext="M_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.18#E1">M</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: modulus of Bessel functions</a>,
<a href="./10.18#E2" title="(10.18.2) ‣ §10.18(i) Definitions ‣ §10.18 Modulus and Phase Functions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="N_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.18#E2">N</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: modulus of derivatives of Bessel functions</a>,
<a href="./10.18#E3" title="(10.18.3) ‣ §10.18(i) Definitions ‣ §10.18 Modulus and Phase Functions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="\theta_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.18#E3">θ</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: phase of Bessel functions</a>,
<a href="./10.1#p2.t1.r3" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./10.1#p2.t1.r3">x</mi></math>: real variable</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m45.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">9.2.22</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E10" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.18.10</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E10.png" altimg-height="28px" altimg-valign="-7px" altimg-width="504px" alttext="(x^{2}-\nu^{2})M_{\nu}\left(x\right)M_{\nu}'\left(x\right)+x^{2}N_{\nu}\left(x%
\right)N_{\nu}'\left(x\right)+x{N_{\nu}^{2}}\left(x\right)=0." display="block"><mrow><mrow><mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><msup><mi href="./10.1#p2.t1.r3">x</mi><mn>2</mn></msup><mo>-</mo><msup><mi href="./10.1#p2.t1.r5">ν</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><msub><mi href="./10.18#E1">M</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><msubsup><mi href="./10.18#E1">M</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′</mo></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mrow><mo>+</mo><mrow><msup><mi href="./10.1#p2.t1.r3">x</mi><mn>2</mn></msup><mo></mo><mrow><msub><mi href="./10.18#E2">N</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><msubsup><mi href="./10.18#E2">N</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′</mo></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mi href="./10.1#p2.t1.r3">x</mi><mo></mo><mrow><msubsup><mi href="./10.18#E2">N</mi><mi href="./10.1#p2.t1.r5">ν</mi><mn>2</mn></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>=</mo><mn>0</mn></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E10.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.18#E1" title="(10.18.1) ‣ §10.18(i) Definitions ‣ §10.18 Modulus and Phase Functions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m35.png" altimg-height="23px" altimg-valign="-7px" altimg-width="64px" alttext="M_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.18#E1">M</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: modulus of Bessel functions</a>,
<a href="./10.18#E2" title="(10.18.2) ‣ §10.18(i) Definitions ‣ §10.18 Modulus and Phase Functions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="N_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.18#E2">N</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: modulus of derivatives of Bessel functions</a>,
<a href="./10.1#p2.t1.r3" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./10.1#p2.t1.r3">x</mi></math>: real variable</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m45.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">9.2.23</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E11" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.18.11</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E11.png" altimg-height="53px" altimg-valign="-21px" altimg-width="506px" alttext="\tan\left(\phi_{\nu}\left(x\right)-\theta_{\nu}\left(x\right)\right)=\frac{M_{%
\nu}\left(x\right)\theta_{\nu}'\left(x\right)}{M_{\nu}'\left(x\right)}=\frac{2%
}{\pi xM_{\nu}\left(x\right)M_{\nu}'\left(x\right)}," display="block"><mrow><mrow><mrow><mi href="./4.14#E4">tan</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><msub><mi href="./10.18#E3">ϕ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow><mo>-</mo><mrow><msub><mi href="./10.18#E3">θ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mfrac><mrow><mrow><msub><mi href="./10.18#E1">M</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><msubsup><mi href="./10.18#E3">θ</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′</mo></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mrow><mrow><msubsup><mi href="./10.18#E1">M</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′</mo></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mfrac><mo>=</mo><mfrac><mn>2</mn><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi href="./10.1#p2.t1.r3">x</mi><mo></mo><mrow><msub><mi href="./10.18#E1">M</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><msubsup><mi href="./10.18#E1">M</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′</mo></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mrow></mfrac></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E11.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./10.18#E1" title="(10.18.1) ‣ §10.18(i) Definitions ‣ §10.18 Modulus and Phase Functions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m35.png" altimg-height="23px" altimg-valign="-7px" altimg-width="64px" alttext="M_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.18#E1">M</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: modulus of Bessel functions</a>,
<a href="./10.18#E3" title="(10.18.3) ‣ §10.18(i) Definitions ‣ §10.18 Modulus and Phase Functions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="23px" altimg-valign="-7px" altimg-width="56px" alttext="\phi_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.18#E3">ϕ</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: phase of derivatives of Bessel functions</a>,
<a href="./10.18#E3" title="(10.18.3) ‣ §10.18(i) Definitions ‣ §10.18 Modulus and Phase Functions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="\theta_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.18#E3">θ</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: phase of Bessel functions</a>,
<a href="./4.14#E4" title="(4.14.4) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="17px" altimg-valign="-2px" altimg-width="47px" alttext="\tan\NVar{z}" display="inline"><mrow><mi href="./4.14#E4">tan</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: tangent function</a>,
<a href="./10.1#p2.t1.r3" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./10.1#p2.t1.r3">x</mi></math>: real variable</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m45.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">9.2.24</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E12" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.18.12</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E12.png" altimg-height="46px" altimg-valign="-16px" altimg-width="358px" alttext="M_{\nu}\left(x\right)N_{\nu}\left(x\right)\sin\left(\phi_{\nu}\left(x\right)-%
\theta_{\nu}\left(x\right)\right)=\frac{2}{\pi x}." display="block"><mrow><mrow><mrow><mrow><msub><mi href="./10.18#E1">M</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./10.18#E2">N</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><msub><mi href="./10.18#E3">ϕ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow><mo>-</mo><mrow><msub><mi href="./10.18#E3">θ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mfrac><mn>2</mn><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi href="./10.1#p2.t1.r3">x</mi></mrow></mfrac></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E12.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./10.18#E1" title="(10.18.1) ‣ §10.18(i) Definitions ‣ §10.18 Modulus and Phase Functions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m35.png" altimg-height="23px" altimg-valign="-7px" altimg-width="64px" alttext="M_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.18#E1">M</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: modulus of Bessel functions</a>,
<a href="./10.18#E2" title="(10.18.2) ‣ §10.18(i) Definitions ‣ §10.18 Modulus and Phase Functions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="N_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.18#E2">N</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: modulus of derivatives of Bessel functions</a>,
<a href="./10.18#E3" title="(10.18.3) ‣ §10.18(i) Definitions ‣ §10.18 Modulus and Phase Functions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="23px" altimg-valign="-7px" altimg-width="56px" alttext="\phi_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.18#E3">ϕ</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: phase of derivatives of Bessel functions</a>,
<a href="./10.18#E3" title="(10.18.3) ‣ §10.18(i) Definitions ‣ §10.18 Modulus and Phase Functions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="\theta_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.18#E3">θ</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: phase of Bessel functions</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./10.1#p2.t1.r3" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./10.1#p2.t1.r3">x</mi></math>: real variable</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m45.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">9.2.24</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E13" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.18.13</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E13.png" altimg-height="51px" altimg-valign="-21px" altimg-width="463px" alttext="x^{2}M_{\nu}''\left(x\right)+xM_{\nu}'\left(x\right)+(x^{2}-\nu^{2})M_{\nu}%
\left(x\right)=\frac{4}{\pi^{2}{{M_{\nu}^{3}}(x)}}," display="block"><mrow><mrow><mrow><mrow><msup><mi href="./10.1#p2.t1.r3">x</mi><mn>2</mn></msup><mo></mo><mrow><msubsup><mi href="./10.18#E1">M</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′′</mo></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mi href="./10.1#p2.t1.r3">x</mi><mo></mo><mrow><msubsup><mi href="./10.18#E1">M</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′</mo></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><msup><mi href="./10.1#p2.t1.r3">x</mi><mn>2</mn></msup><mo>-</mo><msup><mi href="./10.1#p2.t1.r5">ν</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><msub><mi href="./10.18#E1">M</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>=</mo><mfrac><mn>4</mn><mrow><msup><mi href="./3.12#E1">π</mi><mn>2</mn></msup><mo></mo><msubsup><mi href="./10.18#E1">M</mi><mi href="./10.1#p2.t1.r5">ν</mi><mn>3</mn></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo stretchy="false">)</mo></mrow></mrow></mfrac></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E13.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./10.18#E1" title="(10.18.1) ‣ §10.18(i) Definitions ‣ §10.18 Modulus and Phase Functions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m35.png" altimg-height="23px" altimg-valign="-7px" altimg-width="64px" alttext="M_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.18#E1">M</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: modulus of Bessel functions</a>,
<a href="./10.1#p2.t1.r3" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./10.1#p2.t1.r3">x</mi></math>: real variable</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m45.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">9.2.25</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E14" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.18.14</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E14.png" altimg-height="56px" altimg-valign="-21px" altimg-width="281px" alttext="w^{\prime\prime}+\left(1+\frac{\frac{1}{4}-\nu^{2}}{x^{2}}\right)w=\frac{4}{%
\pi^{2}w^{3}}," display="block"><mrow><mrow><mrow><msup><mi>w</mi><mo>′′</mo></msup><mo>+</mo><mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mfrac><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac><mo>-</mo><msup><mi href="./10.1#p2.t1.r5">ν</mi><mn>2</mn></msup></mrow><msup><mi href="./10.1#p2.t1.r3">x</mi><mn>2</mn></msup></mfrac></mrow><mo>)</mo></mrow><mo></mo><mi>w</mi></mrow></mrow><mo>=</mo><mfrac><mn>4</mn><mrow><msup><mi href="./3.12#E1">π</mi><mn>2</mn></msup><mo></mo><msup><mi>w</mi><mn>3</mn></msup></mrow></mfrac></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m60.png" altimg-height="28px" altimg-valign="-7px" altimg-width="129px" alttext="w=x^{\frac{1}{2}}M_{\nu}\left(x\right)" display="inline"><mrow><mi>w</mi><mo>=</mo><mrow><msup><mi href="./10.1#p2.t1.r3">x</mi><mfrac><mn>1</mn><mn>2</mn></mfrac></msup><mo></mo><mrow><msub><mi href="./10.18#E1">M</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E14.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./10.18#E1" title="(10.18.1) ‣ §10.18(i) Definitions ‣ §10.18 Modulus and Phase Functions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m35.png" altimg-height="23px" altimg-valign="-7px" altimg-width="64px" alttext="M_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.18#E1">M</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: modulus of Bessel functions</a>,
<a href="./10.1#p2.t1.r3" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./10.1#p2.t1.r3">x</mi></math>: real variable</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m45.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E15" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.18.15</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E15.png" altimg-height="28px" altimg-valign="-7px" altimg-width="408px" alttext="x^{3}w^{\prime\prime\prime}+x(4x^{2}+1-4\nu^{2})w^{\prime}+(4\nu^{2}-1)w=0," display="block"><mrow><mrow><mrow><mrow><msup><mi href="./10.1#p2.t1.r3">x</mi><mn>3</mn></msup><mo></mo><msup><mi>w</mi><mo>′′′</mo></msup></mrow><mo>+</mo><mrow><mi href="./10.1#p2.t1.r3">x</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mrow><mn>4</mn><mo></mo><msup><mi href="./10.1#p2.t1.r3">x</mi><mn>2</mn></msup></mrow><mo>+</mo><mn>1</mn></mrow><mo>-</mo><mrow><mn>4</mn><mo></mo><msup><mi href="./10.1#p2.t1.r5">ν</mi><mn>2</mn></msup></mrow></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msup><mi>w</mi><mo>′</mo></msup></mrow><mo>+</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>4</mn><mo></mo><msup><mi href="./10.1#p2.t1.r5">ν</mi><mn>2</mn></msup></mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi>w</mi></mrow></mrow><mo>=</mo><mn>0</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m61.png" altimg-height="25px" altimg-valign="-7px" altimg-width="118px" alttext="w=x{M_{\nu}^{2}}\left(x\right)" display="inline"><mrow><mi>w</mi><mo>=</mo><mrow><mi href="./10.1#p2.t1.r3">x</mi><mo></mo><mrow><msubsup><mi href="./10.18#E1">M</mi><mi href="./10.1#p2.t1.r5">ν</mi><mn>2</mn></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E15.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.18#E1" title="(10.18.1) ‣ §10.18(i) Definitions ‣ §10.18 Modulus and Phase Functions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m35.png" altimg-height="23px" altimg-valign="-7px" altimg-width="64px" alttext="M_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.18#E1">M</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: modulus of Bessel functions</a>,
<a href="./10.1#p2.t1.r3" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./10.1#p2.t1.r3">x</mi></math>: real variable</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m45.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">9.2.26</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E16" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.18.16</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E16.png" altimg-height="57px" altimg-valign="-21px" altimg-width="416px" alttext="{\theta_{\nu}'^{2}}\left(x\right)+\frac{1}{2}\frac{\theta_{\nu}'''\left(x%
\right)}{\theta_{\nu}'\left(x\right)}-\frac{3}{4}\left(\frac{\theta_{\nu}''%
\left(x\right)}{\theta_{\nu}'\left(x\right)}\right)^{2}=1-\frac{\nu^{2}-\tfrac%
{1}{4}}{x^{2}}." display="block"><mrow><mrow><mrow><mrow><mrow><mmultiscripts><mi href="./10.18#E3">θ</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′</mo><none></none><mn>2</mn></mmultiscripts><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow><mo>+</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mfrac><mrow><msubsup><mi href="./10.18#E3">θ</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′′′</mo></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow><mrow><msubsup><mi href="./10.18#E3">θ</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′</mo></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mfrac></mrow></mrow><mo>-</mo><mrow><mfrac><mn>3</mn><mn>4</mn></mfrac><mo></mo><msup><mrow><mo>(</mo><mfrac><mrow><msubsup><mi href="./10.18#E3">θ</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′′</mo></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow><mrow><msubsup><mi href="./10.18#E3">θ</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′</mo></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mfrac><mo>)</mo></mrow><mn>2</mn></msup></mrow></mrow><mo>=</mo><mrow><mn>1</mn><mo>-</mo><mfrac><mrow><msup><mi href="./10.1#p2.t1.r5">ν</mi><mn>2</mn></msup><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow><msup><mi href="./10.1#p2.t1.r3">x</mi><mn>2</mn></msup></mfrac></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E16.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.18#E3" title="(10.18.3) ‣ §10.18(i) Definitions ‣ §10.18 Modulus and Phase Functions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="\theta_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.18#E3">θ</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: phase of Bessel functions</a>,
<a href="./10.1#p2.t1.r3" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./10.1#p2.t1.r3">x</mi></math>: real variable</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m45.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">9.2.27</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="iii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§10.18(iii) </span>Asymptotic Expansions for Large Argument</h2>
<div id="SS3.info" class="ltx_metadata ltx_info">
) substitute
into <math class="ltx_Math" altimg="m67.png" altimg-height="32px" altimg-valign="-7px" altimg-width="242px" alttext="{N_{\nu}^{2}}\left(x\right)={H^{(1)}_{\nu}}'\left(x\right){H^{(2)}_{\nu}}'%
\left(x\right)" display="inline"><mrow><mrow><msubsup><mi href="./10.18#E2">N</mi><mi href="./10.1#p2.t1.r5">ν</mi><mn>2</mn></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mmultiscripts><mi href="./10.2#E5">H</mi><mi href="./10.1#p2.t1.r5">ν</mi><mrow><mo href="./10.2#E5" stretchy="false">(</mo><mn>1</mn><mo href="./10.2#E5" stretchy="false">)</mo></mrow><none></none><mo>′</mo></mmultiscripts><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mmultiscripts><mi href="./10.2#E6">H</mi><mi href="./10.1#p2.t1.r5">ν</mi><mrow><mo href="./10.2#E6" stretchy="false">(</mo><mn>2</mn><mo href="./10.2#E6" stretchy="false">)</mo></mrow><none></none><mo>′</mo></mmultiscripts><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mrow></mrow></math> by means of (), except for an arbitrary integer multiple of <math class="ltx_Math" altimg="m50.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>.
Higher terms can be calculated via (). By continuity, the multiple of <math class="ltx_Math" altimg="m50.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math> is independent
of <math class="ltx_Math" altimg="m45.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>, hence it may be determined, e.g. by setting <math class="ltx_Math" altimg="m44.png" altimg-height="27px" altimg-valign="-9px" altimg-width="55px" alttext="\nu=\tfrac{1}{2}" display="inline"><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math> and
referring to (), together with the interlacing properties of the zeros
of <math class="ltx_Math" altimg="m33.png" altimg-height="26px" altimg-valign="-9px" altimg-width="69px" alttext="J_{1/2}\left(z\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>, <math class="ltx_Math" altimg="m39.png" altimg-height="26px" altimg-valign="-9px" altimg-width="70px" alttext="Y_{1/2}\left(z\right)" display="inline"><mrow><msub><mi href="./10.2#E3">Y</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>, and their derivatives
(§</dd>
</dl>
</div>
</div>
<div id="SS3.p1" class="ltx_para">
<p class="ltx_p">As <math class="ltx_Math" altimg="m65.png" altimg-height="13px" altimg-valign="-2px" altimg-width="67px" alttext="x\to\infty" display="inline"><mrow><mi href="./10.1#p2.t1.r3">x</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></math>, with <math class="ltx_Math" altimg="m45.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math> fixed and <math class="ltx_Math" altimg="m43.png" altimg-height="24px" altimg-valign="-6px" altimg-width="73px" alttext="\mu=4\nu^{2}" display="inline"><mrow><mi>μ</mi><mo>=</mo><mrow><mn>4</mn><mo></mo><msup><mi href="./10.1#p2.t1.r5">ν</mi><mn>2</mn></msup></mrow></mrow></math>,</p>
</div>
<div id="SS3.p2" class="ltx_para">
<table id="EGx2" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E17">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.18.17</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m27.png" altimg-height="28px" altimg-valign="-7px" altimg-width="67px" alttext="\displaystyle{M_{\nu}^{2}}\left(x\right)" display="inline"><mrow><msubsup><mi href="./10.18#E1">M</mi><mi href="./10.1#p2.t1.r5">ν</mi><mn>2</mn></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m23.png" altimg-height="53px" altimg-valign="-21px" altimg-width="724px" alttext="\displaystyle\sim\frac{2}{\pi x}\left(1+\frac{1}{2}\frac{\mu-1}{(2x)^{2}}+%
\frac{1\cdot 3}{2\cdot 4}\frac{(\mu-1)(\mu-9)}{(2x)^{4}}+\frac{1\cdot 3\cdot 5%
}{2\cdot 4\cdot 6}\frac{(\mu-1)(\mu-9)(\mu-25)}{(2x)^{6}}+\cdots\right)," display="inline"><mrow><mrow><mi></mi><mo href="./2.1#SS3.p1">∼</mo><mrow><mstyle displaystyle="true"><mfrac><mn>2</mn><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi href="./10.1#p2.t1.r3">x</mi></mrow></mfrac></mstyle><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mrow><mstyle displaystyle="true"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mstyle displaystyle="true"><mfrac><mrow><mi>μ</mi><mo>-</mo><mn>1</mn></mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r3">x</mi></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mfrac></mstyle></mrow><mo>+</mo><mrow><mstyle displaystyle="true"><mfrac><mrow><mn>1</mn><mo>⋅</mo><mn>3</mn></mrow><mrow><mn>2</mn><mo>⋅</mo><mn>4</mn></mrow></mfrac></mstyle><mo></mo><mstyle displaystyle="true"><mfrac><mrow><mrow><mo stretchy="false">(</mo><mrow><mi>μ</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>μ</mi><mo>-</mo><mn>9</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r3">x</mi></mrow><mo stretchy="false">)</mo></mrow><mn>4</mn></msup></mfrac></mstyle></mrow><mo>+</mo><mrow><mstyle displaystyle="true"><mfrac><mrow><mn>1</mn><mo>⋅</mo><mn>3</mn><mo>⋅</mo><mn>5</mn></mrow><mrow><mn>2</mn><mo>⋅</mo><mn>4</mn><mo>⋅</mo><mn>6</mn></mrow></mfrac></mstyle><mo></mo><mstyle displaystyle="true"><mfrac><mrow><mrow><mo stretchy="false">(</mo><mrow><mi>μ</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>μ</mi><mo>-</mo><mn>9</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>μ</mi><mo>-</mo><mn>25</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r3">x</mi></mrow><mo stretchy="false">)</mo></mrow><mn>6</mn></msup></mfrac></mstyle></mrow><mo>+</mo><mi mathvariant="normal">⋯</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E17.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./2.1#SS3.p1" title="§2.1(iii) Asymptotic Expansions ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="12px" altimg-valign="-2px" altimg-width="20px" alttext="\sim" display="inline"><mo href="./2.1#SS3.p1">∼</mo></math>: Poincaré asymptotic expansion</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./10.18#E1" title="(10.18.1) ‣ §10.18(i) Definitions ‣ §10.18 Modulus and Phase Functions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m35.png" altimg-height="23px" altimg-valign="-7px" altimg-width="64px" alttext="M_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.18#E1">M</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: modulus of Bessel functions</a>,
<a href="./10.1#p2.t1.r3" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./10.1#p2.t1.r3">x</mi></math>: real variable</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m45.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">9.2.28</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E18">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.18.18</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m24.png" altimg-height="25px" altimg-valign="-7px" altimg-width="56px" alttext="\displaystyle\theta_{\nu}\left(x\right)" display="inline"><mrow><msub><mi href="./10.18#E3">θ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m22.png" altimg-height="108px" altimg-valign="-21px" altimg-width="669px" alttext="\displaystyle\sim x-\left(\frac{1}{2}\nu+\frac{1}{4}\right)\pi+\frac{\mu-1}{2(%
4x)}+\frac{(\mu-1)(\mu-25)}{6(4x)^{3}}+\frac{(\mu-1)(\mu^{2}-114\mu+1073)}{5(4%
x)^{5}}+\frac{(\mu-1)(5\mu^{3}-1535\mu^{2}+54703\mu-3\;75733)}{14(4x)^{7}}+\cdots." display="inline"><mrow><mi></mi><mo href="./2.1#SS3.p1">∼</mo><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mrow><mi href="./10.1#p2.t1.r3">x</mi><mo>-</mo><mrow><mrow><mo>(</mo><mrow><mrow><mstyle displaystyle="true"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo>+</mo><mstyle displaystyle="true"><mfrac><mn>1</mn><mn>4</mn></mfrac></mstyle></mrow><mo>)</mo></mrow><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow><mo>+</mo><mstyle displaystyle="true"><mfrac><mrow><mi>μ</mi><mo>-</mo><mn>1</mn></mrow><mrow><mn>2</mn><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mn>4</mn><mo></mo><mi href="./10.1#p2.t1.r3">x</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mfrac></mstyle><mo>+</mo><mstyle displaystyle="true"><mfrac><mrow><mrow><mo stretchy="false">(</mo><mrow><mi>μ</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>μ</mi><mo>-</mo><mn>25</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mrow><mn>6</mn><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mn>4</mn><mo></mo><mi href="./10.1#p2.t1.r3">x</mi></mrow><mo stretchy="false">)</mo></mrow><mn>3</mn></msup></mrow></mfrac></mstyle><mo>+</mo><mstyle displaystyle="true"><mfrac><mrow><mrow><mo stretchy="false">(</mo><mrow><mi>μ</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><msup><mi>μ</mi><mn>2</mn></msup><mo>-</mo><mrow><mn>114</mn><mo></mo><mi>μ</mi></mrow></mrow><mo>+</mo><mn>1073</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mrow><mn>5</mn><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mn>4</mn><mo></mo><mi href="./10.1#p2.t1.r3">x</mi></mrow><mo stretchy="false">)</mo></mrow><mn>5</mn></msup></mrow></mfrac></mstyle></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>+</mo><mstyle displaystyle="true"><mfrac><mrow><mrow><mo stretchy="false">(</mo><mrow><mi>μ</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mrow><mrow><mn>5</mn><mo></mo><msup><mi>μ</mi><mn>3</mn></msup></mrow><mo>-</mo><mrow><mn>1535</mn><mo></mo><msup><mi>μ</mi><mn>2</mn></msup></mrow></mrow><mo>+</mo><mrow><mn>54703</mn><mo></mo><mi>μ</mi></mrow></mrow><mo>-</mo><mn>3 75733</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mrow><mn>14</mn><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mn>4</mn><mo></mo><mi href="./10.1#p2.t1.r3">x</mi></mrow><mo stretchy="false">)</mo></mrow><mn>7</mn></msup></mrow></mfrac></mstyle><mo>+</mo><mi mathvariant="normal">⋯</mi><mo>.</mo></mtd></mtr></mtable></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E18.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./2.1#SS3.p1" title="§2.1(iii) Asymptotic Expansions ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="12px" altimg-valign="-2px" altimg-width="20px" alttext="\sim" display="inline"><mo href="./2.1#SS3.p1">∼</mo></math>: Poincaré asymptotic expansion</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./10.18#E3" title="(10.18.3) ‣ §10.18(i) Definitions ‣ §10.18 Modulus and Phase Functions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="\theta_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.18#E3">θ</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: phase of Bessel functions</a>,
<a href="./10.1#p2.t1.r3" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./10.1#p2.t1.r3">x</mi></math>: real variable</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m45.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">9.2.29</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
<p class="ltx_p">Also,</p>
<table id="E19" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.18.19</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E19.png" altimg-height="53px" altimg-valign="-21px" altimg-width="509px" alttext="{N_{\nu}^{2}}\left(x\right)\sim\frac{2}{\pi x}\left(1-\frac{1}{2}\frac{\mu-3}{%
(2x)^{2}}-\frac{1}{2\cdot 4}\frac{(\mu-1)(\mu-45)}{(2x)^{4}}-\cdots\right)," display="block"><mrow><mrow><mrow><msubsup><mi href="./10.18#E2">N</mi><mi href="./10.1#p2.t1.r5">ν</mi><mn>2</mn></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow><mo href="./2.1#SS3.p1">∼</mo><mrow><mfrac><mn>2</mn><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi href="./10.1#p2.t1.r3">x</mi></mrow></mfrac><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mfrac><mrow><mi>μ</mi><mo>-</mo><mn>3</mn></mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r3">x</mi></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mfrac></mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mrow><mn>2</mn><mo>⋅</mo><mn>4</mn></mrow></mfrac><mo></mo><mfrac><mrow><mrow><mo stretchy="false">(</mo><mrow><mi>μ</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>μ</mi><mo>-</mo><mn>45</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r3">x</mi></mrow><mo stretchy="false">)</mo></mrow><mn>4</mn></msup></mfrac></mrow><mo>-</mo><mi mathvariant="normal">⋯</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E19.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./2.1#SS3.p1" title="§2.1(iii) Asymptotic Expansions ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="12px" altimg-valign="-2px" altimg-width="20px" alttext="\sim" display="inline"><mo href="./2.1#SS3.p1">∼</mo></math>: Poincaré asymptotic expansion</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./10.18#E2" title="(10.18.2) ‣ §10.18(i) Definitions ‣ §10.18 Modulus and Phase Functions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="N_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.18#E2">N</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: modulus of derivatives of Bessel functions</a>,
<a href="./10.1#p2.t1.r3" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./10.1#p2.t1.r3">x</mi></math>: real variable</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m45.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">9.2.30</span> (corrected)</span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">the general term in this expansion being</p>
<table id="E20" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.18.20</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E20.png" altimg-height="55px" altimg-valign="-21px" altimg-width="606px" alttext="-\frac{(2k-3)!!}{(2k)!!}\frac{(\mu-1)(\mu-9)\cdots(\mu-(2k-3)^{2})(\mu-(2k+1)(%
2k-1)^{2})}{(2x)^{2k}}," display="block"><mrow><mrow><mo>-</mo><mrow><mfrac><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r2">k</mi></mrow><mo>-</mo><mn>3</mn></mrow><mo stretchy="false">)</mo></mrow><mo href="./front/introduction#Sx4.p1.t1.r15" lspace="0pt" rspace="3.5pt">!!</mo></mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r2">k</mi></mrow><mo stretchy="false">)</mo></mrow><mo href="./front/introduction#Sx4.p1.t1.r15" lspace="0pt" rspace="3.5pt">!!</mo></mrow></mfrac><mo></mo><mfrac><mrow><mrow><mo stretchy="false">(</mo><mrow><mi>μ</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>μ</mi><mo>-</mo><mn>9</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi mathvariant="normal">⋯</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>μ</mi><mo>-</mo><msup><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r2">k</mi></mrow><mo>-</mo><mn>3</mn></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>μ</mi><mo>-</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r2">k</mi></mrow><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r2">k</mi></mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r3">x</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r2">k</mi></mrow></msup></mfrac></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m58.png" altimg-height="20px" altimg-valign="-5px" altimg-width="52px" alttext="k\geq 2" display="inline"><mrow><mi href="./10.1#p2.t1.r2">k</mi><mo>≥</mo><mn>2</mn></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E20.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./front/introduction#Sx4.p1.t1.r15" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m30.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="!!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r15" lspace="0pt" rspace="3.5pt">!!</mo></math>: double factorial (as in <math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="n!!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r15" lspace="0pt" rspace="3.5pt">!!</mo></mrow></math>)</a>,
<a href="./10.1#p2.t1.r2" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m57.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./10.1#p2.t1.r2">k</mi></math>: nonnegative integer</a> and
<a href="./10.1#p2.t1.r3" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./10.1#p2.t1.r3">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and</p>
<table id="E21" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.18.21</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E21.png" altimg-height="54px" altimg-valign="-21px" altimg-width="779px" alttext="\phi_{\nu}\left(x\right)\sim x-\left(\frac{1}{2}\nu-\frac{1}{4}\right)\pi+%
\frac{\mu+3}{2(4x)}+\frac{\mu^{2}+46\mu-63}{6(4x)^{3}}+\frac{\mu^{3}+185\mu^{2%
}-2053\mu+1899}{5(4x)^{5}}+\cdots." display="block"><mrow><mrow><mrow><msub><mi href="./10.18#E3">ϕ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow><mo href="./2.1#SS3.p1">∼</mo><mrow><mrow><mi href="./10.1#p2.t1.r3">x</mi><mo>-</mo><mrow><mrow><mo>(</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow><mo>)</mo></mrow><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow><mo>+</mo><mfrac><mrow><mi>μ</mi><mo>+</mo><mn>3</mn></mrow><mrow><mn>2</mn><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mn>4</mn><mo></mo><mi href="./10.1#p2.t1.r3">x</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mfrac><mo>+</mo><mfrac><mrow><mrow><msup><mi>μ</mi><mn>2</mn></msup><mo>+</mo><mrow><mn>46</mn><mo></mo><mi>μ</mi></mrow></mrow><mo>-</mo><mn>63</mn></mrow><mrow><mn>6</mn><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mn>4</mn><mo></mo><mi href="./10.1#p2.t1.r3">x</mi></mrow><mo stretchy="false">)</mo></mrow><mn>3</mn></msup></mrow></mfrac><mo>+</mo><mfrac><mrow><mrow><mrow><msup><mi>μ</mi><mn>3</mn></msup><mo>+</mo><mrow><mn>185</mn><mo></mo><msup><mi>μ</mi><mn>2</mn></msup></mrow></mrow><mo>-</mo><mrow><mn>2053</mn><mo></mo><mi>μ</mi></mrow></mrow><mo>+</mo><mn>1899</mn></mrow><mrow><mn>5</mn><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mn>4</mn><mo></mo><mi href="./10.1#p2.t1.r3">x</mi></mrow><mo stretchy="false">)</mo></mrow><mn>5</mn></msup></mrow></mfrac><mo>+</mo><mi mathvariant="normal">⋯</mi></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E21.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./2.1#SS3.p1" title="§2.1(iii) Asymptotic Expansions ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="12px" altimg-valign="-2px" altimg-width="20px" alttext="\sim" display="inline"><mo href="./2.1#SS3.p1">∼</mo></math>: Poincaré asymptotic expansion</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./10.18#E3" title="(10.18.3) ‣ §10.18(i) Definitions ‣ §10.18 Modulus and Phase Functions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="23px" altimg-valign="-7px" altimg-width="56px" alttext="\phi_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.18#E3">ϕ</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: phase of derivatives of Bessel functions</a>,
<a href="./10.1#p2.t1.r3" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./10.1#p2.t1.r3">x</mi></math>: real variable</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m45.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">9.2.31</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS3.p3" class="ltx_para">
<p class="ltx_p">The remainder after <math class="ltx_Math" altimg="m57.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./10.1#p2.t1.r2">k</mi></math> terms in () does not exceed the
<math class="ltx_Math" altimg="m32.png" altimg-height="23px" altimg-valign="-7px" altimg-width="65px" alttext="(k+1)" display="inline"><mrow><mo stretchy="false">(</mo><mrow><mi href="./10.1#p2.t1.r2">k</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></math>th term in absolute value and is of the same sign, provided that
<math class="ltx_Math" altimg="m56.png" altimg-height="27px" altimg-valign="-9px" altimg-width="90px" alttext="k>\nu-\tfrac{1}{2}" display="inline"><mrow><mi href="./10.1#p2.t1.r2">k</mi><mo>></mo><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow></math>.</p>
</div>
</section>
</section>
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<span></div>
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<title>DLMF: 10.21 Zeros</title>
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<li class="ltx_tocentry"><a href="#xii"><span class="ltx_tag ltx_tag_subsection">§10.21(xii) </span>Zeros of <math class="ltx_Math" altimg="m168.png" altimg-height="24px" altimg-valign="-7px" altimg-width="155px" alttext="\alpha J_{\nu}\left(x\right)+xJ_{\nu}'\left(x\right)" display="inline"><mrow><mrow><mi>α</mi><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mi href="./10.1#p2.t1.r3">x</mi><mo></mo><mrow><msubsup><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′</mo></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mrow></mrow></math></a></li>
<li class="ltx_tocentry"></li>
<li class="ltx_tocentry"><a href="#xiv"><span class="ltx_tag ltx_tag_subsection">§10.21(xiv) </span><math class="ltx_Math" altimg="m215.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>-Zeros</a></li>
</ul>
<section id="i" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§10.21(i) </span>Distribution</h2>
<div id="SS1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd>
<span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m273.png" altimg-height="22px" altimg-valign="-8px" altimg-width="41px" alttext="j_{\NVar{\nu},\NVar{m}}" display="inline"><msub><mi href="./10.21#SS1.p2">j</mi><mrow><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r1">m</mi></mrow></msub></math>: zeros of the Bessel function <math class="ltx_Math" altimg="m141.png" altimg-height="23px" altimg-valign="-7px" altimg-width="55px" alttext="J_{\nu}\left(x\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math></span>,
<span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m303.png" altimg-height="18px" altimg-valign="-8px" altimg-width="43px" alttext="y_{\NVar{\nu},\NVar{m}}" display="inline"><msub><mi href="./10.21#SS1.p2">y</mi><mrow><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r1">m</mi></mrow></msub></math>: zeros of the Bessel function <math class="ltx_Math" altimg="m157.png" altimg-height="23px" altimg-valign="-7px" altimg-width="56px" alttext="Y_{\nu}\left(x\right)" display="inline"><mrow><msub><mi href="./10.2#E3">Y</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math></span>,
<span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m318.png" altimg-height="26px" altimg-valign="-10px" altimg-width="41px" alttext="{j^{\prime}_{\NVar{\nu},\NVar{m}}}" display="inline"><msubsup><mi href="./10.21#SS1.p2">j</mi><mrow><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r1">m</mi></mrow><mo href="./10.21#SS1.p2">′</mo></msubsup></math>: zeros of the Bessel function derivative <math class="ltx_Math" altimg="m138.png" altimg-height="24px" altimg-valign="-7px" altimg-width="55px" alttext="J_{\nu}'\left(x\right)" display="inline"><mrow><msubsup><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′</mo></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math></span> and
<span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m320.png" altimg-height="26px" altimg-valign="-10px" altimg-width="43px" alttext="{y^{\prime}_{\NVar{\nu},\NVar{m}}}" display="inline"><msubsup><mi href="./10.21#SS1.p2">y</mi><mrow><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r1">m</mi></mrow><mo href="./10.21#SS1.p2">′</mo></msubsup></math>: zeros of the Bessel function derivative <math class="ltx_Math" altimg="m154.png" altimg-height="24px" altimg-valign="-7px" altimg-width="56px" alttext="Y_{\nu}'\left(x\right)" display="inline"><mrow><msubsup><mi href="./10.2#E3">Y</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′</mo></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math></span>
</dd>
<dt>Keywords:</dt>
<dd>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">§9.5</span> (Statements re the zeros of <math class="ltx_Math" altimg="m139.png" altimg-height="24px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\nu}'\left(z\right)" display="inline"><mrow><msubsup><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′</mo></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math> and
<math class="ltx_Math" altimg="m156.png" altimg-height="24px" altimg-valign="-7px" altimg-width="55px" alttext="Y_{\nu}'\left(z\right)" display="inline"><mrow><msubsup><mi href="./10.2#E3">Y</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′</mo></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math> in the first sentence of §9.5 are corrected here.)</span>
</dd>
<dt>Addition (effective with 1.0.5):</dt>
<dd>
The reference to <cite class="ltx_cite ltx_citemacro_citet">Pálmai and Apagyi (</dd>
</dl>
</div>
</div>
<div id="SS1.p1" class="ltx_para">
<p class="ltx_p">The zeros of any cylinder function or its derivative are simple, with the
possible exceptions of <math class="ltx_Math" altimg="m309.png" altimg-height="17px" altimg-valign="-2px" altimg-width="51px" alttext="z=0" display="inline"><mrow><mi href="./10.1#p2.t1.r4">z</mi><mo>=</mo><mn>0</mn></mrow></math> in the case of the functions, and <math class="ltx_Math" altimg="m308.png" altimg-height="20px" altimg-valign="-6px" altimg-width="86px" alttext="z=0,\pm\nu" display="inline"><mrow><mi href="./10.1#p2.t1.r4">z</mi><mo>=</mo><mrow><mn>0</mn><mo>,</mo><mrow><mo>±</mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow></mrow></mrow></math>
in the case of the derivatives.</p>
</div>
<div id="SS1.p2" class="ltx_para">
<p class="ltx_p">If <math class="ltx_Math" altimg="m215.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math> is real, then <math class="ltx_Math" altimg="m142.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\nu}\left(z\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>, <math class="ltx_Math" altimg="m139.png" altimg-height="24px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\nu}'\left(z\right)" display="inline"><mrow><msubsup><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′</mo></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>,
<math class="ltx_Math" altimg="m159.png" altimg-height="23px" altimg-valign="-7px" altimg-width="55px" alttext="Y_{\nu}\left(z\right)" display="inline"><mrow><msub><mi href="./10.2#E3">Y</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>, and <math class="ltx_Math" altimg="m156.png" altimg-height="24px" altimg-valign="-7px" altimg-width="55px" alttext="Y_{\nu}'\left(z\right)" display="inline"><mrow><msubsup><mi href="./10.2#E3">Y</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′</mo></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>, each have an infinite number of
positive real zeros. All of these zeros are simple, provided that <math class="ltx_Math" altimg="m217.png" altimg-height="19px" altimg-valign="-5px" altimg-width="67px" alttext="\nu\geq-1" display="inline"><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>≥</mo><mrow><mo>-</mo><mn>1</mn></mrow></mrow></math>
in the case of <math class="ltx_Math" altimg="m139.png" altimg-height="24px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\nu}'\left(z\right)" display="inline"><mrow><msubsup><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′</mo></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>, and <math class="ltx_Math" altimg="m218.png" altimg-height="27px" altimg-valign="-9px" altimg-width="70px" alttext="\nu\geq-\tfrac{1}{2}" display="inline"><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>≥</mo><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow></math> in the case of
<math class="ltx_Math" altimg="m156.png" altimg-height="24px" altimg-valign="-7px" altimg-width="55px" alttext="Y_{\nu}'\left(z\right)" display="inline"><mrow><msubsup><mi href="./10.2#E3">Y</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′</mo></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>. When all of their zeros are simple, the <math class="ltx_Math" altimg="m284.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./10.1#p2.t1.r1">m</mi></math>th
positive zeros of these functions are
denoted by <math class="ltx_Math" altimg="m277.png" altimg-height="22px" altimg-valign="-8px" altimg-width="40px" alttext="j_{\nu,m}" display="inline"><msub><mi href="./10.21#SS1.p2">j</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi href="./10.1#p2.t1.r1">m</mi></mrow></msub></math>, <math class="ltx_Math" altimg="m319.png" altimg-height="26px" altimg-valign="-10px" altimg-width="40px" alttext="{j^{\prime}_{\nu,m}}" display="inline"><msubsup><mi href="./10.21#SS1.p2">j</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi href="./10.1#p2.t1.r1">m</mi></mrow><mo href="./10.21#SS1.p2">′</mo></msubsup></math>,
<math class="ltx_Math" altimg="m305.png" altimg-height="18px" altimg-valign="-8px" altimg-width="42px" alttext="y_{\nu,m}" display="inline"><msub><mi href="./10.21#SS1.p2">y</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi href="./10.1#p2.t1.r1">m</mi></mrow></msub></math>, and <math class="ltx_Math" altimg="m321.png" altimg-height="26px" altimg-valign="-10px" altimg-width="42px" alttext="{y^{\prime}_{\nu,m}}" display="inline"><msubsup><mi href="./10.21#SS1.p2">y</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi href="./10.1#p2.t1.r1">m</mi></mrow><mo href="./10.21#SS1.p2">′</mo></msubsup></math> respectively, except
that <math class="ltx_Math" altimg="m309.png" altimg-height="17px" altimg-valign="-2px" altimg-width="51px" alttext="z=0" display="inline"><mrow><mi href="./10.1#p2.t1.r4">z</mi><mo>=</mo><mn>0</mn></mrow></math> is counted as the first zero of <math class="ltx_Math" altimg="m129.png" altimg-height="24px" altimg-valign="-7px" altimg-width="53px" alttext="J_{0}'\left(z\right)" display="inline"><mrow><msubsup><mi href="./10.2#E2">J</mi><mn>0</mn><mo>′</mo></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>. Since
<math class="ltx_Math" altimg="m128.png" altimg-height="24px" altimg-valign="-7px" altimg-width="144px" alttext="J_{0}'\left(z\right)=-J_{1}\left(z\right)" display="inline"><mrow><mrow><msubsup><mi href="./10.2#E2">J</mi><mn>0</mn><mo>′</mo></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mo>-</mo><mrow><msub><mi href="./10.2#E2">J</mi><mn>1</mn></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></mrow></math> we have</p>
<table id="E1" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex1" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="3" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">10.21.1</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m100.png" altimg-height="29px" altimg-valign="-10px" altimg-width="36px" alttext="\displaystyle{j^{\prime}_{0,1}}" display="inline"><msubsup><mi href="./10.21#SS1.p2">j</mi><mrow><mn>0</mn><mo href="./10.21#SS1.p2">,</mo><mn>1</mn></mrow><mo href="./10.21#SS1.p2">′</mo></msubsup></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m12.png" altimg-height="22px" altimg-valign="-6px" altimg-width="43px" alttext="\displaystyle=0," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mn>0</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex2" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m101.png" altimg-height="29px" altimg-valign="-10px" altimg-width="42px" alttext="\displaystyle{j^{\prime}_{0,m}}" display="inline"><msubsup><mi href="./10.21#SS1.p2">j</mi><mrow><mn>0</mn><mo href="./10.21#SS1.p2">,</mo><mi href="./10.1#p2.t1.r1">m</mi></mrow><mo href="./10.21#SS1.p2">′</mo></msubsup></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m45.png" altimg-height="24px" altimg-valign="-8px" altimg-width="89px" alttext="\displaystyle=j_{1,m-1}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><msub><mi href="./10.21#SS1.p2">j</mi><mrow><mn>1</mn><mo href="./10.21#SS1.p2">,</mo><mrow><mi href="./10.1#p2.t1.r1">m</mi><mo>-</mo><mn>1</mn></mrow></mrow></msub></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m283.png" altimg-height="20px" altimg-valign="-6px" altimg-width="109px" alttext="m=2,3,\ldots" display="inline"><mrow><mi href="./10.1#p2.t1.r1">m</mi><mo>=</mo><mrow><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.21#SS1.p2" title="§10.21(i) Distribution ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m273.png" altimg-height="22px" altimg-valign="-8px" altimg-width="41px" alttext="j_{\NVar{\nu},\NVar{m}}" display="inline"><msub><mi href="./10.21#SS1.p2">j</mi><mrow><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r1">m</mi></mrow></msub></math>: zeros of the Bessel function <math class="ltx_Math" altimg="m141.png" altimg-height="23px" altimg-valign="-7px" altimg-width="55px" alttext="J_{\nu}\left(x\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math></a>,
<a href="./10.21#SS1.p2" title="§10.21(i) Distribution ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m318.png" altimg-height="26px" altimg-valign="-10px" altimg-width="41px" alttext="{j^{\prime}_{\NVar{\nu},\NVar{m}}}" display="inline"><msubsup><mi href="./10.21#SS1.p2">j</mi><mrow><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r1">m</mi></mrow><mo href="./10.21#SS1.p2">′</mo></msubsup></math>: zeros of the Bessel function derivative <math class="ltx_Math" altimg="m138.png" altimg-height="24px" altimg-valign="-7px" altimg-width="55px" alttext="J_{\nu}'\left(x\right)" display="inline"><mrow><msubsup><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′</mo></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math></a> and
<a href="./10.1#p2.t1.r1" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m284.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./10.1#p2.t1.r1">m</mi></math>: integer</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">9.5.1</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS1.p3" class="ltx_para">
<p class="ltx_p">When <math class="ltx_Math" altimg="m216.png" altimg-height="19px" altimg-valign="-5px" altimg-width="52px" alttext="\nu\geq 0" display="inline"><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>≥</mo><mn>0</mn></mrow></math>, the zeros interlace according to the inequalities
</p>
<table id="E2" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex3" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">10.21.2</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m93.png" altimg-height="24px" altimg-valign="-8px" altimg-width="36px" alttext="\displaystyle j_{\nu,1}" display="inline"><msub><mi href="./10.21#SS1.p2">j</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mn>1</mn></mrow></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m1.png" altimg-height="24px" altimg-valign="-8px" altimg-width="327px" alttext="\displaystyle<j_{\nu+1,1}<j_{\nu,2}<j_{\nu+1,2}<j_{\nu,3}<\cdots," display="inline"><mrow><mrow><mi></mi><mo><</mo><msub><mi href="./10.21#SS1.p2">j</mi><mrow><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mn>1</mn></mrow><mo href="./10.21#SS1.p2">,</mo><mn>1</mn></mrow></msub><mo><</mo><msub><mi href="./10.21#SS1.p2">j</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mn>2</mn></mrow></msub><mo><</mo><msub><mi href="./10.21#SS1.p2">j</mi><mrow><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mn>1</mn></mrow><mo href="./10.21#SS1.p2">,</mo><mn>2</mn></mrow></msub><mo><</mo><msub><mi href="./10.21#SS1.p2">j</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mn>3</mn></mrow></msub><mo><</mo><mi mathvariant="normal">⋯</mi></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex4" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m98.png" altimg-height="20px" altimg-valign="-8px" altimg-width="38px" alttext="\displaystyle y_{\nu,1}" display="inline"><msub><mi href="./10.21#SS1.p2">y</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mn>1</mn></mrow></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m2.png" altimg-height="22px" altimg-valign="-8px" altimg-width="334px" alttext="\displaystyle<y_{\nu+1,1}<y_{\nu,2}<y_{\nu+1,2}<y_{\nu,3}<\cdots," display="inline"><mrow><mrow><mi></mi><mo><</mo><msub><mi href="./10.21#SS1.p2">y</mi><mrow><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mn>1</mn></mrow><mo href="./10.21#SS1.p2">,</mo><mn>1</mn></mrow></msub><mo><</mo><msub><mi href="./10.21#SS1.p2">y</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mn>2</mn></mrow></msub><mo><</mo><msub><mi href="./10.21#SS1.p2">y</mi><mrow><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mn>1</mn></mrow><mo href="./10.21#SS1.p2">,</mo><mn>2</mn></mrow></msub><mo><</mo><msub><mi href="./10.21#SS1.p2">y</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mn>3</mn></mrow></msub><mo><</mo><mi mathvariant="normal">⋯</mi></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.21#SS1.p2" title="§10.21(i) Distribution ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m273.png" altimg-height="22px" altimg-valign="-8px" altimg-width="41px" alttext="j_{\NVar{\nu},\NVar{m}}" display="inline"><msub><mi href="./10.21#SS1.p2">j</mi><mrow><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r1">m</mi></mrow></msub></math>: zeros of the Bessel function <math class="ltx_Math" altimg="m141.png" altimg-height="23px" altimg-valign="-7px" altimg-width="55px" alttext="J_{\nu}\left(x\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math></a>,
<a href="./10.21#SS1.p2" title="§10.21(i) Distribution ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m303.png" altimg-height="18px" altimg-valign="-8px" altimg-width="43px" alttext="y_{\NVar{\nu},\NVar{m}}" display="inline"><msub><mi href="./10.21#SS1.p2">y</mi><mrow><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r1">m</mi></mrow></msub></math>: zeros of the Bessel function <math class="ltx_Math" altimg="m157.png" altimg-height="23px" altimg-valign="-7px" altimg-width="56px" alttext="Y_{\nu}\left(x\right)" display="inline"><mrow><msub><mi href="./10.2#E3">Y</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math></a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m215.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">9.5.2</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E3" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.21.3</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E3.png" altimg-height="29px" altimg-valign="-10px" altimg-width="420px" alttext="\nu\leq{j^{\prime}_{\nu,1}}<y_{\nu,1}<{y^{\prime}_{\nu,1}}<j_{\nu,1}<{j^{%
\prime}_{\nu,2}}<y_{\nu,2}<\cdots." display="block"><mrow><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>≤</mo><msubsup><mi href="./10.21#SS1.p2">j</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mn>1</mn></mrow><mo href="./10.21#SS1.p2">′</mo></msubsup><mo><</mo><msub><mi href="./10.21#SS1.p2">y</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mn>1</mn></mrow></msub><mo><</mo><msubsup><mi href="./10.21#SS1.p2">y</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mn>1</mn></mrow><mo href="./10.21#SS1.p2">′</mo></msubsup><mo><</mo><msub><mi href="./10.21#SS1.p2">j</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mn>1</mn></mrow></msub><mo><</mo><msubsup><mi href="./10.21#SS1.p2">j</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mn>2</mn></mrow><mo href="./10.21#SS1.p2">′</mo></msubsup><mo><</mo><msub><mi href="./10.21#SS1.p2">y</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mn>2</mn></mrow></msub><mo><</mo><mi mathvariant="normal">⋯</mi></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.21#SS1.p2" title="§10.21(i) Distribution ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m273.png" altimg-height="22px" altimg-valign="-8px" altimg-width="41px" alttext="j_{\NVar{\nu},\NVar{m}}" display="inline"><msub><mi href="./10.21#SS1.p2">j</mi><mrow><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r1">m</mi></mrow></msub></math>: zeros of the Bessel function <math class="ltx_Math" altimg="m141.png" altimg-height="23px" altimg-valign="-7px" altimg-width="55px" alttext="J_{\nu}\left(x\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math></a>,
<a href="./10.21#SS1.p2" title="§10.21(i) Distribution ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m303.png" altimg-height="18px" altimg-valign="-8px" altimg-width="43px" alttext="y_{\NVar{\nu},\NVar{m}}" display="inline"><msub><mi href="./10.21#SS1.p2">y</mi><mrow><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r1">m</mi></mrow></msub></math>: zeros of the Bessel function <math class="ltx_Math" altimg="m157.png" altimg-height="23px" altimg-valign="-7px" altimg-width="56px" alttext="Y_{\nu}\left(x\right)" display="inline"><mrow><msub><mi href="./10.2#E3">Y</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math></a>,
<a href="./10.21#SS1.p2" title="§10.21(i) Distribution ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m318.png" altimg-height="26px" altimg-valign="-10px" altimg-width="41px" alttext="{j^{\prime}_{\NVar{\nu},\NVar{m}}}" display="inline"><msubsup><mi href="./10.21#SS1.p2">j</mi><mrow><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r1">m</mi></mrow><mo href="./10.21#SS1.p2">′</mo></msubsup></math>: zeros of the Bessel function derivative <math class="ltx_Math" altimg="m138.png" altimg-height="24px" altimg-valign="-7px" altimg-width="55px" alttext="J_{\nu}'\left(x\right)" display="inline"><mrow><msubsup><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′</mo></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math></a>,
<a href="./10.21#SS1.p2" title="§10.21(i) Distribution ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m320.png" altimg-height="26px" altimg-valign="-10px" altimg-width="43px" alttext="{y^{\prime}_{\NVar{\nu},\NVar{m}}}" display="inline"><msubsup><mi href="./10.21#SS1.p2">y</mi><mrow><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r1">m</mi></mrow><mo href="./10.21#SS1.p2">′</mo></msubsup></math>: zeros of the Bessel function derivative <math class="ltx_Math" altimg="m154.png" altimg-height="24px" altimg-valign="-7px" altimg-width="56px" alttext="Y_{\nu}'\left(x\right)" display="inline"><mrow><msubsup><mi href="./10.2#E3">Y</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′</mo></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math></a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m215.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">9.5.2</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS1.p4" class="ltx_para">
<p class="ltx_p">For an extension see <cite class="ltx_cite ltx_citemacro_citet">Pálmai and Apagyi ()</cite>.</p>
</div>
<div id="SS1.p5" class="ltx_para">
<p class="ltx_p">The positive zeros of any two real distinct cylinder functions of the same
order are interlaced, as are the positive zeros of any real cylinder function
<math class="ltx_Math" altimg="m208.png" altimg-height="23px" altimg-valign="-7px" altimg-width="56px" alttext="\mathscr{C}_{\nu}\left(z\right)" display="inline"><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5.p1">𝒞</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math> and the contiguous function <math class="ltx_Math" altimg="m204.png" altimg-height="23px" altimg-valign="-7px" altimg-width="77px" alttext="\mathscr{C}_{\nu+1}\left(z\right)" display="inline"><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5.p1">𝒞</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>. See
also <cite class="ltx_cite ltx_citemacro_citet">Elbert and Laforgia ()</cite>.
</p>
</div>
<div id="SS1.p6" class="ltx_para">
<p class="ltx_p">When <math class="ltx_Math" altimg="m217.png" altimg-height="19px" altimg-valign="-5px" altimg-width="67px" alttext="\nu\geq-1" display="inline"><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>≥</mo><mrow><mo>-</mo><mn>1</mn></mrow></mrow></math> the zeros of <math class="ltx_Math" altimg="m142.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\nu}\left(z\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math> are all real. If <math class="ltx_Math" altimg="m211.png" altimg-height="18px" altimg-valign="-4px" altimg-width="67px" alttext="\nu<-1" display="inline"><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo><</mo><mrow><mo>-</mo><mn>1</mn></mrow></mrow></math>
and <math class="ltx_Math" altimg="m215.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math> is not an integer, then the number of complex zeros of
<math class="ltx_Math" altimg="m142.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\nu}\left(z\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math> is <math class="ltx_Math" altimg="m121.png" altimg-height="23px" altimg-valign="-7px" altimg-width="62px" alttext="2\left\lfloor-\nu\right\rfloor" display="inline"><mrow><mn>2</mn><mo></mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r16">⌊</mo><mrow><mo>-</mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo href="./front/introduction#Sx4.p1.t1.r16">⌋</mo></mrow></mrow></math>. If <math class="ltx_Math" altimg="m186.png" altimg-height="23px" altimg-valign="-7px" altimg-width="49px" alttext="\left\lfloor-\nu\right\rfloor" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r16">⌊</mo><mrow><mo>-</mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo href="./front/introduction#Sx4.p1.t1.r16">⌋</mo></mrow></math> is odd, then two of
these zeros lie on the imaginary axis.
</p>
</div>
<div id="SS1.p7" class="ltx_para">
<p class="ltx_p">If <math class="ltx_Math" altimg="m216.png" altimg-height="19px" altimg-valign="-5px" altimg-width="52px" alttext="\nu\geq 0" display="inline"><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>≥</mo><mn>0</mn></mrow></math>, then the zeros of <math class="ltx_Math" altimg="m139.png" altimg-height="24px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\nu}'\left(z\right)" display="inline"><mrow><msubsup><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′</mo></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math> are all real.</p>
</div>
<div id="SS1.p8" class="ltx_para">
<p class="ltx_p">For information on the real double zeros of <math class="ltx_Math" altimg="m139.png" altimg-height="24px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\nu}'\left(z\right)" display="inline"><mrow><msubsup><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′</mo></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math> and
<math class="ltx_Math" altimg="m156.png" altimg-height="24px" altimg-valign="-7px" altimg-width="55px" alttext="Y_{\nu}'\left(z\right)" display="inline"><mrow><msubsup><mi href="./10.2#E3">Y</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′</mo></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math> when <math class="ltx_Math" altimg="m211.png" altimg-height="18px" altimg-valign="-4px" altimg-width="67px" alttext="\nu<-1" display="inline"><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo><</mo><mrow><mo>-</mo><mn>1</mn></mrow></mrow></math> and <math class="ltx_Math" altimg="m212.png" altimg-height="27px" altimg-valign="-9px" altimg-width="70px" alttext="\nu<-\tfrac{1}{2}" display="inline"><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo><</mo><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow></math>, respectively, see
<cite class="ltx_cite ltx_citemacro_cite">Döring ()</cite>. The latter reference also
has information on double zeros of the second and third derivatives of
<math class="ltx_Math" altimg="m142.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\nu}\left(z\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math> and <math class="ltx_Math" altimg="m159.png" altimg-height="23px" altimg-valign="-7px" altimg-width="55px" alttext="Y_{\nu}\left(z\right)" display="inline"><mrow><msub><mi href="./10.2#E3">Y</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>.
</p>
</div>
<div id="SS1.p9" class="ltx_para">
<p class="ltx_p">No two of the functions <math class="ltx_Math" altimg="m131.png" altimg-height="23px" altimg-valign="-7px" altimg-width="53px" alttext="J_{0}\left(z\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mn>0</mn></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>, <math class="ltx_Math" altimg="m132.png" altimg-height="23px" altimg-valign="-7px" altimg-width="53px" alttext="J_{1}\left(z\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mn>1</mn></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>,
<math class="ltx_Math" altimg="m133.png" altimg-height="23px" altimg-valign="-7px" altimg-width="89px" alttext="J_{2}\left(z\right),\ldots" display="inline"><mrow><mrow><msub><mi href="./10.2#E2">J</mi><mn>2</mn></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow><mo>,</mo><mi mathvariant="normal">…</mi></mrow></math>, have any common zeros other than <math class="ltx_Math" altimg="m309.png" altimg-height="17px" altimg-valign="-2px" altimg-width="51px" alttext="z=0" display="inline"><mrow><mi href="./10.1#p2.t1.r4">z</mi><mo>=</mo><mn>0</mn></mrow></math>; see
<cite class="ltx_cite ltx_citemacro_citet">Watson ()</cite>.
(In the latter reference <math class="ltx_Math" altimg="m297.png" altimg-height="17px" altimg-valign="-2px" altimg-width="11px" alttext="t" display="inline"><mi>t</mi></math> in () is replaced by <math class="ltx_Math" altimg="m117.png" altimg-height="18px" altimg-valign="-4px" altimg-width="27px" alttext="-t" display="inline"><mrow><mo>-</mo><mi>t</mi></mrow></math>.)
(), and the fact that <math class="ltx_Math" altimg="m240.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="\theta_{\nu}\left(x\right)" display="inline"><mrow><msub><mi href="./10.18#E3">θ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math> is increasing
when <math class="ltx_Math" altimg="m299.png" altimg-height="17px" altimg-valign="-3px" altimg-width="52px" alttext="x>0" display="inline"><mrow><mi href="./10.1#p2.t1.r3">x</mi><mo>></mo><mn>0</mn></mrow></math>, whereas <math class="ltx_Math" altimg="m225.png" altimg-height="23px" altimg-valign="-7px" altimg-width="56px" alttext="\phi_{\nu}\left(x\right)" display="inline"><mrow><msub><mi href="./10.18#E3">ϕ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math> is decreasing when <math class="ltx_Math" altimg="m119.png" altimg-height="17px" altimg-valign="-3px" altimg-width="90px" alttext="0<x<\nu" display="inline"><mrow><mn>0</mn><mo><</mo><mi href="./10.1#p2.t1.r3">x</mi><mo><</mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow></math> and
increasing when <math class="ltx_Math" altimg="m300.png" altimg-height="15px" altimg-valign="-3px" altimg-width="53px" alttext="x>\nu" display="inline"><mrow><mi href="./10.1#p2.t1.r3">x</mi><mo>></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow></math>; compare (</dd>
</dl>
</div>
</div>
<div id="SS2.p1" class="ltx_para">
<p class="ltx_p">If <math class="ltx_Math" altimg="m229.png" altimg-height="16px" altimg-valign="-6px" altimg-width="25px" alttext="\rho_{\nu}" display="inline"><msub><mi href="./10.21#SS2.p1">ρ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub></math> is a zero of the cylinder function
</p>
<table id="E4" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.21.4</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E4.png" altimg-height="25px" altimg-valign="-7px" altimg-width="352px" alttext="\mathscr{C}_{\nu}\left(z\right)=J_{\nu}\left(z\right)\cos\left(\pi t\right)+Y_%
{\nu}\left(z\right)\sin\left(\pi t\right)," display="block"><mrow><mrow><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5.p1">𝒞</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi>t</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mrow><msub><mi href="./10.2#E3">Y</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi>t</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E4.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m134.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./10.2#E3" title="(10.2.3) ‣ Bessel Function of the Second Kind (Weber’s Function) ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m153.png" altimg-height="23px" altimg-valign="-7px" altimg-width="55px" alttext="Y_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E3">Y</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the second kind</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m226.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m181.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./10.2#Px5.p1" title="Cylinder Functions ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m203.png" altimg-height="23px" altimg-valign="-7px" altimg-width="56px" alttext="\mathscr{C}_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5.p1">𝒞</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: cylinder function</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m235.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./10.1#p2.t1.r4" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m312.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./10.1#p2.t1.r4">z</mi></math>: complex variable</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m215.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">9.5.3</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m297.png" altimg-height="17px" altimg-valign="-2px" altimg-width="11px" alttext="t" display="inline"><mi>t</mi></math> is a parameter, then</p>
<table id="E5" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.21.5</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E5.png" altimg-height="26px" altimg-valign="-7px" altimg-width="311px" alttext="\mathscr{C}_{\nu}'\left(\rho_{\nu}\right)=\mathscr{C}_{\nu-1}\left(\rho_{\nu}%
\right)=-\mathscr{C}_{\nu+1}\left(\rho_{\nu}\right)." display="block"><mrow><mrow><mrow><msubsup><mi class="ltx_font_mathscript" href="./10.2#Px5.p1">𝒞</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′</mo></msubsup><mo></mo><mrow><mo>(</mo><msub><mi href="./10.21#SS2.p1">ρ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5.p1">𝒞</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>-</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><msub><mi href="./10.21#SS2.p1">ρ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mo>-</mo><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5.p1">𝒞</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><msub><mi href="./10.21#SS2.p1">ρ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo>)</mo></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E5.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#Px5.p1" title="Cylinder Functions ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m203.png" altimg-height="23px" altimg-valign="-7px" altimg-width="56px" alttext="\mathscr{C}_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5.p1">𝒞</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: cylinder function</a>,
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m215.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a> and
<a href="./10.21#SS2.p1" title="§10.21(ii) Analytic Properties ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m228.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="\rho_{\nu}(t)" display="inline"><mrow><msub><mi href="./10.21#SS2.p1">ρ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math>: zero of cylinder function</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">9.5.4</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">If <math class="ltx_Math" altimg="m232.png" altimg-height="16px" altimg-valign="-5px" altimg-width="26px" alttext="\sigma_{\nu}" display="inline"><msub><mi href="./10.21#SS2.p1">σ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub></math> is a zero of <math class="ltx_Math" altimg="m206.png" altimg-height="24px" altimg-valign="-7px" altimg-width="56px" alttext="\mathscr{C}_{\nu}'\left(z\right)" display="inline"><mrow><msubsup><mi class="ltx_font_mathscript" href="./10.2#Px5.p1">𝒞</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′</mo></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>, then
</p>
<table id="E6" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.21.6</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E6.png" altimg-height="41px" altimg-valign="-16px" altimg-width="351px" alttext="\mathscr{C}_{\nu}\left(\sigma_{\nu}\right)=\frac{\sigma_{\nu}}{\nu}\mathscr{C}%
_{\nu-1}\left(\sigma_{\nu}\right)=\frac{\sigma_{\nu}}{\nu}\mathscr{C}_{\nu+1}%
\left(\sigma_{\nu}\right)." display="block"><mrow><mrow><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5.p1">𝒞</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><msub><mi href="./10.21#SS2.p1">σ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mfrac><msub><mi href="./10.21#SS2.p1">σ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mi href="./10.1#p2.t1.r5">ν</mi></mfrac><mo></mo><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5.p1">𝒞</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>-</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><msub><mi href="./10.21#SS2.p1">σ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mrow><mfrac><msub><mi href="./10.21#SS2.p1">σ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mi href="./10.1#p2.t1.r5">ν</mi></mfrac><mo></mo><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5.p1">𝒞</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><msub><mi href="./10.21#SS2.p1">σ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo>)</mo></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E6.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#Px5.p1" title="Cylinder Functions ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m203.png" altimg-height="23px" altimg-valign="-7px" altimg-width="56px" alttext="\mathscr{C}_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5.p1">𝒞</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: cylinder function</a>,
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m215.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a> and
<a href="./10.21#SS2.p1" title="§10.21(ii) Analytic Properties ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m231.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="\sigma_{\nu}(t)" display="inline"><mrow><msub><mi href="./10.21#SS2.p1">σ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math>: zero of cylinder function</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">9.5.5</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS2.p2" class="ltx_para">
<p class="ltx_p">The parameter <math class="ltx_Math" altimg="m297.png" altimg-height="17px" altimg-valign="-2px" altimg-width="11px" alttext="t" display="inline"><mi>t</mi></math> may be regarded as a continuous variable and <math class="ltx_Math" altimg="m229.png" altimg-height="16px" altimg-valign="-6px" altimg-width="25px" alttext="\rho_{\nu}" display="inline"><msub><mi href="./10.21#SS2.p1">ρ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub></math>,
<math class="ltx_Math" altimg="m232.png" altimg-height="16px" altimg-valign="-5px" altimg-width="26px" alttext="\sigma_{\nu}" display="inline"><msub><mi href="./10.21#SS2.p1">σ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub></math> as functions <math class="ltx_Math" altimg="m228.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="\rho_{\nu}(t)" display="inline"><mrow><msub><mi href="./10.21#SS2.p1">ρ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math>, <math class="ltx_Math" altimg="m231.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="\sigma_{\nu}(t)" display="inline"><mrow><msub><mi href="./10.21#SS2.p1">σ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math> of <math class="ltx_Math" altimg="m297.png" altimg-height="17px" altimg-valign="-2px" altimg-width="11px" alttext="t" display="inline"><mi>t</mi></math>. If <math class="ltx_Math" altimg="m216.png" altimg-height="19px" altimg-valign="-5px" altimg-width="52px" alttext="\nu\geq 0" display="inline"><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>≥</mo><mn>0</mn></mrow></math>
and these functions are fixed by</p>
<table id="E7" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex5" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">10.21.7</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m81.png" altimg-height="25px" altimg-valign="-7px" altimg-width="52px" alttext="\displaystyle\rho_{\nu}(0)" display="inline"><mrow><msub><mi href="./10.21#SS2.p1">ρ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m12.png" altimg-height="22px" altimg-valign="-6px" altimg-width="43px" alttext="\displaystyle=0," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mn>0</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex6" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m82.png" altimg-height="25px" altimg-valign="-7px" altimg-width="53px" alttext="\displaystyle\sigma_{\nu}(0)" display="inline"><mrow><msub><mi href="./10.21#SS2.p1">σ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m47.png" altimg-height="29px" altimg-valign="-10px" altimg-width="63px" alttext="\displaystyle={j^{\prime}_{\nu,1}}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><msubsup><mi href="./10.21#SS1.p2">j</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mn>1</mn></mrow><mo href="./10.21#SS1.p2">′</mo></msubsup></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E7.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.21#SS1.p2" title="§10.21(i) Distribution ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m318.png" altimg-height="26px" altimg-valign="-10px" altimg-width="41px" alttext="{j^{\prime}_{\NVar{\nu},\NVar{m}}}" display="inline"><msubsup><mi href="./10.21#SS1.p2">j</mi><mrow><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r1">m</mi></mrow><mo href="./10.21#SS1.p2">′</mo></msubsup></math>: zeros of the Bessel function derivative <math class="ltx_Math" altimg="m138.png" altimg-height="24px" altimg-valign="-7px" altimg-width="55px" alttext="J_{\nu}'\left(x\right)" display="inline"><mrow><msubsup><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′</mo></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math></a>,
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m215.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>,
<a href="./10.21#SS2.p1" title="§10.21(ii) Analytic Properties ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m228.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="\rho_{\nu}(t)" display="inline"><mrow><msub><mi href="./10.21#SS2.p1">ρ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math>: zero of cylinder function</a> and
<a href="./10.21#SS2.p1" title="§10.21(ii) Analytic Properties ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m231.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="\sigma_{\nu}(t)" display="inline"><mrow><msub><mi href="./10.21#SS2.p1">σ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math>: zero of cylinder function</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">9.5.6</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">then</p>
<table id="E8" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex7" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="3" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">10.21.8</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m94.png" altimg-height="24px" altimg-valign="-8px" altimg-width="42px" alttext="\displaystyle j_{\nu,m}" display="inline"><msub><mi href="./10.21#SS1.p2">j</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi href="./10.1#p2.t1.r1">m</mi></mrow></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m33.png" altimg-height="25px" altimg-valign="-7px" altimg-width="86px" alttext="\displaystyle=\rho_{\nu}(m)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msub><mi href="./10.21#SS2.p1">ρ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./10.1#p2.t1.r1">m</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex8" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m99.png" altimg-height="20px" altimg-valign="-8px" altimg-width="44px" alttext="\displaystyle y_{\nu,m}" display="inline"><msub><mi href="./10.21#SS1.p2">y</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi href="./10.1#p2.t1.r1">m</mi></mrow></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell">
<math class="ltx_Math" altimg="m34.png" altimg-height="29px" altimg-valign="-9px" altimg-width="117px" alttext="\displaystyle=\rho_{\nu}(m-\tfrac{1}{2})" display="inline"><mrow><mi></mi><mo>=</mo><mrow><msub><mi href="./10.21#SS2.p1">ρ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./10.1#p2.t1.r1">m</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></math>,</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m281.png" altimg-height="20px" altimg-valign="-6px" altimg-width="109px" alttext="m=1,2,\ldots" display="inline"><mrow><mi href="./10.1#p2.t1.r1">m</mi><mo>=</mo><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E8.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.21#SS1.p2" title="§10.21(i) Distribution ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m273.png" altimg-height="22px" altimg-valign="-8px" altimg-width="41px" alttext="j_{\NVar{\nu},\NVar{m}}" display="inline"><msub><mi href="./10.21#SS1.p2">j</mi><mrow><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r1">m</mi></mrow></msub></math>: zeros of the Bessel function <math class="ltx_Math" altimg="m141.png" altimg-height="23px" altimg-valign="-7px" altimg-width="55px" alttext="J_{\nu}\left(x\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math></a>,
<a href="./10.21#SS1.p2" title="§10.21(i) Distribution ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m303.png" altimg-height="18px" altimg-valign="-8px" altimg-width="43px" alttext="y_{\NVar{\nu},\NVar{m}}" display="inline"><msub><mi href="./10.21#SS1.p2">y</mi><mrow><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r1">m</mi></mrow></msub></math>: zeros of the Bessel function <math class="ltx_Math" altimg="m157.png" altimg-height="23px" altimg-valign="-7px" altimg-width="56px" alttext="Y_{\nu}\left(x\right)" display="inline"><mrow><msub><mi href="./10.2#E3">Y</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math></a>,
<a href="./10.1#p2.t1.r1" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m284.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./10.1#p2.t1.r1">m</mi></math>: integer</a>,
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m215.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a> and
<a href="./10.21#SS2.p1" title="§10.21(ii) Analytic Properties ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m228.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="\rho_{\nu}(t)" display="inline"><mrow><msub><mi href="./10.21#SS2.p1">ρ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math>: zero of cylinder function</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">9.5.7</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E9" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex9" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="3" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">10.21.9</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m103.png" altimg-height="29px" altimg-valign="-10px" altimg-width="42px" alttext="\displaystyle{j^{\prime}_{\nu,m}}" display="inline"><msubsup><mi href="./10.21#SS1.p2">j</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi href="./10.1#p2.t1.r1">m</mi></mrow><mo href="./10.21#SS1.p2">′</mo></msubsup></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m36.png" altimg-height="25px" altimg-valign="-7px" altimg-width="121px" alttext="\displaystyle=\sigma_{\nu}(m-1)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msub><mi href="./10.21#SS2.p1">σ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./10.1#p2.t1.r1">m</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex10" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m105.png" altimg-height="29px" altimg-valign="-10px" altimg-width="44px" alttext="\displaystyle{y^{\prime}_{\nu,m}}" display="inline"><msubsup><mi href="./10.21#SS1.p2">y</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi href="./10.1#p2.t1.r1">m</mi></mrow><mo href="./10.21#SS1.p2">′</mo></msubsup></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell">
<math class="ltx_Math" altimg="m37.png" altimg-height="29px" altimg-valign="-9px" altimg-width="118px" alttext="\displaystyle=\sigma_{\nu}(m-\tfrac{1}{2})" display="inline"><mrow><mi></mi><mo>=</mo><mrow><msub><mi href="./10.21#SS2.p1">σ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./10.1#p2.t1.r1">m</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></math>,</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m281.png" altimg-height="20px" altimg-valign="-6px" altimg-width="109px" alttext="m=1,2,\ldots" display="inline"><mrow><mi href="./10.1#p2.t1.r1">m</mi><mo>=</mo><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E9.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.21#SS1.p2" title="§10.21(i) Distribution ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m318.png" altimg-height="26px" altimg-valign="-10px" altimg-width="41px" alttext="{j^{\prime}_{\NVar{\nu},\NVar{m}}}" display="inline"><msubsup><mi href="./10.21#SS1.p2">j</mi><mrow><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r1">m</mi></mrow><mo href="./10.21#SS1.p2">′</mo></msubsup></math>: zeros of the Bessel function derivative <math class="ltx_Math" altimg="m138.png" altimg-height="24px" altimg-valign="-7px" altimg-width="55px" alttext="J_{\nu}'\left(x\right)" display="inline"><mrow><msubsup><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′</mo></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math></a>,
<a href="./10.21#SS1.p2" title="§10.21(i) Distribution ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m320.png" altimg-height="26px" altimg-valign="-10px" altimg-width="43px" alttext="{y^{\prime}_{\NVar{\nu},\NVar{m}}}" display="inline"><msubsup><mi href="./10.21#SS1.p2">y</mi><mrow><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r1">m</mi></mrow><mo href="./10.21#SS1.p2">′</mo></msubsup></math>: zeros of the Bessel function derivative <math class="ltx_Math" altimg="m154.png" altimg-height="24px" altimg-valign="-7px" altimg-width="56px" alttext="Y_{\nu}'\left(x\right)" display="inline"><mrow><msubsup><mi href="./10.2#E3">Y</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′</mo></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math></a>,
<a href="./10.1#p2.t1.r1" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m284.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./10.1#p2.t1.r1">m</mi></math>: integer</a>,
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m215.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a> and
<a href="./10.21#SS2.p1" title="§10.21(ii) Analytic Properties ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m231.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="\sigma_{\nu}(t)" display="inline"><mrow><msub><mi href="./10.21#SS2.p1">σ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math>: zero of cylinder function</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">9.5.8</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E10" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex11" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">10.21.10</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m77.png" altimg-height="26px" altimg-valign="-7px" altimg-width="68px" alttext="\displaystyle\mathscr{C}_{\nu}'\left(\rho_{\nu}\right)" display="inline"><mrow><msubsup><mi class="ltx_font_mathscript" href="./10.2#Px5.p1">𝒞</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′</mo></msubsup><mo></mo><mrow><mo>(</mo><msub><mi href="./10.21#SS2.p1">ρ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m31.png" altimg-height="60px" altimg-valign="-21px" altimg-width="145px" alttext="\displaystyle=\left(\frac{\rho_{\nu}}{2}\frac{\mathrm{d}\rho_{\nu}}{\mathrm{d}%
t}\right)^{-\frac{1}{2}}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><msup><mrow><mo>(</mo><mrow><mstyle displaystyle="true"><mfrac><msub><mi href="./10.21#SS2.p1">ρ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mn>2</mn></mfrac></mstyle><mo></mo><mstyle displaystyle="true"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><msub><mi href="./10.21#SS2.p1">ρ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi>t</mi></mrow></mfrac></mstyle></mrow><mo>)</mo></mrow><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex12" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m78.png" altimg-height="25px" altimg-valign="-7px" altimg-width="69px" alttext="\displaystyle\mathscr{C}_{\nu}\left(\sigma_{\nu}\right)" display="inline"><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5.p1">𝒞</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><msub><mi href="./10.21#SS2.p1">σ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m32.png" altimg-height="61px" altimg-valign="-21px" altimg-width="191px" alttext="\displaystyle=\left(\frac{\sigma_{\nu}^{2}-\nu^{2}}{2\sigma_{\nu}}\frac{%
\mathrm{d}\sigma_{\nu}}{\mathrm{d}t}\right)^{-\frac{1}{2}}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><msup><mrow><mo>(</mo><mrow><mstyle displaystyle="true"><mfrac><mrow><msubsup><mi href="./10.21#SS2.p1">σ</mi><mi href="./10.1#p2.t1.r5">ν</mi><mn>2</mn></msubsup><mo>-</mo><msup><mi href="./10.1#p2.t1.r5">ν</mi><mn>2</mn></msup></mrow><mrow><mn>2</mn><mo></mo><msub><mi href="./10.21#SS2.p1">σ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub></mrow></mfrac></mstyle><mo></mo><mstyle displaystyle="true"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><msub><mi href="./10.21#SS2.p1">σ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi>t</mi></mrow></mfrac></mstyle></mrow><mo>)</mo></mrow><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E10.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#Px5.p1" title="Cylinder Functions ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m203.png" altimg-height="23px" altimg-valign="-7px" altimg-width="56px" alttext="\mathscr{C}_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5.p1">𝒞</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: cylinder function</a>,
<a href="./1.4#E4" title="(1.4.4) ‣ §1.4(iii) Derivatives ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m183.png" altimg-height="29px" altimg-valign="-9px" altimg-width="27px" alttext="\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: derivative of <math class="ltx_Math" altimg="m266.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m301.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m215.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>,
<a href="./10.21#SS2.p1" title="§10.21(ii) Analytic Properties ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m228.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="\rho_{\nu}(t)" display="inline"><mrow><msub><mi href="./10.21#SS2.p1">ρ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math>: zero of cylinder function</a> and
<a href="./10.21#SS2.p1" title="§10.21(ii) Analytic Properties ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m231.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="\sigma_{\nu}(t)" display="inline"><mrow><msub><mi href="./10.21#SS2.p1">σ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math>: zero of cylinder function</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">9.5.9</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E11" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.21.11</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E11.png" altimg-height="58px" altimg-valign="-21px" altimg-width="678px" alttext="2\rho_{\nu}^{2}\frac{\mathrm{d}\rho_{\nu}}{\mathrm{d}t}\frac{{\mathrm{d}}^{3}%
\rho_{\nu}}{{\mathrm{d}t}^{3}}-3\rho_{\nu}^{2}\*\left(\frac{{\mathrm{d}}^{2}%
\rho_{\nu}}{{\mathrm{d}t}^{2}}\right)^{2}-4\pi^{2}\rho_{\nu}^{2}\*\left(\frac{%
\mathrm{d}\rho_{\nu}}{\mathrm{d}t}\right)^{2}+(4\rho_{\nu}^{2}+1-4\nu^{2})%
\left(\frac{\mathrm{d}\rho_{\nu}}{\mathrm{d}t}\right)^{4}=0." display="block"><mrow><mrow><mrow><mrow><mrow><mn>2</mn><mo></mo><msubsup><mi href="./10.21#SS2.p1">ρ</mi><mi href="./10.1#p2.t1.r5">ν</mi><mn>2</mn></msubsup><mo></mo><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><msub><mi href="./10.21#SS2.p1">ρ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi>t</mi></mrow></mfrac><mo></mo><mfrac><mrow><mpadded lspace="-1.7pt" width="-4.5pt"><msup><mo href="./1.4#E4">d</mo><mn>3</mn></msup></mpadded><msub><mi href="./10.21#SS2.p1">ρ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub></mrow><msup><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi>t</mi></mrow><mn>3</mn></msup></mfrac></mrow><mo>-</mo><mrow><mn>3</mn><mo></mo><msubsup><mi href="./10.21#SS2.p1">ρ</mi><mi href="./10.1#p2.t1.r5">ν</mi><mn>2</mn></msubsup><mo></mo><msup><mrow><mo>(</mo><mfrac><mrow><mpadded lspace="-1.7pt" width="-4.5pt"><msup><mo href="./1.4#E4">d</mo><mn>2</mn></msup></mpadded><msub><mi href="./10.21#SS2.p1">ρ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub></mrow><msup><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi>t</mi></mrow><mn>2</mn></msup></mfrac><mo>)</mo></mrow><mn>2</mn></msup></mrow><mo>-</mo><mrow><mn>4</mn><mo></mo><msup><mi href="./3.12#E1">π</mi><mn>2</mn></msup><mo></mo><msubsup><mi href="./10.21#SS2.p1">ρ</mi><mi href="./10.1#p2.t1.r5">ν</mi><mn>2</mn></msubsup><mo></mo><msup><mrow><mo>(</mo><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><msub><mi href="./10.21#SS2.p1">ρ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi>t</mi></mrow></mfrac><mo>)</mo></mrow><mn>2</mn></msup></mrow></mrow><mo>+</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mrow><mn>4</mn><mo></mo><msubsup><mi href="./10.21#SS2.p1">ρ</mi><mi href="./10.1#p2.t1.r5">ν</mi><mn>2</mn></msubsup></mrow><mo>+</mo><mn>1</mn></mrow><mo>-</mo><mrow><mn>4</mn><mo></mo><msup><mi href="./10.1#p2.t1.r5">ν</mi><mn>2</mn></msup></mrow></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msup><mrow><mo>(</mo><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><msub><mi href="./10.21#SS2.p1">ρ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi>t</mi></mrow></mfrac><mo>)</mo></mrow><mn>4</mn></msup></mrow></mrow><mo>=</mo><mn>0</mn></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E11.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m226.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#E4" title="(1.4.4) ‣ §1.4(iii) Derivatives ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m183.png" altimg-height="29px" altimg-valign="-9px" altimg-width="27px" alttext="\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: derivative of <math class="ltx_Math" altimg="m266.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m301.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m215.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a> and
<a href="./10.21#SS2.p1" title="§10.21(ii) Analytic Properties ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m228.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="\rho_{\nu}(t)" display="inline"><mrow><msub><mi href="./10.21#SS2.p1">ρ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math>: zero of cylinder function</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS2.p3" class="ltx_para">
<p class="ltx_p">The functions <math class="ltx_Math" altimg="m228.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="\rho_{\nu}(t)" display="inline"><mrow><msub><mi href="./10.21#SS2.p1">ρ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math> and <math class="ltx_Math" altimg="m231.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="\sigma_{\nu}(t)" display="inline"><mrow><msub><mi href="./10.21#SS2.p1">σ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math> are related to the inverses of
the phase functions <math class="ltx_Math" altimg="m240.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="\theta_{\nu}\left(x\right)" display="inline"><mrow><msub><mi href="./10.18#E3">θ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math> and <math class="ltx_Math" altimg="m225.png" altimg-height="23px" altimg-valign="-7px" altimg-width="56px" alttext="\phi_{\nu}\left(x\right)" display="inline"><mrow><msub><mi href="./10.18#E3">ϕ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math> defined in
§: if <math class="ltx_Math" altimg="m216.png" altimg-height="19px" altimg-valign="-5px" altimg-width="52px" alttext="\nu\geq 0" display="inline"><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>≥</mo><mn>0</mn></mrow></math>, then</p>
<table id="E12" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex13" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="3" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">10.21.12</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m91.png" altimg-height="26px" altimg-valign="-8px" altimg-width="80px" alttext="\displaystyle\theta_{\nu}\left(j_{\nu,m}\right)" display="inline"><mrow><msub><mi href="./10.18#E3">θ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><msub><mi href="./10.21#SS1.p2">j</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi href="./10.1#p2.t1.r1">m</mi></mrow></msub><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m4.png" altimg-height="29px" altimg-valign="-9px" altimg-width="115px" alttext="\displaystyle=(m-\tfrac{1}{2})\pi," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./10.1#p2.t1.r1">m</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex14" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m92.png" altimg-height="26px" altimg-valign="-8px" altimg-width="82px" alttext="\displaystyle\theta_{\nu}\left(y_{\nu,m}\right)" display="inline"><mrow><msub><mi href="./10.18#E3">θ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><msub><mi href="./10.21#SS1.p2">y</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi href="./10.1#p2.t1.r1">m</mi></mrow></msub><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m3.png" altimg-height="25px" altimg-valign="-7px" altimg-width="112px" alttext="\displaystyle=(m-1)\pi," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./10.1#p2.t1.r1">m</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m281.png" altimg-height="20px" altimg-valign="-6px" altimg-width="109px" alttext="m=1,2,\ldots" display="inline"><mrow><mi href="./10.1#p2.t1.r1">m</mi><mo>=</mo><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E12.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m226.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./10.18#E3" title="(10.18.3) ‣ §10.18(i) Definitions ‣ §10.18 Modulus and Phase Functions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m239.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="\theta_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.18#E3">θ</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: phase of Bessel functions</a>,
<a href="./10.21#SS1.p2" title="§10.21(i) Distribution ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m273.png" altimg-height="22px" altimg-valign="-8px" altimg-width="41px" alttext="j_{\NVar{\nu},\NVar{m}}" display="inline"><msub><mi href="./10.21#SS1.p2">j</mi><mrow><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r1">m</mi></mrow></msub></math>: zeros of the Bessel function <math class="ltx_Math" altimg="m141.png" altimg-height="23px" altimg-valign="-7px" altimg-width="55px" alttext="J_{\nu}\left(x\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math></a>,
<a href="./10.21#SS1.p2" title="§10.21(i) Distribution ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m303.png" altimg-height="18px" altimg-valign="-8px" altimg-width="43px" alttext="y_{\NVar{\nu},\NVar{m}}" display="inline"><msub><mi href="./10.21#SS1.p2">y</mi><mrow><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r1">m</mi></mrow></msub></math>: zeros of the Bessel function <math class="ltx_Math" altimg="m157.png" altimg-height="23px" altimg-valign="-7px" altimg-width="56px" alttext="Y_{\nu}\left(x\right)" display="inline"><mrow><msub><mi href="./10.2#E3">Y</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math></a>,
<a href="./10.1#p2.t1.r1" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m284.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./10.1#p2.t1.r1">m</mi></math>: integer</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m215.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E13" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex15" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="3" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">10.21.13</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m79.png" altimg-height="30px" altimg-valign="-10px" altimg-width="86px" alttext="\displaystyle\phi_{\nu}\left({j^{\prime}_{\nu,m}}\right)" display="inline"><mrow><msub><mi href="./10.18#E3">ϕ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><msubsup><mi href="./10.21#SS1.p2">j</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi href="./10.1#p2.t1.r1">m</mi></mrow><mo href="./10.21#SS1.p2">′</mo></msubsup><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m4.png" altimg-height="29px" altimg-valign="-9px" altimg-width="115px" alttext="\displaystyle=(m-\tfrac{1}{2})\pi," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./10.1#p2.t1.r1">m</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex16" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m80.png" altimg-height="30px" altimg-valign="-10px" altimg-width="87px" alttext="\displaystyle\phi_{\nu}\left({y^{\prime}_{\nu,m}}\right)" display="inline"><mrow><msub><mi href="./10.18#E3">ϕ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><msubsup><mi href="./10.21#SS1.p2">y</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi href="./10.1#p2.t1.r1">m</mi></mrow><mo href="./10.21#SS1.p2">′</mo></msubsup><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m46.png" altimg-height="18px" altimg-valign="-6px" altimg-width="62px" alttext="\displaystyle=m\pi," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./10.1#p2.t1.r1">m</mi><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m281.png" altimg-height="20px" altimg-valign="-6px" altimg-width="109px" alttext="m=1,2,\ldots" display="inline"><mrow><mi href="./10.1#p2.t1.r1">m</mi><mo>=</mo><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E13.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m226.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./10.18#E3" title="(10.18.3) ‣ §10.18(i) Definitions ‣ §10.18 Modulus and Phase Functions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m224.png" altimg-height="23px" altimg-valign="-7px" altimg-width="56px" alttext="\phi_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.18#E3">ϕ</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: phase of derivatives of Bessel functions</a>,
<a href="./10.21#SS1.p2" title="§10.21(i) Distribution ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m318.png" altimg-height="26px" altimg-valign="-10px" altimg-width="41px" alttext="{j^{\prime}_{\NVar{\nu},\NVar{m}}}" display="inline"><msubsup><mi href="./10.21#SS1.p2">j</mi><mrow><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r1">m</mi></mrow><mo href="./10.21#SS1.p2">′</mo></msubsup></math>: zeros of the Bessel function derivative <math class="ltx_Math" altimg="m138.png" altimg-height="24px" altimg-valign="-7px" altimg-width="55px" alttext="J_{\nu}'\left(x\right)" display="inline"><mrow><msubsup><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′</mo></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math></a>,
<a href="./10.21#SS1.p2" title="§10.21(i) Distribution ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m320.png" altimg-height="26px" altimg-valign="-10px" altimg-width="43px" alttext="{y^{\prime}_{\NVar{\nu},\NVar{m}}}" display="inline"><msubsup><mi href="./10.21#SS1.p2">y</mi><mrow><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r1">m</mi></mrow><mo href="./10.21#SS1.p2">′</mo></msubsup></math>: zeros of the Bessel function derivative <math class="ltx_Math" altimg="m154.png" altimg-height="24px" altimg-valign="-7px" altimg-width="56px" alttext="Y_{\nu}'\left(x\right)" display="inline"><mrow><msubsup><mi href="./10.2#E3">Y</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′</mo></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math></a>,
<a href="./10.1#p2.t1.r1" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m284.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./10.1#p2.t1.r1">m</mi></math>: integer</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m215.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS2.p4" class="ltx_para">
<p class="ltx_p">For sign properties of the forward differences that are defined by</p>
<table id="E14" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex17" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">10.21.14</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m54.png" altimg-height="25px" altimg-valign="-7px" altimg-width="66px" alttext="\displaystyle\Delta\rho_{\nu}(t)" display="inline"><mrow><mi href="./3.6#SS1.p1" mathvariant="normal">Δ</mi><mrow><msub><mi href="./10.21#SS2.p1">ρ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m35.png" altimg-height="25px" altimg-valign="-7px" altimg-width="177px" alttext="\displaystyle=\rho_{\nu}(t+1)-\rho_{\nu}(t)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><msub><mi href="./10.21#SS2.p1">ρ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>-</mo><mrow><msub><mi href="./10.21#SS2.p1">ρ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex18" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m55.png" altimg-height="28px" altimg-valign="-7px" altimg-width="74px" alttext="\displaystyle\Delta^{2}\rho_{\nu}(t)" display="inline"><mrow><msup><mi href="./3.6#SS1.p1" mathvariant="normal">Δ</mi><mn>2</mn></msup><mrow><msub><mi href="./10.21#SS2.p1">ρ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m14.png" altimg-height="25px" altimg-valign="-7px" altimg-width="246px" alttext="\displaystyle=\Delta\rho_{\nu}(t+1)-\Delta\rho_{\nu}(t),\ldots," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mrow><mi href="./3.6#SS1.p1" mathvariant="normal">Δ</mi><mrow><msub><mi href="./10.21#SS2.p1">ρ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>-</mo><mrow><mi href="./3.6#SS1.p1" mathvariant="normal">Δ</mi><mrow><msub><mi href="./10.21#SS2.p1">ρ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E14.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.6#SS1.p1" title="§3.6(i) Introduction ‣ §3.6 Linear Difference Equations ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m163.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="\Delta" display="inline"><mi href="./3.6#SS1.p1" mathvariant="normal">Δ</mi></math>: forward difference operator</a>,
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m215.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a> and
<a href="./10.21#SS2.p1" title="§10.21(ii) Analytic Properties ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m228.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="\rho_{\nu}(t)" display="inline"><mrow><msub><mi href="./10.21#SS2.p1">ρ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math>: zero of cylinder function</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">when <math class="ltx_Math" altimg="m296.png" altimg-height="20px" altimg-valign="-6px" altimg-width="118px" alttext="t=1,2,3,\ldots" display="inline"><mrow><mi>t</mi><mo>=</mo><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>, and similarly for <math class="ltx_Math" altimg="m231.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="\sigma_{\nu}(t)" display="inline"><mrow><msub><mi href="./10.21#SS2.p1">σ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math>, see
<cite class="ltx_cite ltx_citemacro_citet">Lorch and Szegő ()</cite>.</p>
</div>
<div id="SS2.p5" class="ltx_para">
<p class="ltx_p">Some information on the distribution of <math class="ltx_Math" altimg="m228.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="\rho_{\nu}(t)" display="inline"><mrow><msub><mi href="./10.21#SS2.p1">ρ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math> and <math class="ltx_Math" altimg="m231.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="\sigma_{\nu}(t)" display="inline"><mrow><msub><mi href="./10.21#SS2.p1">σ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math> for real values
of <math class="ltx_Math" altimg="m215.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math> and <math class="ltx_Math" altimg="m297.png" altimg-height="17px" altimg-valign="-2px" altimg-width="11px" alttext="t" display="inline"><mi>t</mi></math> is given in <cite class="ltx_cite ltx_citemacro_citet">Muldoon and Spigler (</dd>
</dl>
</div>
</div>
<div id="SS3.p1" class="ltx_para">
<table id="EGx1" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E15">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.21.15</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m51.png" altimg-height="25px" altimg-valign="-7px" altimg-width="56px" alttext="\displaystyle J_{\nu}\left(z\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m25.png" altimg-height="66px" altimg-valign="-28px" altimg-width="254px" alttext="\displaystyle=\frac{(\tfrac{1}{2}z)^{\nu}}{\Gamma\left(\nu+1\right)}\prod_{k=1%
}^{\infty}\left(1-\frac{z^{2}}{{j_{\nu,k}^{2}}}\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><mfrac><msup><mrow><mo stretchy="false">(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo stretchy="false">)</mo></mrow><mi href="./10.1#p2.t1.r5">ν</mi></msup><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mfrac></mstyle><mo></mo><mrow><mstyle displaystyle="true"><munderover><mo largeop="true" movablelimits="false" symmetric="true">∏</mo><mrow><mi href="./10.1#p2.t1.r2">k</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover></mstyle><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mstyle displaystyle="true"><mfrac><msup><mi href="./10.1#p2.t1.r4">z</mi><mn>2</mn></msup><msubsup><mi href="./10.21#SS1.p2">j</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi href="./10.1#p2.t1.r2">k</mi></mrow><mn>2</mn></msubsup></mfrac></mstyle></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m216.png" altimg-height="19px" altimg-valign="-5px" altimg-width="52px" alttext="\nu\geq 0" display="inline"><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>≥</mo><mn>0</mn></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E15.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m134.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m164.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./10.21#SS1.p2" title="§10.21(i) Distribution ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m273.png" altimg-height="22px" altimg-valign="-8px" altimg-width="41px" alttext="j_{\NVar{\nu},\NVar{m}}" display="inline"><msub><mi href="./10.21#SS1.p2">j</mi><mrow><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r1">m</mi></mrow></msub></math>: zeros of the Bessel function <math class="ltx_Math" altimg="m141.png" altimg-height="23px" altimg-valign="-7px" altimg-width="55px" alttext="J_{\nu}\left(x\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math></a>,
<a href="./10.1#p2.t1.r2" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m279.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./10.1#p2.t1.r2">k</mi></math>: nonnegative integer</a>,
<a href="./10.1#p2.t1.r4" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m312.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./10.1#p2.t1.r4">z</mi></math>: complex variable</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m215.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">9.5.10</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E16">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.21.16</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m49.png" altimg-height="26px" altimg-valign="-7px" altimg-width="56px" alttext="\displaystyle J_{\nu}'\left(z\right)" display="inline"><mrow><msubsup><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′</mo></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m24.png" altimg-height="66px" altimg-valign="-28px" altimg-width="241px" alttext="\displaystyle=\frac{(\tfrac{1}{2}z)^{\nu-1}}{2\Gamma\left(\nu\right)}\prod_{k=%
1}^{\infty}\left(1-\frac{z^{2}}{{{j^{\prime}_{\nu,k}}^{\mspace{-16.0mu }2}}}%
\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><mfrac><msup><mrow><mo stretchy="false">(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>-</mo><mn>1</mn></mrow></msup><mrow><mn>2</mn><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r5">ν</mi><mo>)</mo></mrow></mrow></mrow></mfrac></mstyle><mo></mo><mrow><mstyle displaystyle="true"><munderover><mo largeop="true" movablelimits="false" symmetric="true">∏</mo><mrow><mi href="./10.1#p2.t1.r2">k</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover></mstyle><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mstyle displaystyle="true"><mfrac><msup><mi href="./10.1#p2.t1.r4">z</mi><mn>2</mn></msup><mmultiscripts><mi href="./10.21#SS1.p2">j</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi href="./10.1#p2.t1.r2">k</mi></mrow><mo href="./10.21#SS1.p2">′</mo><none></none><mpadded lspace="-8.9pt" width="-8.9pt"><mn>2</mn></mpadded></mmultiscripts></mfrac></mstyle></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m213.png" altimg-height="17px" altimg-valign="-3px" altimg-width="52px" alttext="\nu>0" display="inline"><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>></mo><mn>0</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E16.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m134.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m164.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./10.21#SS1.p2" title="§10.21(i) Distribution ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m318.png" altimg-height="26px" altimg-valign="-10px" altimg-width="41px" alttext="{j^{\prime}_{\NVar{\nu},\NVar{m}}}" display="inline"><msubsup><mi href="./10.21#SS1.p2">j</mi><mrow><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r1">m</mi></mrow><mo href="./10.21#SS1.p2">′</mo></msubsup></math>: zeros of the Bessel function derivative <math class="ltx_Math" altimg="m138.png" altimg-height="24px" altimg-valign="-7px" altimg-width="55px" alttext="J_{\nu}'\left(x\right)" display="inline"><mrow><msubsup><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′</mo></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math></a>,
<a href="./10.1#p2.t1.r2" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m279.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./10.1#p2.t1.r2">k</mi></math>: nonnegative integer</a>,
<a href="./10.1#p2.t1.r4" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m312.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./10.1#p2.t1.r4">z</mi></math>: complex variable</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m215.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">9.5.11</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
</div>
</section>
<section id="iv" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§10.21(iv) </span>Monotonicity Properties</h2>
<div id="SS4.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="SS4.p1" class="ltx_para">
<p class="ltx_p">Any positive zero <math class="ltx_Math" altimg="m262.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="c" display="inline"><mi href="./10.21#SS4.p1">c</mi></math> of the cylinder function <math class="ltx_Math" altimg="m207.png" altimg-height="23px" altimg-valign="-7px" altimg-width="58px" alttext="\mathscr{C}_{\nu}\left(x\right)" display="inline"><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5.p1">𝒞</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math> and any
positive zero <math class="ltx_Math" altimg="m264.png" altimg-height="19px" altimg-valign="-2px" altimg-width="19px" alttext="c^{\prime}" display="inline"><msup><mi href="./10.21#SS4.p1">c</mi><mo>′</mo></msup></math> of <math class="ltx_Math" altimg="m205.png" altimg-height="24px" altimg-valign="-7px" altimg-width="58px" alttext="\mathscr{C}_{\nu}'\left(x\right)" display="inline"><mrow><msubsup><mi class="ltx_font_mathscript" href="./10.2#Px5.p1">𝒞</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′</mo></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math> such that <math class="ltx_Math" altimg="m263.png" altimg-height="24px" altimg-valign="-7px" altimg-width="67px" alttext="c^{\prime}>|\nu|" display="inline"><mrow><msup><mi href="./10.21#SS4.p1">c</mi><mo>′</mo></msup><mo>></mo><mrow><mo stretchy="false">|</mo><mi href="./10.1#p2.t1.r5">ν</mi><mo stretchy="false">|</mo></mrow></mrow></math> are definable
as continuous and increasing functions of <math class="ltx_Math" altimg="m215.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>:</p>
<table id="E17" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.21.17</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E17.png" altimg-height="53px" altimg-valign="-20px" altimg-width="300px" alttext="\frac{\mathrm{d}c}{\mathrm{d}\nu}=2c\int_{0}^{\infty}K_{0}(2c\sinh t)e^{-2\nu t%
}\mathrm{d}t," display="block"><mrow><mrow><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi href="./10.21#SS4.p1">c</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow></mfrac><mo>=</mo><mrow><mn>2</mn><mo></mo><mi href="./10.21#SS4.p1">c</mi><mo></mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><mrow><mrow><msub><mi href="./10.25#E3">K</mi><mn>0</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mn>2</mn><mo></mo><mi href="./10.21#SS4.p1">c</mi><mo></mo><mrow><mi href="./4.28#E1">sinh</mi><mo></mo><mi>t</mi></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi><mo></mo><mi>t</mi></mrow></mrow></msup><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E17.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#E4" title="(1.4.4) ‣ §1.4(iii) Derivatives ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m183.png" altimg-height="29px" altimg-valign="-9px" altimg-width="27px" alttext="\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: derivative of <math class="ltx_Math" altimg="m266.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m301.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m201.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m301.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m202.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./4.28#E1" title="(4.28.1) ‣ §4.28 Definitions and Periodicity ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m236.png" altimg-height="18px" altimg-valign="-2px" altimg-width="53px" alttext="\sinh\NVar{z}" display="inline"><mrow><mi href="./4.28#E1">sinh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: hyperbolic sine function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m184.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./10.25#E3" title="(10.25.3) ‣ §10.25(ii) Standard Solutions ‣ §10.25 Definitions ‣ Modified Bessel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m145.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="K_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.25#E3">K</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: modified Bessel function of the second kind</a>,
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m215.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a> and
<a href="./10.21#SS4.p1" title="§10.21(iv) Monotonicity Properties ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m262.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="c" display="inline"><mi href="./10.21#SS4.p1">c</mi></math>: zero of cylinder function</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E18" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.21.18</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E18.png" altimg-height="52px" altimg-valign="-20px" altimg-width="523px" alttext="\frac{\mathrm{d}c^{\prime}}{\mathrm{d}\nu}=\frac{2c^{\prime}}{{c^{\prime}}^{2}%
-\nu^{2}}\*\int_{0}^{\infty}({c^{\prime}}^{2}\cosh(2t)-\nu^{2})\*K_{0}(2c^{%
\prime}\sinh t)e^{-2\nu t}\mathrm{d}t," display="block"><mrow><mrow><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><msup><mi href="./10.21#SS4.p1">c</mi><mo>′</mo></msup></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow></mfrac><mo>=</mo><mrow><mfrac><mrow><mn>2</mn><mo></mo><msup><mi href="./10.21#SS4.p1">c</mi><mo>′</mo></msup></mrow><mrow><mmultiscripts><mi href="./10.21#SS4.p1">c</mi><none></none><mo>′</mo><none></none><mn>2</mn></mmultiscripts><mo>-</mo><msup><mi href="./10.1#p2.t1.r5">ν</mi><mn>2</mn></msup></mrow></mfrac><mo></mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mmultiscripts><mi href="./10.21#SS4.p1">c</mi><none></none><mo>′</mo><none></none><mn>2</mn></mmultiscripts><mo></mo><mrow><mi href="./4.28#E2">cosh</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mn>2</mn><mo></mo><mi>t</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>-</mo><msup><mi href="./10.1#p2.t1.r5">ν</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><msub><mi href="./10.25#E3">K</mi><mn>0</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mn>2</mn><mo></mo><msup><mi href="./10.21#SS4.p1">c</mi><mo>′</mo></msup><mo></mo><mrow><mi href="./4.28#E1">sinh</mi><mo></mo><mi>t</mi></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi><mo></mo><mi>t</mi></mrow></mrow></msup><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E18.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#E4" title="(1.4.4) ‣ §1.4(iii) Derivatives ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m183.png" altimg-height="29px" altimg-valign="-9px" altimg-width="27px" alttext="\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: derivative of <math class="ltx_Math" altimg="m266.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m301.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m201.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m301.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m202.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./4.28#E2" title="(4.28.2) ‣ §4.28 Definitions and Periodicity ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m182.png" altimg-height="18px" altimg-valign="-2px" altimg-width="56px" alttext="\cosh\NVar{z}" display="inline"><mrow><mi href="./4.28#E2">cosh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: hyperbolic cosine function</a>,
<a href="./4.28#E1" title="(4.28.1) ‣ §4.28 Definitions and Periodicity ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m236.png" altimg-height="18px" altimg-valign="-2px" altimg-width="53px" alttext="\sinh\NVar{z}" display="inline"><mrow><mi href="./4.28#E1">sinh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: hyperbolic sine function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m184.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./10.25#E3" title="(10.25.3) ‣ §10.25(ii) Standard Solutions ‣ §10.25 Definitions ‣ Modified Bessel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m145.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="K_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.25#E3">K</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: modified Bessel function of the second kind</a>,
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m215.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a> and
<a href="./10.21#SS4.p1" title="§10.21(iv) Monotonicity Properties ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m262.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="c" display="inline"><mi href="./10.21#SS4.p1">c</mi></math>: zero of cylinder function</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m144.png" altimg-height="21px" altimg-valign="-5px" altimg-width="30px" alttext="K_{0}" display="inline"><msub><mi href="./10.25#E3">K</mi><mn>0</mn></msub></math> is defined in §.</p>
</div>
<div id="SS4.p2" class="ltx_para">
<p class="ltx_p">In particular, <math class="ltx_Math" altimg="m277.png" altimg-height="22px" altimg-valign="-8px" altimg-width="40px" alttext="j_{\nu,m}" display="inline"><msub><mi href="./10.21#SS1.p2">j</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi href="./10.1#p2.t1.r1">m</mi></mrow></msub></math>, <math class="ltx_Math" altimg="m305.png" altimg-height="18px" altimg-valign="-8px" altimg-width="42px" alttext="y_{\nu,m}" display="inline"><msub><mi href="./10.21#SS1.p2">y</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi href="./10.1#p2.t1.r1">m</mi></mrow></msub></math>,
<math class="ltx_Math" altimg="m275.png" altimg-height="26px" altimg-valign="-10px" altimg-width="40px" alttext="j_{\nu,m}'" display="inline"><msubsup><mi href="./10.21#SS1.p2">j</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi href="./10.1#p2.t1.r1">m</mi></mrow><mo>′</mo></msubsup></math>, and <math class="ltx_Math" altimg="m304.png" altimg-height="26px" altimg-valign="-10px" altimg-width="42px" alttext="y_{\nu,m}'" display="inline"><msubsup><mi href="./10.21#SS1.p2">y</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi href="./10.1#p2.t1.r1">m</mi></mrow><mo>′</mo></msubsup></math> are increasing functions
of <math class="ltx_Math" altimg="m215.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math> when <math class="ltx_Math" altimg="m216.png" altimg-height="19px" altimg-valign="-5px" altimg-width="52px" alttext="\nu\geq 0" display="inline"><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>≥</mo><mn>0</mn></mrow></math>. It is also true that the positive zeros <math class="ltx_Math" altimg="m272.png" altimg-height="23px" altimg-valign="-7px" altimg-width="24px" alttext="j^{\prime\prime}_{\nu}" display="inline"><msubsup><mi>j</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′′</mo></msubsup></math> and
<math class="ltx_Math" altimg="m270.png" altimg-height="23px" altimg-valign="-7px" altimg-width="28px" alttext="j^{\prime\prime\prime}_{\nu}" display="inline"><msubsup><mi>j</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′′′</mo></msubsup></math> of <math class="ltx_Math" altimg="m136.png" altimg-height="24px" altimg-valign="-7px" altimg-width="58px" alttext="J_{\nu}''\left(x\right)" display="inline"><mrow><msubsup><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′′</mo></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math> and <math class="ltx_Math" altimg="m135.png" altimg-height="24px" altimg-valign="-7px" altimg-width="62px" alttext="J_{\nu}'''\left(x\right)" display="inline"><mrow><msubsup><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′′′</mo></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>, respectively,
are increasing functions of <math class="ltx_Math" altimg="m215.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math> when <math class="ltx_Math" altimg="m213.png" altimg-height="17px" altimg-valign="-3px" altimg-width="52px" alttext="\nu>0" display="inline"><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>></mo><mn>0</mn></mrow></math>, provided that in the latter
case <math class="ltx_Math" altimg="m269.png" altimg-height="27px" altimg-valign="-7px" altimg-width="82px" alttext="j^{\prime\prime\prime}_{\nu}>\sqrt{3}" display="inline"><mrow><msubsup><mi>j</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′′′</mo></msubsup><mo>></mo><msqrt><mn>3</mn></msqrt></mrow></math> when <math class="ltx_Math" altimg="m118.png" altimg-height="17px" altimg-valign="-3px" altimg-width="88px" alttext="0<\nu<1" display="inline"><mrow><mn>0</mn><mo><</mo><mi href="./10.1#p2.t1.r5">ν</mi><mo><</mo><mn>1</mn></mrow></math>.</p>
</div>
<div id="SS4.p3" class="ltx_para">
<p class="ltx_p"><math class="ltx_Math" altimg="m276.png" altimg-height="24px" altimg-valign="-8px" altimg-width="62px" alttext="j_{\nu,m}/\nu" display="inline"><mrow><msub><mi href="./10.21#SS1.p2">j</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi href="./10.1#p2.t1.r1">m</mi></mrow></msub><mo>/</mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow></math> and <math class="ltx_Math" altimg="m274.png" altimg-height="26px" altimg-valign="-10px" altimg-width="62px" alttext="j_{\nu,m}'/\nu" display="inline"><mrow><msubsup><mi href="./10.21#SS1.p2">j</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi href="./10.1#p2.t1.r1">m</mi></mrow><mo>′</mo></msubsup><mo>/</mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow></math> are decreasing
functions of <math class="ltx_Math" altimg="m215.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math> when <math class="ltx_Math" altimg="m213.png" altimg-height="17px" altimg-valign="-3px" altimg-width="52px" alttext="\nu>0" display="inline"><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>></mo><mn>0</mn></mrow></math> for <math class="ltx_Math" altimg="m280.png" altimg-height="20px" altimg-valign="-6px" altimg-width="128px" alttext="m=1,2,3,\ldots" display="inline"><mrow><mi href="./10.1#p2.t1.r1">m</mi><mo>=</mo><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>.</p>
</div>
<div id="SS4.p4" class="ltx_para">
<p class="ltx_p">For further monotonicity properties see <cite class="ltx_cite ltx_citemacro_citet">Elbert (</dd>
</dl>
</div>
</div>
<div id="SS5.p1" class="ltx_para">
<p class="ltx_p">For bounds for the smallest real or purely imaginary zeros of <math class="ltx_Math" altimg="m141.png" altimg-height="23px" altimg-valign="-7px" altimg-width="55px" alttext="J_{\nu}\left(x\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>
when <math class="ltx_Math" altimg="m215.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math> is real see <cite class="ltx_cite ltx_citemacro_citet">Ismail and Muldoon (</dd>
</dl>
</div>
</div>
<div id="SS6.p1" class="ltx_para">
<p class="ltx_p">If <math class="ltx_Math" altimg="m215.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math> <math class="ltx_Math" altimg="m108.png" altimg-height="23px" altimg-valign="-7px" altimg-width="51px" alttext="(\geq 0)" display="inline"><mrow><mo stretchy="false">(</mo><mrow><mi></mi><mo>≥</mo><mn>0</mn></mrow><mo stretchy="false">)</mo></mrow></math> is fixed, <math class="ltx_Math" altimg="m209.png" altimg-height="24px" altimg-valign="-6px" altimg-width="73px" alttext="\mu=4\nu^{2}" display="inline"><mrow><mi>μ</mi><mo>=</mo><mrow><mn>4</mn><mo></mo><msup><mi href="./10.1#p2.t1.r5">ν</mi><mn>2</mn></msup></mrow></mrow></math>, and <math class="ltx_Math" altimg="m285.png" altimg-height="13px" altimg-valign="-2px" altimg-width="73px" alttext="m\to\infty" display="inline"><mrow><mi href="./10.1#p2.t1.r1">m</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></math>, then</p>
</div>
<div id="SS6.p2" class="ltx_para">
<table id="E19" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.21.19</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_Math ltx_eqn_cell ltx_align_center"><math class="ltx_Math" alttext="j_{\nu,m},y_{\nu,m}\sim a-\frac{\mu-1}{8a}-\frac{4(\mu-1)(7\mu-31)}{3(8a)^{3}}%
-\frac{32(\mu-1)(83\mu^{2}-982\mu+3779)}{15(8a)^{5}}-\frac{64(\mu-1)(6949\mu^{%
3}-1\;53855\mu^{2}+15\;85743\mu-62\;77237)}{105(8a)^{7}}-\cdots," display="block"><mrow><mrow><msub><mi href="./10.21#SS1.p2">j</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi href="./10.1#p2.t1.r1">m</mi></mrow></msub><mo>,</mo><msub><mi href="./10.21#SS1.p2">y</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi href="./10.1#p2.t1.r1">m</mi></mrow></msub></mrow><mo href="./2.1#SS3.p1">∼</mo><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mi>a</mi><mo>-</mo><mfrac><mrow><mi>μ</mi><mo>-</mo><mn>1</mn></mrow><mrow><mn>8</mn><mo></mo><mi>a</mi></mrow></mfrac><mo>-</mo><mfrac><mrow><mn>4</mn><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>μ</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>7</mn><mo></mo><mi>μ</mi></mrow><mo>-</mo><mn>31</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mrow><mn>3</mn><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mn>8</mn><mo></mo><mi>a</mi></mrow><mo stretchy="false">)</mo></mrow><mn>3</mn></msup></mrow></mfrac><mo>-</mo><mfrac><mrow><mn>32</mn><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>μ</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mrow><mn>83</mn><mo></mo><msup><mi>μ</mi><mn>2</mn></msup></mrow><mo>-</mo><mrow><mn>982</mn><mo></mo><mi>μ</mi></mrow></mrow><mo>+</mo><mn>3779</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mrow><mn>15</mn><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mn>8</mn><mo></mo><mi>a</mi></mrow><mo stretchy="false">)</mo></mrow><mn>5</mn></msup></mrow></mfrac></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>-</mo><mfrac><mrow><mn>64</mn><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>μ</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mrow><mrow><mn>6949</mn><mo></mo><msup><mi>μ</mi><mn>3</mn></msup></mrow><mo>-</mo><mrow><mn>1 53855</mn><mo></mo><msup><mi>μ</mi><mn>2</mn></msup></mrow></mrow><mo>+</mo><mrow><mn>15 85743</mn><mo></mo><mi>μ</mi></mrow></mrow><mo>-</mo><mn>62 77237</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mrow><mn>105</mn><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mn>8</mn><mo></mo><mi>a</mi></mrow><mo stretchy="false">)</mo></mrow><mn>7</mn></msup></mrow></mfrac><mo>-</mo><mi mathvariant="normal">⋯</mi><mo>,</mo></mtd></mtr></mtable></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E19.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./2.1#SS3.p1" title="§2.1(iii) Asymptotic Expansions ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m234.png" altimg-height="12px" altimg-valign="-2px" altimg-width="20px" alttext="\sim" display="inline"><mo href="./2.1#SS3.p1">∼</mo></math>: Poincaré asymptotic expansion</a>,
<a href="./10.21#SS1.p2" title="§10.21(i) Distribution ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m273.png" altimg-height="22px" altimg-valign="-8px" altimg-width="41px" alttext="j_{\NVar{\nu},\NVar{m}}" display="inline"><msub><mi href="./10.21#SS1.p2">j</mi><mrow><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r1">m</mi></mrow></msub></math>: zeros of the Bessel function <math class="ltx_Math" altimg="m141.png" altimg-height="23px" altimg-valign="-7px" altimg-width="55px" alttext="J_{\nu}\left(x\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math></a>,
<a href="./10.21#SS1.p2" title="§10.21(i) Distribution ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m303.png" altimg-height="18px" altimg-valign="-8px" altimg-width="43px" alttext="y_{\NVar{\nu},\NVar{m}}" display="inline"><msub><mi href="./10.21#SS1.p2">y</mi><mrow><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r1">m</mi></mrow></msub></math>: zeros of the Bessel function <math class="ltx_Math" altimg="m157.png" altimg-height="23px" altimg-valign="-7px" altimg-width="56px" alttext="Y_{\nu}\left(x\right)" display="inline"><mrow><msub><mi href="./10.2#E3">Y</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math></a>,
<a href="./10.1#p2.t1.r1" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m284.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./10.1#p2.t1.r1">m</mi></math>: integer</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m215.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">9.5.12</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m245.png" altimg-height="27px" altimg-valign="-9px" altimg-width="172px" alttext="a=(m+\tfrac{1}{2}\nu-\tfrac{1}{4})\pi" display="inline"><mrow><mi>a</mi><mo>=</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mi href="./10.1#p2.t1.r1">m</mi><mo>+</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow></mrow><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow></math> for <math class="ltx_Math" altimg="m277.png" altimg-height="22px" altimg-valign="-8px" altimg-width="40px" alttext="j_{\nu,m}" display="inline"><msub><mi href="./10.21#SS1.p2">j</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi href="./10.1#p2.t1.r1">m</mi></mrow></msub></math>,
<math class="ltx_Math" altimg="m246.png" altimg-height="27px" altimg-valign="-9px" altimg-width="172px" alttext="a=(m+\tfrac{1}{2}\nu-\tfrac{3}{4})\pi" display="inline"><mrow><mi>a</mi><mo>=</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mi href="./10.1#p2.t1.r1">m</mi><mo>+</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow></mrow><mo>-</mo><mfrac><mn>3</mn><mn>4</mn></mfrac></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow></math> for <math class="ltx_Math" altimg="m305.png" altimg-height="18px" altimg-valign="-8px" altimg-width="42px" alttext="y_{\nu,m}" display="inline"><msub><mi href="./10.21#SS1.p2">y</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi href="./10.1#p2.t1.r1">m</mi></mrow></msub></math>. With
<math class="ltx_Math" altimg="m247.png" altimg-height="27px" altimg-valign="-9px" altimg-width="161px" alttext="a=(t+\tfrac{1}{2}\nu-\tfrac{1}{4})\pi" display="inline"><mrow><mi>a</mi><mo>=</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mi>t</mi><mo>+</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow></mrow><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow></math>, the right-hand side is the
asymptotic expansion of <math class="ltx_Math" altimg="m228.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="\rho_{\nu}(t)" display="inline"><mrow><msub><mi href="./10.21#SS2.p1">ρ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math> for large <math class="ltx_Math" altimg="m297.png" altimg-height="17px" altimg-valign="-2px" altimg-width="11px" alttext="t" display="inline"><mi>t</mi></math>.</p>
<table id="E20" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.21.20</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_Math ltx_eqn_cell ltx_align_center"><math class="ltx_Math" alttext="j_{\nu,m}',y_{\nu,m}'\sim b-\frac{\mu+3}{8b}-\frac{4(7\mu^{2}+82\mu-9)}{3(8b)^%
{3}}-\frac{32(83\mu^{3}+2075\mu^{2}-3039\mu+3537)}{15(8b)^{5}}-\frac{64(6949%
\mu^{4}+2\;96492\mu^{3}-12\;48002\mu^{2}+74\;14380\mu-58\;53627)}{105(8b)^{7}}%
-\cdots," display="block"><mrow><mrow><msubsup><mi href="./10.21#SS1.p2">j</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi href="./10.1#p2.t1.r1">m</mi></mrow><mo>′</mo></msubsup><mo>,</mo><msubsup><mi href="./10.21#SS1.p2">y</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi href="./10.1#p2.t1.r1">m</mi></mrow><mo>′</mo></msubsup></mrow><mo href="./2.1#SS3.p1">∼</mo><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mi>b</mi><mo>-</mo><mfrac><mrow><mi>μ</mi><mo>+</mo><mn>3</mn></mrow><mrow><mn>8</mn><mo></mo><mi>b</mi></mrow></mfrac><mo>-</mo><mfrac><mrow><mn>4</mn><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mrow><mn>7</mn><mo></mo><msup><mi>μ</mi><mn>2</mn></msup></mrow><mo>+</mo><mrow><mn>82</mn><mo></mo><mi>μ</mi></mrow></mrow><mo>-</mo><mn>9</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mrow><mn>3</mn><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mn>8</mn><mo></mo><mi>b</mi></mrow><mo stretchy="false">)</mo></mrow><mn>3</mn></msup></mrow></mfrac><mo>-</mo><mfrac><mrow><mn>32</mn><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mrow><mrow><mn>83</mn><mo></mo><msup><mi>μ</mi><mn>3</mn></msup></mrow><mo>+</mo><mrow><mn>2075</mn><mo></mo><msup><mi>μ</mi><mn>2</mn></msup></mrow></mrow><mo>-</mo><mrow><mn>3039</mn><mo></mo><mi>μ</mi></mrow></mrow><mo>+</mo><mn>3537</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mrow><mn>15</mn><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mn>8</mn><mo></mo><mi>b</mi></mrow><mo stretchy="false">)</mo></mrow><mn>5</mn></msup></mrow></mfrac></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>-</mo><mfrac><mrow><mn>64</mn><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mrow><mrow><mrow><mn>6949</mn><mo></mo><msup><mi>μ</mi><mn>4</mn></msup></mrow><mo>+</mo><mrow><mn>2 96492</mn><mo></mo><msup><mi>μ</mi><mn>3</mn></msup></mrow></mrow><mo>-</mo><mrow><mn>12 48002</mn><mo></mo><msup><mi>μ</mi><mn>2</mn></msup></mrow></mrow><mo>+</mo><mrow><mn>74 14380</mn><mo></mo><mi>μ</mi></mrow></mrow><mo>-</mo><mn>58 53627</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mrow><mn>105</mn><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mn>8</mn><mo></mo><mi>b</mi></mrow><mo stretchy="false">)</mo></mrow><mn>7</mn></msup></mrow></mfrac><mo>-</mo><mi mathvariant="normal">⋯</mi><mo>,</mo></mtd></mtr></mtable></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E20.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./2.1#SS3.p1" title="§2.1(iii) Asymptotic Expansions ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m234.png" altimg-height="12px" altimg-valign="-2px" altimg-width="20px" alttext="\sim" display="inline"><mo href="./2.1#SS3.p1">∼</mo></math>: Poincaré asymptotic expansion</a>,
<a href="./10.21#SS1.p2" title="§10.21(i) Distribution ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m273.png" altimg-height="22px" altimg-valign="-8px" altimg-width="41px" alttext="j_{\NVar{\nu},\NVar{m}}" display="inline"><msub><mi href="./10.21#SS1.p2">j</mi><mrow><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r1">m</mi></mrow></msub></math>: zeros of the Bessel function <math class="ltx_Math" altimg="m141.png" altimg-height="23px" altimg-valign="-7px" altimg-width="55px" alttext="J_{\nu}\left(x\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math></a>,
<a href="./10.21#SS1.p2" title="§10.21(i) Distribution ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m303.png" altimg-height="18px" altimg-valign="-8px" altimg-width="43px" alttext="y_{\NVar{\nu},\NVar{m}}" display="inline"><msub><mi href="./10.21#SS1.p2">y</mi><mrow><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r1">m</mi></mrow></msub></math>: zeros of the Bessel function <math class="ltx_Math" altimg="m157.png" altimg-height="23px" altimg-valign="-7px" altimg-width="56px" alttext="Y_{\nu}\left(x\right)" display="inline"><mrow><msub><mi href="./10.2#E3">Y</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math></a>,
<a href="./10.1#p2.t1.r1" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m284.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./10.1#p2.t1.r1">m</mi></math>: integer</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m215.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">9.5.13</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS6.p3" class="ltx_para">
<p class="ltx_p">where <math class="ltx_Math" altimg="m253.png" altimg-height="27px" altimg-valign="-9px" altimg-width="170px" alttext="b=(m+\tfrac{1}{2}\nu-\tfrac{3}{4})\pi" display="inline"><mrow><mi>b</mi><mo>=</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mi href="./10.1#p2.t1.r1">m</mi><mo>+</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow></mrow><mo>-</mo><mfrac><mn>3</mn><mn>4</mn></mfrac></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow></math> for <math class="ltx_Math" altimg="m275.png" altimg-height="26px" altimg-valign="-10px" altimg-width="40px" alttext="j_{\nu,m}'" display="inline"><msubsup><mi href="./10.21#SS1.p2">j</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi href="./10.1#p2.t1.r1">m</mi></mrow><mo>′</mo></msubsup></math>,
<math class="ltx_Math" altimg="m252.png" altimg-height="27px" altimg-valign="-9px" altimg-width="170px" alttext="b=(m+\tfrac{1}{2}\nu-\tfrac{1}{4})\pi" display="inline"><mrow><mi>b</mi><mo>=</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mi href="./10.1#p2.t1.r1">m</mi><mo>+</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow></mrow><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow></math> for <math class="ltx_Math" altimg="m304.png" altimg-height="26px" altimg-valign="-10px" altimg-width="42px" alttext="y_{\nu,m}'" display="inline"><msubsup><mi href="./10.21#SS1.p2">y</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi href="./10.1#p2.t1.r1">m</mi></mrow><mo>′</mo></msubsup></math>, and
<math class="ltx_Math" altimg="m254.png" altimg-height="27px" altimg-valign="-9px" altimg-width="159px" alttext="b=(t+\tfrac{1}{2}\nu+\tfrac{1}{4})\pi" display="inline"><mrow><mi>b</mi><mo>=</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mi>t</mi><mo>+</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo>+</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow></math> for <math class="ltx_Math" altimg="m231.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="\sigma_{\nu}(t)" display="inline"><mrow><msub><mi href="./10.21#SS2.p1">σ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math>.</p>
</div>
<div id="SS6.p4" class="ltx_para">
<p class="ltx_p">For the next three terms in ()</cite>.
</p>
</div>
<div id="SS6.p6" class="ltx_para">
<p class="ltx_p">For the <math class="ltx_Math" altimg="m284.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./10.1#p2.t1.r1">m</mi></math>th positive zero <math class="ltx_Math" altimg="m271.png" altimg-height="26px" altimg-valign="-10px" altimg-width="40px" alttext="j^{\prime\prime}_{\nu,m}" display="inline"><msubsup><mi>j</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>,</mo><mi href="./10.1#p2.t1.r1">m</mi></mrow><mo>′′</mo></msubsup></math> of <math class="ltx_Math" altimg="m136.png" altimg-height="24px" altimg-valign="-7px" altimg-width="58px" alttext="J_{\nu}''\left(x\right)" display="inline"><mrow><msubsup><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′′</mo></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>
<cite class="ltx_cite ltx_citemacro_citet">Wong and Lang ()</cite> gives the corresponding expansion
</p>
<table id="E21" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.21.21</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E21.png" altimg-height="55px" altimg-valign="-21px" altimg-width="410px" alttext="j^{\prime\prime}_{\nu,m}\sim c-\frac{\mu+7}{8c}-\frac{28\mu^{2}+424\mu+1724}{3%
(8c)^{3}}-\cdots," display="block"><mrow><mrow><msubsup><mi>j</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>,</mo><mi href="./10.1#p2.t1.r1">m</mi></mrow><mo>′′</mo></msubsup><mo href="./2.1#SS3.p1">∼</mo><mrow><mi href="./10.21#SS4.p1">c</mi><mo>-</mo><mfrac><mrow><mi>μ</mi><mo>+</mo><mn>7</mn></mrow><mrow><mn>8</mn><mo></mo><mi href="./10.21#SS4.p1">c</mi></mrow></mfrac><mo>-</mo><mfrac><mrow><mrow><mn>28</mn><mo></mo><msup><mi>μ</mi><mn>2</mn></msup></mrow><mo>+</mo><mrow><mn>424</mn><mo></mo><mi>μ</mi></mrow><mo>+</mo><mn>1724</mn></mrow><mrow><mn>3</mn><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mn>8</mn><mo></mo><mi href="./10.21#SS4.p1">c</mi></mrow><mo stretchy="false">)</mo></mrow><mn>3</mn></msup></mrow></mfrac><mo>-</mo><mi mathvariant="normal">⋯</mi></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E21.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./2.1#SS3.p1" title="§2.1(iii) Asymptotic Expansions ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m234.png" altimg-height="12px" altimg-valign="-2px" altimg-width="20px" alttext="\sim" display="inline"><mo href="./2.1#SS3.p1">∼</mo></math>: Poincaré asymptotic expansion</a>,
<a href="./10.1#p2.t1.r1" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m284.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./10.1#p2.t1.r1">m</mi></math>: integer</a>,
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m215.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a> and
<a href="./10.21#SS4.p1" title="§10.21(iv) Monotonicity Properties ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m262.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="c" display="inline"><mi href="./10.21#SS4.p1">c</mi></math>: zero of cylinder function</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m259.png" altimg-height="27px" altimg-valign="-9px" altimg-width="170px" alttext="c=(m+\tfrac{1}{2}\nu-\tfrac{1}{4})\pi" display="inline"><mrow><mi href="./10.21#SS4.p1">c</mi><mo>=</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mi href="./10.1#p2.t1.r1">m</mi><mo>+</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow></mrow><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow></math> if <math class="ltx_Math" altimg="m118.png" altimg-height="17px" altimg-valign="-3px" altimg-width="88px" alttext="0<\nu<1" display="inline"><mrow><mn>0</mn><mo><</mo><mi href="./10.1#p2.t1.r5">ν</mi><mo><</mo><mn>1</mn></mrow></math>, and
<math class="ltx_Math" altimg="m260.png" altimg-height="27px" altimg-valign="-9px" altimg-width="170px" alttext="c=(m+\tfrac{1}{2}\nu-\tfrac{5}{4})\pi" display="inline"><mrow><mi href="./10.21#SS4.p1">c</mi><mo>=</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mi href="./10.1#p2.t1.r1">m</mi><mo>+</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow></mrow><mo>-</mo><mfrac><mn>5</mn><mn>4</mn></mfrac></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow></math> if <math class="ltx_Math" altimg="m214.png" altimg-height="17px" altimg-valign="-3px" altimg-width="52px" alttext="\nu>1" display="inline"><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>></mo><mn>1</mn></mrow></math>. An error bound is
included for the case <math class="ltx_Math" altimg="m219.png" altimg-height="27px" altimg-valign="-9px" altimg-width="55px" alttext="\nu\geq\tfrac{3}{2}" display="inline"><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>≥</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow></math>.</p>
</div>
</section>
<section id="vii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§10.21(vii) </span>Asymptotic Expansions for Large Order</h2>
<div id="SS7.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="SS7.p1" class="ltx_para">
<p class="ltx_p">Let <math class="ltx_Math" altimg="m207.png" altimg-height="23px" altimg-valign="-7px" altimg-width="58px" alttext="\mathscr{C}_{\nu}\left(x\right)" display="inline"><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5.p1">𝒞</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>, <math class="ltx_Math" altimg="m228.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="\rho_{\nu}(t)" display="inline"><mrow><msub><mi href="./10.21#SS2.p1">ρ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math>, and <math class="ltx_Math" altimg="m231.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="\sigma_{\nu}(t)" display="inline"><mrow><msub><mi href="./10.21#SS2.p1">σ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math> be defined as in
§ and <math class="ltx_Math" altimg="m147.png" altimg-height="23px" altimg-valign="-7px" altimg-width="56px" alttext="M\left(x\right)" display="inline"><mrow><mi href="./9.8#E3">M</mi><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>, <math class="ltx_Math" altimg="m238.png" altimg-height="23px" altimg-valign="-7px" altimg-width="44px" alttext="\theta\left(x\right)" display="inline"><mrow><mi href="./9.8#E4">θ</mi><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>,
<math class="ltx_Math" altimg="m149.png" altimg-height="23px" altimg-valign="-7px" altimg-width="53px" alttext="N\left(x\right)" display="inline"><mrow><mi href="./9.8#E7">N</mi><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>, and <math class="ltx_Math" altimg="m223.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\phi\left(x\right)" display="inline"><mrow><mi href="./9.8#E8">ϕ</mi><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math> denote the
modulus and phase functions for the Airy functions and their derivatives as in
§.</p>
</div>
<div id="SS7.p2" class="ltx_para">
<p class="ltx_p">As <math class="ltx_Math" altimg="m220.png" altimg-height="13px" altimg-valign="-2px" altimg-width="66px" alttext="\nu\to\infty" display="inline"><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></math> with <math class="ltx_Math" altimg="m297.png" altimg-height="17px" altimg-valign="-2px" altimg-width="11px" alttext="t" display="inline"><mi>t</mi></math> <math class="ltx_Math" altimg="m107.png" altimg-height="23px" altimg-valign="-7px" altimg-width="51px" alttext="(>0)" display="inline"><mrow><mo stretchy="false">(</mo><mrow><mi></mi><mo>></mo><mn>0</mn></mrow><mo stretchy="false">)</mo></mrow></math> fixed,</p>
<table id="E22" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.21.22</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E22.png" altimg-height="64px" altimg-valign="-28px" altimg-width="178px" alttext="\rho_{\nu}(t)\sim\nu\sum_{k=0}^{\infty}\frac{\alpha_{k}}{\nu^{2k/3}}," display="block"><mrow><mrow><mrow><msub><mi href="./10.21#SS2.p1">ρ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo href="./2.1#SS3.p1">∼</mo><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./10.1#p2.t1.r2">k</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mfrac><msub><mi href="./10.21#E24">α</mi><mi href="./10.1#p2.t1.r2">k</mi></msub><msup><mi href="./10.1#p2.t1.r5">ν</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r2">k</mi></mrow><mo>/</mo><mn>3</mn></mrow></msup></mfrac></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E22.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./2.1#SS3.p1" title="§2.1(iii) Asymptotic Expansions ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m234.png" altimg-height="12px" altimg-valign="-2px" altimg-width="20px" alttext="\sim" display="inline"><mo href="./2.1#SS3.p1">∼</mo></math>: Poincaré asymptotic expansion</a>,
<a href="./10.1#p2.t1.r2" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m279.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./10.1#p2.t1.r2">k</mi></math>: nonnegative integer</a>,
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m215.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>,
<a href="./10.21#SS2.p1" title="§10.21(ii) Analytic Properties ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m228.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="\rho_{\nu}(t)" display="inline"><mrow><msub><mi href="./10.21#SS2.p1">ρ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math>: zero of cylinder function</a> and
<a href="./10.21#E24" title="(10.21.24) ‣ §10.21(vii) Asymptotic Expansions for Large Order ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m171.png" altimg-height="13px" altimg-valign="-2px" altimg-width="17px" alttext="\alpha" display="inline"><mi href="./10.21#E24">α</mi></math>: coefficients</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E23" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.21.23</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E23.png" altimg-height="72px" altimg-valign="-36px" altimg-width="325px" alttext="\mathscr{C}_{\nu}'\left(\rho_{\nu}(t)\right)\sim\frac{(2/\nu)^{\frac{2}{3}}}{%
\pi M\left(-2^{\frac{1}{3}}\alpha\right)}\sum_{k=0}^{\infty}\frac{\beta_{k}}{%
\nu^{2k/3}}," display="block"><mrow><mrow><mrow><msubsup><mi class="ltx_font_mathscript" href="./10.2#Px5.p1">𝒞</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′</mo></msubsup><mo></mo><mrow><mo>(</mo><mrow><msub><mi href="./10.21#SS2.p1">ρ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo>)</mo></mrow></mrow><mo href="./2.1#SS3.p1">∼</mo><mrow><mfrac><msup><mrow><mo stretchy="false">(</mo><mrow><mn>2</mn><mo>/</mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo stretchy="false">)</mo></mrow><mfrac><mn>2</mn><mn>3</mn></mfrac></msup><mrow><mi href="./3.12#E1">π</mi><mo></mo><mrow><mi href="./9.8#E3">M</mi><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mrow><msup><mn>2</mn><mfrac><mn>1</mn><mn>3</mn></mfrac></msup><mo></mo><mi href="./10.21#E24">α</mi></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mfrac><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./10.1#p2.t1.r2">k</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mfrac><msub><mi href="./10.21#E26">β</mi><mi href="./10.1#p2.t1.r2">k</mi></msub><msup><mi href="./10.1#p2.t1.r5">ν</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r2">k</mi></mrow><mo>/</mo><mn>3</mn></mrow></msup></mfrac></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E23.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./2.1#SS3.p1" title="§2.1(iii) Asymptotic Expansions ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m234.png" altimg-height="12px" altimg-valign="-2px" altimg-width="20px" alttext="\sim" display="inline"><mo href="./2.1#SS3.p1">∼</mo></math>: Poincaré asymptotic expansion</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m226.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./10.2#Px5.p1" title="Cylinder Functions ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m203.png" altimg-height="23px" altimg-valign="-7px" altimg-width="56px" alttext="\mathscr{C}_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5.p1">𝒞</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: cylinder function</a>,
<a href="./9.8#E3" title="(9.8.3) ‣ §9.8(i) Definitions ‣ §9.8 Modulus and Phase ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m146.png" altimg-height="23px" altimg-valign="-7px" altimg-width="55px" alttext="M\left(\NVar{z}\right)" display="inline"><mrow><mi href="./9.8#E3">M</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./9.1#p2.t1.r3">z</mi><mo>)</mo></mrow></mrow></math>: Airy modulus function</a>,
<a href="./10.1#p2.t1.r2" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m279.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./10.1#p2.t1.r2">k</mi></math>: nonnegative integer</a>,
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m215.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>,
<a href="./10.21#SS2.p1" title="§10.21(ii) Analytic Properties ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m228.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="\rho_{\nu}(t)" display="inline"><mrow><msub><mi href="./10.21#SS2.p1">ρ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math>: zero of cylinder function</a>,
<a href="./10.21#E24" title="(10.21.24) ‣ §10.21(vii) Asymptotic Expansions for Large Order ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m171.png" altimg-height="13px" altimg-valign="-2px" altimg-width="17px" alttext="\alpha" display="inline"><mi href="./10.21#E24">α</mi></math>: coefficients</a> and
<a href="./10.21#E26" title="(10.21.26) ‣ §10.21(vii) Asymptotic Expansions for Large Order ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m177.png" altimg-height="21px" altimg-valign="-6px" altimg-width="17px" alttext="\beta" display="inline"><mi href="./10.21#E26">β</mi></math>: coefficients</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m171.png" altimg-height="13px" altimg-valign="-2px" altimg-width="17px" alttext="\alpha" display="inline"><mi href="./10.21#E24">α</mi></math> is given by</p>
<table id="E24" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.21.24</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E24.png" altimg-height="41px" altimg-valign="-15px" altimg-width="145px" alttext="\theta\left(-2^{\frac{1}{3}}\alpha\right)=\pi t," display="block"><mrow><mrow><mrow><mi href="./9.8#E4">θ</mi><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mrow><msup><mn>2</mn><mfrac><mn>1</mn><mn>3</mn></mfrac></msup><mo></mo><mi href="./10.21#E24">α</mi></mrow></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi>t</mi></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E24.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text"><math class="ltx_Math" altimg="m171.png" altimg-height="13px" altimg-valign="-2px" altimg-width="17px" alttext="\alpha" display="inline"><mi href="./10.21#E24">α</mi></math>: coefficients (locally)</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m226.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a> and
<a href="./9.8#E4" title="(9.8.4) ‣ §9.8(i) Definitions ‣ §9.8 Modulus and Phase ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m237.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="\theta\left(\NVar{z}\right)" display="inline"><mrow><mi href="./9.8#E4">θ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./9.1#p2.t1.r3">z</mi><mo>)</mo></mrow></mrow></math>: Airy phase function</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and</p>
<table id="E25" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex19" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="6" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">10.21.25</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m62.png" altimg-height="17px" altimg-valign="-5px" altimg-width="28px" alttext="\displaystyle\alpha_{0}" display="inline"><msub><mi href="./10.21#E24">α</mi><mn>0</mn></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m13.png" altimg-height="22px" altimg-valign="-6px" altimg-width="43px" alttext="\displaystyle=1," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mn>1</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex20" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m63.png" altimg-height="17px" altimg-valign="-5px" altimg-width="28px" alttext="\displaystyle\alpha_{1}" display="inline"><msub><mi href="./10.21#E24">α</mi><mn>1</mn></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m15.png" altimg-height="18px" altimg-valign="-6px" altimg-width="45px" alttext="\displaystyle=\alpha," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mi href="./10.21#E24">α</mi></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex21" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m64.png" altimg-height="17px" altimg-valign="-5px" altimg-width="28px" alttext="\displaystyle\alpha_{2}" display="inline"><msub><mi href="./10.21#E24">α</mi><mn>2</mn></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m40.png" altimg-height="30px" altimg-valign="-9px" altimg-width="75px" alttext="\displaystyle=\tfrac{3}{10}\alpha^{2}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mfrac><mn>3</mn><mn>10</mn></mfrac><mo></mo><msup><mi href="./10.21#E24">α</mi><mn>2</mn></msup></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex22" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m65.png" altimg-height="17px" altimg-valign="-5px" altimg-width="28px" alttext="\displaystyle\alpha_{3}" display="inline"><msub><mi href="./10.21#E24">α</mi><mn>3</mn></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m5.png" altimg-height="30px" altimg-valign="-9px" altimg-width="143px" alttext="\displaystyle=-\tfrac{1}{350}\alpha^{3}+\tfrac{1}{70}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>350</mn></mfrac><mo></mo><msup><mi href="./10.21#E24">α</mi><mn>3</mn></msup></mrow></mrow><mo>+</mo><mfrac><mn>1</mn><mn>70</mn></mfrac></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex23" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m66.png" altimg-height="17px" altimg-valign="-5px" altimg-width="28px" alttext="\displaystyle\alpha_{4}" display="inline"><msub><mi href="./10.21#E24">α</mi><mn>4</mn></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m8.png" altimg-height="30px" altimg-valign="-9px" altimg-width="188px" alttext="\displaystyle=-\tfrac{479}{63000}\alpha^{4}-\tfrac{1}{3150}\alpha," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mo>-</mo><mrow><mfrac><mn>479</mn><mn>63000</mn></mfrac><mo></mo><msup><mi href="./10.21#E24">α</mi><mn>4</mn></msup></mrow></mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>3150</mn></mfrac><mo></mo><mi href="./10.21#E24">α</mi></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex24" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m67.png" altimg-height="17px" altimg-valign="-5px" altimg-width="28px" alttext="\displaystyle\alpha_{5}" display="inline"><msub><mi href="./10.21#E24">α</mi><mn>5</mn></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m39.png" altimg-height="30px" altimg-valign="-9px" altimg-width="222px" alttext="\displaystyle=\tfrac{20231}{80\;85000}\alpha^{5}-\tfrac{551}{1\;61700}\alpha^{%
2}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mfrac><mn>20231</mn><mn>80 85000</mn></mfrac><mo></mo><msup><mi href="./10.21#E24">α</mi><mn>5</mn></msup></mrow><mo>-</mo><mrow><mfrac><mn>551</mn><mn>1 61700</mn></mfrac><mo></mo><msup><mi href="./10.21#E24">α</mi><mn>2</mn></msup></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E25.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./10.21#E24" title="(10.21.24) ‣ §10.21(vii) Asymptotic Expansions for Large Order ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m171.png" altimg-height="13px" altimg-valign="-2px" altimg-width="17px" alttext="\alpha" display="inline"><mi href="./10.21#E24">α</mi></math>: coefficients</a></dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E26" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex25" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="5" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">10.21.26</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m72.png" altimg-height="23px" altimg-valign="-6px" altimg-width="26px" alttext="\displaystyle\beta_{0}" display="inline"><msub><mi href="./10.21#E26">β</mi><mn>0</mn></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m13.png" altimg-height="22px" altimg-valign="-6px" altimg-width="43px" alttext="\displaystyle=1," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mn>1</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex26" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m73.png" altimg-height="23px" altimg-valign="-6px" altimg-width="26px" alttext="\displaystyle\beta_{1}" display="inline"><msub><mi href="./10.21#E26">β</mi><mn>1</mn></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m10.png" altimg-height="29px" altimg-valign="-9px" altimg-width="74px" alttext="\displaystyle=-\tfrac{4}{5}\alpha," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo>-</mo><mrow><mfrac><mn>4</mn><mn>5</mn></mfrac><mo></mo><mi href="./10.21#E24">α</mi></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex27" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m74.png" altimg-height="23px" altimg-valign="-6px" altimg-width="26px" alttext="\displaystyle\beta_{2}" display="inline"><msub><mi href="./10.21#E26">β</mi><mn>2</mn></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m38.png" altimg-height="30px" altimg-valign="-9px" altimg-width="75px" alttext="\displaystyle=\tfrac{18}{35}\alpha^{2}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mfrac><mn>18</mn><mn>35</mn></mfrac><mo></mo><msup><mi href="./10.21#E24">α</mi><mn>2</mn></msup></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex28" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m75.png" altimg-height="23px" altimg-valign="-6px" altimg-width="26px" alttext="\displaystyle\beta_{3}" display="inline"><msub><mi href="./10.21#E26">β</mi><mn>3</mn></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m11.png" altimg-height="30px" altimg-valign="-9px" altimg-width="159px" alttext="\displaystyle=-\tfrac{88}{315}\alpha^{3}-\tfrac{11}{1575}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mo>-</mo><mrow><mfrac><mn>88</mn><mn>315</mn></mfrac><mo></mo><msup><mi href="./10.21#E24">α</mi><mn>3</mn></msup></mrow></mrow><mo>-</mo><mfrac><mn>11</mn><mn>1575</mn></mfrac></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex29" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m76.png" altimg-height="23px" altimg-valign="-6px" altimg-width="26px" alttext="\displaystyle\beta_{4}" display="inline"><msub><mi href="./10.21#E26">β</mi><mn>4</mn></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m42.png" altimg-height="30px" altimg-valign="-9px" altimg-width="205px" alttext="\displaystyle=\tfrac{79586}{6\;06375}\alpha^{4}+\tfrac{9824}{6\;06375}\alpha." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mfrac><mn>79586</mn><mn>6 06375</mn></mfrac><mo></mo><msup><mi href="./10.21#E24">α</mi><mn>4</mn></msup></mrow><mo>+</mo><mrow><mfrac><mn>9824</mn><mn>6 06375</mn></mfrac><mo></mo><mi href="./10.21#E24">α</mi></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E26.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text"><math class="ltx_Math" altimg="m177.png" altimg-height="21px" altimg-valign="-6px" altimg-width="17px" alttext="\beta" display="inline"><mi href="./10.21#E26">β</mi></math>: coefficients (locally)</span></dd>
<dt>Symbols:</dt>
<dd><a href="./10.21#E24" title="(10.21.24) ‣ §10.21(vii) Asymptotic Expansions for Large Order ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m171.png" altimg-height="13px" altimg-valign="-2px" altimg-width="17px" alttext="\alpha" display="inline"><mi href="./10.21#E24">α</mi></math>: coefficients</a></dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS7.p3" class="ltx_para">
<p class="ltx_p">As <math class="ltx_Math" altimg="m220.png" altimg-height="13px" altimg-valign="-2px" altimg-width="66px" alttext="\nu\to\infty" display="inline"><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></math> with <math class="ltx_Math" altimg="m297.png" altimg-height="17px" altimg-valign="-2px" altimg-width="11px" alttext="t" display="inline"><mi>t</mi></math> <math class="ltx_Math" altimg="m106.png" altimg-height="27px" altimg-valign="-9px" altimg-width="69px" alttext="(>-\tfrac{1}{6})" display="inline"><mrow><mo stretchy="false">(</mo><mrow><mi></mi><mo>></mo><mrow><mo>-</mo><mfrac><mn>1</mn><mn>6</mn></mfrac></mrow></mrow><mo stretchy="false">)</mo></mrow></math> fixed,</p>
<table id="E27" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.21.27</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E27.png" altimg-height="64px" altimg-valign="-28px" altimg-width="179px" alttext="\sigma_{\nu}(t)\sim\nu\sum_{k=0}^{\infty}\frac{\alpha^{\prime}_{k}}{\nu^{2k/3}}," display="block"><mrow><mrow><mrow><msub><mi href="./10.21#SS2.p1">σ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo href="./2.1#SS3.p1">∼</mo><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./10.1#p2.t1.r2">k</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mfrac><msubsup><mi href="./10.21#E29">α</mi><mi href="./10.1#p2.t1.r2">k</mi><mo>′</mo></msubsup><msup><mi href="./10.1#p2.t1.r5">ν</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r2">k</mi></mrow><mo>/</mo><mn>3</mn></mrow></msup></mfrac></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E27.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./2.1#SS3.p1" title="§2.1(iii) Asymptotic Expansions ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m234.png" altimg-height="12px" altimg-valign="-2px" altimg-width="20px" alttext="\sim" display="inline"><mo href="./2.1#SS3.p1">∼</mo></math>: Poincaré asymptotic expansion</a>,
<a href="./10.1#p2.t1.r2" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m279.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./10.1#p2.t1.r2">k</mi></math>: nonnegative integer</a>,
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m215.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>,
<a href="./10.21#SS2.p1" title="§10.21(ii) Analytic Properties ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m231.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="\sigma_{\nu}(t)" display="inline"><mrow><msub><mi href="./10.21#SS2.p1">σ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math>: zero of cylinder function</a> and
<a href="./10.21#E29" title="(10.21.29) ‣ §10.21(vii) Asymptotic Expansions for Large Order ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m174.png" altimg-height="19px" altimg-valign="-2px" altimg-width="23px" alttext="\alpha^{\prime}" display="inline"><msup><mi href="./10.21#E29">α</mi><mo>′</mo></msup></math>: coefficients</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E28" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.21.28</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E28.png" altimg-height="72px" altimg-valign="-36px" altimg-width="328px" alttext="\mathscr{C}_{\nu}\left(\sigma_{\nu}(t)\right)\sim\frac{(2/\nu)^{\frac{1}{3}}}{%
\pi N\left(-2^{\frac{1}{3}}\alpha^{\prime}\right)}\sum_{k=0}^{\infty}\frac{%
\beta^{\prime}_{k}}{\nu^{2k/3}}," display="block"><mrow><mrow><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5.p1">𝒞</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mrow><msub><mi href="./10.21#SS2.p1">σ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo>)</mo></mrow></mrow><mo href="./2.1#SS3.p1">∼</mo><mrow><mfrac><msup><mrow><mo stretchy="false">(</mo><mrow><mn>2</mn><mo>/</mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo stretchy="false">)</mo></mrow><mfrac><mn>1</mn><mn>3</mn></mfrac></msup><mrow><mi href="./3.12#E1">π</mi><mo></mo><mrow><mi href="./9.8#E7">N</mi><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mrow><msup><mn>2</mn><mfrac><mn>1</mn><mn>3</mn></mfrac></msup><mo></mo><msup><mi href="./10.21#E29">α</mi><mo>′</mo></msup></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mfrac><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./10.1#p2.t1.r2">k</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mfrac><msubsup><mi href="./10.21#E26">β</mi><mi href="./10.1#p2.t1.r2">k</mi><mo>′</mo></msubsup><msup><mi href="./10.1#p2.t1.r5">ν</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r2">k</mi></mrow><mo>/</mo><mn>3</mn></mrow></msup></mfrac></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E28.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./2.1#SS3.p1" title="§2.1(iii) Asymptotic Expansions ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m234.png" altimg-height="12px" altimg-valign="-2px" altimg-width="20px" alttext="\sim" display="inline"><mo href="./2.1#SS3.p1">∼</mo></math>: Poincaré asymptotic expansion</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m226.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./10.2#Px5.p1" title="Cylinder Functions ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m203.png" altimg-height="23px" altimg-valign="-7px" altimg-width="56px" alttext="\mathscr{C}_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5.p1">𝒞</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: cylinder function</a>,
<a href="./9.8#E7" title="(9.8.7) ‣ §9.8(i) Definitions ‣ §9.8 Modulus and Phase ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m148.png" altimg-height="23px" altimg-valign="-7px" altimg-width="51px" alttext="N\left(\NVar{z}\right)" display="inline"><mrow><mi href="./9.8#E7">N</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./9.1#p2.t1.r3">z</mi><mo>)</mo></mrow></mrow></math>: Airy modulus function</a>,
<a href="./10.1#p2.t1.r2" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m279.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./10.1#p2.t1.r2">k</mi></math>: nonnegative integer</a>,
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m215.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>,
<a href="./10.21#SS2.p1" title="§10.21(ii) Analytic Properties ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m231.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="\sigma_{\nu}(t)" display="inline"><mrow><msub><mi href="./10.21#SS2.p1">σ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math>: zero of cylinder function</a>,
<a href="./10.21#E26" title="(10.21.26) ‣ §10.21(vii) Asymptotic Expansions for Large Order ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m177.png" altimg-height="21px" altimg-valign="-6px" altimg-width="17px" alttext="\beta" display="inline"><mi href="./10.21#E26">β</mi></math>: coefficients</a> and
<a href="./10.21#E29" title="(10.21.29) ‣ §10.21(vii) Asymptotic Expansions for Large Order ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m174.png" altimg-height="19px" altimg-valign="-2px" altimg-width="23px" alttext="\alpha^{\prime}" display="inline"><msup><mi href="./10.21#E29">α</mi><mo>′</mo></msup></math>: coefficients</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m174.png" altimg-height="19px" altimg-valign="-2px" altimg-width="23px" alttext="\alpha^{\prime}" display="inline"><msup><mi href="./10.21#E29">α</mi><mo>′</mo></msup></math> is given by
</p>
<table id="E29" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.21.29</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E29.png" altimg-height="41px" altimg-valign="-15px" altimg-width="153px" alttext="\phi\left(-2^{\frac{1}{3}}\alpha^{\prime}\right)=\pi t," display="block"><mrow><mrow><mrow><mi href="./9.8#E8">ϕ</mi><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mrow><msup><mn>2</mn><mfrac><mn>1</mn><mn>3</mn></mfrac></msup><mo></mo><msup><mi href="./10.21#E29">α</mi><mo>′</mo></msup></mrow></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi>t</mi></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E29.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text"><math class="ltx_Math" altimg="m174.png" altimg-height="19px" altimg-valign="-2px" altimg-width="23px" alttext="\alpha^{\prime}" display="inline"><msup><mi href="./10.21#E29">α</mi><mo>′</mo></msup></math>: coefficients (locally)</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m226.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a> and
<a href="./9.8#E8" title="(9.8.8) ‣ §9.8(i) Definitions ‣ §9.8 Modulus and Phase ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m222.png" altimg-height="23px" altimg-valign="-7px" altimg-width="45px" alttext="\phi\left(\NVar{z}\right)" display="inline"><mrow><mi href="./9.8#E8">ϕ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./9.1#p2.t1.r3">z</mi><mo>)</mo></mrow></mrow></math>: Airy phase function</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and</p>
<table id="E30" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex30" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="5" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">10.21.30</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m57.png" altimg-height="26px" altimg-valign="-7px" altimg-width="28px" alttext="\displaystyle\alpha^{\prime}_{0}" display="inline"><msubsup><mi href="./10.21#E29">α</mi><mn>0</mn><mo>′</mo></msubsup></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m13.png" altimg-height="22px" altimg-valign="-6px" altimg-width="43px" alttext="\displaystyle=1," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mn>1</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex31" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m58.png" altimg-height="26px" altimg-valign="-7px" altimg-width="28px" alttext="\displaystyle\alpha^{\prime}_{1}" display="inline"><msubsup><mi href="./10.21#E29">α</mi><mn>1</mn><mo>′</mo></msubsup></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m16.png" altimg-height="25px" altimg-valign="-6px" altimg-width="51px" alttext="\displaystyle=\alpha^{\prime}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><msup><mi href="./10.21#E29">α</mi><mo>′</mo></msup></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex32" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m59.png" altimg-height="26px" altimg-valign="-7px" altimg-width="28px" alttext="\displaystyle\alpha^{\prime}_{2}" display="inline"><msubsup><mi href="./10.21#E29">α</mi><mn>2</mn><mo>′</mo></msubsup></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m41.png" altimg-height="32px" altimg-valign="-9px" altimg-width="165px" alttext="\displaystyle=\tfrac{3}{10}{\alpha^{\prime}}^{2}-\tfrac{1}{10}{\alpha^{\prime}%
}^{-1}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mfrac><mn>3</mn><mn>10</mn></mfrac><mo></mo><mmultiscripts><mi href="./10.21#E29">α</mi><none></none><mo>′</mo><none></none><mn>2</mn></mmultiscripts></mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>10</mn></mfrac><mo></mo><mmultiscripts><mi href="./10.21#E29">α</mi><none></none><mo>′</mo><none></none><mrow><mo>-</mo><mn>1</mn></mrow></mmultiscripts></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex33" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m60.png" altimg-height="26px" altimg-valign="-7px" altimg-width="28px" alttext="\displaystyle\alpha^{\prime}_{3}" display="inline"><msubsup><mi href="./10.21#E29">α</mi><mn>3</mn><mo>′</mo></msubsup></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m6.png" altimg-height="32px" altimg-valign="-9px" altimg-width="242px" alttext="\displaystyle=-\tfrac{1}{350}{\alpha^{\prime}}^{3}-\tfrac{1}{25}-\tfrac{1}{200%
}{\alpha^{\prime}}^{-3}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>350</mn></mfrac><mo></mo><mmultiscripts><mi href="./10.21#E29">α</mi><none></none><mo>′</mo><none></none><mn>3</mn></mmultiscripts></mrow></mrow><mo>-</mo><mfrac><mn>1</mn><mn>25</mn></mfrac><mo>-</mo><mrow><mfrac><mn>1</mn><mn>200</mn></mfrac><mo></mo><mmultiscripts><mi href="./10.21#E29">α</mi><none></none><mo>′</mo><none></none><mrow><mo>-</mo><mn>3</mn></mrow></mmultiscripts></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex34" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m61.png" altimg-height="26px" altimg-valign="-7px" altimg-width="28px" alttext="\displaystyle\alpha^{\prime}_{4}" display="inline"><msubsup><mi href="./10.21#E29">α</mi><mn>4</mn><mo>′</mo></msubsup></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m9.png" altimg-height="32px" altimg-valign="-9px" altimg-width="409px" alttext="\displaystyle=-\tfrac{479}{63000}{\alpha^{\prime}}^{4}+\tfrac{509}{31500}%
\alpha^{\prime}+\tfrac{1}{1500}{\alpha^{\prime}}^{-2}-\tfrac{1}{2000}{\alpha^{%
\prime}}^{-5}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mrow><mo>-</mo><mrow><mfrac><mn>479</mn><mn>63000</mn></mfrac><mo></mo><mmultiscripts><mi href="./10.21#E29">α</mi><none></none><mo>′</mo><none></none><mn>4</mn></mmultiscripts></mrow></mrow><mo>+</mo><mrow><mfrac><mn>509</mn><mn>31500</mn></mfrac><mo></mo><msup><mi href="./10.21#E29">α</mi><mo>′</mo></msup></mrow><mo>+</mo><mrow><mfrac><mn>1</mn><mn>1500</mn></mfrac><mo></mo><mmultiscripts><mi href="./10.21#E29">α</mi><none></none><mo>′</mo><none></none><mrow><mo>-</mo><mn>2</mn></mrow></mmultiscripts></mrow></mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2000</mn></mfrac><mo></mo><mmultiscripts><mi href="./10.21#E29">α</mi><none></none><mo>′</mo><none></none><mrow><mo>-</mo><mn>5</mn></mrow></mmultiscripts></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E30.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./10.21#E29" title="(10.21.29) ‣ §10.21(vii) Asymptotic Expansions for Large Order ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m174.png" altimg-height="19px" altimg-valign="-2px" altimg-width="23px" alttext="\alpha^{\prime}" display="inline"><msup><mi href="./10.21#E29">α</mi><mo>′</mo></msup></math>: coefficients</a></dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E31" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex35" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="4" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">10.21.31</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m68.png" altimg-height="26px" altimg-valign="-7px" altimg-width="26px" alttext="\displaystyle\beta^{\prime}_{0}" display="inline"><msubsup><mi href="./10.21#E26">β</mi><mn>0</mn><mo>′</mo></msubsup></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m13.png" altimg-height="22px" altimg-valign="-6px" altimg-width="43px" alttext="\displaystyle=1," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mn>1</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex36" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m69.png" altimg-height="26px" altimg-valign="-7px" altimg-width="26px" alttext="\displaystyle\beta^{\prime}_{1}" display="inline"><msubsup><mi href="./10.21#E26">β</mi><mn>1</mn><mo>′</mo></msubsup></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m7.png" altimg-height="29px" altimg-valign="-9px" altimg-width="79px" alttext="\displaystyle=-\tfrac{1}{5}\alpha^{\prime}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>5</mn></mfrac><mo></mo><msup><mi href="./10.21#E29">α</mi><mo>′</mo></msup></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex37" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m70.png" altimg-height="26px" altimg-valign="-7px" altimg-width="26px" alttext="\displaystyle\beta^{\prime}_{2}" display="inline"><msubsup><mi href="./10.21#E26">β</mi><mn>2</mn><mo>′</mo></msubsup></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m44.png" altimg-height="32px" altimg-valign="-9px" altimg-width="181px" alttext="\displaystyle=\tfrac{9}{350}{\alpha^{\prime}}^{2}+\tfrac{1}{100}{\alpha^{%
\prime}}^{-1}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mfrac><mn>9</mn><mn>350</mn></mfrac><mo></mo><mmultiscripts><mi href="./10.21#E29">α</mi><none></none><mo>′</mo><none></none><mn>2</mn></mmultiscripts></mrow><mo>+</mo><mrow><mfrac><mn>1</mn><mn>100</mn></mfrac><mo></mo><mmultiscripts><mi href="./10.21#E29">α</mi><none></none><mo>′</mo><none></none><mrow><mo>-</mo><mn>1</mn></mrow></mmultiscripts></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex38" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m71.png" altimg-height="26px" altimg-valign="-7px" altimg-width="26px" alttext="\displaystyle\beta^{\prime}_{3}" display="inline"><msubsup><mi href="./10.21#E26">β</mi><mn>3</mn><mo>′</mo></msubsup></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m43.png" altimg-height="32px" altimg-valign="-9px" altimg-width="266px" alttext="\displaystyle=\tfrac{89}{15750}{\alpha^{\prime}}^{3}-\tfrac{47}{4500}+\tfrac{1%
}{3000}{\alpha^{\prime}}^{-3}." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mrow><mfrac><mn>89</mn><mn>15750</mn></mfrac><mo></mo><mmultiscripts><mi href="./10.21#E29">α</mi><none></none><mo>′</mo><none></none><mn>3</mn></mmultiscripts></mrow><mo>-</mo><mfrac><mn>47</mn><mn>4500</mn></mfrac></mrow><mo>+</mo><mrow><mfrac><mn>1</mn><mn>3000</mn></mfrac><mo></mo><mmultiscripts><mi href="./10.21#E29">α</mi><none></none><mo>′</mo><none></none><mrow><mo>-</mo><mn>3</mn></mrow></mmultiscripts></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E31.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.21#E26" title="(10.21.26) ‣ §10.21(vii) Asymptotic Expansions for Large Order ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m177.png" altimg-height="21px" altimg-valign="-6px" altimg-width="17px" alttext="\beta" display="inline"><mi href="./10.21#E26">β</mi></math>: coefficients</a> and
<a href="./10.21#E29" title="(10.21.29) ‣ §10.21(vii) Asymptotic Expansions for Large Order ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m174.png" altimg-height="19px" altimg-valign="-2px" altimg-width="23px" alttext="\alpha^{\prime}" display="inline"><msup><mi href="./10.21#E29">α</mi><mo>′</mo></msup></math>: coefficients</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS7.p4" class="ltx_para">
<p class="ltx_p">In particular, with the notation as below,</p>
<table id="E32" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.21.32</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E32.png" altimg-height="64px" altimg-valign="-28px" altimg-width="171px" alttext="j_{\nu,m}\sim\nu\sum_{k=0}^{\infty}\frac{\alpha_{k}}{\nu^{2k/3}}," display="block"><mrow><mrow><msub><mi href="./10.21#SS1.p2">j</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi href="./10.1#p2.t1.r1">m</mi></mrow></msub><mo href="./2.1#SS3.p1">∼</mo><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./10.1#p2.t1.r2">k</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mfrac><msub><mi href="./10.21#E24">α</mi><mi href="./10.1#p2.t1.r2">k</mi></msub><msup><mi href="./10.1#p2.t1.r5">ν</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r2">k</mi></mrow><mo>/</mo><mn>3</mn></mrow></msup></mfrac></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E32.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./2.1#SS3.p1" title="§2.1(iii) Asymptotic Expansions ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m234.png" altimg-height="12px" altimg-valign="-2px" altimg-width="20px" alttext="\sim" display="inline"><mo href="./2.1#SS3.p1">∼</mo></math>: Poincaré asymptotic expansion</a>,
<a href="./10.21#SS1.p2" title="§10.21(i) Distribution ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m273.png" altimg-height="22px" altimg-valign="-8px" altimg-width="41px" alttext="j_{\NVar{\nu},\NVar{m}}" display="inline"><msub><mi href="./10.21#SS1.p2">j</mi><mrow><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r1">m</mi></mrow></msub></math>: zeros of the Bessel function <math class="ltx_Math" altimg="m141.png" altimg-height="23px" altimg-valign="-7px" altimg-width="55px" alttext="J_{\nu}\left(x\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math></a>,
<a href="./10.1#p2.t1.r1" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m284.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./10.1#p2.t1.r1">m</mi></math>: integer</a>,
<a href="./10.1#p2.t1.r2" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m279.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./10.1#p2.t1.r2">k</mi></math>: nonnegative integer</a>,
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m215.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a> and
<a href="./10.21#E24" title="(10.21.24) ‣ §10.21(vii) Asymptotic Expansions for Large Order ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m171.png" altimg-height="13px" altimg-valign="-2px" altimg-width="17px" alttext="\alpha" display="inline"><mi href="./10.21#E24">α</mi></math>: coefficients</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E33" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.21.33</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E33.png" altimg-height="64px" altimg-valign="-28px" altimg-width="172px" alttext="y_{\nu,m}\sim\nu\sum_{k=0}^{\infty}\frac{\alpha_{k}}{\nu^{2k/3}}," display="block"><mrow><mrow><msub><mi href="./10.21#SS1.p2">y</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi href="./10.1#p2.t1.r1">m</mi></mrow></msub><mo href="./2.1#SS3.p1">∼</mo><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./10.1#p2.t1.r2">k</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mfrac><msub><mi href="./10.21#E24">α</mi><mi href="./10.1#p2.t1.r2">k</mi></msub><msup><mi href="./10.1#p2.t1.r5">ν</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r2">k</mi></mrow><mo>/</mo><mn>3</mn></mrow></msup></mfrac></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E33.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./2.1#SS3.p1" title="§2.1(iii) Asymptotic Expansions ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m234.png" altimg-height="12px" altimg-valign="-2px" altimg-width="20px" alttext="\sim" display="inline"><mo href="./2.1#SS3.p1">∼</mo></math>: Poincaré asymptotic expansion</a>,
<a href="./10.21#SS1.p2" title="§10.21(i) Distribution ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m303.png" altimg-height="18px" altimg-valign="-8px" altimg-width="43px" alttext="y_{\NVar{\nu},\NVar{m}}" display="inline"><msub><mi href="./10.21#SS1.p2">y</mi><mrow><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r1">m</mi></mrow></msub></math>: zeros of the Bessel function <math class="ltx_Math" altimg="m157.png" altimg-height="23px" altimg-valign="-7px" altimg-width="56px" alttext="Y_{\nu}\left(x\right)" display="inline"><mrow><msub><mi href="./10.2#E3">Y</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math></a>,
<a href="./10.1#p2.t1.r1" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m284.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./10.1#p2.t1.r1">m</mi></math>: integer</a>,
<a href="./10.1#p2.t1.r2" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m279.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./10.1#p2.t1.r2">k</mi></math>: nonnegative integer</a>,
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m215.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a> and
<a href="./10.21#E24" title="(10.21.24) ‣ §10.21(vii) Asymptotic Expansions for Large Order ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m171.png" altimg-height="13px" altimg-valign="-2px" altimg-width="17px" alttext="\alpha" display="inline"><mi href="./10.21#E24">α</mi></math>: coefficients</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E34" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.21.34</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E34.png" altimg-height="64px" altimg-valign="-28px" altimg-width="338px" alttext="J_{\nu}'\left(j_{\nu,m}\right)\sim(-1)^{m}\frac{(2/\nu)^{\frac{2}{3}}}{\pi M%
\left(a_{m}\right)}\sum_{k=0}^{\infty}\frac{\beta_{k}}{\nu^{2k/3}}," display="block"><mrow><mrow><mrow><msubsup><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′</mo></msubsup><mo></mo><mrow><mo>(</mo><msub><mi href="./10.21#SS1.p2">j</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi href="./10.1#p2.t1.r1">m</mi></mrow></msub><mo>)</mo></mrow></mrow><mo href="./2.1#SS3.p1">∼</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./10.1#p2.t1.r1">m</mi></msup><mo></mo><mfrac><msup><mrow><mo stretchy="false">(</mo><mrow><mn>2</mn><mo>/</mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo stretchy="false">)</mo></mrow><mfrac><mn>2</mn><mn>3</mn></mfrac></msup><mrow><mi href="./3.12#E1">π</mi><mo></mo><mrow><mi href="./9.8#E3">M</mi><mo></mo><mrow><mo>(</mo><msub><mi href="./9.9#SS1.p1">a</mi><mi href="./10.1#p2.t1.r1">m</mi></msub><mo>)</mo></mrow></mrow></mrow></mfrac><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./10.1#p2.t1.r2">k</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mfrac><msub><mi href="./10.21#E26">β</mi><mi href="./10.1#p2.t1.r2">k</mi></msub><msup><mi href="./10.1#p2.t1.r5">ν</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r2">k</mi></mrow><mo>/</mo><mn>3</mn></mrow></msup></mfrac></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E34.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m134.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./2.1#SS3.p1" title="§2.1(iii) Asymptotic Expansions ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m234.png" altimg-height="12px" altimg-valign="-2px" altimg-width="20px" alttext="\sim" display="inline"><mo href="./2.1#SS3.p1">∼</mo></math>: Poincaré asymptotic expansion</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m226.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./9.8#E3" title="(9.8.3) ‣ §9.8(i) Definitions ‣ §9.8 Modulus and Phase ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m146.png" altimg-height="23px" altimg-valign="-7px" altimg-width="55px" alttext="M\left(\NVar{z}\right)" display="inline"><mrow><mi href="./9.8#E3">M</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./9.1#p2.t1.r3">z</mi><mo>)</mo></mrow></mrow></math>: Airy modulus function</a>,
<a href="./9.9#SS1.p1" title="§9.9(i) Distribution and Notation ‣ §9.9 Zeros ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m250.png" altimg-height="16px" altimg-valign="-5px" altimg-width="25px" alttext="a_{\NVar{k}}" display="inline"><msub><mi href="./9.9#SS1.p1">a</mi><mi class="ltx_nvar" href="./9.1#p2.t1.r1">k</mi></msub></math>: <math class="ltx_Math" altimg="m279.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./9.1#p2.t1.r1">k</mi></math>th zero of Airy <math class="ltx_Math" altimg="m193.png" altimg-height="18px" altimg-valign="-2px" altimg-width="25px" alttext="\mathrm{Ai}" display="inline"><mi href="./9.2#i">Ai</mi></math></a>,
<a href="./10.21#SS1.p2" title="§10.21(i) Distribution ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m273.png" altimg-height="22px" altimg-valign="-8px" altimg-width="41px" alttext="j_{\NVar{\nu},\NVar{m}}" display="inline"><msub><mi href="./10.21#SS1.p2">j</mi><mrow><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r1">m</mi></mrow></msub></math>: zeros of the Bessel function <math class="ltx_Math" altimg="m141.png" altimg-height="23px" altimg-valign="-7px" altimg-width="55px" alttext="J_{\nu}\left(x\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math></a>,
<a href="./10.1#p2.t1.r1" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m284.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./10.1#p2.t1.r1">m</mi></math>: integer</a>,
<a href="./10.1#p2.t1.r2" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m279.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./10.1#p2.t1.r2">k</mi></math>: nonnegative integer</a>,
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m215.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a> and
<a href="./10.21#E26" title="(10.21.26) ‣ §10.21(vii) Asymptotic Expansions for Large Order ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m177.png" altimg-height="21px" altimg-valign="-6px" altimg-width="17px" alttext="\beta" display="inline"><mi href="./10.21#E26">β</mi></math>: coefficients</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E35" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.21.35</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E35.png" altimg-height="64px" altimg-valign="-28px" altimg-width="359px" alttext="Y_{\nu}'\left(y_{\nu,m}\right)\sim(-1)^{m-1}\frac{(2/\nu)^{\frac{2}{3}}}{\pi M%
\left(b_{m}\right)}\sum_{k=0}^{\infty}\frac{\beta_{k}}{\nu^{2k/3}}," display="block"><mrow><mrow><mrow><msubsup><mi href="./10.2#E3">Y</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′</mo></msubsup><mo></mo><mrow><mo>(</mo><msub><mi href="./10.21#SS1.p2">y</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi href="./10.1#p2.t1.r1">m</mi></mrow></msub><mo>)</mo></mrow></mrow><mo href="./2.1#SS3.p1">∼</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mrow><mi href="./10.1#p2.t1.r1">m</mi><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mfrac><msup><mrow><mo stretchy="false">(</mo><mrow><mn>2</mn><mo>/</mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo stretchy="false">)</mo></mrow><mfrac><mn>2</mn><mn>3</mn></mfrac></msup><mrow><mi href="./3.12#E1">π</mi><mo></mo><mrow><mi href="./9.8#E3">M</mi><mo></mo><mrow><mo>(</mo><msub><mi href="./9.9#SS1.p1">b</mi><mi href="./10.1#p2.t1.r1">m</mi></msub><mo>)</mo></mrow></mrow></mrow></mfrac><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./10.1#p2.t1.r2">k</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mfrac><msub><mi href="./10.21#E26">β</mi><mi href="./10.1#p2.t1.r2">k</mi></msub><msup><mi href="./10.1#p2.t1.r5">ν</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r2">k</mi></mrow><mo>/</mo><mn>3</mn></mrow></msup></mfrac></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E35.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E3" title="(10.2.3) ‣ Bessel Function of the Second Kind (Weber’s Function) ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m153.png" altimg-height="23px" altimg-valign="-7px" altimg-width="55px" alttext="Y_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E3">Y</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the second kind</a>,
<a href="./2.1#SS3.p1" title="§2.1(iii) Asymptotic Expansions ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m234.png" altimg-height="12px" altimg-valign="-2px" altimg-width="20px" alttext="\sim" display="inline"><mo href="./2.1#SS3.p1">∼</mo></math>: Poincaré asymptotic expansion</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m226.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./9.8#E3" title="(9.8.3) ‣ §9.8(i) Definitions ‣ §9.8 Modulus and Phase ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m146.png" altimg-height="23px" altimg-valign="-7px" altimg-width="55px" alttext="M\left(\NVar{z}\right)" display="inline"><mrow><mi href="./9.8#E3">M</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./9.1#p2.t1.r3">z</mi><mo>)</mo></mrow></mrow></math>: Airy modulus function</a>,
<a href="./9.9#SS1.p1" title="§9.9(i) Distribution and Notation ‣ §9.9 Zeros ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m257.png" altimg-height="21px" altimg-valign="-5px" altimg-width="23px" alttext="b_{\NVar{k}}" display="inline"><msub><mi href="./9.9#SS1.p1">b</mi><mi class="ltx_nvar" href="./9.1#p2.t1.r1">k</mi></msub></math>: <math class="ltx_Math" altimg="m279.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./9.1#p2.t1.r1">k</mi></math>th zero of Airy <math class="ltx_Math" altimg="m199.png" altimg-height="18px" altimg-valign="-2px" altimg-width="24px" alttext="\mathrm{Bi}" display="inline"><mi href="./9.2#i">Bi</mi></math></a>,
<a href="./10.21#SS1.p2" title="§10.21(i) Distribution ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m303.png" altimg-height="18px" altimg-valign="-8px" altimg-width="43px" alttext="y_{\NVar{\nu},\NVar{m}}" display="inline"><msub><mi href="./10.21#SS1.p2">y</mi><mrow><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r1">m</mi></mrow></msub></math>: zeros of the Bessel function <math class="ltx_Math" altimg="m157.png" altimg-height="23px" altimg-valign="-7px" altimg-width="56px" alttext="Y_{\nu}\left(x\right)" display="inline"><mrow><msub><mi href="./10.2#E3">Y</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math></a>,
<a href="./10.1#p2.t1.r1" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m284.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./10.1#p2.t1.r1">m</mi></math>: integer</a>,
<a href="./10.1#p2.t1.r2" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m279.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./10.1#p2.t1.r2">k</mi></math>: nonnegative integer</a>,
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m215.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a> and
<a href="./10.21#E26" title="(10.21.26) ‣ §10.21(vii) Asymptotic Expansions for Large Order ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m177.png" altimg-height="21px" altimg-valign="-6px" altimg-width="17px" alttext="\beta" display="inline"><mi href="./10.21#E26">β</mi></math>: coefficients</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and</p>
<table id="E36" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.21.36</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E36.png" altimg-height="64px" altimg-valign="-28px" altimg-width="171px" alttext="j_{\nu,m}'\sim\nu\sum_{k=0}^{\infty}\frac{\alpha^{\prime}_{k}}{\nu^{2k/3}}," display="block"><mrow><mrow><msubsup><mi href="./10.21#SS1.p2">j</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi href="./10.1#p2.t1.r1">m</mi></mrow><mo>′</mo></msubsup><mo href="./2.1#SS3.p1">∼</mo><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./10.1#p2.t1.r2">k</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mfrac><msubsup><mi href="./10.21#E29">α</mi><mi href="./10.1#p2.t1.r2">k</mi><mo>′</mo></msubsup><msup><mi href="./10.1#p2.t1.r5">ν</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r2">k</mi></mrow><mo>/</mo><mn>3</mn></mrow></msup></mfrac></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E36.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./2.1#SS3.p1" title="§2.1(iii) Asymptotic Expansions ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m234.png" altimg-height="12px" altimg-valign="-2px" altimg-width="20px" alttext="\sim" display="inline"><mo href="./2.1#SS3.p1">∼</mo></math>: Poincaré asymptotic expansion</a>,
<a href="./10.21#SS1.p2" title="§10.21(i) Distribution ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m273.png" altimg-height="22px" altimg-valign="-8px" altimg-width="41px" alttext="j_{\NVar{\nu},\NVar{m}}" display="inline"><msub><mi href="./10.21#SS1.p2">j</mi><mrow><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r1">m</mi></mrow></msub></math>: zeros of the Bessel function <math class="ltx_Math" altimg="m141.png" altimg-height="23px" altimg-valign="-7px" altimg-width="55px" alttext="J_{\nu}\left(x\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math></a>,
<a href="./10.1#p2.t1.r1" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m284.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./10.1#p2.t1.r1">m</mi></math>: integer</a>,
<a href="./10.1#p2.t1.r2" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m279.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./10.1#p2.t1.r2">k</mi></math>: nonnegative integer</a>,
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m215.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a> and
<a href="./10.21#E29" title="(10.21.29) ‣ §10.21(vii) Asymptotic Expansions for Large Order ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m174.png" altimg-height="19px" altimg-valign="-2px" altimg-width="23px" alttext="\alpha^{\prime}" display="inline"><msup><mi href="./10.21#E29">α</mi><mo>′</mo></msup></math>: coefficients</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E37" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.21.37</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E37.png" altimg-height="64px" altimg-valign="-28px" altimg-width="172px" alttext="y_{\nu,m}'\sim\nu\sum_{k=0}^{\infty}\frac{\alpha^{\prime}_{k}}{\nu^{2k/3}}," display="block"><mrow><mrow><msubsup><mi href="./10.21#SS1.p2">y</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi href="./10.1#p2.t1.r1">m</mi></mrow><mo>′</mo></msubsup><mo href="./2.1#SS3.p1">∼</mo><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./10.1#p2.t1.r2">k</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mfrac><msubsup><mi href="./10.21#E29">α</mi><mi href="./10.1#p2.t1.r2">k</mi><mo>′</mo></msubsup><msup><mi href="./10.1#p2.t1.r5">ν</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r2">k</mi></mrow><mo>/</mo><mn>3</mn></mrow></msup></mfrac></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E37.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./2.1#SS3.p1" title="§2.1(iii) Asymptotic Expansions ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m234.png" altimg-height="12px" altimg-valign="-2px" altimg-width="20px" alttext="\sim" display="inline"><mo href="./2.1#SS3.p1">∼</mo></math>: Poincaré asymptotic expansion</a>,
<a href="./10.21#SS1.p2" title="§10.21(i) Distribution ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m303.png" altimg-height="18px" altimg-valign="-8px" altimg-width="43px" alttext="y_{\NVar{\nu},\NVar{m}}" display="inline"><msub><mi href="./10.21#SS1.p2">y</mi><mrow><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r1">m</mi></mrow></msub></math>: zeros of the Bessel function <math class="ltx_Math" altimg="m157.png" altimg-height="23px" altimg-valign="-7px" altimg-width="56px" alttext="Y_{\nu}\left(x\right)" display="inline"><mrow><msub><mi href="./10.2#E3">Y</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math></a>,
<a href="./10.1#p2.t1.r1" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m284.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./10.1#p2.t1.r1">m</mi></math>: integer</a>,
<a href="./10.1#p2.t1.r2" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m279.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./10.1#p2.t1.r2">k</mi></math>: nonnegative integer</a>,
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m215.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a> and
<a href="./10.21#E29" title="(10.21.29) ‣ §10.21(vii) Asymptotic Expansions for Large Order ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m174.png" altimg-height="19px" altimg-valign="-2px" altimg-width="23px" alttext="\alpha^{\prime}" display="inline"><msup><mi href="./10.21#E29">α</mi><mo>′</mo></msup></math>: coefficients</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E38" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.21.38</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E38.png" altimg-height="64px" altimg-valign="-28px" altimg-width="358px" alttext="J_{\nu}\left(j_{\nu,m}'\right)\sim(-1)^{m-1}\frac{(2/\nu)^{\frac{1}{3}}}{\pi N%
\left(a^{\prime}_{m}\right)}\sum_{k=0}^{\infty}\frac{\beta^{\prime}_{k}}{\nu^{%
2k/3}}," display="block"><mrow><mrow><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><msubsup><mi href="./10.21#SS1.p2">j</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi href="./10.1#p2.t1.r1">m</mi></mrow><mo>′</mo></msubsup><mo>)</mo></mrow></mrow><mo href="./2.1#SS3.p1">∼</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mrow><mi href="./10.1#p2.t1.r1">m</mi><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mfrac><msup><mrow><mo stretchy="false">(</mo><mrow><mn>2</mn><mo>/</mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo stretchy="false">)</mo></mrow><mfrac><mn>1</mn><mn>3</mn></mfrac></msup><mrow><mi href="./3.12#E1">π</mi><mo></mo><mrow><mi href="./9.8#E7">N</mi><mo></mo><mrow><mo>(</mo><msubsup><mi href="./9.9#SS1.p1">a</mi><mi href="./10.1#p2.t1.r1">m</mi><mo href="./9.9#SS1.p1">′</mo></msubsup><mo>)</mo></mrow></mrow></mrow></mfrac><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./10.1#p2.t1.r2">k</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mfrac><msubsup><mi href="./10.21#E26">β</mi><mi href="./10.1#p2.t1.r2">k</mi><mo>′</mo></msubsup><msup><mi href="./10.1#p2.t1.r5">ν</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r2">k</mi></mrow><mo>/</mo><mn>3</mn></mrow></msup></mfrac></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E38.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m134.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./2.1#SS3.p1" title="§2.1(iii) Asymptotic Expansions ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m234.png" altimg-height="12px" altimg-valign="-2px" altimg-width="20px" alttext="\sim" display="inline"><mo href="./2.1#SS3.p1">∼</mo></math>: Poincaré asymptotic expansion</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m226.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./9.8#E7" title="(9.8.7) ‣ §9.8(i) Definitions ‣ §9.8 Modulus and Phase ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m148.png" altimg-height="23px" altimg-valign="-7px" altimg-width="51px" alttext="N\left(\NVar{z}\right)" display="inline"><mrow><mi href="./9.8#E7">N</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./9.1#p2.t1.r3">z</mi><mo>)</mo></mrow></mrow></math>: Airy modulus function</a>,
<a href="./10.21#SS1.p2" title="§10.21(i) Distribution ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m273.png" altimg-height="22px" altimg-valign="-8px" altimg-width="41px" alttext="j_{\NVar{\nu},\NVar{m}}" display="inline"><msub><mi href="./10.21#SS1.p2">j</mi><mrow><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r1">m</mi></mrow></msub></math>: zeros of the Bessel function <math class="ltx_Math" altimg="m141.png" altimg-height="23px" altimg-valign="-7px" altimg-width="55px" alttext="J_{\nu}\left(x\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math></a>,
<a href="./9.9#SS1.p1" title="§9.9(i) Distribution and Notation ‣ §9.9 Zeros ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m248.png" altimg-height="24px" altimg-valign="-8px" altimg-width="25px" alttext="a^{\prime}_{\NVar{k}}" display="inline"><msubsup><mi href="./9.9#SS1.p1">a</mi><mi class="ltx_nvar" href="./9.1#p2.t1.r1">k</mi><mo href="./9.9#SS1.p1">′</mo></msubsup></math>: <math class="ltx_Math" altimg="m279.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./9.1#p2.t1.r1">k</mi></math>th zero of Airy <math class="ltx_Math" altimg="m190.png" altimg-height="20px" altimg-valign="-2px" altimg-width="30px" alttext="\mathrm{Ai}'" display="inline"><msup><mi href="./9.2#i">Ai</mi><mo>′</mo></msup></math></a>,
<a href="./10.1#p2.t1.r1" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m284.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./10.1#p2.t1.r1">m</mi></math>: integer</a>,
<a href="./10.1#p2.t1.r2" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m279.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./10.1#p2.t1.r2">k</mi></math>: nonnegative integer</a>,
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m215.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a> and
<a href="./10.21#E26" title="(10.21.26) ‣ §10.21(vii) Asymptotic Expansions for Large Order ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m177.png" altimg-height="21px" altimg-valign="-6px" altimg-width="17px" alttext="\beta" display="inline"><mi href="./10.21#E26">β</mi></math>: coefficients</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E39" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.21.39</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E39.png" altimg-height="64px" altimg-valign="-28px" altimg-width="358px" alttext="Y_{\nu}\left(y_{\nu,m}'\right)\sim(-1)^{m-1}\frac{(2/\nu)^{\frac{1}{3}}}{\pi N%
\left(b^{\prime}_{m}\right)}\sum_{k=0}^{\infty}\frac{\beta^{\prime}_{k}}{\nu^{%
2k/3}}." display="block"><mrow><mrow><mrow><msub><mi href="./10.2#E3">Y</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><msubsup><mi href="./10.21#SS1.p2">y</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi href="./10.1#p2.t1.r1">m</mi></mrow><mo>′</mo></msubsup><mo>)</mo></mrow></mrow><mo href="./2.1#SS3.p1">∼</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mrow><mi href="./10.1#p2.t1.r1">m</mi><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mfrac><msup><mrow><mo stretchy="false">(</mo><mrow><mn>2</mn><mo>/</mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo stretchy="false">)</mo></mrow><mfrac><mn>1</mn><mn>3</mn></mfrac></msup><mrow><mi href="./3.12#E1">π</mi><mo></mo><mrow><mi href="./9.8#E7">N</mi><mo></mo><mrow><mo>(</mo><msubsup><mi href="./9.9#SS1.p1">b</mi><mi href="./10.1#p2.t1.r1">m</mi><mo href="./9.9#SS1.p1">′</mo></msubsup><mo>)</mo></mrow></mrow></mrow></mfrac><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./10.1#p2.t1.r2">k</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mfrac><msubsup><mi href="./10.21#E26">β</mi><mi href="./10.1#p2.t1.r2">k</mi><mo>′</mo></msubsup><msup><mi href="./10.1#p2.t1.r5">ν</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r2">k</mi></mrow><mo>/</mo><mn>3</mn></mrow></msup></mfrac></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E39.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E3" title="(10.2.3) ‣ Bessel Function of the Second Kind (Weber’s Function) ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m153.png" altimg-height="23px" altimg-valign="-7px" altimg-width="55px" alttext="Y_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E3">Y</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the second kind</a>,
<a href="./2.1#SS3.p1" title="§2.1(iii) Asymptotic Expansions ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m234.png" altimg-height="12px" altimg-valign="-2px" altimg-width="20px" alttext="\sim" display="inline"><mo href="./2.1#SS3.p1">∼</mo></math>: Poincaré asymptotic expansion</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m226.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./9.8#E7" title="(9.8.7) ‣ §9.8(i) Definitions ‣ §9.8 Modulus and Phase ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m148.png" altimg-height="23px" altimg-valign="-7px" altimg-width="51px" alttext="N\left(\NVar{z}\right)" display="inline"><mrow><mi href="./9.8#E7">N</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./9.1#p2.t1.r3">z</mi><mo>)</mo></mrow></mrow></math>: Airy modulus function</a>,
<a href="./10.21#SS1.p2" title="§10.21(i) Distribution ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m303.png" altimg-height="18px" altimg-valign="-8px" altimg-width="43px" alttext="y_{\NVar{\nu},\NVar{m}}" display="inline"><msub><mi href="./10.21#SS1.p2">y</mi><mrow><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r1">m</mi></mrow></msub></math>: zeros of the Bessel function <math class="ltx_Math" altimg="m157.png" altimg-height="23px" altimg-valign="-7px" altimg-width="56px" alttext="Y_{\nu}\left(x\right)" display="inline"><mrow><msub><mi href="./10.2#E3">Y</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math></a>,
<a href="./9.9#SS1.p1" title="§9.9(i) Distribution and Notation ‣ §9.9 Zeros ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m255.png" altimg-height="24px" altimg-valign="-8px" altimg-width="23px" alttext="b^{\prime}_{\NVar{k}}" display="inline"><msubsup><mi href="./9.9#SS1.p1">b</mi><mi class="ltx_nvar" href="./9.1#p2.t1.r1">k</mi><mo href="./9.9#SS1.p1">′</mo></msubsup></math>: <math class="ltx_Math" altimg="m279.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./9.1#p2.t1.r1">k</mi></math>th zero of Airy <math class="ltx_Math" altimg="m197.png" altimg-height="20px" altimg-valign="-2px" altimg-width="30px" alttext="\mathrm{Bi}'" display="inline"><msup><mi href="./9.2#i">Bi</mi><mo>′</mo></msup></math></a>,
<a href="./10.1#p2.t1.r1" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m284.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./10.1#p2.t1.r1">m</mi></math>: integer</a>,
<a href="./10.1#p2.t1.r2" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m279.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./10.1#p2.t1.r2">k</mi></math>: nonnegative integer</a>,
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m215.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a> and
<a href="./10.21#E26" title="(10.21.26) ‣ §10.21(vii) Asymptotic Expansions for Large Order ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m177.png" altimg-height="21px" altimg-valign="-6px" altimg-width="17px" alttext="\beta" display="inline"><mi href="./10.21#E26">β</mi></math>: coefficients</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Here <math class="ltx_Math" altimg="m251.png" altimg-height="16px" altimg-valign="-5px" altimg-width="30px" alttext="a_{m}" display="inline"><msub><mi href="./9.9#SS1.p1">a</mi><mi href="./10.1#p2.t1.r1">m</mi></msub></math>, <math class="ltx_Math" altimg="m258.png" altimg-height="21px" altimg-valign="-5px" altimg-width="28px" alttext="b_{m}" display="inline"><msub><mi href="./9.9#SS1.p1">b</mi><mi href="./10.1#p2.t1.r1">m</mi></msub></math>, <math class="ltx_Math" altimg="m249.png" altimg-height="23px" altimg-valign="-7px" altimg-width="30px" alttext="a^{\prime}_{m}" display="inline"><msubsup><mi href="./9.9#SS1.p1">a</mi><mi href="./10.1#p2.t1.r1">m</mi><mo href="./9.9#SS1.p1">′</mo></msubsup></math>, <math class="ltx_Math" altimg="m256.png" altimg-height="23px" altimg-valign="-7px" altimg-width="28px" alttext="b^{\prime}_{m}" display="inline"><msubsup><mi href="./9.9#SS1.p1">b</mi><mi href="./10.1#p2.t1.r1">m</mi><mo href="./9.9#SS1.p1">′</mo></msubsup></math> are
the <math class="ltx_Math" altimg="m284.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./10.1#p2.t1.r1">m</mi></math>th negative zeros of
<math class="ltx_Math" altimg="m195.png" altimg-height="23px" altimg-valign="-7px" altimg-width="55px" alttext="\mathrm{Ai}\left(x\right)" display="inline"><mrow><mi href="./9.2#i">Ai</mi><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>, <math class="ltx_Math" altimg="m200.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="\mathrm{Bi}\left(x\right)" display="inline"><mrow><mi href="./9.2#i">Bi</mi><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>, <math class="ltx_Math" altimg="m191.png" altimg-height="25px" altimg-valign="-7px" altimg-width="61px" alttext="\mathrm{Ai}'\left(x\right)" display="inline"><mrow><msup><mi href="./9.2#i">Ai</mi><mo>′</mo></msup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>, <math class="ltx_Math" altimg="m198.png" altimg-height="25px" altimg-valign="-7px" altimg-width="60px" alttext="\mathrm{Bi}'\left(x\right)" display="inline"><mrow><msup><mi href="./9.2#i">Bi</mi><mo>′</mo></msup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>, respectively
(§), <math class="ltx_Math" altimg="m176.png" altimg-height="16px" altimg-valign="-5px" altimg-width="27px" alttext="\alpha_{k}" display="inline"><msub><mi href="./10.21#E24">α</mi><mi href="./10.1#p2.t1.r2">k</mi></msub></math>, <math class="ltx_Math" altimg="m179.png" altimg-height="21px" altimg-valign="-6px" altimg-width="25px" alttext="\beta_{k}" display="inline"><msub><mi href="./10.21#E26">β</mi><mi href="./10.1#p2.t1.r2">k</mi></msub></math>, <math class="ltx_Math" altimg="m175.png" altimg-height="24px" altimg-valign="-8px" altimg-width="27px" alttext="\alpha^{\prime}_{k}" display="inline"><msubsup><mi href="./10.21#E29">α</mi><mi href="./10.1#p2.t1.r2">k</mi><mo>′</mo></msubsup></math>, <math class="ltx_Math" altimg="m178.png" altimg-height="24px" altimg-valign="-8px" altimg-width="25px" alttext="\beta^{\prime}_{k}" display="inline"><msubsup><mi href="./10.21#E26">β</mi><mi href="./10.1#p2.t1.r2">k</mi><mo>′</mo></msubsup></math> are given by
(), with <math class="ltx_Math" altimg="m169.png" altimg-height="26px" altimg-valign="-5px" altimg-width="120px" alttext="\alpha=-2^{-\frac{1}{3}}a_{m}" display="inline"><mrow><mi href="./10.21#E24">α</mi><mo>=</mo><mrow><mo>-</mo><mrow><msup><mn>2</mn><mrow><mo>-</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></mrow></msup><mo></mo><msub><mi href="./9.9#SS1.p1">a</mi><mi href="./10.1#p2.t1.r1">m</mi></msub></mrow></mrow></mrow></math> in the case of
<math class="ltx_Math" altimg="m277.png" altimg-height="22px" altimg-valign="-8px" altimg-width="40px" alttext="j_{\nu,m}" display="inline"><msub><mi href="./10.21#SS1.p2">j</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi href="./10.1#p2.t1.r1">m</mi></mrow></msub></math> and <math class="ltx_Math" altimg="m137.png" altimg-height="24px" altimg-valign="-8px" altimg-width="80px" alttext="J_{\nu}'\left(j_{\nu,m}\right)" display="inline"><mrow><msubsup><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′</mo></msubsup><mo></mo><mrow><mo>(</mo><msub><mi href="./10.21#SS1.p2">j</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi href="./10.1#p2.t1.r1">m</mi></mrow></msub><mo>)</mo></mrow></mrow></math>,
<math class="ltx_Math" altimg="m170.png" altimg-height="26px" altimg-valign="-5px" altimg-width="118px" alttext="\alpha=-2^{-\frac{1}{3}}b_{m}" display="inline"><mrow><mi href="./10.21#E24">α</mi><mo>=</mo><mrow><mo>-</mo><mrow><msup><mn>2</mn><mrow><mo>-</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></mrow></msup><mo></mo><msub><mi href="./9.9#SS1.p1">b</mi><mi href="./10.1#p2.t1.r1">m</mi></msub></mrow></mrow></mrow></math> in the case of <math class="ltx_Math" altimg="m305.png" altimg-height="18px" altimg-valign="-8px" altimg-width="42px" alttext="y_{\nu,m}" display="inline"><msub><mi href="./10.21#SS1.p2">y</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi href="./10.1#p2.t1.r1">m</mi></mrow></msub></math> and
<math class="ltx_Math" altimg="m155.png" altimg-height="24px" altimg-valign="-8px" altimg-width="82px" alttext="Y_{\nu}'\left(y_{\nu,m}\right)" display="inline"><mrow><msubsup><mi href="./10.2#E3">Y</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′</mo></msubsup><mo></mo><mrow><mo>(</mo><msub><mi href="./10.21#SS1.p2">y</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi href="./10.1#p2.t1.r1">m</mi></mrow></msub><mo>)</mo></mrow></mrow></math>, <math class="ltx_Math" altimg="m172.png" altimg-height="27px" altimg-valign="-7px" altimg-width="125px" alttext="\alpha^{\prime}=-2^{-\frac{1}{3}}a^{\prime}_{m}" display="inline"><mrow><msup><mi href="./10.21#E29">α</mi><mo>′</mo></msup><mo>=</mo><mrow><mo>-</mo><mrow><msup><mn>2</mn><mrow><mo>-</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></mrow></msup><mo></mo><msubsup><mi href="./9.9#SS1.p1">a</mi><mi href="./10.1#p2.t1.r1">m</mi><mo href="./9.9#SS1.p1">′</mo></msubsup></mrow></mrow></mrow></math> in
the case of <math class="ltx_Math" altimg="m275.png" altimg-height="26px" altimg-valign="-10px" altimg-width="40px" alttext="j_{\nu,m}'" display="inline"><msubsup><mi href="./10.21#SS1.p2">j</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi href="./10.1#p2.t1.r1">m</mi></mrow><mo>′</mo></msubsup></math> and <math class="ltx_Math" altimg="m140.png" altimg-height="28px" altimg-valign="-10px" altimg-width="83px" alttext="J_{\nu}\left(j_{\nu,m}'\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><msubsup><mi href="./10.21#SS1.p2">j</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi href="./10.1#p2.t1.r1">m</mi></mrow><mo>′</mo></msubsup><mo>)</mo></mrow></mrow></math>,
<math class="ltx_Math" altimg="m173.png" altimg-height="27px" altimg-valign="-7px" altimg-width="123px" alttext="\alpha^{\prime}=-2^{-\frac{1}{3}}b^{\prime}_{m}" display="inline"><mrow><msup><mi href="./10.21#E29">α</mi><mo>′</mo></msup><mo>=</mo><mrow><mo>-</mo><mrow><msup><mn>2</mn><mrow><mo>-</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></mrow></msup><mo></mo><msubsup><mi href="./9.9#SS1.p1">b</mi><mi href="./10.1#p2.t1.r1">m</mi><mo href="./9.9#SS1.p1">′</mo></msubsup></mrow></mrow></mrow></math> in the case of <math class="ltx_Math" altimg="m304.png" altimg-height="26px" altimg-valign="-10px" altimg-width="42px" alttext="y_{\nu,m}'" display="inline"><msubsup><mi href="./10.21#SS1.p2">y</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi href="./10.1#p2.t1.r1">m</mi></mrow><mo>′</mo></msubsup></math> and
<math class="ltx_Math" altimg="m158.png" altimg-height="28px" altimg-valign="-10px" altimg-width="85px" alttext="Y_{\nu}\left(y_{\nu,m}'\right)" display="inline"><mrow><msub><mi href="./10.2#E3">Y</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><msubsup><mi href="./10.21#SS1.p2">y</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi href="./10.1#p2.t1.r1">m</mi></mrow><mo>′</mo></msubsup><mo>)</mo></mrow></mrow></math>.
</p>
</div>
<div id="SS7.p5" class="ltx_para">
<p class="ltx_p">For error bounds for ()</cite>.</p>
</div>
<div id="SS7.p6" class="ltx_para">
<p class="ltx_p">For the first zeros rounded numerical values of the coefficients are given by</p>
</div>
<div id="SS7.p7" class="ltx_para">
<table id="E40" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex39" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="8" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">10.21.40</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m93.png" altimg-height="24px" altimg-valign="-8px" altimg-width="36px" alttext="\displaystyle j_{\nu,1}" display="inline"><msub><mi href="./10.21#SS1.p2">j</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mn>1</mn></mrow></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m90.png" altimg-height="29px" altimg-valign="-6px" altimg-width="719px" alttext="\displaystyle\sim\nu+1.85575\;71\nu^{\frac{1}{3}}+1.03315\;0\nu^{-\frac{1}{3}}%
-0.00397\nu^{-1}-0.0908\nu^{-\frac{5}{3}}+0.043\nu^{-\frac{7}{3}}+\cdots," display="inline"><mrow><mrow><mi></mi><mo href="./2.1#SS3.p1">∼</mo><mrow><mrow><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mrow><mn>1.85575 71</mn><mo></mo><msup><mi href="./10.1#p2.t1.r5">ν</mi><mfrac><mn>1</mn><mn>3</mn></mfrac></msup></mrow><mo>+</mo><mrow><mn>1.03315 0</mn><mo></mo><msup><mi href="./10.1#p2.t1.r5">ν</mi><mrow><mo>-</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></mrow></msup></mrow></mrow><mo>-</mo><mrow><mn>0.00397</mn><mo></mo><msup><mi href="./10.1#p2.t1.r5">ν</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></mrow><mo>-</mo><mrow><mn>0.0908</mn><mo></mo><msup><mi href="./10.1#p2.t1.r5">ν</mi><mrow><mo>-</mo><mfrac><mn>5</mn><mn>3</mn></mfrac></mrow></msup></mrow></mrow><mo>+</mo><mrow><mn>0.043</mn><mo></mo><msup><mi href="./10.1#p2.t1.r5">ν</mi><mrow><mo>-</mo><mfrac><mn>7</mn><mn>3</mn></mfrac></mrow></msup></mrow><mo>+</mo><mi mathvariant="normal">⋯</mi></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex40" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m98.png" altimg-height="20px" altimg-valign="-8px" altimg-width="38px" alttext="\displaystyle y_{\nu,1}" display="inline"><msub><mi href="./10.21#SS1.p2">y</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mn>1</mn></mrow></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m88.png" altimg-height="29px" altimg-valign="-6px" altimg-width="719px" alttext="\displaystyle\sim\nu+0.93157\;68\nu^{\frac{1}{3}}+0.26035\;1\nu^{-\frac{1}{3}}%
+0.01198\nu^{-1}-0.0060\nu^{-\frac{5}{3}}-0.001\nu^{-\frac{7}{3}}+\cdots," display="inline"><mrow><mrow><mi></mi><mo href="./2.1#SS3.p1">∼</mo><mrow><mrow><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mrow><mn>0.93157 68</mn><mo></mo><msup><mi href="./10.1#p2.t1.r5">ν</mi><mfrac><mn>1</mn><mn>3</mn></mfrac></msup></mrow><mo>+</mo><mrow><mn>0.26035 1</mn><mo></mo><msup><mi href="./10.1#p2.t1.r5">ν</mi><mrow><mo>-</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></mrow></msup></mrow><mo>+</mo><mrow><mn>0.01198</mn><mo></mo><msup><mi href="./10.1#p2.t1.r5">ν</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></mrow></mrow><mo>-</mo><mrow><mn>0.0060</mn><mo></mo><msup><mi href="./10.1#p2.t1.r5">ν</mi><mrow><mo>-</mo><mfrac><mn>5</mn><mn>3</mn></mfrac></mrow></msup></mrow><mo>-</mo><mrow><mn>0.001</mn><mo></mo><msup><mi href="./10.1#p2.t1.r5">ν</mi><mrow><mo>-</mo><mfrac><mn>7</mn><mn>3</mn></mfrac></mrow></msup></mrow></mrow><mo>+</mo><mi mathvariant="normal">⋯</mi></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex41" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m48.png" altimg-height="27px" altimg-valign="-8px" altimg-width="76px" alttext="\displaystyle J_{\nu}'\left(j_{\nu,1}\right)" display="inline"><mrow><msubsup><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′</mo></msubsup><mo></mo><mrow><mo>(</mo><msub><mi href="./10.21#SS1.p2">j</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mn>1</mn></mrow></msub><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m86.png" altimg-height="31px" altimg-valign="-7px" altimg-width="758px" alttext="\displaystyle\sim-1.11310\;28\nu^{-\frac{2}{3}}\div(1+1.48460\;6\nu^{-\frac{2}%
{3}}+0.43294\nu^{-\frac{4}{3}}-0.1943\nu^{-2}+0.019\nu^{-\frac{8}{3}}+\cdots)," display="inline"><mrow><mrow><mi></mi><mo href="./2.1#SS3.p1">∼</mo><mrow><mo>-</mo><mrow><mrow><mn>1.11310 28</mn><mo></mo><msup><mi href="./10.1#p2.t1.r5">ν</mi><mrow><mo>-</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></mrow></msup></mrow><mo>÷</mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mrow><mn>1</mn><mo>+</mo><mrow><mn>1.48460 6</mn><mo></mo><msup><mi href="./10.1#p2.t1.r5">ν</mi><mrow><mo>-</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></mrow></msup></mrow><mo>+</mo><mrow><mn>0.43294</mn><mo></mo><msup><mi href="./10.1#p2.t1.r5">ν</mi><mrow><mo>-</mo><mfrac><mn>4</mn><mn>3</mn></mfrac></mrow></msup></mrow></mrow><mo>-</mo><mrow><mn>0.1943</mn><mo></mo><msup><mi href="./10.1#p2.t1.r5">ν</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></mrow></mrow><mo>+</mo><mrow><mn>0.019</mn><mo></mo><msup><mi href="./10.1#p2.t1.r5">ν</mi><mrow><mo>-</mo><mfrac><mn>8</mn><mn>3</mn></mfrac></mrow></msup></mrow><mo>+</mo><mi mathvariant="normal">⋯</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex42" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m52.png" altimg-height="27px" altimg-valign="-8px" altimg-width="78px" alttext="\displaystyle Y_{\nu}'\left(y_{\nu,1}\right)" display="inline"><mrow><msubsup><mi href="./10.2#E3">Y</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′</mo></msubsup><mo></mo><mrow><mo>(</mo><msub><mi href="./10.21#SS1.p2">y</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mn>1</mn></mrow></msub><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m85.png" altimg-height="31px" altimg-valign="-7px" altimg-width="742px" alttext="\displaystyle\sim 0.95554\;86\nu^{-\frac{2}{3}}\div(1+0.74526\;1\nu^{-\frac{2}%
{3}}+0.10910\nu^{-\frac{4}{3}}-0.0185\nu^{-2}-0.003\nu^{-\frac{8}{3}}+\cdots)," display="inline"><mrow><mrow><mi></mi><mo href="./2.1#SS3.p1">∼</mo><mrow><mrow><mn>0.95554 86</mn><mo></mo><msup><mi href="./10.1#p2.t1.r5">ν</mi><mrow><mo>-</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></mrow></msup></mrow><mo>÷</mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mrow><mn>1</mn><mo>+</mo><mrow><mn>0.74526 1</mn><mo></mo><msup><mi href="./10.1#p2.t1.r5">ν</mi><mrow><mo>-</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></mrow></msup></mrow><mo>+</mo><mrow><mn>0.10910</mn><mo></mo><msup><mi href="./10.1#p2.t1.r5">ν</mi><mrow><mo>-</mo><mfrac><mn>4</mn><mn>3</mn></mfrac></mrow></msup></mrow></mrow><mo>-</mo><mrow><mn>0.0185</mn><mo></mo><msup><mi href="./10.1#p2.t1.r5">ν</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></mrow><mo>-</mo><mrow><mn>0.003</mn><mo></mo><msup><mi href="./10.1#p2.t1.r5">ν</mi><mrow><mo>-</mo><mfrac><mn>8</mn><mn>3</mn></mfrac></mrow></msup></mrow></mrow><mo>+</mo><mi mathvariant="normal">⋯</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex43" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m102.png" altimg-height="29px" altimg-valign="-10px" altimg-width="36px" alttext="\displaystyle{j^{\prime}_{\nu,1}}" display="inline"><msubsup><mi href="./10.21#SS1.p2">j</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mn>1</mn></mrow><mo href="./10.21#SS1.p2">′</mo></msubsup></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m87.png" altimg-height="29px" altimg-valign="-6px" altimg-width="613px" alttext="\displaystyle\sim\nu+0.80861\;65\nu^{\frac{1}{3}}+0.07249\;0\nu^{-\frac{1}{3}}%
-0.05097\nu^{-1}+0.0094\nu^{-\frac{5}{3}}+\cdots," display="inline"><mrow><mrow><mi></mi><mo href="./2.1#SS3.p1">∼</mo><mrow><mrow><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mrow><mn>0.80861 65</mn><mo></mo><msup><mi href="./10.1#p2.t1.r5">ν</mi><mfrac><mn>1</mn><mn>3</mn></mfrac></msup></mrow><mo>+</mo><mrow><mn>0.07249 0</mn><mo></mo><msup><mi href="./10.1#p2.t1.r5">ν</mi><mrow><mo>-</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></mrow></msup></mrow></mrow><mo>-</mo><mrow><mn>0.05097</mn><mo></mo><msup><mi href="./10.1#p2.t1.r5">ν</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></mrow></mrow><mo>+</mo><mrow><mn>0.0094</mn><mo></mo><msup><mi href="./10.1#p2.t1.r5">ν</mi><mrow><mo>-</mo><mfrac><mn>5</mn><mn>3</mn></mfrac></mrow></msup></mrow><mo>+</mo><mi mathvariant="normal">⋯</mi></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex44" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m104.png" altimg-height="29px" altimg-valign="-10px" altimg-width="38px" alttext="\displaystyle{y^{\prime}_{\nu,1}}" display="inline"><msubsup><mi href="./10.21#SS1.p2">y</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mn>1</mn></mrow><mo href="./10.21#SS1.p2">′</mo></msubsup></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m89.png" altimg-height="27px" altimg-valign="-4px" altimg-width="613px" alttext="\displaystyle\sim\nu+1.82109\;80\nu^{\frac{1}{3}}+0.94000\;7\nu^{-\frac{1}{3}}%
-0.05808\nu^{-1}-0.0540\nu^{-\frac{5}{3}}+\cdots." display="inline"><mrow><mrow><mi></mi><mo href="./2.1#SS3.p1">∼</mo><mrow><mrow><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mrow><mn>1.82109 80</mn><mo></mo><msup><mi href="./10.1#p2.t1.r5">ν</mi><mfrac><mn>1</mn><mn>3</mn></mfrac></msup></mrow><mo>+</mo><mrow><mn>0.94000 7</mn><mo></mo><msup><mi href="./10.1#p2.t1.r5">ν</mi><mrow><mo>-</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></mrow></msup></mrow></mrow><mo>-</mo><mrow><mn>0.05808</mn><mo></mo><msup><mi href="./10.1#p2.t1.r5">ν</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></mrow><mo>-</mo><mrow><mn>0.0540</mn><mo></mo><msup><mi href="./10.1#p2.t1.r5">ν</mi><mrow><mo>-</mo><mfrac><mn>5</mn><mn>3</mn></mfrac></mrow></msup></mrow></mrow><mo>+</mo><mi mathvariant="normal">⋯</mi></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex45" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m50.png" altimg-height="30px" altimg-valign="-10px" altimg-width="79px" alttext="\displaystyle J_{\nu}\left(j_{\nu,1}'\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><msubsup><mi href="./10.21#SS1.p2">j</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mn>1</mn></mrow><mo>′</mo></msubsup><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m84.png" altimg-height="31px" altimg-valign="-7px" altimg-width="612px" alttext="\displaystyle\sim 0.67488\;51\nu^{-\frac{1}{3}}(1-0.16172\;3\nu^{-\frac{2}{3}}%
+0.02918\nu^{-\frac{4}{3}}-0.0068\nu^{-2}+\cdots)," display="inline"><mrow><mrow><mi></mi><mo href="./2.1#SS3.p1">∼</mo><mrow><mn>0.67488 51</mn><mo></mo><msup><mi href="./10.1#p2.t1.r5">ν</mi><mrow><mo>-</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></mrow></msup><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mrow><mrow><mn>1</mn><mo>-</mo><mrow><mn>0.16172 3</mn><mo></mo><msup><mi href="./10.1#p2.t1.r5">ν</mi><mrow><mo>-</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></mrow></msup></mrow></mrow><mo>+</mo><mrow><mn>0.02918</mn><mo></mo><msup><mi href="./10.1#p2.t1.r5">ν</mi><mrow><mo>-</mo><mfrac><mn>4</mn><mn>3</mn></mfrac></mrow></msup></mrow></mrow><mo>-</mo><mrow><mn>0.0068</mn><mo></mo><msup><mi href="./10.1#p2.t1.r5">ν</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></mrow></mrow><mo>+</mo><mi mathvariant="normal">⋯</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex46" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m53.png" altimg-height="30px" altimg-valign="-10px" altimg-width="81px" alttext="\displaystyle Y_{\nu}\left(y_{\nu,1}'\right)" display="inline"><mrow><msub><mi href="./10.2#E3">Y</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><msubsup><mi href="./10.21#SS1.p2">y</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mn>1</mn></mrow><mo>′</mo></msubsup><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m83.png" altimg-height="31px" altimg-valign="-7px" altimg-width="612px" alttext="\displaystyle\sim 0.57319\;40\nu^{-\frac{1}{3}}(1-0.36422\;0\nu^{-\frac{2}{3}}%
+0.09077\nu^{-\frac{4}{3}}+0.0237\nu^{-2}+\cdots)." display="inline"><mrow><mrow><mi></mi><mo href="./2.1#SS3.p1">∼</mo><mrow><mn>0.57319 40</mn><mo></mo><msup><mi href="./10.1#p2.t1.r5">ν</mi><mrow><mo>-</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></mrow></msup><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>1</mn><mo>-</mo><mrow><mn>0.36422 0</mn><mo></mo><msup><mi href="./10.1#p2.t1.r5">ν</mi><mrow><mo>-</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></mrow></msup></mrow></mrow><mo>+</mo><mrow><mn>0.09077</mn><mo></mo><msup><mi href="./10.1#p2.t1.r5">ν</mi><mrow><mo>-</mo><mfrac><mn>4</mn><mn>3</mn></mfrac></mrow></msup></mrow><mo>+</mo><mrow><mn>0.0237</mn><mo></mo><msup><mi href="./10.1#p2.t1.r5">ν</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></mrow><mo>+</mo><mi mathvariant="normal">⋯</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E40.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m134.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./10.2#E3" title="(10.2.3) ‣ Bessel Function of the Second Kind (Weber’s Function) ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m153.png" altimg-height="23px" altimg-valign="-7px" altimg-width="55px" alttext="Y_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E3">Y</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the second kind</a>,
<a href="./2.1#SS3.p1" title="§2.1(iii) Asymptotic Expansions ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m234.png" altimg-height="12px" altimg-valign="-2px" altimg-width="20px" alttext="\sim" display="inline"><mo href="./2.1#SS3.p1">∼</mo></math>: Poincaré asymptotic expansion</a>,
<a href="./10.21#SS1.p2" title="§10.21(i) Distribution ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m273.png" altimg-height="22px" altimg-valign="-8px" altimg-width="41px" alttext="j_{\NVar{\nu},\NVar{m}}" display="inline"><msub><mi href="./10.21#SS1.p2">j</mi><mrow><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r1">m</mi></mrow></msub></math>: zeros of the Bessel function <math class="ltx_Math" altimg="m141.png" altimg-height="23px" altimg-valign="-7px" altimg-width="55px" alttext="J_{\nu}\left(x\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math></a>,
<a href="./10.21#SS1.p2" title="§10.21(i) Distribution ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m303.png" altimg-height="18px" altimg-valign="-8px" altimg-width="43px" alttext="y_{\NVar{\nu},\NVar{m}}" display="inline"><msub><mi href="./10.21#SS1.p2">y</mi><mrow><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r1">m</mi></mrow></msub></math>: zeros of the Bessel function <math class="ltx_Math" altimg="m157.png" altimg-height="23px" altimg-valign="-7px" altimg-width="56px" alttext="Y_{\nu}\left(x\right)" display="inline"><mrow><msub><mi href="./10.2#E3">Y</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math></a>,
<a href="./10.21#SS1.p2" title="§10.21(i) Distribution ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m318.png" altimg-height="26px" altimg-valign="-10px" altimg-width="41px" alttext="{j^{\prime}_{\NVar{\nu},\NVar{m}}}" display="inline"><msubsup><mi href="./10.21#SS1.p2">j</mi><mrow><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r1">m</mi></mrow><mo href="./10.21#SS1.p2">′</mo></msubsup></math>: zeros of the Bessel function derivative <math class="ltx_Math" altimg="m138.png" altimg-height="24px" altimg-valign="-7px" altimg-width="55px" alttext="J_{\nu}'\left(x\right)" display="inline"><mrow><msubsup><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′</mo></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math></a>,
<a href="./10.21#SS1.p2" title="§10.21(i) Distribution ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m320.png" altimg-height="26px" altimg-valign="-10px" altimg-width="43px" alttext="{y^{\prime}_{\NVar{\nu},\NVar{m}}}" display="inline"><msubsup><mi href="./10.21#SS1.p2">y</mi><mrow><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r1">m</mi></mrow><mo href="./10.21#SS1.p2">′</mo></msubsup></math>: zeros of the Bessel function derivative <math class="ltx_Math" altimg="m154.png" altimg-height="24px" altimg-valign="-7px" altimg-width="56px" alttext="Y_{\nu}'\left(x\right)" display="inline"><mrow><msubsup><mi href="./10.2#E3">Y</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′</mo></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math></a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m215.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">9.5.14--9.5.21</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS7.p8" class="ltx_para">
<p class="ltx_p">For numerical coefficients for <math class="ltx_Math" altimg="m282.png" altimg-height="20px" altimg-valign="-6px" altimg-width="115px" alttext="m=2,3,4,5" display="inline"><mrow><mi href="./10.1#p2.t1.r1">m</mi><mo>=</mo><mrow><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>5</mn></mrow></mrow></math> see Olver (1951, Tables 3–6).
</p>
</div>
<div id="SS7.p9" class="ltx_para">
<p class="ltx_p">The expansions () become
progressively weaker as <math class="ltx_Math" altimg="m284.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./10.1#p2.t1.r1">m</mi></math> increases. The approximations that follow in
§</dd>
</dl>
</div>
</div>
<div id="SS8.p1" class="ltx_para">
<p class="ltx_p">As <math class="ltx_Math" altimg="m220.png" altimg-height="13px" altimg-valign="-2px" altimg-width="66px" alttext="\nu\to\infty" display="inline"><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></math> the following four approximations hold uniformly for
<math class="ltx_Math" altimg="m281.png" altimg-height="20px" altimg-valign="-6px" altimg-width="109px" alttext="m=1,2,\ldots" display="inline"><mrow><mi href="./10.1#p2.t1.r1">m</mi><mo>=</mo><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>:
</p>
<table id="E41" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.21.41</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E41.png" altimg-height="54px" altimg-valign="-21px" altimg-width="398px" alttext="j_{\nu,m}=\nu z(\zeta)+\frac{z(\zeta)(h(\zeta))^{2}B_{0}(\zeta)}{2\nu}+O\left(%
\frac{1}{\nu^{3}}\right)," display="block"><mrow><mrow><msub><mi href="./10.21#SS1.p2">j</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi href="./10.1#p2.t1.r1">m</mi></mrow></msub><mo>=</mo><mrow><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo></mo><mrow><mi href="./10.21#SS8.p1">z</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>ζ</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>+</mo><mfrac><mrow><mrow><mi href="./10.21#SS8.p1">z</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>ζ</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./10.21#E45">h</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>ζ</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo></mo><mrow><msub><mi href="./10.20#E11">B</mi><mn>0</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>ζ</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow></mfrac><mo>+</mo><mrow><mi href="./2.1#E3">O</mi><mo></mo><mrow><mo>(</mo><mfrac><mn>1</mn><msup><mi href="./10.1#p2.t1.r5">ν</mi><mn>3</mn></msup></mfrac><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m243.png" altimg-height="26px" altimg-valign="-6px" altimg-width="103px" alttext="\zeta=\nu^{-\frac{2}{3}}a_{m}" display="inline"><mrow><mi>ζ</mi><mo>=</mo><mrow><msup><mi href="./10.1#p2.t1.r5">ν</mi><mrow><mo>-</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></mrow></msup><mo></mo><msub><mi href="./9.9#SS1.p1">a</mi><mi href="./10.1#p2.t1.r1">m</mi></msub></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E41.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./2.1#E3" title="(2.1.3) ‣ §2.1(i) Asymptotic and Order Symbols ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m151.png" altimg-height="23px" altimg-valign="-7px" altimg-width="50px" alttext="O\left(\NVar{x}\right)" display="inline"><mrow><mi href="./2.1#E3">O</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>)</mo></mrow></mrow></math>: order not exceeding</a>,
<a href="./9.9#SS1.p1" title="§9.9(i) Distribution and Notation ‣ §9.9 Zeros ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m250.png" altimg-height="16px" altimg-valign="-5px" altimg-width="25px" alttext="a_{\NVar{k}}" display="inline"><msub><mi href="./9.9#SS1.p1">a</mi><mi class="ltx_nvar" href="./9.1#p2.t1.r1">k</mi></msub></math>: <math class="ltx_Math" altimg="m279.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./9.1#p2.t1.r1">k</mi></math>th zero of Airy <math class="ltx_Math" altimg="m193.png" altimg-height="18px" altimg-valign="-2px" altimg-width="25px" alttext="\mathrm{Ai}" display="inline"><mi href="./9.2#i">Ai</mi></math></a>,
<a href="./10.21#SS1.p2" title="§10.21(i) Distribution ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m273.png" altimg-height="22px" altimg-valign="-8px" altimg-width="41px" alttext="j_{\NVar{\nu},\NVar{m}}" display="inline"><msub><mi href="./10.21#SS1.p2">j</mi><mrow><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r1">m</mi></mrow></msub></math>: zeros of the Bessel function <math class="ltx_Math" altimg="m141.png" altimg-height="23px" altimg-valign="-7px" altimg-width="55px" alttext="J_{\nu}\left(x\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math></a>,
<a href="./10.1#p2.t1.r1" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m284.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./10.1#p2.t1.r1">m</mi></math>: integer</a>,
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m215.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>,
<a href="./10.20#E11" title="(10.20.11) ‣ Interval 0 < z < 1 ‣ §10.20(i) Real Variables ‣ §10.20 Uniform Asymptotic Expansions for Large Order ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m124.png" altimg-height="23px" altimg-valign="-7px" altimg-width="55px" alttext="B_{k}(\zeta)" display="inline"><mrow><msub><mi href="./10.20#E11">B</mi><mi href="./10.1#p2.t1.r2">k</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./10.20#SS1.p1">ζ</mi><mo stretchy="false">)</mo></mrow></mrow></math>: coefficients</a>,
<a href="./10.21#SS8.p1" title="§10.21(viii) Uniform Asymptotic Approximations for Large Order ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m307.png" altimg-height="23px" altimg-valign="-7px" altimg-width="40px" alttext="z(\zeta)" display="inline"><mrow><mi href="./10.21#SS8.p1">z</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>ζ</mi><mo stretchy="false">)</mo></mrow></mrow></math>: inverse of <math class="ltx_Math" altimg="m241.png" altimg-height="23px" altimg-valign="-7px" altimg-width="40px" alttext="\zeta(z)" display="inline"><mrow><mi>ζ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./10.21#SS8.p1">z</mi><mo stretchy="false">)</mo></mrow></mrow></math></a> and
<a href="./10.21#E45" title="(10.21.45) ‣ §10.21(viii) Uniform Asymptotic Approximations for Large Order ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m267.png" altimg-height="23px" altimg-valign="-7px" altimg-width="41px" alttext="h(\zeta)" display="inline"><mrow><mi href="./10.21#E45">h</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>ζ</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">9.5.22</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E42" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.21.42</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E42.png" altimg-height="54px" altimg-valign="-21px" altimg-width="379px" alttext="J_{\nu}'\left(j_{\nu,m}\right)=-\frac{2}{\nu^{\frac{2}{3}}}\frac{\mathrm{Ai}'%
\left(a_{m}\right)}{z(\zeta)h(\zeta)}\left(1+O\left(\frac{1}{\nu^{2}}\right)%
\right)," display="block"><mrow><mrow><mrow><msubsup><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′</mo></msubsup><mo></mo><mrow><mo>(</mo><msub><mi href="./10.21#SS1.p2">j</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi href="./10.1#p2.t1.r1">m</mi></mrow></msub><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mo>-</mo><mrow><mfrac><mn>2</mn><msup><mi href="./10.1#p2.t1.r5">ν</mi><mfrac><mn>2</mn><mn>3</mn></mfrac></msup></mfrac><mo></mo><mfrac><mrow><msup><mi href="./9.2#i">Ai</mi><mo>′</mo></msup><mo></mo><mrow><mo>(</mo><msub><mi href="./9.9#SS1.p1">a</mi><mi href="./10.1#p2.t1.r1">m</mi></msub><mo>)</mo></mrow></mrow><mrow><mrow><mi href="./10.21#SS8.p1">z</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>ζ</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mi href="./10.21#E45">h</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>ζ</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></mfrac><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mrow><mi href="./2.1#E3">O</mi><mo></mo><mrow><mo>(</mo><mfrac><mn>1</mn><msup><mi href="./10.1#p2.t1.r5">ν</mi><mn>2</mn></msup></mfrac><mo>)</mo></mrow></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m243.png" altimg-height="26px" altimg-valign="-6px" altimg-width="103px" alttext="\zeta=\nu^{-\frac{2}{3}}a_{m}" display="inline"><mrow><mi>ζ</mi><mo>=</mo><mrow><msup><mi href="./10.1#p2.t1.r5">ν</mi><mrow><mo>-</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></mrow></msup><mo></mo><msub><mi href="./9.9#SS1.p1">a</mi><mi href="./10.1#p2.t1.r1">m</mi></msub></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E42.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./9.2#i" title="§9.2(i) Airy’s Equation ‣ §9.2 Differential Equation ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m194.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="\mathrm{Ai}\left(\NVar{z}\right)" display="inline"><mrow><mi href="./9.2#i">Ai</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./9.1#p2.t1.r3">z</mi><mo>)</mo></mrow></mrow></math>: Airy function</a>,
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m134.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./2.1#E3" title="(2.1.3) ‣ §2.1(i) Asymptotic and Order Symbols ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m151.png" altimg-height="23px" altimg-valign="-7px" altimg-width="50px" alttext="O\left(\NVar{x}\right)" display="inline"><mrow><mi href="./2.1#E3">O</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>)</mo></mrow></mrow></math>: order not exceeding</a>,
<a href="./9.9#SS1.p1" title="§9.9(i) Distribution and Notation ‣ §9.9 Zeros ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m250.png" altimg-height="16px" altimg-valign="-5px" altimg-width="25px" alttext="a_{\NVar{k}}" display="inline"><msub><mi href="./9.9#SS1.p1">a</mi><mi class="ltx_nvar" href="./9.1#p2.t1.r1">k</mi></msub></math>: <math class="ltx_Math" altimg="m279.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./9.1#p2.t1.r1">k</mi></math>th zero of Airy <math class="ltx_Math" altimg="m193.png" altimg-height="18px" altimg-valign="-2px" altimg-width="25px" alttext="\mathrm{Ai}" display="inline"><mi href="./9.2#i">Ai</mi></math></a>,
<a href="./10.21#SS1.p2" title="§10.21(i) Distribution ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m273.png" altimg-height="22px" altimg-valign="-8px" altimg-width="41px" alttext="j_{\NVar{\nu},\NVar{m}}" display="inline"><msub><mi href="./10.21#SS1.p2">j</mi><mrow><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r1">m</mi></mrow></msub></math>: zeros of the Bessel function <math class="ltx_Math" altimg="m141.png" altimg-height="23px" altimg-valign="-7px" altimg-width="55px" alttext="J_{\nu}\left(x\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math></a>,
<a href="./10.1#p2.t1.r1" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m284.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./10.1#p2.t1.r1">m</mi></math>: integer</a>,
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m215.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>,
<a href="./10.21#SS8.p1" title="§10.21(viii) Uniform Asymptotic Approximations for Large Order ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m307.png" altimg-height="23px" altimg-valign="-7px" altimg-width="40px" alttext="z(\zeta)" display="inline"><mrow><mi href="./10.21#SS8.p1">z</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>ζ</mi><mo stretchy="false">)</mo></mrow></mrow></math>: inverse of <math class="ltx_Math" altimg="m241.png" altimg-height="23px" altimg-valign="-7px" altimg-width="40px" alttext="\zeta(z)" display="inline"><mrow><mi>ζ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./10.21#SS8.p1">z</mi><mo stretchy="false">)</mo></mrow></mrow></math></a> and
<a href="./10.21#E45" title="(10.21.45) ‣ §10.21(viii) Uniform Asymptotic Approximations for Large Order ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m267.png" altimg-height="23px" altimg-valign="-7px" altimg-width="41px" alttext="h(\zeta)" display="inline"><mrow><mi href="./10.21#E45">h</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>ζ</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">9.5.23</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E43" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.21.43</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E43.png" altimg-height="54px" altimg-valign="-21px" altimg-width="389px" alttext="j_{\nu,m}'=\nu z(\zeta)+\frac{z(\zeta)(h(\zeta))^{2}C_{0}(\zeta)}{2\zeta\nu}+O%
\left(\frac{1}{\nu}\right)," display="block"><mrow><mrow><msubsup><mi href="./10.21#SS1.p2">j</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi href="./10.1#p2.t1.r1">m</mi></mrow><mo>′</mo></msubsup><mo>=</mo><mrow><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo></mo><mrow><mi href="./10.21#SS8.p1">z</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>ζ</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>+</mo><mfrac><mrow><mrow><mi href="./10.21#SS8.p1">z</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>ζ</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./10.21#E45">h</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>ζ</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo></mo><mrow><msub><mi href="./10.20#E12">C</mi><mn>0</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>ζ</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mrow><mn>2</mn><mo></mo><mi>ζ</mi><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow></mfrac><mo>+</mo><mrow><mi href="./2.1#E3">O</mi><mo></mo><mrow><mo>(</mo><mfrac><mn>1</mn><mi href="./10.1#p2.t1.r5">ν</mi></mfrac><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m242.png" altimg-height="27px" altimg-valign="-7px" altimg-width="103px" alttext="\zeta=\nu^{-\frac{2}{3}}a^{\prime}_{m}" display="inline"><mrow><mi>ζ</mi><mo>=</mo><mrow><msup><mi href="./10.1#p2.t1.r5">ν</mi><mrow><mo>-</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></mrow></msup><mo></mo><msubsup><mi href="./9.9#SS1.p1">a</mi><mi href="./10.1#p2.t1.r1">m</mi><mo href="./9.9#SS1.p1">′</mo></msubsup></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E43.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./2.1#E3" title="(2.1.3) ‣ §2.1(i) Asymptotic and Order Symbols ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m151.png" altimg-height="23px" altimg-valign="-7px" altimg-width="50px" alttext="O\left(\NVar{x}\right)" display="inline"><mrow><mi href="./2.1#E3">O</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>)</mo></mrow></mrow></math>: order not exceeding</a>,
<a href="./10.21#SS1.p2" title="§10.21(i) Distribution ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m273.png" altimg-height="22px" altimg-valign="-8px" altimg-width="41px" alttext="j_{\NVar{\nu},\NVar{m}}" display="inline"><msub><mi href="./10.21#SS1.p2">j</mi><mrow><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r1">m</mi></mrow></msub></math>: zeros of the Bessel function <math class="ltx_Math" altimg="m141.png" altimg-height="23px" altimg-valign="-7px" altimg-width="55px" alttext="J_{\nu}\left(x\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math></a>,
<a href="./9.9#SS1.p1" title="§9.9(i) Distribution and Notation ‣ §9.9 Zeros ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m248.png" altimg-height="24px" altimg-valign="-8px" altimg-width="25px" alttext="a^{\prime}_{\NVar{k}}" display="inline"><msubsup><mi href="./9.9#SS1.p1">a</mi><mi class="ltx_nvar" href="./9.1#p2.t1.r1">k</mi><mo href="./9.9#SS1.p1">′</mo></msubsup></math>: <math class="ltx_Math" altimg="m279.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./9.1#p2.t1.r1">k</mi></math>th zero of Airy <math class="ltx_Math" altimg="m190.png" altimg-height="20px" altimg-valign="-2px" altimg-width="30px" alttext="\mathrm{Ai}'" display="inline"><msup><mi href="./9.2#i">Ai</mi><mo>′</mo></msup></math></a>,
<a href="./10.1#p2.t1.r1" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m284.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./10.1#p2.t1.r1">m</mi></math>: integer</a>,
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m215.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>,
<a href="./10.20#E12" title="(10.20.12) ‣ Interval 0 < z < 1 ‣ §10.20(i) Real Variables ‣ §10.20 Uniform Asymptotic Expansions for Large Order ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m126.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="C_{k}(\zeta)" display="inline"><mrow><msub><mi href="./10.20#E12">C</mi><mi href="./10.1#p2.t1.r2">k</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./10.20#SS1.p1">ζ</mi><mo stretchy="false">)</mo></mrow></mrow></math>: coefficients</a>,
<a href="./10.21#SS8.p1" title="§10.21(viii) Uniform Asymptotic Approximations for Large Order ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m307.png" altimg-height="23px" altimg-valign="-7px" altimg-width="40px" alttext="z(\zeta)" display="inline"><mrow><mi href="./10.21#SS8.p1">z</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>ζ</mi><mo stretchy="false">)</mo></mrow></mrow></math>: inverse of <math class="ltx_Math" altimg="m241.png" altimg-height="23px" altimg-valign="-7px" altimg-width="40px" alttext="\zeta(z)" display="inline"><mrow><mi>ζ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./10.21#SS8.p1">z</mi><mo stretchy="false">)</mo></mrow></mrow></math></a> and
<a href="./10.21#E45" title="(10.21.45) ‣ §10.21(viii) Uniform Asymptotic Approximations for Large Order ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m267.png" altimg-height="23px" altimg-valign="-7px" altimg-width="41px" alttext="h(\zeta)" display="inline"><mrow><mi href="./10.21#E45">h</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>ζ</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">9.5.24</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E44" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.21.44</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E44.png" altimg-height="53px" altimg-valign="-21px" altimg-width="371px" alttext="J_{\nu}\left(j_{\nu,m}'\right)=\frac{h(\zeta)\mathrm{Ai}\left(a^{\prime}_{m}%
\right)}{\nu^{\frac{1}{3}}}\left(1+O\left(\frac{1}{\nu^{\frac{4}{3}}}\right)%
\right)," display="block"><mrow><mrow><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><msubsup><mi href="./10.21#SS1.p2">j</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi href="./10.1#p2.t1.r1">m</mi></mrow><mo>′</mo></msubsup><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mfrac><mrow><mrow><mi href="./10.21#E45">h</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>ζ</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mi href="./9.2#i">Ai</mi><mo></mo><mrow><mo>(</mo><msubsup><mi href="./9.9#SS1.p1">a</mi><mi href="./10.1#p2.t1.r1">m</mi><mo href="./9.9#SS1.p1">′</mo></msubsup><mo>)</mo></mrow></mrow></mrow><msup><mi href="./10.1#p2.t1.r5">ν</mi><mfrac><mn>1</mn><mn>3</mn></mfrac></msup></mfrac><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mrow><mi href="./2.1#E3">O</mi><mo></mo><mrow><mo>(</mo><mfrac><mn>1</mn><msup><mi href="./10.1#p2.t1.r5">ν</mi><mfrac><mn>4</mn><mn>3</mn></mfrac></msup></mfrac><mo>)</mo></mrow></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m242.png" altimg-height="27px" altimg-valign="-7px" altimg-width="103px" alttext="\zeta=\nu^{-\frac{2}{3}}a^{\prime}_{m}" display="inline"><mrow><mi>ζ</mi><mo>=</mo><mrow><msup><mi href="./10.1#p2.t1.r5">ν</mi><mrow><mo>-</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></mrow></msup><mo></mo><msubsup><mi href="./9.9#SS1.p1">a</mi><mi href="./10.1#p2.t1.r1">m</mi><mo href="./9.9#SS1.p1">′</mo></msubsup></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E44.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./9.2#i" title="§9.2(i) Airy’s Equation ‣ §9.2 Differential Equation ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m194.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="\mathrm{Ai}\left(\NVar{z}\right)" display="inline"><mrow><mi href="./9.2#i">Ai</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./9.1#p2.t1.r3">z</mi><mo>)</mo></mrow></mrow></math>: Airy function</a>,
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m134.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./2.1#E3" title="(2.1.3) ‣ §2.1(i) Asymptotic and Order Symbols ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m151.png" altimg-height="23px" altimg-valign="-7px" altimg-width="50px" alttext="O\left(\NVar{x}\right)" display="inline"><mrow><mi href="./2.1#E3">O</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>)</mo></mrow></mrow></math>: order not exceeding</a>,
<a href="./10.21#SS1.p2" title="§10.21(i) Distribution ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m273.png" altimg-height="22px" altimg-valign="-8px" altimg-width="41px" alttext="j_{\NVar{\nu},\NVar{m}}" display="inline"><msub><mi href="./10.21#SS1.p2">j</mi><mrow><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r1">m</mi></mrow></msub></math>: zeros of the Bessel function <math class="ltx_Math" altimg="m141.png" altimg-height="23px" altimg-valign="-7px" altimg-width="55px" alttext="J_{\nu}\left(x\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math></a>,
<a href="./9.9#SS1.p1" title="§9.9(i) Distribution and Notation ‣ §9.9 Zeros ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m248.png" altimg-height="24px" altimg-valign="-8px" altimg-width="25px" alttext="a^{\prime}_{\NVar{k}}" display="inline"><msubsup><mi href="./9.9#SS1.p1">a</mi><mi class="ltx_nvar" href="./9.1#p2.t1.r1">k</mi><mo href="./9.9#SS1.p1">′</mo></msubsup></math>: <math class="ltx_Math" altimg="m279.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./9.1#p2.t1.r1">k</mi></math>th zero of Airy <math class="ltx_Math" altimg="m190.png" altimg-height="20px" altimg-valign="-2px" altimg-width="30px" alttext="\mathrm{Ai}'" display="inline"><msup><mi href="./9.2#i">Ai</mi><mo>′</mo></msup></math></a>,
<a href="./10.1#p2.t1.r1" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m284.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./10.1#p2.t1.r1">m</mi></math>: integer</a>,
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m215.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a> and
<a href="./10.21#E45" title="(10.21.45) ‣ §10.21(viii) Uniform Asymptotic Approximations for Large Order ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m267.png" altimg-height="23px" altimg-valign="-7px" altimg-width="41px" alttext="h(\zeta)" display="inline"><mrow><mi href="./10.21#E45">h</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>ζ</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">9.5.25</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Here <math class="ltx_Math" altimg="m251.png" altimg-height="16px" altimg-valign="-5px" altimg-width="30px" alttext="a_{m}" display="inline"><msub><mi href="./9.9#SS1.p1">a</mi><mi href="./10.1#p2.t1.r1">m</mi></msub></math> and <math class="ltx_Math" altimg="m249.png" altimg-height="23px" altimg-valign="-7px" altimg-width="30px" alttext="a^{\prime}_{m}" display="inline"><msubsup><mi href="./9.9#SS1.p1">a</mi><mi href="./10.1#p2.t1.r1">m</mi><mo href="./9.9#SS1.p1">′</mo></msubsup></math> denote respectively the zeros of the Airy function
<math class="ltx_Math" altimg="m196.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="\mathrm{Ai}\left(z\right)" display="inline"><mrow><mi href="./9.2#i">Ai</mi><mo></mo><mrow><mo>(</mo><mi href="./10.21#SS8.p1">z</mi><mo>)</mo></mrow></mrow></math> and its derivative <math class="ltx_Math" altimg="m192.png" altimg-height="25px" altimg-valign="-7px" altimg-width="59px" alttext="\mathrm{Ai}'\left(z\right)" display="inline"><mrow><msup><mi href="./9.2#i">Ai</mi><mo>′</mo></msup><mo></mo><mrow><mo>(</mo><mi href="./10.21#SS8.p1">z</mi><mo>)</mo></mrow></mrow></math>; see §. Next,
<math class="ltx_Math" altimg="m307.png" altimg-height="23px" altimg-valign="-7px" altimg-width="40px" alttext="z(\zeta)" display="inline"><mrow><mi href="./10.21#SS8.p1">z</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>ζ</mi><mo stretchy="false">)</mo></mrow></mrow></math> is the inverse of the function <math class="ltx_Math" altimg="m244.png" altimg-height="23px" altimg-valign="-7px" altimg-width="77px" alttext="\zeta=\zeta(z)" display="inline"><mrow><mi>ζ</mi><mo>=</mo><mrow><mi>ζ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./10.21#SS8.p1">z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></math> defined by
(). <math class="ltx_Math" altimg="m123.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="B_{0}(\zeta)" display="inline"><mrow><msub><mi href="./10.20#E11">B</mi><mn>0</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>ζ</mi><mo stretchy="false">)</mo></mrow></mrow></math> and <math class="ltx_Math" altimg="m125.png" altimg-height="23px" altimg-valign="-7px" altimg-width="53px" alttext="C_{0}(\zeta)" display="inline"><mrow><msub><mi href="./10.20#E12">C</mi><mn>0</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>ζ</mi><mo stretchy="false">)</mo></mrow></mrow></math> are defined by
() with <math class="ltx_Math" altimg="m278.png" altimg-height="18px" altimg-valign="-2px" altimg-width="52px" alttext="k=0" display="inline"><mrow><mi href="./10.1#p2.t1.r2">k</mi><mo>=</mo><mn>0</mn></mrow></math>. Lastly,</p>
<table id="E45" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.21.45</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E45.png" altimg-height="37px" altimg-valign="-9px" altimg-width="206px" alttext="h(\zeta)=\left(4\zeta/(1-z^{2})\right)^{\frac{1}{4}}." display="block"><mrow><mrow><mrow><mi href="./10.21#E45">h</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>ζ</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><msup><mrow><mo>(</mo><mrow><mrow><mn>4</mn><mo></mo><mi>ζ</mi></mrow><mo>/</mo><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><msup><mi href="./10.21#SS8.p1">z</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>)</mo></mrow><mfrac><mn>1</mn><mn>4</mn></mfrac></msup></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E45.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text"><math class="ltx_Math" altimg="m267.png" altimg-height="23px" altimg-valign="-7px" altimg-width="41px" alttext="h(\zeta)" display="inline"><mrow><mi href="./10.21#E45">h</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>ζ</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function (locally)</span></dd>
<dt>Symbols:</dt>
<dd><a href="./10.21#SS8.p1" title="§10.21(viii) Uniform Asymptotic Approximations for Large Order ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m307.png" altimg-height="23px" altimg-valign="-7px" altimg-width="40px" alttext="z(\zeta)" display="inline"><mrow><mi href="./10.21#SS8.p1">z</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>ζ</mi><mo stretchy="false">)</mo></mrow></mrow></math>: inverse of <math class="ltx_Math" altimg="m241.png" altimg-height="23px" altimg-valign="-7px" altimg-width="40px" alttext="\zeta(z)" display="inline"><mrow><mi>ζ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./10.21#SS8.p1">z</mi><mo stretchy="false">)</mo></mrow></mrow></math></a></dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">9.5.36</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">(Note: If the term <math class="ltx_Math" altimg="m306.png" altimg-height="25px" altimg-valign="-7px" altimg-width="207px" alttext="z(\zeta)(h(\zeta))^{2}C_{0}(\zeta)/(2\zeta\nu)" display="inline"><mrow><mrow><mrow><mi href="./10.21#SS8.p1">z</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>ζ</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./10.21#E45">h</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>ζ</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo></mo><mrow><msub><mi href="./10.20#E12">C</mi><mn>0</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>ζ</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>/</mo><mrow><mo stretchy="false">(</mo><mrow><mn>2</mn><mo></mo><mi>ζ</mi><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></math> in
() is omitted, then the uniform character of the error
term <math class="ltx_Math" altimg="m150.png" altimg-height="23px" altimg-valign="-7px" altimg-width="71px" alttext="O(\ifrac{1}{\nu})" display="inline"><mrow><mi href="./2.1#E3">O</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>/</mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></math> is destroyed.)</p>
</div>
<div id="SS8.p2" class="ltx_para">
<p class="ltx_p">Corresponding uniform approximations for <math class="ltx_Math" altimg="m305.png" altimg-height="18px" altimg-valign="-8px" altimg-width="42px" alttext="y_{\nu,m}" display="inline"><msub><mi href="./10.21#SS1.p2">y</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi href="./10.1#p2.t1.r1">m</mi></mrow></msub></math>,
<math class="ltx_Math" altimg="m155.png" altimg-height="24px" altimg-valign="-8px" altimg-width="82px" alttext="Y_{\nu}'\left(y_{\nu,m}\right)" display="inline"><mrow><msubsup><mi href="./10.2#E3">Y</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′</mo></msubsup><mo></mo><mrow><mo>(</mo><msub><mi href="./10.21#SS1.p2">y</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi href="./10.1#p2.t1.r1">m</mi></mrow></msub><mo>)</mo></mrow></mrow></math>, <math class="ltx_Math" altimg="m304.png" altimg-height="26px" altimg-valign="-10px" altimg-width="42px" alttext="y_{\nu,m}'" display="inline"><msubsup><mi href="./10.21#SS1.p2">y</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi href="./10.1#p2.t1.r1">m</mi></mrow><mo>′</mo></msubsup></math>, and
<math class="ltx_Math" altimg="m160.png" altimg-height="28px" altimg-valign="-10px" altimg-width="85px" alttext="Y_{\nu}\left({y^{\prime}_{\nu,m}}\right)" display="inline"><mrow><msub><mi href="./10.2#E3">Y</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><msubsup><mi href="./10.21#SS1.p2">y</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi href="./10.1#p2.t1.r1">m</mi></mrow><mo href="./10.21#SS1.p2">′</mo></msubsup><mo>)</mo></mrow></mrow></math>, are obtained from
() by changing
the symbols <math class="ltx_Math" altimg="m268.png" altimg-height="20px" altimg-valign="-6px" altimg-width="14px" alttext="j" display="inline"><mi>j</mi></math>, <math class="ltx_Math" altimg="m127.png" altimg-height="18px" altimg-valign="-2px" altimg-width="17px" alttext="J" display="inline"><mi href="./10.2#E2">J</mi></math>, <math class="ltx_Math" altimg="m193.png" altimg-height="18px" altimg-valign="-2px" altimg-width="25px" alttext="\mathrm{Ai}" display="inline"><mi href="./9.2#i">Ai</mi></math>, <math class="ltx_Math" altimg="m190.png" altimg-height="20px" altimg-valign="-2px" altimg-width="30px" alttext="\mathrm{Ai}'" display="inline"><msup><mi href="./9.2#i">Ai</mi><mo>′</mo></msup></math>, <math class="ltx_Math" altimg="m251.png" altimg-height="16px" altimg-valign="-5px" altimg-width="30px" alttext="a_{m}" display="inline"><msub><mi href="./9.9#SS1.p1">a</mi><mi href="./10.1#p2.t1.r1">m</mi></msub></math>, and <math class="ltx_Math" altimg="m249.png" altimg-height="23px" altimg-valign="-7px" altimg-width="30px" alttext="a^{\prime}_{m}" display="inline"><msubsup><mi href="./9.9#SS1.p1">a</mi><mi href="./10.1#p2.t1.r1">m</mi><mo href="./9.9#SS1.p1">′</mo></msubsup></math> to <math class="ltx_Math" altimg="m302.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./10.1#p2.t1.r3">y</mi></math>,
<math class="ltx_Math" altimg="m152.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="Y" display="inline"><mi href="./10.2#E3">Y</mi></math>, <math class="ltx_Math" altimg="m116.png" altimg-height="19px" altimg-valign="-4px" altimg-width="39px" alttext="-\mathrm{Bi}" display="inline"><mrow><mo>-</mo><mi href="./9.2#i">Bi</mi></mrow></math>, <math class="ltx_Math" altimg="m115.png" altimg-height="22px" altimg-valign="-4px" altimg-width="45px" alttext="-\mathrm{Bi}'" display="inline"><mrow><mo>-</mo><msup><mi href="./9.2#i">Bi</mi><mo>′</mo></msup></mrow></math>, <math class="ltx_Math" altimg="m258.png" altimg-height="21px" altimg-valign="-5px" altimg-width="28px" alttext="b_{m}" display="inline"><msub><mi href="./9.9#SS1.p1">b</mi><mi href="./10.1#p2.t1.r1">m</mi></msub></math>, and <math class="ltx_Math" altimg="m256.png" altimg-height="23px" altimg-valign="-7px" altimg-width="28px" alttext="b^{\prime}_{m}" display="inline"><msubsup><mi href="./9.9#SS1.p1">b</mi><mi href="./10.1#p2.t1.r1">m</mi><mo href="./9.9#SS1.p1">′</mo></msubsup></math>, respectively.</p>
</div>
<div id="SS8.p3" class="ltx_para">
<p class="ltx_p">For derivations and further information, including extensions to uniform
asymptotic expansions, see <cite class="ltx_cite ltx_citemacro_citet">Olver (</dd>
</dl>
</div>
</div>
<div id="SS9.p1" class="ltx_para">
<p class="ltx_p">This subsection describes the distribution in <math class="ltx_Math" altimg="m188.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\mathbb{C}" display="inline"><mi href="./front/introduction#Sx4.p1.t1.r1">ℂ</mi></math> of the zeros of the
principal branches of the Bessel functions of the second and third kinds, and
their derivatives, in the case when the order is a positive integer <math class="ltx_Math" altimg="m291.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./10.1#p2.t1.r1">n</mi></math>. For
further information, including uniform asymptotic expansions, extensions to
other branches of the functions and their derivatives, and extensions to
half-integer values of the order, see <cite class="ltx_cite ltx_citemacro_citet">Olver ()</cite>.</p>
</div>
<section id="Px1" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Zeros of <math class="ltx_Math" altimg="m162.png" altimg-height="23px" altimg-valign="-7px" altimg-width="68px" alttext="Y_{n}\left(nz\right)" display="inline"><mrow><msub><mi href="./10.2#E3">Y</mi><mi href="./10.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi href="./10.1#p2.t1.r1">n</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></math> and <math class="ltx_Math" altimg="m161.png" altimg-height="24px" altimg-valign="-7px" altimg-width="68px" alttext="Y_{n}'\left(nz\right)" display="inline"><mrow><msubsup><mi href="./10.2#E3">Y</mi><mi href="./10.1#p2.t1.r1">n</mi><mo>′</mo></msubsup><mo></mo><mrow><mo>(</mo><mrow><mi href="./10.1#p2.t1.r1">n</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></math>
</h3>
<div id="Px1.info" class="ltx_metadata ltx_info">
the two continuous curves that join the points <math class="ltx_Math" altimg="m227.png" altimg-height="18px" altimg-valign="-4px" altimg-width="30px" alttext="\pm 1" display="inline"><mrow><mo>±</mo><mn>1</mn></mrow></math>
are the boundaries of <math class="ltx_Math" altimg="m189.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="\mathbf{K}" display="inline"><mi href="./10.20#F3" mathvariant="bold">K</mi></math>, that is, the eye-shaped domain depicted
in Figure . These curves therefore intersect the
imaginary axis at the points <math class="ltx_Math" altimg="m310.png" altimg-height="19px" altimg-valign="-4px" altimg-width="71px" alttext="z=\pm\mathrm{i}c" display="inline"><mrow><mi href="./10.1#p2.t1.r4">z</mi><mo>=</mo><mrow><mo>±</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./10.21#SS4.p1">c</mi></mrow></mrow></mrow></math>, where <math class="ltx_Math" altimg="m261.png" altimg-height="17px" altimg-valign="-2px" altimg-width="131px" alttext="c=0.66274\ldots" display="inline"><mrow><mi href="./10.21#SS4.p1">c</mi><mo>=</mo><mrow><mn>0.66274</mn><mo></mo><mi mathvariant="normal">…</mi></mrow></mrow></math>.</p>
</div>
<div id="Px1.p2" class="ltx_para">
<p class="ltx_p">The first set of zeros of the principal value of <math class="ltx_Math" altimg="m162.png" altimg-height="23px" altimg-valign="-7px" altimg-width="68px" alttext="Y_{n}\left(nz\right)" display="inline"><mrow><msub><mi href="./10.2#E3">Y</mi><mi href="./10.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi href="./10.1#p2.t1.r1">n</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></math> are the
points <math class="ltx_Math" altimg="m311.png" altimg-height="24px" altimg-valign="-8px" altimg-width="102px" alttext="z=y_{n,m}/n" display="inline"><mrow><mi href="./10.1#p2.t1.r4">z</mi><mo>=</mo><mrow><msub><mi href="./10.21#SS1.p2">y</mi><mrow><mi href="./10.1#p2.t1.r1">n</mi><mo href="./10.21#SS1.p2">,</mo><mi href="./10.1#p2.t1.r1">m</mi></mrow></msub><mo>/</mo><mi href="./10.1#p2.t1.r1">n</mi></mrow></mrow></math>, <math class="ltx_Math" altimg="m281.png" altimg-height="20px" altimg-valign="-6px" altimg-width="109px" alttext="m=1,2,\ldots" display="inline"><mrow><mi href="./10.1#p2.t1.r1">m</mi><mo>=</mo><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>, on the positive real axis
(§). Secondly, there is a conjugate pair of infinite strings
of zeros with asymptotes <math class="ltx_Math" altimg="m166.png" altimg-height="23px" altimg-valign="-7px" altimg-width="109px" alttext="\Im z=\pm\mathrm{i}a/n" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℑ</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo>=</mo><mrow><mo>±</mo><mrow><mrow><mi mathvariant="normal">i</mi><mo></mo><mi>a</mi></mrow><mo>/</mo><mi href="./10.1#p2.t1.r1">n</mi></mrow></mrow></mrow></math>, where</p>
<table id="E46" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.21.46</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E46.png" altimg-height="29px" altimg-valign="-9px" altimg-width="216px" alttext="a=\tfrac{1}{2}\ln 3=0.54931\ldots." display="block"><mrow><mrow><mi>a</mi><mo>=</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mn>3</mn></mrow></mrow><mo>=</mo><mrow><mn>0.54931</mn><mo></mo><mi mathvariant="normal">…</mi></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E46.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m187.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a></dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Lastly, there are two conjugate sets, with <math class="ltx_Math" altimg="m291.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./10.1#p2.t1.r1">n</mi></math> zeros in each set, that are
asymptotically close to the boundary of <math class="ltx_Math" altimg="m189.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="\mathbf{K}" display="inline"><mi href="./10.20#F3" mathvariant="bold">K</mi></math> as <math class="ltx_Math" altimg="m292.png" altimg-height="13px" altimg-valign="-2px" altimg-width="67px" alttext="n\to\infty" display="inline"><mrow><mi href="./10.1#p2.t1.r1">n</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></math>. Figures
plot the actual zeros for <math class="ltx_Math" altimg="m287.png" altimg-height="20px" altimg-valign="-6px" altimg-width="72px" alttext="n=1,5" display="inline"><mrow><mi href="./10.1#p2.t1.r1">n</mi><mo>=</mo><mrow><mn>1</mn><mo>,</mo><mn>5</mn></mrow></mrow></math>, and <math class="ltx_Math" altimg="m120.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="10" display="inline"><mn>10</mn></math>, respectively.
</p>
</div>
<div id="Px1.p3" class="ltx_para">
<p class="ltx_p">The zeros of <math class="ltx_Math" altimg="m161.png" altimg-height="24px" altimg-valign="-7px" altimg-width="68px" alttext="Y_{n}'\left(nz\right)" display="inline"><mrow><msubsup><mi href="./10.2#E3">Y</mi><mi href="./10.1#p2.t1.r1">n</mi><mo>′</mo></msubsup><mo></mo><mrow><mo>(</mo><mrow><mi href="./10.1#p2.t1.r1">n</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></math> have a similar pattern to those of
<math class="ltx_Math" altimg="m162.png" altimg-height="23px" altimg-valign="-7px" altimg-width="68px" alttext="Y_{n}\left(nz\right)" display="inline"><mrow><msub><mi href="./10.2#E3">Y</mi><mi href="./10.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi href="./10.1#p2.t1.r1">n</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></math>.</p>
</div>
<figure id="Px1.fig1" class="ltx_figure">
<table style="width:100%;">
<tr>
<td class="ltx_subfigure">
<figure id="F1" class="ltx_figure"><figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 10.21.1: </span>Zeros <math class="ltx_Math" altimg="m180.png" altimg-height="13px" altimg-valign="-2px" altimg-width="43px" alttext="\bullet\bullet\bullet" display="inline"><mrow><mo>∙</mo><mo></mo><mo>∙</mo><mo></mo><mo>∙</mo></mrow></math> of <math class="ltx_Math" altimg="m162.png" altimg-height="23px" altimg-valign="-7px" altimg-width="68px" alttext="Y_{n}\left(nz\right)" display="inline"><mrow><msub><mi href="./10.2#E3">Y</mi><mi href="./10.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi href="./10.1#p2.t1.r1">n</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></math> in <math class="ltx_Math" altimg="m322.png" altimg-height="23px" altimg-valign="-7px" altimg-width="93px" alttext="|\operatorname{ph}z|\leq\pi" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo stretchy="false">|</mo></mrow><mo>≤</mo><mi href="./3.12#E1">π</mi></mrow></math>.
Case <math class="ltx_Math" altimg="m289.png" altimg-height="17px" altimg-valign="-2px" altimg-width="53px" alttext="n=1" display="inline"><mrow><mi href="./10.1#p2.t1.r1">n</mi><mo>=</mo><mn>1</mn></mrow></math>, <math class="ltx_Math" altimg="m111.png" altimg-height="20px" altimg-valign="-5px" altimg-width="148px" alttext="-1.6\leq\Re z\leq 2.6" display="inline"><mrow><mrow><mo>-</mo><mn>1.6</mn></mrow><mo>≤</mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo>≤</mo><mn>2.6</mn></mrow></math>.
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E3" title="(10.2.3) ‣ Bessel Function of the Second Kind (Weber’s Function) ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m153.png" altimg-height="23px" altimg-valign="-7px" altimg-width="55px" alttext="Y_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E3">Y</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the second kind</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m226.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.9#E7" title="(1.9.7) ‣ Modulus and Phase ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m221.png" altimg-height="21px" altimg-valign="-6px" altimg-width="26px" alttext="\operatorname{ph}" display="inline"><mi href="./1.9#E7">ph</mi></math>: phase</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m167.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./10.1#p2.t1.r1" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m291.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./10.1#p2.t1.r1">n</mi></math>: integer</a> and
<a href="./10.1#p2.t1.r4" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m312.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./10.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</figure>
</td>
<td class="ltx_subfigure">
<figure id="F2" class="ltx_figure"><figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 10.21.2: </span>Zeros <math class="ltx_Math" altimg="m180.png" altimg-height="13px" altimg-valign="-2px" altimg-width="43px" alttext="\bullet\bullet\bullet" display="inline"><mrow><mo>∙</mo><mo></mo><mo>∙</mo><mo></mo><mo>∙</mo></mrow></math> of <math class="ltx_Math" altimg="m315.png" altimg-height="29px" altimg-valign="-7px" altimg-width="85px" alttext="{H^{(1)}_{n}}\left(nz\right)" display="inline"><mrow><msubsup><mi href="./10.2#E5">H</mi><mi href="./10.1#p2.t1.r1">n</mi><mrow><mo href="./10.2#E5" stretchy="false">(</mo><mn>1</mn><mo href="./10.2#E5" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mrow><mi href="./10.1#p2.t1.r1">n</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></math> in <math class="ltx_Math" altimg="m322.png" altimg-height="23px" altimg-valign="-7px" altimg-width="93px" alttext="|\operatorname{ph}z|\leq\pi" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo stretchy="false">|</mo></mrow><mo>≤</mo><mi href="./3.12#E1">π</mi></mrow></math>.
Case <math class="ltx_Math" altimg="m289.png" altimg-height="17px" altimg-valign="-2px" altimg-width="53px" alttext="n=1" display="inline"><mrow><mi href="./10.1#p2.t1.r1">n</mi><mo>=</mo><mn>1</mn></mrow></math>, <math class="ltx_Math" altimg="m114.png" altimg-height="20px" altimg-valign="-5px" altimg-width="148px" alttext="-2.8\leq\Re z\leq 1.4" display="inline"><mrow><mrow><mo>-</mo><mn>2.8</mn></mrow><mo>≤</mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo>≤</mo><mn>1.4</mn></mrow></math>.
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E5" title="(10.2.5) ‣ Bessel Functions of the Third Kind (Hankel Functions) ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m313.png" altimg-height="29px" altimg-valign="-7px" altimg-width="73px" alttext="{H^{(1)}_{\NVar{\nu}}}\left(\NVar{z}\right)" display="inline"><mrow><msubsup><mi href="./10.2#E5">H</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi><mrow><mo href="./10.2#E5" stretchy="false">(</mo><mn>1</mn><mo href="./10.2#E5" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the third kind (or Hankel function)</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m226.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.9#E7" title="(1.9.7) ‣ Modulus and Phase ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m221.png" altimg-height="21px" altimg-valign="-6px" altimg-width="26px" alttext="\operatorname{ph}" display="inline"><mi href="./1.9#E7">ph</mi></math>: phase</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m167.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./10.1#p2.t1.r1" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m291.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./10.1#p2.t1.r1">n</mi></math>: integer</a> and
<a href="./10.1#p2.t1.r4" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m312.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./10.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</figure>
</td>
</tr>
</table>
</figure>
<figure id="Px1.fig2" class="ltx_figure">
<table style="width:100%;">
<tr>
<td class="ltx_subfigure">
<figure id="F3" class="ltx_figure"><figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 10.21.3: </span>Zeros <math class="ltx_Math" altimg="m180.png" altimg-height="13px" altimg-valign="-2px" altimg-width="43px" alttext="\bullet\bullet\bullet" display="inline"><mrow><mo>∙</mo><mo></mo><mo>∙</mo><mo></mo><mo>∙</mo></mrow></math> of <math class="ltx_Math" altimg="m162.png" altimg-height="23px" altimg-valign="-7px" altimg-width="68px" alttext="Y_{n}\left(nz\right)" display="inline"><mrow><msub><mi href="./10.2#E3">Y</mi><mi href="./10.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi href="./10.1#p2.t1.r1">n</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></math> in <math class="ltx_Math" altimg="m322.png" altimg-height="23px" altimg-valign="-7px" altimg-width="93px" alttext="|\operatorname{ph}z|\leq\pi" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo stretchy="false">|</mo></mrow><mo>≤</mo><mi href="./3.12#E1">π</mi></mrow></math>.
Case <math class="ltx_Math" altimg="m290.png" altimg-height="17px" altimg-valign="-2px" altimg-width="53px" alttext="n=5" display="inline"><mrow><mi href="./10.1#p2.t1.r1">n</mi><mo>=</mo><mn>5</mn></mrow></math>, <math class="ltx_Math" altimg="m113.png" altimg-height="20px" altimg-valign="-5px" altimg-width="148px" alttext="-2.6\leq\Re z\leq 1.6" display="inline"><mrow><mrow><mo>-</mo><mn>2.6</mn></mrow><mo>≤</mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo>≤</mo><mn>1.6</mn></mrow></math>.
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E3" title="(10.2.3) ‣ Bessel Function of the Second Kind (Weber’s Function) ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m153.png" altimg-height="23px" altimg-valign="-7px" altimg-width="55px" alttext="Y_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E3">Y</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the second kind</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m226.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.9#E7" title="(1.9.7) ‣ Modulus and Phase ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m221.png" altimg-height="21px" altimg-valign="-6px" altimg-width="26px" alttext="\operatorname{ph}" display="inline"><mi href="./1.9#E7">ph</mi></math>: phase</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m167.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./10.1#p2.t1.r1" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m291.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./10.1#p2.t1.r1">n</mi></math>: integer</a> and
<a href="./10.1#p2.t1.r4" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m312.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./10.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</figure>
</td>
<td class="ltx_subfigure">
<figure id="F4" class="ltx_figure"><figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 10.21.4: </span>Zeros <math class="ltx_Math" altimg="m180.png" altimg-height="13px" altimg-valign="-2px" altimg-width="43px" alttext="\bullet\bullet\bullet" display="inline"><mrow><mo>∙</mo><mo></mo><mo>∙</mo><mo></mo><mo>∙</mo></mrow></math> of <math class="ltx_Math" altimg="m315.png" altimg-height="29px" altimg-valign="-7px" altimg-width="85px" alttext="{H^{(1)}_{n}}\left(nz\right)" display="inline"><mrow><msubsup><mi href="./10.2#E5">H</mi><mi href="./10.1#p2.t1.r1">n</mi><mrow><mo href="./10.2#E5" stretchy="false">(</mo><mn>1</mn><mo href="./10.2#E5" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mrow><mi href="./10.1#p2.t1.r1">n</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></math> in <math class="ltx_Math" altimg="m322.png" altimg-height="23px" altimg-valign="-7px" altimg-width="93px" alttext="|\operatorname{ph}z|\leq\pi" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo stretchy="false">|</mo></mrow><mo>≤</mo><mi href="./3.12#E1">π</mi></mrow></math>.
Case <math class="ltx_Math" altimg="m290.png" altimg-height="17px" altimg-valign="-2px" altimg-width="53px" alttext="n=5" display="inline"><mrow><mi href="./10.1#p2.t1.r1">n</mi><mo>=</mo><mn>5</mn></mrow></math>, <math class="ltx_Math" altimg="m113.png" altimg-height="20px" altimg-valign="-5px" altimg-width="148px" alttext="-2.6\leq\Re z\leq 1.6" display="inline"><mrow><mrow><mo>-</mo><mn>2.6</mn></mrow><mo>≤</mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo>≤</mo><mn>1.6</mn></mrow></math>.
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E5" title="(10.2.5) ‣ Bessel Functions of the Third Kind (Hankel Functions) ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m313.png" altimg-height="29px" altimg-valign="-7px" altimg-width="73px" alttext="{H^{(1)}_{\NVar{\nu}}}\left(\NVar{z}\right)" display="inline"><mrow><msubsup><mi href="./10.2#E5">H</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi><mrow><mo href="./10.2#E5" stretchy="false">(</mo><mn>1</mn><mo href="./10.2#E5" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the third kind (or Hankel function)</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m226.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.9#E7" title="(1.9.7) ‣ Modulus and Phase ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m221.png" altimg-height="21px" altimg-valign="-6px" altimg-width="26px" alttext="\operatorname{ph}" display="inline"><mi href="./1.9#E7">ph</mi></math>: phase</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m167.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./10.1#p2.t1.r1" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m291.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./10.1#p2.t1.r1">n</mi></math>: integer</a> and
<a href="./10.1#p2.t1.r4" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m312.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./10.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</figure>
</td>
</tr>
</table>
</figure>
<figure id="Px1.fig3" class="ltx_figure">
<table style="width:100%;">
<tr>
<td class="ltx_subfigure">
<figure id="F5" class="ltx_figure"><figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 10.21.5: </span>Zeros <math class="ltx_Math" altimg="m180.png" altimg-height="13px" altimg-valign="-2px" altimg-width="43px" alttext="\bullet\bullet\bullet" display="inline"><mrow><mo>∙</mo><mo></mo><mo>∙</mo><mo></mo><mo>∙</mo></mrow></math> of <math class="ltx_Math" altimg="m162.png" altimg-height="23px" altimg-valign="-7px" altimg-width="68px" alttext="Y_{n}\left(nz\right)" display="inline"><mrow><msub><mi href="./10.2#E3">Y</mi><mi href="./10.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi href="./10.1#p2.t1.r1">n</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></math> in <math class="ltx_Math" altimg="m322.png" altimg-height="23px" altimg-valign="-7px" altimg-width="93px" alttext="|\operatorname{ph}z|\leq\pi" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo stretchy="false">|</mo></mrow><mo>≤</mo><mi href="./3.12#E1">π</mi></mrow></math>.
Case <math class="ltx_Math" altimg="m288.png" altimg-height="17px" altimg-valign="-2px" altimg-width="63px" alttext="n=10" display="inline"><mrow><mi href="./10.1#p2.t1.r1">n</mi><mo>=</mo><mn>10</mn></mrow></math>, <math class="ltx_Math" altimg="m112.png" altimg-height="20px" altimg-valign="-5px" altimg-width="148px" alttext="-2.3\leq\Re z\leq 1.9" display="inline"><mrow><mrow><mo>-</mo><mn>2.3</mn></mrow><mo>≤</mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo>≤</mo><mn>1.9</mn></mrow></math>.
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E3" title="(10.2.3) ‣ Bessel Function of the Second Kind (Weber’s Function) ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m153.png" altimg-height="23px" altimg-valign="-7px" altimg-width="55px" alttext="Y_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E3">Y</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the second kind</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m226.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.9#E7" title="(1.9.7) ‣ Modulus and Phase ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m221.png" altimg-height="21px" altimg-valign="-6px" altimg-width="26px" alttext="\operatorname{ph}" display="inline"><mi href="./1.9#E7">ph</mi></math>: phase</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m167.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./10.1#p2.t1.r1" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m291.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./10.1#p2.t1.r1">n</mi></math>: integer</a> and
<a href="./10.1#p2.t1.r4" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m312.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./10.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</figure>
</td>
<td class="ltx_subfigure">
<figure id="F6" class="ltx_figure"><figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 10.21.6: </span>Zeros <math class="ltx_Math" altimg="m180.png" altimg-height="13px" altimg-valign="-2px" altimg-width="43px" alttext="\bullet\bullet\bullet" display="inline"><mrow><mo>∙</mo><mo></mo><mo>∙</mo><mo></mo><mo>∙</mo></mrow></math> of <math class="ltx_Math" altimg="m315.png" altimg-height="29px" altimg-valign="-7px" altimg-width="85px" alttext="{H^{(1)}_{n}}\left(nz\right)" display="inline"><mrow><msubsup><mi href="./10.2#E5">H</mi><mi href="./10.1#p2.t1.r1">n</mi><mrow><mo href="./10.2#E5" stretchy="false">(</mo><mn>1</mn><mo href="./10.2#E5" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mrow><mi href="./10.1#p2.t1.r1">n</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></math> in <math class="ltx_Math" altimg="m322.png" altimg-height="23px" altimg-valign="-7px" altimg-width="93px" alttext="|\operatorname{ph}z|\leq\pi" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo stretchy="false">|</mo></mrow><mo>≤</mo><mi href="./3.12#E1">π</mi></mrow></math>.
Case <math class="ltx_Math" altimg="m288.png" altimg-height="17px" altimg-valign="-2px" altimg-width="63px" alttext="n=10" display="inline"><mrow><mi href="./10.1#p2.t1.r1">n</mi><mo>=</mo><mn>10</mn></mrow></math>, <math class="ltx_Math" altimg="m112.png" altimg-height="20px" altimg-valign="-5px" altimg-width="148px" alttext="-2.3\leq\Re z\leq 1.9" display="inline"><mrow><mrow><mo>-</mo><mn>2.3</mn></mrow><mo>≤</mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo>≤</mo><mn>1.9</mn></mrow></math>.
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E5" title="(10.2.5) ‣ Bessel Functions of the Third Kind (Hankel Functions) ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m313.png" altimg-height="29px" altimg-valign="-7px" altimg-width="73px" alttext="{H^{(1)}_{\NVar{\nu}}}\left(\NVar{z}\right)" display="inline"><mrow><msubsup><mi href="./10.2#E5">H</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi><mrow><mo href="./10.2#E5" stretchy="false">(</mo><mn>1</mn><mo href="./10.2#E5" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the third kind (or Hankel function)</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m226.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.9#E7" title="(1.9.7) ‣ Modulus and Phase ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m221.png" altimg-height="21px" altimg-valign="-6px" altimg-width="26px" alttext="\operatorname{ph}" display="inline"><mi href="./1.9#E7">ph</mi></math>: phase</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m167.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./10.1#p2.t1.r1" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m291.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./10.1#p2.t1.r1">n</mi></math>: integer</a> and
<a href="./10.1#p2.t1.r4" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m312.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./10.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</figure>
</td>
</tr>
</table>
</figure>
</section>
<section id="Px2" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Zeros of <math class="ltx_Math" altimg="m315.png" altimg-height="29px" altimg-valign="-7px" altimg-width="85px" alttext="{H^{(1)}_{n}}\left(nz\right)" display="inline"><mrow><msubsup><mi href="./10.2#E5">H</mi><mi href="./10.1#p2.t1.r1">n</mi><mrow><mo href="./10.2#E5" stretchy="false">(</mo><mn>1</mn><mo href="./10.2#E5" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mrow><mi href="./10.1#p2.t1.r1">n</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></math>, <math class="ltx_Math" altimg="m317.png" altimg-height="29px" altimg-valign="-7px" altimg-width="85px" alttext="{H^{(2)}_{n}}\left(nz\right)" display="inline"><mrow><msubsup><mi href="./10.2#E6">H</mi><mi href="./10.1#p2.t1.r1">n</mi><mrow><mo href="./10.2#E6" stretchy="false">(</mo><mn>2</mn><mo href="./10.2#E6" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mrow><mi href="./10.1#p2.t1.r1">n</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></math>,
<math class="ltx_Math" altimg="m314.png" altimg-height="32px" altimg-valign="-7px" altimg-width="90px" alttext="{H^{(1)}_{n}}'\left(nz\right)" display="inline"><mrow><mmultiscripts><mi href="./10.2#E5">H</mi><mi href="./10.1#p2.t1.r1">n</mi><mrow><mo href="./10.2#E5" stretchy="false">(</mo><mn>1</mn><mo href="./10.2#E5" stretchy="false">)</mo></mrow><none></none><mo>′</mo></mmultiscripts><mo></mo><mrow><mo>(</mo><mrow><mi href="./10.1#p2.t1.r1">n</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></math>, <math class="ltx_Math" altimg="m316.png" altimg-height="32px" altimg-valign="-7px" altimg-width="90px" alttext="{H^{(2)}_{n}}'\left(nz\right)" display="inline"><mrow><mmultiscripts><mi href="./10.2#E6">H</mi><mi href="./10.1#p2.t1.r1">n</mi><mrow><mo href="./10.2#E6" stretchy="false">(</mo><mn>2</mn><mo href="./10.2#E6" stretchy="false">)</mo></mrow><none></none><mo>′</mo></mmultiscripts><mo></mo><mrow><mo>(</mo><mrow><mi href="./10.1#p2.t1.r1">n</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></math>
</h3>
<div id="Px2.info" class="ltx_metadata ltx_info">
the continuous curve that joins the points <math class="ltx_Math" altimg="m227.png" altimg-height="18px" altimg-valign="-4px" altimg-width="30px" alttext="\pm 1" display="inline"><mrow><mo>±</mo><mn>1</mn></mrow></math> is
the lower boundary of <math class="ltx_Math" altimg="m189.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="\mathbf{K}" display="inline"><mi href="./10.20#F3" mathvariant="bold">K</mi></math>.</p>
</div>
<div id="Px2.p2" class="ltx_para">
<p class="ltx_p">The first set of zeros of the principal value of <math class="ltx_Math" altimg="m315.png" altimg-height="29px" altimg-valign="-7px" altimg-width="85px" alttext="{H^{(1)}_{n}}\left(nz\right)" display="inline"><mrow><msubsup><mi href="./10.2#E5">H</mi><mi href="./10.1#p2.t1.r1">n</mi><mrow><mo href="./10.2#E5" stretchy="false">(</mo><mn>1</mn><mo href="./10.2#E5" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mrow><mi href="./10.1#p2.t1.r1">n</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></math> is an
infinite string with asymptote <math class="ltx_Math" altimg="m165.png" altimg-height="23px" altimg-valign="-7px" altimg-width="109px" alttext="\Im z=-\mathrm{i}d/n" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℑ</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo>=</mo><mrow><mo>-</mo><mrow><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./10.21#E47">d</mi></mrow><mo>/</mo><mi href="./10.1#p2.t1.r1">n</mi></mrow></mrow></mrow></math>, where</p>
<table id="E47" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.21.47</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E47.png" altimg-height="29px" altimg-valign="-9px" altimg-width="216px" alttext="d=\tfrac{1}{2}\ln 2=0.34657\ldots." display="block"><mrow><mrow><mi href="./10.21#E47">d</mi><mo>=</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mn>2</mn></mrow></mrow><mo>=</mo><mrow><mn>0.34657</mn><mo></mo><mi mathvariant="normal">…</mi></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E47.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text"><math class="ltx_Math" altimg="m265.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="d" display="inline"><mi href="./10.21#E47">d</mi></math> (locally)</span></dd>
<dt>Symbols:</dt>
<dd><a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m187.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a></dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">The only other set comprises <math class="ltx_Math" altimg="m291.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./10.1#p2.t1.r1">n</mi></math> zeros that are asymptotically close to the
lower boundary of <math class="ltx_Math" altimg="m189.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="\mathbf{K}" display="inline"><mi href="./10.20#F3" mathvariant="bold">K</mi></math> as <math class="ltx_Math" altimg="m292.png" altimg-height="13px" altimg-valign="-2px" altimg-width="67px" alttext="n\to\infty" display="inline"><mrow><mi href="./10.1#p2.t1.r1">n</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></math>. Figures
plot the actual zeros for <math class="ltx_Math" altimg="m287.png" altimg-height="20px" altimg-valign="-6px" altimg-width="72px" alttext="n=1,5" display="inline"><mrow><mi href="./10.1#p2.t1.r1">n</mi><mo>=</mo><mrow><mn>1</mn><mo>,</mo><mn>5</mn></mrow></mrow></math>, and <math class="ltx_Math" altimg="m120.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="10" display="inline"><mn>10</mn></math>, respectively.</p>
</div>
<div id="Px2.p3" class="ltx_para">
<p class="ltx_p">The zeros of <math class="ltx_Math" altimg="m314.png" altimg-height="32px" altimg-valign="-7px" altimg-width="90px" alttext="{H^{(1)}_{n}}'\left(nz\right)" display="inline"><mrow><mmultiscripts><mi href="./10.2#E5">H</mi><mi href="./10.1#p2.t1.r1">n</mi><mrow><mo href="./10.2#E5" stretchy="false">(</mo><mn>1</mn><mo href="./10.2#E5" stretchy="false">)</mo></mrow><none></none><mo>′</mo></mmultiscripts><mo></mo><mrow><mo>(</mo><mrow><mi href="./10.1#p2.t1.r1">n</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></math> have a similar pattern to those of
<math class="ltx_Math" altimg="m315.png" altimg-height="29px" altimg-valign="-7px" altimg-width="85px" alttext="{H^{(1)}_{n}}\left(nz\right)" display="inline"><mrow><msubsup><mi href="./10.2#E5">H</mi><mi href="./10.1#p2.t1.r1">n</mi><mrow><mo href="./10.2#E5" stretchy="false">(</mo><mn>1</mn><mo href="./10.2#E5" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mrow><mi href="./10.1#p2.t1.r1">n</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></math>. The zeros of <math class="ltx_Math" altimg="m317.png" altimg-height="29px" altimg-valign="-7px" altimg-width="85px" alttext="{H^{(2)}_{n}}\left(nz\right)" display="inline"><mrow><msubsup><mi href="./10.2#E6">H</mi><mi href="./10.1#p2.t1.r1">n</mi><mrow><mo href="./10.2#E6" stretchy="false">(</mo><mn>2</mn><mo href="./10.2#E6" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mrow><mi href="./10.1#p2.t1.r1">n</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></math> and
<math class="ltx_Math" altimg="m316.png" altimg-height="32px" altimg-valign="-7px" altimg-width="90px" alttext="{H^{(2)}_{n}}'\left(nz\right)" display="inline"><mrow><mmultiscripts><mi href="./10.2#E6">H</mi><mi href="./10.1#p2.t1.r1">n</mi><mrow><mo href="./10.2#E6" stretchy="false">(</mo><mn>2</mn><mo href="./10.2#E6" stretchy="false">)</mo></mrow><none></none><mo>′</mo></mmultiscripts><mo></mo><mrow><mo>(</mo><mrow><mi href="./10.1#p2.t1.r1">n</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></math> are the complex conjugates of the zeros of
<math class="ltx_Math" altimg="m315.png" altimg-height="29px" altimg-valign="-7px" altimg-width="85px" alttext="{H^{(1)}_{n}}\left(nz\right)" display="inline"><mrow><msubsup><mi href="./10.2#E5">H</mi><mi href="./10.1#p2.t1.r1">n</mi><mrow><mo href="./10.2#E5" stretchy="false">(</mo><mn>1</mn><mo href="./10.2#E5" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mrow><mi href="./10.1#p2.t1.r1">n</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></math> and <math class="ltx_Math" altimg="m314.png" altimg-height="32px" altimg-valign="-7px" altimg-width="90px" alttext="{H^{(1)}_{n}}'\left(nz\right)" display="inline"><mrow><mmultiscripts><mi href="./10.2#E5">H</mi><mi href="./10.1#p2.t1.r1">n</mi><mrow><mo href="./10.2#E5" stretchy="false">(</mo><mn>1</mn><mo href="./10.2#E5" stretchy="false">)</mo></mrow><none></none><mo>′</mo></mmultiscripts><mo></mo><mrow><mo>(</mo><mrow><mi href="./10.1#p2.t1.r1">n</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></math>, respectively.</p>
</div>
</section>
<section id="Px3" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Zeros of <math class="ltx_Math" altimg="m130.png" altimg-height="23px" altimg-valign="-7px" altimg-width="132px" alttext="J_{0}\left(z\right)-\mathrm{i}J_{1}\left(z\right)" display="inline"><mrow><mrow><msub><mi href="./10.2#E2">J</mi><mn>0</mn></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow><mo>-</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mn>1</mn></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></mrow></math> and
<math class="ltx_Math" altimg="m143.png" altimg-height="23px" altimg-valign="-7px" altimg-width="156px" alttext="J_{n}\left(z\right)-\mathrm{i}J_{n+1}\left(z\right)" display="inline"><mrow><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow><mo>-</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mrow><mi href="./10.1#p2.t1.r1">n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></mrow></math>
</h3>
<div id="Px3.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="SS10.p1" class="ltx_para">
<p class="ltx_p">Throughout this subsection we assume <math class="ltx_Math" altimg="m216.png" altimg-height="19px" altimg-valign="-5px" altimg-width="52px" alttext="\nu\geq 0" display="inline"><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>≥</mo><mn>0</mn></mrow></math>, <math class="ltx_Math" altimg="m299.png" altimg-height="17px" altimg-valign="-3px" altimg-width="52px" alttext="x>0" display="inline"><mrow><mi href="./10.1#p2.t1.r3">x</mi><mo>></mo><mn>0</mn></mrow></math>, <math class="ltx_Math" altimg="m185.png" altimg-height="18px" altimg-valign="-3px" altimg-width="52px" alttext="\lambda>1" display="inline"><mrow><mi>λ</mi><mo>></mo><mn>1</mn></mrow></math>, and we
denote <math class="ltx_Math" altimg="m122.png" altimg-height="20px" altimg-valign="-2px" altimg-width="34px" alttext="4\nu^{2}" display="inline"><mrow><mn>4</mn><mo></mo><msup><mi href="./10.1#p2.t1.r5">ν</mi><mn>2</mn></msup></mrow></math> by <math class="ltx_Math" altimg="m210.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\mu" display="inline"><mi>μ</mi></math>.</p>
</div>
<div id="SS10.p2" class="ltx_para">
<p class="ltx_p">The zeros of the functions</p>
<table id="E48" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.21.48</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E48.png" altimg-height="25px" altimg-valign="-7px" altimg-width="266px" alttext="J_{\nu}\left(x\right)Y_{\nu}\left(\lambda x\right)-Y_{\nu}\left(x\right)J_{\nu%
}\left(\lambda x\right)" display="block"><mrow><mrow><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./10.2#E3">Y</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi>λ</mi><mo></mo><mi href="./10.1#p2.t1.r3">x</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>-</mo><mrow><mrow><msub><mi href="./10.2#E3">Y</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi>λ</mi><mo></mo><mi href="./10.1#p2.t1.r3">x</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E48.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m134.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./10.2#E3" title="(10.2.3) ‣ Bessel Function of the Second Kind (Weber’s Function) ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m153.png" altimg-height="23px" altimg-valign="-7px" altimg-width="55px" alttext="Y_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E3">Y</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the second kind</a>,
<a href="./10.1#p2.t1.r3" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m301.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./10.1#p2.t1.r3">x</mi></math>: real variable</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m215.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">9.5.27</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and</p>
<table id="E49" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.21.49</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E49.png" altimg-height="26px" altimg-valign="-7px" altimg-width="266px" alttext="J_{\nu}'\left(x\right)Y_{\nu}'\left(\lambda x\right)-Y_{\nu}'\left(x\right)J_{%
\nu}'\left(\lambda x\right)" display="block"><mrow><mrow><mrow><msubsup><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′</mo></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><msubsup><mi href="./10.2#E3">Y</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′</mo></msubsup><mo></mo><mrow><mo>(</mo><mrow><mi>λ</mi><mo></mo><mi href="./10.1#p2.t1.r3">x</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>-</mo><mrow><mrow><msubsup><mi href="./10.2#E3">Y</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′</mo></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><msubsup><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′</mo></msubsup><mo></mo><mrow><mo>(</mo><mrow><mi>λ</mi><mo></mo><mi href="./10.1#p2.t1.r3">x</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E49.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m134.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./10.2#E3" title="(10.2.3) ‣ Bessel Function of the Second Kind (Weber’s Function) ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m153.png" altimg-height="23px" altimg-valign="-7px" altimg-width="55px" alttext="Y_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E3">Y</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the second kind</a>,
<a href="./10.1#p2.t1.r3" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m301.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./10.1#p2.t1.r3">x</mi></math>: real variable</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m215.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">9.5.30</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">are simple and the asymptotic expansion of the <math class="ltx_Math" altimg="m284.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./10.1#p2.t1.r1">m</mi></math>th positive zero as
<math class="ltx_Math" altimg="m285.png" altimg-height="13px" altimg-valign="-2px" altimg-width="73px" alttext="m\to\infty" display="inline"><mrow><mi href="./10.1#p2.t1.r1">m</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></math> is given by</p>
<table id="E50" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.21.50</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E50.png" altimg-height="50px" altimg-valign="-16px" altimg-width="345px" alttext="\alpha+\frac{p}{\alpha}+\frac{q-p^{2}}{\alpha^{3}}+\frac{r-4pq+2p^{3}}{\alpha^%
{5}}+\cdots," display="block"><mrow><mrow><mi href="./10.21#E51">α</mi><mo>+</mo><mfrac><mi href="./10.21#E51">p</mi><mi href="./10.21#E51">α</mi></mfrac><mo>+</mo><mfrac><mrow><mi href="./10.21#E51">q</mi><mo>-</mo><msup><mi href="./10.21#E51">p</mi><mn>2</mn></msup></mrow><msup><mi href="./10.21#E51">α</mi><mn>3</mn></msup></mfrac><mo>+</mo><mfrac><mrow><mrow><mi href="./10.21#E51">r</mi><mo>-</mo><mrow><mn>4</mn><mo></mo><mi href="./10.21#E51">p</mi><mo></mo><mi href="./10.21#E51">q</mi></mrow></mrow><mo>+</mo><mrow><mn>2</mn><mo></mo><msup><mi href="./10.21#E51">p</mi><mn>3</mn></msup></mrow></mrow><msup><mi href="./10.21#E51">α</mi><mn>5</mn></msup></mfrac><mo>+</mo><mi mathvariant="normal">⋯</mi></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E50.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.21#E51" title="(10.21.51) ‣ §10.21(x) Cross-Products ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m171.png" altimg-height="13px" altimg-valign="-2px" altimg-width="17px" alttext="\alpha" display="inline"><mi href="./10.21#E51">α</mi></math></a>,
<a href="./10.21#E51" title="(10.21.51) ‣ §10.21(x) Cross-Products ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m293.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="p" display="inline"><mi href="./10.21#E51">p</mi></math></a>,
<a href="./10.21#E51" title="(10.21.51) ‣ §10.21(x) Cross-Products ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m294.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="q" display="inline"><mi href="./10.21#E51">q</mi></math></a> and
<a href="./10.21#E51" title="(10.21.51) ‣ §10.21(x) Cross-Products ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m295.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="r" display="inline"><mi href="./10.21#E51">r</mi></math></a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">9.5.28</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where, in the case of (),</p>
<table id="E51" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex47" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="4" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">10.21.51</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m56.png" altimg-height="14px" altimg-valign="-2px" altimg-width="19px" alttext="\displaystyle\alpha" display="inline"><mi href="./10.21#E51">α</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m30.png" altimg-height="43px" altimg-valign="-17px" altimg-width="83px" alttext="\displaystyle=\frac{m\pi}{\lambda-1}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mstyle displaystyle="true"><mfrac><mrow><mi href="./10.1#p2.t1.r1">m</mi><mo></mo><mi href="./3.12#E1">π</mi></mrow><mrow><mi>λ</mi><mo>-</mo><mn>1</mn></mrow></mfrac></mstyle></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex48" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m95.png" altimg-height="18px" altimg-valign="-6px" altimg-width="16px" alttext="\displaystyle p" display="inline"><mi href="./10.21#E51">p</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m29.png" altimg-height="46px" altimg-valign="-16px" altimg-width="84px" alttext="\displaystyle=\frac{\mu-1}{8\lambda}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mstyle displaystyle="true"><mfrac><mrow><mi>μ</mi><mo>-</mo><mn>1</mn></mrow><mrow><mn>8</mn><mo></mo><mi>λ</mi></mrow></mfrac></mstyle></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex49" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m96.png" altimg-height="18px" altimg-valign="-6px" altimg-width="16px" alttext="\displaystyle q" display="inline"><mi href="./10.21#E51">q</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m18.png" altimg-height="55px" altimg-valign="-21px" altimg-width="241px" alttext="\displaystyle=\frac{(\mu-1)(\mu-25)(\lambda^{3}-1)}{6(4\lambda)^{3}(\lambda-1)}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mstyle displaystyle="true"><mfrac><mrow><mrow><mo stretchy="false">(</mo><mrow><mi>μ</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>μ</mi><mo>-</mo><mn>25</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><msup><mi>λ</mi><mn>3</mn></msup><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mrow><mn>6</mn><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mn>4</mn><mo></mo><mi>λ</mi></mrow><mo stretchy="false">)</mo></mrow><mn>3</mn></msup><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>λ</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow></mfrac></mstyle></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex50" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m97.png" altimg-height="14px" altimg-valign="-2px" altimg-width="16px" alttext="\displaystyle r" display="inline"><mi href="./10.21#E51">r</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m19.png" altimg-height="55px" altimg-valign="-21px" altimg-width="336px" alttext="\displaystyle=\frac{(\mu-1)(\mu^{2}-114\mu+1073)(\lambda^{5}-1)}{5(4\lambda)^{%
5}(\lambda-1)}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mstyle displaystyle="true"><mfrac><mrow><mrow><mo stretchy="false">(</mo><mrow><mi>μ</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><msup><mi>μ</mi><mn>2</mn></msup><mo>-</mo><mrow><mn>114</mn><mo></mo><mi>μ</mi></mrow></mrow><mo>+</mo><mn>1073</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><msup><mi>λ</mi><mn>5</mn></msup><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mrow><mn>5</mn><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mn>4</mn><mo></mo><mi>λ</mi></mrow><mo stretchy="false">)</mo></mrow><mn>5</mn></msup><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>λ</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow></mfrac></mstyle></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E51.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd>
<span class="ltx_text"><math class="ltx_Math" altimg="m171.png" altimg-height="13px" altimg-valign="-2px" altimg-width="17px" alttext="\alpha" display="inline"><mi href="./10.21#E51">α</mi></math> (locally)</span>,
<span class="ltx_text"><math class="ltx_Math" altimg="m293.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="p" display="inline"><mi href="./10.21#E51">p</mi></math> (locally)</span>,
<span class="ltx_text"><math class="ltx_Math" altimg="m294.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="q" display="inline"><mi href="./10.21#E51">q</mi></math> (locally)</span> and
<span class="ltx_text"><math class="ltx_Math" altimg="m295.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="r" display="inline"><mi href="./10.21#E51">r</mi></math> (locally)</span>
</dd>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m226.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a> and
<a href="./10.1#p2.t1.r1" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m284.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./10.1#p2.t1.r1">m</mi></math>: integer</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">9.5.29</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and, in the case of (),</p>
<table id="E52" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex51" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="4" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">10.21.52</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m56.png" altimg-height="14px" altimg-valign="-2px" altimg-width="19px" alttext="\displaystyle\alpha" display="inline"><mi href="./10.21#E51">α</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m26.png" altimg-height="49px" altimg-valign="-17px" altimg-width="117px" alttext="\displaystyle=\frac{(m-1)\pi}{\lambda-1}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mstyle displaystyle="true"><mfrac><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./10.1#p2.t1.r1">m</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi href="./3.12#E1">π</mi></mrow><mrow><mi>λ</mi><mo>-</mo><mn>1</mn></mrow></mfrac></mstyle></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex52" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m95.png" altimg-height="18px" altimg-valign="-6px" altimg-width="16px" alttext="\displaystyle p" display="inline"><mi href="./10.21#E51">p</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m28.png" altimg-height="46px" altimg-valign="-16px" altimg-width="84px" alttext="\displaystyle=\frac{\mu+3}{8\lambda}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mstyle displaystyle="true"><mfrac><mrow><mi>μ</mi><mo>+</mo><mn>3</mn></mrow><mrow><mn>8</mn><mo></mo><mi>λ</mi></mrow></mfrac></mstyle></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex53" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m96.png" altimg-height="18px" altimg-valign="-6px" altimg-width="16px" alttext="\displaystyle q" display="inline"><mi href="./10.21#E51">q</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m20.png" altimg-height="55px" altimg-valign="-21px" altimg-width="245px" alttext="\displaystyle=\frac{(\mu^{2}+46\mu-63)(\lambda^{3}-1)}{6(4\lambda)^{3}(\lambda%
-1)}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mstyle displaystyle="true"><mfrac><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><msup><mi>μ</mi><mn>2</mn></msup><mo>+</mo><mrow><mn>46</mn><mo></mo><mi>μ</mi></mrow></mrow><mo>-</mo><mn>63</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><msup><mi>λ</mi><mn>3</mn></msup><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mrow><mn>6</mn><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mn>4</mn><mo></mo><mi>λ</mi></mrow><mo stretchy="false">)</mo></mrow><mn>3</mn></msup><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>λ</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow></mfrac></mstyle></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex54" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m97.png" altimg-height="14px" altimg-valign="-2px" altimg-width="16px" alttext="\displaystyle r" display="inline"><mi href="./10.21#E51">r</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m22.png" altimg-height="55px" altimg-valign="-21px" altimg-width="360px" alttext="\displaystyle=\frac{(\mu^{3}+185\mu^{2}-2053\mu+1899)(\lambda^{5}-1)}{5(4%
\lambda)^{5}(\lambda-1)}." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mstyle displaystyle="true"><mfrac><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mrow><msup><mi>μ</mi><mn>3</mn></msup><mo>+</mo><mrow><mn>185</mn><mo></mo><msup><mi>μ</mi><mn>2</mn></msup></mrow></mrow><mo>-</mo><mrow><mn>2053</mn><mo></mo><mi>μ</mi></mrow></mrow><mo>+</mo><mn>1899</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><msup><mi>λ</mi><mn>5</mn></msup><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mrow><mn>5</mn><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mn>4</mn><mo></mo><mi>λ</mi></mrow><mo stretchy="false">)</mo></mrow><mn>5</mn></msup><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>λ</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow></mfrac></mstyle></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E52.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m226.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./10.1#p2.t1.r1" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m284.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./10.1#p2.t1.r1">m</mi></math>: integer</a>,
<a href="./10.21#E51" title="(10.21.51) ‣ §10.21(x) Cross-Products ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m171.png" altimg-height="13px" altimg-valign="-2px" altimg-width="17px" alttext="\alpha" display="inline"><mi href="./10.21#E51">α</mi></math></a>,
<a href="./10.21#E51" title="(10.21.51) ‣ §10.21(x) Cross-Products ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m293.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="p" display="inline"><mi href="./10.21#E51">p</mi></math></a>,
<a href="./10.21#E51" title="(10.21.51) ‣ §10.21(x) Cross-Products ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m294.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="q" display="inline"><mi href="./10.21#E51">q</mi></math></a> and
<a href="./10.21#E51" title="(10.21.51) ‣ §10.21(x) Cross-Products ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m295.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="r" display="inline"><mi href="./10.21#E51">r</mi></math></a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">9.5.31</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS10.p3" class="ltx_para">
<p class="ltx_p">The asymptotic expansion of the large positive zeros (not necessarily the
<math class="ltx_Math" altimg="m284.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./10.1#p2.t1.r1">m</mi></math>th) of the function</p>
<table id="E53" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.21.53</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E53.png" altimg-height="26px" altimg-valign="-7px" altimg-width="266px" alttext="J_{\nu}'\left(x\right)Y_{\nu}\left(\lambda x\right)-Y_{\nu}'\left(x\right)J_{%
\nu}\left(\lambda x\right)" display="block"><mrow><mrow><mrow><msubsup><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′</mo></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./10.2#E3">Y</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi>λ</mi><mo></mo><mi href="./10.1#p2.t1.r3">x</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>-</mo><mrow><mrow><msubsup><mi href="./10.2#E3">Y</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′</mo></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi>λ</mi><mo></mo><mi href="./10.1#p2.t1.r3">x</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E53.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m134.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./10.2#E3" title="(10.2.3) ‣ Bessel Function of the Second Kind (Weber’s Function) ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m153.png" altimg-height="23px" altimg-valign="-7px" altimg-width="55px" alttext="Y_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E3">Y</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the second kind</a>,
<a href="./10.1#p2.t1.r3" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m301.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./10.1#p2.t1.r3">x</mi></math>: real variable</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m215.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">9.5.32</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">is given by (), where</p>
</div>
<div id="SS10.p4" class="ltx_para">
<table id="E54" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex55" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="4" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">10.21.54</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m56.png" altimg-height="14px" altimg-valign="-2px" altimg-width="19px" alttext="\displaystyle\alpha" display="inline"><mi href="./10.21#E51">α</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m27.png" altimg-height="52px" altimg-valign="-17px" altimg-width="120px" alttext="\displaystyle=\frac{(m-\tfrac{1}{2})\pi}{\lambda-1}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mstyle displaystyle="true"><mfrac><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./10.1#p2.t1.r1">m</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi href="./3.12#E1">π</mi></mrow><mrow><mi>λ</mi><mo>-</mo><mn>1</mn></mrow></mfrac></mstyle></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex56" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m95.png" altimg-height="18px" altimg-valign="-6px" altimg-width="16px" alttext="\displaystyle p" display="inline"><mi href="./10.21#E51">p</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m17.png" altimg-height="53px" altimg-valign="-21px" altimg-width="197px" alttext="\displaystyle=\frac{(\mu+3)\lambda-(\mu-1)}{8\lambda(\lambda-1)}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mstyle displaystyle="true"><mfrac><mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mi>μ</mi><mo>+</mo><mn>3</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi>λ</mi></mrow><mo>-</mo><mrow><mo stretchy="false">(</mo><mrow><mi>μ</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mrow><mn>8</mn><mo></mo><mi>λ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>λ</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow></mfrac></mstyle></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex57" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m96.png" altimg-height="18px" altimg-valign="-6px" altimg-width="16px" alttext="\displaystyle q" display="inline"><mi href="./10.21#E51">q</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m21.png" altimg-height="55px" altimg-valign="-21px" altimg-width="353px" alttext="\displaystyle=\frac{(\mu^{2}+46\mu-63)\lambda^{3}-(\mu-1)(\mu-25)}{6(4\lambda)%
^{3}(\lambda-1)}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mstyle displaystyle="true"><mfrac><mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><msup><mi>μ</mi><mn>2</mn></msup><mo>+</mo><mrow><mn>46</mn><mo></mo><mi>μ</mi></mrow></mrow><mo>-</mo><mn>63</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msup><mi>λ</mi><mn>3</mn></msup></mrow><mo>-</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mi>μ</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>μ</mi><mo>-</mo><mn>25</mn></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mrow><mn>6</mn><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mn>4</mn><mo></mo><mi>λ</mi></mrow><mo stretchy="false">)</mo></mrow><mn>3</mn></msup><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>λ</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow></mfrac></mstyle></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex58" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m97.png" altimg-height="14px" altimg-valign="-2px" altimg-width="16px" alttext="\displaystyle r" display="inline"><mi href="./10.21#E51">r</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m23.png" altimg-height="55px" altimg-valign="-21px" altimg-width="563px" alttext="\displaystyle=\frac{(\mu^{3}+185\mu^{2}-2053\mu+1899)\lambda^{5}-(\mu-1)(\mu^{%
2}-114\mu+1073)}{5(4\lambda)^{5}(\lambda-1)}." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mstyle displaystyle="true"><mfrac><mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mrow><msup><mi>μ</mi><mn>3</mn></msup><mo>+</mo><mrow><mn>185</mn><mo></mo><msup><mi>μ</mi><mn>2</mn></msup></mrow></mrow><mo>-</mo><mrow><mn>2053</mn><mo></mo><mi>μ</mi></mrow></mrow><mo>+</mo><mn>1899</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msup><mi>λ</mi><mn>5</mn></msup></mrow><mo>-</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mi>μ</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><msup><mi>μ</mi><mn>2</mn></msup><mo>-</mo><mrow><mn>114</mn><mo></mo><mi>μ</mi></mrow></mrow><mo>+</mo><mn>1073</mn></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mrow><mn>5</mn><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mn>4</mn><mo></mo><mi>λ</mi></mrow><mo stretchy="false">)</mo></mrow><mn>5</mn></msup><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>λ</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow></mfrac></mstyle></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E54.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m226.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./10.1#p2.t1.r1" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m284.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./10.1#p2.t1.r1">m</mi></math>: integer</a>,
<a href="./10.21#E51" title="(10.21.51) ‣ §10.21(x) Cross-Products ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m171.png" altimg-height="13px" altimg-valign="-2px" altimg-width="17px" alttext="\alpha" display="inline"><mi href="./10.21#E51">α</mi></math></a>,
<a href="./10.21#E51" title="(10.21.51) ‣ §10.21(x) Cross-Products ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m293.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="p" display="inline"><mi href="./10.21#E51">p</mi></math></a>,
<a href="./10.21#E51" title="(10.21.51) ‣ §10.21(x) Cross-Products ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m294.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="q" display="inline"><mi href="./10.21#E51">q</mi></math></a> and
<a href="./10.21#E51" title="(10.21.51) ‣ §10.21(x) Cross-Products ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m295.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="r" display="inline"><mi href="./10.21#E51">r</mi></math></a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">9.5.33</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS10.p5" class="ltx_para">
<p class="ltx_p">Higher coefficients in the asymptotic expansions in this subsection can be
obtained by expressing the cross-products in terms of the modulus and phase
functions (§</dd>
</dl>
</div>
</div>
<div id="SS11.p1" class="ltx_para">
<p class="ltx_p">The Riccati–Bessel functions are
<math class="ltx_Math" altimg="m109.png" altimg-height="29px" altimg-valign="-9px" altimg-width="120px" alttext="(\tfrac{1}{2}\pi x)^{\frac{1}{2}}J_{\nu}\left(x\right)" display="inline"><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi href="./10.1#p2.t1.r3">x</mi></mrow><mo stretchy="false">)</mo></mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></msup><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mrow></math> and
<math class="ltx_Math" altimg="m110.png" altimg-height="29px" altimg-valign="-9px" altimg-width="120px" alttext="(\tfrac{1}{2}\pi x)^{\frac{1}{2}}Y_{\nu}\left(x\right)" display="inline"><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi href="./10.1#p2.t1.r3">x</mi></mrow><mo stretchy="false">)</mo></mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></msup><mo></mo><mrow><msub><mi href="./10.2#E3">Y</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mrow></math>. Except possibly for
<math class="ltx_Math" altimg="m298.png" altimg-height="17px" altimg-valign="-2px" altimg-width="52px" alttext="x=0" display="inline"><mrow><mi href="./10.1#p2.t1.r3">x</mi><mo>=</mo><mn>0</mn></mrow></math> their zeros are the same as those of <math class="ltx_Math" altimg="m141.png" altimg-height="23px" altimg-valign="-7px" altimg-width="55px" alttext="J_{\nu}\left(x\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math> and
<math class="ltx_Math" altimg="m157.png" altimg-height="23px" altimg-valign="-7px" altimg-width="56px" alttext="Y_{\nu}\left(x\right)" display="inline"><mrow><msub><mi href="./10.2#E3">Y</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>, respectively. For information on the zeros of the
derivatives of Riccati–Bessel functions, and also on zeros of their
cross-products, see <cite class="ltx_cite ltx_citemacro_citet">Boyer (.</p>
</div>
</section>
<section id="xii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§10.21(xii) </span>Zeros of <math class="ltx_Math" altimg="m168.png" altimg-height="24px" altimg-valign="-7px" altimg-width="155px" alttext="\alpha J_{\nu}\left(x\right)+xJ_{\nu}'\left(x\right)" display="inline"><mrow><mrow><mi>α</mi><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mi href="./10.1#p2.t1.r3">x</mi><mo></mo><mrow><msubsup><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′</mo></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mrow></mrow></math>
</h2>
<div id="SS12.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="SS12.p1" class="ltx_para">
<p class="ltx_p">For properties of the positive zeros of the function
<math class="ltx_Math" altimg="m168.png" altimg-height="24px" altimg-valign="-7px" altimg-width="155px" alttext="\alpha J_{\nu}\left(x\right)+xJ_{\nu}'\left(x\right)" display="inline"><mrow><mrow><mi>α</mi><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mi href="./10.1#p2.t1.r3">x</mi><mo></mo><mrow><msubsup><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′</mo></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mrow></mrow></math>, with <math class="ltx_Math" altimg="m171.png" altimg-height="13px" altimg-valign="-2px" altimg-width="17px" alttext="\alpha" display="inline"><mi>α</mi></math> and
<math class="ltx_Math" altimg="m215.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math> real, see <cite class="ltx_cite ltx_citemacro_citet">Landau (</dd>
</dl>
</div>
</div>
<div id="SS13.p1" class="ltx_para">
<p class="ltx_p">The <em class="ltx_emph ltx_font_italic">Rayleigh function</em> <math class="ltx_Math" altimg="m233.png" altimg-height="23px" altimg-valign="-7px" altimg-width="56px" alttext="\sigma_{n}\left(\nu\right)" display="inline"><mrow><msub><mi href="./10.21#E55">σ</mi><mi href="./10.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r5">ν</mi><mo>)</mo></mrow></mrow></math> is defined by</p>
<table id="E55" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.21.55</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E55.png" altimg-height="64px" altimg-valign="-27px" altimg-width="207px" alttext="\sigma_{n}\left(\nu\right)=\sum_{m=1}^{\infty}(j_{\nu,m})^{-2n}," display="block"><mrow><mrow><mrow><msub><mi href="./10.21#E55">σ</mi><mi href="./10.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r5">ν</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./10.1#p2.t1.r1">m</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover><msup><mrow><mo stretchy="false">(</mo><msub><mi>j</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>,</mo><mi href="./10.1#p2.t1.r1">m</mi></mrow></msub><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r1">n</mi></mrow></mrow></msup></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m286.png" altimg-height="20px" altimg-valign="-6px" altimg-width="123px" alttext="n=1,2,3,\dots" display="inline"><mrow><mi href="./10.1#p2.t1.r1">n</mi><mo>=</mo><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E55.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m230.png" altimg-height="23px" altimg-valign="-7px" altimg-width="56px" alttext="\sigma_{\NVar{n}}\left(\NVar{\nu}\right)" display="inline"><mrow><msub><mi href="./10.21#E55">σ</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi><mo>)</mo></mrow></mrow></math>: Rayleigh function</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./10.1#p2.t1.r1" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m284.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./10.1#p2.t1.r1">m</mi></math>: integer</a>,
<a href="./10.1#p2.t1.r1" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m291.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./10.1#p2.t1.r1">n</mi></math>: integer</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m215.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">For properties, computation, and generalizations see <cite class="ltx_cite ltx_citemacro_citet">Kapitsa (, §§15.5, 15.51)</cite>.
</p>
</div>
</section>
<section id="xiv" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§10.21(xiv) </span><math class="ltx_Math" altimg="m215.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>-Zeros</h2>
<div id="SS14.info" class="ltx_metadata ltx_info">
</div>
</div>
</body></text>
</html>
</page>
<page>
<!DOCTYPE html><html>
<head>
<title>DLMF: 10.23 Sums</title>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
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<div class="ltx_page_navlogo"></dd>
</dl>
</div>
</div>
<div id="SS1.p1" class="ltx_para">
<table id="E1" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.23.1</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E1.png" altimg-height="64px" altimg-valign="-28px" altimg-width="432px" alttext="\mathscr{C}_{\nu}\left(\lambda z\right)=\lambda^{\pm\nu}\sum_{k=0}^{\infty}%
\frac{(\mp 1)^{k}(\lambda^{2}-1)^{k}(\tfrac{1}{2}z)^{k}}{k!}\mathscr{C}_{\nu%
\pm k}\left(z\right)," display="block"><mrow><mrow><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5.p1">𝒞</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi>λ</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msup><mi>λ</mi><mrow><mo>±</mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow></msup><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./10.1#p2.t1.r2">k</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><mfrac><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>∓</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./10.1#p2.t1.r2">k</mi></msup><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><msup><mi>λ</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./10.1#p2.t1.r2">k</mi></msup><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo stretchy="false">)</mo></mrow><mi href="./10.1#p2.t1.r2">k</mi></msup></mrow><mrow><mi href="./10.1#p2.t1.r2">k</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mfrac><mo></mo><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5.p1">𝒞</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>±</mo><mi href="./10.1#p2.t1.r2">k</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m70.png" altimg-height="25px" altimg-valign="-7px" altimg-width="107px" alttext="|\lambda^{2}-1|<1" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mrow><msup><mi>λ</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">|</mo></mrow><mo><</mo><mn>1</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#Px5.p1" title="Cylinder Functions ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m36.png" altimg-height="23px" altimg-valign="-7px" altimg-width="56px" alttext="\mathscr{C}_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5.p1">𝒞</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: cylinder function</a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m7.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m54.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./10.1#p2.t1.r2" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./10.1#p2.t1.r2">k</mi></math>: nonnegative integer</a>,
<a href="./10.1#p2.t1.r4" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./10.1#p2.t1.r4">z</mi></math>: complex variable</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">9.1.74</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">If <math class="ltx_Math" altimg="m35.png" altimg-height="18px" altimg-valign="-2px" altimg-width="60px" alttext="\mathscr{C}=J" display="inline"><mrow><mi class="ltx_font_mathscript" href="./10.2#Px5.p1">𝒞</mi><mo>=</mo><mi href="./10.2#E2">J</mi></mrow></math> and the upper signs are taken, then the restriction
on <math class="ltx_Math" altimg="m30.png" altimg-height="18px" altimg-valign="-2px" altimg-width="16px" alttext="\lambda" display="inline"><mi>λ</mi></math> is unnecessary.</p>
</div>
</section>
<section id="ii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§10.23(ii) </span>Addition Theorems</h2>
<div id="SS2.info" class="ltx_metadata ltx_info">
) by taking <math class="ltx_Math" altimg="m23.png" altimg-height="20px" altimg-valign="-6px" altimg-width="53px" alttext="\chi=0" display="inline"><mrow><mi>χ</mi><mo>=</mo><mn>0</mn></mrow></math> and <math class="ltx_Math" altimg="m22.png" altimg-height="20px" altimg-valign="-6px" altimg-width="75px" alttext="\alpha=0,\pi" display="inline"><mrow><mi>α</mi><mo>=</mo><mrow><mn>0</mn><mo>,</mo><mi href="./3.12#E1">π</mi></mrow></mrow></math>.</dd>
<dt>Addition (effective with 1.0.7):</dt>
<dd>
The cross-reference to §</dd>
</dl>
</div>
</div>
<div id="Px1.p1" class="ltx_para">
<table id="E2" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.23.2</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E2.png" altimg-height="66px" altimg-valign="-30px" altimg-width="310px" alttext="\mathscr{C}_{\nu}\left(u\pm v\right)=\sum_{k=-\infty}^{\infty}\mathscr{C}_{\nu%
\mp k}\left(u\right)J_{k}\left(v\right)," display="block"><mrow><mrow><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5.p1">𝒞</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi>u</mi><mo>±</mo><mi>v</mi></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./10.1#p2.t1.r2">k</mi><mo>=</mo><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5.p1">𝒞</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>∓</mo><mi href="./10.1#p2.t1.r2">k</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r2">k</mi></msub><mo></mo><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m72.png" altimg-height="23px" altimg-valign="-7px" altimg-width="75px" alttext="|v|<|u|" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mi>v</mi><mo stretchy="false">|</mo></mrow><mo><</mo><mrow><mo stretchy="false">|</mo><mi>u</mi><mo stretchy="false">|</mo></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m14.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./10.2#Px5.p1" title="Cylinder Functions ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m36.png" altimg-height="23px" altimg-valign="-7px" altimg-width="56px" alttext="\mathscr{C}_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5.p1">𝒞</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: cylinder function</a>,
<a href="./10.1#p2.t1.r2" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./10.1#p2.t1.r2">k</mi></math>: nonnegative integer</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">9.1.75</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">The restriction <math class="ltx_Math" altimg="m72.png" altimg-height="23px" altimg-valign="-7px" altimg-width="75px" alttext="|v|<|u|" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mi>v</mi><mo stretchy="false">|</mo></mrow><mo><</mo><mrow><mo stretchy="false">|</mo><mi>u</mi><mo stretchy="false">|</mo></mrow></mrow></math> is unnecessary when <math class="ltx_Math" altimg="m35.png" altimg-height="18px" altimg-valign="-2px" altimg-width="60px" alttext="\mathscr{C}=J" display="inline"><mrow><mi class="ltx_font_mathscript" href="./10.2#Px5.p1">𝒞</mi><mo>=</mo><mi href="./10.2#E2">J</mi></mrow></math> and
<math class="ltx_Math" altimg="m38.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math> is an integer. Special cases are:</p>
<table id="E3" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.23.3</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E3.png" altimg-height="64px" altimg-valign="-28px" altimg-width="224px" alttext="{J_{0}^{2}}\left(z\right)+2\sum_{k=1}^{\infty}{J_{k}^{2}}\left(z\right)=1," display="block"><mrow><mrow><mrow><mrow><msubsup><mi href="./10.2#E2">J</mi><mn>0</mn><mn>2</mn></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow><mo>+</mo><mrow><mn>2</mn><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./10.1#p2.t1.r2">k</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><msubsup><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r2">k</mi><mn>2</mn></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>=</mo><mn>1</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m14.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./10.1#p2.t1.r2" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./10.1#p2.t1.r2">k</mi></math>: nonnegative integer</a> and
<a href="./10.1#p2.t1.r4" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./10.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">9.1.76</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E4" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.23.4</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E4.png" altimg-height="132px" altimg-valign="-28px" altimg-width="719px" alttext="\sum_{k=0}^{2n}(-1)^{k}J_{k}\left(z\right)J_{2n-k}\left(z\right)\\
+2\sum_{k=1}^{\infty}J_{k}\left(z\right)J_{2n+k}\left(z\right)=0," display="block"><mrow><mrow><mrow><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./10.1#p2.t1.r2">k</mi><mo>=</mo><mn>0</mn></mrow><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r1">n</mi></mrow></munderover><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./10.1#p2.t1.r2">k</mi></msup><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r2">k</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r1">n</mi></mrow><mo>-</mo><mi href="./10.1#p2.t1.r2">k</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>+</mo><mrow><mn>2</mn><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./10.1#p2.t1.r2">k</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r2">k</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mi href="./10.1#p2.t1.r2">k</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></mrow></mrow></mrow><mo>=</mo><mn>0</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m58.png" altimg-height="19px" altimg-valign="-5px" altimg-width="53px" alttext="n\geq 1" display="inline"><mrow><mi href="./10.1#p2.t1.r1">n</mi><mo>≥</mo><mn>1</mn></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E4.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m14.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./10.1#p2.t1.r1" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m57.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./10.1#p2.t1.r1">n</mi></math>: integer</a>,
<a href="./10.1#p2.t1.r2" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./10.1#p2.t1.r2">k</mi></math>: nonnegative integer</a> and
<a href="./10.1#p2.t1.r4" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./10.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">9.1.77</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E5" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.23.5</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E5.png" altimg-height="64px" altimg-valign="-28px" altimg-width="511px" alttext="\sum_{k=0}^{n}J_{k}\left(z\right)J_{n-k}\left(z\right)+2\sum_{k=1}^{\infty}(-1%
)^{k}J_{k}\left(z\right)J_{n+k}\left(z\right)=J_{n}\left(2z\right)." display="block"><mrow><mrow><mrow><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./10.1#p2.t1.r2">k</mi><mo>=</mo><mn>0</mn></mrow><mi href="./10.1#p2.t1.r1">n</mi></munderover><mrow><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r2">k</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mrow><mi href="./10.1#p2.t1.r1">n</mi><mo>-</mo><mi href="./10.1#p2.t1.r2">k</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>+</mo><mrow><mn>2</mn><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./10.1#p2.t1.r2">k</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./10.1#p2.t1.r2">k</mi></msup><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r2">k</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mrow><mi href="./10.1#p2.t1.r1">n</mi><mo>+</mo><mi href="./10.1#p2.t1.r2">k</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></mrow></mrow></mrow><mo>=</mo><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E5.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m14.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./10.1#p2.t1.r1" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m57.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./10.1#p2.t1.r1">n</mi></math>: integer</a>,
<a href="./10.1#p2.t1.r2" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./10.1#p2.t1.r2">k</mi></math>: nonnegative integer</a> and
<a href="./10.1#p2.t1.r4" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./10.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">9.1.78</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px2" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Graf’s and Gegenbauer’s Addition Theorems</h3>
<div id="Px2.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px2.p1" class="ltx_para">
<p class="ltx_p">Define</p>
<table id="E6" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex1" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="3" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">10.23.6</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m6.png" altimg-height="14px" altimg-valign="-2px" altimg-width="21px" alttext="\displaystyle w" display="inline"><mi>w</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m1.png" altimg-height="31px" altimg-valign="-6px" altimg-width="219px" alttext="\displaystyle=\sqrt{u^{2}+v^{2}-2uv\cos\alpha}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><msqrt><mrow><mrow><msup><mi>u</mi><mn>2</mn></msup><mo>+</mo><msup><mi>v</mi><mn>2</mn></msup></mrow><mo>-</mo><mrow><mn>2</mn><mo></mo><mi>u</mi><mo></mo><mi>v</mi><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi>α</mi></mrow></mrow></mrow></msqrt></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex2" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m4.png" altimg-height="19px" altimg-valign="-4px" altimg-width="98px" alttext="\displaystyle u-v\cos\alpha" display="inline"><mrow><mi>u</mi><mo>-</mo><mrow><mi>v</mi><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi>α</mi></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m2.png" altimg-height="18px" altimg-valign="-6px" altimg-width="93px" alttext="\displaystyle=w\cos\chi," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi>w</mi><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi>χ</mi></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex3" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m5.png" altimg-height="19px" altimg-valign="-2px" altimg-width="60px" alttext="\displaystyle v\sin\alpha" display="inline"><mrow><mi>v</mi><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi>α</mi></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m3.png" altimg-height="23px" altimg-valign="-6px" altimg-width="91px" alttext="\displaystyle=w\sin\chi," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi>w</mi><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi>χ</mi></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E6.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m26.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a> and
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m45.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">the branches being continuous and chosen so that <math class="ltx_Math" altimg="m64.png" altimg-height="13px" altimg-valign="-2px" altimg-width="61px" alttext="w\to u" display="inline"><mrow><mi>w</mi><mo>→</mo><mi>u</mi></mrow></math> and <math class="ltx_Math" altimg="m25.png" altimg-height="20px" altimg-valign="-6px" altimg-width="58px" alttext="\chi\to 0" display="inline"><mrow><mi>χ</mi><mo>→</mo><mn>0</mn></mrow></math> as
<math class="ltx_Math" altimg="m62.png" altimg-height="17px" altimg-valign="-2px" altimg-width="56px" alttext="v\to 0" display="inline"><mrow><mi>v</mi><mo>→</mo><mn>0</mn></mrow></math>. If <math class="ltx_Math" altimg="m60.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="u" display="inline"><mi>u</mi></math>, <math class="ltx_Math" altimg="m61.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="v" display="inline"><mi>v</mi></math> are real and positive and <math class="ltx_Math" altimg="m11.png" altimg-height="19px" altimg-valign="-5px" altimg-width="92px" alttext="0\leq\alpha\leq\pi" display="inline"><mrow><mn>0</mn><mo>≤</mo><mi>α</mi><mo>≤</mo><mi href="./3.12#E1">π</mi></mrow></math>, then
<math class="ltx_Math" altimg="m63.png" altimg-height="13px" altimg-valign="-2px" altimg-width="19px" alttext="w" display="inline"><mi>w</mi></math> and <math class="ltx_Math" altimg="m24.png" altimg-height="16px" altimg-valign="-6px" altimg-width="17px" alttext="\chi" display="inline"><mi>χ</mi></math> are real and nonnegative, and the geometrical relationship is
shown in Figure </dd>
</dl>
</div>
</div>
</figure>
<div id="Px2.p2" class="ltx_para">
<table id="E7" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.23.7</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E7.png" altimg-height="66px" altimg-valign="-30px" altimg-width="436px" alttext="\mathscr{C}_{\nu}\left(w\right)\selection{\cos\\
\sin}(\nu\chi)=\sum_{k=-\infty}^{\infty}\mathscr{C}_{\nu+k}\left(u\right)J_{k}%
\left(v\right)\selection{\cos\\
\sin}(k\alpha)," display="block"><mrow><mrow><mrow><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5.p1">𝒞</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi>w</mi><mo>)</mo></mrow></mrow><mo></mo><mtable displaystyle="true" rowspacing="0pt"><mtr><mtd columnalign="center"><mi href="./4.14#E2">cos</mi></mtd></mtr><mtr><mtd columnalign="center"><mi href="./4.14#E1">sin</mi></mtd></mtr></mtable><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo></mo><mi>χ</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./10.1#p2.t1.r2">k</mi><mo>=</mo><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5.p1">𝒞</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mi href="./10.1#p2.t1.r2">k</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r2">k</mi></msub><mo></mo><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow></mrow><mo></mo><mtable displaystyle="true" rowspacing="0pt"><mtr><mtd columnalign="center"><mi href="./4.14#E2">cos</mi></mtd></mtr><mtr><mtd columnalign="center"><mi href="./4.14#E1">sin</mi></mtd></mtr></mtable><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./10.1#p2.t1.r2">k</mi><mo></mo><mi>α</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m71.png" altimg-height="25px" altimg-valign="-7px" altimg-width="114px" alttext="|ve^{\pm i\alpha}|<|u|" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mrow><mi>v</mi><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>±</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi>α</mi></mrow></mrow></msup></mrow><mo stretchy="false">|</mo></mrow><mo><</mo><mrow><mo stretchy="false">|</mo><mi>u</mi><mo stretchy="false">|</mo></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E7.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m14.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m26.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./10.2#Px5.p1" title="Cylinder Functions ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m36.png" altimg-height="23px" altimg-valign="-7px" altimg-width="56px" alttext="\mathscr{C}_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5.p1">𝒞</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: cylinder function</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m34.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m45.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./10.1#p2.t1.r2" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./10.1#p2.t1.r2">k</mi></math>: nonnegative integer</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">9.1.79</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E8" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.23.8</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E8.png" altimg-height="64px" altimg-valign="-28px" altimg-width="518px" alttext="\frac{\mathscr{C}_{\nu}\left(w\right)}{w^{\nu}}=2^{\nu}\Gamma\left(\nu\right)%
\*\sum_{k=0}^{\infty}(\nu+k)\frac{\mathscr{C}_{\nu+k}\left(u\right)}{u^{\nu}}%
\frac{J_{\nu+k}\left(v\right)}{v^{\nu}}C^{(\nu)}_{k}\left(\cos\alpha\right)," display="block"><mrow><mrow><mfrac><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5.p1">𝒞</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi>w</mi><mo>)</mo></mrow></mrow><msup><mi>w</mi><mi href="./10.1#p2.t1.r5">ν</mi></msup></mfrac><mo>=</mo><mrow><msup><mn>2</mn><mi href="./10.1#p2.t1.r5">ν</mi></msup><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r5">ν</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./10.1#p2.t1.r2">k</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mi href="./10.1#p2.t1.r2">k</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mfrac><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5.p1">𝒞</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mi href="./10.1#p2.t1.r2">k</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow></mrow><msup><mi>u</mi><mi href="./10.1#p2.t1.r5">ν</mi></msup></mfrac><mo></mo><mfrac><mrow><msub><mi href="./10.2#E2">J</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mi href="./10.1#p2.t1.r2">k</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow></mrow><msup><mi>v</mi><mi href="./10.1#p2.t1.r5">ν</mi></msup></mfrac><mo></mo><mrow><msubsup><mi href="./18.3#T1.t1.r3">C</mi><mi href="./10.1#p2.t1.r2">k</mi><mrow><mo href="./18.3#T1.t1.r3" stretchy="false">(</mo><mi href="./10.1#p2.t1.r5">ν</mi><mo href="./18.3#T1.t1.r3" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi>α</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m41.png" altimg-height="21px" altimg-valign="-6px" altimg-width="118px" alttext="\nu\neq 0,-1,\dots" display="inline"><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>≠</mo><mrow><mn>0</mn><mo>,</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>, <math class="ltx_Math" altimg="m71.png" altimg-height="25px" altimg-valign="-7px" altimg-width="114px" alttext="|ve^{\pm i\alpha}|<|u|" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mrow><mi>v</mi><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>±</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi>α</mi></mrow></mrow></msup></mrow><mo stretchy="false">|</mo></mrow><mo><</mo><mrow><mo stretchy="false">|</mo><mi>u</mi><mo stretchy="false">|</mo></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E8.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m14.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m21.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m26.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./10.2#Px5.p1" title="Cylinder Functions ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m36.png" altimg-height="23px" altimg-valign="-7px" altimg-width="56px" alttext="\mathscr{C}_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5.p1">𝒞</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: cylinder function</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m34.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./18.3#T1.t1.r3" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m12.png" altimg-height="29px" altimg-valign="-7px" altimg-width="73px" alttext="C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msubsup><mi href="./18.3#T1.t1.r3">C</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r3" stretchy="false">(</mo><mi class="ltx_nvar">λ</mi><mo href="./18.3#T1.t1.r3" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: ultraspherical (or Gegenbauer) polynomial</a>,
<a href="./10.1#p2.t1.r2" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./10.1#p2.t1.r2">k</mi></math>: nonnegative integer</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">9.1.80</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m13.png" altimg-height="30px" altimg-valign="-8px" altimg-width="104px" alttext="C^{(\nu)}_{k}\left(\cos\alpha\right)" display="inline"><mrow><msubsup><mi href="./18.3#T1.t1.r3">C</mi><mi href="./10.1#p2.t1.r2">k</mi><mrow><mo href="./18.3#T1.t1.r3" stretchy="false">(</mo><mi href="./10.1#p2.t1.r5">ν</mi><mo href="./18.3#T1.t1.r3" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi>α</mi></mrow><mo>)</mo></mrow></mrow></math> is Gegenbauer’s polynomial
(§). The restriction <math class="ltx_Math" altimg="m71.png" altimg-height="25px" altimg-valign="-7px" altimg-width="114px" alttext="|ve^{\pm i\alpha}|<|u|" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mrow><mi>v</mi><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>±</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi>α</mi></mrow></mrow></msup></mrow><mo stretchy="false">|</mo></mrow><mo><</mo><mrow><mo stretchy="false">|</mo><mi>u</mi><mo stretchy="false">|</mo></mrow></mrow></math> is
unnecessary in () when <math class="ltx_Math" altimg="m35.png" altimg-height="18px" altimg-valign="-2px" altimg-width="60px" alttext="\mathscr{C}=J" display="inline"><mrow><mi class="ltx_font_mathscript" href="./10.2#Px5.p1">𝒞</mi><mo>=</mo><mi href="./10.2#E2">J</mi></mrow></math> and <math class="ltx_Math" altimg="m38.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>
is an integer, and in () when <math class="ltx_Math" altimg="m35.png" altimg-height="18px" altimg-valign="-2px" altimg-width="60px" alttext="\mathscr{C}=J" display="inline"><mrow><mi class="ltx_font_mathscript" href="./10.2#Px5.p1">𝒞</mi><mo>=</mo><mi href="./10.2#E2">J</mi></mrow></math>.</p>
</div>
<div id="Px2.p3" class="ltx_para">
<p class="ltx_p">The degenerate form of () when <math class="ltx_Math" altimg="m59.png" altimg-height="13px" altimg-valign="-2px" altimg-width="62px" alttext="u=\infty" display="inline"><mrow><mi>u</mi><mo>=</mo><mi mathvariant="normal">∞</mi></mrow></math> is given by</p>
<table id="E9" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.23.9</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E9.png" altimg-height="64px" altimg-valign="-28px" altimg-width="445px" alttext="e^{iv\cos\alpha}=\frac{\Gamma\left(\nu\right)}{(\tfrac{1}{2}v)^{\nu}}\*\sum_{k%
=0}^{\infty}(\nu+k)i^{k}J_{\nu+k}\left(v\right)C^{(\nu)}_{k}\left(\cos\alpha%
\right)," display="block"><mrow><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mo></mo><mi>v</mi><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi>α</mi></mrow></mrow></msup><mo>=</mo><mrow><mfrac><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r5">ν</mi><mo>)</mo></mrow></mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi>v</mi></mrow><mo stretchy="false">)</mo></mrow><mi href="./10.1#p2.t1.r5">ν</mi></msup></mfrac><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./10.1#p2.t1.r2">k</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mi href="./10.1#p2.t1.r2">k</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msup><mi mathvariant="normal">i</mi><mi href="./10.1#p2.t1.r2">k</mi></msup><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mi href="./10.1#p2.t1.r2">k</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><msubsup><mi href="./18.3#T1.t1.r3">C</mi><mi href="./10.1#p2.t1.r2">k</mi><mrow><mo href="./18.3#T1.t1.r3" stretchy="false">(</mo><mi href="./10.1#p2.t1.r5">ν</mi><mo href="./18.3#T1.t1.r3" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi>α</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m41.png" altimg-height="21px" altimg-valign="-6px" altimg-width="118px" alttext="\nu\neq 0,-1,\dots" display="inline"><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>≠</mo><mrow><mn>0</mn><mo>,</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E9.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m14.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m21.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m26.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m34.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./18.3#T1.t1.r3" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m12.png" altimg-height="29px" altimg-valign="-7px" altimg-width="73px" alttext="C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msubsup><mi href="./18.3#T1.t1.r3">C</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r3" stretchy="false">(</mo><mi class="ltx_nvar">λ</mi><mo href="./18.3#T1.t1.r3" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: ultraspherical (or Gegenbauer) polynomial</a>,
<a href="./10.1#p2.t1.r2" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./10.1#p2.t1.r2">k</mi></math>: nonnegative integer</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">9.1.81</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="Px2.p4" class="ltx_para">
<p class="ltx_p">See also §</dd>
</dl>
</div>
</div>
<div id="Px4.p1" class="ltx_para">
<table id="E10" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.23.10</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E10.png" altimg-height="64px" altimg-valign="-28px" altimg-width="288px" alttext="f(z)=a_{0}J_{0}\left(z\right)+2\sum_{k=1}^{\infty}a_{k}J_{k}\left(z\right)," display="block"><mrow><mrow><mrow><mi href="./10.23#Px6.p1">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mrow><msub><mi href="./10.23#E11">a</mi><mn>0</mn></msub><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mn>0</mn></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mn>2</mn><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./10.1#p2.t1.r2">k</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><msub><mi href="./10.23#E11">a</mi><mi href="./10.1#p2.t1.r2">k</mi></msub><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r2">k</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m73.png" altimg-height="23px" altimg-valign="-7px" altimg-width="61px" alttext="|z|<c" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mi href="./10.1#p2.t1.r4">z</mi><mo stretchy="false">|</mo></mrow><mo><</mo><mi>c</mi></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E10.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m14.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./10.1#p2.t1.r2" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./10.1#p2.t1.r2">k</mi></math>: nonnegative integer</a>,
<a href="./10.1#p2.t1.r4" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./10.1#p2.t1.r4">z</mi></math>: complex variable</a>,
<a href="./10.23#E11" title="(10.23.11) ‣ Neumann’s Expansion ‣ §10.23(iii) Series Expansions of Arbitrary Functions ‣ §10.23 Sums ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m46.png" altimg-height="16px" altimg-valign="-5px" altimg-width="25px" alttext="a_{k}" display="inline"><msub><mi href="./10.23#E11">a</mi><mi href="./10.1#p2.t1.r2">k</mi></msub></math></a> and
<a href="./10.23#Px6.p1" title="Fourier–Bessel Expansion ‣ §10.23(iii) Series Expansions of Arbitrary Functions ‣ §10.23 Sums ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="23px" altimg-valign="-7px" altimg-width="39px" alttext="f(t)" display="inline"><mrow><mi href="./10.23#Px6.p1">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math></a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">9.1.82</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m47.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="c" display="inline"><mi>c</mi></math> is the distance of the nearest singularity of the analytic function
<math class="ltx_Math" altimg="m49.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(z)" display="inline"><mrow><mi href="./10.23#Px6.p1">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> from <math class="ltx_Math" altimg="m68.png" altimg-height="17px" altimg-valign="-2px" altimg-width="51px" alttext="z=0" display="inline"><mrow><mi href="./10.1#p2.t1.r4">z</mi><mo>=</mo><mn>0</mn></mrow></math>,</p>
<table id="E11" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.23.11</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E11.png" altimg-height="54px" altimg-valign="-24px" altimg-width="260px" alttext="a_{k}=\frac{1}{2\pi i}\int_{|z|=c^{\prime}}f(t)O_{k}\left(t\right)\mathrm{d}t," display="block"><mrow><mrow><msub><mi href="./10.23#E11">a</mi><mi href="./10.1#p2.t1.r2">k</mi></msub><mo>=</mo><mrow><mfrac><mn>1</mn><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi></mrow></mfrac><mo></mo><mrow><msub><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><mrow><mo stretchy="false">|</mo><mi href="./10.1#p2.t1.r4">z</mi><mo stretchy="false">|</mo></mrow><mo>=</mo><msup><mi>c</mi><mo>′</mo></msup></mrow></msub><mrow><mrow><mi href="./10.23#Px6.p1">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./10.23#E12">O</mi><mi href="./10.1#p2.t1.r2">k</mi></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m9.png" altimg-height="19px" altimg-valign="-3px" altimg-width="90px" alttext="0<c^{\prime}<c" display="inline"><mrow><mn>0</mn><mo><</mo><msup><mi>c</mi><mo>′</mo></msup><mo><</mo><mi>c</mi></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E11.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text"><math class="ltx_Math" altimg="m46.png" altimg-height="16px" altimg-valign="-5px" altimg-width="25px" alttext="a_{k}" display="inline"><msub><mi href="./10.23#E11">a</mi><mi href="./10.1#p2.t1.r2">k</mi></msub></math> (locally)</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./10.23#E12" title="(10.23.12) ‣ Neumann’s Expansion ‣ §10.23(iii) Series Expansions of Arbitrary Functions ‣ §10.23 Sums ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m16.png" altimg-height="23px" altimg-valign="-7px" altimg-width="61px" alttext="O_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.23#E12">O</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: Neumann’s polynomial</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m42.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m33.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m67.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./10.1#p2.t1.r2" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./10.1#p2.t1.r2">k</mi></math>: nonnegative integer</a>,
<a href="./10.1#p2.t1.r4" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./10.1#p2.t1.r4">z</mi></math>: complex variable</a> and
<a href="./10.23#Px6.p1" title="Fourier–Bessel Expansion ‣ §10.23(iii) Series Expansions of Arbitrary Functions ‣ §10.23 Sums ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="23px" altimg-valign="-7px" altimg-width="39px" alttext="f(t)" display="inline"><mrow><mi href="./10.23#Px6.p1">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math></a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">9.1.83</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and <math class="ltx_Math" altimg="m17.png" altimg-height="23px" altimg-valign="-7px" altimg-width="55px" alttext="O_{k}\left(t\right)" display="inline"><mrow><msub><mi href="./10.23#E12">O</mi><mi href="./10.1#p2.t1.r2">k</mi></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></math> is <em class="ltx_emph ltx_font_italic">Neumann’s polynomial</em>, defined by the generating
function:</p>
<table id="E12" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.23.12</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E12.png" altimg-height="64px" altimg-valign="-28px" altimg-width="364px" alttext="\frac{1}{t-z}=J_{0}\left(z\right)O_{0}\left(t\right)+2\sum_{k=1}^{\infty}J_{k}%
\left(z\right)O_{k}\left(t\right)," display="block"><mrow><mrow><mfrac><mn>1</mn><mrow><mi>t</mi><mo>-</mo><mi href="./10.1#p2.t1.r4">z</mi></mrow></mfrac><mo>=</mo><mrow><mrow><mrow><msub><mi href="./10.2#E2">J</mi><mn>0</mn></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./10.23#E12">O</mi><mn>0</mn></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mn>2</mn><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./10.1#p2.t1.r2">k</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r2">k</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./10.23#E12">O</mi><mi href="./10.1#p2.t1.r2">k</mi></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m74.png" altimg-height="23px" altimg-valign="-7px" altimg-width="70px" alttext="|z|<|t|" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mi href="./10.1#p2.t1.r4">z</mi><mo stretchy="false">|</mo></mrow><mo><</mo><mrow><mo stretchy="false">|</mo><mi>t</mi><mo stretchy="false">|</mo></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E12.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m16.png" altimg-height="23px" altimg-valign="-7px" altimg-width="61px" alttext="O_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.23#E12">O</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: Neumann’s polynomial</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m14.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./10.1#p2.t1.r1" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m57.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./10.1#p2.t1.r1">n</mi></math>: integer</a>,
<a href="./10.1#p2.t1.r2" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./10.1#p2.t1.r2">k</mi></math>: nonnegative integer</a>,
<a href="./10.1#p2.t1.r3" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m67.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./10.1#p2.t1.r3">x</mi></math>: real variable</a> and
<a href="./10.1#p2.t1.r4" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./10.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">9.1.84</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="Px4.p2" class="ltx_para">
<p class="ltx_p"><math class="ltx_Math" altimg="m18.png" altimg-height="23px" altimg-valign="-7px" altimg-width="56px" alttext="O_{n}\left(t\right)" display="inline"><mrow><msub><mi href="./10.23#E12">O</mi><mi href="./10.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></math> is a polynomial of degree <math class="ltx_Math" altimg="m55.png" altimg-height="18px" altimg-valign="-4px" altimg-width="51px" alttext="n+1" display="inline"><mrow><mi href="./10.1#p2.t1.r1">n</mi><mo>+</mo><mn>1</mn></mrow></math> in <math class="ltx_Math" altimg="m28.png" altimg-height="23px" altimg-valign="-7px" altimg-width="157px" alttext="\ifrac{1}{t}:O_{0}\left(t\right)=1/t" display="inline"><mrow><mrow><mn>1</mn><mo>/</mo><mi>t</mi></mrow><mo>:</mo><mrow><mrow><msub><mi href="./10.23#E12">O</mi><mn>0</mn></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mn>1</mn><mo>/</mo><mi>t</mi></mrow></mrow></mrow></math> and</p>
<table id="E13" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.23.13</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E13.png" altimg-height="71px" altimg-valign="-28px" altimg-width="376px" alttext="O_{n}\left(t\right)=\frac{1}{4}\sum_{k=0}^{\left\lfloor n/2\right\rfloor}\frac%
{(n-k-1)!n}{k!}\left(\frac{2}{t}\right)^{n-2k+1}," display="block"><mrow><mrow><mrow><msub><mi href="./10.23#E12">O</mi><mi href="./10.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./10.1#p2.t1.r2">k</mi><mo>=</mo><mn>0</mn></mrow><mrow><mo href="./front/introduction#Sx4.p1.t1.r16">⌊</mo><mrow><mi href="./10.1#p2.t1.r1">n</mi><mo>/</mo><mn>2</mn></mrow><mo href="./front/introduction#Sx4.p1.t1.r16">⌋</mo></mrow></munderover><mrow><mfrac><mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./10.1#p2.t1.r1">n</mi><mo>-</mo><mi href="./10.1#p2.t1.r2">k</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow><mo></mo><mi href="./10.1#p2.t1.r1">n</mi></mrow><mrow><mi href="./10.1#p2.t1.r2">k</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mfrac><mo></mo><msup><mrow><mo>(</mo><mfrac><mn>2</mn><mi>t</mi></mfrac><mo>)</mo></mrow><mrow><mrow><mi href="./10.1#p2.t1.r1">n</mi><mo>-</mo><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r2">k</mi></mrow></mrow><mo>+</mo><mn>1</mn></mrow></msup></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m56.png" altimg-height="20px" altimg-valign="-6px" altimg-width="104px" alttext="n=1,2,\ldots" display="inline"><mrow><mi href="./10.1#p2.t1.r1">n</mi><mo>=</mo><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E13.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.23#E12" title="(10.23.12) ‣ Neumann’s Expansion ‣ §10.23(iii) Series Expansions of Arbitrary Functions ‣ §10.23 Sums ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m16.png" altimg-height="23px" altimg-valign="-7px" altimg-width="61px" alttext="O_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.23#E12">O</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: Neumann’s polynomial</a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m7.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m54.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./front/introduction#Sx4.p1.t1.r16" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m31.png" altimg-height="23px" altimg-valign="-7px" altimg-width="33px" alttext="\left\lfloor\NVar{x}\right\rfloor" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r16">⌊</mo><mi class="ltx_nvar">x</mi><mo href="./front/introduction#Sx4.p1.t1.r16">⌋</mo></mrow></math>: floor of <math class="ltx_Math" altimg="m67.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./10.1#p2.t1.r1" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m57.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./10.1#p2.t1.r1">n</mi></math>: integer</a> and
<a href="./10.1#p2.t1.r2" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./10.1#p2.t1.r2">k</mi></math>: nonnegative integer</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">9.1.85</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="Px4.p3" class="ltx_para">
<p class="ltx_p">For the more general form of expansion</p>
<table id="E14" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.23.14</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E14.png" altimg-height="64px" altimg-valign="-28px" altimg-width="321px" alttext="z^{\nu}f(z)=a_{0}J_{\nu}\left(z\right)+2\sum_{k=1}^{\infty}a_{k}J_{\nu+k}\left%
(z\right)" display="block"><mrow><mrow><msup><mi href="./10.1#p2.t1.r4">z</mi><mi href="./10.1#p2.t1.r5">ν</mi></msup><mo></mo><mrow><mi href="./10.23#Px6.p1">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>=</mo><mrow><mrow><msub><mi href="./10.23#E11">a</mi><mn>0</mn></msub><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mn>2</mn><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./10.1#p2.t1.r2">k</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><msub><mi href="./10.23#E11">a</mi><mi href="./10.1#p2.t1.r2">k</mi></msub><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mi href="./10.1#p2.t1.r2">k</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></mrow></mrow></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E14.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m14.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./10.1#p2.t1.r2" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./10.1#p2.t1.r2">k</mi></math>: nonnegative integer</a>,
<a href="./10.1#p2.t1.r4" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./10.1#p2.t1.r4">z</mi></math>: complex variable</a>,
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>,
<a href="./10.23#E11" title="(10.23.11) ‣ Neumann’s Expansion ‣ §10.23(iii) Series Expansions of Arbitrary Functions ‣ §10.23 Sums ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m46.png" altimg-height="16px" altimg-valign="-5px" altimg-width="25px" alttext="a_{k}" display="inline"><msub><mi href="./10.23#E11">a</mi><mi href="./10.1#p2.t1.r2">k</mi></msub></math></a> and
<a href="./10.23#Px6.p1" title="Fourier–Bessel Expansion ‣ §10.23(iii) Series Expansions of Arbitrary Functions ‣ §10.23 Sums ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="23px" altimg-valign="-7px" altimg-width="39px" alttext="f(t)" display="inline"><mrow><mi href="./10.23#Px6.p1">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math></a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">9.1.86</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">see <cite class="ltx_cite ltx_citemacro_citet">Watson (</dd>
</dl>
</div>
</div>
<div id="Px5.p1" class="ltx_para">
<table id="E15" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.23.15</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E15.png" altimg-height="64px" altimg-valign="-28px" altimg-width="355px" alttext="(\tfrac{1}{2}z)^{\nu}=\sum_{k=0}^{\infty}\frac{(\nu+2k)\Gamma\left(\nu+k\right%
)}{k!}J_{\nu+2k}\left(z\right)," display="block"><mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo stretchy="false">)</mo></mrow><mi href="./10.1#p2.t1.r5">ν</mi></msup><mo>=</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./10.1#p2.t1.r2">k</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><mfrac><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r2">k</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mi href="./10.1#p2.t1.r2">k</mi></mrow><mo>)</mo></mrow></mrow></mrow><mrow><mi href="./10.1#p2.t1.r2">k</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mfrac><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r2">k</mi></mrow></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m40.png" altimg-height="21px" altimg-valign="-6px" altimg-width="153px" alttext="\nu\neq 0,-1,-2,\dots" display="inline"><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>≠</mo><mrow><mn>0</mn><mo>,</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo>,</mo><mrow><mo>-</mo><mn>2</mn></mrow><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E15.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m14.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m21.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m7.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m54.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./10.1#p2.t1.r2" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./10.1#p2.t1.r2">k</mi></math>: nonnegative integer</a>,
<a href="./10.1#p2.t1.r4" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./10.1#p2.t1.r4">z</mi></math>: complex variable</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">9.1.87</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E16" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.23.16</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E16.png" altimg-height="64px" altimg-valign="-28px" altimg-width="462px" alttext="Y_{0}\left(z\right)=\frac{2}{\pi}\left(\ln\left(\tfrac{1}{2}z\right)+\gamma%
\right)J_{0}\left(z\right)-\frac{4}{\pi}\sum_{k=1}^{\infty}(-1)^{k}\frac{J_{2k%
}\left(z\right)}{k}," display="block"><mrow><mrow><mrow><msub><mi href="./10.2#E3">Y</mi><mn>0</mn></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mfrac><mn>2</mn><mi href="./3.12#E1">π</mi></mfrac><mo></mo><mrow><mo>(</mo><mrow><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><mo>(</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow><mo>+</mo><mi href="./5.2#E3">γ</mi></mrow><mo>)</mo></mrow><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mn>0</mn></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow><mo>-</mo><mrow><mfrac><mn>4</mn><mi href="./3.12#E1">π</mi></mfrac><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./10.1#p2.t1.r2">k</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./10.1#p2.t1.r2">k</mi></msup><mo></mo><mfrac><mrow><msub><mi href="./10.2#E2">J</mi><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r2">k</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow><mi href="./10.1#p2.t1.r2">k</mi></mfrac></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E16.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m14.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./10.2#E3" title="(10.2.3) ‣ Bessel Function of the Second Kind (Weber’s Function) ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="23px" altimg-valign="-7px" altimg-width="55px" alttext="Y_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E3">Y</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the second kind</a>,
<a href="./5.2#E3" title="(5.2.3) ‣ §5.2(ii) Euler’s Constant ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m27.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\gamma" display="inline"><mi href="./5.2#E3">γ</mi></math>: Euler’s constant</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m42.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m32.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a>,
<a href="./10.1#p2.t1.r2" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./10.1#p2.t1.r2">k</mi></math>: nonnegative integer</a> and
<a href="./10.1#p2.t1.r4" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./10.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">9.1.89</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E17" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.23.17</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_Math ltx_eqn_cell ltx_align_center"><math class="ltx_Math" alttext="Y_{n}\left(z\right)=-\frac{n!(\tfrac{1}{2}z)^{-n}}{\pi}\sum_{k=0}^{n-1}\frac{(%
\tfrac{1}{2}z)^{k}J_{k}\left(z\right)}{k!(n-k)}+\frac{2}{\pi}\left(\ln\left(%
\tfrac{1}{2}z\right)-\psi\left(n+1\right)\right)J_{n}\left(z\right)-\frac{2}{%
\pi}\sum_{k=1}^{\infty}(-1)^{k}\frac{(n+2k)J_{n+2k}\left(z\right)}{k(n+k)}," display="block"><mrow><mrow><msub><mi href="./10.2#E3">Y</mi><mi href="./10.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mrow><mrow><mo>-</mo><mrow><mfrac><mrow><mrow><mi href="./10.1#p2.t1.r1">n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mi href="./10.1#p2.t1.r1">n</mi></mrow></msup></mrow><mi href="./3.12#E1">π</mi></mfrac><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./10.1#p2.t1.r2">k</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi href="./10.1#p2.t1.r1">n</mi><mo>-</mo><mn>1</mn></mrow></munderover><mfrac><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo stretchy="false">)</mo></mrow><mi href="./10.1#p2.t1.r2">k</mi></msup><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r2">k</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow><mrow><mrow><mi href="./10.1#p2.t1.r2">k</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./10.1#p2.t1.r1">n</mi><mo>-</mo><mi href="./10.1#p2.t1.r2">k</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mfrac></mrow></mrow></mrow><mo>+</mo><mrow><mfrac><mn>2</mn><mi href="./3.12#E1">π</mi></mfrac><mo></mo><mrow><mo>(</mo><mrow><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><mo>(</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow><mo>-</mo><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./10.1#p2.t1.r1">n</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mrow><mo>)</mo></mrow><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>-</mo><mrow><mfrac><mn>2</mn><mi href="./3.12#E1">π</mi></mfrac><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./10.1#p2.t1.r2">k</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./10.1#p2.t1.r2">k</mi></msup><mo></mo><mfrac><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./10.1#p2.t1.r1">n</mi><mo>+</mo><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r2">k</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mrow><mi href="./10.1#p2.t1.r1">n</mi><mo>+</mo><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r2">k</mi></mrow></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow><mrow><mi href="./10.1#p2.t1.r2">k</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./10.1#p2.t1.r1">n</mi><mo>+</mo><mi href="./10.1#p2.t1.r2">k</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mfrac></mrow></mrow></mrow><mo>,</mo></mtd></mtr></mtable></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E17.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m14.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./10.2#E3" title="(10.2.3) ‣ Bessel Function of the Second Kind (Weber’s Function) ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="23px" altimg-valign="-7px" altimg-width="55px" alttext="Y_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E3">Y</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the second kind</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m42.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./5.2#E2" title="(5.2.2) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="\psi\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: psi (or digamma) function</a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m7.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m54.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m32.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a>,
<a href="./10.1#p2.t1.r1" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m57.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./10.1#p2.t1.r1">n</mi></math>: integer</a>,
<a href="./10.1#p2.t1.r2" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./10.1#p2.t1.r2">k</mi></math>: nonnegative integer</a> and
<a href="./10.1#p2.t1.r4" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./10.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">9.1.88</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m27.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\gamma" display="inline"><mi href="./5.2#E3">γ</mi></math> is Euler’s constant and
<math class="ltx_Math" altimg="m44.png" altimg-height="24px" altimg-valign="-7px" altimg-width="284px" alttext="\psi\left(n+1\right)=\Gamma'\left(n+1\right)/\Gamma\left(n+1\right)" display="inline"><mrow><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./10.1#p2.t1.r1">n</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mrow><msup><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo>′</mo></msup><mo></mo><mrow><mo>(</mo><mrow><mi href="./10.1#p2.t1.r1">n</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mo>/</mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./10.1#p2.t1.r1">n</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mrow></mrow></math> (§</dd>
</dl>
</div>
</div>
<div id="Px6.p1" class="ltx_para">
<p class="ltx_p">Assume <math class="ltx_Math" altimg="m48.png" altimg-height="23px" altimg-valign="-7px" altimg-width="39px" alttext="f(t)" display="inline"><mrow><mi href="./10.23#Px6.p1">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math> satisfies
</p>
<table id="E18" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.23.18</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E18.png" altimg-height="56px" altimg-valign="-20px" altimg-width="174px" alttext="\int_{0}^{1}t^{\frac{1}{2}}|f(t)|\mathrm{d}t<\infty," display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mn>1</mn></msubsup><mrow><msup><mi>t</mi><mfrac><mn>1</mn><mn>2</mn></mfrac></msup><mo></mo><mrow><mo stretchy="false">|</mo><mrow><mi href="./10.23#Px6.p1">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow><mo><</mo><mi mathvariant="normal">∞</mi></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E18.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m33.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m67.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a> and
<a href="./10.23#Px6.p1" title="Fourier–Bessel Expansion ‣ §10.23(iii) Series Expansions of Arbitrary Functions ‣ §10.23 Sums ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="23px" altimg-valign="-7px" altimg-width="39px" alttext="f(t)" display="inline"><mrow><mi href="./10.23#Px6.p1">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and define</p>
<table id="E19" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.23.19</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E19.png" altimg-height="56px" altimg-valign="-21px" altimg-width="368px" alttext="a_{m}=\frac{2}{(J_{\nu+1}(j_{\nu,m}))^{2}}\int_{0}^{1}tf(t)J_{\nu}\left(j_{\nu%
,m}t\right)\mathrm{d}t," display="block"><mrow><mrow><msub><mi href="./10.23#E11">a</mi><mi href="./10.1#p2.t1.r1">m</mi></msub><mo>=</mo><mrow><mfrac><mn>2</mn><msup><mrow><mo stretchy="false">(</mo><mrow><msub><mi href="./10.2#E2">J</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo stretchy="false">(</mo><msub><mi href="./10.21#SS1.p2">j</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi href="./10.1#p2.t1.r1">m</mi></mrow></msub><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mfrac><mo></mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mn>1</mn></msubsup><mrow><mi>t</mi><mo></mo><mrow><mi href="./10.23#Px6.p1">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mrow><msub><mi>j</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>,</mo><mi href="./10.1#p2.t1.r1">m</mi></mrow></msub><mo></mo><mi>t</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m39.png" altimg-height="27px" altimg-valign="-9px" altimg-width="70px" alttext="\nu\geq-\tfrac{1}{2}" display="inline"><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>≥</mo><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E19.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m14.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m33.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m67.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./10.21#SS1.p2" title="§10.21(i) Distribution ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="22px" altimg-valign="-8px" altimg-width="41px" alttext="j_{\NVar{\nu},\NVar{m}}" display="inline"><msub><mi href="./10.21#SS1.p2">j</mi><mrow><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r1">m</mi></mrow></msub></math>: zeros of the Bessel function <math class="ltx_Math" altimg="m15.png" altimg-height="23px" altimg-valign="-7px" altimg-width="55px" alttext="J_{\nu}\left(x\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math></a>,
<a href="./10.1#p2.t1.r1" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./10.1#p2.t1.r1">m</mi></math>: integer</a>,
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>,
<a href="./10.23#E11" title="(10.23.11) ‣ Neumann’s Expansion ‣ §10.23(iii) Series Expansions of Arbitrary Functions ‣ §10.23 Sums ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m46.png" altimg-height="16px" altimg-valign="-5px" altimg-width="25px" alttext="a_{k}" display="inline"><msub><mi href="./10.23#E11">a</mi><mi href="./10.1#p2.t1.r2">k</mi></msub></math></a> and
<a href="./10.23#Px6.p1" title="Fourier–Bessel Expansion ‣ §10.23(iii) Series Expansions of Arbitrary Functions ‣ §10.23 Sums ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="23px" altimg-valign="-7px" altimg-width="39px" alttext="f(t)" display="inline"><mrow><mi href="./10.23#Px6.p1">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m51.png" altimg-height="22px" altimg-valign="-8px" altimg-width="40px" alttext="j_{\nu,m}" display="inline"><msub><mi href="./10.21#SS1.p2">j</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi href="./10.1#p2.t1.r1">m</mi></mrow></msub></math> is as in §. If <math class="ltx_Math" altimg="m10.png" altimg-height="17px" altimg-valign="-3px" altimg-width="89px" alttext="0<x<1" display="inline"><mrow><mn>0</mn><mo><</mo><mi href="./10.1#p2.t1.r3">x</mi><mo><</mo><mn>1</mn></mrow></math>, then</p>
<table id="E20" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.23.20</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E20.png" altimg-height="64px" altimg-valign="-27px" altimg-width="350px" alttext="\tfrac{1}{2}f(x-)+\tfrac{1}{2}f(x+)=\sum_{m=1}^{\infty}a_{m}J_{\nu}\left(j_{%
\nu,m}x\right)," display="block"><mrow><mrow><mrow><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mrow><mi href="./10.23#Px6.p1">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./10.1#p2.t1.r3">x</mi><mo>-</mo></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mrow><mi href="./10.23#Px6.p1">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./10.1#p2.t1.r3">x</mi><mo>+</mo></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow><mo>=</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./10.1#p2.t1.r1">m</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><msub><mi href="./10.23#E11">a</mi><mi href="./10.1#p2.t1.r1">m</mi></msub><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mrow><msub><mi>j</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>,</mo><mi href="./10.1#p2.t1.r1">m</mi></mrow></msub><mo></mo><mi href="./10.1#p2.t1.r3">x</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E20.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m14.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./10.1#p2.t1.r1" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./10.1#p2.t1.r1">m</mi></math>: integer</a>,
<a href="./10.1#p2.t1.r3" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m67.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./10.1#p2.t1.r3">x</mi></math>: real variable</a>,
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>,
<a href="./10.23#E11" title="(10.23.11) ‣ Neumann’s Expansion ‣ §10.23(iii) Series Expansions of Arbitrary Functions ‣ §10.23 Sums ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m46.png" altimg-height="16px" altimg-valign="-5px" altimg-width="25px" alttext="a_{k}" display="inline"><msub><mi href="./10.23#E11">a</mi><mi href="./10.1#p2.t1.r2">k</mi></msub></math></a> and
<a href="./10.23#Px6.p1" title="Fourier–Bessel Expansion ‣ §10.23(iii) Series Expansions of Arbitrary Functions ‣ §10.23 Sums ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="23px" altimg-valign="-7px" altimg-width="39px" alttext="f(t)" display="inline"><mrow><mi href="./10.23#Px6.p1">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">provided that <math class="ltx_Math" altimg="m48.png" altimg-height="23px" altimg-valign="-7px" altimg-width="39px" alttext="f(t)" display="inline"><mrow><mi href="./10.23#Px6.p1">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math> is of bounded variation (§) on an
interval <math class="ltx_Math" altimg="m20.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="[a,b]" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">[</mo><mi>a</mi><mo href="./front/introduction#Sx4.p1.t1.r29">,</mo><mi>b</mi><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">]</mo></mrow></math> with <math class="ltx_Math" altimg="m8.png" altimg-height="18px" altimg-valign="-3px" altimg-width="161px" alttext="0<a<x<b<1" display="inline"><mrow><mn>0</mn><mo><</mo><mi>a</mi><mo><</mo><mi href="./10.1#p2.t1.r3">x</mi><mo><</mo><mi>b</mi><mo><</mo><mn>1</mn></mrow></math>. This result is proved in
<cite class="ltx_cite ltx_citemacro_citet">Watson (, Chapter 18)</cite> and further information is provided in
this reference, including the behavior of the series near <math class="ltx_Math" altimg="m65.png" altimg-height="17px" altimg-valign="-2px" altimg-width="52px" alttext="x=0" display="inline"><mrow><mi href="./10.1#p2.t1.r3">x</mi><mo>=</mo><mn>0</mn></mrow></math> and <math class="ltx_Math" altimg="m66.png" altimg-height="17px" altimg-valign="-2px" altimg-width="52px" alttext="x=1" display="inline"><mrow><mi href="./10.1#p2.t1.r3">x</mi><mo>=</mo><mn>1</mn></mrow></math>.</p>
</div>
<div id="Px6.p2" class="ltx_para">
<p class="ltx_p">As an example,</p>
<table id="E21" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.23.21</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E21.png" altimg-height="64px" altimg-valign="-27px" altimg-width="234px" alttext="x^{\nu}=\sum_{m=1}^{\infty}\frac{2\!J_{\nu}\left(j_{\nu,m}x\right)}{j_{\nu,m}J%
_{\nu+1}\left(j_{\nu,m}\right)}," display="block"><mrow><mrow><msup><mi href="./10.1#p2.t1.r3">x</mi><mi href="./10.1#p2.t1.r5">ν</mi></msup><mo>=</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./10.1#p2.t1.r1">m</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mfrac><mrow><mpadded width="-1.7pt"><mn>2</mn></mpadded><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mrow><msub><mi>j</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>,</mo><mi href="./10.1#p2.t1.r1">m</mi></mrow></msub><mo></mo><mi href="./10.1#p2.t1.r3">x</mi></mrow><mo>)</mo></mrow></mrow></mrow><mrow><msub><mi href="./10.21#SS1.p2">j</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi href="./10.1#p2.t1.r1">m</mi></mrow></msub><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><msub><mi>j</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>,</mo><mi href="./10.1#p2.t1.r1">m</mi></mrow></msub><mo>)</mo></mrow></mrow></mrow></mfrac></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m37.png" altimg-height="20px" altimg-valign="-6px" altimg-width="145px" alttext="\nu>0,0\leq x<1" display="inline"><mrow><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>></mo><mn>0</mn></mrow><mo>,</mo><mrow><mn>0</mn><mo>≤</mo><mi href="./10.1#p2.t1.r3">x</mi><mo><</mo><mn>1</mn></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E21.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m14.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./10.21#SS1.p2" title="§10.21(i) Distribution ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="22px" altimg-valign="-8px" altimg-width="41px" alttext="j_{\NVar{\nu},\NVar{m}}" display="inline"><msub><mi href="./10.21#SS1.p2">j</mi><mrow><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi><mo href="./10.21#SS1.p2">,</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r1">m</mi></mrow></msub></math>: zeros of the Bessel function <math class="ltx_Math" altimg="m15.png" altimg-height="23px" altimg-valign="-7px" altimg-width="55px" alttext="J_{\nu}\left(x\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math></a>,
<a href="./10.1#p2.t1.r1" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./10.1#p2.t1.r1">m</mi></math>: integer</a>,
<a href="./10.1#p2.t1.r3" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m67.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./10.1#p2.t1.r3">x</mi></math>: real variable</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">(Note that when <math class="ltx_Math" altimg="m66.png" altimg-height="17px" altimg-valign="-2px" altimg-width="52px" alttext="x=1" display="inline"><mrow><mi href="./10.1#p2.t1.r3">x</mi><mo>=</mo><mn>1</mn></mrow></math> the left-hand side is 1 and the right-hand side is 0.)</p>
</div>
</section>
<section id="Px7" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Other Series Expansions</h3>
<div id="Px7.info" class="ltx_metadata ltx_info">
</div>
</div>
</body></text>
</html>
</page>
<page>
<!DOCTYPE html><html>
<head>
<title>DLMF: 10.22 Integrals</title>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<!--GOOGLE BOOTSTRAP--></head>
<text><body class="color_default textfont_default titlefont_default fontsize_default navbar_default">
<div class="ltx_page_navbar">
<div class="ltx_page_navlogo"></li>
<li class="ltx_tocentry"><a href="#iii"><span class="ltx_tag ltx_tag_subsection">§10.22(iii) </span>Integrals over the Interval <math class="ltx_Math" altimg="m89.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="(x,\infty)" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi mathvariant="normal">∞</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math></a></li>
<li class="ltx_tocentry"><a href="#iv"><span class="ltx_tag ltx_tag_subsection">§10.22(iv) </span>Integrals over the Interval <math class="ltx_Math" altimg="m88.png" altimg-height="23px" altimg-valign="-7px" altimg-width="59px" alttext="(0,\infty)" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mn>0</mn><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi mathvariant="normal">∞</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math></a></li>
<li class="ltx_tocentry"></dd>
</dl>
</div>
</div>
<div id="SS1.p1" class="ltx_para">
<p class="ltx_p">In this subsection <math class="ltx_Math" altimg="m180.png" altimg-height="23px" altimg-valign="-7px" altimg-width="56px" alttext="\mathscr{C}_{\nu}\left(z\right)" display="inline"><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5.p1">𝒞</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math> and <math class="ltx_Math" altimg="m182.png" altimg-height="24px" altimg-valign="-8px" altimg-width="56px" alttext="\mathscr{D}_{\mu}(z)" display="inline"><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5">𝒟</mi><mi>μ</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> denote
cylinder functions(§) of orders <math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math> and <math class="ltx_Math" altimg="m196.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\mu" display="inline"><mi>μ</mi></math>,
respectively, not necessarily distinct.</p>
<table id="E1" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex1" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">10.22.1</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m52.png" altimg-height="51px" altimg-valign="-19px" altimg-width="146px" alttext="\displaystyle\int z^{\nu+1}\mathscr{C}_{\nu}\left(z\right)\mathrm{d}z" display="inline"><mstyle displaystyle="true"><mrow><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><msup><mi href="./10.1#p2.t1.r4">z</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mn>1</mn></mrow></msup><mo></mo><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5.p1">𝒞</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./10.1#p2.t1.r4">z</mi></mrow></mrow></mrow></mstyle></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m44.png" altimg-height="28px" altimg-valign="-7px" altimg-width="148px" alttext="\displaystyle=z^{\nu+1}\mathscr{C}_{\nu+1}\left(z\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msup><mi href="./10.1#p2.t1.r4">z</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mn>1</mn></mrow></msup><mo></mo><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5.p1">𝒞</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex2" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m50.png" altimg-height="51px" altimg-valign="-19px" altimg-width="159px" alttext="\displaystyle\int z^{-\nu+1}\mathscr{C}_{\nu}\left(z\right)\mathrm{d}z" display="inline"><mstyle displaystyle="true"><mrow><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><msup><mi href="./10.1#p2.t1.r4">z</mi><mrow><mrow><mo>-</mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo>+</mo><mn>1</mn></mrow></msup><mo></mo><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5.p1">𝒞</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./10.1#p2.t1.r4">z</mi></mrow></mrow></mrow></mstyle></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m4.png" altimg-height="28px" altimg-valign="-7px" altimg-width="177px" alttext="\displaystyle=-z^{-\nu+1}\mathscr{C}_{\nu-1}\left(z\right)." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo>-</mo><mrow><msup><mi href="./10.1#p2.t1.r4">z</mi><mrow><mrow><mo>-</mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo>+</mo><mn>1</mn></mrow></msup><mo></mo><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5.p1">𝒞</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>-</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#Px5.p1" title="Cylinder Functions ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m178.png" altimg-height="23px" altimg-valign="-7px" altimg-width="56px" alttext="\mathscr{C}_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5.p1">𝒞</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: cylinder function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m163.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./10.1#p2.t1.r4" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m258.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./10.1#p2.t1.r4">z</mi></math>: complex variable</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">11.3.20, 11.3.21</span> (modified)</span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E2" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.22.2</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E2.png" altimg-height="51px" altimg-valign="-19px" altimg-width="634px" alttext="\int z^{\nu}\mathscr{C}_{\nu}\left(z\right)\mathrm{d}z=\pi^{\frac{1}{2}}2^{\nu%
-1}\Gamma\left(\nu+\tfrac{1}{2}\right)\*z\left(\mathscr{C}_{\nu}\left(z\right)%
\mathbf{H}_{\nu-1}\left(z\right)-\mathscr{C}_{\nu-1}\left(z\right)\mathbf{H}_{%
\nu}\left(z\right)\right)," display="block"><mrow><mrow><mrow><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><msup><mi href="./10.1#p2.t1.r4">z</mi><mi href="./10.1#p2.t1.r5">ν</mi></msup><mo></mo><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5.p1">𝒞</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./10.1#p2.t1.r4">z</mi></mrow></mrow></mrow><mo>=</mo><mrow><msup><mi href="./3.12#E1">π</mi><mfrac><mn>1</mn><mn>2</mn></mfrac></msup><mo></mo><msup><mn>2</mn><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle></mrow><mo>)</mo></mrow></mrow><mo></mo><mi href="./10.1#p2.t1.r4">z</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5.p1">𝒞</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./11.2#E1" mathvariant="bold">H</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>-</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow><mo>-</mo><mrow><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5.p1">𝒞</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>-</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./11.2#E1" mathvariant="bold">H</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m207.png" altimg-height="27px" altimg-valign="-9px" altimg-width="70px" alttext="\nu\neq-\tfrac{1}{2}" display="inline"><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>≠</mo><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m114.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./11.2#E1" title="(11.2.1) ‣ §11.2(i) Power-Series Expansions ‣ §11.2 Definitions ‣ Struve and Modified Struve Functions ‣ Chapter 11 Struve and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m174.png" altimg-height="23px" altimg-valign="-7px" altimg-width="61px" alttext="\mathbf{H}_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./11.2#E1" mathvariant="bold">H</mi><mi class="ltx_nvar" href="./11.1#p1.t1.r3">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./11.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: Struve function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m211.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./10.2#Px5.p1" title="Cylinder Functions ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m178.png" altimg-height="23px" altimg-valign="-7px" altimg-width="56px" alttext="\mathscr{C}_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5.p1">𝒞</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: cylinder function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m163.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./10.1#p2.t1.r4" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m258.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./10.1#p2.t1.r4">z</mi></math>: complex variable</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">11.17</span> (Case <math class="ltx_Math" altimg="m202.png" altimg-height="17px" altimg-valign="-2px" altimg-width="52px" alttext="\nu=0" display="inline"><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>=</mo><mn>0</mn></mrow></math> only)</span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">For the Struve function <math class="ltx_Math" altimg="m175.png" altimg-height="23px" altimg-valign="-7px" altimg-width="61px" alttext="\mathbf{H}_{\nu}\left(z\right)" display="inline"><mrow><msub><mi href="./11.2#E1" mathvariant="bold">H</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math> see §.</p>
<table id="E3" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex3" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="4" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">10.22.3</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m47.png" altimg-height="51px" altimg-valign="-19px" altimg-width="150px" alttext="\displaystyle\int e^{iz}z^{\nu}\mathscr{C}_{\nu}\left(z\right)\mathrm{d}z" display="inline"><mstyle displaystyle="true"><mrow><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow></msup><mo></mo><msup><mi href="./10.1#p2.t1.r4">z</mi><mi href="./10.1#p2.t1.r5">ν</mi></msup><mo></mo><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5.p1">𝒞</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./10.1#p2.t1.r4">z</mi></mrow></mrow></mrow></mstyle></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m27.png" altimg-height="52px" altimg-valign="-17px" altimg-width="273px" alttext="\displaystyle=\frac{e^{iz}z^{\nu+1}}{2\nu+1}(\mathscr{C}_{\nu}\left(z\right)-i%
\mathscr{C}_{\nu+1}\left(z\right))," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><mfrac><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow></msup><mo></mo><msup><mi href="./10.1#p2.t1.r4">z</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><mrow><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo>+</mo><mn>1</mn></mrow></mfrac></mstyle><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5.p1">𝒞</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow><mo>-</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5.p1">𝒞</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m207.png" altimg-height="27px" altimg-valign="-9px" altimg-width="70px" alttext="\nu\neq-\tfrac{1}{2}" display="inline"><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>≠</mo><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow></math>,</span></td></tr>
<tr id="Ex4" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m46.png" altimg-height="51px" altimg-valign="-19px" altimg-width="163px" alttext="\displaystyle\int e^{iz}z^{-\nu}\mathscr{C}_{\nu}\left(z\right)\mathrm{d}z" display="inline"><mstyle displaystyle="true"><mrow><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow></msup><mo></mo><msup><mi href="./10.1#p2.t1.r4">z</mi><mrow><mo>-</mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow></msup><mo></mo><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5.p1">𝒞</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./10.1#p2.t1.r4">z</mi></mrow></mrow></mrow></mstyle></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m26.png" altimg-height="52px" altimg-valign="-17px" altimg-width="286px" alttext="\displaystyle=\frac{e^{iz}z^{-\nu+1}}{1-2\nu}(\mathscr{C}_{\nu}\left(z\right)+%
i\mathscr{C}_{\nu-1}\left(z\right))," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><mfrac><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow></msup><mo></mo><msup><mi href="./10.1#p2.t1.r4">z</mi><mrow><mrow><mo>-</mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo>+</mo><mn>1</mn></mrow></msup></mrow><mrow><mn>1</mn><mo>-</mo><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow></mrow></mfrac></mstyle><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5.p1">𝒞</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5.p1">𝒞</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>-</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m208.png" altimg-height="27px" altimg-valign="-9px" altimg-width="55px" alttext="\nu\neq\tfrac{1}{2}" display="inline"><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>≠</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#Px5.p1" title="Cylinder Functions ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m178.png" altimg-height="23px" altimg-valign="-7px" altimg-width="56px" alttext="\mathscr{C}_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5.p1">𝒞</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: cylinder function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m177.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m163.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./10.1#p2.t1.r4" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m258.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./10.1#p2.t1.r4">z</mi></math>: complex variable</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">11.3.9--11.3.11</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<section id="Px1" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Products</h3>
<div id="Px1.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px1.p1" class="ltx_para">
<table id="E4" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.22.4</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E4.png" altimg-height="51px" altimg-valign="-19px" altimg-width="591px" alttext="\int z\mathscr{C}_{\mu}\left(az\right)\mathscr{D}_{\mu}(bz)\mathrm{d}z=\frac{z%
\left(a\mathscr{C}_{\mu+1}\left(az\right)\mathscr{D}_{\mu}(bz)-b\mathscr{C}_{%
\mu}\left(az\right)\mathscr{D}_{\mu+1}(bz)\right)}{a^{2}-b^{2}}," display="block"><mrow><mrow><mrow><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><mi href="./10.1#p2.t1.r4">z</mi><mo></mo><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5.p1">𝒞</mi><mi>μ</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi>a</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5">𝒟</mi><mi>μ</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>b</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./10.1#p2.t1.r4">z</mi></mrow></mrow></mrow><mo>=</mo><mfrac><mrow><mi href="./10.1#p2.t1.r4">z</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mi>a</mi><mo></mo><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5.p1">𝒞</mi><mrow><mi>μ</mi><mo>+</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mrow><mi>a</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5">𝒟</mi><mi>μ</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>b</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>-</mo><mrow><mi>b</mi><mo></mo><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5.p1">𝒞</mi><mi>μ</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi>a</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5">𝒟</mi><mrow><mi>μ</mi><mo>+</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>b</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow><mo>)</mo></mrow></mrow><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>-</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></mfrac></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m230.png" altimg-height="24px" altimg-valign="-6px" altimg-width="68px" alttext="a^{2}\neq b^{2}" display="inline"><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>≠</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E4.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#Px5.p1" title="Cylinder Functions ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m178.png" altimg-height="23px" altimg-valign="-7px" altimg-width="56px" alttext="\mathscr{C}_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5.p1">𝒞</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: cylinder function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m163.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./10.1#p2.t1.r4" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m258.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./10.1#p2.t1.r4">z</mi></math>: complex variable</a> and
<a href="./10.2#Px5" title="Cylinder Functions ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m183.png" altimg-height="23px" altimg-valign="-7px" altimg-width="55px" alttext="\mathscr{D}_{\nu}(z)" display="inline"><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5">𝒟</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: cylinder function</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="EGx1" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E5">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.22.5</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m48.png" altimg-height="51px" altimg-valign="-19px" altimg-width="190px" alttext="\displaystyle\int z\mathscr{C}_{\mu}\left(az\right)\mathscr{D}_{\mu}(az)%
\mathrm{d}z" display="inline"><mstyle displaystyle="true"><mrow><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><mi href="./10.1#p2.t1.r4">z</mi><mo></mo><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5.p1">𝒞</mi><mi>μ</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi>a</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5">𝒟</mi><mi>μ</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>a</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./10.1#p2.t1.r4">z</mi></mrow></mrow></mrow></mstyle></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m41.png" altimg-height="30px" altimg-valign="-9px" altimg-width="614px" alttext="\displaystyle=\tfrac{1}{4}z^{2}\left(2\mathscr{C}_{\mu}\left(az\right)\mathscr%
{D}_{\mu}(az)-\mathscr{C}_{\mu-1}\left(az\right)\mathscr{D}_{\mu+1}(az)-%
\mathscr{C}_{\mu+1}\left(az\right)\mathscr{D}_{\mu-1}(az)\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac><mo></mo><msup><mi href="./10.1#p2.t1.r4">z</mi><mn>2</mn></msup><mo></mo><mrow><mo>(</mo><mrow><mrow><mn>2</mn><mo></mo><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5.p1">𝒞</mi><mi>μ</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi>a</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5">𝒟</mi><mi>μ</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>a</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>-</mo><mrow><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5.p1">𝒞</mi><mrow><mi>μ</mi><mo>-</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mrow><mi>a</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5">𝒟</mi><mrow><mi>μ</mi><mo>+</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>a</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>-</mo><mrow><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5.p1">𝒞</mi><mrow><mi>μ</mi><mo>+</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mrow><mi>a</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5">𝒟</mi><mrow><mi>μ</mi><mo>-</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>a</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E5.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#Px5.p1" title="Cylinder Functions ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m178.png" altimg-height="23px" altimg-valign="-7px" altimg-width="56px" alttext="\mathscr{C}_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5.p1">𝒞</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: cylinder function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m163.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./10.1#p2.t1.r4" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m258.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./10.1#p2.t1.r4">z</mi></math>: complex variable</a> and
<a href="./10.2#Px5" title="Cylinder Functions ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m183.png" altimg-height="23px" altimg-valign="-7px" altimg-width="55px" alttext="\mathscr{D}_{\nu}(z)" display="inline"><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5">𝒟</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: cylinder function</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E6">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.22.6</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m53.png" altimg-height="50px" altimg-valign="-19px" altimg-width="184px" alttext="\displaystyle\int\mathscr{C}_{\mu}\left(az\right)\mathscr{D}_{\nu}(az)\frac{%
\mathrm{d}z}{z}" display="inline"><mstyle displaystyle="true"><mrow><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5.p1">𝒞</mi><mi>μ</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi>a</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5">𝒟</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>a</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mfrac><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mi href="./10.1#p2.t1.r4">z</mi></mfrac></mrow></mrow></mstyle></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m2.png" altimg-height="51px" altimg-valign="-20px" altimg-width="568px" alttext="\displaystyle=-\frac{az(\mathscr{C}_{\mu+1}\left(az\right)\mathscr{D}_{\nu}(az%
)-\mathscr{C}_{\mu}\left(az\right)\mathscr{D}_{\nu+1}(az))}{\mu^{2}-\nu^{2}}+%
\frac{\mathscr{C}_{\mu}\left(az\right)\mathscr{D}_{\nu}(az)}{\mu+\nu}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mo>-</mo><mstyle displaystyle="true"><mfrac><mrow><mi>a</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5.p1">𝒞</mi><mrow><mi>μ</mi><mo>+</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mrow><mi>a</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5">𝒟</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>a</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>-</mo><mrow><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5.p1">𝒞</mi><mi>μ</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi>a</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5">𝒟</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>a</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow><mrow><msup><mi>μ</mi><mn>2</mn></msup><mo>-</mo><msup><mi href="./10.1#p2.t1.r5">ν</mi><mn>2</mn></msup></mrow></mfrac></mstyle></mrow><mo>+</mo><mstyle displaystyle="true"><mfrac><mrow><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5.p1">𝒞</mi><mi>μ</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi>a</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5">𝒟</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>a</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mrow><mi>μ</mi><mo>+</mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow></mfrac></mstyle></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m198.png" altimg-height="24px" altimg-valign="-6px" altimg-width="72px" alttext="\mu^{2}\neq\nu^{2}" display="inline"><mrow><msup><mi>μ</mi><mn>2</mn></msup><mo>≠</mo><msup><mi href="./10.1#p2.t1.r5">ν</mi><mn>2</mn></msup></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E6.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#Px5.p1" title="Cylinder Functions ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m178.png" altimg-height="23px" altimg-valign="-7px" altimg-width="56px" alttext="\mathscr{C}_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5.p1">𝒞</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: cylinder function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m163.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./10.1#p2.t1.r4" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m258.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./10.1#p2.t1.r4">z</mi></math>: complex variable</a>,
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a> and
<a href="./10.2#Px5" title="Cylinder Functions ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m183.png" altimg-height="23px" altimg-valign="-7px" altimg-width="55px" alttext="\mathscr{D}_{\nu}(z)" display="inline"><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5">𝒟</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: cylinder function</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
<table id="E7" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex5" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="4" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">10.22.7</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m51.png" altimg-height="51px" altimg-valign="-19px" altimg-width="241px" alttext="\displaystyle\int z^{\mu+\nu+1}\mathscr{C}_{\mu}\left(az\right)\mathscr{D}_{%
\nu}(az)\mathrm{d}z" display="inline"><mstyle displaystyle="true"><mrow><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><msup><mi href="./10.1#p2.t1.r4">z</mi><mrow><mi>μ</mi><mo>+</mo><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mn>1</mn></mrow></msup><mo></mo><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5.p1">𝒞</mi><mi>μ</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi>a</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5">𝒟</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>a</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./10.1#p2.t1.r4">z</mi></mrow></mrow></mrow></mstyle></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m29.png" altimg-height="55px" altimg-valign="-21px" altimg-width="491px" alttext="\displaystyle=\frac{z^{\mu+\nu+2}}{2(\mu+\nu+1)}\*\left(\mathscr{C}_{\mu}\left%
(az\right)\mathscr{D}_{\nu}(az)+\mathscr{C}_{\mu+1}\left(az\right)\mathscr{D}_%
{\nu+1}(az)\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><mfrac><msup><mi href="./10.1#p2.t1.r4">z</mi><mrow><mi>μ</mi><mo>+</mo><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mn>2</mn></mrow></msup><mrow><mn>2</mn><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>μ</mi><mo>+</mo><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow></mfrac></mstyle><mo></mo><mrow><mo>(</mo><mrow><mrow><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5.p1">𝒞</mi><mi>μ</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi>a</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5">𝒟</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>a</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5.p1">𝒞</mi><mrow><mi>μ</mi><mo>+</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mrow><mi>a</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5">𝒟</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>a</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m189.png" altimg-height="21px" altimg-valign="-6px" altimg-width="104px" alttext="\mu+\nu\neq-1" display="inline"><mrow><mrow><mi>μ</mi><mo>+</mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo>≠</mo><mrow><mo>-</mo><mn>1</mn></mrow></mrow></math>,</span></td></tr>
<tr id="Ex6" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m49.png" altimg-height="51px" altimg-valign="-19px" altimg-width="254px" alttext="\displaystyle\int z^{-\mu-\nu+1}\mathscr{C}_{\mu}\left(az\right)\mathscr{D}_{%
\nu}(az)\mathrm{d}z" display="inline"><mstyle displaystyle="true"><mrow><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><msup><mi href="./10.1#p2.t1.r4">z</mi><mrow><mrow><mrow><mo>-</mo><mi>μ</mi></mrow><mo>-</mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo>+</mo><mn>1</mn></mrow></msup><mo></mo><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5.p1">𝒞</mi><mi>μ</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi>a</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5">𝒟</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>a</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./10.1#p2.t1.r4">z</mi></mrow></mrow></mrow></mstyle></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m28.png" altimg-height="55px" altimg-valign="-21px" altimg-width="492px" alttext="\displaystyle=\frac{z^{-\mu-\nu+2}}{2(1-\mu-\nu)}\*\left(\mathscr{C}_{\mu}%
\left(az\right)\mathscr{D}_{\nu}(az)+\mathscr{C}_{\mu-1}\left(az\right)%
\mathscr{D}_{\nu-1}(az)\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><mfrac><msup><mi href="./10.1#p2.t1.r4">z</mi><mrow><mrow><mrow><mo>-</mo><mi>μ</mi></mrow><mo>-</mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo>+</mo><mn>2</mn></mrow></msup><mrow><mn>2</mn><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><mi>μ</mi><mo>-</mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mfrac></mstyle><mo></mo><mrow><mo>(</mo><mrow><mrow><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5.p1">𝒞</mi><mi>μ</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi>a</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5">𝒟</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>a</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5.p1">𝒞</mi><mrow><mi>μ</mi><mo>-</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mrow><mi>a</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5">𝒟</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>-</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>a</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m188.png" altimg-height="21px" altimg-valign="-6px" altimg-width="88px" alttext="\mu+\nu\neq 1" display="inline"><mrow><mrow><mi>μ</mi><mo>+</mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo>≠</mo><mn>1</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E7.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#Px5.p1" title="Cylinder Functions ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m178.png" altimg-height="23px" altimg-valign="-7px" altimg-width="56px" alttext="\mathscr{C}_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5.p1">𝒞</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: cylinder function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m163.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./10.1#p2.t1.r4" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m258.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./10.1#p2.t1.r4">z</mi></math>: complex variable</a>,
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a> and
<a href="./10.2#Px5" title="Cylinder Functions ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m183.png" altimg-height="23px" altimg-valign="-7px" altimg-width="55px" alttext="\mathscr{D}_{\nu}(z)" display="inline"><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5">𝒟</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: cylinder function</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">11.3.30, 11.3.31</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
</section>
<section id="ii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§10.22(ii) </span>Integrals over Finite Intervals</h2>
<div id="SS2.info" class="ltx_metadata ltx_info">
)
construct the expansion of the left-hand side in powers of <math class="ltx_Math" altimg="m258.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./10.1#p2.t1.r4">z</mi></math> by use of
(). Next, the result
<math class="ltx_Math" altimg="m165.png" altimg-height="31px" altimg-valign="-9px" altimg-width="458px" alttext="\int_{0}^{2\pi}J_{2\nu}\left(2z\sin\theta\right)e^{\pm 2\mathrm{i}\mu\theta}%
\mathrm{d}\theta=\pi e^{\pm\mathrm{i}\mu\pi}J_{\nu+\mu}\left(z\right)J_{\nu-%
\mu}\left(z\right)" display="inline"><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi></mrow></msubsup><mrow><mrow><msub><mi href="./10.2#E2">J</mi><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r4">z</mi><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi>θ</mi></mrow></mrow><mo>)</mo></mrow></mrow><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>±</mo><mrow><mn>2</mn><mo></mo><mi mathvariant="normal">i</mi><mo></mo><mi>μ</mi><mo></mo><mi>θ</mi></mrow></mrow></msup><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>θ</mi></mrow></mrow></mrow><mo>=</mo><mrow><mi href="./3.12#E1">π</mi><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>±</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi>μ</mi><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow></msup><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mi>μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>-</mo><mi>μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></mrow></math>, <math class="ltx_Math" altimg="m145.png" altimg-height="27px" altimg-valign="-9px" altimg-width="85px" alttext="\Re\nu>-\tfrac{1}{2}" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo>></mo><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow></math>, is proved in a similar
manner with the aid of () replace <math class="ltx_Math" altimg="m220.png" altimg-height="18px" altimg-valign="-2px" altimg-width="14px" alttext="\theta" display="inline"><mi>θ</mi></math> by
<math class="ltx_Math" altimg="m219.png" altimg-height="27px" altimg-valign="-9px" altimg-width="63px" alttext="\tfrac{1}{2}\pi-\theta" display="inline"><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo>-</mo><mi>θ</mi></mrow></math> and set <math class="ltx_Math" altimg="m195.png" altimg-height="16px" altimg-valign="-6px" altimg-width="55px" alttext="\mu=n" display="inline"><mrow><mi>μ</mi><mo>=</mo><mi href="./10.1#p2.t1.r1">n</mi></mrow></math> in () and let <math class="ltx_Math" altimg="m209.png" altimg-height="17px" altimg-valign="-2px" altimg-width="56px" alttext="\nu\to 0" display="inline"><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>→</mo><mn>0</mn></mrow></math>.
For (), page 374 of this reference lacks a factor <math class="ltx_Math" altimg="m161.png" altimg-height="27px" altimg-valign="-9px" altimg-width="17px" alttext="\frac{1}{2}" display="inline"><mfrac><mn>1</mn><mn>2</mn></mfrac></math>
on the right-hand side.)
The verification of () with <math class="ltx_Math" altimg="m248.png" altimg-height="19px" altimg-valign="-5px" altimg-width="53px" alttext="n\geq 1" display="inline"><mrow><mi href="./10.1#p2.t1.r1">n</mi><mo>≥</mo><mn>1</mn></mrow></math> it follows by differentiation and use
of () that the left-hand side equals <math class="ltx_Math" altimg="m167.png" altimg-height="28px" altimg-valign="-9px" altimg-width="218px" alttext="\int_{0}^{x}t^{-1}{J_{n}^{2}}\left(t\right)\mathrm{d}t-\tfrac{1}{2}{J_{n}^{2}}%
\left(x\right)" display="inline"><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi href="./10.1#p2.t1.r3">x</mi></msubsup><mrow><msup><mi>t</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mrow><msubsup><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r1">n</mi><mn>2</mn></msubsup><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mrow><msubsup><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r1">n</mi><mn>2</mn></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mrow></mrow></math>;
application of <cite class="ltx_cite ltx_citemacro_citet">Watson () are needed when <math class="ltx_Math" altimg="m245.png" altimg-height="17px" altimg-valign="-2px" altimg-width="53px" alttext="n=0" display="inline"><mrow><mi href="./10.1#p2.t1.r1">n</mi><mo>=</mo><mn>0</mn></mrow></math>. For
() replace
<math class="ltx_Math" altimg="m251.png" altimg-height="17px" altimg-valign="-2px" altimg-width="11px" alttext="t" display="inline"><mi>t</mi></math> by <math class="ltx_Math" altimg="m257.png" altimg-height="18px" altimg-valign="-4px" altimg-width="46px" alttext="z-t" display="inline"><mrow><mi href="./10.1#p2.t1.r4">z</mi><mo>-</mo><mi>t</mi></mrow></math>, substitute for <math class="ltx_Math" altimg="m252.png" altimg-height="18px" altimg-valign="-2px" altimg-width="23px" alttext="t^{\alpha}" display="inline"><msup><mi>t</mi><mi href="./10.22#Px5.p1">α</mi></msup></math> via () (with <math class="ltx_Math" altimg="m258.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./10.1#p2.t1.r4">z</mi></math>
replaced by <math class="ltx_Math" altimg="m251.png" altimg-height="17px" altimg-valign="-2px" altimg-width="11px" alttext="t" display="inline"><mi>t</mi></math>, and <math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math> replaced by <math class="ltx_Math" altimg="m151.png" altimg-height="13px" altimg-valign="-2px" altimg-width="17px" alttext="\alpha" display="inline"><mi href="./10.22#Px5.p1">α</mi></math>), and then apply
() after replacing <math class="ltx_Math" altimg="m179.png" altimg-height="24px" altimg-valign="-8px" altimg-width="88px" alttext="\mathscr{C}_{\mu\pm 1}\left(az\right)" display="inline"><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5.p1">𝒞</mi><mrow><mi>μ</mi><mo>±</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mrow><mi>a</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></math> and
<math class="ltx_Math" altimg="m181.png" altimg-height="24px" altimg-valign="-8px" altimg-width="85px" alttext="\mathscr{D}_{\mu\pm 1}(bz)" display="inline"><mrow><msub><mi class="ltx_font_mathscript" href="./10.2#Px5">𝒟</mi><mrow><mi>μ</mi><mo>±</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>b</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></math> by <math class="ltx_Math" altimg="m186.png" altimg-height="26px" altimg-valign="-10px" altimg-width="83px" alttext="\mp\mathscr{C}_{\mu}'\left(az\right)" display="inline"><mrow><mo>∓</mo><mrow><msubsup><mi class="ltx_font_mathscript" href="./10.2#Px5.p1">𝒞</mi><mi>μ</mi><mo>′</mo></msubsup><mo></mo><mrow><mo>(</mo><mrow><mi>a</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></mrow></math> and
<math class="ltx_Math" altimg="m187.png" altimg-height="26px" altimg-valign="-10px" altimg-width="80px" alttext="\mp\mathscr{D}_{\mu}^{\prime}(bz)" display="inline"><mrow><mo>∓</mo><mrow><msubsup><mi class="ltx_font_mathscript" href="./10.2#Px5">𝒟</mi><mi>μ</mi><mo>′</mo></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>b</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></math>, respectively, by means of
(</dd>
</dl>
</div>
</div>
<div id="SS2.p1" class="ltx_para">
<p class="ltx_p">Throughout this subsection <math class="ltx_Math" altimg="m254.png" altimg-height="17px" altimg-valign="-3px" altimg-width="52px" alttext="x>0" display="inline"><mrow><mi href="./10.1#p2.t1.r3">x</mi><mo>></mo><mn>0</mn></mrow></math>.</p>
</div>
<div id="SS2.p2" class="ltx_para">
<table id="E8" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.22.8</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E8.png" altimg-height="64px" altimg-valign="-28px" altimg-width="289px" alttext="\int_{0}^{x}J_{\nu}\left(t\right)\mathrm{d}t=2\sum_{k=0}^{\infty}J_{\nu+2k+1}%
\left(x\right)," display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi href="./10.1#p2.t1.r3">x</mi></msubsup><mrow><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow><mo>=</mo><mrow><mn>2</mn><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./10.1#p2.t1.r2">k</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><msub><mi href="./10.2#E2">J</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r2">k</mi></mrow><mo>+</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m143.png" altimg-height="19px" altimg-valign="-4px" altimg-width="82px" alttext="\Re\nu>-1" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo>></mo><mrow><mo>-</mo><mn>1</mn></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E8.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m163.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m128.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./10.1#p2.t1.r2" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m239.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./10.1#p2.t1.r2">k</mi></math>: nonnegative integer</a>,
<a href="./10.1#p2.t1.r3" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./10.1#p2.t1.r3">x</mi></math>: real variable</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">11.1.2</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E9" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.22.9</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_Math ltx_eqn_cell ltx_align_center"><math class="ltx_Math" alttext="\int_{0}^{x}J_{2n}\left(t\right)\mathrm{d}t=\int_{0}^{x}J_{0}\left(t\right)%
\mathrm{d}t-2\sum_{k=0}^{n-1}J_{2k+1}\left(x\right),\quad\int_{0}^{x}J_{2n+1}%
\left(t\right)\mathrm{d}t=1-J_{0}\left(x\right)-2\sum_{k=1}^{n}J_{2k}\left(x%
\right)," display="block"><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi href="./10.1#p2.t1.r3">x</mi></msubsup><mrow><mrow><msub><mi href="./10.2#E2">J</mi><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r1">n</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow><mo>=</mo><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi href="./10.1#p2.t1.r3">x</mi></msubsup><mrow><mrow><msub><mi href="./10.2#E2">J</mi><mn>0</mn></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow><mo>-</mo><mrow><mn>2</mn><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./10.1#p2.t1.r2">k</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi href="./10.1#p2.t1.r1">n</mi><mo>-</mo><mn>1</mn></mrow></munderover><mrow><msub><mi href="./10.2#E2">J</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r2">k</mi></mrow><mo>+</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mrow></mrow></mrow></mrow><mo rspace="12.5pt">,</mo></mtd></mtr><mtr><mtd><mspace width="0em"></mspace><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi href="./10.1#p2.t1.r3">x</mi></msubsup><mrow><mrow><msub><mi href="./10.2#E2">J</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow><mo>=</mo><mrow><mn>1</mn><mo>-</mo><mrow><msub><mi href="./10.2#E2">J</mi><mn>0</mn></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow><mo>-</mo><mrow><mn>2</mn><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./10.1#p2.t1.r2">k</mi><mo>=</mo><mn>1</mn></mrow><mi href="./10.1#p2.t1.r1">n</mi></munderover><mrow><msub><mi href="./10.2#E2">J</mi><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r2">k</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mtd></mtr></mtable></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m244.png" altimg-height="20px" altimg-valign="-6px" altimg-width="104px" alttext="n=0,1,\dots" display="inline"><mrow><mi href="./10.1#p2.t1.r1">n</mi><mo>=</mo><mrow><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E9.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m163.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./10.1#p2.t1.r1" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m247.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./10.1#p2.t1.r1">n</mi></math>: integer</a>,
<a href="./10.1#p2.t1.r2" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m239.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./10.1#p2.t1.r2">k</mi></math>: nonnegative integer</a> and
<a href="./10.1#p2.t1.r3" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./10.1#p2.t1.r3">x</mi></math>: real variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">11.1.3, 11.1.4</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E10" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.22.10</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E10.png" altimg-height="64px" altimg-valign="-28px" altimg-width="762px" alttext="\int_{0}^{x}t^{\mu}J_{\nu}\left(t\right)\mathrm{d}t=x^{\mu}\frac{\Gamma\left(%
\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{1}{2}\right)}{\Gamma\left(\frac{1}{2}\nu-%
\frac{1}{2}\mu+\frac{1}{2}\right)}\*\sum_{k=0}^{\infty}\frac{(\nu+2k+1)\Gamma%
\left(\frac{1}{2}\nu-\frac{1}{2}\mu+\frac{1}{2}+k\right)}{\Gamma\left(\frac{1}%
{2}\nu+\frac{1}{2}\mu+\frac{3}{2}+k\right)}J_{\nu+2k+1}\left(x\right)," display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi href="./10.1#p2.t1.r3">x</mi></msubsup><mrow><msup><mi>t</mi><mi>μ</mi></msup><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow><mo>=</mo><mrow><msup><mi href="./10.1#p2.t1.r3">x</mi><mi>μ</mi></msup><mo></mo><mfrac><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo>+</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi>μ</mi></mrow><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi>μ</mi></mrow></mrow><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow></mfrac><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./10.1#p2.t1.r2">k</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><mfrac><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r2">k</mi></mrow><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi>μ</mi></mrow></mrow><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>+</mo><mi href="./10.1#p2.t1.r2">k</mi></mrow><mo>)</mo></mrow></mrow></mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo>+</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi>μ</mi></mrow><mo>+</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mo>+</mo><mi href="./10.1#p2.t1.r2">k</mi></mrow><mo>)</mo></mrow></mrow></mfrac><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r2">k</mi></mrow><mo>+</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m122.png" altimg-height="23px" altimg-valign="-7px" altimg-width="152px" alttext="\Re(\mu+\nu+1)>0" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>μ</mi><mo>+</mo><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>></mo><mn>0</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E10.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m114.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m163.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m128.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./10.1#p2.t1.r2" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m239.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./10.1#p2.t1.r2">k</mi></math>: nonnegative integer</a>,
<a href="./10.1#p2.t1.r3" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./10.1#p2.t1.r3">x</mi></math>: real variable</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">11.1.1</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="EGx2" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E11">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.22.11</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m81.png" altimg-height="52px" altimg-valign="-20px" altimg-width="143px" alttext="\displaystyle\int_{0}^{x}\frac{1-J_{0}\left(t\right)}{t}\mathrm{d}t" display="inline"><mrow><mstyle displaystyle="true"><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi href="./10.1#p2.t1.r3">x</mi></msubsup></mstyle><mrow><mstyle displaystyle="true"><mfrac><mrow><mn>1</mn><mo>-</mo><mrow><msub><mi href="./10.2#E2">J</mi><mn>0</mn></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></mrow><mi>t</mi></mfrac></mstyle><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m20.png" altimg-height="64px" altimg-valign="-28px" altimg-width="336px" alttext="\displaystyle=\frac{1}{2}\sum_{k=1}^{\infty}\frac{\psi\left(k+1\right)-\psi%
\left(1\right)}{k!}(\tfrac{1}{2}x)^{k}J_{k}\left(x\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mrow><mstyle displaystyle="true"><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./10.1#p2.t1.r2">k</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover></mstyle><mrow><mstyle displaystyle="true"><mfrac><mrow><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./10.1#p2.t1.r2">k</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mo>-</mo><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></mrow><mrow><mi href="./10.1#p2.t1.r2">k</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mfrac></mstyle><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./10.1#p2.t1.r3">x</mi></mrow><mo stretchy="false">)</mo></mrow><mi href="./10.1#p2.t1.r2">k</mi></msup><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r2">k</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E11.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./5.2#E2" title="(5.2.2) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m212.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="\psi\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: psi (or digamma) function</a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m241.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m163.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./10.1#p2.t1.r2" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m239.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./10.1#p2.t1.r2">k</mi></math>: nonnegative integer</a> and
<a href="./10.1#p2.t1.r3" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./10.1#p2.t1.r3">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E12">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.22.12</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m86.png" altimg-height="52px" altimg-valign="-20px" altimg-width="157px" alttext="\displaystyle x\int_{0}^{x}\frac{1-J_{0}\left(t\right)}{t}\mathrm{d}t" display="inline"><mrow><mi href="./10.1#p2.t1.r3">x</mi><mo></mo><mrow><mstyle displaystyle="true"><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi href="./10.1#p2.t1.r3">x</mi></msubsup></mstyle><mrow><mstyle displaystyle="true"><mfrac><mrow><mn>1</mn><mo>-</mo><mrow><msub><mi href="./10.2#E2">J</mi><mn>0</mn></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></mrow><mi>t</mi></mfrac></mstyle><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m7.png" altimg-height="64px" altimg-valign="-28px" altimg-width="379px" alttext="\displaystyle=2\sum_{k=0}^{\infty}(2k+3)(\psi\left(k+2\right)-\psi\left(1%
\right))J_{2k+3}\left(x\right)" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mn>2</mn><mo></mo><mrow><mstyle displaystyle="true"><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./10.1#p2.t1.r2">k</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover></mstyle><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r2">k</mi></mrow><mo>+</mo><mn>3</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./10.1#p2.t1.r2">k</mi><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mo>-</mo><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r2">k</mi></mrow><mo>+</mo><mn>3</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mrow></mrow></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m43.png" altimg-height="64px" altimg-valign="-28px" altimg-width="543px" alttext="\displaystyle=x-2\!J_{1}\left(x\right)+2\sum_{k=0}^{\infty}(2k+5)\*(\psi\left(%
k+3\right)-\psi\left(1\right)-1)J_{2k+5}\left(x\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mi href="./10.1#p2.t1.r3">x</mi><mo>-</mo><mrow><mpadded width="-1.7pt"><mn>2</mn></mpadded><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mn>1</mn></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>+</mo><mrow><mn>2</mn><mo></mo><mrow><mstyle displaystyle="true"><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./10.1#p2.t1.r2">k</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover></mstyle><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r2">k</mi></mrow><mo>+</mo><mn>5</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./10.1#p2.t1.r2">k</mi><mo>+</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mo>-</mo><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r2">k</mi></mrow><mo>+</mo><mn>5</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E12.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./5.2#E2" title="(5.2.2) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m212.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="\psi\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: psi (or digamma) function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m163.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./10.1#p2.t1.r2" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m239.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./10.1#p2.t1.r2">k</mi></math>: nonnegative integer</a> and
<a href="./10.1#p2.t1.r3" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./10.1#p2.t1.r3">x</mi></math>: real variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">11.1.19</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m213.png" altimg-height="24px" altimg-valign="-7px" altimg-width="179px" alttext="\psi\left(x\right)=\Gamma'\left(x\right)/\Gamma\left(x\right)" display="inline"><mrow><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mrow><msup><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo>′</mo></msup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow><mo>/</mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mrow></mrow></math> (§</dd>
</dl>
</div>
</div>
<div id="Px2.p1" class="ltx_para">
<table id="EGx3" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E13">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.22.13</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m55.png" altimg-height="58px" altimg-valign="-20px" altimg-width="268px" alttext="\displaystyle\int_{0}^{\frac{1}{2}\pi}J_{2\nu}\left(2z\cos\theta\right)\cos%
\left(2\mu\theta\right)\mathrm{d}\theta" display="inline"><mrow><mstyle displaystyle="true"><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow></msubsup></mstyle><mrow><mrow><msub><mi href="./10.2#E2">J</mi><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r4">z</mi><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi>θ</mi></mrow></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><mi>μ</mi><mo></mo><mi>θ</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>θ</mi></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m38.png" altimg-height="29px" altimg-valign="-9px" altimg-width="208px" alttext="\displaystyle=\tfrac{1}{2}\pi J_{\nu+\mu}\left(z\right)J_{\nu-\mu}\left(z%
\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mi>μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>-</mo><mi>μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m145.png" altimg-height="27px" altimg-valign="-9px" altimg-width="85px" alttext="\Re\nu>-\tfrac{1}{2}" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo>></mo><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E13.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m211.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m154.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m163.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m128.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./10.1#p2.t1.r1" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m247.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./10.1#p2.t1.r1">n</mi></math>: integer</a>,
<a href="./10.1#p2.t1.r4" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m258.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./10.1#p2.t1.r4">z</mi></math>: complex variable</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">11.4.7</span> (Case <math class="ltx_Math" altimg="m203.png" altimg-height="13px" altimg-valign="-2px" altimg-width="54px" alttext="\nu=n" display="inline"><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>=</mo><mi href="./10.1#p2.t1.r1">n</mi></mrow></math>, <math class="ltx_Math" altimg="m191.png" altimg-height="20px" altimg-valign="-6px" altimg-width="53px" alttext="\mu=0" display="inline"><mrow><mi>μ</mi><mo>=</mo><mn>0</mn></mrow></math>.)</span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E14">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.22.14</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m77.png" altimg-height="53px" altimg-valign="-20px" altimg-width="254px" alttext="\displaystyle\int_{0}^{\pi}J_{2\nu}\left(2z\sin\theta\right)\cos\left(2\mu%
\theta\right)\mathrm{d}\theta" display="inline"><mrow><mstyle displaystyle="true"><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi href="./3.12#E1">π</mi></msubsup></mstyle><mrow><mrow><msub><mi href="./10.2#E2">J</mi><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r4">z</mi><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi>θ</mi></mrow></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><mi>μ</mi><mo></mo><mi>θ</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>θ</mi></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m31.png" altimg-height="26px" altimg-valign="-8px" altimg-width="271px" alttext="\displaystyle=\pi\cos\left(\mu\pi\right)J_{\nu+\mu}\left(z\right)J_{\nu-\mu}%
\left(z\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./3.12#E1">π</mi><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mi>μ</mi><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mi>μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>-</mo><mi>μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m145.png" altimg-height="27px" altimg-valign="-9px" altimg-width="85px" alttext="\Re\nu>-\tfrac{1}{2}" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo>></mo><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E14.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m211.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m154.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m163.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m128.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m216.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./10.1#p2.t1.r1" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m247.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./10.1#p2.t1.r1">n</mi></math>: integer</a>,
<a href="./10.1#p2.t1.r4" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m258.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./10.1#p2.t1.r4">z</mi></math>: complex variable</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">11.4.8</span> (Case <math class="ltx_Math" altimg="m202.png" altimg-height="17px" altimg-valign="-2px" altimg-width="52px" alttext="\nu=0" display="inline"><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>=</mo><mn>0</mn></mrow></math>, <math class="ltx_Math" altimg="m195.png" altimg-height="16px" altimg-valign="-6px" altimg-width="55px" alttext="\mu=n" display="inline"><mrow><mi>μ</mi><mo>=</mo><mi href="./10.1#p2.t1.r1">n</mi></mrow></math>.)</span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E15">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.22.15</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m78.png" altimg-height="53px" altimg-valign="-20px" altimg-width="245px" alttext="\displaystyle\int_{0}^{\pi}J_{2\nu}\left(2z\sin\theta\right)\sin(2\mu\theta)%
\mathrm{d}\theta" display="inline"><mrow><mstyle displaystyle="true"><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi href="./3.12#E1">π</mi></msubsup></mstyle><mrow><mrow><msub><mi href="./10.2#E2">J</mi><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r4">z</mi><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi>θ</mi></mrow></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mn>2</mn><mo></mo><mi>μ</mi><mo></mo><mi>θ</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>θ</mi></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m32.png" altimg-height="26px" altimg-valign="-8px" altimg-width="263px" alttext="\displaystyle=\pi\sin(\mu\pi)J_{\nu+\mu}\left(z\right)J_{\nu-\mu}\left(z\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./3.12#E1">π</mi><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>μ</mi><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mi>μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>-</mo><mi>μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m143.png" altimg-height="19px" altimg-valign="-4px" altimg-width="82px" alttext="\Re\nu>-1" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo>></mo><mrow><mo>-</mo><mn>1</mn></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E15.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m211.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m163.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m128.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m216.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./10.1#p2.t1.r4" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m258.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./10.1#p2.t1.r4">z</mi></math>: complex variable</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E16">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.22.16</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m54.png" altimg-height="58px" altimg-valign="-20px" altimg-width="250px" alttext="\displaystyle\int_{0}^{\frac{1}{2}\pi}J_{0}\left(2z\sin\theta\right)\cos(2n%
\theta)\mathrm{d}\theta" display="inline"><mrow><mstyle displaystyle="true"><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow></msubsup></mstyle><mrow><mrow><msub><mi href="./10.2#E2">J</mi><mn>0</mn></msub><mo></mo><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r4">z</mi><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi>θ</mi></mrow></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r1">n</mi><mo></mo><mi>θ</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>θ</mi></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m39.png" altimg-height="30px" altimg-valign="-9px" altimg-width="112px" alttext="\displaystyle=\tfrac{1}{2}\pi{J_{n}^{2}}\left(z\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mrow><msubsup><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r1">n</mi><mn>2</mn></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m243.png" altimg-height="20px" altimg-valign="-6px" altimg-width="123px" alttext="n=0,1,2,\ldots" display="inline"><mrow><mi href="./10.1#p2.t1.r1">n</mi><mo>=</mo><mrow><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E16.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m211.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m154.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m163.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m216.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./10.1#p2.t1.r1" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m247.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./10.1#p2.t1.r1">n</mi></math>: integer</a> and
<a href="./10.1#p2.t1.r4" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m258.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./10.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
<table id="E17" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.22.17</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_Math ltx_eqn_cell ltx_align_center"><math class="ltx_Math" alttext="\int_{0}^{\frac{1}{2}\pi}Y_{2\nu}\left(2z\cos\theta\right)\cos(2\mu\theta)%
\mathrm{d}\theta=\tfrac{1}{2}\pi\cot(2\nu\pi)J_{\nu+\mu}\left(z\right)J_{\nu-%
\mu}\left(z\right)-\tfrac{1}{2}\pi\csc(2\nu\pi)J_{\mu-\nu}\left(z\right)J_{-%
\mu-\nu}\left(z\right)," display="block"><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow></msubsup><mrow><mrow><msub><mi href="./10.2#E3">Y</mi><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r4">z</mi><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi>θ</mi></mrow></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mn>2</mn><mo></mo><mi>μ</mi><mo></mo><mi>θ</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>θ</mi></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>=</mo><mrow><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mrow><mi href="./4.14#E7">cot</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mi>μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>-</mo><mi>μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow><mo>-</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mrow><mi href="./4.14#E5">csc</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mrow><mi>μ</mi><mo>-</mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mrow><mrow><mo>-</mo><mi>μ</mi></mrow><mo>-</mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mtd></mtr></mtable></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m93.png" altimg-height="27px" altimg-valign="-9px" altimg-width="124px" alttext="-\tfrac{1}{2}<\Re\nu<\tfrac{1}{2}" display="inline"><mrow><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo><</mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo><</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E17.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./10.2#E3" title="(10.2.3) ‣ Bessel Function of the Second Kind (Weber’s Function) ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m112.png" altimg-height="23px" altimg-valign="-7px" altimg-width="55px" alttext="Y_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E3">Y</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the second kind</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m211.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.14#E5" title="(4.14.5) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m157.png" altimg-height="13px" altimg-valign="-2px" altimg-width="43px" alttext="\csc\NVar{z}" display="inline"><mrow><mi href="./4.14#E5">csc</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosecant function</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m154.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./4.14#E7" title="(4.14.7) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m156.png" altimg-height="17px" altimg-valign="-2px" altimg-width="44px" alttext="\cot\NVar{z}" display="inline"><mrow><mi href="./4.14#E7">cot</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cotangent function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m163.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m128.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./10.1#p2.t1.r4" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m258.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./10.1#p2.t1.r4">z</mi></math>: complex variable</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E18" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.22.18</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E18.png" altimg-height="58px" altimg-valign="-20px" altimg-width="423px" alttext="\int_{0}^{\frac{1}{2}\pi}Y_{0}\left(2z\sin\theta\right)\cos\left(2n\theta%
\right)\mathrm{d}\theta=\tfrac{1}{2}\pi J_{n}\left(z\right)Y_{n}\left(z\right)," display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow></msubsup><mrow><mrow><msub><mi href="./10.2#E3">Y</mi><mn>0</mn></msub><mo></mo><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r4">z</mi><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi>θ</mi></mrow></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r1">n</mi><mo></mo><mi>θ</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>θ</mi></mrow></mrow></mrow><mo>=</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./10.2#E3">Y</mi><mi href="./10.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m242.png" altimg-height="20px" altimg-valign="-6px" altimg-width="123px" alttext="n=0,1,2,\dots" display="inline"><mrow><mi href="./10.1#p2.t1.r1">n</mi><mo>=</mo><mrow><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E18.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./10.2#E3" title="(10.2.3) ‣ Bessel Function of the Second Kind (Weber’s Function) ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m112.png" altimg-height="23px" altimg-valign="-7px" altimg-width="55px" alttext="Y_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E3">Y</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the second kind</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m211.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m154.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m163.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m216.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./10.1#p2.t1.r1" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m247.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./10.1#p2.t1.r1">n</mi></math>: integer</a> and
<a href="./10.1#p2.t1.r4" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m258.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./10.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">11.4.9</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E19" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.22.19</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E19.png" altimg-height="58px" altimg-valign="-20px" altimg-width="625px" alttext="\int_{0}^{\frac{1}{2}\pi}J_{\mu}\left(z\sin\theta\right)(\sin\theta)^{\mu+1}(%
\cos\theta)^{2\nu+1}\mathrm{d}\theta=2^{\nu}\Gamma\left(\nu+1\right)z^{-\nu-1}%
J_{\mu+\nu+1}\left(z\right)," display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow></msubsup><mrow><mrow><msub><mi href="./10.2#E2">J</mi><mi>μ</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi href="./10.1#p2.t1.r4">z</mi><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi>θ</mi></mrow></mrow><mo>)</mo></mrow></mrow><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi>θ</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mi>μ</mi><mo>+</mo><mn>1</mn></mrow></msup><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi>θ</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo>+</mo><mn>1</mn></mrow></msup><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>θ</mi></mrow></mrow></mrow><mo>=</mo><mrow><msup><mn>2</mn><mi href="./10.1#p2.t1.r5">ν</mi></msup><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mo></mo><msup><mi href="./10.1#p2.t1.r4">z</mi><mrow><mrow><mo>-</mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mrow><mi>μ</mi><mo>+</mo><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m131.png" altimg-height="21px" altimg-valign="-6px" altimg-width="83px" alttext="\Re\mu>-1" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>μ</mi></mrow><mo>></mo><mrow><mo>-</mo><mn>1</mn></mrow></mrow></math>, <math class="ltx_Math" altimg="m143.png" altimg-height="19px" altimg-valign="-4px" altimg-width="82px" alttext="\Re\nu>-1" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo>></mo><mrow><mo>-</mo><mn>1</mn></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E19.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m114.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m211.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m154.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m163.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m128.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m216.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./10.1#p2.t1.r4" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m258.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./10.1#p2.t1.r4">z</mi></math>: complex variable</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">11.4.10</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="EGx4" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E20">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.22.20</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m56.png" altimg-height="58px" altimg-valign="-20px" altimg-width="306px" alttext="\displaystyle\int_{0}^{\frac{1}{2}\pi}J_{\mu}\left(z\sin\theta\right)(\sin%
\theta)^{\mu}(\cos\theta)^{2\mu}\mathrm{d}\theta" display="inline"><mrow><mstyle displaystyle="true"><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow></msubsup></mstyle><mrow><mrow><msub><mi href="./10.2#E2">J</mi><mi>μ</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi href="./10.1#p2.t1.r4">z</mi><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi>θ</mi></mrow></mrow><mo>)</mo></mrow></mrow><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi>θ</mi></mrow><mo stretchy="false">)</mo></mrow><mi>μ</mi></msup><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi>θ</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mn>2</mn><mo></mo><mi>μ</mi></mrow></msup><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>θ</mi></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m34.png" altimg-height="33px" altimg-valign="-10px" altimg-width="291px" alttext="\displaystyle=\pi^{\frac{1}{2}}2^{\mu-1}z^{-\mu}\*\Gamma\left(\mu+\tfrac{1}{2}%
\right){J_{\mu}^{2}}\left(\tfrac{1}{2}z\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msup><mi href="./3.12#E1">π</mi><mfrac><mn>1</mn><mn>2</mn></mfrac></msup><mo></mo><msup><mn>2</mn><mrow><mi>μ</mi><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><msup><mi href="./10.1#p2.t1.r4">z</mi><mrow><mo>-</mo><mi>μ</mi></mrow></msup><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi>μ</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><msubsup><mi href="./10.2#E2">J</mi><mi>μ</mi><mn>2</mn></msubsup><mo></mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m135.png" altimg-height="27px" altimg-valign="-9px" altimg-width="85px" alttext="\Re\mu>-\tfrac{1}{2}" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>μ</mi></mrow><mo>></mo><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E20.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m114.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m211.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m154.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m163.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m128.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m216.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a> and
<a href="./10.1#p2.t1.r4" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m258.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./10.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E21">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.22.21</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m60.png" altimg-height="58px" altimg-valign="-20px" altimg-width="306px" alttext="\displaystyle\int_{0}^{\frac{1}{2}\pi}Y_{\mu}\left(z\sin\theta\right)(\sin%
\theta)^{\mu}(\cos\theta)^{2\mu}\mathrm{d}\theta" display="inline"><mrow><mstyle displaystyle="true"><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow></msubsup></mstyle><mrow><mrow><msub><mi href="./10.2#E3">Y</mi><mi>μ</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi href="./10.1#p2.t1.r4">z</mi><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi>θ</mi></mrow></mrow><mo>)</mo></mrow></mrow><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi>θ</mi></mrow><mo stretchy="false">)</mo></mrow><mi>μ</mi></msup><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi>θ</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mn>2</mn><mo></mo><mi>μ</mi></mrow></msup><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>θ</mi></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m33.png" altimg-height="33px" altimg-valign="-9px" altimg-width="361px" alttext="\displaystyle=\pi^{\frac{1}{2}}2^{\mu-1}z^{-\mu}\*\Gamma\left(\mu+\tfrac{1}{2}%
\right)J_{\mu}\left(\tfrac{1}{2}z\right)Y_{\mu}\left(\tfrac{1}{2}z\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msup><mi href="./3.12#E1">π</mi><mfrac><mn>1</mn><mn>2</mn></mfrac></msup><mo></mo><msup><mn>2</mn><mrow><mi>μ</mi><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><msup><mi href="./10.1#p2.t1.r4">z</mi><mrow><mo>-</mo><mi>μ</mi></mrow></msup><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi>μ</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mi>μ</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./10.2#E3">Y</mi><mi>μ</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m135.png" altimg-height="27px" altimg-valign="-9px" altimg-width="85px" alttext="\Re\mu>-\tfrac{1}{2}" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>μ</mi></mrow><mo>></mo><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E21.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./10.2#E3" title="(10.2.3) ‣ Bessel Function of the Second Kind (Weber’s Function) ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m112.png" altimg-height="23px" altimg-valign="-7px" altimg-width="55px" alttext="Y_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E3">Y</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the second kind</a>,
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m114.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m211.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m154.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m163.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m128.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m216.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a> and
<a href="./10.1#p2.t1.r4" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m258.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./10.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
<table id="E22" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.22.22</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_Math ltx_eqn_cell ltx_align_center"><math class="ltx_Math" alttext="\int_{0}^{\frac{1}{2}\pi}J_{\mu}\left(z{\sin^{2}}\theta\right)J_{\nu}\left(z{%
\cos^{2}}\theta\right)(\sin\theta)^{2\mu+1}(\cos\theta)^{2\nu+1}\mathrm{d}%
\theta=\frac{\Gamma\left(\mu+\tfrac{1}{2}\right)\Gamma\left(\nu+\tfrac{1}{2}%
\right)J_{\mu+\nu+\frac{1}{2}}\left(z\right)}{(8\pi z)^{\frac{1}{2}}\Gamma%
\left(\mu+\nu+1\right)}," display="block"><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow></msubsup><mrow><mrow><msub><mi href="./10.2#E2">J</mi><mi>μ</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi href="./10.1#p2.t1.r4">z</mi><mo></mo><msup><mi href="./4.14#E1">sin</mi><mn>2</mn></msup><mo></mo><mi>θ</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi href="./10.1#p2.t1.r4">z</mi><mo></mo><msup><mi href="./4.14#E2">cos</mi><mn>2</mn></msup><mo></mo><mi>θ</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi>θ</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mrow><mn>2</mn><mo></mo><mi>μ</mi></mrow><mo>+</mo><mn>1</mn></mrow></msup><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi>θ</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo>+</mo><mn>1</mn></mrow></msup><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>θ</mi></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>=</mo><mfrac><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi>μ</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mrow><mi>μ</mi><mo>+</mo><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mn>8</mn><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo stretchy="false">)</mo></mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></msup><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi>μ</mi><mo>+</mo><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mrow></mfrac><mo>,</mo></mtd></mtr></mtable></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m134.png" altimg-height="27px" altimg-valign="-9px" altimg-width="175px" alttext="\Re\mu>-\tfrac{1}{2},\Re\nu>-\tfrac{1}{2}" display="inline"><mrow><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>μ</mi></mrow><mo>></mo><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow><mo>,</mo><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo>></mo><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E22.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m114.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m211.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m154.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m163.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m128.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m216.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./10.1#p2.t1.r4" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m258.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./10.1#p2.t1.r4">z</mi></math>: complex variable</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="EGx5" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E23">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.22.23</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m58.png" altimg-height="58px" altimg-valign="-20px" altimg-width="409px" alttext="\displaystyle\int_{0}^{\frac{1}{2}\pi}J_{\mu}\left(z{\sin^{2}}\theta\right)J_{%
\nu}\left(z{\cos^{2}}\theta\right)(\sin\theta)^{2\alpha-1}\sec\theta\mathrm{d}\theta" display="inline"><mrow><mstyle displaystyle="true"><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow></msubsup></mstyle><mrow><mrow><msub><mi href="./10.2#E2">J</mi><mi>μ</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi href="./10.1#p2.t1.r4">z</mi><mo></mo><mrow><msup><mi href="./4.14#E1">sin</mi><mn>2</mn></msup><mo></mo><mi>θ</mi></mrow></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi href="./10.1#p2.t1.r4">z</mi><mo></mo><mrow><msup><mi href="./4.14#E2">cos</mi><mn>2</mn></msup><mo></mo><mi>θ</mi></mrow></mrow><mo>)</mo></mrow></mrow><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi>θ</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mrow><mn>2</mn><mo></mo><mi href="./10.22#Px5.p1">α</mi></mrow><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mrow><mi href="./4.14#E6">sec</mi><mo></mo><mi>θ</mi></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>θ</mi></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m14.png" altimg-height="55px" altimg-valign="-21px" altimg-width="361px" alttext="\displaystyle=\frac{(\mu+\nu+\alpha)\Gamma\left(\mu+\alpha\right)2^{\alpha-1}}%
{\nu\Gamma\left(\mu+1\right)z^{\alpha}}J_{\mu+\nu+\alpha}\left(z\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><mfrac><mrow><mrow><mo stretchy="false">(</mo><mrow><mi>μ</mi><mo>+</mo><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mi href="./10.22#Px5.p1">α</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi>μ</mi><mo>+</mo><mi href="./10.22#Px5.p1">α</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><msup><mn>2</mn><mrow><mi href="./10.22#Px5.p1">α</mi><mo>-</mo><mn>1</mn></mrow></msup></mrow><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi>μ</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mo></mo><msup><mi href="./10.1#p2.t1.r4">z</mi><mi href="./10.22#Px5.p1">α</mi></msup></mrow></mfrac></mstyle><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mrow><mi>μ</mi><mo>+</mo><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mi href="./10.22#Px5.p1">α</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m118.png" altimg-height="23px" altimg-valign="-7px" altimg-width="120px" alttext="\Re(\mu+\alpha)>0" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>μ</mi><mo>+</mo><mi href="./10.22#Px5.p1">α</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>></mo><mn>0</mn></mrow></math>, <math class="ltx_Math" altimg="m148.png" altimg-height="18px" altimg-valign="-3px" altimg-width="66px" alttext="\Re\nu>0" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo>></mo><mn>0</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E23.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m114.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m211.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m154.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m163.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m128.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./4.14#E6" title="(4.14.6) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m215.png" altimg-height="13px" altimg-valign="-2px" altimg-width="43px" alttext="\sec\NVar{z}" display="inline"><mrow><mi href="./4.14#E6">sec</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: secant function</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m216.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./10.1#p2.t1.r4" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m258.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./10.1#p2.t1.r4">z</mi></math>: complex variable</a>,
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a> and
<a href="./10.22#Px5.p1" title="Fractional Integral ‣ §10.22(ii) Integrals over Finite Intervals ‣ §10.22 Integrals ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m151.png" altimg-height="13px" altimg-valign="-2px" altimg-width="17px" alttext="\alpha" display="inline"><mi href="./10.22#Px5.p1">α</mi></math>: positive integer</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">11.4.11</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E24">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.22.24</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m59.png" altimg-height="58px" altimg-valign="-20px" altimg-width="314px" alttext="\displaystyle\int_{0}^{\frac{1}{2}\pi}J_{\mu}\left(z{\sin^{2}}\theta\right)J_{%
\nu}\left(z{\cos^{2}}\theta\right)\cot\theta\mathrm{d}\theta" display="inline"><mrow><mstyle displaystyle="true"><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow></msubsup></mstyle><mrow><mrow><msub><mi href="./10.2#E2">J</mi><mi>μ</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi href="./10.1#p2.t1.r4">z</mi><mo></mo><msup><mi href="./4.14#E1">sin</mi><mn>2</mn></msup><mo></mo><mi>θ</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi href="./10.1#p2.t1.r4">z</mi><mo></mo><msup><mi href="./4.14#E2">cos</mi><mn>2</mn></msup><mo></mo><mi>θ</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./4.14#E7">cot</mi><mo></mo><mrow><mi>θ</mi><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>θ</mi></mrow></mrow></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m37.png" altimg-height="30px" altimg-valign="-9px" altimg-width="154px" alttext="\displaystyle=\tfrac{1}{2}\mu^{-1}J_{\mu+\nu}\left(z\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><msup><mi>μ</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mrow><mi>μ</mi><mo>+</mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m137.png" altimg-height="21px" altimg-valign="-6px" altimg-width="154px" alttext="\Re\mu>0,\Re\nu>-1" display="inline"><mrow><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>μ</mi></mrow><mo>></mo><mn>0</mn></mrow><mo>,</mo><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo>></mo><mrow><mo>-</mo><mn>1</mn></mrow></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E24.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m211.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m154.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./4.14#E7" title="(4.14.7) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m156.png" altimg-height="17px" altimg-valign="-2px" altimg-width="44px" alttext="\cot\NVar{z}" display="inline"><mrow><mi href="./4.14#E7">cot</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cotangent function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m163.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m128.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m216.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./10.1#p2.t1.r4" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m258.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./10.1#p2.t1.r4">z</mi></math>: complex variable</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E25">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.22.25</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m57.png" altimg-height="58px" altimg-valign="-20px" altimg-width="350px" alttext="\displaystyle\int_{0}^{\frac{1}{2}\pi}J_{\mu}\left(z\sin\theta\right)I_{\nu}%
\left(z\cos\theta\right)(\tan\theta)^{\mu+1}\mathrm{d}\theta" display="inline"><mrow><mstyle displaystyle="true"><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow></msubsup></mstyle><mrow><mrow><msub><mi href="./10.2#E2">J</mi><mi>μ</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi href="./10.1#p2.t1.r4">z</mi><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi>θ</mi></mrow></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./10.25#E2">I</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi href="./10.1#p2.t1.r4">z</mi><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi>θ</mi></mrow></mrow><mo>)</mo></mrow></mrow><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./4.14#E4">tan</mi><mo></mo><mi>θ</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mi>μ</mi><mo>+</mo><mn>1</mn></mrow></msup><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>θ</mi></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m23.png" altimg-height="59px" altimg-valign="-24px" altimg-width="250px" alttext="\displaystyle=\frac{\Gamma\left(\tfrac{1}{2}\nu-\tfrac{1}{2}\mu\right)(\tfrac{%
1}{2}z)^{\mu}}{2\!\Gamma\left(\tfrac{1}{2}\nu+\tfrac{1}{2}\mu+1\right)}J_{\nu}%
\left(z\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><mfrac><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi>μ</mi></mrow></mrow><mo>)</mo></mrow></mrow><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo stretchy="false">)</mo></mrow><mi>μ</mi></msup></mrow><mrow><mpadded width="-1.7pt"><mn>2</mn></mpadded><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo>+</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi>μ</mi></mrow><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mrow></mfrac></mstyle><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m149.png" altimg-height="21px" altimg-valign="-6px" altimg-width="135px" alttext="\Re\nu>\Re\mu>-1" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo>></mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>μ</mi></mrow><mo>></mo><mrow><mo>-</mo><mn>1</mn></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E25.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m114.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m211.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m154.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m163.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./10.25#E2" title="(10.25.2) ‣ §10.25(ii) Standard Solutions ‣ §10.25 Definitions ‣ Modified Bessel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m100.png" altimg-height="23px" altimg-valign="-7px" altimg-width="52px" alttext="I_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.25#E2">I</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: modified Bessel function of the first kind</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m128.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m216.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./4.14#E4" title="(4.14.4) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="17px" altimg-valign="-2px" altimg-width="47px" alttext="\tan\NVar{z}" display="inline"><mrow><mi href="./4.14#E4">tan</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: tangent function</a>,
<a href="./10.1#p2.t1.r4" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m258.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./10.1#p2.t1.r4">z</mi></math>: complex variable</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
<p class="ltx_p">For <math class="ltx_Math" altimg="m101.png" altimg-height="21px" altimg-valign="-5px" altimg-width="23px" alttext="I_{\nu}" display="inline"><msub><mi href="./10.25#E2">I</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub></math> see §.</p>
<table id="E26" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.22.26</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_Math ltx_eqn_cell ltx_align_center"><math class="ltx_Math" alttext="\int_{0}^{\frac{1}{2}\pi}J_{\mu}\left(z\sin\theta\right)J_{\nu}\left(\zeta\cos%
\theta\right)(\sin\theta)^{\mu+1}(\cos\theta)^{\nu+1}\mathrm{d}\theta=\frac{z^%
{\mu}\zeta^{\nu}J_{\mu+\nu+1}\left(\sqrt{\zeta^{2}+z^{2}}\right)}{(\zeta^{2}+z%
^{2})^{\frac{1}{2}(\mu+\nu+1)}}," display="block"><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow></msubsup><mrow><mrow><msub><mi href="./10.2#E2">J</mi><mi>μ</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi href="./10.1#p2.t1.r4">z</mi><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi>θ</mi></mrow></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi>ζ</mi><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi>θ</mi></mrow></mrow><mo>)</mo></mrow></mrow><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi>θ</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mi>μ</mi><mo>+</mo><mn>1</mn></mrow></msup><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi>θ</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mn>1</mn></mrow></msup><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>θ</mi></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>=</mo><mfrac><mrow><msup><mi href="./10.1#p2.t1.r4">z</mi><mi>μ</mi></msup><mo></mo><msup><mi>ζ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msup><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mrow><mi>μ</mi><mo>+</mo><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><msqrt><mrow><msup><mi>ζ</mi><mn>2</mn></msup><mo>+</mo><msup><mi href="./10.1#p2.t1.r4">z</mi><mn>2</mn></msup></mrow></msqrt><mo>)</mo></mrow></mrow></mrow><msup><mrow><mo stretchy="false">(</mo><mrow><msup><mi>ζ</mi><mn>2</mn></msup><mo>+</mo><msup><mi href="./10.1#p2.t1.r4">z</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>μ</mi><mo>+</mo><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow></msup></mfrac><mo>,</mo></mtd></mtr></mtable></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m130.png" altimg-height="21px" altimg-valign="-6px" altimg-width="169px" alttext="\Re\mu>-1,\Re\nu>-1" display="inline"><mrow><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>μ</mi></mrow><mo>></mo><mrow><mo>-</mo><mn>1</mn></mrow></mrow><mo>,</mo><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo>></mo><mrow><mo>-</mo><mn>1</mn></mrow></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E26.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m211.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m154.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m163.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m128.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m216.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./10.1#p2.t1.r4" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m258.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./10.1#p2.t1.r4">z</mi></math>: complex variable</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px3" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Products</h3>
<div id="Px3.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px3.p1" class="ltx_para">
<table id="EGx6" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E27">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.22.27</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m84.png" altimg-height="53px" altimg-valign="-20px" altimg-width="135px" alttext="\displaystyle\int_{0}^{x}t{J_{\nu-1}^{2}}\left(t\right)\mathrm{d}t" display="inline"><mrow><mstyle displaystyle="true"><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi href="./10.1#p2.t1.r3">x</mi></msubsup></mstyle><mrow><mi>t</mi><mo></mo><mrow><msubsup><mi href="./10.2#E2">J</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>-</mo><mn>1</mn></mrow><mn>2</mn></msubsup><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m8.png" altimg-height="64px" altimg-valign="-28px" altimg-width="230px" alttext="\displaystyle=2\sum_{k=0}^{\infty}(\nu+2k){J_{\nu+2k}^{2}}\left(x\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mn>2</mn><mo></mo><mrow><mstyle displaystyle="true"><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./10.1#p2.t1.r2">k</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover></mstyle><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r2">k</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><msubsup><mi href="./10.2#E2">J</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r2">k</mi></mrow></mrow><mn>2</mn></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m148.png" altimg-height="18px" altimg-valign="-3px" altimg-width="66px" alttext="\Re\nu>0" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo>></mo><mn>0</mn></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E27.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m163.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m128.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./10.1#p2.t1.r2" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m239.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./10.1#p2.t1.r2">k</mi></math>: nonnegative integer</a>,
<a href="./10.1#p2.t1.r3" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./10.1#p2.t1.r3">x</mi></math>: real variable</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">11.3.32</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E28">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.22.28</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m82.png" altimg-height="53px" altimg-valign="-20px" altimg-width="249px" alttext="\displaystyle\int_{0}^{x}t\left({J_{\nu-1}^{2}}\left(t\right)-{J_{\nu+1}^{2}}%
\left(t\right)\right)\mathrm{d}t" display="inline"><mrow><mstyle displaystyle="true"><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi href="./10.1#p2.t1.r3">x</mi></msubsup></mstyle><mrow><mi>t</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><msubsup><mi href="./10.2#E2">J</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>-</mo><mn>1</mn></mrow><mn>2</mn></msubsup><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow><mo>-</mo><mrow><msubsup><mi href="./10.2#E2">J</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mn>1</mn></mrow><mn>2</mn></msubsup><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></mrow><mo>)</mo></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m6.png" altimg-height="28px" altimg-valign="-7px" altimg-width="109px" alttext="\displaystyle=2\nu{J_{\nu}^{2}}\left(x\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi><mo></mo><mrow><msubsup><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi><mn>2</mn></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m148.png" altimg-height="18px" altimg-valign="-3px" altimg-width="66px" alttext="\Re\nu>0" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo>></mo><mn>0</mn></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E28.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m163.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m128.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./10.1#p2.t1.r3" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./10.1#p2.t1.r3">x</mi></math>: real variable</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">11.3.33</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E29">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.22.29</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m83.png" altimg-height="53px" altimg-valign="-20px" altimg-width="116px" alttext="\displaystyle\int_{0}^{x}t{J_{0}^{2}}\left(t\right)\mathrm{d}t" display="inline"><mrow><mstyle displaystyle="true"><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi href="./10.1#p2.t1.r3">x</mi></msubsup></mstyle><mrow><mi>t</mi><mo></mo><mrow><msubsup><mi href="./10.2#E2">J</mi><mn>0</mn><mn>2</mn></msubsup><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m40.png" altimg-height="30px" altimg-valign="-9px" altimg-width="219px" alttext="\displaystyle=\tfrac{1}{2}x^{2}\left({J_{0}^{2}}\left(x\right)+{J_{1}^{2}}%
\left(x\right)\right)." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><msup><mi href="./10.1#p2.t1.r3">x</mi><mn>2</mn></msup><mo></mo><mrow><mo>(</mo><mrow><mrow><msubsup><mi href="./10.2#E2">J</mi><mn>0</mn><mn>2</mn></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow><mo>+</mo><mrow><msubsup><mi href="./10.2#E2">J</mi><mn>1</mn><mn>2</mn></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E29.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m163.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a> and
<a href="./10.1#p2.t1.r3" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./10.1#p2.t1.r3">x</mi></math>: real variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">11.3.34</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
<table id="E30" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.22.30</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E30.png" altimg-height="66px" altimg-valign="-30px" altimg-width="578px" alttext="\int_{0}^{x}J_{n}\left(t\right)J_{n+1}\left(t\right)\mathrm{d}t=\tfrac{1}{2}%
\left(1-{J_{0}^{2}}\left(x\right)\right)-\sum_{k=1}^{n}{J_{k}^{2}}\left(x%
\right)=\sum_{k=n+1}^{\infty}{J_{k}^{2}}\left(x\right)," display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi href="./10.1#p2.t1.r3">x</mi></msubsup><mrow><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mrow><mi href="./10.1#p2.t1.r1">n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow><mo>=</mo><mrow><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mrow><msubsup><mi href="./10.2#E2">J</mi><mn>0</mn><mn>2</mn></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mrow><mo>)</mo></mrow></mrow><mo>-</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./10.1#p2.t1.r2">k</mi><mo>=</mo><mn>1</mn></mrow><mi href="./10.1#p2.t1.r1">n</mi></munderover><mrow><msubsup><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r2">k</mi><mn>2</mn></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>=</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./10.1#p2.t1.r2">k</mi><mo>=</mo><mrow><mi href="./10.1#p2.t1.r1">n</mi><mo>+</mo><mn>1</mn></mrow></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><msubsup><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r2">k</mi><mn>2</mn></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m243.png" altimg-height="20px" altimg-valign="-6px" altimg-width="123px" alttext="n=0,1,2,\ldots" display="inline"><mrow><mi href="./10.1#p2.t1.r1">n</mi><mo>=</mo><mrow><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E30.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m163.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./10.1#p2.t1.r1" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m247.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./10.1#p2.t1.r1">n</mi></math>: integer</a>,
<a href="./10.1#p2.t1.r2" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m239.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./10.1#p2.t1.r2">k</mi></math>: nonnegative integer</a> and
<a href="./10.1#p2.t1.r3" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./10.1#p2.t1.r3">x</mi></math>: real variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">11.3.35</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px4" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Convolutions</h3>
<div id="Px4.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px4.p1" class="ltx_para">
<table id="E31" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.22.31</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E31.png" altimg-height="64px" altimg-valign="-28px" altimg-width="445px" alttext="\int_{0}^{x}J_{\mu}\left(t\right)J_{\nu}\left(x-t\right)\mathrm{d}t=2\sum_{k=0%
}^{\infty}(-1)^{k}J_{\mu+\nu+2k+1}\left(x\right)," display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi href="./10.1#p2.t1.r3">x</mi></msubsup><mrow><mrow><msub><mi href="./10.2#E2">J</mi><mi>μ</mi></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi href="./10.1#p2.t1.r3">x</mi><mo>-</mo><mi>t</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow><mo>=</mo><mrow><mn>2</mn><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./10.1#p2.t1.r2">k</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./10.1#p2.t1.r2">k</mi></msup><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mrow><mi>μ</mi><mo>+</mo><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r2">k</mi></mrow><mo>+</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m130.png" altimg-height="21px" altimg-valign="-6px" altimg-width="169px" alttext="\Re\mu>-1,\Re\nu>-1" display="inline"><mrow><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>μ</mi></mrow><mo>></mo><mrow><mo>-</mo><mn>1</mn></mrow></mrow><mo>,</mo><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo>></mo><mrow><mo>-</mo><mn>1</mn></mrow></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E31.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m163.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m128.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./10.1#p2.t1.r2" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m239.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./10.1#p2.t1.r2">k</mi></math>: nonnegative integer</a>,
<a href="./10.1#p2.t1.r3" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./10.1#p2.t1.r3">x</mi></math>: real variable</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">11.3.37</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="EGx7" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E32">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.22.32</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m80.png" altimg-height="53px" altimg-valign="-20px" altimg-width="214px" alttext="\displaystyle\int_{0}^{x}J_{\nu}\left(t\right)J_{1-\nu}\left(x-t\right)\mathrm%
{d}t" display="inline"><mrow><mstyle displaystyle="true"><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi href="./10.1#p2.t1.r3">x</mi></msubsup></mstyle><mrow><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mrow><mn>1</mn><mo>-</mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mrow><mi href="./10.1#p2.t1.r3">x</mi><mo>-</mo><mi>t</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m10.png" altimg-height="25px" altimg-valign="-7px" altimg-width="149px" alttext="\displaystyle=J_{0}\left(x\right)-\cos x," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><msub><mi href="./10.2#E2">J</mi><mn>0</mn></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow><mo>-</mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi href="./10.1#p2.t1.r3">x</mi></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m91.png" altimg-height="19px" altimg-valign="-4px" altimg-width="118px" alttext="-1<\Re\nu<2" display="inline"><mrow><mrow><mo>-</mo><mn>1</mn></mrow><mo><</mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo><</mo><mn>2</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E32.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m154.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m163.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m128.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./10.1#p2.t1.r3" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./10.1#p2.t1.r3">x</mi></math>: real variable</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">11.3.38</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E33">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.22.33</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m79.png" altimg-height="53px" altimg-valign="-20px" altimg-width="206px" alttext="\displaystyle\int_{0}^{x}J_{\nu}\left(t\right)J_{-\nu}\left(x-t\right)\mathrm{%
d}t" display="inline"><mrow><mstyle displaystyle="true"><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi href="./10.1#p2.t1.r3">x</mi></msubsup></mstyle><mrow><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mrow><mo>-</mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mrow><mi href="./10.1#p2.t1.r3">x</mi><mo>-</mo><mi>t</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m35.png" altimg-height="23px" altimg-valign="-6px" altimg-width="72px" alttext="\displaystyle=\sin x," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi href="./10.1#p2.t1.r3">x</mi></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m264.png" altimg-height="23px" altimg-valign="-7px" altimg-width="77px" alttext="|\Re\nu|<1" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo stretchy="false">|</mo></mrow><mo><</mo><mn>1</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E33.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m163.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m128.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m216.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./10.1#p2.t1.r3" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./10.1#p2.t1.r3">x</mi></math>: real variable</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">11.3.39</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
<table id="E34" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.22.34</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E34.png" altimg-height="52px" altimg-valign="-20px" altimg-width="333px" alttext="\int_{0}^{x}t^{-1}J_{\mu}\left(t\right)J_{\nu}\left(x-t\right)\mathrm{d}t=%
\frac{J_{\mu+\nu}\left(x\right)}{\mu}," display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi href="./10.1#p2.t1.r3">x</mi></msubsup><mrow><msup><mi>t</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mi>μ</mi></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi href="./10.1#p2.t1.r3">x</mi><mo>-</mo><mi>t</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow><mo>=</mo><mfrac><mrow><msub><mi href="./10.2#E2">J</mi><mrow><mi>μ</mi><mo>+</mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow><mi>μ</mi></mfrac></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m137.png" altimg-height="21px" altimg-valign="-6px" altimg-width="154px" alttext="\Re\mu>0,\Re\nu>-1" display="inline"><mrow><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>μ</mi></mrow><mo>></mo><mn>0</mn></mrow><mo>,</mo><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo>></mo><mrow><mo>-</mo><mn>1</mn></mrow></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E34.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m163.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m128.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./10.1#p2.t1.r3" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./10.1#p2.t1.r3">x</mi></math>: real variable</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">11.3.40</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E35" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.22.35</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E35.png" altimg-height="53px" altimg-valign="-21px" altimg-width="372px" alttext="\int_{0}^{x}\frac{J_{\mu}\left(t\right)J_{\nu}\left(x-t\right)\mathrm{d}t}{t(x%
-t)}=\frac{(\mu+\nu)J_{\mu+\nu}\left(x\right)}{\mu\nu x}," display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi href="./10.1#p2.t1.r3">x</mi></msubsup><mfrac><mrow><mrow><msub><mi href="./10.2#E2">J</mi><mi>μ</mi></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi href="./10.1#p2.t1.r3">x</mi><mo>-</mo><mi>t</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow><mrow><mi>t</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./10.1#p2.t1.r3">x</mi><mo>-</mo><mi>t</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mfrac></mrow><mo>=</mo><mfrac><mrow><mrow><mo stretchy="false">(</mo><mrow><mi>μ</mi><mo>+</mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mrow><mi>μ</mi><mo>+</mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mrow><mrow><mi>μ</mi><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi><mo></mo><mi href="./10.1#p2.t1.r3">x</mi></mrow></mfrac></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m138.png" altimg-height="21px" altimg-valign="-6px" altimg-width="138px" alttext="\Re\mu>0,\Re\nu>0" display="inline"><mrow><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>μ</mi></mrow><mo>></mo><mn>0</mn></mrow><mo>,</mo><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo>></mo><mn>0</mn></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E35.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m163.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m128.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./10.1#p2.t1.r3" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./10.1#p2.t1.r3">x</mi></math>: real variable</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">11.3.41</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px5" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Fractional Integral</h3>
<div id="Px5.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px5.p1" class="ltx_para">
<table id="E36" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.22.36</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E36.png" altimg-height="64px" altimg-valign="-28px" altimg-width="485px" alttext="\frac{1}{\Gamma\left(\alpha\right)}\int_{0}^{x}(x-t)^{\alpha-1}J_{\nu}\left(t%
\right)\mathrm{d}t=2^{\alpha}\sum_{k=0}^{\infty}\frac{(\alpha)_{k}}{k!}J_{\nu+%
\alpha+2k}\left(x\right)," display="block"><mrow><mrow><mrow><mfrac><mn>1</mn><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi href="./10.22#Px5.p1">α</mi><mo>)</mo></mrow></mrow></mfrac><mo></mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi href="./10.1#p2.t1.r3">x</mi></msubsup><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./10.1#p2.t1.r3">x</mi><mo>-</mo><mi>t</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mi href="./10.22#Px5.p1">α</mi><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow><mo>=</mo><mrow><msup><mn>2</mn><mi href="./10.22#Px5.p1">α</mi></msup><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./10.1#p2.t1.r2">k</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><mfrac><msub><mrow><mo stretchy="false">(</mo><mi href="./10.22#Px5.p1">α</mi><mo stretchy="false">)</mo></mrow><mi href="./10.1#p2.t1.r2">k</mi></msub><mrow><mi href="./10.1#p2.t1.r2">k</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mfrac><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mi href="./10.22#Px5.p1">α</mi><mo>+</mo><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r2">k</mi></mrow></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m127.png" altimg-height="21px" altimg-valign="-6px" altimg-width="139px" alttext="\Re\alpha>0,\Re\nu\geq 0" display="inline"><mrow><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./10.22#Px5.p1">α</mi></mrow><mo>></mo><mn>0</mn></mrow><mo>,</mo><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo>≥</mo><mn>0</mn></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E36.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m114.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m241.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m163.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m128.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./10.1#p2.t1.r2" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m239.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./10.1#p2.t1.r2">k</mi></math>: nonnegative integer</a>,
<a href="./10.1#p2.t1.r3" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./10.1#p2.t1.r3">x</mi></math>: real variable</a>,
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a> and
<a href="./10.22#Px5.p1" title="Fractional Integral ‣ §10.22(ii) Integrals over Finite Intervals ‣ §10.22 Integrals ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m151.png" altimg-height="13px" altimg-valign="-2px" altimg-width="17px" alttext="\alpha" display="inline"><mi href="./10.22#Px5.p1">α</mi></math>: positive integer</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">11.2.3 and 11.2.4</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">When <math class="ltx_Math" altimg="m150.png" altimg-height="20px" altimg-valign="-6px" altimg-width="167px" alttext="\alpha=m=1,2,3,\ldots" display="inline"><mrow><mrow><mi href="./10.22#Px5.p1">α</mi><mo>=</mo><mi href="./10.1#p2.t1.r1">m</mi><mo>=</mo><mn>1</mn></mrow><mo>,</mo><mrow><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math> the left-hand side of () is
the <math class="ltx_Math" altimg="m240.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./10.1#p2.t1.r1">m</mi></math>th repeated integral of <math class="ltx_Math" altimg="m104.png" altimg-height="23px" altimg-valign="-7px" altimg-width="55px" alttext="J_{\nu}\left(x\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>
(§§</dd>
</dl>
</div>
</div>
<div id="Px6.p1" class="ltx_para">
<p class="ltx_p">If <math class="ltx_Math" altimg="m204.png" altimg-height="18px" altimg-valign="-4px" altimg-width="67px" alttext="\nu>-1" display="inline"><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>></mo><mrow><mo>-</mo><mn>1</mn></mrow></mrow></math>, then</p>
<table id="E37" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.22.37</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E37.png" altimg-height="56px" altimg-valign="-20px" altimg-width="407px" alttext="\int_{0}^{1}tJ_{\nu}\left(j_{\nu,\ell}t\right)J_{\nu}\left(j_{\nu,m}t\right)%
\mathrm{d}t=\tfrac{1}{2}\left(J_{\nu}'\left(j_{\nu,\ell}\right)\right)^{2}%
\delta_{\ell,m}," display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mn>1</mn></msubsup><mrow><mi>t</mi><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mrow><msub><mi>j</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>,</mo><mi mathvariant="normal">ℓ</mi></mrow></msub><mo></mo><mi>t</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mrow><msub><mi>j</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>,</mo><mi href="./10.1#p2.t1.r1">m</mi></mrow></msub><mo></mo><mi>t</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow><mo>=</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><msup><mrow><mo>(</mo><mrow><msubsup><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′</mo></msubsup><mo></mo><mrow><mo>(</mo><msub><mi>j</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>,</mo><mi mathvariant="normal">ℓ</mi></mrow></msub><mo>)</mo></mrow></mrow><mo>)</mo></mrow><mn>2</mn></msup><mo></mo><msub><mi href="./front/introduction#Sx4.p1.t1.r4">δ</mi><mrow><mi mathvariant="normal">ℓ</mi><mo href="./front/introduction#Sx4.p1.t1.r4">,</mo><mi href="./10.1#p2.t1.r1">m</mi></mrow></msub></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E37.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./front/introduction#Sx4.p1.t1.r4" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m158.png" altimg-height="23px" altimg-valign="-8px" altimg-width="35px" alttext="\delta_{\NVar{j},\NVar{k}}" display="inline"><msub><mi href="./front/introduction#Sx4.p1.t1.r4">δ</mi><mrow><mi class="ltx_nvar">j</mi><mo href="./front/introduction#Sx4.p1.t1.r4">,</mo><mi class="ltx_nvar">k</mi></mrow></msub></math>: Kronecker delta</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m163.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./10.1#p2.t1.r1" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m240.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./10.1#p2.t1.r1">m</mi></math>: integer</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">11.4.5</span></span>
</dd>
<dt>Clarification (effective with 1.0.17):</dt>
<dd>
The Kronecker delta symbol has been moved furthest to the right.
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m237.png" altimg-height="22px" altimg-valign="-8px" altimg-width="33px" alttext="j_{\nu,\ell}" display="inline"><msub><mi>j</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>,</mo><mi mathvariant="normal">ℓ</mi></mrow></msub></math> and <math class="ltx_Math" altimg="m238.png" altimg-height="22px" altimg-valign="-8px" altimg-width="40px" alttext="j_{\nu,m}" display="inline"><msub><mi>j</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>,</mo><mi href="./10.1#p2.t1.r1">m</mi></mrow></msub></math> are zeros of <math class="ltx_Math" altimg="m104.png" altimg-height="23px" altimg-valign="-7px" altimg-width="55px" alttext="J_{\nu}\left(x\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>
(§), and <math class="ltx_Math" altimg="m159.png" altimg-height="23px" altimg-valign="-8px" altimg-width="40px" alttext="\delta_{\ell,m}" display="inline"><msub><mi href="./front/introduction#Sx4.p1.t1.r4">δ</mi><mrow><mi mathvariant="normal">ℓ</mi><mo href="./front/introduction#Sx4.p1.t1.r4">,</mo><mi href="./10.1#p2.t1.r1">m</mi></mrow></msub></math> is Kronecker’s symbol.</p>
</div>
<div id="Px6.p2" class="ltx_para">
<p class="ltx_p">Also, if <math class="ltx_Math" altimg="m221.png" altimg-height="21px" altimg-valign="-6px" altimg-width="52px" alttext="a,b,\nu" display="inline"><mrow><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow></math> are real constants with <math class="ltx_Math" altimg="m234.png" altimg-height="21px" altimg-valign="-6px" altimg-width="49px" alttext="b\neq 0" display="inline"><mrow><mi>b</mi><mo>≠</mo><mn>0</mn></mrow></math> and <math class="ltx_Math" altimg="m204.png" altimg-height="18px" altimg-valign="-4px" altimg-width="67px" alttext="\nu>-1" display="inline"><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>></mo><mrow><mo>-</mo><mn>1</mn></mrow></mrow></math>, then</p>
<table id="E38" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.22.38</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E38.png" altimg-height="56px" altimg-valign="-22px" altimg-width="517px" alttext="\int_{0}^{1}tJ_{\nu}\left(\alpha_{\ell}t\right)J_{\nu}\left(\alpha_{m}t\right)%
\mathrm{d}t=\left(\frac{a^{2}}{b^{2}}+\alpha_{\ell}^{2}-\nu^{2}\right)\frac{(J%
_{\nu}\left(\alpha_{\ell}\right))^{2}}{2\alpha_{\ell}^{2}}\delta_{\ell,m}," display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mn>1</mn></msubsup><mrow><mi>t</mi><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mrow><msub><mi>α</mi><mi mathvariant="normal">ℓ</mi></msub><mo></mo><mi>t</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mrow><msub><mi>α</mi><mi href="./10.1#p2.t1.r1">m</mi></msub><mo></mo><mi>t</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow><mo>=</mo><mrow><mrow><mo>(</mo><mrow><mrow><mfrac><msup><mi>a</mi><mn>2</mn></msup><msup><mi>b</mi><mn>2</mn></msup></mfrac><mo>+</mo><msubsup><mi>α</mi><mi mathvariant="normal">ℓ</mi><mn>2</mn></msubsup></mrow><mo>-</mo><msup><mi href="./10.1#p2.t1.r5">ν</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow><mo></mo><mfrac><msup><mrow><mo stretchy="false">(</mo><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><msub><mi>α</mi><mi mathvariant="normal">ℓ</mi></msub><mo>)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mrow><mn>2</mn><mo></mo><msubsup><mi>α</mi><mi mathvariant="normal">ℓ</mi><mn>2</mn></msubsup></mrow></mfrac><mo></mo><msub><mi href="./front/introduction#Sx4.p1.t1.r4">δ</mi><mrow><mi mathvariant="normal">ℓ</mi><mo href="./front/introduction#Sx4.p1.t1.r4">,</mo><mi href="./10.1#p2.t1.r1">m</mi></mrow></msub></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E38.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./front/introduction#Sx4.p1.t1.r4" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m158.png" altimg-height="23px" altimg-valign="-8px" altimg-width="35px" alttext="\delta_{\NVar{j},\NVar{k}}" display="inline"><msub><mi href="./front/introduction#Sx4.p1.t1.r4">δ</mi><mrow><mi class="ltx_nvar">j</mi><mo href="./front/introduction#Sx4.p1.t1.r4">,</mo><mi class="ltx_nvar">k</mi></mrow></msub></math>: Kronecker delta</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m163.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./10.1#p2.t1.r1" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m240.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./10.1#p2.t1.r1">m</mi></math>: integer</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">11.4.5</span></span>
</dd>
<dt>Clarification (effective with 1.0.17):</dt>
<dd>
The Kronecker delta symbol has been moved furthest to the right.
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m152.png" altimg-height="16px" altimg-valign="-5px" altimg-width="25px" alttext="\alpha_{\ell}" display="inline"><msub><mi>α</mi><mi mathvariant="normal">ℓ</mi></msub></math> and <math class="ltx_Math" altimg="m153.png" altimg-height="16px" altimg-valign="-5px" altimg-width="32px" alttext="\alpha_{m}" display="inline"><msub><mi>α</mi><mi href="./10.1#p2.t1.r1">m</mi></msub></math> are positive zeros of
<math class="ltx_Math" altimg="m226.png" altimg-height="24px" altimg-valign="-7px" altimg-width="161px" alttext="aJ_{\nu}\left(x\right)+bxJ_{\nu}'\left(x\right)" display="inline"><mrow><mrow><mi>a</mi><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mi>b</mi><mo></mo><mi href="./10.1#p2.t1.r3">x</mi><mo></mo><mrow><msubsup><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′</mo></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mrow></mrow></math>. (Compare
()).
</p>
</div>
</section>
</section>
<section id="iii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§10.22(iii) </span>Integrals over the Interval <math class="ltx_Math" altimg="m89.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="(x,\infty)" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi mathvariant="normal">∞</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math>
</h2>
<div id="SS3.info" class="ltx_metadata ltx_info">
) with <math class="ltx_Math" altimg="m202.png" altimg-height="17px" altimg-valign="-2px" altimg-width="52px" alttext="\nu=0" display="inline"><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>=</mo><mn>0</mn></mrow></math> and <math class="ltx_Math" altimg="m196.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\mu" display="inline"><mi>μ</mi></math> replaced by <math class="ltx_Math" altimg="m190.png" altimg-height="20px" altimg-valign="-6px" altimg-width="51px" alttext="\mu-1" display="inline"><mrow><mi>μ</mi><mo>-</mo><mn>1</mn></mrow></math>, split the
integration range at <math class="ltx_Math" altimg="m250.png" altimg-height="17px" altimg-valign="-2px" altimg-width="49px" alttext="t=x" display="inline"><mrow><mi>t</mi><mo>=</mo><mi href="./10.1#p2.t1.r3">x</mi></mrow></math> and take limits as <math class="ltx_Math" altimg="m197.png" altimg-height="20px" altimg-valign="-6px" altimg-width="57px" alttext="\mu\to 0" display="inline"><mrow><mi>μ</mi><mo>→</mo><mn>0</mn></mrow></math>; for the second
result substitute into the first result by () with <math class="ltx_Math" altimg="m202.png" altimg-height="17px" altimg-valign="-2px" altimg-width="52px" alttext="\nu=0" display="inline"><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>=</mo><mn>0</mn></mrow></math> for the term-by-term
integration.</dd>
</dl>
</div>
</div>
<div id="SS3.p1" class="ltx_para">
<p class="ltx_p">When <math class="ltx_Math" altimg="m254.png" altimg-height="17px" altimg-valign="-3px" altimg-width="52px" alttext="x>0" display="inline"><mrow><mi href="./10.1#p2.t1.r3">x</mi><mo>></mo><mn>0</mn></mrow></math></p>
<table id="E39" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.22.39</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E39.png" altimg-height="64px" altimg-valign="-28px" altimg-width="600px" alttext="\int_{x}^{\infty}\frac{J_{0}\left(t\right)}{t}\mathrm{d}t+\gamma+\ln\left(%
\tfrac{1}{2}x\right)=\int_{0}^{x}\frac{1-J_{0}\left(t\right)}{t}\mathrm{d}t=%
\sum_{k=1}^{\infty}(-1)^{k-1}\frac{(\frac{1}{2}x)^{2k}}{2k(k!)^{2}}," display="block"><mrow><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi href="./10.1#p2.t1.r3">x</mi><mi mathvariant="normal">∞</mi></msubsup><mrow><mfrac><mrow><msub><mi href="./10.2#E2">J</mi><mn>0</mn></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow><mi>t</mi></mfrac><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow><mo>+</mo><mi href="./5.2#E3">γ</mi><mo>+</mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><mo>(</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./10.1#p2.t1.r3">x</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi href="./10.1#p2.t1.r3">x</mi></msubsup><mrow><mfrac><mrow><mn>1</mn><mo>-</mo><mrow><msub><mi href="./10.2#E2">J</mi><mn>0</mn></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></mrow><mi>t</mi></mfrac><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow><mo>=</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./10.1#p2.t1.r2">k</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mrow><mi href="./10.1#p2.t1.r2">k</mi><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mfrac><msup><mrow><mo stretchy="false">(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./10.1#p2.t1.r3">x</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r2">k</mi></mrow></msup><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r2">k</mi><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./10.1#p2.t1.r2">k</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow></mfrac></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E39.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./5.2#E3" title="(5.2.3) ‣ §5.2(ii) Euler’s Constant ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m162.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\gamma" display="inline"><mi href="./5.2#E3">γ</mi></math>: Euler’s constant</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m241.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m163.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m170.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a>,
<a href="./10.1#p2.t1.r2" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m239.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./10.1#p2.t1.r2">k</mi></math>: nonnegative integer</a> and
<a href="./10.1#p2.t1.r3" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./10.1#p2.t1.r3">x</mi></math>: real variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">11.1.20</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E40" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.22.40</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E40.png" altimg-height="64px" altimg-valign="-28px" altimg-width="793px" alttext="\int_{x}^{\infty}\frac{Y_{0}\left(t\right)}{t}\mathrm{d}t=-\frac{1}{\pi}\left(%
\ln\left(\tfrac{1}{2}x\right)+\gamma\right)^{2}+\frac{\pi}{6}+\frac{2}{\pi}%
\sum_{k=1}^{\infty}(-1)^{k}\*\left(\psi\left(k+1\right)+\frac{1}{2k}-\ln\left(%
\tfrac{1}{2}x\right)\right)\frac{(\tfrac{1}{2}x)^{2k}}{2k(k!)^{2}}," display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi href="./10.1#p2.t1.r3">x</mi><mi mathvariant="normal">∞</mi></msubsup><mrow><mfrac><mrow><msub><mi href="./10.2#E3">Y</mi><mn>0</mn></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow><mi>t</mi></mfrac><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow><mo>=</mo><mrow><mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mi href="./3.12#E1">π</mi></mfrac><mo></mo><msup><mrow><mo>(</mo><mrow><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><mo>(</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./10.1#p2.t1.r3">x</mi></mrow><mo>)</mo></mrow></mrow><mo>+</mo><mi href="./5.2#E3">γ</mi></mrow><mo>)</mo></mrow><mn>2</mn></msup></mrow></mrow><mo>+</mo><mfrac><mi href="./3.12#E1">π</mi><mn>6</mn></mfrac><mo>+</mo><mrow><mfrac><mn>2</mn><mi href="./3.12#E1">π</mi></mfrac><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./10.1#p2.t1.r2">k</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./10.1#p2.t1.r2">k</mi></msup><mo></mo><mrow><mo>(</mo><mrow><mrow><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./10.1#p2.t1.r2">k</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mo>+</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r2">k</mi></mrow></mfrac></mrow><mo>-</mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><mo>(</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./10.1#p2.t1.r3">x</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>)</mo></mrow><mo></mo><mfrac><msup><mrow><mo stretchy="false">(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./10.1#p2.t1.r3">x</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r2">k</mi></mrow></msup><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r2">k</mi><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./10.1#p2.t1.r2">k</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow></mfrac></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E40.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E3" title="(10.2.3) ‣ Bessel Function of the Second Kind (Weber’s Function) ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m112.png" altimg-height="23px" altimg-valign="-7px" altimg-width="55px" alttext="Y_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E3">Y</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the second kind</a>,
<a href="./5.2#E3" title="(5.2.3) ‣ §5.2(ii) Euler’s Constant ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m162.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\gamma" display="inline"><mi href="./5.2#E3">γ</mi></math>: Euler’s constant</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m211.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./5.2#E2" title="(5.2.2) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m212.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="\psi\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: psi (or digamma) function</a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m241.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m163.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m170.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a>,
<a href="./10.1#p2.t1.r2" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m239.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./10.1#p2.t1.r2">k</mi></math>: nonnegative integer</a> and
<a href="./10.1#p2.t1.r3" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./10.1#p2.t1.r3">x</mi></math>: real variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">11.1.21</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m162.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\gamma" display="inline"><mi href="./5.2#E3">γ</mi></math> is Euler’s constant (§).</p>
</div>
</section>
<section id="iv" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§10.22(iv) </span>Integrals over the Interval <math class="ltx_Math" altimg="m88.png" altimg-height="23px" altimg-valign="-7px" altimg-width="59px" alttext="(0,\infty)" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mn>0</mn><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi mathvariant="normal">∞</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math>
</h2>
<div id="SS4.info" class="ltx_metadata ltx_info">
, §13.53)</cite> obtained by setting <math class="ltx_Math" altimg="m194.png" altimg-height="21px" altimg-valign="-6px" altimg-width="88px" alttext="\mu=b=0" display="inline"><mrow><mi>μ</mi><mo>=</mo><mi>b</mi><mo>=</mo><mn>0</mn></mrow></math>,
<math class="ltx_Math" altimg="m214.png" altimg-height="20px" altimg-valign="-6px" altimg-width="87px" alttext="\rho=\nu+1" display="inline"><mrow><mi>ρ</mi><mo>=</mo><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mn>1</mn></mrow></mrow></math>, and subsequently replacing <math class="ltx_Math" altimg="m239.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./10.1#p2.t1.r2">k</mi></math> by <math class="ltx_Math" altimg="m233.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="b" display="inline"><mi>b</mi></math>. For
() with respect
to <math class="ltx_Math" altimg="m196.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\mu" display="inline"><mi>μ</mi></math> and use () with <math class="ltx_Math" altimg="m245.png" altimg-height="17px" altimg-valign="-2px" altimg-width="53px" alttext="n=0" display="inline"><mrow><mi href="./10.1#p2.t1.r1">n</mi><mo>=</mo><mn>0</mn></mrow></math>. For () with <math class="ltx_Math" altimg="m168.png" altimg-height="21px" altimg-valign="-6px" altimg-width="124px" alttext="\lambda=\nu-\mu-1" display="inline"><mrow><mi>λ</mi><mo>=</mo><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>-</mo><mi>μ</mi><mo>-</mo><mn>1</mn></mrow></mrow></math> and () with <math class="ltx_Math" altimg="m193.png" altimg-height="20px" altimg-valign="-6px" altimg-width="91px" alttext="\mu=\nu=0" display="inline"><mrow><mi>μ</mi><mo>=</mo><mi href="./10.1#p2.t1.r5">ν</mi><mo>=</mo><mn>0</mn></mrow></math>, then let <math class="ltx_Math" altimg="m169.png" altimg-height="18px" altimg-valign="-2px" altimg-width="57px" alttext="\lambda\to 1" display="inline"><mrow><mi>λ</mi><mo>→</mo><mn>1</mn></mrow></math>. For
() set <math class="ltx_Math" altimg="m223.png" altimg-height="18px" altimg-valign="-2px" altimg-width="50px" alttext="a=b" display="inline"><mrow><mi>a</mi><mo>=</mo><mi>b</mi></mrow></math>
in (), differentiate with respect to <math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math> and apply
() with <math class="ltx_Math" altimg="m245.png" altimg-height="17px" altimg-valign="-2px" altimg-width="53px" alttext="n=0" display="inline"><mrow><mi href="./10.1#p2.t1.r1">n</mi><mo>=</mo><mn>0</mn></mrow></math>. For
(, p. 429, Eqs. (3),(4), with <math class="ltx_Math" altimg="m192.png" altimg-height="20px" altimg-valign="-6px" altimg-width="88px" alttext="\mu=\nu+1" display="inline"><mrow><mi>μ</mi><mo>=</mo><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mn>1</mn></mrow></mrow></math> in (3))</cite>. For
(</dd>
</dl>
</div>
</div>
<div id="SS4.p1" class="ltx_para">
<table id="EGx8" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E41">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.22.41</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m66.png" altimg-height="53px" altimg-valign="-20px" altimg-width="115px" alttext="\displaystyle\int_{0}^{\infty}J_{\nu}\left(t\right)\mathrm{d}t" display="inline"><mrow><mstyle displaystyle="true"><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup></mstyle><mrow><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m5.png" altimg-height="22px" altimg-valign="-6px" altimg-width="43px" alttext="\displaystyle=1," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mn>1</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m143.png" altimg-height="19px" altimg-valign="-4px" altimg-width="82px" alttext="\Re\nu>-1" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo>></mo><mrow><mo>-</mo><mn>1</mn></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E41.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m163.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m128.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">11.4.17</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E42">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.22.42</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m69.png" altimg-height="53px" altimg-valign="-20px" altimg-width="115px" alttext="\displaystyle\int_{0}^{\infty}Y_{\nu}\left(t\right)\mathrm{d}t" display="inline"><mrow><mstyle displaystyle="true"><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup></mstyle><mrow><mrow><msub><mi href="./10.2#E3">Y</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m3.png" altimg-height="29px" altimg-valign="-9px" altimg-width="141px" alttext="\displaystyle=-\tan\left(\tfrac{1}{2}\nu\pi\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo>-</mo><mrow><mi href="./4.14#E4">tan</mi><mo></mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m264.png" altimg-height="23px" altimg-valign="-7px" altimg-width="77px" alttext="|\Re\nu|<1" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo stretchy="false">|</mo></mrow><mo><</mo><mn>1</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E42.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E3" title="(10.2.3) ‣ Bessel Function of the Second Kind (Weber’s Function) ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m112.png" altimg-height="23px" altimg-valign="-7px" altimg-width="55px" alttext="Y_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E3">Y</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the second kind</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m211.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m163.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m128.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./4.14#E4" title="(4.14.4) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="17px" altimg-valign="-2px" altimg-width="47px" alttext="\tan\NVar{z}" display="inline"><mrow><mi href="./4.14#E4">tan</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: tangent function</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">11.4.20, 11.4.21</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
<table id="EGx9" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E43">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.22.43</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m75.png" altimg-height="53px" altimg-valign="-20px" altimg-width="133px" alttext="\displaystyle\int_{0}^{\infty}t^{\mu}J_{\nu}\left(t\right)\mathrm{d}t" display="inline"><mrow><mstyle displaystyle="true"><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup></mstyle><mrow><msup><mi>t</mi><mi>μ</mi></msup><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m9.png" altimg-height="59px" altimg-valign="-24px" altimg-width="202px" alttext="\displaystyle=2^{\mu}\frac{\Gamma\left(\tfrac{1}{2}\nu+\tfrac{1}{2}\mu+\tfrac{%
1}{2}\right)}{\Gamma\left(\tfrac{1}{2}\nu-\tfrac{1}{2}\mu+\tfrac{1}{2}\right)}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msup><mn>2</mn><mi>μ</mi></msup><mo></mo><mstyle displaystyle="true"><mfrac><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo>+</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi>μ</mi></mrow><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi>μ</mi></mrow></mrow><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow></mfrac></mstyle></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m119.png" altimg-height="23px" altimg-valign="-7px" altimg-width="134px" alttext="\Re(\mu+\nu)>-1" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>μ</mi><mo>+</mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>></mo><mrow><mo>-</mo><mn>1</mn></mrow></mrow></math>, <math class="ltx_Math" altimg="m129.png" altimg-height="27px" altimg-valign="-9px" altimg-width="70px" alttext="\Re\mu<\tfrac{1}{2}" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>μ</mi></mrow><mo><</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E43.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m114.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m163.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m128.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">11.4.16</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E44">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.22.44</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m76.png" altimg-height="53px" altimg-valign="-20px" altimg-width="133px" alttext="\displaystyle\int_{0}^{\infty}t^{\mu}Y_{\nu}\left(t\right)\mathrm{d}t" display="inline"><mrow><mstyle displaystyle="true"><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup></mstyle><mrow><msup><mi>t</mi><mi>μ</mi></msup><mo></mo><mrow><msub><mi href="./10.2#E3">Y</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m22.png" altimg-height="46px" altimg-valign="-16px" altimg-width="487px" alttext="\displaystyle=\frac{2^{\mu}}{\pi}\Gamma\left(\tfrac{1}{2}\mu+\tfrac{1}{2}\nu+%
\tfrac{1}{2}\right)\Gamma\left(\tfrac{1}{2}\mu-\tfrac{1}{2}\nu+\tfrac{1}{2}%
\right)\sin\left(\tfrac{1}{2}\mu-\tfrac{1}{2}\nu\right)\pi," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><mfrac><msup><mn>2</mn><mi>μ</mi></msup><mi href="./3.12#E1">π</mi></mfrac></mstyle><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi>μ</mi></mrow><mo>+</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi>μ</mi></mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow></mrow><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi>μ</mi></mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow></mrow><mo>)</mo></mrow></mrow><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m125.png" altimg-height="23px" altimg-valign="-7px" altimg-width="134px" alttext="\Re(\mu\pm\nu)>-1" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>μ</mi><mo>±</mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>></mo><mrow><mo>-</mo><mn>1</mn></mrow></mrow></math>, <math class="ltx_Math" altimg="m129.png" altimg-height="27px" altimg-valign="-9px" altimg-width="70px" alttext="\Re\mu<\tfrac{1}{2}" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>μ</mi></mrow><mo><</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E44.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E3" title="(10.2.3) ‣ Bessel Function of the Second Kind (Weber’s Function) ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m112.png" altimg-height="23px" altimg-valign="-7px" altimg-width="55px" alttext="Y_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E3">Y</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the second kind</a>,
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m114.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m211.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m163.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m128.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m216.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">11.4.19</span> (modified)</span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
<table id="E45" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.22.45</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E45.png" altimg-height="59px" altimg-valign="-24px" altimg-width="327px" alttext="\int_{0}^{\infty}\frac{1-J_{0}\left(t\right)}{t^{\mu}}\mathrm{d}t=-\frac{\pi%
\sec\left(\frac{1}{2}\mu\pi\right)}{2^{\mu}{\Gamma^{2}}\left(\frac{1}{2}\mu+%
\frac{1}{2}\right)}," display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><mrow><mfrac><mrow><mn>1</mn><mo>-</mo><mrow><msub><mi href="./10.2#E2">J</mi><mn>0</mn></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></mrow><msup><mi>t</mi><mi>μ</mi></msup></mfrac><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow><mo>=</mo><mrow><mo>-</mo><mfrac><mrow><mi href="./3.12#E1">π</mi><mo></mo><mrow><mi href="./4.14#E6">sec</mi><mo></mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi>μ</mi><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo>)</mo></mrow></mrow></mrow><mrow><msup><mn>2</mn><mi>μ</mi></msup><mo></mo><mrow><msup><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mn>2</mn></msup><mo></mo><mrow><mo>(</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi>μ</mi></mrow><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow></mrow></mfrac></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m96.png" altimg-height="21px" altimg-valign="-6px" altimg-width="104px" alttext="1<\Re\mu<3" display="inline"><mrow><mn>1</mn><mo><</mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>μ</mi></mrow><mo><</mo><mn>3</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E45.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m114.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m211.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m163.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m128.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a> and
<a href="./4.14#E6" title="(4.14.6) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m215.png" altimg-height="13px" altimg-valign="-2px" altimg-width="43px" alttext="\sec\NVar{z}" display="inline"><mrow><mi href="./4.14#E6">sec</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: secant function</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">11.4.18</span> (modified)</span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E46" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.22.46</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E46.png" altimg-height="55px" altimg-valign="-21px" altimg-width="399px" alttext="\int_{0}^{\infty}\frac{t^{\nu+1}J_{\nu}\left(at\right)}{(t^{2}+b^{2})^{\mu+1}}%
\mathrm{d}t=\frac{a^{\mu}b^{\nu-\mu}}{2^{\mu}\Gamma\left(\mu+1\right)}K_{\nu-%
\mu}\left(ab\right)," display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><mrow><mfrac><mrow><msup><mi>t</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mn>1</mn></mrow></msup><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi>a</mi><mo></mo><mi>t</mi></mrow><mo>)</mo></mrow></mrow></mrow><msup><mrow><mo stretchy="false">(</mo><mrow><msup><mi>t</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mrow><mi>μ</mi><mo>+</mo><mn>1</mn></mrow></msup></mfrac><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow><mo>=</mo><mrow><mfrac><mrow><msup><mi>a</mi><mi>μ</mi></msup><mo></mo><msup><mi>b</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>-</mo><mi>μ</mi></mrow></msup></mrow><mrow><msup><mn>2</mn><mi>μ</mi></msup><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi>μ</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mrow></mfrac><mo></mo><mrow><msub><mi href="./10.25#E3">K</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>-</mo><mi>μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mrow><mi>a</mi><mo></mo><mi>b</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m225.png" altimg-height="17px" altimg-valign="-3px" altimg-width="51px" alttext="a>0" display="inline"><mrow><mi>a</mi><mo>></mo><mn>0</mn></mrow></math>,
<math class="ltx_Math" altimg="m116.png" altimg-height="18px" altimg-valign="-3px" altimg-width="64px" alttext="\Re b>0" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>b</mi></mrow><mo>></mo><mn>0</mn></mrow></math>,
<math class="ltx_Math" altimg="m90.png" altimg-height="27px" altimg-valign="-9px" altimg-width="182px" alttext="-1<\Re\nu<2\Re\mu+\tfrac{3}{2}" display="inline"><mrow><mrow><mo>-</mo><mn>1</mn></mrow><mo><</mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo><</mo><mrow><mrow><mn>2</mn><mo></mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>μ</mi></mrow></mrow><mo>+</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E46.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m114.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m163.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./10.25#E3" title="(10.25.3) ‣ §10.25(ii) Standard Solutions ‣ §10.25 Definitions ‣ Modified Bessel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m107.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="K_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.25#E3">K</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: modified Bessel function of the second kind</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m128.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">11.4.44</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E47" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.22.47</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E47.png" altimg-height="52px" altimg-valign="-20px" altimg-width="299px" alttext="\int_{0}^{\infty}\frac{t^{\nu}Y_{\nu}\left(at\right)}{t^{2}+b^{2}}\mathrm{d}t=%
-b^{\nu-1}K_{\nu}\left(ab\right)," display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><mrow><mfrac><mrow><msup><mi>t</mi><mi href="./10.1#p2.t1.r5">ν</mi></msup><mo></mo><mrow><msub><mi href="./10.2#E3">Y</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi>a</mi><mo></mo><mi>t</mi></mrow><mo>)</mo></mrow></mrow></mrow><mrow><msup><mi>t</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></mfrac><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow><mo>=</mo><mrow><mo>-</mo><mrow><msup><mi>b</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mrow><msub><mi href="./10.25#E3">K</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi>a</mi><mo></mo><mi>b</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m224.png" altimg-height="27px" altimg-valign="-9px" altimg-width="248px" alttext="a>0,\Re b>0,-\tfrac{1}{2}<\Re\nu<\tfrac{5}{2}" display="inline"><mrow><mrow><mi>a</mi><mo>></mo><mn>0</mn></mrow><mo>,</mo><mrow><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>b</mi></mrow><mo>></mo><mn>0</mn></mrow><mo>,</mo><mrow><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo><</mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo><</mo><mfrac><mn>5</mn><mn>2</mn></mfrac></mrow></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E47.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E3" title="(10.2.3) ‣ Bessel Function of the Second Kind (Weber’s Function) ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m112.png" altimg-height="23px" altimg-valign="-7px" altimg-width="55px" alttext="Y_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E3">Y</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the second kind</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m163.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./10.25#E3" title="(10.25.3) ‣ §10.25(ii) Standard Solutions ‣ §10.25 Definitions ‣ Modified Bessel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m107.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="K_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.25#E3">K</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: modified Bessel function of the second kind</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m128.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">11.4.46</span> (is the special case <math class="ltx_Math" altimg="m202.png" altimg-height="17px" altimg-valign="-2px" altimg-width="52px" alttext="\nu=0" display="inline"><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>=</mo><mn>0</mn></mrow></math>.)</span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">For <math class="ltx_Math" altimg="m108.png" altimg-height="21px" altimg-valign="-5px" altimg-width="31px" alttext="K_{\nu}" display="inline"><msub><mi href="./10.25#E3">K</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub></math> see §.</p>
<table id="E48" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.22.48</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E48.png" altimg-height="53px" altimg-valign="-20px" altimg-width="668px" alttext="\int_{0}^{\infty}J_{\mu}\left(x\cosh\phi\right)(\cosh\phi)^{1-\mu}(\sinh\phi)^%
{2\nu+1}\mathrm{d}\phi=2^{\nu}\Gamma\left(\nu+1\right)x^{-\nu-1}J_{\mu-\nu-1}%
\left(x\right)," display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><mrow><mrow><msub><mi href="./10.2#E2">J</mi><mi>μ</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi href="./10.1#p2.t1.r3">x</mi><mo></mo><mrow><mi href="./4.28#E2">cosh</mi><mo></mo><mi>ϕ</mi></mrow></mrow><mo>)</mo></mrow></mrow><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./4.28#E2">cosh</mi><mo></mo><mi>ϕ</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>-</mo><mi>μ</mi></mrow></msup><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./4.28#E1">sinh</mi><mo></mo><mi>ϕ</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo>+</mo><mn>1</mn></mrow></msup><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>ϕ</mi></mrow></mrow></mrow><mo>=</mo><mrow><msup><mn>2</mn><mi href="./10.1#p2.t1.r5">ν</mi></msup><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mo></mo><msup><mi href="./10.1#p2.t1.r3">x</mi><mrow><mrow><mo>-</mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mrow><mi>μ</mi><mo>-</mo><mi href="./10.1#p2.t1.r5">ν</mi><mo>-</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m253.png" altimg-height="27px" altimg-valign="-9px" altimg-width="273px" alttext="x>0,\Re\nu>-1,\Re\mu>2\Re\nu+\tfrac{1}{2}" display="inline"><mrow><mrow><mi href="./10.1#p2.t1.r3">x</mi><mo>></mo><mn>0</mn></mrow><mo>,</mo><mrow><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo>></mo><mrow><mo>-</mo><mn>1</mn></mrow></mrow><mo>,</mo><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>μ</mi></mrow><mo>></mo><mrow><mrow><mn>2</mn><mo></mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow></mrow><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E48.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m114.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.28#E2" title="(4.28.2) ‣ §4.28 Definitions and Periodicity ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m155.png" altimg-height="18px" altimg-valign="-2px" altimg-width="56px" alttext="\cosh\NVar{z}" display="inline"><mrow><mi href="./4.28#E2">cosh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: hyperbolic cosine function</a>,
<a href="./4.28#E1" title="(4.28.1) ‣ §4.28 Definitions and Periodicity ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m217.png" altimg-height="18px" altimg-valign="-2px" altimg-width="53px" alttext="\sinh\NVar{z}" display="inline"><mrow><mi href="./4.28#E1">sinh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: hyperbolic sine function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m163.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m128.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./10.1#p2.t1.r3" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./10.1#p2.t1.r3">x</mi></math>: real variable</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E49" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.22.49</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E49.png" altimg-height="56px" altimg-valign="-21px" altimg-width="669px" alttext="\int_{0}^{\infty}t^{\mu-1}e^{-at}J_{\nu}\left(bt\right)\mathrm{d}t=\frac{(%
\tfrac{1}{2}b)^{\nu}}{a^{\mu+\nu}}\Gamma\left(\mu+\nu\right)\*\mathbf{F}\left(%
\frac{\mu+\nu}{2},\frac{\mu+\nu+1}{2};\nu+1;-\frac{b^{2}}{a^{2}}\right)," display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><mrow><msup><mi>t</mi><mrow><mi>μ</mi><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mi>a</mi><mo></mo><mi>t</mi></mrow></mrow></msup><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi>b</mi><mo></mo><mi>t</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow><mo>=</mo><mrow><mfrac><msup><mrow><mo stretchy="false">(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi>b</mi></mrow><mo stretchy="false">)</mo></mrow><mi href="./10.1#p2.t1.r5">ν</mi></msup><msup><mi>a</mi><mrow><mi>μ</mi><mo>+</mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow></msup></mfrac><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi>μ</mi><mo>+</mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./15.2#E2" mathvariant="bold">F</mi><mo></mo><mrow><mo>(</mo><mfrac><mrow><mi>μ</mi><mo>+</mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mn>2</mn></mfrac><mo>,</mo><mfrac><mrow><mi>μ</mi><mo>+</mo><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mn>1</mn></mrow><mn>2</mn></mfrac><mo>;</mo><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mn>1</mn></mrow><mo>;</mo><mrow><mo>-</mo><mfrac><msup><mi>b</mi><mn>2</mn></msup><msup><mi>a</mi><mn>2</mn></msup></mfrac></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m120.png" altimg-height="23px" altimg-valign="-7px" altimg-width="244px" alttext="\Re(\mu+\nu)>0,\Re(a\pm ib)>0" display="inline"><mrow><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>μ</mi><mo>+</mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>></mo><mn>0</mn></mrow><mo>,</mo><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>a</mi><mo>±</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi>b</mi></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>></mo><mn>0</mn></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E49.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m114.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m177.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m163.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m128.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./15.2#E2" title="(15.2.2) ‣ §15.2(i) Gauss Series ‣ §15.2 Definitions and Analytical Properties ‣ Properties ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m172.png" altimg-height="23px" altimg-valign="-7px" altimg-width="102px" alttext="\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E2" mathvariant="bold">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m173.png" altimg-height="39px" altimg-valign="-15px" altimg-width="90px" alttext="\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E2" mathvariant="bold">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi></mfrac><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: Olver’s hypergeometric function</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E50" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.22.50</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_Math ltx_eqn_cell ltx_align_center"><math class="ltx_Math" alttext="\int_{0}^{\infty}t^{\mu-1}e^{-at}Y_{\nu}\left(bt\right)\mathrm{d}t=\cot(\nu\pi%
)\frac{(\tfrac{1}{2}b)^{\nu}\Gamma\left(\mu+\nu\right)}{(a^{2}+b^{2})^{\frac{1%
}{2}(\mu+\nu)}}\*\mathbf{F}\left(\frac{\mu+\nu}{2},\frac{1-\mu+\nu}{2};\nu+1;%
\frac{b^{2}}{a^{2}+b^{2}}\right)-\csc(\nu\pi)\frac{(\tfrac{1}{2}b)^{-\nu}%
\Gamma\left(\mu-\nu\right)}{(a^{2}+b^{2})^{\frac{1}{2}(\mu-\nu)}}\*\mathbf{F}%
\left(\frac{\mu-\nu}{2},\frac{1-\mu-\nu}{2};1-\nu;\frac{b^{2}}{a^{2}+b^{2}}%
\right)," display="block"><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><mrow><msup><mi>t</mi><mrow><mi>μ</mi><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mi>a</mi><mo></mo><mi>t</mi></mrow></mrow></msup><mo></mo><mrow><msub><mi href="./10.2#E3">Y</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi>b</mi><mo></mo><mi>t</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>=</mo><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mrow><mrow><mi href="./4.14#E7">cot</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mfrac><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi>b</mi></mrow><mo stretchy="false">)</mo></mrow><mi href="./10.1#p2.t1.r5">ν</mi></msup><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi>μ</mi><mo>+</mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo>)</mo></mrow></mrow></mrow><msup><mrow><mo stretchy="false">(</mo><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>μ</mi><mo>+</mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></msup></mfrac><mo></mo><mrow><mi href="./15.2#E2" mathvariant="bold">F</mi><mo></mo><mrow><mo>(</mo><mfrac><mrow><mi>μ</mi><mo>+</mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mn>2</mn></mfrac><mo>,</mo><mfrac><mrow><mrow><mn>1</mn><mo>-</mo><mi>μ</mi></mrow><mo>+</mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mn>2</mn></mfrac><mo>;</mo><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mn>1</mn></mrow><mo>;</mo><mfrac><msup><mi>b</mi><mn>2</mn></msup><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></mfrac><mo>)</mo></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>-</mo><mrow><mrow><mi href="./4.14#E5">csc</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mfrac><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi>b</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow></msup><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi>μ</mi><mo>-</mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo>)</mo></mrow></mrow></mrow><msup><mrow><mo stretchy="false">(</mo><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>μ</mi><mo>-</mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></msup></mfrac><mo></mo><mrow><mi href="./15.2#E2" mathvariant="bold">F</mi><mo></mo><mrow><mo>(</mo><mfrac><mrow><mi>μ</mi><mo>-</mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mn>2</mn></mfrac><mo>,</mo><mfrac><mrow><mn>1</mn><mo>-</mo><mi>μ</mi><mo>-</mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mn>2</mn></mfrac><mo>;</mo><mrow><mn>1</mn><mo>-</mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo>;</mo><mfrac><msup><mi>b</mi><mn>2</mn></msup><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></mfrac><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mtd></mtr></mtable></mrow></mtd></mtr></mtable></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m140.png" altimg-height="23px" altimg-valign="-7px" altimg-width="219px" alttext="\Re\mu>|\Re\nu|,\Re(a\pm ib)>0" display="inline"><mrow><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>μ</mi></mrow><mo>></mo><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo stretchy="false">|</mo></mrow></mrow><mo>,</mo><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>a</mi><mo>±</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi>b</mi></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>></mo><mn>0</mn></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E50.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E3" title="(10.2.3) ‣ Bessel Function of the Second Kind (Weber’s Function) ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m112.png" altimg-height="23px" altimg-valign="-7px" altimg-width="55px" alttext="Y_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E3">Y</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the second kind</a>,
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m114.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m211.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.14#E5" title="(4.14.5) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m157.png" altimg-height="13px" altimg-valign="-2px" altimg-width="43px" alttext="\csc\NVar{z}" display="inline"><mrow><mi href="./4.14#E5">csc</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosecant function</a>,
<a href="./4.14#E7" title="(4.14.7) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m156.png" altimg-height="17px" altimg-valign="-2px" altimg-width="44px" alttext="\cot\NVar{z}" display="inline"><mrow><mi href="./4.14#E7">cot</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cotangent function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m177.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m163.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m128.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./15.2#E2" title="(15.2.2) ‣ §15.2(i) Gauss Series ‣ §15.2 Definitions and Analytical Properties ‣ Properties ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m172.png" altimg-height="23px" altimg-valign="-7px" altimg-width="102px" alttext="\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E2" mathvariant="bold">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m173.png" altimg-height="39px" altimg-valign="-15px" altimg-width="90px" alttext="\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E2" mathvariant="bold">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi></mfrac><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: Olver’s hypergeometric function</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">For the hypergeometric function <math class="ltx_Math" altimg="m171.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\mathbf{F}" display="inline"><mi href="./15.2#E2" mathvariant="bold">F</mi></math> see §.</p>
<table id="EGx10" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E51">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.22.51</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m65.png" altimg-height="53px" altimg-valign="-20px" altimg-width="267px" alttext="\displaystyle\int_{0}^{\infty}J_{\nu}\left(bt\right)\exp\left(-p^{2}t^{2}%
\right)t^{\nu+1}\mathrm{d}t" display="inline"><mrow><mstyle displaystyle="true"><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup></mstyle><mrow><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi>b</mi><mo></mo><mi>t</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mrow><msup><mi>p</mi><mn>2</mn></msup><mo></mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mrow><mo>)</mo></mrow></mrow><mo></mo><msup><mi>t</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mn>1</mn></mrow></msup><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m25.png" altimg-height="54px" altimg-valign="-21px" altimg-width="231px" alttext="\displaystyle=\frac{b^{\nu}}{(2p^{2})^{\nu+1}}\exp\left(-\frac{b^{2}}{4p^{2}}%
\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><mfrac><msup><mi>b</mi><mi href="./10.1#p2.t1.r5">ν</mi></msup><msup><mrow><mo stretchy="false">(</mo><mrow><mn>2</mn><mo></mo><msup><mi>p</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mn>1</mn></mrow></msup></mfrac></mstyle><mo></mo><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mstyle displaystyle="true"><mfrac><msup><mi>b</mi><mn>2</mn></msup><mrow><mn>4</mn><mo></mo><msup><mi>p</mi><mn>2</mn></msup></mrow></mfrac></mstyle></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m143.png" altimg-height="19px" altimg-valign="-4px" altimg-width="82px" alttext="\Re\nu>-1" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo>></mo><mrow><mo>-</mo><mn>1</mn></mrow></mrow></math>, <math class="ltx_Math" altimg="m126.png" altimg-height="25px" altimg-valign="-7px" altimg-width="90px" alttext="\Re(p^{2})>0" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mrow><mo stretchy="false">(</mo><msup><mi>p</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow></mrow><mo>></mo><mn>0</mn></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E51.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E19" title="(4.2.19) ‣ §4.2(iii) The Exponential Function ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m160.png" altimg-height="16px" altimg-valign="-6px" altimg-width="48px" alttext="\exp\NVar{z}" display="inline"><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: exponential function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m163.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m128.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">11.4.29</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E52">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.22.52</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m64.png" altimg-height="53px" altimg-valign="-20px" altimg-width="220px" alttext="\displaystyle\int_{0}^{\infty}J_{\nu}\left(bt\right)\exp(-p^{2}t^{2})\mathrm{d}t" display="inline"><mrow><mstyle displaystyle="true"><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup></mstyle><mrow><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi>b</mi><mo></mo><mi>t</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mrow><msup><mi>p</mi><mn>2</mn></msup><mo></mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m24.png" altimg-height="54px" altimg-valign="-21px" altimg-width="294px" alttext="\displaystyle=\frac{\sqrt{\pi}}{2p}\exp\left(-\frac{b^{2}}{8p^{2}}\right)I_{%
\ifrac{\nu}{2}}\left(\frac{b^{2}}{8p^{2}}\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><mfrac><msqrt><mi href="./3.12#E1">π</mi></msqrt><mrow><mn>2</mn><mo></mo><mi>p</mi></mrow></mfrac></mstyle><mo></mo><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mstyle displaystyle="true"><mfrac><msup><mi>b</mi><mn>2</mn></msup><mrow><mn>8</mn><mo></mo><msup><mi>p</mi><mn>2</mn></msup></mrow></mfrac></mstyle></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./10.25#E2">I</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>/</mo><mn>2</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac><msup><mi>b</mi><mn>2</mn></msup><mrow><mn>8</mn><mo></mo><msup><mi>p</mi><mn>2</mn></msup></mrow></mfrac></mstyle><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m142.png" altimg-height="25px" altimg-valign="-7px" altimg-width="176px" alttext="\Re\nu>-1,\Re(p^{2})>0" display="inline"><mrow><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo>></mo><mrow><mo>-</mo><mn>1</mn></mrow></mrow><mo>,</mo><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mrow><mo stretchy="false">(</mo><msup><mi>p</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow></mrow><mo>></mo><mn>0</mn></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E52.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m211.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E19" title="(4.2.19) ‣ §4.2(iii) The Exponential Function ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m160.png" altimg-height="16px" altimg-valign="-6px" altimg-width="48px" alttext="\exp\NVar{z}" display="inline"><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: exponential function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m163.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./10.25#E2" title="(10.25.2) ‣ §10.25(ii) Standard Solutions ‣ §10.25 Definitions ‣ Modified Bessel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m100.png" altimg-height="23px" altimg-valign="-7px" altimg-width="52px" alttext="I_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.25#E2">I</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: modified Bessel function of the first kind</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m128.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
<table id="E53" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.22.53</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_Math ltx_eqn_cell ltx_align_center"><math class="ltx_Math" alttext="\int_{0}^{\infty}Y_{2\nu}\left(bt\right)\exp\left(-p^{2}t^{2}\right)\mathrm{d}%
t=-\frac{\sqrt{\pi}}{2p}\exp\left(-\frac{b^{2}}{8p^{2}}\right)\left(I_{\nu}%
\left(\frac{b^{2}}{8p^{2}}\right)\tan\left(\nu\pi\right)+\frac{1}{\pi}K_{\nu}%
\left(\frac{b^{2}}{8p^{2}}\right)\sec\left(\nu\pi\right)\right)," display="block"><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><mrow><mrow><msub><mi href="./10.2#E3">Y</mi><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mrow><mi>b</mi><mo></mo><mi>t</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mrow><msup><mi>p</mi><mn>2</mn></msup><mo></mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>=</mo><mrow><mo>-</mo><mrow><mfrac><msqrt><mi href="./3.12#E1">π</mi></msqrt><mrow><mn>2</mn><mo></mo><mi>p</mi></mrow></mfrac><mo></mo><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mfrac><msup><mi>b</mi><mn>2</mn></msup><mrow><mn>8</mn><mo></mo><msup><mi>p</mi><mn>2</mn></msup></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mo>(</mo><mrow><mrow><mrow><msub><mi href="./10.25#E2">I</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mfrac><msup><mi>b</mi><mn>2</mn></msup><mrow><mn>8</mn><mo></mo><msup><mi>p</mi><mn>2</mn></msup></mrow></mfrac><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./4.14#E4">tan</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mfrac><mn>1</mn><mi href="./3.12#E1">π</mi></mfrac><mo></mo><mrow><msub><mi href="./10.25#E3">K</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mfrac><msup><mi>b</mi><mn>2</mn></msup><mrow><mn>8</mn><mo></mo><msup><mi>p</mi><mn>2</mn></msup></mrow></mfrac><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./4.14#E6">sec</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mtd></mtr></mtable></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m265.png" altimg-height="27px" altimg-valign="-9px" altimg-width="80px" alttext="|\Re\nu|<\tfrac{1}{2}" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo stretchy="false">|</mo></mrow><mo><</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math>, <math class="ltx_Math" altimg="m126.png" altimg-height="25px" altimg-valign="-7px" altimg-width="90px" alttext="\Re(p^{2})>0" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mrow><mo stretchy="false">(</mo><msup><mi>p</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow></mrow><mo>></mo><mn>0</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E53.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E3" title="(10.2.3) ‣ Bessel Function of the Second Kind (Weber’s Function) ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m112.png" altimg-height="23px" altimg-valign="-7px" altimg-width="55px" alttext="Y_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E3">Y</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the second kind</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m211.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E19" title="(4.2.19) ‣ §4.2(iii) The Exponential Function ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m160.png" altimg-height="16px" altimg-valign="-6px" altimg-width="48px" alttext="\exp\NVar{z}" display="inline"><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: exponential function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m163.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./10.25#E2" title="(10.25.2) ‣ §10.25(ii) Standard Solutions ‣ §10.25 Definitions ‣ Modified Bessel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m100.png" altimg-height="23px" altimg-valign="-7px" altimg-width="52px" alttext="I_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.25#E2">I</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: modified Bessel function of the first kind</a>,
<a href="./10.25#E3" title="(10.25.3) ‣ §10.25(ii) Standard Solutions ‣ §10.25 Definitions ‣ Modified Bessel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m107.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="K_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.25#E3">K</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: modified Bessel function of the second kind</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m128.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./4.14#E6" title="(4.14.6) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m215.png" altimg-height="13px" altimg-valign="-2px" altimg-width="43px" alttext="\sec\NVar{z}" display="inline"><mrow><mi href="./4.14#E6">sec</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: secant function</a>,
<a href="./4.14#E4" title="(4.14.4) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="17px" altimg-valign="-2px" altimg-width="47px" alttext="\tan\NVar{z}" display="inline"><mrow><mi href="./4.14#E4">tan</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: tangent function</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">11.4.30</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">For <math class="ltx_Math" altimg="m99.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="I" display="inline"><mi href="./10.25#E2">I</mi></math> and <math class="ltx_Math" altimg="m106.png" altimg-height="18px" altimg-valign="-2px" altimg-width="23px" alttext="K" display="inline"><mi href="./10.25#E3">K</mi></math> see
§.</p>
<table id="E54" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.22.54</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_Math ltx_eqn_cell ltx_align_center"><math class="ltx_Math" alttext="\int_{0}^{\infty}J_{\nu}\left(bt\right)\exp\left(-p^{2}t^{2}\right)t^{\mu-1}%
\mathrm{d}t=\frac{(\tfrac{1}{2}b/p)^{\nu}\Gamma\left(\tfrac{1}{2}\nu+\tfrac{1}%
{2}\mu\right)}{2p^{\mu}}\exp\left(-\frac{b^{2}}{4p^{2}}\right)\*{\mathbf{M}}%
\left(\tfrac{1}{2}\nu-\tfrac{1}{2}\mu+1,\nu+1,\frac{b^{2}}{4p^{2}}\right)," display="block"><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><mrow><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi>b</mi><mo></mo><mi>t</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mrow><msup><mi>p</mi><mn>2</mn></msup><mo></mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mrow><mo>)</mo></mrow></mrow><mo></mo><msup><mi>t</mi><mrow><mi>μ</mi><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>=</mo><mrow><mfrac><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi>b</mi></mrow><mo>/</mo><mi>p</mi></mrow><mo stretchy="false">)</mo></mrow><mi href="./10.1#p2.t1.r5">ν</mi></msup><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo>+</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi>μ</mi></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mrow><mn>2</mn><mo></mo><msup><mi>p</mi><mi>μ</mi></msup></mrow></mfrac><mo></mo><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mfrac><msup><mi>b</mi><mn>2</mn></msup><mrow><mn>4</mn><mo></mo><msup><mi>p</mi><mn>2</mn></msup></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./13.2#E3" mathvariant="bold">M</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo>-</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi>μ</mi></mrow></mrow><mo>+</mo><mn>1</mn></mrow><mo>,</mo><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mn>1</mn></mrow><mo>,</mo><mfrac><msup><mi>b</mi><mn>2</mn></msup><mrow><mn>4</mn><mo></mo><msup><mi>p</mi><mn>2</mn></msup></mrow></mfrac><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mtd></mtr></mtable></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m121.png" altimg-height="23px" altimg-valign="-7px" altimg-width="118px" alttext="\Re(\mu+\nu)>0" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>μ</mi><mo>+</mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>></mo><mn>0</mn></mrow></math>, <math class="ltx_Math" altimg="m126.png" altimg-height="25px" altimg-valign="-7px" altimg-width="90px" alttext="\Re(p^{2})>0" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mrow><mo stretchy="false">(</mo><msup><mi>p</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow></mrow><mo>></mo><mn>0</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E54.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m114.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./13.2#E3" title="(13.2.3) ‣ Standard Solutions ‣ §13.2(i) Differential Equation ‣ §13.2 Definitions and Basic Properties ‣ Kummer Functions ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m263.png" altimg-height="23px" altimg-valign="-7px" altimg-width="92px" alttext="{\mathbf{M}}\left(\NVar{a},\NVar{b},\NVar{z}\right)" display="inline"><mrow><mi href="./13.2#E3" mathvariant="bold">M</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">a</mi><mo>,</mo><mi class="ltx_nvar">b</mi><mo>,</mo><mi class="ltx_nvar" href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Olver’s confluent hypergeometric function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E19" title="(4.2.19) ‣ §4.2(iii) The Exponential Function ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m160.png" altimg-height="16px" altimg-valign="-6px" altimg-width="48px" alttext="\exp\NVar{z}" display="inline"><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: exponential function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m163.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m128.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">11.4.28</span> (modified)</span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">For the confluent hypergeometric function <math class="ltx_Math" altimg="m262.png" altimg-height="18px" altimg-valign="-2px" altimg-width="26px" alttext="{\mathbf{M}}" display="inline"><mi href="./13.2#E3" mathvariant="bold">M</mi></math> see §</dd>
</dl>
</div>
</div>
<div id="Px7.p1" class="ltx_para">
<table id="E55" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.22.55</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E55.png" altimg-height="52px" altimg-valign="-21px" altimg-width="446px" alttext="\int_{0}^{\infty}t^{-1}J_{\nu+2\ell+1}\left(t\right)J_{\nu+2m+1}\left(t\right)%
\mathrm{d}t=\frac{\delta_{\ell,m}}{2(2\ell+\nu+1)}," display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><mrow><msup><mi>t</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mrow><mn>2</mn><mo></mo><mi mathvariant="normal">ℓ</mi></mrow><mo>+</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r1">m</mi></mrow><mo>+</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow><mo>=</mo><mfrac><msub><mi href="./front/introduction#Sx4.p1.t1.r4">δ</mi><mrow><mi mathvariant="normal">ℓ</mi><mo href="./front/introduction#Sx4.p1.t1.r4">,</mo><mi href="./10.1#p2.t1.r1">m</mi></mrow></msub><mrow><mn>2</mn><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>2</mn><mo></mo><mi mathvariant="normal">ℓ</mi></mrow><mo>+</mo><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow></mfrac></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m201.png" altimg-height="19px" altimg-valign="-4px" altimg-width="142px" alttext="\nu+\ell+m>-1" display="inline"><mrow><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mi mathvariant="normal">ℓ</mi><mo>+</mo><mi href="./10.1#p2.t1.r1">m</mi></mrow><mo>></mo><mrow><mo>-</mo><mn>1</mn></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E55.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./front/introduction#Sx4.p1.t1.r4" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m158.png" altimg-height="23px" altimg-valign="-8px" altimg-width="35px" alttext="\delta_{\NVar{j},\NVar{k}}" display="inline"><msub><mi href="./front/introduction#Sx4.p1.t1.r4">δ</mi><mrow><mi class="ltx_nvar">j</mi><mo href="./front/introduction#Sx4.p1.t1.r4">,</mo><mi class="ltx_nvar">k</mi></mrow></msub></math>: Kronecker delta</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m163.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./10.1#p2.t1.r1" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m240.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./10.1#p2.t1.r1">m</mi></math>: integer</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">11.4.6</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px8" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Weber–Schafheitlin Discontinuous Integrals, including Special Cases</h3>
<div id="Px8.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px8.p1" class="ltx_para">
<table id="E56" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.22.56</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_Math ltx_eqn_cell ltx_align_center"><math class="ltx_Math" alttext="\int_{0}^{\infty}\frac{J_{\mu}\left(at\right)J_{\nu}\left(bt\right)}{t^{%
\lambda}}\mathrm{d}t=\frac{a^{\mu}\Gamma\left(\frac{1}{2}\nu+\frac{1}{2}\mu-%
\frac{1}{2}\lambda+\frac{1}{2}\right)}{2^{\lambda}b^{\mu-\lambda+1}\Gamma\left%
(\frac{1}{2}\nu-\frac{1}{2}\mu+\frac{1}{2}\lambda+\frac{1}{2}\right)}\*\mathbf%
{F}\left(\tfrac{1}{2}(\mu+\nu-\lambda+1),\tfrac{1}{2}(\mu-\nu-\lambda+1);\mu+1%
;\frac{a^{2}}{b^{2}}\right)," display="block"><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><mrow><mfrac><mrow><mrow><msub><mi href="./10.2#E2">J</mi><mi>μ</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi>a</mi><mo></mo><mi>t</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi>b</mi><mo></mo><mi>t</mi></mrow><mo>)</mo></mrow></mrow></mrow><msup><mi>t</mi><mi>λ</mi></msup></mfrac><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>=</mo><mrow><mfrac><mrow><msup><mi>a</mi><mi>μ</mi></msup><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo>+</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi>μ</mi></mrow></mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi>λ</mi></mrow></mrow><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow></mrow><mrow><msup><mn>2</mn><mi>λ</mi></msup><mo></mo><msup><mi>b</mi><mrow><mrow><mi>μ</mi><mo>-</mo><mi>λ</mi></mrow><mo>+</mo><mn>1</mn></mrow></msup><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi>μ</mi></mrow></mrow><mo>+</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi>λ</mi></mrow><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow></mrow></mfrac><mo></mo><mrow><mi href="./15.2#E2" mathvariant="bold">F</mi><mo></mo><mrow><mo>(</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mrow><mi>μ</mi><mo>+</mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo>-</mo><mi>λ</mi></mrow><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mi>μ</mi><mo>-</mo><mi href="./10.1#p2.t1.r5">ν</mi><mo>-</mo><mi>λ</mi></mrow><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>;</mo><mrow><mi>μ</mi><mo>+</mo><mn>1</mn></mrow><mo>;</mo><mfrac><msup><mi>a</mi><mn>2</mn></msup><msup><mi>b</mi><mn>2</mn></msup></mfrac><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mtd></mtr></mtable></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m94.png" altimg-height="18px" altimg-valign="-3px" altimg-width="86px" alttext="0<a<b" display="inline"><mrow><mn>0</mn><mo><</mo><mi>a</mi><mo><</mo><mi>b</mi></mrow></math>, <math class="ltx_Math" altimg="m123.png" altimg-height="23px" altimg-valign="-7px" altimg-width="221px" alttext="\Re(\mu+\nu+1)>\Re\lambda>-1" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>μ</mi><mo>+</mo><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>></mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>λ</mi></mrow><mo>></mo><mrow><mo>-</mo><mn>1</mn></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E56.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m114.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m163.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m128.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./15.2#E2" title="(15.2.2) ‣ §15.2(i) Gauss Series ‣ §15.2 Definitions and Analytical Properties ‣ Properties ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m172.png" altimg-height="23px" altimg-valign="-7px" altimg-width="102px" alttext="\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E2" mathvariant="bold">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m173.png" altimg-height="39px" altimg-valign="-15px" altimg-width="90px" alttext="\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E2" mathvariant="bold">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi></mfrac><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: Olver’s hypergeometric function</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">11.4.33, 11.4.34</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">If <math class="ltx_Math" altimg="m95.png" altimg-height="18px" altimg-valign="-3px" altimg-width="86px" alttext="0<b<a" display="inline"><mrow><mn>0</mn><mo><</mo><mi>b</mi><mo><</mo><mi>a</mi></mrow></math>, then interchange <math class="ltx_Math" altimg="m227.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi>a</mi></math> and <math class="ltx_Math" altimg="m233.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="b" display="inline"><mi>b</mi></math>, and also <math class="ltx_Math" altimg="m196.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\mu" display="inline"><mi>μ</mi></math> and <math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>. If <math class="ltx_Math" altimg="m232.png" altimg-height="18px" altimg-valign="-2px" altimg-width="50px" alttext="b=a" display="inline"><mrow><mi>b</mi><mo>=</mo><mi>a</mi></mrow></math>,
then</p>
<table id="EGx11" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E57">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.22.57</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m70.png" altimg-height="52px" altimg-valign="-20px" altimg-width="188px" alttext="\displaystyle\int_{0}^{\infty}\frac{J_{\mu}\left(at\right)J_{\nu}\left(at%
\right)}{t^{\lambda}}\mathrm{d}t" display="inline"><mrow><mstyle displaystyle="true"><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup></mstyle><mrow><mstyle displaystyle="true"><mfrac><mrow><mrow><msub><mi href="./10.2#E2">J</mi><mi>μ</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi>a</mi><mo></mo><mi>t</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi>a</mi><mo></mo><mi>t</mi></mrow><mo>)</mo></mrow></mrow></mrow><msup><mi>t</mi><mi>λ</mi></msup></mfrac></mstyle><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m13.png" altimg-height="59px" altimg-valign="-24px" altimg-width="632px" alttext="\displaystyle=\frac{(\frac{1}{2}a)^{\lambda-1}\Gamma\left(\frac{1}{2}\mu+\frac%
{1}{2}\nu-\frac{1}{2}\lambda+\frac{1}{2}\right)\Gamma\left(\lambda\right)}{2%
\Gamma\left(\frac{1}{2}\lambda+\frac{1}{2}\nu-\frac{1}{2}\mu+\frac{1}{2}\right%
)\Gamma\left(\frac{1}{2}\lambda+\frac{1}{2}\mu-\frac{1}{2}\nu+\frac{1}{2}%
\right)\Gamma\left(\frac{1}{2}\lambda+\frac{1}{2}\mu+\frac{1}{2}\nu+\frac{1}{2%
}\right)}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mstyle displaystyle="true"><mfrac><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi>a</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mi>λ</mi><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi>μ</mi></mrow><mo>+</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow></mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi>λ</mi></mrow></mrow><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi>λ</mi><mo>)</mo></mrow></mrow></mrow><mrow><mn>2</mn><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi>λ</mi></mrow><mo>+</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow></mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi>μ</mi></mrow></mrow><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi>λ</mi></mrow><mo>+</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi>μ</mi></mrow></mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow></mrow><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi>λ</mi></mrow><mo>+</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi>μ</mi></mrow><mo>+</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow></mrow></mfrac></mstyle></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m124.png" altimg-height="23px" altimg-valign="-7px" altimg-width="205px" alttext="\Re(\mu+\nu+1)>\Re\lambda>0" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>μ</mi><mo>+</mo><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>></mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>λ</mi></mrow><mo>></mo><mn>0</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E57.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m114.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m163.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m128.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E58">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.22.58</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m71.png" altimg-height="52px" altimg-valign="-20px" altimg-width="186px" alttext="\displaystyle\int_{0}^{\infty}\frac{J_{\nu}\left(at\right)J_{\nu}\left(bt%
\right)}{t^{\lambda}}\mathrm{d}t" display="inline"><mrow><mstyle displaystyle="true"><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup></mstyle><mrow><mstyle displaystyle="true"><mfrac><mrow><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi>a</mi><mo></mo><mi>t</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi>b</mi><mo></mo><mi>t</mi></mrow><mo>)</mo></mrow></mrow></mrow><msup><mi>t</mi><mi>λ</mi></msup></mfrac></mstyle><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m15.png" altimg-height="61px" altimg-valign="-26px" altimg-width="702px" alttext="\displaystyle=\frac{(ab)^{\nu}\Gamma\left(\nu-\frac{1}{2}\lambda+\frac{1}{2}%
\right)}{2^{\lambda}(a^{2}+b^{2})^{\nu-\frac{1}{2}\lambda+\frac{1}{2}}\Gamma%
\left(\frac{1}{2}\lambda+\frac{1}{2}\right)}\mathbf{F}\left(\frac{2\nu+1-%
\lambda}{4},\frac{2\nu+3-\lambda}{4};\nu+1;\frac{4a^{2}b^{2}}{(a^{2}+b^{2})^{2%
}}\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><mfrac><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mi>a</mi><mo></mo><mi>b</mi></mrow><mo stretchy="false">)</mo></mrow><mi href="./10.1#p2.t1.r5">ν</mi></msup><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi>λ</mi></mrow></mrow><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow></mrow><mrow><msup><mn>2</mn><mi>λ</mi></msup><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mrow><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi>λ</mi></mrow></mrow><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi>λ</mi></mrow><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow></mrow></mfrac></mstyle><mo></mo><mrow><mi href="./15.2#E2" mathvariant="bold">F</mi><mo></mo><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac><mrow><mrow><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo>+</mo><mn>1</mn></mrow><mo>-</mo><mi>λ</mi></mrow><mn>4</mn></mfrac></mstyle><mo>,</mo><mstyle displaystyle="true"><mfrac><mrow><mrow><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo>+</mo><mn>3</mn></mrow><mo>-</mo><mi>λ</mi></mrow><mn>4</mn></mfrac></mstyle><mo>;</mo><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mn>1</mn></mrow><mo>;</mo><mstyle displaystyle="true"><mfrac><mrow><mn>4</mn><mo></mo><msup><mi>a</mi><mn>2</mn></msup><mo></mo><msup><mi>b</mi><mn>2</mn></msup></mrow><msup><mrow><mo stretchy="false">(</mo><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mfrac></mstyle><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m229.png" altimg-height="21px" altimg-valign="-6px" altimg-width="50px" alttext="a\neq b" display="inline"><mrow><mi>a</mi><mo>≠</mo><mi>b</mi></mrow></math>, <math class="ltx_Math" altimg="m117.png" altimg-height="23px" altimg-valign="-7px" altimg-width="194px" alttext="\Re(2\nu+1)>\Re\lambda>-1" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>></mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>λ</mi></mrow><mo>></mo><mrow><mo>-</mo><mn>1</mn></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E58.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m114.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m163.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m128.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./15.2#E2" title="(15.2.2) ‣ §15.2(i) Gauss Series ‣ §15.2 Definitions and Analytical Properties ‣ Properties ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m172.png" altimg-height="23px" altimg-valign="-7px" altimg-width="102px" alttext="\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E2" mathvariant="bold">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m173.png" altimg-height="39px" altimg-valign="-15px" altimg-width="90px" alttext="\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E2" mathvariant="bold">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi></mfrac><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: Olver’s hypergeometric function</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
</div>
<div id="Px8.p2" class="ltx_para">
<p class="ltx_p">When <math class="ltx_Math" altimg="m131.png" altimg-height="21px" altimg-valign="-6px" altimg-width="83px" alttext="\Re\mu>-1" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>μ</mi></mrow><mo>></mo><mrow><mo>-</mo><mn>1</mn></mrow></mrow></math></p>
<table id="E59" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.22.59</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E59.png" altimg-height="123px" altimg-valign="-55px" altimg-width="569px" alttext="\int_{0}^{\infty}e^{ibt}J_{\mu}\left(at\right)\mathrm{d}t=\begin{cases}\dfrac{%
\exp\left(i\mu\operatorname{arcsin}\left(b/a\right)\right)}{(a^{2}-b^{2})^{%
\frac{1}{2}}},&0\leq b<a,\\
\dfrac{ia^{\mu}\exp\left(\frac{1}{2}\mu\pi i\right)}{(b^{2}-a^{2})^{\frac{1}{2%
}}\left(b+(b^{2}-a^{2})^{\frac{1}{2}}\right)^{\mu}},&0<a<b.\end{cases}" display="block"><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mo></mo><mi>b</mi><mo></mo><mi>t</mi></mrow></msup><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mi>μ</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi>a</mi><mo></mo><mi>t</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow><mo>=</mo><mrow><mo>{</mo><mtable columnspacing="5pt" displaystyle="true" rowspacing="0pt"><mtr><mtd columnalign="left"><mrow><mfrac><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mrow><mo>(</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi>μ</mi><mo></mo><mrow><mi href="./4.23#SS2.p1">arcsin</mi><mo></mo><mrow><mo>(</mo><mrow><mi>b</mi><mo>/</mo><mi>a</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>)</mo></mrow></mrow><msup><mrow><mo stretchy="false">(</mo><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>-</mo><msup><mi>b</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></msup></mfrac><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mrow><mn>0</mn><mo>≤</mo><mi>b</mi><mo><</mo><mi>a</mi></mrow><mo>,</mo></mrow></mtd></mtr><mtr><mtd columnalign="left"><mrow><mfrac><mrow><mi mathvariant="normal">i</mi><mo></mo><msup><mi>a</mi><mi>μ</mi></msup><mo></mo><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi>μ</mi><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi></mrow><mo>)</mo></mrow></mrow></mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><msup><mi>b</mi><mn>2</mn></msup><mo>-</mo><msup><mi>a</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></msup><mo></mo><msup><mrow><mo>(</mo><mrow><mi>b</mi><mo>+</mo><msup><mrow><mo stretchy="false">(</mo><mrow><msup><mi>b</mi><mn>2</mn></msup><mo>-</mo><msup><mi>a</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></msup></mrow><mo>)</mo></mrow><mi>μ</mi></msup></mrow></mfrac><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mrow><mn>0</mn><mo><</mo><mi>a</mi><mo><</mo><mi>b</mi></mrow><mo>.</mo></mrow></mtd></mtr></mtable></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E59.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m211.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E19" title="(4.2.19) ‣ §4.2(iii) The Exponential Function ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m160.png" altimg-height="16px" altimg-valign="-6px" altimg-width="48px" alttext="\exp\NVar{z}" display="inline"><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: exponential function</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m177.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m163.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a> and
<a href="./4.23#SS2.p1" title="§4.23(ii) Principal Values ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m210.png" altimg-height="18px" altimg-valign="-2px" altimg-width="69px" alttext="\operatorname{arcsin}\NVar{z}" display="inline"><mrow><mi href="./4.23#SS2.p1">arcsin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: arcsine function</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">11.4.37, 11.4.38</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="Px8.p3" class="ltx_para">
<table id="E60" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.22.60</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E60.png" altimg-height="85px" altimg-valign="-37px" altimg-width="702px" alttext="\int_{0}^{\infty}e^{ibt}Y_{0}\left(at\right)\mathrm{d}t=\begin{cases}(2i/\pi)(%
a^{2}-b^{2})^{-\frac{1}{2}}\operatorname{arcsin}\left(b/a\right),&0\leq b<a,\\
(b^{2}-a^{2})^{-\frac{1}{2}}\left(-1+\dfrac{2i}{\pi}\ln\left(\dfrac{a}{b+(b^{2%
}-a^{2})^{\frac{1}{2}}}\right)\right),&0<a<b.\end{cases}" display="block"><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mo></mo><mi>b</mi><mo></mo><mi>t</mi></mrow></msup><mo></mo><mrow><msub><mi href="./10.2#E3">Y</mi><mn>0</mn></msub><mo></mo><mrow><mo>(</mo><mrow><mi>a</mi><mo></mo><mi>t</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow><mo>=</mo><mrow><mo>{</mo><mtable columnspacing="5pt" displaystyle="true" rowspacing="0pt"><mtr><mtd columnalign="left"><mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>2</mn><mo></mo><mi mathvariant="normal">i</mi></mrow><mo>/</mo><mi href="./3.12#E1">π</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>-</mo><msup><mi>b</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup><mo></mo><mrow><mi href="./4.23#SS2.p1">arcsin</mi><mo></mo><mrow><mo>(</mo><mrow><mi>b</mi><mo>/</mo><mi>a</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mrow><mn>0</mn><mo>≤</mo><mi>b</mi><mo><</mo><mi>a</mi></mrow><mo>,</mo></mrow></mtd></mtr><mtr><mtd columnalign="left"><mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><msup><mi>b</mi><mn>2</mn></msup><mo>-</mo><msup><mi>a</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup><mo></mo><mrow><mo>(</mo><mrow><mrow><mo>-</mo><mn>1</mn></mrow><mo>+</mo><mrow><mfrac><mrow><mn>2</mn><mo></mo><mi mathvariant="normal">i</mi></mrow><mi href="./3.12#E1">π</mi></mfrac><mo></mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><mo>(</mo><mfrac><mi>a</mi><mrow><mi>b</mi><mo>+</mo><msup><mrow><mo stretchy="false">(</mo><mrow><msup><mi>b</mi><mn>2</mn></msup><mo>-</mo><msup><mi>a</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></msup></mrow></mfrac><mo>)</mo></mrow></mrow></mrow></mrow><mo>)</mo></mrow></mrow><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mrow><mn>0</mn><mo><</mo><mi>a</mi><mo><</mo><mi>b</mi></mrow><mo>.</mo></mrow></mtd></mtr></mtable></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E60.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E3" title="(10.2.3) ‣ Bessel Function of the Second Kind (Weber’s Function) ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m112.png" altimg-height="23px" altimg-valign="-7px" altimg-width="55px" alttext="Y_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E3">Y</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the second kind</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m211.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m177.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m163.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./4.23#SS2.p1" title="§4.23(ii) Principal Values ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m210.png" altimg-height="18px" altimg-valign="-2px" altimg-width="69px" alttext="\operatorname{arcsin}\NVar{z}" display="inline"><mrow><mi href="./4.23#SS2.p1">arcsin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: arcsine function</a> and
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m170.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">11.4.40</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="Px8.p4" class="ltx_para">
<p class="ltx_p">When <math class="ltx_Math" altimg="m139.png" altimg-height="21px" altimg-valign="-6px" altimg-width="67px" alttext="\Re\mu>0" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>μ</mi></mrow><mo>></mo><mn>0</mn></mrow></math>,</p>
<table id="E61" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.22.61</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E61.png" altimg-height="101px" altimg-valign="-45px" altimg-width="568px" alttext="\int_{0}^{\infty}t^{-1}e^{ibt}J_{\mu}\left(at\right)\mathrm{d}t=\begin{cases}(%
1/\mu)\exp\left(i\mu\operatorname{arcsin}\left(b/a\right)\right),&0\leq b\leq a%
,\\
\dfrac{a^{\mu}\exp\left(\frac{1}{2}\mu\pi i\right)}{\mu\left(b+(b^{2}-a^{2})^{%
\frac{1}{2}}\right)^{\mu}},&0<a\leq b.\end{cases}" display="block"><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><mrow><msup><mi>t</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mo></mo><mi>b</mi><mo></mo><mi>t</mi></mrow></msup><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mi>μ</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi>a</mi><mo></mo><mi>t</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow><mo>=</mo><mrow><mo>{</mo><mtable columnspacing="5pt" displaystyle="true" rowspacing="0pt"><mtr><mtd columnalign="left"><mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>/</mo><mi>μ</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mrow><mo>(</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi>μ</mi><mo></mo><mrow><mi href="./4.23#SS2.p1">arcsin</mi><mo></mo><mrow><mo>(</mo><mrow><mi>b</mi><mo>/</mo><mi>a</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mrow><mn>0</mn><mo>≤</mo><mi>b</mi><mo>≤</mo><mi>a</mi></mrow><mo>,</mo></mrow></mtd></mtr><mtr><mtd columnalign="left"><mrow><mfrac><mrow><msup><mi>a</mi><mi>μ</mi></msup><mo></mo><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi>μ</mi><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi></mrow><mo>)</mo></mrow></mrow></mrow><mrow><mi>μ</mi><mo></mo><msup><mrow><mo>(</mo><mrow><mi>b</mi><mo>+</mo><msup><mrow><mo stretchy="false">(</mo><mrow><msup><mi>b</mi><mn>2</mn></msup><mo>-</mo><msup><mi>a</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></msup></mrow><mo>)</mo></mrow><mi>μ</mi></msup></mrow></mfrac><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mrow><mn>0</mn><mo><</mo><mi>a</mi><mo>≤</mo><mi>b</mi></mrow><mo>.</mo></mrow></mtd></mtr></mtable></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E61.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m211.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E19" title="(4.2.19) ‣ §4.2(iii) The Exponential Function ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m160.png" altimg-height="16px" altimg-valign="-6px" altimg-width="48px" alttext="\exp\NVar{z}" display="inline"><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: exponential function</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m177.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m163.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a> and
<a href="./4.23#SS2.p1" title="§4.23(ii) Principal Values ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m210.png" altimg-height="18px" altimg-valign="-2px" altimg-width="69px" alttext="\operatorname{arcsin}\NVar{z}" display="inline"><mrow><mi href="./4.23#SS2.p1">arcsin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: arcsine function</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">11.4.35, 11.4.36</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="Px8.p5" class="ltx_para">
<p class="ltx_p">When <math class="ltx_Math" altimg="m149.png" altimg-height="21px" altimg-valign="-6px" altimg-width="135px" alttext="\Re\nu>\Re\mu>-1" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo>></mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>μ</mi></mrow><mo>></mo><mrow><mo>-</mo><mn>1</mn></mrow></mrow></math>,</p>
<table id="E62" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.22.62</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E62.png" altimg-height="83px" altimg-valign="-36px" altimg-width="622px" alttext="\int_{0}^{\infty}t^{\mu-\nu+1}J_{\mu}\left(at\right)J_{\nu}\left(bt\right)%
\mathrm{d}t=\begin{cases}0,&0<b<a,\\
\dfrac{2^{\mu-\nu+1}a^{\mu}(b^{2}-a^{2})^{\nu-\mu-1}}{b^{\nu}\Gamma\left(\nu-%
\mu\right)},&0<a\leq b.\end{cases}" display="block"><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><mrow><msup><mi>t</mi><mrow><mrow><mi>μ</mi><mo>-</mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo>+</mo><mn>1</mn></mrow></msup><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mi>μ</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi>a</mi><mo></mo><mi>t</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi>b</mi><mo></mo><mi>t</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow><mo>=</mo><mrow><mo>{</mo><mtable columnspacing="5pt" displaystyle="true" rowspacing="0pt"><mtr><mtd columnalign="left"><mrow><mn>0</mn><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mrow><mn>0</mn><mo><</mo><mi>b</mi><mo><</mo><mi>a</mi></mrow><mo>,</mo></mrow></mtd></mtr><mtr><mtd columnalign="left"><mrow><mfrac><mrow><msup><mn>2</mn><mrow><mrow><mi>μ</mi><mo>-</mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo>+</mo><mn>1</mn></mrow></msup><mo></mo><msup><mi>a</mi><mi>μ</mi></msup><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><msup><mi>b</mi><mn>2</mn></msup><mo>-</mo><msup><mi>a</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>-</mo><mi>μ</mi><mo>-</mo><mn>1</mn></mrow></msup></mrow><mrow><msup><mi>b</mi><mi href="./10.1#p2.t1.r5">ν</mi></msup><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>-</mo><mi>μ</mi></mrow><mo>)</mo></mrow></mrow></mrow></mfrac><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mrow><mn>0</mn><mo><</mo><mi>a</mi><mo>≤</mo><mi>b</mi></mrow><mo>.</mo></mrow></mtd></mtr></mtable></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E62.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m114.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m163.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">11.4.41</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="Px8.p6" class="ltx_para">
<p class="ltx_p">When <math class="ltx_Math" altimg="m139.png" altimg-height="21px" altimg-valign="-6px" altimg-width="67px" alttext="\Re\mu>0" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>μ</mi></mrow><mo>></mo><mn>0</mn></mrow></math>,</p>
<table id="E63" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.22.63</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E63.png" altimg-height="92px" altimg-valign="-40px" altimg-width="449px" alttext="\int_{0}^{\infty}J_{\mu}\left(at\right)J_{\mu-1}\left(bt\right)\mathrm{d}t=%
\begin{cases}b^{\mu-1}a^{-\mu},&0<b<a,\\
(2b)^{-1},&b=a(>0),\\
0,&0<a<b.\end{cases}" display="block"><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><mrow><mrow><msub><mi href="./10.2#E2">J</mi><mi>μ</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi>a</mi><mo></mo><mi>t</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mrow><mi>μ</mi><mo>-</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mrow><mi>b</mi><mo></mo><mi>t</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow><mo>=</mo><mrow><mo>{</mo><mtable columnspacing="5pt" displaystyle="true" rowspacing="0pt"><mtr><mtd columnalign="left"><mrow><mrow><msup><mi>b</mi><mrow><mi>μ</mi><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><msup><mi>a</mi><mrow><mo>-</mo><mi>μ</mi></mrow></msup></mrow><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mrow><mn>0</mn><mo><</mo><mi>b</mi><mo><</mo><mi>a</mi></mrow><mo>,</mo></mrow></mtd></mtr><mtr><mtd columnalign="left"><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mn>2</mn><mo></mo><mi>b</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mrow><mi>b</mi><mo>=</mo><mrow><mi>a</mi><mspace width="veryverythickmathspace"></mspace><mrow><mo stretchy="false">(</mo><mrow><mi></mi><mo>></mo><mn>0</mn></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>,</mo></mrow></mtd></mtr><mtr><mtd columnalign="left"><mrow><mn>0</mn><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mrow><mn>0</mn><mo><</mo><mi>a</mi><mo><</mo><mi>b</mi></mrow><mo>.</mo></mrow></mtd></mtr></mtable></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E63.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m163.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">11.4.42</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="Px8.p7" class="ltx_para">
<p class="ltx_p">When <math class="ltx_Math" altimg="m242.png" altimg-height="20px" altimg-valign="-6px" altimg-width="123px" alttext="n=0,1,2,\dots" display="inline"><mrow><mi href="./10.1#p2.t1.r1">n</mi><mo>=</mo><mrow><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math> and <math class="ltx_Math" altimg="m136.png" altimg-height="21px" altimg-valign="-6px" altimg-width="119px" alttext="\Re\mu>-n-1" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>μ</mi></mrow><mo>></mo><mrow><mrow><mo>-</mo><mi href="./10.1#p2.t1.r1">n</mi></mrow><mo>-</mo><mn>1</mn></mrow></mrow></math>,</p>
<table id="E64" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.22.64</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E64.png" altimg-height="113px" altimg-valign="-51px" altimg-width="801px" alttext="\int_{0}^{\infty}J_{\mu+2n+1}\left(at\right)J_{\mu}\left(bt\right)\mathrm{d}t=%
\begin{cases}\dfrac{b^{\mu}\Gamma\left(\mu+n+1\right)}{a^{\mu+1}n!}\mathbf{F}%
\left(-n,\mu+n+1;\mu+1;\dfrac{b^{2}}{a^{2}}\right),&0<b<a,\\
(-1)^{n}/(2a),&b=a(>0),\\
0,&0<a<b.\end{cases}" display="block"><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><mrow><mrow><msub><mi href="./10.2#E2">J</mi><mrow><mi>μ</mi><mo>+</mo><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mrow><mi>a</mi><mo></mo><mi>t</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mi>μ</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi>b</mi><mo></mo><mi>t</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow><mo>=</mo><mrow><mo>{</mo><mtable columnspacing="5pt" displaystyle="true" rowspacing="0pt"><mtr><mtd columnalign="left"><mrow><mrow><mfrac><mrow><msup><mi>b</mi><mi>μ</mi></msup><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi>μ</mi><mo>+</mo><mi href="./10.1#p2.t1.r1">n</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mrow><mrow><msup><mi>a</mi><mrow><mi>μ</mi><mo>+</mo><mn>1</mn></mrow></msup><mo></mo><mrow><mi href="./10.1#p2.t1.r1">n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mrow></mfrac><mo></mo><mrow><mi href="./15.2#E2" mathvariant="bold">F</mi><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mi href="./10.1#p2.t1.r1">n</mi></mrow><mo>,</mo><mrow><mi>μ</mi><mo>+</mo><mi href="./10.1#p2.t1.r1">n</mi><mo>+</mo><mn>1</mn></mrow><mo>;</mo><mrow><mi>μ</mi><mo>+</mo><mn>1</mn></mrow><mo>;</mo><mfrac><msup><mi>b</mi><mn>2</mn></msup><msup><mi>a</mi><mn>2</mn></msup></mfrac><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mrow><mn>0</mn><mo><</mo><mi>b</mi><mo><</mo><mi>a</mi></mrow><mo>,</mo></mrow></mtd></mtr><mtr><mtd columnalign="left"><mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./10.1#p2.t1.r1">n</mi></msup><mo>/</mo><mrow><mo stretchy="false">(</mo><mrow><mn>2</mn><mo></mo><mi>a</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mrow><mi>b</mi><mo>=</mo><mrow><mi>a</mi><mspace width="veryverythickmathspace"></mspace><mrow><mo stretchy="false">(</mo><mrow><mi></mi><mo>></mo><mn>0</mn></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>,</mo></mrow></mtd></mtr><mtr><mtd columnalign="left"><mrow><mn>0</mn><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mrow><mn>0</mn><mo><</mo><mi>a</mi><mo><</mo><mi>b</mi></mrow><mo>.</mo></mrow></mtd></mtr></mtable></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E64.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m114.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m241.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m163.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./15.2#E2" title="(15.2.2) ‣ §15.2(i) Gauss Series ‣ §15.2 Definitions and Analytical Properties ‣ Properties ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m172.png" altimg-height="23px" altimg-valign="-7px" altimg-width="102px" alttext="\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E2" mathvariant="bold">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m173.png" altimg-height="39px" altimg-valign="-15px" altimg-width="90px" alttext="\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E2" mathvariant="bold">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi></mfrac><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: Olver’s hypergeometric function</a> and
<a href="./10.1#p2.t1.r1" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m247.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./10.1#p2.t1.r1">n</mi></math>: integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E65" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.22.65</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E65.png" altimg-height="65px" altimg-valign="-27px" altimg-width="599px" alttext="\int_{0}^{\infty}J_{0}\left(at\right)\left(J_{0}\left(bt\right)-J_{0}\left(ct%
\right)\right)\frac{\mathrm{d}t}{t}=\begin{cases}0,&0\leq b<a,0<c\leq a,\\
\ln\left(c/a\right),&0\leq b<a\leq c.\end{cases}" display="block"><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><mrow><mrow><msub><mi href="./10.2#E2">J</mi><mn>0</mn></msub><mo></mo><mrow><mo>(</mo><mrow><mi>a</mi><mo></mo><mi>t</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mo>(</mo><mrow><mrow><msub><mi href="./10.2#E2">J</mi><mn>0</mn></msub><mo></mo><mrow><mo>(</mo><mrow><mi>b</mi><mo></mo><mi>t</mi></mrow><mo>)</mo></mrow></mrow><mo>-</mo><mrow><msub><mi href="./10.2#E2">J</mi><mn>0</mn></msub><mo></mo><mrow><mo>(</mo><mrow><mi>c</mi><mo></mo><mi>t</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>)</mo></mrow><mo></mo><mfrac><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow><mi>t</mi></mfrac></mrow></mrow><mo>=</mo><mrow><mo>{</mo><mtable columnspacing="5pt" displaystyle="true" rowspacing="0pt"><mtr><mtd columnalign="left"><mrow><mn>0</mn><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mrow><mrow><mn>0</mn><mo>≤</mo><mi>b</mi><mo><</mo><mi>a</mi></mrow><mo>,</mo><mrow><mn>0</mn><mo><</mo><mi>c</mi><mo>≤</mo><mi>a</mi></mrow></mrow><mo>,</mo></mrow></mtd></mtr><mtr><mtd columnalign="left"><mrow><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><mo>(</mo><mrow><mi>c</mi><mo>/</mo><mi>a</mi></mrow><mo>)</mo></mrow></mrow><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mrow><mn>0</mn><mo>≤</mo><mi>b</mi><mo><</mo><mi>a</mi><mo>≤</mo><mi>c</mi></mrow><mo>.</mo></mrow></mtd></mtr></mtable></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E65.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m163.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a> and
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m170.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">11.4.43</span> (Case <math class="ltx_Math" altimg="m231.png" altimg-height="18px" altimg-valign="-2px" altimg-width="49px" alttext="b=0" display="inline"><mrow><mi>b</mi><mo>=</mo><mn>0</mn></mrow></math>.)</span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px9" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Other Double Products</h3>
<div id="Px9.info" class="ltx_metadata ltx_info">
) <math class="ltx_Math" altimg="m222.png" altimg-height="21px" altimg-valign="-6px" altimg-width="50px" alttext="a,b,c" display="inline"><mrow><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi></mrow></math> are positive
constants.</p>
<table id="EGx12" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E66">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.22.66</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m72.png" altimg-height="53px" altimg-valign="-20px" altimg-width="220px" alttext="\displaystyle\int_{0}^{\infty}e^{-at}J_{\nu}\left(bt\right)J_{\nu}\left(ct%
\right)\mathrm{d}t" display="inline"><mrow><mstyle displaystyle="true"><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup></mstyle><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mi>a</mi><mo></mo><mi>t</mi></mrow></mrow></msup><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi>b</mi><mo></mo><mi>t</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi>c</mi><mo></mo><mi>t</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m21.png" altimg-height="57px" altimg-valign="-24px" altimg-width="292px" alttext="\displaystyle=\frac{1}{\pi(bc)^{\frac{1}{2}}}\*Q_{\nu-\frac{1}{2}}\left(\frac{%
a^{2}+b^{2}+c^{2}}{2bc}\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><mfrac><mn>1</mn><mrow><mi href="./3.12#E1">π</mi><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi>b</mi><mo></mo><mi>c</mi></mrow><mo stretchy="false">)</mo></mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></msup></mrow></mfrac></mstyle><mo></mo><mrow><msub><mi href="./14.2#SS2.p2">Q</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msub><mo></mo><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup><mo>+</mo><msup><mi>c</mi><mn>2</mn></msup></mrow><mrow><mn>2</mn><mo></mo><mi>b</mi><mo></mo><mi>c</mi></mrow></mfrac></mstyle><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m145.png" altimg-height="27px" altimg-valign="-9px" altimg-width="85px" alttext="\Re\nu>-\tfrac{1}{2}" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo>></mo><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E66.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m211.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m177.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m163.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m128.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./14.2#SS2.p2" title="§14.2(ii) Associated Legendre Equation ‣ §14.2 Differential Equations ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m111.png" altimg-height="25px" altimg-valign="-7px" altimg-width="140px" alttext="Q_{\NVar{\nu}}\left(\NVar{z}\right)=Q^{0}_{\nu}\left(z\right)" display="inline"><mrow><mrow><msub><mi href="./14.2#SS2.p2">Q</mi><mi class="ltx_nvar" href="./14.1#p1.t1.r4">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./14.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msubsup><mi href="./14.21#SS1.p1">Q</mi><mi href="./14.1#p1.t1.r4">ν</mi><mn>0</mn></msubsup><mo></mo><mrow><mo>(</mo><mi href="./14.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow></math>: Legendre function of the second kind</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E67">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.22.67</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m74.png" altimg-height="53px" altimg-valign="-20px" altimg-width="291px" alttext="\displaystyle\int_{0}^{\infty}t\exp(-p^{2}t^{2})J_{\nu}\left(at\right)J_{\nu}%
\left(bt\right)\mathrm{d}t" display="inline"><mrow><mstyle displaystyle="true"><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup></mstyle><mrow><mi>t</mi><mo></mo><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mrow><msup><mi>p</mi><mn>2</mn></msup><mo></mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi>a</mi><mo></mo><mi>t</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi>b</mi><mo></mo><mi>t</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m18.png" altimg-height="54px" altimg-valign="-21px" altimg-width="306px" alttext="\displaystyle=\frac{1}{2p^{2}}\exp\left(-\frac{a^{2}+b^{2}}{4p^{2}}\right)I_{%
\nu}\left(\frac{ab}{2p^{2}}\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><mfrac><mn>1</mn><mrow><mn>2</mn><mo></mo><msup><mi>p</mi><mn>2</mn></msup></mrow></mfrac></mstyle><mo></mo><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mstyle displaystyle="true"><mfrac><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow><mrow><mn>4</mn><mo></mo><msup><mi>p</mi><mn>2</mn></msup></mrow></mfrac></mstyle></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./10.25#E2">I</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac><mrow><mi>a</mi><mo></mo><mi>b</mi></mrow><mrow><mn>2</mn><mo></mo><msup><mi>p</mi><mn>2</mn></msup></mrow></mfrac></mstyle><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m142.png" altimg-height="25px" altimg-valign="-7px" altimg-width="176px" alttext="\Re\nu>-1,\Re(p^{2})>0" display="inline"><mrow><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo>></mo><mrow><mo>-</mo><mn>1</mn></mrow></mrow><mo>,</mo><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mrow><mo stretchy="false">(</mo><msup><mi>p</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow></mrow><mo>></mo><mn>0</mn></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E67.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E19" title="(4.2.19) ‣ §4.2(iii) The Exponential Function ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m160.png" altimg-height="16px" altimg-valign="-6px" altimg-width="48px" alttext="\exp\NVar{z}" display="inline"><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: exponential function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m163.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./10.25#E2" title="(10.25.2) ‣ §10.25(ii) Standard Solutions ‣ §10.25 Definitions ‣ Modified Bessel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m100.png" altimg-height="23px" altimg-valign="-7px" altimg-width="52px" alttext="I_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.25#E2">I</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: modified Bessel function of the first kind</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m128.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E68">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.22.68</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m73.png" altimg-height="53px" altimg-valign="-20px" altimg-width="292px" alttext="\displaystyle\int_{0}^{\infty}t\exp(-p^{2}t^{2})J_{0}\left(at\right)Y_{0}\left%
(at\right)\mathrm{d}t" display="inline"><mrow><mstyle displaystyle="true"><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup></mstyle><mrow><mi>t</mi><mo></mo><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mrow><msup><mi>p</mi><mn>2</mn></msup><mo></mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mn>0</mn></msub><mo></mo><mrow><mo>(</mo><mrow><mi>a</mi><mo></mo><mi>t</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./10.2#E3">Y</mi><mn>0</mn></msub><mo></mo><mrow><mo>(</mo><mrow><mi>a</mi><mo></mo><mi>t</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m1.png" altimg-height="54px" altimg-valign="-21px" altimg-width="308px" alttext="\displaystyle=-\frac{1}{2\pi p^{2}}\exp\left(-\frac{a^{2}}{2p^{2}}\right)K_{0}%
\left(\frac{a^{2}}{2p^{2}}\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo>-</mo><mrow><mstyle displaystyle="true"><mfrac><mn>1</mn><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><msup><mi>p</mi><mn>2</mn></msup></mrow></mfrac></mstyle><mo></mo><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mstyle displaystyle="true"><mfrac><msup><mi>a</mi><mn>2</mn></msup><mrow><mn>2</mn><mo></mo><msup><mi>p</mi><mn>2</mn></msup></mrow></mfrac></mstyle></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./10.25#E3">K</mi><mn>0</mn></msub><mo></mo><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac><msup><mi>a</mi><mn>2</mn></msup><mrow><mn>2</mn><mo></mo><msup><mi>p</mi><mn>2</mn></msup></mrow></mfrac></mstyle><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m126.png" altimg-height="25px" altimg-valign="-7px" altimg-width="90px" alttext="\Re(p^{2})>0" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mrow><mo stretchy="false">(</mo><msup><mi>p</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow></mrow><mo>></mo><mn>0</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E68.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./10.2#E3" title="(10.2.3) ‣ Bessel Function of the Second Kind (Weber’s Function) ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m112.png" altimg-height="23px" altimg-valign="-7px" altimg-width="55px" alttext="Y_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E3">Y</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the second kind</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m211.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E19" title="(4.2.19) ‣ §4.2(iii) The Exponential Function ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m160.png" altimg-height="16px" altimg-valign="-6px" altimg-width="48px" alttext="\exp\NVar{z}" display="inline"><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: exponential function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m163.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./10.25#E3" title="(10.25.3) ‣ §10.25(ii) Standard Solutions ‣ §10.25 Definitions ‣ Modified Bessel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m107.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="K_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.25#E3">K</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: modified Bessel function of the second kind</a> and
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m128.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
<p class="ltx_p">For the associated Legendre function <math class="ltx_Math" altimg="m109.png" altimg-height="21px" altimg-valign="-6px" altimg-width="20px" alttext="Q" display="inline"><mi href="./14.2#SS2.p2">Q</mi></math> see §with <math class="ltx_Math" altimg="m191.png" altimg-height="20px" altimg-valign="-6px" altimg-width="53px" alttext="\mu=0" display="inline"><mrow><mi>μ</mi><mo>=</mo><mn>0</mn></mrow></math>. For <math class="ltx_Math" altimg="m99.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="I" display="inline"><mi href="./10.25#E2">I</mi></math> and <math class="ltx_Math" altimg="m106.png" altimg-height="18px" altimg-valign="-2px" altimg-width="23px" alttext="K" display="inline"><mi href="./10.25#E3">K</mi></math> see §.</p>
<table id="EGx13" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E69">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.22.69</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m63.png" altimg-height="53px" altimg-valign="-20px" altimg-width="230px" alttext="\displaystyle\int_{0}^{\infty}J_{\nu}\left(at\right)J_{\nu}\left(bt\right)%
\frac{t\mathrm{d}t}{t^{2}-z^{2}}" display="inline"><mrow><mstyle displaystyle="true"><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup></mstyle><mrow><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi>a</mi><mo></mo><mi>t</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi>b</mi><mo></mo><mi>t</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mstyle displaystyle="true"><mfrac><mrow><mi>t</mi><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow><mrow><msup><mi>t</mi><mn>2</mn></msup><mo>-</mo><msup><mi href="./10.1#p2.t1.r4">z</mi><mn>2</mn></msup></mrow></mfrac></mstyle></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m30.png" altimg-height="65px" altimg-valign="-27px" altimg-width="335px" alttext="\displaystyle=\left\{\begin{array}[]{ll}\frac{1}{2}\pi iJ_{\nu}\left(bz\right)%
{H^{(1)}_{\nu}}\left(az\right),&a>b\\
\frac{1}{2}\pi iJ_{\nu}\left(az\right){H^{(1)}_{\nu}}\left(bz\right),&b>a\end{%
array}\right\}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo>{</mo><mtable columnspacing="5pt" rowspacing="0pt"><mtr><mtd columnalign="left"><mrow><mrow><mstyle displaystyle="true"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi>b</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><msubsup><mi href="./10.2#E5">H</mi><mi href="./10.1#p2.t1.r5">ν</mi><mrow><mo href="./10.2#E5" stretchy="false">(</mo><mn>1</mn><mo href="./10.2#E5" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mrow><mi>a</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mi>a</mi><mo>></mo><mi>b</mi></mrow></mtd></mtr><mtr><mtd columnalign="left"><mrow><mrow><mstyle displaystyle="true"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi>a</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><msubsup><mi href="./10.2#E5">H</mi><mi href="./10.1#p2.t1.r5">ν</mi><mrow><mo href="./10.2#E5" stretchy="false">(</mo><mn>1</mn><mo href="./10.2#E5" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mrow><mi>b</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mi>b</mi><mo>></mo><mi>a</mi></mrow></mtd></mtr></mtable><mo>}</mo></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m141.png" altimg-height="21px" altimg-valign="-6px" altimg-width="152px" alttext="\Re\nu>-1,\Im z>0" display="inline"><mrow><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo>></mo><mrow><mo>-</mo><mn>1</mn></mrow></mrow><mo>,</mo><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℑ</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo>></mo><mn>0</mn></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E69.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./10.2#E5" title="(10.2.5) ‣ Bessel Functions of the Third Kind (Hankel Functions) ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m259.png" altimg-height="29px" altimg-valign="-7px" altimg-width="73px" alttext="{H^{(1)}_{\NVar{\nu}}}\left(\NVar{z}\right)" display="inline"><mrow><msubsup><mi href="./10.2#E5">H</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi><mrow><mo href="./10.2#E5" stretchy="false">(</mo><mn>1</mn><mo href="./10.2#E5" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the third kind (or Hankel function)</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m211.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m115.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Im" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℑ</mi><mo></mo></mrow></math>: imaginary part</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m163.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m128.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./10.1#p2.t1.r4" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m258.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./10.1#p2.t1.r4">z</mi></math>: complex variable</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E70">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.22.70</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m67.png" altimg-height="53px" altimg-valign="-20px" altimg-width="251px" alttext="\displaystyle\int_{0}^{\infty}Y_{\nu}\left(at\right)J_{\nu+1}\left(bt\right)%
\frac{t\mathrm{d}t}{t^{2}-z^{2}}" display="inline"><mrow><mstyle displaystyle="true"><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup></mstyle><mrow><mrow><msub><mi href="./10.2#E3">Y</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi>a</mi><mo></mo><mi>t</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mrow><mi>b</mi><mo></mo><mi>t</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mstyle displaystyle="true"><mfrac><mrow><mi>t</mi><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow><mrow><msup><mi>t</mi><mn>2</mn></msup><mo>-</mo><msup><mi href="./10.1#p2.t1.r4">z</mi><mn>2</mn></msup></mrow></mfrac></mstyle></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m19.png" altimg-height="46px" altimg-valign="-16px" altimg-width="224px" alttext="\displaystyle=\frac{1}{2}\pi J_{\nu+1}\left(bz\right){H^{(1)}_{\nu}}\left(az%
\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mrow><mi>b</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><msubsup><mi href="./10.2#E5">H</mi><mi href="./10.1#p2.t1.r5">ν</mi><mrow><mo href="./10.2#E5" stretchy="false">(</mo><mn>1</mn><mo href="./10.2#E5" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mrow><mi>a</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m228.png" altimg-height="20px" altimg-valign="-5px" altimg-width="86px" alttext="a\geq b>0" display="inline"><mrow><mi>a</mi><mo>≥</mo><mi>b</mi><mo>></mo><mn>0</mn></mrow></math>, <math class="ltx_Math" altimg="m146.png" altimg-height="27px" altimg-valign="-9px" altimg-width="155px" alttext="\Re\nu>-\tfrac{3}{2},\Im z>0" display="inline"><mrow><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo>></mo><mrow><mo>-</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow></mrow><mo>,</mo><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℑ</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo>></mo><mn>0</mn></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E70.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./10.2#E3" title="(10.2.3) ‣ Bessel Function of the Second Kind (Weber’s Function) ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m112.png" altimg-height="23px" altimg-valign="-7px" altimg-width="55px" alttext="Y_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E3">Y</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the second kind</a>,
<a href="./10.2#E5" title="(10.2.5) ‣ Bessel Functions of the Third Kind (Hankel Functions) ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m259.png" altimg-height="29px" altimg-valign="-7px" altimg-width="73px" alttext="{H^{(1)}_{\NVar{\nu}}}\left(\NVar{z}\right)" display="inline"><mrow><msubsup><mi href="./10.2#E5">H</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi><mrow><mo href="./10.2#E5" stretchy="false">(</mo><mn>1</mn><mo href="./10.2#E5" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the third kind (or Hankel function)</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m211.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m115.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Im" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℑ</mi><mo></mo></mrow></math>: imaginary part</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m163.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m128.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./10.1#p2.t1.r4" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m258.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./10.1#p2.t1.r4">z</mi></math>: complex variable</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
<p class="ltx_p">Equation () also remains valid if the order <math class="ltx_Math" altimg="m199.png" altimg-height="18px" altimg-valign="-4px" altimg-width="50px" alttext="\nu+1" display="inline"><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mn>1</mn></mrow></math> of the
<math class="ltx_Math" altimg="m102.png" altimg-height="18px" altimg-valign="-2px" altimg-width="17px" alttext="J" display="inline"><mi href="./10.2#E2">J</mi></math> functions on both sides is replaced by <math class="ltx_Math" altimg="m200.png" altimg-height="18px" altimg-valign="-4px" altimg-width="96px" alttext="\nu+2n-3" display="inline"><mrow><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r1">n</mi></mrow></mrow><mo>-</mo><mn>3</mn></mrow></math>, <math class="ltx_Math" altimg="m246.png" altimg-height="20px" altimg-valign="-6px" altimg-width="104px" alttext="n=1,2,\dots" display="inline"><mrow><mi href="./10.1#p2.t1.r1">n</mi><mo>=</mo><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>,
and the constraint <math class="ltx_Math" altimg="m144.png" altimg-height="27px" altimg-valign="-9px" altimg-width="85px" alttext="\Re\nu>-\frac{3}{2}" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo>></mo><mrow><mo>-</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow></mrow></math> is replaced by
<math class="ltx_Math" altimg="m147.png" altimg-height="27px" altimg-valign="-9px" altimg-width="121px" alttext="\Re\nu>-n+\frac{1}{2}" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo>></mo><mrow><mrow><mo>-</mo><mi href="./10.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow></math>.</p>
</div>
<div id="Px9.p2" class="ltx_para">
<p class="ltx_p">See also §) <math class="ltx_Math" altimg="m222.png" altimg-height="21px" altimg-valign="-6px" altimg-width="50px" alttext="a,b,c" display="inline"><mrow><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi></mrow></math> are positive
constants.</p>
<table id="EGx14" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E71">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.22.71</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m61.png" altimg-height="53px" altimg-valign="-20px" altimg-width="282px" alttext="\displaystyle\int_{0}^{\infty}J_{\mu}\left(at\right)J_{\nu}\left(bt\right)J_{%
\nu}\left(ct\right)t^{1-\mu}\mathrm{d}t" display="inline"><mrow><mstyle displaystyle="true"><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup></mstyle><mrow><mrow><msub><mi href="./10.2#E2">J</mi><mi>μ</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi>a</mi><mo></mo><mi>t</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi>b</mi><mo></mo><mi>t</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi>c</mi><mo></mo><mi>t</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><msup><mi>t</mi><mrow><mn>1</mn><mo>-</mo><mi>μ</mi></mrow></msup><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m16.png" altimg-height="60px" altimg-valign="-24px" altimg-width="296px" alttext="\displaystyle=\frac{(bc)^{\mu-1}(\sin\phi)^{\mu-\frac{1}{2}}}{(2\pi)^{\frac{1}%
{2}}a^{\mu}}\mathsf{P}^{\frac{1}{2}-\mu}_{\nu-\frac{1}{2}}(\cos\phi)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><mfrac><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mi>b</mi><mo></mo><mi>c</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mi>μ</mi><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi>ϕ</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mi>μ</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup></mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo stretchy="false">)</mo></mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></msup><mo></mo><msup><mi>a</mi><mi>μ</mi></msup></mrow></mfrac></mstyle><mo></mo><mrow><msubsup><mi href="./14.3#E1" mathvariant="sans-serif">P</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>-</mo><mi>μ</mi></mrow></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi>ϕ</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m133.png" altimg-height="27px" altimg-valign="-9px" altimg-width="587px" alttext="\Re\mu>-\tfrac{1}{2},\Re\nu>-1,|b-c|<a<b+c,\cos\phi=(b^{2}+c^{2}-a^{2})/(2bc)" display="inline"><mrow><mrow><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>μ</mi></mrow><mo>></mo><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow><mo>,</mo><mrow><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo>></mo><mrow><mo>-</mo><mn>1</mn></mrow></mrow><mo>,</mo><mrow><mrow><mo stretchy="false">|</mo><mrow><mi>b</mi><mo>-</mo><mi>c</mi></mrow><mo stretchy="false">|</mo></mrow><mo><</mo><mi>a</mi><mo><</mo><mrow><mi>b</mi><mo>+</mo><mi>c</mi></mrow></mrow></mrow></mrow><mo>,</mo><mrow><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi>ϕ</mi></mrow><mo>=</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><msup><mi>b</mi><mn>2</mn></msup><mo>+</mo><msup><mi>c</mi><mn>2</mn></msup></mrow><mo>-</mo><msup><mi>a</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mo>/</mo><mrow><mo stretchy="false">(</mo><mrow><mn>2</mn><mo></mo><mi>b</mi><mo></mo><mi>c</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E71.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./14.3#E1" title="(14.3.1) ‣ Ferrers Function of the First Kind ‣ §14.3(i) Interval - 1 < x < 1 ‣ §14.3 Definitions and Hypergeometric Representations ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m185.png" altimg-height="23px" altimg-valign="-7px" altimg-width="58px" alttext="\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msubsup><mi href="./14.3#E1" mathvariant="sans-serif">P</mi><mi class="ltx_nvar" href="./14.1#p1.t1.r4">ν</mi><mi class="ltx_nvar" href="./14.1#p1.t1.r4">μ</mi></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./14.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow></math>: Ferrers function of the first kind</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m211.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m154.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m163.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m128.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m216.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E72">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.22.72</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m61.png" altimg-height="53px" altimg-valign="-20px" altimg-width="282px" alttext="\displaystyle\int_{0}^{\infty}J_{\mu}\left(at\right)J_{\nu}\left(bt\right)J_{%
\nu}\left(ct\right)t^{1-\mu}\mathrm{d}t" display="inline"><mrow><mstyle displaystyle="true"><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup></mstyle><mrow><mrow><msub><mi href="./10.2#E2">J</mi><mi>μ</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi>a</mi><mo></mo><mi>t</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi>b</mi><mo></mo><mi>t</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi>c</mi><mo></mo><mi>t</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><msup><mi>t</mi><mrow><mn>1</mn><mo>-</mo><mi>μ</mi></mrow></msup><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m17.png" altimg-height="61px" altimg-valign="-25px" altimg-width="391px" alttext="\displaystyle=\frac{(bc)^{\mu-1}\cos(\nu\pi)(\sinh\chi)^{\mu-\frac{1}{2}}}{(%
\frac{1}{2}\pi^{3})^{\frac{1}{2}}a^{\mu}}Q^{\frac{1}{2}-\mu}_{\nu-\frac{1}{2}}%
(\cosh\chi)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><mfrac><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mi>b</mi><mo></mo><mi>c</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mi>μ</mi><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./4.28#E1">sinh</mi><mo></mo><mi>χ</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mi>μ</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup></mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><msup><mi href="./3.12#E1">π</mi><mn>3</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></msup><mo></mo><msup><mi>a</mi><mi>μ</mi></msup></mrow></mfrac></mstyle><mo></mo><mrow><msubsup><mi href="./14.21#SS1.p1">Q</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>-</mo><mi>μ</mi></mrow></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./4.28#E2">cosh</mi><mo></mo><mi>χ</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m132.png" altimg-height="27px" altimg-valign="-9px" altimg-width="520px" alttext="\Re\mu>-\tfrac{1}{2},\Re\nu>-1,a>b+c,\cosh\chi=(a^{2}-b^{2}-c^{2})/(2bc)" display="inline"><mrow><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>μ</mi></mrow><mo>></mo><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow><mo>,</mo><mrow><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo>></mo><mrow><mo>-</mo><mn>1</mn></mrow></mrow><mo>,</mo><mrow><mrow><mi>a</mi><mo>></mo><mrow><mi>b</mi><mo>+</mo><mi>c</mi></mrow></mrow><mo>,</mo><mrow><mrow><mi href="./4.28#E2">cosh</mi><mo></mo><mi>χ</mi></mrow><mo>=</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>-</mo><msup><mi>b</mi><mn>2</mn></msup><mo>-</mo><msup><mi>c</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mo>/</mo><mrow><mo stretchy="false">(</mo><mrow><mn>2</mn><mo></mo><mi>b</mi><mo></mo><mi>c</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E72.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./14.21#SS1.p1" title="§14.21(i) Associated Legendre Equation ‣ §14.21 Definitions and Basic Properties ‣ Complex Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m110.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="Q^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msubsup><mi href="./14.21#SS1.p1">Q</mi><mi class="ltx_nvar" href="./14.1#p1.t1.r4">ν</mi><mi class="ltx_nvar" href="./14.1#p1.t1.r4">μ</mi></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./14.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: associated Legendre function of the second kind</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m211.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m154.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.28#E2" title="(4.28.2) ‣ §4.28 Definitions and Periodicity ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m155.png" altimg-height="18px" altimg-valign="-2px" altimg-width="56px" alttext="\cosh\NVar{z}" display="inline"><mrow><mi href="./4.28#E2">cosh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: hyperbolic cosine function</a>,
<a href="./4.28#E1" title="(4.28.1) ‣ §4.28 Definitions and Periodicity ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m217.png" altimg-height="18px" altimg-valign="-2px" altimg-width="53px" alttext="\sinh\NVar{z}" display="inline"><mrow><mi href="./4.28#E1">sinh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: hyperbolic sine function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m163.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m128.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
<p class="ltx_p">For the Ferrers function <math class="ltx_Math" altimg="m184.png" altimg-height="18px" altimg-valign="-2px" altimg-width="17px" alttext="\mathsf{P}" display="inline"><mi href="./14.2#SS2.p2" mathvariant="sans-serif">P</mi></math> and the associated Legendre function <math class="ltx_Math" altimg="m109.png" altimg-height="21px" altimg-valign="-6px" altimg-width="20px" alttext="Q" display="inline"><mi href="./14.2#SS2.p2">Q</mi></math>, see
§§), <math class="ltx_Math" altimg="m222.png" altimg-height="21px" altimg-valign="-6px" altimg-width="50px" alttext="a,b,c" display="inline"><mrow><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi></mrow></math> are positive
constants and
</p>
<table id="E73" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex7" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">10.22.73</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m45.png" altimg-height="19px" altimg-valign="-2px" altimg-width="21px" alttext="\displaystyle A" display="inline"><mi href="./10.22#Ex7">A</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m42.png" altimg-height="25px" altimg-valign="-7px" altimg-width="217px" alttext="\displaystyle=s(s-a)(s-b)(s-c)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./10.22#Ex8">s</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./10.22#Ex8">s</mi><mo>-</mo><mi>a</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./10.22#Ex8">s</mi><mo>-</mo><mi>b</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./10.22#Ex8">s</mi><mo>-</mo><mi>c</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex8" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m85.png" altimg-height="14px" altimg-valign="-2px" altimg-width="15px" alttext="\displaystyle s" display="inline"><mi href="./10.22#Ex8">s</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m36.png" altimg-height="29px" altimg-valign="-9px" altimg-width="137px" alttext="\displaystyle=\tfrac{1}{2}(a+b+c)." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>a</mi><mo>+</mo><mi>b</mi><mo>+</mo><mi>c</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E73.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.22#Ex7" title="(10.22.73) ‣ Triple Products ‣ §10.22(iv) Integrals over the Interval ( 0 , ∞ ) ‣ §10.22 Integrals ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m97.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="A" display="inline"><mi href="./10.22#Ex7">A</mi></math>: area</a> and
<a href="./10.22#Ex8" title="(10.22.73) ‣ Triple Products ‣ §10.22(iv) Integrals over the Interval ( 0 , ∞ ) ‣ §10.22 Integrals ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m249.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="s" display="inline"><mi href="./10.22#Ex8">s</mi></math>: sum</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">(Thus if <math class="ltx_Math" altimg="m222.png" altimg-height="21px" altimg-valign="-6px" altimg-width="50px" alttext="a,b,c" display="inline"><mrow><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi></mrow></math> are the sides of a triangle, then <math class="ltx_Math" altimg="m98.png" altimg-height="23px" altimg-valign="-2px" altimg-width="32px" alttext="A^{\frac{1}{2}}" display="inline"><msup><mi href="./10.22#Ex7">A</mi><mfrac><mn>1</mn><mn>2</mn></mfrac></msup></math> is the
area of the triangle.)
</p>
</div>
<div id="Px10.p3" class="ltx_para">
<p class="ltx_p">If <math class="ltx_Math" altimg="m145.png" altimg-height="27px" altimg-valign="-9px" altimg-width="85px" alttext="\Re\nu>-\tfrac{1}{2}" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo>></mo><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow></math>, then</p>
<table id="EGx15" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E74">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.22.74</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m62.png" altimg-height="53px" altimg-valign="-20px" altimg-width="281px" alttext="\displaystyle\int_{0}^{\infty}J_{\nu}\left(at\right)J_{\nu}\left(bt\right)J_{%
\nu}\left(ct\right)t^{1-\nu}\mathrm{d}t" display="inline"><mrow><mstyle displaystyle="true"><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup></mstyle><mrow><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi>a</mi><mo></mo><mi>t</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi>b</mi><mo></mo><mi>t</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi>c</mi><mo></mo><mi>t</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><msup><mi>t</mi><mrow><mn>1</mn><mo>-</mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow></msup><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m12.png" altimg-height="90px" altimg-valign="-39px" altimg-width="294px" alttext="\displaystyle=\begin{cases}\dfrac{2^{\nu-1}A^{\nu-\frac{1}{2}}}{\pi^{\frac{1}{%
2}}(abc)^{\nu}\Gamma\left(\nu+\frac{1}{2}\right)},&A>0,\\
0,&A\leq 0.\end{cases}" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mo>{</mo><mtable columnspacing="5pt" rowspacing="0pt"><mtr><mtd columnalign="left"><mrow><mstyle displaystyle="true"><mfrac><mrow><msup><mn>2</mn><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><msup><mi href="./10.22#Ex7">A</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup></mrow><mrow><msup><mi href="./3.12#E1">π</mi><mfrac><mn>1</mn><mn>2</mn></mfrac></msup><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi>a</mi><mo></mo><mi>b</mi><mo></mo><mi>c</mi></mrow><mo stretchy="false">)</mo></mrow><mi href="./10.1#p2.t1.r5">ν</mi></msup><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow></mrow></mfrac></mstyle><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mrow><mi href="./10.22#Ex7">A</mi><mo>></mo><mn>0</mn></mrow><mo>,</mo></mrow></mtd></mtr><mtr><mtd columnalign="left"><mrow><mn>0</mn><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mrow><mi href="./10.22#Ex7">A</mi><mo>≤</mo><mn>0</mn></mrow><mo>.</mo></mrow></mtd></mtr></mtable></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E74.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m114.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m211.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m163.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a> and
<a href="./10.22#Ex7" title="(10.22.73) ‣ Triple Products ‣ §10.22(iv) Integrals over the Interval ( 0 , ∞ ) ‣ §10.22 Integrals ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m97.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="A" display="inline"><mi href="./10.22#Ex7">A</mi></math>: area</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tr class="ltx_eqn_row ltx_align_baseline"><td class="ltx_eqn_cell ltx_align_left" style="white-space:normal;" colspan="5">If <math class="ltx_Math" altimg="m266.png" altimg-height="27px" altimg-valign="-9px" altimg-width="66px" alttext="|\nu|<\tfrac{1}{2}" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mi href="./10.1#p2.t1.r5">ν</mi><mo stretchy="false">|</mo></mrow><mo><</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math>, then</td></tr>
<tbody id="E75">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.22.75</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m68.png" altimg-height="53px" altimg-valign="-20px" altimg-width="281px" alttext="\displaystyle\int_{0}^{\infty}Y_{\nu}\left(at\right)J_{\nu}\left(bt\right)J_{%
\nu}\left(ct\right)t^{1+\nu}\mathrm{d}t" display="inline"><mrow><mstyle displaystyle="true"><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup></mstyle><mrow><mrow><msub><mi href="./10.2#E3">Y</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi>a</mi><mo></mo><mi>t</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi>b</mi><mo></mo><mi>t</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi>c</mi><mo></mo><mi>t</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><msup><mi>t</mi><mrow><mn>1</mn><mo>+</mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow></msup><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m11.png" altimg-height="146px" altimg-valign="-67px" altimg-width="403px" alttext="\displaystyle=\begin{cases}-\dfrac{(abc)^{\nu}(-A)^{-\nu-\frac{1}{2}}}{\pi^{%
\frac{1}{2}}2^{\nu+1}\Gamma\left(\frac{1}{2}-\nu\right)},&0<a<|b-c|,\\
0,&|b-c|<a<b+c,\\
\dfrac{(abc)^{\nu}(-A)^{-\nu-\frac{1}{2}}}{\pi^{\frac{1}{2}}2^{\nu+1}\Gamma%
\left(\frac{1}{2}-\nu\right)},&a>b+c.\end{cases}" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mo>{</mo><mtable columnspacing="5pt" rowspacing="0pt"><mtr><mtd columnalign="left"><mrow><mrow><mo>-</mo><mstyle displaystyle="true"><mfrac><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mi>a</mi><mo></mo><mi>b</mi><mo></mo><mi>c</mi></mrow><mo stretchy="false">)</mo></mrow><mi href="./10.1#p2.t1.r5">ν</mi></msup><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mi href="./10.22#Ex7">A</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mrow><mo>-</mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup></mrow><mrow><msup><mi href="./3.12#E1">π</mi><mfrac><mn>1</mn><mn>2</mn></mfrac></msup><mo></mo><msup><mn>2</mn><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mn>1</mn></mrow></msup><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>-</mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo>)</mo></mrow></mrow></mrow></mfrac></mstyle></mrow><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mrow><mn>0</mn><mo><</mo><mi>a</mi><mo><</mo><mrow><mo stretchy="false">|</mo><mrow><mi>b</mi><mo>-</mo><mi>c</mi></mrow><mo stretchy="false">|</mo></mrow></mrow><mo>,</mo></mrow></mtd></mtr><mtr><mtd columnalign="left"><mrow><mn>0</mn><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mrow><mrow><mo stretchy="false">|</mo><mrow><mi>b</mi><mo>-</mo><mi>c</mi></mrow><mo stretchy="false">|</mo></mrow><mo><</mo><mi>a</mi><mo><</mo><mrow><mi>b</mi><mo>+</mo><mi>c</mi></mrow></mrow><mo>,</mo></mrow></mtd></mtr><mtr><mtd columnalign="left"><mrow><mstyle displaystyle="true"><mfrac><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mi>a</mi><mo></mo><mi>b</mi><mo></mo><mi>c</mi></mrow><mo stretchy="false">)</mo></mrow><mi href="./10.1#p2.t1.r5">ν</mi></msup><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mi href="./10.22#Ex7">A</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mrow><mo>-</mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup></mrow><mrow><msup><mi href="./3.12#E1">π</mi><mfrac><mn>1</mn><mn>2</mn></mfrac></msup><mo></mo><msup><mn>2</mn><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mn>1</mn></mrow></msup><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>-</mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo>)</mo></mrow></mrow></mrow></mfrac></mstyle><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mrow><mi>a</mi><mo>></mo><mrow><mi>b</mi><mo>+</mo><mi>c</mi></mrow></mrow><mo>.</mo></mrow></mtd></mtr></mtable></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E75.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./10.2#E3" title="(10.2.3) ‣ Bessel Function of the Second Kind (Weber’s Function) ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m112.png" altimg-height="23px" altimg-valign="-7px" altimg-width="55px" alttext="Y_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E3">Y</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the second kind</a>,
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m114.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m211.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m163.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a> and
<a href="./10.22#Ex7" title="(10.22.73) ‣ Triple Products ‣ §10.22(iv) Integrals over the Interval ( 0 , ∞ ) ‣ §10.22 Integrals ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m97.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="A" display="inline"><mi href="./10.22#Ex7">A</mi></math>: area</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
</div>
<div id="Px10.p4" class="ltx_para">
<p class="ltx_p">Additional infinite integrals over the product of three Bessel functions
(including modified Bessel functions) are given in
<cite class="ltx_cite ltx_citemacro_citet">Gervois and Navelet (</dd>
</dl>
</div>
</div>
<div id="SS5.p1" class="ltx_para">
<p class="ltx_p">The <em class="ltx_emph ltx_font_italic">Hankel transform</em> (or <em class="ltx_emph ltx_font_italic">Bessel transform</em>) of a function
<math class="ltx_Math" altimg="m235.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="f(x)" display="inline"><mrow><mi href="./10.22#SS5.p1">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo stretchy="false">)</mo></mrow></mrow></math> is defined as
</p>
<table id="E76" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.22.76</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E76.png" altimg-height="53px" altimg-valign="-20px" altimg-width="291px" alttext="g(y)=\int_{0}^{\infty}f(x)J_{\nu}\left(xy\right)(xy)^{\frac{1}{2}}\mathrm{d}x." display="block"><mrow><mrow><mrow><mi href="./10.22#SS5.p1">g</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./10.1#p2.t1.r3">y</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><mrow><mrow><mi href="./10.22#SS5.p1">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi href="./10.1#p2.t1.r3">x</mi><mo></mo><mi href="./10.1#p2.t1.r3">y</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./10.1#p2.t1.r3">x</mi><mo></mo><mi href="./10.1#p2.t1.r3">y</mi></mrow><mo stretchy="false">)</mo></mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></msup><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./10.1#p2.t1.r3">x</mi></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E76.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m163.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./10.1#p2.t1.r3" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./10.1#p2.t1.r3">x</mi></math>: real variable</a>,
<a href="./10.1#p2.t1.r3" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m256.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./10.1#p2.t1.r3">y</mi></math>: real variable</a>,
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>,
<a href="./10.22#SS5.p1" title="§10.22(v) Hankel Transform ‣ §10.22 Integrals ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m235.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="f(x)" display="inline"><mrow><mi href="./10.22#SS5.p1">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function</a> and
<a href="./10.22#SS5.p1" title="§10.22(v) Hankel Transform ‣ §10.22 Integrals ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m236.png" altimg-height="23px" altimg-valign="-7px" altimg-width="40px" alttext="g(y)" display="inline"><mrow><mi href="./10.22#SS5.p1">g</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./10.1#p2.t1.r3">y</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS5.p2" class="ltx_para">
<p class="ltx_p"><em class="ltx_emph ltx_font_italic">Hankel’s inversion theorem</em> is given by</p>
<table id="E77" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.22.77</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E77.png" altimg-height="53px" altimg-valign="-20px" altimg-width="291px" alttext="f(y)=\int_{0}^{\infty}g(x)J_{\nu}\left(xy\right)(xy)^{\frac{1}{2}}\mathrm{d}x." display="block"><mrow><mrow><mrow><mi href="./10.22#SS5.p1">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./10.1#p2.t1.r3">y</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><mrow><mrow><mi href="./10.22#SS5.p1">g</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi href="./10.1#p2.t1.r3">x</mi><mo></mo><mi href="./10.1#p2.t1.r3">y</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./10.1#p2.t1.r3">x</mi><mo></mo><mi href="./10.1#p2.t1.r3">y</mi></mrow><mo stretchy="false">)</mo></mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></msup><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./10.1#p2.t1.r3">x</mi></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E77.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m163.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./10.1#p2.t1.r3" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m255.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./10.1#p2.t1.r3">x</mi></math>: real variable</a>,
<a href="./10.1#p2.t1.r3" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m256.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./10.1#p2.t1.r3">y</mi></math>: real variable</a>,
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>,
<a href="./10.22#SS5.p1" title="§10.22(v) Hankel Transform ‣ §10.22 Integrals ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m235.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="f(x)" display="inline"><mrow><mi href="./10.22#SS5.p1">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function</a> and
<a href="./10.22#SS5.p1" title="§10.22(v) Hankel Transform ‣ §10.22 Integrals ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m236.png" altimg-height="23px" altimg-valign="-7px" altimg-width="40px" alttext="g(y)" display="inline"><mrow><mi href="./10.22#SS5.p1">g</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./10.1#p2.t1.r3">y</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Sufficient conditions for the validity of () are that
<math class="ltx_Math" altimg="m166.png" altimg-height="28px" altimg-valign="-9px" altimg-width="157px" alttext="\int_{0}^{\infty}|f(x)|\mathrm{d}x<\infty" display="inline"><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./10.22#SS5.p1">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./10.1#p2.t1.r3">x</mi></mrow></mrow></mrow><mo><</mo><mi mathvariant="normal">∞</mi></mrow></math> when <math class="ltx_Math" altimg="m206.png" altimg-height="27px" altimg-valign="-9px" altimg-width="70px" alttext="\nu\geq-\tfrac{1}{2}" display="inline"><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>≥</mo><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow></math>, or that
<math class="ltx_Math" altimg="m166.png" altimg-height="28px" altimg-valign="-9px" altimg-width="157px" alttext="\int_{0}^{\infty}|f(x)|\mathrm{d}x<\infty" display="inline"><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./10.22#SS5.p1">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./10.1#p2.t1.r3">x</mi></mrow></mrow></mrow><mo><</mo><mi mathvariant="normal">∞</mi></mrow></math> and
<math class="ltx_Math" altimg="m164.png" altimg-height="31px" altimg-valign="-9px" altimg-width="194px" alttext="\int_{0}^{1}x^{\nu+\frac{1}{2}}|f(x)|\mathrm{d}x<\infty" display="inline"><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mn>1</mn></msubsup><mrow><msup><mi href="./10.1#p2.t1.r3">x</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup><mo></mo><mrow><mo stretchy="false">|</mo><mrow><mi href="./10.22#SS5.p1">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./10.1#p2.t1.r3">x</mi></mrow></mrow></mrow><mo><</mo><mi mathvariant="normal">∞</mi></mrow></math> when
<math class="ltx_Math" altimg="m92.png" altimg-height="27px" altimg-valign="-9px" altimg-width="122px" alttext="-1<\nu<-\tfrac{1}{2}" display="inline"><mrow><mrow><mo>-</mo><mn>1</mn></mrow><mo><</mo><mi href="./10.1#p2.t1.r5">ν</mi><mo><</mo><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow></math>; see
<cite class="ltx_cite ltx_citemacro_citet">Titchmarsh (</dd>
</dl>
</div>
</div>
<div id="SS6.p1" class="ltx_para">
<p class="ltx_p">For collections of integrals of the functions <math class="ltx_Math" altimg="m105.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\nu}\left(z\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>,
<math class="ltx_Math" altimg="m113.png" altimg-height="23px" altimg-valign="-7px" altimg-width="55px" alttext="Y_{\nu}\left(z\right)" display="inline"><mrow><msub><mi href="./10.2#E3">Y</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>, <math class="ltx_Math" altimg="m260.png" altimg-height="29px" altimg-valign="-7px" altimg-width="73px" alttext="{H^{(1)}_{\nu}}\left(z\right)" display="inline"><mrow><msubsup><mi href="./10.2#E5">H</mi><mi href="./10.1#p2.t1.r5">ν</mi><mrow><mo href="./10.2#E5" stretchy="false">(</mo><mn>1</mn><mo href="./10.2#E5" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>, and <math class="ltx_Math" altimg="m261.png" altimg-height="29px" altimg-valign="-7px" altimg-width="73px" alttext="{H^{(2)}_{\nu}}\left(z\right)" display="inline"><mrow><msubsup><mi href="./10.2#E6">H</mi><mi href="./10.1#p2.t1.r5">ν</mi><mrow><mo href="./10.2#E6" stretchy="false">(</mo><mn>2</mn><mo href="./10.2#E6" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>,
including integrals with respect to the order, see
<cite class="ltx_cite ltx_citemacro_citet">Andrews<span class="ltx_text ltx_bib_etal"> et al.</span> (</div>
</div>
</body></text>
</html>
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<page>
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<head>
<title>DLMF: 10.24 Functions of Imaginary Order</title>
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<div class="ltx_page_navbar">
<div class="ltx_page_navlogo"></dd>
</dl>
</div>
</div>
<div id="p1" class="ltx_para">
<p class="ltx_p">With <math class="ltx_Math" altimg="m50.png" altimg-height="13px" altimg-valign="-2px" altimg-width="52px" alttext="z=x" display="inline"><mrow><mi href="./10.1#p2.t1.r4">z</mi><mo>=</mo><mi href="./10.1#p2.t1.r3">x</mi></mrow></math> and <math class="ltx_Math" altimg="m32.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math> replaced by <math class="ltx_Math" altimg="m46.png" altimg-height="17px" altimg-valign="-2px" altimg-width="22px" alttext="i\nu" display="inline"><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow></math>, Bessel’s equation ()
becomes</p>
<table id="E1" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.24.1</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E1.png" altimg-height="51px" altimg-valign="-18px" altimg-width="294px" alttext="x^{2}\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}x}^{2}}+x\frac{\mathrm{d}w}{\mathrm{d%
}x}+(x^{2}+\nu^{2})w=0." display="block"><mrow><mrow><mrow><mrow><msup><mi href="./10.1#p2.t1.r3">x</mi><mn>2</mn></msup><mo></mo><mfrac><mrow><mpadded lspace="-1.7pt" width="-4.5pt"><msup><mo href="./1.4#E4">d</mo><mn>2</mn></msup></mpadded><mi>w</mi></mrow><msup><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi href="./10.1#p2.t1.r3">x</mi></mrow><mn>2</mn></msup></mfrac></mrow><mo>+</mo><mrow><mi href="./10.1#p2.t1.r3">x</mi><mo></mo><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi>w</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi href="./10.1#p2.t1.r3">x</mi></mrow></mfrac></mrow><mo>+</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><msup><mi href="./10.1#p2.t1.r3">x</mi><mn>2</mn></msup><mo>+</mo><msup><mi href="./10.1#p2.t1.r5">ν</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi>w</mi></mrow></mrow><mo>=</mo><mn>0</mn></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#E4" title="(1.4.4) ‣ §1.4(iii) Derivatives ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="29px" altimg-valign="-9px" altimg-width="27px" alttext="\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: derivative of <math class="ltx_Math" altimg="m45.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m47.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./10.1#p2.t1.r3" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./10.1#p2.t1.r3">x</mi></math>: real variable</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m32.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">For <math class="ltx_Math" altimg="m33.png" altimg-height="18px" altimg-valign="-3px" altimg-width="54px" alttext="\nu\in\mathbb{R}" display="inline"><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mi href="./front/introduction#Sx4.p2.t1.r14">ℝ</mi></mrow></math> and <math class="ltx_Math" altimg="m47.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./10.1#p2.t1.r3">x</mi></math> <math class="ltx_Math" altimg="m26.png" altimg-height="23px" altimg-valign="-7px" altimg-width="77px" alttext="\in(0,\infty)" display="inline"><mrow><mi></mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mn>0</mn><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi mathvariant="normal">∞</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></mrow></math> define</p>
<table id="E2" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex1" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">10.24.2</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m10.png" altimg-height="30px" altimg-valign="-7px" altimg-width="57px" alttext="\displaystyle\widetilde{J}_{\nu}\left(x\right)" display="inline"><mrow><msub><mover accent="true"><mi href="./10.24#Ex1">J</mi><mo href="./10.24#Ex1">~</mo></mover><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m3.png" altimg-height="29px" altimg-valign="-9px" altimg-width="216px" alttext="\displaystyle=\operatorname{sech}\left(\tfrac{1}{2}\pi\nu\right)\Re(J_{i\nu}%
\left(x\right))," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mi href="./4.28#E6">sech</mi><mo></mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><msub><mi href="./10.2#E2">J</mi><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex2" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m12.png" altimg-height="30px" altimg-valign="-7px" altimg-width="58px" alttext="\displaystyle\widetilde{Y}_{\nu}\left(x\right)" display="inline"><mrow><msub><mover accent="true"><mi href="./10.24#Ex2">Y</mi><mo href="./10.24#Ex2">~</mo></mover><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m4.png" altimg-height="29px" altimg-valign="-9px" altimg-width="217px" alttext="\displaystyle=\operatorname{sech}\left(\tfrac{1}{2}\pi\nu\right)\Re(Y_{i\nu}%
\left(x\right))," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mi href="./4.28#E6">sech</mi><mo></mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><msub><mi href="./10.2#E3">Y</mi><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m15.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./10.24#Ex1" title="(10.24.2) ‣ §10.24 Functions of Imaginary Order ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m41.png" altimg-height="28px" altimg-valign="-7px" altimg-width="55px" alttext="\widetilde{J}_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mover accent="true"><mi href="./10.24#Ex1">J</mi><mo href="./10.24#Ex1">~</mo></mover><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: Bessel function of imaginary order</a>,
<a href="./10.2#E3" title="(10.2.3) ‣ Bessel Function of the Second Kind (Weber’s Function) ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m17.png" altimg-height="23px" altimg-valign="-7px" altimg-width="55px" alttext="Y_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E3">Y</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the second kind</a>,
<a href="./10.24#Ex2" title="(10.24.2) ‣ §10.24 Functions of Imaginary Order ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="28px" altimg-valign="-7px" altimg-width="56px" alttext="\widetilde{Y}_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mover accent="true"><mi href="./10.24#Ex2">Y</mi><mo href="./10.24#Ex2">~</mo></mover><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: Bessel function of imaginary order</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m36.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.28#E6" title="(4.28.6) ‣ §4.28 Definitions and Periodicity ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m35.png" altimg-height="18px" altimg-valign="-2px" altimg-width="54px" alttext="\operatorname{sech}\NVar{z}" display="inline"><mrow><mi href="./4.28#E6">sech</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: hyperbolic secant function</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./10.1#p2.t1.r3" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./10.1#p2.t1.r3">x</mi></math>: real variable</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m32.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and</p>
<table id="E3" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.24.3</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E3.png" altimg-height="60px" altimg-valign="-21px" altimg-width="273px" alttext="\Gamma\left(1+i\nu\right)=\left(\frac{\pi\nu}{\sinh(\pi\nu)}\right)^{\frac{1}{%
2}}e^{i\gamma_{\nu}}," display="block"><mrow><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msup><mrow><mo>(</mo><mfrac><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mrow><mi href="./4.28#E1">sinh</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mfrac><mo>)</mo></mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></msup><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mo></mo><msub><mi href="./10.24#p1">γ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub></mrow></msup></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m18.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m36.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m28.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./4.28#E1" title="(4.28.1) ‣ §4.28 Definitions and Periodicity ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="18px" altimg-valign="-2px" altimg-width="53px" alttext="\sinh\NVar{z}" display="inline"><mrow><mi href="./4.28#E1">sinh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: hyperbolic sine function</a>,
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m32.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a> and
<a href="./10.24#p1" title="§10.24 Functions of Imaginary Order ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m25.png" altimg-height="16px" altimg-valign="-6px" altimg-width="25px" alttext="\gamma_{\nu}" display="inline"><msub><mi href="./10.24#p1">γ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub></math>: parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m25.png" altimg-height="16px" altimg-valign="-6px" altimg-width="25px" alttext="\gamma_{\nu}" display="inline"><msub><mi href="./10.24#p1">γ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub></math> is real and continuous with <math class="ltx_Math" altimg="m24.png" altimg-height="20px" altimg-valign="-6px" altimg-width="60px" alttext="\gamma_{0}=0" display="inline"><mrow><msub><mi href="./10.24#p1">γ</mi><mn>0</mn></msub><mo>=</mo><mn>0</mn></mrow></math>; compare
(). Then</p>
<table id="E4" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex3" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">10.24.4</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m9.png" altimg-height="30px" altimg-valign="-7px" altimg-width="70px" alttext="\displaystyle\widetilde{J}_{-\nu}\left(x\right)" display="inline"><mrow><msub><mover accent="true"><mi href="./10.24#Ex1">J</mi><mo href="./10.24#Ex1">~</mo></mover><mrow><mo>-</mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m7.png" altimg-height="30px" altimg-valign="-7px" altimg-width="87px" alttext="\displaystyle=\widetilde{J}_{\nu}\left(x\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msub><mover accent="true"><mi href="./10.24#Ex1">J</mi><mo href="./10.24#Ex1">~</mo></mover><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex4" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m11.png" altimg-height="30px" altimg-valign="-7px" altimg-width="70px" alttext="\displaystyle\widetilde{Y}_{-\nu}\left(x\right)" display="inline"><mrow><msub><mover accent="true"><mi href="./10.24#Ex2">Y</mi><mo href="./10.24#Ex2">~</mo></mover><mrow><mo>-</mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m8.png" altimg-height="30px" altimg-valign="-7px" altimg-width="88px" alttext="\displaystyle=\widetilde{Y}_{\nu}\left(x\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msub><mover accent="true"><mi href="./10.24#Ex2">Y</mi><mo href="./10.24#Ex2">~</mo></mover><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E4.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.24#Ex1" title="(10.24.2) ‣ §10.24 Functions of Imaginary Order ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m41.png" altimg-height="28px" altimg-valign="-7px" altimg-width="55px" alttext="\widetilde{J}_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mover accent="true"><mi href="./10.24#Ex1">J</mi><mo href="./10.24#Ex1">~</mo></mover><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: Bessel function of imaginary order</a>,
<a href="./10.24#Ex2" title="(10.24.2) ‣ §10.24 Functions of Imaginary Order ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="28px" altimg-valign="-7px" altimg-width="56px" alttext="\widetilde{Y}_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mover accent="true"><mi href="./10.24#Ex2">Y</mi><mo href="./10.24#Ex2">~</mo></mover><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: Bessel function of imaginary order</a>,
<a href="./10.1#p2.t1.r3" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./10.1#p2.t1.r3">x</mi></math>: real variable</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m32.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and <math class="ltx_Math" altimg="m42.png" altimg-height="28px" altimg-valign="-7px" altimg-width="55px" alttext="\widetilde{J}_{\nu}\left(x\right)" display="inline"><mrow><msub><mover accent="true"><mi href="./10.24#Ex1">J</mi><mo href="./10.24#Ex1">~</mo></mover><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>, <math class="ltx_Math" altimg="m44.png" altimg-height="28px" altimg-valign="-7px" altimg-width="56px" alttext="\widetilde{Y}_{\nu}\left(x\right)" display="inline"><mrow><msub><mover accent="true"><mi href="./10.24#Ex2">Y</mi><mo href="./10.24#Ex2">~</mo></mover><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math> are linearly
independent solutions of ():</p>
<table id="E5" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.24.5</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E5.png" altimg-height="30px" altimg-valign="-7px" altimg-width="254px" alttext="\mathscr{W}\{\widetilde{J}_{\nu}\left(x\right),\widetilde{Y}_{\nu}\left(x%
\right)\}=2/(\pi x)." display="block"><mrow><mrow><mrow><mi class="ltx_font_mathscript" href="./1.13#E4">𝒲</mi><mo></mo><mrow><mo stretchy="false">{</mo><mrow><msub><mover accent="true"><mi href="./10.24#Ex1">J</mi><mo href="./10.24#Ex1">~</mo></mover><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow><mo>,</mo><mrow><msub><mover accent="true"><mi href="./10.24#Ex2">Y</mi><mo href="./10.24#Ex2">~</mo></mover><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow><mo stretchy="false">}</mo></mrow></mrow><mo>=</mo><mrow><mn>2</mn><mo>/</mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi href="./10.1#p2.t1.r3">x</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E5.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.24#Ex1" title="(10.24.2) ‣ §10.24 Functions of Imaginary Order ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m41.png" altimg-height="28px" altimg-valign="-7px" altimg-width="55px" alttext="\widetilde{J}_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mover accent="true"><mi href="./10.24#Ex1">J</mi><mo href="./10.24#Ex1">~</mo></mover><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: Bessel function of imaginary order</a>,
<a href="./10.24#Ex2" title="(10.24.2) ‣ §10.24 Functions of Imaginary Order ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="28px" altimg-valign="-7px" altimg-width="56px" alttext="\widetilde{Y}_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mover accent="true"><mi href="./10.24#Ex2">Y</mi><mo href="./10.24#Ex2">~</mo></mover><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: Bessel function of imaginary order</a>,
<a href="./1.13#E4" title="(1.13.4) ‣ Wronskian ‣ §1.13(i) Existence of Solutions ‣ §1.13 Differential Equations ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="18px" altimg-valign="-2px" altimg-width="26px" alttext="\mathscr{W}" display="inline"><mi class="ltx_font_mathscript" href="./1.13#E4">𝒲</mi></math>: Wronskian</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m36.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./10.1#p2.t1.r3" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./10.1#p2.t1.r3">x</mi></math>: real variable</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m32.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="p2" class="ltx_para">
<p class="ltx_p">As <math class="ltx_Math" altimg="m49.png" altimg-height="17px" altimg-valign="-4px" altimg-width="82px" alttext="x\to+\infty" display="inline"><mrow><mi href="./10.1#p2.t1.r3">x</mi><mo>→</mo><mrow><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></mrow></math>, with <math class="ltx_Math" altimg="m32.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math> fixed,</p>
<table id="E6" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex5" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">10.24.6</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m10.png" altimg-height="30px" altimg-valign="-7px" altimg-width="57px" alttext="\displaystyle\widetilde{J}_{\nu}\left(x\right)" display="inline"><mrow><msub><mover accent="true"><mi href="./10.24#Ex1">J</mi><mo href="./10.24#Ex1">~</mo></mover><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m5.png" altimg-height="41px" altimg-valign="-15px" altimg-width="330px" alttext="\displaystyle=\sqrt{2/(\pi x)}\cos\left(x-\tfrac{1}{4}\pi\right)+O\left(x^{-%
\frac{3}{2}}\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><msqrt><mrow><mn>2</mn><mo>/</mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi href="./10.1#p2.t1.r3">x</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></msqrt><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./10.1#p2.t1.r3">x</mi><mo>-</mo><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mi href="./2.1#E3">O</mi><mo></mo><mrow><mo>(</mo><msup><mi href="./10.1#p2.t1.r3">x</mi><mrow><mo>-</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow></msup><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex6" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m12.png" altimg-height="30px" altimg-valign="-7px" altimg-width="58px" alttext="\displaystyle\widetilde{Y}_{\nu}\left(x\right)" display="inline"><mrow><msub><mover accent="true"><mi href="./10.24#Ex2">Y</mi><mo href="./10.24#Ex2">~</mo></mover><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m6.png" altimg-height="41px" altimg-valign="-15px" altimg-width="328px" alttext="\displaystyle=\sqrt{2/(\pi x)}\sin\left(x-\tfrac{1}{4}\pi\right)+O\left(x^{-%
\frac{3}{2}}\right)." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><msqrt><mrow><mn>2</mn><mo>/</mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi href="./10.1#p2.t1.r3">x</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></msqrt><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./10.1#p2.t1.r3">x</mi><mo>-</mo><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mi href="./2.1#E3">O</mi><mo></mo><mrow><mo>(</mo><msup><mi href="./10.1#p2.t1.r3">x</mi><mrow><mo>-</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow></msup><mo>)</mo></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E6.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.24#Ex1" title="(10.24.2) ‣ §10.24 Functions of Imaginary Order ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m41.png" altimg-height="28px" altimg-valign="-7px" altimg-width="55px" alttext="\widetilde{J}_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mover accent="true"><mi href="./10.24#Ex1">J</mi><mo href="./10.24#Ex1">~</mo></mover><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: Bessel function of imaginary order</a>,
<a href="./10.24#Ex2" title="(10.24.2) ‣ §10.24 Functions of Imaginary Order ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="28px" altimg-valign="-7px" altimg-width="56px" alttext="\widetilde{Y}_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mover accent="true"><mi href="./10.24#Ex2">Y</mi><mo href="./10.24#Ex2">~</mo></mover><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: Bessel function of imaginary order</a>,
<a href="./2.1#E3" title="(2.1.3) ‣ §2.1(i) Asymptotic and Order Symbols ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m16.png" altimg-height="23px" altimg-valign="-7px" altimg-width="50px" alttext="O\left(\NVar{x}\right)" display="inline"><mrow><mi href="./2.1#E3">O</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>)</mo></mrow></mrow></math>: order not exceeding</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m36.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./10.1#p2.t1.r3" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./10.1#p2.t1.r3">x</mi></math>: real variable</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m32.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="p3" class="ltx_para">
<p class="ltx_p">As <math class="ltx_Math" altimg="m48.png" altimg-height="18px" altimg-valign="-4px" altimg-width="72px" alttext="x\to 0+" display="inline"><mrow><mi href="./10.1#p2.t1.r3">x</mi><mo>→</mo><mrow><mn>0</mn><mo>+</mo></mrow></mrow></math>, with <math class="ltx_Math" altimg="m32.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math> fixed,</p>
<table id="EGx1" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E7">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.24.7</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m10.png" altimg-height="30px" altimg-valign="-7px" altimg-width="57px" alttext="\displaystyle\widetilde{J}_{\nu}\left(x\right)" display="inline"><mrow><msub><mover accent="true"><mi href="./10.24#Ex1">J</mi><mo href="./10.24#Ex1">~</mo></mover><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m2.png" altimg-height="72px" altimg-valign="-27px" altimg-width="444px" alttext="\displaystyle=\left(\frac{2\tanh\left(\frac{1}{2}\pi\nu\right)}{\pi\nu}\right)%
^{\frac{1}{2}}\cos\left(\nu\ln\left(\tfrac{1}{2}x\right)-\gamma_{\nu}\right)+O%
\left(x^{2}\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><msup><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac><mrow><mn>2</mn><mo></mo><mrow><mi href="./4.28#E4">tanh</mi><mo></mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo>)</mo></mrow></mrow></mrow><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow></mfrac></mstyle><mo>)</mo></mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></msup><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo></mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./10.1#p2.t1.r3">x</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>-</mo><msub><mi href="./10.24#p1">γ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub></mrow><mo>)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mi href="./2.1#E3">O</mi><mo></mo><mrow><mo>(</mo><msup><mi href="./10.1#p2.t1.r3">x</mi><mn>2</mn></msup><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E7.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.24#Ex1" title="(10.24.2) ‣ §10.24 Functions of Imaginary Order ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m41.png" altimg-height="28px" altimg-valign="-7px" altimg-width="55px" alttext="\widetilde{J}_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mover accent="true"><mi href="./10.24#Ex1">J</mi><mo href="./10.24#Ex1">~</mo></mover><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: Bessel function of imaginary order</a>,
<a href="./2.1#E3" title="(2.1.3) ‣ §2.1(i) Asymptotic and Order Symbols ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m16.png" altimg-height="23px" altimg-valign="-7px" altimg-width="50px" alttext="O\left(\NVar{x}\right)" display="inline"><mrow><mi href="./2.1#E3">O</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>)</mo></mrow></mrow></math>: order not exceeding</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m36.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./4.28#E4" title="(4.28.4) ‣ §4.28 Definitions and Periodicity ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m40.png" altimg-height="18px" altimg-valign="-2px" altimg-width="58px" alttext="\tanh\NVar{z}" display="inline"><mrow><mi href="./4.28#E4">tanh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: hyperbolic tangent function</a>,
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m27.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a>,
<a href="./10.1#p2.t1.r3" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./10.1#p2.t1.r3">x</mi></math>: real variable</a>,
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m32.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a> and
<a href="./10.24#p1" title="§10.24 Functions of Imaginary Order ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m25.png" altimg-height="16px" altimg-valign="-6px" altimg-width="25px" alttext="\gamma_{\nu}" display="inline"><msub><mi href="./10.24#p1">γ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub></math>: parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E8">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.24.8</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m12.png" altimg-height="30px" altimg-valign="-7px" altimg-width="58px" alttext="\displaystyle\widetilde{Y}_{\nu}\left(x\right)" display="inline"><mrow><msub><mover accent="true"><mi href="./10.24#Ex2">Y</mi><mo href="./10.24#Ex2">~</mo></mover><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m1.png" altimg-height="72px" altimg-valign="-27px" altimg-width="443px" alttext="\displaystyle=\left(\frac{2\coth\left(\frac{1}{2}\pi\nu\right)}{\pi\nu}\right)%
^{\frac{1}{2}}\*\sin\left(\nu\ln\left(\tfrac{1}{2}x\right)-\gamma_{\nu}\right)%
+O\left(x^{2}\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><msup><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac><mrow><mn>2</mn><mo></mo><mrow><mi href="./4.28#E7">coth</mi><mo></mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo>)</mo></mrow></mrow></mrow><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow></mfrac></mstyle><mo>)</mo></mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></msup><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo></mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./10.1#p2.t1.r3">x</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>-</mo><msub><mi href="./10.24#p1">γ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub></mrow><mo>)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mi href="./2.1#E3">O</mi><mo></mo><mrow><mo>(</mo><msup><mi href="./10.1#p2.t1.r3">x</mi><mn>2</mn></msup><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m31.png" altimg-height="17px" altimg-valign="-3px" altimg-width="52px" alttext="\nu>0" display="inline"><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>></mo><mn>0</mn></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E8.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.24#Ex2" title="(10.24.2) ‣ §10.24 Functions of Imaginary Order ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="28px" altimg-valign="-7px" altimg-width="56px" alttext="\widetilde{Y}_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mover accent="true"><mi href="./10.24#Ex2">Y</mi><mo href="./10.24#Ex2">~</mo></mover><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: Bessel function of imaginary order</a>,
<a href="./2.1#E3" title="(2.1.3) ‣ §2.1(i) Asymptotic and Order Symbols ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m16.png" altimg-height="23px" altimg-valign="-7px" altimg-width="50px" alttext="O\left(\NVar{x}\right)" display="inline"><mrow><mi href="./2.1#E3">O</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>)</mo></mrow></mrow></math>: order not exceeding</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m36.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.28#E7" title="(4.28.7) ‣ §4.28 Definitions and Periodicity ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m21.png" altimg-height="18px" altimg-valign="-2px" altimg-width="55px" alttext="\coth\NVar{z}" display="inline"><mrow><mi href="./4.28#E7">coth</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: hyperbolic cotangent function</a>,
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m27.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./10.1#p2.t1.r3" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./10.1#p2.t1.r3">x</mi></math>: real variable</a>,
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m32.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a> and
<a href="./10.24#p1" title="§10.24 Functions of Imaginary Order ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m25.png" altimg-height="16px" altimg-valign="-6px" altimg-width="25px" alttext="\gamma_{\nu}" display="inline"><msub><mi href="./10.24#p1">γ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub></math>: parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
<p class="ltx_p">and</p>
<table id="E9" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.24.9</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E9.png" altimg-height="46px" altimg-valign="-16px" altimg-width="423px" alttext="\widetilde{Y}_{0}\left(x\right)=Y_{0}\left(x\right)=\frac{2}{\pi}\left(\ln%
\left(\tfrac{1}{2}x\right)+\gamma\right)+O\left(x^{2}\ln x\right)," display="block"><mrow><mrow><mrow><msub><mover accent="true"><mi href="./10.24#Ex2">Y</mi><mo href="./10.24#Ex2">~</mo></mover><mn>0</mn></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msub><mi href="./10.2#E3">Y</mi><mn>0</mn></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mfrac><mn>2</mn><mi href="./3.12#E1">π</mi></mfrac><mo></mo><mrow><mo>(</mo><mrow><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><mo>(</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./10.1#p2.t1.r3">x</mi></mrow><mo>)</mo></mrow></mrow><mo>+</mo><mi href="./5.2#E3">γ</mi></mrow><mo>)</mo></mrow></mrow><mo>+</mo><mrow><mi href="./2.1#E3">O</mi><mo></mo><mrow><mo>(</mo><mrow><msup><mi href="./10.1#p2.t1.r3">x</mi><mn>2</mn></msup><mo></mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi href="./10.1#p2.t1.r3">x</mi></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E9.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E3" title="(10.2.3) ‣ Bessel Function of the Second Kind (Weber’s Function) ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m17.png" altimg-height="23px" altimg-valign="-7px" altimg-width="55px" alttext="Y_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E3">Y</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the second kind</a>,
<a href="./10.24#Ex2" title="(10.24.2) ‣ §10.24 Functions of Imaginary Order ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="28px" altimg-valign="-7px" altimg-width="56px" alttext="\widetilde{Y}_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mover accent="true"><mi href="./10.24#Ex2">Y</mi><mo href="./10.24#Ex2">~</mo></mover><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: Bessel function of imaginary order</a>,
<a href="./2.1#E3" title="(2.1.3) ‣ §2.1(i) Asymptotic and Order Symbols ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m16.png" altimg-height="23px" altimg-valign="-7px" altimg-width="50px" alttext="O\left(\NVar{x}\right)" display="inline"><mrow><mi href="./2.1#E3">O</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>)</mo></mrow></mrow></math>: order not exceeding</a>,
<a href="./5.2#E3" title="(5.2.3) ‣ §5.2(ii) Euler’s Constant ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m23.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\gamma" display="inline"><mi href="./5.2#E3">γ</mi></math>: Euler’s constant</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m36.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m27.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a> and
<a href="./10.1#p2.t1.r3" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./10.1#p2.t1.r3">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m23.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\gamma" display="inline"><mi href="./5.2#E3">γ</mi></math> denotes Euler’s constant §), when <math class="ltx_Math" altimg="m47.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./10.1#p2.t1.r3">x</mi></math> is large
<math class="ltx_Math" altimg="m42.png" altimg-height="28px" altimg-valign="-7px" altimg-width="55px" alttext="\widetilde{J}_{\nu}\left(x\right)" display="inline"><mrow><msub><mover accent="true"><mi href="./10.24#Ex1">J</mi><mo href="./10.24#Ex1">~</mo></mover><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math> and <math class="ltx_Math" altimg="m44.png" altimg-height="28px" altimg-valign="-7px" altimg-width="56px" alttext="\widetilde{Y}_{\nu}\left(x\right)" display="inline"><mrow><msub><mover accent="true"><mi href="./10.24#Ex2">Y</mi><mo href="./10.24#Ex2">~</mo></mover><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math> comprise a numerically
satisfactory pair of solutions of (), when <math class="ltx_Math" altimg="m47.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./10.1#p2.t1.r3">x</mi></math> is small either
<math class="ltx_Math" altimg="m42.png" altimg-height="28px" altimg-valign="-7px" altimg-width="55px" alttext="\widetilde{J}_{\nu}\left(x\right)" display="inline"><mrow><msub><mover accent="true"><mi href="./10.24#Ex1">J</mi><mo href="./10.24#Ex1">~</mo></mover><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math> and
<math class="ltx_Math" altimg="m39.png" altimg-height="30px" altimg-valign="-9px" altimg-width="147px" alttext="\tanh(\tfrac{1}{2}\pi\nu)\widetilde{Y}_{\nu}\left(x\right)" display="inline"><mrow><mrow><mi href="./4.28#E4">tanh</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><msub><mover accent="true"><mi href="./10.24#Ex2">Y</mi><mo href="./10.24#Ex2">~</mo></mover><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mrow></math> or
<math class="ltx_Math" altimg="m42.png" altimg-height="28px" altimg-valign="-7px" altimg-width="55px" alttext="\widetilde{J}_{\nu}\left(x\right)" display="inline"><mrow><msub><mover accent="true"><mi href="./10.24#Ex1">J</mi><mo href="./10.24#Ex1">~</mo></mover><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math> and <math class="ltx_Math" altimg="m44.png" altimg-height="28px" altimg-valign="-7px" altimg-width="56px" alttext="\widetilde{Y}_{\nu}\left(x\right)" display="inline"><mrow><msub><mover accent="true"><mi href="./10.24#Ex2">Y</mi><mo href="./10.24#Ex2">~</mo></mover><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math> comprise a
numerically satisfactory pair depending whether <math class="ltx_Math" altimg="m34.png" altimg-height="21px" altimg-valign="-6px" altimg-width="52px" alttext="\nu\neq 0" display="inline"><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>≠</mo><mn>0</mn></mrow></math> or <math class="ltx_Math" altimg="m30.png" altimg-height="17px" altimg-valign="-2px" altimg-width="52px" alttext="\nu=0" display="inline"><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>=</mo><mn>0</mn></mrow></math>.
</p>
</div>
<div id="p5" class="ltx_para">
<p class="ltx_p">For graphs of <math class="ltx_Math" altimg="m42.png" altimg-height="28px" altimg-valign="-7px" altimg-width="55px" alttext="\widetilde{J}_{\nu}\left(x\right)" display="inline"><mrow><msub><mover accent="true"><mi href="./10.24#Ex1">J</mi><mo href="./10.24#Ex1">~</mo></mover><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math> and <math class="ltx_Math" altimg="m44.png" altimg-height="28px" altimg-valign="-7px" altimg-width="56px" alttext="\widetilde{Y}_{\nu}\left(x\right)" display="inline"><mrow><msub><mover accent="true"><mi href="./10.24#Ex2">Y</mi><mo href="./10.24#Ex2">~</mo></mover><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math> see
§.</p>
</div>
<div id="p6" class="ltx_para">
<p class="ltx_p">For mathematical properties and applications of <math class="ltx_Math" altimg="m42.png" altimg-height="28px" altimg-valign="-7px" altimg-width="55px" alttext="\widetilde{J}_{\nu}\left(x\right)" display="inline"><mrow><msub><mover accent="true"><mi href="./10.24#Ex1">J</mi><mo href="./10.24#Ex1">~</mo></mover><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math> and
<math class="ltx_Math" altimg="m44.png" altimg-height="28px" altimg-valign="-7px" altimg-width="56px" alttext="\widetilde{Y}_{\nu}\left(x\right)" display="inline"><mrow><msub><mover accent="true"><mi href="./10.24#Ex2">Y</mi><mo href="./10.24#Ex2">~</mo></mover><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>, including zeros and uniform asymptotic expansions
for large <math class="ltx_Math" altimg="m32.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>, see <cite class="ltx_cite ltx_citemacro_citet">Dunster ()</cite>. In this reference
<math class="ltx_Math" altimg="m42.png" altimg-height="28px" altimg-valign="-7px" altimg-width="55px" alttext="\widetilde{J}_{\nu}\left(x\right)" display="inline"><mrow><msub><mover accent="true"><mi href="./10.24#Ex1">J</mi><mo href="./10.24#Ex1">~</mo></mover><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math> and <math class="ltx_Math" altimg="m44.png" altimg-height="28px" altimg-valign="-7px" altimg-width="56px" alttext="\widetilde{Y}_{\nu}\left(x\right)" display="inline"><mrow><msub><mover accent="true"><mi href="./10.24#Ex2">Y</mi><mo href="./10.24#Ex2">~</mo></mover><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math> are denoted
respectively by <math class="ltx_Math" altimg="m13.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="F_{i\nu}{(x)}" display="inline"><mrow><msub><mi>F</mi><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo stretchy="false">)</mo></mrow></mrow></math> and <math class="ltx_Math" altimg="m14.png" altimg-height="23px" altimg-valign="-7px" altimg-width="62px" alttext="G_{i\nu}{(x)}" display="inline"><mrow><msub><mi>G</mi><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo stretchy="false">)</mo></mrow></mrow></math>.
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<title>DLMF: 10.40 Asymptotic Expansions for Large Argument</title>
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<div class="ltx_page_navlogo">) is a consequence of the fact that <math class="ltx_Math" altimg="m17.png" altimg-height="23px" altimg-valign="-7px" altimg-width="114px" alttext="I_{\nu}\left(x\right)K_{\nu}\left(x\right)" display="inline"><mrow><mrow><msub><mi href="./10.25#E2">I</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./10.25#E3">K</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mrow></math> and <math class="ltx_Math" altimg="m15.png" altimg-height="24px" altimg-valign="-7px" altimg-width="114px" alttext="I_{\nu}'\left(x\right)K_{\nu}'\left(x\right)" display="inline"><mrow><mrow><msubsup><mi href="./10.25#E2">I</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′</mo></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><msubsup><mi href="./10.25#E3">K</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′</mo></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mrow></math> satisfy the
same differential equations as <math class="ltx_Math" altimg="m60.png" altimg-height="29px" altimg-valign="-7px" altimg-width="354px" alttext="{M_{\nu}^{2}}\left(x\right)=|{H^{(1)}_{\nu}}\left(x\right)|^{2}={H^{(1)}_{\nu}%
}\left(x\right){H^{(2)}_{\nu}}\left(x\right)" display="inline"><mrow><mrow><msubsup><mi href="./10.18#E1">M</mi><mi href="./10.1#p2.t1.r5">ν</mi><mn>2</mn></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow><mo>=</mo><msup><mrow><mo stretchy="false">|</mo><mrow><msubsup><mi href="./10.2#E5">H</mi><mi href="./10.1#p2.t1.r5">ν</mi><mrow><mo href="./10.2#E5" stretchy="false">(</mo><mn>1</mn><mo href="./10.2#E5" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow><mn>2</mn></msup><mo>=</mo><mrow><mrow><msubsup><mi href="./10.2#E5">H</mi><mi href="./10.1#p2.t1.r5">ν</mi><mrow><mo href="./10.2#E5" stretchy="false">(</mo><mn>1</mn><mo href="./10.2#E5" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><msubsup><mi href="./10.2#E6">H</mi><mi href="./10.1#p2.t1.r5">ν</mi><mrow><mo href="./10.2#E6" stretchy="false">(</mo><mn>2</mn><mo href="./10.2#E6" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mrow></mrow></math> and
<math class="ltx_Math" altimg="m61.png" altimg-height="32px" altimg-valign="-7px" altimg-width="368px" alttext="{N_{\nu}^{2}}\left(x\right)=|{H^{(1)}_{\nu}}'\left(x\right)|^{2}={H^{(1)}_{\nu%
}}'\left(x\right){H^{(2)}_{\nu}}'\left(x\right)" display="inline"><mrow><mrow><msubsup><mi href="./10.18#E2">N</mi><mi href="./10.1#p2.t1.r5">ν</mi><mn>2</mn></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow><mo>=</mo><msup><mrow><mo stretchy="false">|</mo><mrow><mmultiscripts><mi href="./10.2#E5">H</mi><mi href="./10.1#p2.t1.r5">ν</mi><mrow><mo href="./10.2#E5" stretchy="false">(</mo><mn>1</mn><mo href="./10.2#E5" stretchy="false">)</mo></mrow><none></none><mo>′</mo></mmultiscripts><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow><mn>2</mn></msup><mo>=</mo><mrow><mrow><mmultiscripts><mi href="./10.2#E5">H</mi><mi href="./10.1#p2.t1.r5">ν</mi><mrow><mo href="./10.2#E5" stretchy="false">(</mo><mn>1</mn><mo href="./10.2#E5" stretchy="false">)</mo></mrow><none></none><mo>′</mo></mmultiscripts><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mmultiscripts><mi href="./10.2#E6">H</mi><mi href="./10.1#p2.t1.r5">ν</mi><mrow><mo href="./10.2#E6" stretchy="false">(</mo><mn>2</mn><mo href="./10.2#E6" stretchy="false">)</mo></mrow><none></none><mo>′</mo></mmultiscripts><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mrow></mrow></math>, respectively, except for
replacement of <math class="ltx_Math" altimg="m56.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./10.1#p2.t1.r3">x</mi></math> by <math class="ltx_Math" altimg="m50.png" altimg-height="17px" altimg-valign="-2px" altimg-width="23px" alttext="ix" display="inline"><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./10.1#p2.t1.r3">x</mi></mrow></math>. For the statement concerning the accuracy of
(,
as <math class="ltx_Math" altimg="m59.png" altimg-height="13px" altimg-valign="-2px" altimg-width="65px" alttext="z\to\infty" display="inline"><mrow><mi href="./10.1#p2.t1.r4">z</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></math> with <math class="ltx_Math" altimg="m39.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math> fixed,
</p>
<table id="EGx1" class="ltx_equationgroup ltx_eqn_table">
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<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.40.1</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m3.png" altimg-height="25px" altimg-valign="-7px" altimg-width="54px" alttext="\displaystyle I_{\nu}\left(z\right)" display="inline"><mrow><msub><mi href="./10.25#E2">I</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m7.png" altimg-height="64px" altimg-valign="-28px" altimg-width="232px" alttext="\displaystyle\sim\frac{e^{z}}{(2\pi z)^{\frac{1}{2}}}\sum_{k=0}^{\infty}(-1)^{%
k}\frac{a_{k}(\nu)}{z^{k}}," display="inline"><mrow><mrow><mi></mi><mo href="./2.1#SS3.p1">∼</mo><mrow><mstyle displaystyle="true"><mfrac><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mi href="./10.1#p2.t1.r4">z</mi></msup><msup><mrow><mo stretchy="false">(</mo><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo stretchy="false">)</mo></mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></msup></mfrac></mstyle><mo></mo><mrow><mstyle displaystyle="true"><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./10.1#p2.t1.r2">k</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover></mstyle><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./10.1#p2.t1.r2">k</mi></msup><mo></mo><mstyle displaystyle="true"><mfrac><mrow><msub><mi href="./10.17#E1">a</mi><mi href="./10.1#p2.t1.r2">k</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./10.1#p2.t1.r5">ν</mi><mo stretchy="false">)</mo></mrow></mrow><msup><mi href="./10.1#p2.t1.r4">z</mi><mi href="./10.1#p2.t1.r2">k</mi></msup></mfrac></mstyle></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
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<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m65.png" altimg-height="27px" altimg-valign="-9px" altimg-width="140px" alttext="|\operatorname{ph}z|\leq\tfrac{1}{2}\pi-\delta" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo stretchy="false">|</mo></mrow><mo>≤</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo>-</mo><mi href="./10.1#p2.t1.r6">δ</mi></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./2.1#SS3.p1" title="§2.1(iii) Asymptotic Expansions ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m46.png" altimg-height="12px" altimg-valign="-2px" altimg-width="20px" alttext="\sim" display="inline"><mo href="./2.1#SS3.p1">∼</mo></math>: Poincaré asymptotic expansion</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./10.25#E2" title="(10.25.2) ‣ §10.25(ii) Standard Solutions ‣ §10.25 Definitions ‣ Modified Bessel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m14.png" altimg-height="23px" altimg-valign="-7px" altimg-width="52px" alttext="I_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.25#E2">I</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: modified Bessel function of the first kind</a>,
<a href="./1.9#E7" title="(1.9.7) ‣ Modulus and Phase ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m40.png" altimg-height="21px" altimg-valign="-6px" altimg-width="26px" alttext="\operatorname{ph}" display="inline"><mi href="./1.9#E7">ph</mi></math>: phase</a>,
<a href="./10.1#p2.t1.r2" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./10.1#p2.t1.r2">k</mi></math>: nonnegative integer</a>,
<a href="./10.1#p2.t1.r4" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./10.1#p2.t1.r4">z</mi></math>: complex variable</a>,
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>,
<a href="./10.1#p2.t1.r6" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="18px" altimg-valign="-2px" altimg-width="14px" alttext="\delta" display="inline"><mi href="./10.1#p2.t1.r6">δ</mi></math>: small positive constant</a> and
<a href="./10.17#E1" title="(10.17.1) ‣ §10.17(i) Hankel’s Expansions ‣ §10.17 Asymptotic Expansions for Large Argument ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="23px" altimg-valign="-7px" altimg-width="51px" alttext="a_{k}(\nu)" display="inline"><mrow><msub><mi href="./10.17#E1">a</mi><mi href="./10.1#p2.t1.r2">k</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./10.1#p2.t1.r5">ν</mi><mo stretchy="false">)</mo></mrow></mrow></math>: expansion</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">9.7.1</span></span>
</dd>
</dl>
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</div>
</td></tr>
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<tbody id="E2">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.40.2</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m4.png" altimg-height="25px" altimg-valign="-7px" altimg-width="62px" alttext="\displaystyle K_{\nu}\left(z\right)" display="inline"><mrow><msub><mi href="./10.25#E3">K</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m8.png" altimg-height="64px" altimg-valign="-28px" altimg-width="211px" alttext="\displaystyle\sim\left(\frac{\pi}{2z}\right)^{\frac{1}{2}}e^{-z}\sum_{k=0}^{%
\infty}\frac{a_{k}(\nu)}{z^{k}}," display="inline"><mrow><mrow><mi></mi><mo href="./2.1#SS3.p1">∼</mo><mrow><msup><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac><mi href="./3.12#E1">π</mi><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow></mfrac></mstyle><mo>)</mo></mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></msup><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mi href="./10.1#p2.t1.r4">z</mi></mrow></msup><mo></mo><mrow><mstyle displaystyle="true"><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./10.1#p2.t1.r2">k</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover></mstyle><mstyle displaystyle="true"><mfrac><mrow><msub><mi href="./10.17#E1">a</mi><mi href="./10.1#p2.t1.r2">k</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./10.1#p2.t1.r5">ν</mi><mo stretchy="false">)</mo></mrow></mrow><msup><mi href="./10.1#p2.t1.r4">z</mi><mi href="./10.1#p2.t1.r2">k</mi></msup></mfrac></mstyle></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m66.png" altimg-height="27px" altimg-valign="-9px" altimg-width="140px" alttext="|\operatorname{ph}z|\leq\tfrac{3}{2}\pi-\delta" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo stretchy="false">|</mo></mrow><mo>≤</mo><mrow><mrow><mfrac><mn>3</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo>-</mo><mi href="./10.1#p2.t1.r6">δ</mi></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./2.1#SS3.p1" title="§2.1(iii) Asymptotic Expansions ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m46.png" altimg-height="12px" altimg-valign="-2px" altimg-width="20px" alttext="\sim" display="inline"><mo href="./2.1#SS3.p1">∼</mo></math>: Poincaré asymptotic expansion</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./10.25#E3" title="(10.25.3) ‣ §10.25(ii) Standard Solutions ‣ §10.25 Definitions ‣ Modified Bessel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="K_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.25#E3">K</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: modified Bessel function of the second kind</a>,
<a href="./1.9#E7" title="(1.9.7) ‣ Modulus and Phase ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m40.png" altimg-height="21px" altimg-valign="-6px" altimg-width="26px" alttext="\operatorname{ph}" display="inline"><mi href="./1.9#E7">ph</mi></math>: phase</a>,
<a href="./10.1#p2.t1.r2" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./10.1#p2.t1.r2">k</mi></math>: nonnegative integer</a>,
<a href="./10.1#p2.t1.r4" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./10.1#p2.t1.r4">z</mi></math>: complex variable</a>,
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>,
<a href="./10.1#p2.t1.r6" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="18px" altimg-valign="-2px" altimg-width="14px" alttext="\delta" display="inline"><mi href="./10.1#p2.t1.r6">δ</mi></math>: small positive constant</a> and
<a href="./10.17#E1" title="(10.17.1) ‣ §10.17(i) Hankel’s Expansions ‣ §10.17 Asymptotic Expansions for Large Argument ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="23px" altimg-valign="-7px" altimg-width="51px" alttext="a_{k}(\nu)" display="inline"><mrow><msub><mi href="./10.17#E1">a</mi><mi href="./10.1#p2.t1.r2">k</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./10.1#p2.t1.r5">ν</mi><mo stretchy="false">)</mo></mrow></mrow></math>: expansion</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">9.7.2</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
<table id="E3" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.40.3</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E3.png" altimg-height="64px" altimg-valign="-28px" altimg-width="284px" alttext="I_{\nu}'\left(z\right)\sim\frac{e^{z}}{(2\pi z)^{\frac{1}{2}}}\sum_{k=0}^{%
\infty}(-1)^{k}\frac{b_{k}(\nu)}{z^{k}}," display="block"><mrow><mrow><mrow><msubsup><mi href="./10.25#E2">I</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′</mo></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow><mo href="./2.1#SS3.p1">∼</mo><mrow><mfrac><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mi href="./10.1#p2.t1.r4">z</mi></msup><msup><mrow><mo stretchy="false">(</mo><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo stretchy="false">)</mo></mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></msup></mfrac><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./10.1#p2.t1.r2">k</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./10.1#p2.t1.r2">k</mi></msup><mo></mo><mfrac><mrow><msub><mi href="./10.17#E8">b</mi><mi href="./10.1#p2.t1.r2">k</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./10.1#p2.t1.r5">ν</mi><mo stretchy="false">)</mo></mrow></mrow><msup><mi href="./10.1#p2.t1.r4">z</mi><mi href="./10.1#p2.t1.r2">k</mi></msup></mfrac></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m65.png" altimg-height="27px" altimg-valign="-9px" altimg-width="140px" alttext="|\operatorname{ph}z|\leq\tfrac{1}{2}\pi-\delta" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo stretchy="false">|</mo></mrow><mo>≤</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo>-</mo><mi href="./10.1#p2.t1.r6">δ</mi></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./2.1#SS3.p1" title="§2.1(iii) Asymptotic Expansions ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m46.png" altimg-height="12px" altimg-valign="-2px" altimg-width="20px" alttext="\sim" display="inline"><mo href="./2.1#SS3.p1">∼</mo></math>: Poincaré asymptotic expansion</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./10.25#E2" title="(10.25.2) ‣ §10.25(ii) Standard Solutions ‣ §10.25 Definitions ‣ Modified Bessel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m14.png" altimg-height="23px" altimg-valign="-7px" altimg-width="52px" alttext="I_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.25#E2">I</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: modified Bessel function of the first kind</a>,
<a href="./1.9#E7" title="(1.9.7) ‣ Modulus and Phase ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m40.png" altimg-height="21px" altimg-valign="-6px" altimg-width="26px" alttext="\operatorname{ph}" display="inline"><mi href="./1.9#E7">ph</mi></math>: phase</a>,
<a href="./10.1#p2.t1.r2" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./10.1#p2.t1.r2">k</mi></math>: nonnegative integer</a>,
<a href="./10.1#p2.t1.r4" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./10.1#p2.t1.r4">z</mi></math>: complex variable</a>,
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>,
<a href="./10.1#p2.t1.r6" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="18px" altimg-valign="-2px" altimg-width="14px" alttext="\delta" display="inline"><mi href="./10.1#p2.t1.r6">δ</mi></math>: small positive constant</a> and
<a href="./10.17#E8" title="(10.17.8) ‣ §10.17(ii) Asymptotic Expansions of Derivatives ‣ §10.17 Asymptotic Expansions for Large Argument ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="23px" altimg-valign="-7px" altimg-width="49px" alttext="b_{k}(\nu)" display="inline"><mrow><msub><mi href="./10.17#E8">b</mi><mi href="./10.1#p2.t1.r2">k</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./10.1#p2.t1.r5">ν</mi><mo stretchy="false">)</mo></mrow></mrow></math>: expansion</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">9.7.3</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E4" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.40.4</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E4.png" altimg-height="64px" altimg-valign="-28px" altimg-width="289px" alttext="K_{\nu}'\left(z\right)\sim-\left(\frac{\pi}{2z}\right)^{\frac{1}{2}}e^{-z}\sum%
_{k=0}^{\infty}\frac{b_{k}(\nu)}{z^{k}}," display="block"><mrow><mrow><mrow><msubsup><mi href="./10.25#E3">K</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′</mo></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow><mo href="./2.1#SS3.p1">∼</mo><mrow><mo>-</mo><mrow><msup><mrow><mo>(</mo><mfrac><mi href="./3.12#E1">π</mi><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow></mfrac><mo>)</mo></mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></msup><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mi href="./10.1#p2.t1.r4">z</mi></mrow></msup><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./10.1#p2.t1.r2">k</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mfrac><mrow><msub><mi href="./10.17#E8">b</mi><mi href="./10.1#p2.t1.r2">k</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./10.1#p2.t1.r5">ν</mi><mo stretchy="false">)</mo></mrow></mrow><msup><mi href="./10.1#p2.t1.r4">z</mi><mi href="./10.1#p2.t1.r2">k</mi></msup></mfrac></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m66.png" altimg-height="27px" altimg-valign="-9px" altimg-width="140px" alttext="|\operatorname{ph}z|\leq\tfrac{3}{2}\pi-\delta" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo stretchy="false">|</mo></mrow><mo>≤</mo><mrow><mrow><mfrac><mn>3</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo>-</mo><mi href="./10.1#p2.t1.r6">δ</mi></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E4.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./2.1#SS3.p1" title="§2.1(iii) Asymptotic Expansions ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m46.png" altimg-height="12px" altimg-valign="-2px" altimg-width="20px" alttext="\sim" display="inline"><mo href="./2.1#SS3.p1">∼</mo></math>: Poincaré asymptotic expansion</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./10.25#E3" title="(10.25.3) ‣ §10.25(ii) Standard Solutions ‣ §10.25 Definitions ‣ Modified Bessel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="K_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.25#E3">K</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: modified Bessel function of the second kind</a>,
<a href="./1.9#E7" title="(1.9.7) ‣ Modulus and Phase ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m40.png" altimg-height="21px" altimg-valign="-6px" altimg-width="26px" alttext="\operatorname{ph}" display="inline"><mi href="./1.9#E7">ph</mi></math>: phase</a>,
<a href="./10.1#p2.t1.r2" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./10.1#p2.t1.r2">k</mi></math>: nonnegative integer</a>,
<a href="./10.1#p2.t1.r4" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./10.1#p2.t1.r4">z</mi></math>: complex variable</a>,
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>,
<a href="./10.1#p2.t1.r6" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="18px" altimg-valign="-2px" altimg-width="14px" alttext="\delta" display="inline"><mi href="./10.1#p2.t1.r6">δ</mi></math>: small positive constant</a> and
<a href="./10.17#E8" title="(10.17.8) ‣ §10.17(ii) Asymptotic Expansions of Derivatives ‣ §10.17 Asymptotic Expansions for Large Argument ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="23px" altimg-valign="-7px" altimg-width="49px" alttext="b_{k}(\nu)" display="inline"><mrow><msub><mi href="./10.17#E8">b</mi><mi href="./10.1#p2.t1.r2">k</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./10.1#p2.t1.r5">ν</mi><mo stretchy="false">)</mo></mrow></mrow></math>: expansion</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">9.7.4</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS1.p2" class="ltx_para">
<p class="ltx_p">Corresponding expansions for <math class="ltx_Math" altimg="m18.png" altimg-height="23px" altimg-valign="-7px" altimg-width="52px" alttext="I_{\nu}\left(z\right)" display="inline"><mrow><msub><mi href="./10.25#E2">I</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>, <math class="ltx_Math" altimg="m21.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="K_{\nu}\left(z\right)" display="inline"><mrow><msub><mi href="./10.25#E3">K</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>,
<math class="ltx_Math" altimg="m16.png" altimg-height="24px" altimg-valign="-7px" altimg-width="52px" alttext="I_{\nu}'\left(z\right)" display="inline"><mrow><msubsup><mi href="./10.25#E2">I</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′</mo></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>, and <math class="ltx_Math" altimg="m20.png" altimg-height="24px" altimg-valign="-7px" altimg-width="60px" alttext="K_{\nu}'\left(z\right)" display="inline"><mrow><msubsup><mi href="./10.25#E3">K</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′</mo></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math> for other ranges of <math class="ltx_Math" altimg="m41.png" altimg-height="21px" altimg-valign="-6px" altimg-width="40px" alttext="\operatorname{ph}z" display="inline"><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow></math> are
obtainable by combining () with <math class="ltx_Math" altimg="m53.png" altimg-height="17px" altimg-valign="-2px" altimg-width="58px" alttext="m=0" display="inline"><mrow><mi href="./10.1#p2.t1.r1">m</mi><mo>=</mo><mn>0</mn></mrow></math> yields the following more general (and more
accurate) version of ():</p>
<table id="E5" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.40.5</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E5.png" altimg-height="64px" altimg-valign="-28px" altimg-width="516px" alttext="I_{\nu}\left(z\right)\sim\frac{e^{z}}{(2\pi z)^{\frac{1}{2}}}\sum_{k=0}^{%
\infty}(-1)^{k}\frac{a_{k}(\nu)}{z^{k}}\pm ie^{\pm\nu\pi i}\frac{e^{-z}}{(2\pi
z%
)^{\frac{1}{2}}}\sum_{k=0}^{\infty}\frac{a_{k}(\nu)}{z^{k}}," display="block"><mrow><mrow><mrow><msub><mi href="./10.25#E2">I</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow><mo href="./2.1#SS3.p1">∼</mo><mrow><mrow><mfrac><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mi href="./10.1#p2.t1.r4">z</mi></msup><msup><mrow><mo stretchy="false">(</mo><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo stretchy="false">)</mo></mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></msup></mfrac><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./10.1#p2.t1.r2">k</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./10.1#p2.t1.r2">k</mi></msup><mo></mo><mfrac><mrow><msub><mi href="./10.17#E1">a</mi><mi href="./10.1#p2.t1.r2">k</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./10.1#p2.t1.r5">ν</mi><mo stretchy="false">)</mo></mrow></mrow><msup><mi href="./10.1#p2.t1.r4">z</mi><mi href="./10.1#p2.t1.r2">k</mi></msup></mfrac></mrow></mrow></mrow><mo>±</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>±</mo><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi></mrow></mrow></msup><mo></mo><mfrac><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mi href="./10.1#p2.t1.r4">z</mi></mrow></msup><msup><mrow><mo stretchy="false">(</mo><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo stretchy="false">)</mo></mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></msup></mfrac><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./10.1#p2.t1.r2">k</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mfrac><mrow><msub><mi href="./10.17#E1">a</mi><mi href="./10.1#p2.t1.r2">k</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./10.1#p2.t1.r5">ν</mi><mo stretchy="false">)</mo></mrow></mrow><msup><mi href="./10.1#p2.t1.r4">z</mi><mi href="./10.1#p2.t1.r2">k</mi></msup></mfrac></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m11.png" altimg-height="27px" altimg-valign="-9px" altimg-width="245px" alttext="-\tfrac{1}{2}\pi+\delta\leq\pm\operatorname{ph}z\leq\tfrac{3}{2}\pi-\delta" display="inline"><mrow><mrow><mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow><mo>+</mo><mi href="./10.1#p2.t1.r6">δ</mi></mrow><mo>≤</mo><mrow><mo>±</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow></mrow><mo>≤</mo><mrow><mrow><mfrac><mn>3</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo>-</mo><mi href="./10.1#p2.t1.r6">δ</mi></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E5.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./2.1#SS3.p1" title="§2.1(iii) Asymptotic Expansions ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m46.png" altimg-height="12px" altimg-valign="-2px" altimg-width="20px" alttext="\sim" display="inline"><mo href="./2.1#SS3.p1">∼</mo></math>: Poincaré asymptotic expansion</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./10.25#E2" title="(10.25.2) ‣ §10.25(ii) Standard Solutions ‣ §10.25 Definitions ‣ Modified Bessel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m14.png" altimg-height="23px" altimg-valign="-7px" altimg-width="52px" alttext="I_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.25#E2">I</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: modified Bessel function of the first kind</a>,
<a href="./1.9#E7" title="(1.9.7) ‣ Modulus and Phase ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m40.png" altimg-height="21px" altimg-valign="-6px" altimg-width="26px" alttext="\operatorname{ph}" display="inline"><mi href="./1.9#E7">ph</mi></math>: phase</a>,
<a href="./10.1#p2.t1.r2" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./10.1#p2.t1.r2">k</mi></math>: nonnegative integer</a>,
<a href="./10.1#p2.t1.r4" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./10.1#p2.t1.r4">z</mi></math>: complex variable</a>,
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>,
<a href="./10.1#p2.t1.r6" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="18px" altimg-valign="-2px" altimg-width="14px" alttext="\delta" display="inline"><mi href="./10.1#p2.t1.r6">δ</mi></math>: small positive constant</a> and
<a href="./10.17#E1" title="(10.17.1) ‣ §10.17(i) Hankel’s Expansions ‣ §10.17 Asymptotic Expansions for Large Argument ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="23px" altimg-valign="-7px" altimg-width="51px" alttext="a_{k}(\nu)" display="inline"><mrow><msub><mi href="./10.17#E1">a</mi><mi href="./10.1#p2.t1.r2">k</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./10.1#p2.t1.r5">ν</mi><mo stretchy="false">)</mo></mrow></mrow></math>: expansion</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<section id="Px1" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Products</h3>
<div id="Px1.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px1.p1" class="ltx_para">
<p class="ltx_p">With <math class="ltx_Math" altimg="m38.png" altimg-height="24px" altimg-valign="-6px" altimg-width="73px" alttext="\mu=4\nu^{2}" display="inline"><mrow><mi>μ</mi><mo>=</mo><mrow><mn>4</mn><mo></mo><msup><mi href="./10.1#p2.t1.r5">ν</mi><mn>2</mn></msup></mrow></mrow></math> and fixed,</p>
<table id="EGx2" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E6">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.40.6</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m2.png" altimg-height="25px" altimg-valign="-7px" altimg-width="113px" alttext="\displaystyle I_{\nu}\left(z\right)K_{\nu}\left(z\right)" display="inline"><mrow><mrow><msub><mi href="./10.25#E2">I</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./10.25#E3">K</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m6.png" altimg-height="53px" altimg-valign="-21px" altimg-width="433px" alttext="\displaystyle\sim\frac{1}{2z}\left(1-\frac{1}{2}\frac{\mu-1}{(2z)^{2}}+\frac{1%
\cdot 3}{2\cdot 4}\frac{(\mu-1)(\mu-9)}{(2z)^{4}}-\cdots\right)," display="inline"><mrow><mrow><mi></mi><mo href="./2.1#SS3.p1">∼</mo><mrow><mstyle displaystyle="true"><mfrac><mn>1</mn><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow></mfrac></mstyle><mo></mo><mrow><mo>(</mo><mrow><mrow><mrow><mn>1</mn><mo>-</mo><mrow><mstyle displaystyle="true"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mstyle displaystyle="true"><mfrac><mrow><mi>μ</mi><mo>-</mo><mn>1</mn></mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mfrac></mstyle></mrow></mrow><mo>+</mo><mrow><mstyle displaystyle="true"><mfrac><mrow><mn>1</mn><mo>⋅</mo><mn>3</mn></mrow><mrow><mn>2</mn><mo>⋅</mo><mn>4</mn></mrow></mfrac></mstyle><mo></mo><mstyle displaystyle="true"><mfrac><mrow><mrow><mo stretchy="false">(</mo><mrow><mi>μ</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>μ</mi><mo>-</mo><mn>9</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo stretchy="false">)</mo></mrow><mn>4</mn></msup></mfrac></mstyle></mrow></mrow><mo>-</mo><mi mathvariant="normal">⋯</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E6.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./2.1#SS3.p1" title="§2.1(iii) Asymptotic Expansions ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m46.png" altimg-height="12px" altimg-valign="-2px" altimg-width="20px" alttext="\sim" display="inline"><mo href="./2.1#SS3.p1">∼</mo></math>: Poincaré asymptotic expansion</a>,
<a href="./10.25#E2" title="(10.25.2) ‣ §10.25(ii) Standard Solutions ‣ §10.25 Definitions ‣ Modified Bessel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m14.png" altimg-height="23px" altimg-valign="-7px" altimg-width="52px" alttext="I_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.25#E2">I</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: modified Bessel function of the first kind</a>,
<a href="./10.25#E3" title="(10.25.3) ‣ §10.25(ii) Standard Solutions ‣ §10.25 Definitions ‣ Modified Bessel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="K_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.25#E3">K</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: modified Bessel function of the second kind</a>,
<a href="./10.1#p2.t1.r4" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./10.1#p2.t1.r4">z</mi></math>: complex variable</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">9.7.5</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E7">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.40.7</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m1.png" altimg-height="26px" altimg-valign="-7px" altimg-width="113px" alttext="\displaystyle I_{\nu}'\left(z\right)K_{\nu}'\left(z\right)" display="inline"><mrow><mrow><msubsup><mi href="./10.25#E2">I</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′</mo></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><msubsup><mi href="./10.25#E3">K</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′</mo></msubsup><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m5.png" altimg-height="53px" altimg-valign="-21px" altimg-width="459px" alttext="\displaystyle\sim-\frac{1}{2z}\left(1+\frac{1}{2}\frac{\mu-3}{(2z)^{2}}-\frac{%
1}{2\cdot 4}\frac{(\mu-1)(\mu-45)}{(2z)^{4}}+\cdots\right)," display="inline"><mrow><mrow><mi></mi><mo href="./2.1#SS3.p1">∼</mo><mrow><mo>-</mo><mrow><mstyle displaystyle="true"><mfrac><mn>1</mn><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow></mfrac></mstyle><mo></mo><mrow><mo>(</mo><mrow><mrow><mrow><mn>1</mn><mo>+</mo><mrow><mstyle displaystyle="true"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mstyle displaystyle="true"><mfrac><mrow><mi>μ</mi><mo>-</mo><mn>3</mn></mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mfrac></mstyle></mrow></mrow><mo>-</mo><mrow><mstyle displaystyle="true"><mfrac><mn>1</mn><mrow><mn>2</mn><mo>⋅</mo><mn>4</mn></mrow></mfrac></mstyle><mo></mo><mstyle displaystyle="true"><mfrac><mrow><mrow><mo stretchy="false">(</mo><mrow><mi>μ</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>μ</mi><mo>-</mo><mn>45</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo stretchy="false">)</mo></mrow><mn>4</mn></msup></mfrac></mstyle></mrow></mrow><mo>+</mo><mi mathvariant="normal">⋯</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E7.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./2.1#SS3.p1" title="§2.1(iii) Asymptotic Expansions ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m46.png" altimg-height="12px" altimg-valign="-2px" altimg-width="20px" alttext="\sim" display="inline"><mo href="./2.1#SS3.p1">∼</mo></math>: Poincaré asymptotic expansion</a>,
<a href="./10.25#E2" title="(10.25.2) ‣ §10.25(ii) Standard Solutions ‣ §10.25 Definitions ‣ Modified Bessel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m14.png" altimg-height="23px" altimg-valign="-7px" altimg-width="52px" alttext="I_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.25#E2">I</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: modified Bessel function of the first kind</a>,
<a href="./10.25#E3" title="(10.25.3) ‣ §10.25(ii) Standard Solutions ‣ §10.25 Definitions ‣ Modified Bessel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="K_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.25#E3">K</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: modified Bessel function of the second kind</a>,
<a href="./10.1#p2.t1.r4" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./10.1#p2.t1.r4">z</mi></math>: complex variable</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">9.7.6</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
<p class="ltx_p">as <math class="ltx_Math" altimg="m59.png" altimg-height="13px" altimg-valign="-2px" altimg-width="65px" alttext="z\to\infty" display="inline"><mrow><mi href="./10.1#p2.t1.r4">z</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></math> in <math class="ltx_Math" altimg="m65.png" altimg-height="27px" altimg-valign="-9px" altimg-width="140px" alttext="|\operatorname{ph}z|\leq\tfrac{1}{2}\pi-\delta" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo stretchy="false">|</mo></mrow><mo>≤</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo>-</mo><mi href="./10.1#p2.t1.r6">δ</mi></mrow></mrow></math>. The general terms
in ().
</p>
</div>
</section>
<section id="Px2" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">
<math class="ltx_Math" altimg="m39.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>-Derivative</h3>
<div id="Px2.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px2.p1" class="ltx_para">
<p class="ltx_p">For fixed <math class="ltx_Math" altimg="m39.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>,</p>
<table id="E8" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.40.8</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E8.png" altimg-height="64px" altimg-valign="-28px" altimg-width="307px" alttext="\frac{\partial K_{\nu}\left(z\right)}{\partial\nu}\sim\left(\frac{\pi}{2z}%
\right)^{\frac{1}{2}}\frac{\nu e^{-z}}{z}\sum_{k=0}^{\infty}\frac{\alpha_{k}(%
\nu)}{(8z)^{k}}," display="block"><mrow><mrow><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mrow><msub><mi href="./10.25#E3">K</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow></mfrac><mo href="./2.1#SS3.p1">∼</mo><mrow><msup><mrow><mo>(</mo><mfrac><mi href="./3.12#E1">π</mi><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow></mfrac><mo>)</mo></mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></msup><mo></mo><mfrac><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mi href="./10.1#p2.t1.r4">z</mi></mrow></msup></mrow><mi href="./10.1#p2.t1.r4">z</mi></mfrac><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./10.1#p2.t1.r2">k</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mfrac><mrow><msub><mi>α</mi><mi href="./10.1#p2.t1.r2">k</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./10.1#p2.t1.r5">ν</mi><mo stretchy="false">)</mo></mrow></mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mn>8</mn><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo stretchy="false">)</mo></mrow><mi href="./10.1#p2.t1.r2">k</mi></msup></mfrac></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E8.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./2.1#SS3.p1" title="§2.1(iii) Asymptotic Expansions ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m46.png" altimg-height="12px" altimg-valign="-2px" altimg-width="20px" alttext="\sim" display="inline"><mo href="./2.1#SS3.p1">∼</mo></math>: Poincaré asymptotic expansion</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./10.25#E3" title="(10.25.3) ‣ §10.25(ii) Standard Solutions ‣ §10.25 Definitions ‣ Modified Bessel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="K_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.25#E3">K</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: modified Bessel function of the second kind</a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m33.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m49.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m56.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m42.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m56.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./10.1#p2.t1.r2" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./10.1#p2.t1.r2">k</mi></math>: nonnegative integer</a>,
<a href="./10.1#p2.t1.r4" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./10.1#p2.t1.r4">z</mi></math>: complex variable</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">as <math class="ltx_Math" altimg="m59.png" altimg-height="13px" altimg-valign="-2px" altimg-width="65px" alttext="z\to\infty" display="inline"><mrow><mi href="./10.1#p2.t1.r4">z</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></math> in <math class="ltx_Math" altimg="m66.png" altimg-height="27px" altimg-valign="-9px" altimg-width="140px" alttext="|\operatorname{ph}z|\leq\tfrac{3}{2}\pi-\delta" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo stretchy="false">|</mo></mrow><mo>≤</mo><mrow><mrow><mfrac><mn>3</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo>-</mo><mi href="./10.1#p2.t1.r6">δ</mi></mrow></mrow></math>. Here
<math class="ltx_Math" altimg="m26.png" altimg-height="23px" altimg-valign="-7px" altimg-width="89px" alttext="\alpha_{0}(\nu)=1" display="inline"><mrow><mrow><msub><mi>α</mi><mn>0</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./10.1#p2.t1.r5">ν</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mn>1</mn></mrow></math> and</p>
<table id="E9" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.40.9</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_Math ltx_eqn_cell ltx_align_center"><math class="ltx_Math" alttext="\alpha_{k}(\nu)=\frac{(4\nu^{2}-1^{2})(4\nu^{2}-3^{2})\cdots(4\nu^{2}-(2k+1)^{%
2})}{(k+1)!}\*\left(\frac{1}{4\nu^{2}-1^{2}}+\frac{1}{4\nu^{2}-3^{2}}+\cdots+%
\frac{1}{4\nu^{2}-(2k+1)^{2}}\right)." display="block"><mrow><mrow><msub><mi>α</mi><mi href="./10.1#p2.t1.r2">k</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./10.1#p2.t1.r5">ν</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mfrac><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>4</mn><mo></mo><msup><mi href="./10.1#p2.t1.r5">ν</mi><mn>2</mn></msup></mrow><mo>-</mo><msup><mn>1</mn><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>4</mn><mo></mo><msup><mi href="./10.1#p2.t1.r5">ν</mi><mn>2</mn></msup></mrow><mo>-</mo><msup><mn>3</mn><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi mathvariant="normal">⋯</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>4</mn><mo></mo><msup><mi href="./10.1#p2.t1.r5">ν</mi><mn>2</mn></msup></mrow><mo>-</mo><msup><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r2">k</mi></mrow><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow></mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./10.1#p2.t1.r2">k</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mfrac></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>×</mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mrow><mrow><mn>4</mn><mo></mo><msup><mi href="./10.1#p2.t1.r5">ν</mi><mn>2</mn></msup></mrow><mo>-</mo><msup><mn>1</mn><mn>2</mn></msup></mrow></mfrac><mo>+</mo><mfrac><mn>1</mn><mrow><mrow><mn>4</mn><mo></mo><msup><mi href="./10.1#p2.t1.r5">ν</mi><mn>2</mn></msup></mrow><mo>-</mo><msup><mn>3</mn><mn>2</mn></msup></mrow></mfrac><mo>+</mo><mi mathvariant="normal">⋯</mi><mo>+</mo><mfrac><mn>1</mn><mrow><mrow><mn>4</mn><mo></mo><msup><mi href="./10.1#p2.t1.r5">ν</mi><mn>2</mn></msup></mrow><mo>-</mo><msup><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r2">k</mi></mrow><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow></mfrac></mrow><mo>)</mo></mrow><mo>.</mo></mtd></mtr></mtable></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E9.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m9.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m55.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./10.1#p2.t1.r2" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./10.1#p2.t1.r2">k</mi></math>: nonnegative integer</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
</section>
<section id="ii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§10.40(ii) </span>Error Bounds for Real Argument and Order</h2>
<div id="SS2.info" class="ltx_metadata ltx_info">
) assume that <math class="ltx_Math" altimg="m57.png" altimg-height="17px" altimg-valign="-3px" altimg-width="51px" alttext="z>0" display="inline"><mrow><mi href="./10.1#p2.t1.r4">z</mi><mo>></mo><mn>0</mn></mrow></math> and the sum is
truncated when <math class="ltx_Math" altimg="m51.png" altimg-height="19px" altimg-valign="-4px" altimg-width="84px" alttext="k=\ell-1" display="inline"><mrow><mi href="./10.1#p2.t1.r2">k</mi><mo>=</mo><mrow><mi mathvariant="normal">ℓ</mi><mo>-</mo><mn>1</mn></mrow></mrow></math>. Then the remainder term does not exceed the first
neglected term in absolute value and has the same sign provided that
<math class="ltx_Math" altimg="m31.png" altimg-height="27px" altimg-valign="-9px" altimg-width="170px" alttext="\ell\geq\max(|\nu|-\tfrac{1}{2},1)" display="inline"><mrow><mi mathvariant="normal">ℓ</mi><mo>≥</mo><mrow><mi>max</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mo stretchy="false">|</mo><mi href="./10.1#p2.t1.r5">ν</mi><mo stretchy="false">|</mo></mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mrow></mrow></math>.</p>
</div>
<div id="SS2.p2" class="ltx_para">
<p class="ltx_p">For the error term in () write
</p>
<table id="E10" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.40.10</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E10.png" altimg-height="68px" altimg-valign="-28px" altimg-width="399px" alttext="K_{\nu}\left(z\right)=\left(\frac{\pi}{2z}\right)^{\frac{1}{2}}e^{-z}\left(%
\sum_{k=0}^{\ell-1}\frac{a_{k}(\nu)}{z^{k}}+R_{\ell}(\nu,z)\right)," display="block"><mrow><mrow><mrow><msub><mi href="./10.25#E3">K</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msup><mrow><mo>(</mo><mfrac><mi href="./3.12#E1">π</mi><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow></mfrac><mo>)</mo></mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></msup><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mi href="./10.1#p2.t1.r4">z</mi></mrow></msup><mo></mo><mrow><mo>(</mo><mrow><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./10.1#p2.t1.r2">k</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi mathvariant="normal">ℓ</mi><mo>-</mo><mn>1</mn></mrow></munderover><mfrac><mrow><msub><mi href="./10.17#E1">a</mi><mi href="./10.1#p2.t1.r2">k</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./10.1#p2.t1.r5">ν</mi><mo stretchy="false">)</mo></mrow></mrow><msup><mi href="./10.1#p2.t1.r4">z</mi><mi href="./10.1#p2.t1.r2">k</mi></msup></mfrac></mrow><mo>+</mo><mrow><msub><mi href="./10.40#E11">R</mi><mi mathvariant="normal">ℓ</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./10.1#p2.t1.r5">ν</mi><mo>,</mo><mi href="./10.1#p2.t1.r4">z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m30.png" altimg-height="21px" altimg-valign="-6px" altimg-width="100px" alttext="\ell=1,2,\ldots" display="inline"><mrow><mi mathvariant="normal">ℓ</mi><mo>=</mo><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E10.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./10.25#E3" title="(10.25.3) ‣ §10.25(ii) Standard Solutions ‣ §10.25 Definitions ‣ Modified Bessel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="K_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.25#E3">K</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: modified Bessel function of the second kind</a>,
<a href="./10.1#p2.t1.r2" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./10.1#p2.t1.r2">k</mi></math>: nonnegative integer</a>,
<a href="./10.1#p2.t1.r4" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./10.1#p2.t1.r4">z</mi></math>: complex variable</a>,
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>,
<a href="./10.17#E1" title="(10.17.1) ‣ §10.17(i) Hankel’s Expansions ‣ §10.17 Asymptotic Expansions for Large Argument ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="23px" altimg-valign="-7px" altimg-width="51px" alttext="a_{k}(\nu)" display="inline"><mrow><msub><mi href="./10.17#E1">a</mi><mi href="./10.1#p2.t1.r2">k</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./10.1#p2.t1.r5">ν</mi><mo stretchy="false">)</mo></mrow></mrow></math>: expansion</a> and
<a href="./10.40#E11" title="(10.40.11) ‣ §10.40(iii) Error Bounds for Complex Argument and Order ‣ §10.40 Asymptotic Expansions for Large Argument ‣ Modified Bessel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m23.png" altimg-height="23px" altimg-valign="-7px" altimg-width="72px" alttext="R_{\ell}(\nu,z)" display="inline"><mrow><msub><mi href="./10.40#E11">R</mi><mi mathvariant="normal">ℓ</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./10.1#p2.t1.r5">ν</mi><mo>,</mo><mi href="./10.1#p2.t1.r4">z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: remainder</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Then</p>
<table id="E11" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.40.11</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E11.png" altimg-height="30px" altimg-valign="-9px" altimg-width="496px" alttext="|R_{\ell}(\nu,z)|\leq 2|a_{\ell}(\nu)|\mathcal{V}_{z,\infty}\left(t^{-\ell}%
\right)\*\exp\left(|\nu^{2}-\tfrac{1}{4}|\mathcal{V}_{z,\infty}\left(t^{-1}%
\right)\right)," display="block"><mrow><mrow><mrow><mo stretchy="false">|</mo><mrow><msub><mi href="./10.40#E11">R</mi><mi mathvariant="normal">ℓ</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./10.1#p2.t1.r5">ν</mi><mo>,</mo><mi href="./10.1#p2.t1.r4">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow><mo>≤</mo><mrow><mn>2</mn><mo></mo><mrow><mo stretchy="false">|</mo><mrow><msub><mi href="./10.17#E1">a</mi><mi mathvariant="normal">ℓ</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./10.1#p2.t1.r5">ν</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow><mo></mo><mrow><msub><mi class="ltx_font_mathcaligraphic" href="./1.4#E33">𝒱</mi><mrow><mi href="./10.1#p2.t1.r4">z</mi><mo>,</mo><mi mathvariant="normal">∞</mi></mrow></msub><mo></mo><mrow><mo>(</mo><msup><mi>t</mi><mrow><mo>-</mo><mi mathvariant="normal">ℓ</mi></mrow></msup><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mo stretchy="false">|</mo><mrow><msup><mi href="./10.1#p2.t1.r5">ν</mi><mn>2</mn></msup><mo>-</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>4</mn></mfrac></mstyle></mrow><mo stretchy="false">|</mo></mrow><mo></mo><mrow><msub><mi class="ltx_font_mathcaligraphic" href="./1.4#E33">𝒱</mi><mrow><mi href="./10.1#p2.t1.r4">z</mi><mo>,</mo><mi mathvariant="normal">∞</mi></mrow></msub><mo></mo><mrow><mo>(</mo><msup><mi>t</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo>)</mo></mrow></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E11.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text"><math class="ltx_Math" altimg="m23.png" altimg-height="23px" altimg-valign="-7px" altimg-width="72px" alttext="R_{\ell}(\nu,z)" display="inline"><mrow><msub><mi href="./10.40#E11">R</mi><mi mathvariant="normal">ℓ</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./10.1#p2.t1.r5">ν</mi><mo>,</mo><mi href="./10.1#p2.t1.r4">z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: remainder (locally)</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./4.2#E19" title="(4.2.19) ‣ §4.2(iii) The Exponential Function ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m32.png" altimg-height="16px" altimg-valign="-6px" altimg-width="48px" alttext="\exp\NVar{z}" display="inline"><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: exponential function</a>,
<a href="./1.4#E33" title="(1.4.33) ‣ Functions of Bounded Variation ‣ §1.4(v) Definite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m35.png" altimg-height="24px" altimg-valign="-8px" altimg-width="69px" alttext="\mathcal{V}_{\NVar{a,b}}\left(\NVar{f}\right)" display="inline"><mrow><msub><mi class="ltx_font_mathcaligraphic" href="./1.4#E33">𝒱</mi><mrow><mi class="ltx_nvar">a</mi><mo class="ltx_nvar">,</mo><mi class="ltx_nvar">b</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">f</mi><mo>)</mo></mrow></mrow></math>: total variation</a>,
<a href="./10.1#p2.t1.r4" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./10.1#p2.t1.r4">z</mi></math>: complex variable</a>,
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a> and
<a href="./10.17#E1" title="(10.17.1) ‣ §10.17(i) Hankel’s Expansions ‣ §10.17 Asymptotic Expansions for Large Argument ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="23px" altimg-valign="-7px" altimg-width="51px" alttext="a_{k}(\nu)" display="inline"><mrow><msub><mi href="./10.17#E1">a</mi><mi href="./10.1#p2.t1.r2">k</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./10.1#p2.t1.r5">ν</mi><mo stretchy="false">)</mo></mrow></mrow></math>: expansion</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m34.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="\mathcal{V}" display="inline"><mi class="ltx_font_mathcaligraphic" href="./1.4#E33">𝒱</mi></math> denotes the variational operator (§),
and the paths of variation are subject to the condition that <math class="ltx_Math" altimg="m62.png" altimg-height="23px" altimg-valign="-7px" altimg-width="37px" alttext="|\Re t|" display="inline"><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>t</mi></mrow><mo stretchy="false">|</mo></mrow></math>
changes monotonically. Bounds for <math class="ltx_Math" altimg="m36.png" altimg-height="27px" altimg-valign="-9px" altimg-width="95px" alttext="\mathcal{V}_{z,\infty}\left(t^{-\ell}\right)" display="inline"><mrow><msub><mi class="ltx_font_mathcaligraphic" href="./1.4#E33">𝒱</mi><mrow><mi href="./10.1#p2.t1.r4">z</mi><mo>,</mo><mi mathvariant="normal">∞</mi></mrow></msub><mo></mo><mrow><mo>(</mo><msup><mi>t</mi><mrow><mo>-</mo><mi mathvariant="normal">ℓ</mi></mrow></msup><mo>)</mo></mrow></mrow></math> are
given by</p>
<table id="E12" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.40.12</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E12.png" altimg-height="92px" altimg-valign="-40px" altimg-width="417px" alttext="\mathcal{V}_{z,\infty}\left(t^{-\ell}\right)\leq\begin{cases}|z|^{-\ell},&|%
\operatorname{ph}z|\leq\tfrac{1}{2}\pi,\\
\chi(\ell)|z|^{-\ell},&\tfrac{1}{2}\pi\leq|\operatorname{ph}z|\leq\pi,\\
2\chi(\ell)|\Re z|^{-\ell},&\pi\leq|\operatorname{ph}z|<\tfrac{3}{2}\pi,\end{cases}" display="block"><mrow><mrow><msub><mi class="ltx_font_mathcaligraphic" href="./1.4#E33">𝒱</mi><mrow><mi href="./10.1#p2.t1.r4">z</mi><mo>,</mo><mi mathvariant="normal">∞</mi></mrow></msub><mo></mo><mrow><mo>(</mo><msup><mi>t</mi><mrow><mo>-</mo><mi mathvariant="normal">ℓ</mi></mrow></msup><mo>)</mo></mrow></mrow><mo>≤</mo><mrow><mo>{</mo><mtable columnspacing="5pt" displaystyle="true" rowspacing="0pt"><mtr><mtd columnalign="left"><mrow><msup><mrow><mo stretchy="false">|</mo><mi href="./10.1#p2.t1.r4">z</mi><mo stretchy="false">|</mo></mrow><mrow><mo>-</mo><mi mathvariant="normal">ℓ</mi></mrow></msup><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo stretchy="false">|</mo></mrow><mo>≤</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow><mo>,</mo></mrow></mtd></mtr><mtr><mtd columnalign="left"><mrow><mrow><mrow><mi>χ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi mathvariant="normal">ℓ</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><msup><mrow><mo stretchy="false">|</mo><mi href="./10.1#p2.t1.r4">z</mi><mo stretchy="false">|</mo></mrow><mrow><mo>-</mo><mi mathvariant="normal">ℓ</mi></mrow></msup></mrow><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mrow><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo>≤</mo><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo stretchy="false">|</mo></mrow><mo>≤</mo><mi href="./3.12#E1">π</mi></mrow><mo>,</mo></mrow></mtd></mtr><mtr><mtd columnalign="left"><mrow><mrow><mn>2</mn><mo></mo><mrow><mi>χ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi mathvariant="normal">ℓ</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><msup><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo stretchy="false">|</mo></mrow><mrow><mo>-</mo><mi mathvariant="normal">ℓ</mi></mrow></msup></mrow><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mrow><mi href="./3.12#E1">π</mi><mo>≤</mo><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo stretchy="false">|</mo></mrow><mo><</mo><mrow><mstyle displaystyle="false"><mfrac><mn>3</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow><mo>,</mo></mrow></mtd></mtr></mtable></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E12.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.9#E7" title="(1.9.7) ‣ Modulus and Phase ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m40.png" altimg-height="21px" altimg-valign="-6px" altimg-width="26px" alttext="\operatorname{ph}" display="inline"><mi href="./1.9#E7">ph</mi></math>: phase</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m25.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./1.4#E33" title="(1.4.33) ‣ Functions of Bounded Variation ‣ §1.4(v) Definite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m35.png" altimg-height="24px" altimg-valign="-8px" altimg-width="69px" alttext="\mathcal{V}_{\NVar{a,b}}\left(\NVar{f}\right)" display="inline"><mrow><msub><mi class="ltx_font_mathcaligraphic" href="./1.4#E33">𝒱</mi><mrow><mi class="ltx_nvar">a</mi><mo class="ltx_nvar">,</mo><mi class="ltx_nvar">b</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">f</mi><mo>)</mo></mrow></mrow></math>: total variation</a> and
<a href="./10.1#p2.t1.r4" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./10.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>Correction (effective with 1.0.11):</dt>
<dd>
Originally the third constraint <math class="ltx_Math" altimg="m44.png" altimg-height="27px" altimg-valign="-9px" altimg-width="144px" alttext="\pi\leq|\operatorname{ph}z|<\frac{3}{2}\pi" display="inline"><mrow><mi href="./3.12#E1">π</mi><mo>≤</mo><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo stretchy="false">|</mo></mrow><mo><</mo><mrow><mfrac><mn>3</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow></math> was written incorrectly as
<math class="ltx_Math" altimg="m45.png" altimg-height="27px" altimg-valign="-9px" altimg-width="144px" alttext="\pi\leq|\operatorname{ph}z|\leq\frac{3}{2}\pi" display="inline"><mrow><mi href="./3.12#E1">π</mi><mo>≤</mo><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo stretchy="false">|</mo></mrow><mo>≤</mo><mrow><mfrac><mn>3</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow></math>.
<p><span class="ltx_font_italic">Suggested 2014-11-05 by Gergő Nemes</span></p>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m27.png" altimg-height="30px" altimg-valign="-9px" altimg-width="287px" alttext="\chi(\ell)=\pi^{\frac{1}{2}}\Gamma\left(\tfrac{1}{2}\ell+1\right)/\Gamma\left(%
\tfrac{1}{2}\ell+\tfrac{1}{2}\right)" display="inline"><mrow><mrow><mi>χ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi mathvariant="normal">ℓ</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mrow><msup><mi href="./3.12#E1">π</mi><mfrac><mn>1</mn><mn>2</mn></mfrac></msup><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi mathvariant="normal">ℓ</mi></mrow><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mrow><mo>/</mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi mathvariant="normal">ℓ</mi></mrow><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow></mrow></mrow></math>; see
§), with <math class="ltx_Math" altimg="m53.png" altimg-height="17px" altimg-valign="-2px" altimg-width="58px" alttext="m=0" display="inline"><mrow><mi href="./10.1#p2.t1.r1">m</mi><mo>=</mo><mn>0</mn></mrow></math>, and
()
</p>
<table id="E13" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.40.13</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E13.png" altimg-height="67px" altimg-valign="-28px" altimg-width="558px" alttext="R_{\ell}(\nu,z)=(-1)^{\ell}2\cos(\nu\pi)\*\left(\sum_{k=0}^{m-1}\frac{a_{k}(%
\nu)}{z^{k}}G_{\ell-k}\left(2z\right)+R_{m,\ell}(\nu,z)\right)," display="block"><mrow><mrow><mrow><msub><mi href="./10.40#E14">R</mi><mi mathvariant="normal">ℓ</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./10.1#p2.t1.r5">ν</mi><mo>,</mo><mi href="./10.1#p2.t1.r4">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi mathvariant="normal">ℓ</mi></msup><mo></mo><mn>2</mn><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo>(</mo><mrow><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./10.1#p2.t1.r2">k</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi href="./10.1#p2.t1.r1">m</mi><mo>-</mo><mn>1</mn></mrow></munderover><mrow><mfrac><mrow><msub><mi href="./10.17#E1">a</mi><mi href="./10.1#p2.t1.r2">k</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./10.1#p2.t1.r5">ν</mi><mo stretchy="false">)</mo></mrow></mrow><msup><mi href="./10.1#p2.t1.r4">z</mi><mi href="./10.1#p2.t1.r2">k</mi></msup></mfrac><mo></mo><mrow><msub><mi href="./9.7#E22">G</mi><mrow><mi mathvariant="normal">ℓ</mi><mo>-</mo><mi href="./10.1#p2.t1.r2">k</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>+</mo><mrow><msub><mi href="./10.40#E14">R</mi><mrow><mi href="./10.1#p2.t1.r1">m</mi><mo>,</mo><mi mathvariant="normal">ℓ</mi></mrow></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./10.1#p2.t1.r5">ν</mi><mo>,</mo><mi href="./10.1#p2.t1.r4">z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E13.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m28.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./9.7#E22" title="(9.7.22) ‣ §9.7(v) Exponentially-Improved Expansions ‣ §9.7 Asymptotic Expansions ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m12.png" altimg-height="24px" altimg-valign="-8px" altimg-width="58px" alttext="G_{\NVar{p}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./9.7#E22">G</mi><mi class="ltx_nvar">p</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./9.1#p2.t1.r3">z</mi><mo>)</mo></mrow></mrow></math>: rescaled terminant function</a>,
<a href="./10.1#p2.t1.r1" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./10.1#p2.t1.r1">m</mi></math>: integer</a>,
<a href="./10.1#p2.t1.r2" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./10.1#p2.t1.r2">k</mi></math>: nonnegative integer</a>,
<a href="./10.1#p2.t1.r4" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./10.1#p2.t1.r4">z</mi></math>: complex variable</a>,
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>,
<a href="./10.17#E1" title="(10.17.1) ‣ §10.17(i) Hankel’s Expansions ‣ §10.17 Asymptotic Expansions for Large Argument ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="23px" altimg-valign="-7px" altimg-width="51px" alttext="a_{k}(\nu)" display="inline"><mrow><msub><mi href="./10.17#E1">a</mi><mi href="./10.1#p2.t1.r2">k</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./10.1#p2.t1.r5">ν</mi><mo stretchy="false">)</mo></mrow></mrow></math>: expansion</a> and
<a href="./10.40#E14" title="(10.40.14) ‣ §10.40(iv) Exponentially-Improved Expansions ‣ §10.40 Asymptotic Expansions for Large Argument ‣ Modified Bessel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m24.png" altimg-height="24px" altimg-valign="-8px" altimg-width="90px" alttext="R_{m,\ell}(\nu,z)" display="inline"><mrow><msub><mi href="./10.40#E14">R</mi><mrow><mi href="./10.1#p2.t1.r1">m</mi><mo>,</mo><mi mathvariant="normal">ℓ</mi></mrow></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./10.1#p2.t1.r5">ν</mi><mo>,</mo><mi href="./10.1#p2.t1.r4">z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: remainder</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m13.png" altimg-height="24px" altimg-valign="-8px" altimg-width="58px" alttext="G_{p}\left(z\right)" display="inline"><mrow><msub><mi href="./9.7#E22">G</mi><mi>p</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math> is given by (). If <math class="ltx_Math" altimg="m59.png" altimg-height="13px" altimg-valign="-2px" altimg-width="65px" alttext="z\to\infty" display="inline"><mrow><mi href="./10.1#p2.t1.r4">z</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></math> with
<math class="ltx_Math" altimg="m63.png" altimg-height="23px" altimg-valign="-7px" altimg-width="79px" alttext="|\ell-2|z||" display="inline"><mrow><mo stretchy="false">|</mo><mrow><mi mathvariant="normal">ℓ</mi><mo>-</mo><mrow><mn>2</mn><mo></mo><mrow><mo stretchy="false">|</mo><mi href="./10.1#p2.t1.r4">z</mi><mo stretchy="false">|</mo></mrow></mrow></mrow><mo stretchy="false">|</mo></mrow></math> bounded and <math class="ltx_Math" altimg="m54.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./10.1#p2.t1.r1">m</mi></math> <math class="ltx_Math" altimg="m10.png" altimg-height="23px" altimg-valign="-7px" altimg-width="51px" alttext="(\geq 0)" display="inline"><mrow><mo stretchy="false">(</mo><mrow><mi></mi><mo>≥</mo><mn>0</mn></mrow><mo stretchy="false">)</mo></mrow></math> fixed, then</p>
<table id="E14" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.40.14</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E14.png" altimg-height="41px" altimg-valign="-15px" altimg-width="257px" alttext="R_{m,\ell}(\nu,z)=O\left(e^{-2|z|}z^{-m}\right)," display="block"><mrow><mrow><mrow><msub><mi href="./10.40#E14">R</mi><mrow><mi href="./10.1#p2.t1.r1">m</mi><mo>,</mo><mi mathvariant="normal">ℓ</mi></mrow></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./10.1#p2.t1.r5">ν</mi><mo>,</mo><mi href="./10.1#p2.t1.r4">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mi href="./2.1#E3">O</mi><mo></mo><mrow><mo>(</mo><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mn>2</mn><mo></mo><mrow><mo stretchy="false">|</mo><mi href="./10.1#p2.t1.r4">z</mi><mo stretchy="false">|</mo></mrow></mrow></mrow></msup><mo></mo><msup><mi href="./10.1#p2.t1.r4">z</mi><mrow><mo>-</mo><mi href="./10.1#p2.t1.r1">m</mi></mrow></msup></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m64.png" altimg-height="23px" altimg-valign="-7px" altimg-width="93px" alttext="|\operatorname{ph}z|\leq\pi" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mi href="./10.1#p2.t1.r4">z</mi></mrow><mo stretchy="false">|</mo></mrow><mo>≤</mo><mi href="./3.12#E1">π</mi></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E14.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text"><math class="ltx_Math" altimg="m24.png" altimg-height="24px" altimg-valign="-8px" altimg-width="90px" alttext="R_{m,\ell}(\nu,z)" display="inline"><mrow><msub><mi href="./10.40#E14">R</mi><mrow><mi href="./10.1#p2.t1.r1">m</mi><mo>,</mo><mi mathvariant="normal">ℓ</mi></mrow></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./10.1#p2.t1.r5">ν</mi><mo>,</mo><mi href="./10.1#p2.t1.r4">z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: remainder (locally)</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./2.1#E3" title="(2.1.3) ‣ §2.1(i) Asymptotic and Order Symbols ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="23px" altimg-valign="-7px" altimg-width="50px" alttext="O\left(\NVar{x}\right)" display="inline"><mrow><mi href="./2.1#E3">O</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>)</mo></mrow></mrow></math>: order not exceeding</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./1.9#E7" title="(1.9.7) ‣ Modulus and Phase ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m40.png" altimg-height="21px" altimg-valign="-6px" altimg-width="26px" alttext="\operatorname{ph}" display="inline"><mi href="./1.9#E7">ph</mi></math>: phase</a>,
<a href="./10.1#p2.t1.r1" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./10.1#p2.t1.r1">m</mi></math>: integer</a>,
<a href="./10.1#p2.t1.r4" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./10.1#p2.t1.r4">z</mi></math>: complex variable</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS4.p2" class="ltx_para">
<p class="ltx_p">For higher re-expansions of the remainder term see
<cite class="ltx_cite ltx_citemacro_citet">Olde Daalhuis and Olver (</div>
</div>
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<title>DLMF: 10.68 Modulus and Phase Functions</title>
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<div id="SS1.p1" class="ltx_para">
<table id="E1" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.68.1</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E1.png" altimg-height="29px" altimg-valign="-7px" altimg-width="290px" alttext="M_{\nu}\left(x\right)e^{i\!\theta_{\nu}\left(x\right)}=\operatorname{ber}_{\nu%
}x+i\operatorname{bei}_{\nu}x," display="block"><mrow><mrow><mrow><mrow><msub><mi href="./10.18#E1">M</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mpadded width="-1.7pt"><mi mathvariant="normal">i</mi></mpadded><mo></mo><mrow><msub><mi href="./10.18#E3">θ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mrow></msup></mrow><mo>=</mo><mrow><mrow><msub><mi href="./10.61#E1">ber</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mi href="./10.1#p2.t1.r3">x</mi></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mrow><msub><mi href="./10.61#E1">bei</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mi href="./10.1#p2.t1.r3">x</mi></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.61#E1" title="(10.61.1) ‣ §10.61(i) Definitions ‣ §10.61 Definitions and Basic Properties ‣ Kelvin Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m62.png" altimg-height="23px" altimg-valign="-7px" altimg-width="70px" alttext="\operatorname{bei}_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.61#E1">bei</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: Kelvin function</a>,
<a href="./10.61#E1" title="(10.61.1) ‣ §10.61(i) Definitions ‣ §10.61 Definitions and Basic Properties ‣ Kelvin Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m64.png" altimg-height="23px" altimg-valign="-7px" altimg-width="73px" alttext="\operatorname{ber}_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.61#E1">ber</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: Kelvin function</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m57.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./10.18#E1" title="(10.18.1) ‣ §10.18(i) Definitions ‣ §10.18 Modulus and Phase Functions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="23px" altimg-valign="-7px" altimg-width="64px" alttext="M_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.18#E1">M</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: modulus of Bessel functions</a>,
<a href="./10.18#E3" title="(10.18.3) ‣ §10.18(i) Definitions ‣ §10.18 Modulus and Phase Functions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="\theta_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.18#E3">θ</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: phase of Bessel functions</a>,
<a href="./10.1#p2.t1.r3" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./10.1#p2.t1.r3">x</mi></math>: real variable</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E2" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.68.2</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E2.png" altimg-height="29px" altimg-valign="-7px" altimg-width="285px" alttext="N_{\nu}\left(x\right)e^{i\!\phi_{\nu}\left(x\right)}=\operatorname{ker}_{\nu}x%
+i\operatorname{kei}_{\nu}x," display="block"><mrow><mrow><mrow><mrow><msub><mi href="./10.18#E2">N</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mpadded width="-1.7pt"><mi mathvariant="normal">i</mi></mpadded><mo></mo><mrow><msub><mi href="./10.18#E3">ϕ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mrow></msup></mrow><mo>=</mo><mrow><mrow><msub><mi href="./10.61#E2">ker</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mi href="./10.1#p2.t1.r3">x</mi></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mrow><msub><mi href="./10.61#E2">kei</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mi href="./10.1#p2.t1.r3">x</mi></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.61#E2" title="(10.61.2) ‣ §10.61(i) Definitions ‣ §10.61 Definitions and Basic Properties ‣ Kelvin Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="23px" altimg-valign="-7px" altimg-width="69px" alttext="\operatorname{kei}_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.61#E2">kei</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: Kelvin function</a>,
<a href="./10.61#E2" title="(10.61.2) ‣ §10.61(i) Definitions ‣ §10.61 Definitions and Basic Properties ‣ Kelvin Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m68.png" altimg-height="23px" altimg-valign="-7px" altimg-width="71px" alttext="\operatorname{ker}_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.61#E2">ker</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: Kelvin function</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m57.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./10.18#E2" title="(10.18.2) ‣ §10.18(i) Definitions ‣ §10.18 Modulus and Phase Functions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="N_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.18#E2">N</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: modulus of derivatives of Bessel functions</a>,
<a href="./10.18#E3" title="(10.18.3) ‣ §10.18(i) Definitions ‣ §10.18 Modulus and Phase Functions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m73.png" altimg-height="23px" altimg-valign="-7px" altimg-width="56px" alttext="\phi_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.18#E3">ϕ</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: phase of derivatives of Bessel functions</a>,
<a href="./10.1#p2.t1.r3" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./10.1#p2.t1.r3">x</mi></math>: real variable</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m48.png" altimg-height="23px" altimg-valign="-7px" altimg-width="117px" alttext="M_{\nu}\left(x\right)\,(>0)" display="inline"><mrow><mpadded width="+1.7pt"><mrow><msub><mi href="./10.18#E1">M</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mpadded><mspace width="veryverythickmathspace"></mspace><mrow><mo stretchy="false">(</mo><mrow><mi></mi><mo>></mo><mn>0</mn></mrow><mo stretchy="false">)</mo></mrow></mrow></math>, <math class="ltx_Math" altimg="m51.png" altimg-height="23px" altimg-valign="-7px" altimg-width="114px" alttext="N_{\nu}\left(x\right)\,(>0)" display="inline"><mrow><mpadded width="+1.7pt"><mrow><msub><mi href="./10.18#E2">N</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mpadded><mspace width="veryverythickmathspace"></mspace><mrow><mo stretchy="false">(</mo><mrow><mi></mi><mo>></mo><mn>0</mn></mrow><mo stretchy="false">)</mo></mrow></mrow></math>,
<math class="ltx_Math" altimg="m79.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="\theta_{\nu}\left(x\right)" display="inline"><mrow><msub><mi href="./10.18#E3">θ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>, and <math class="ltx_Math" altimg="m74.png" altimg-height="23px" altimg-valign="-7px" altimg-width="56px" alttext="\phi_{\nu}\left(x\right)" display="inline"><mrow><msub><mi href="./10.18#E3">ϕ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math> are continuous real
functions of <math class="ltx_Math" altimg="m82.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./10.1#p2.t1.r3">x</mi></math> and <math class="ltx_Math" altimg="m59.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>, with the branches of <math class="ltx_Math" altimg="m79.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="\theta_{\nu}\left(x\right)" display="inline"><mrow><msub><mi href="./10.18#E3">θ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math> and
<math class="ltx_Math" altimg="m74.png" altimg-height="23px" altimg-valign="-7px" altimg-width="56px" alttext="\phi_{\nu}\left(x\right)" display="inline"><mrow><msub><mi href="./10.18#E3">ϕ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math> chosen to satisfy () as <math class="ltx_Math" altimg="m83.png" altimg-height="13px" altimg-valign="-2px" altimg-width="67px" alttext="x\to\infty" display="inline"><mrow><mi href="./10.1#p2.t1.r3">x</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></math>. (See also §</dd>
</dl>
</div>
</div>
<div id="SS2.p1" class="ltx_para">
<table id="E3" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex1" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">10.68.3</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m36.png" altimg-height="22px" altimg-valign="-5px" altimg-width="59px" alttext="\displaystyle\operatorname{ber}_{\nu}x" display="inline"><mrow><msub><mi href="./10.61#E1">ber</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mi href="./10.1#p2.t1.r3">x</mi></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m9.png" altimg-height="25px" altimg-valign="-7px" altimg-width="178px" alttext="\displaystyle=M_{\nu}\left(x\right)\cos\theta_{\nu}\left(x\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><msub><mi href="./10.18#E1">M</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><msub><mi href="./10.18#E3">θ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex2" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m34.png" altimg-height="22px" altimg-valign="-5px" altimg-width="57px" alttext="\displaystyle\operatorname{bei}_{\nu}x" display="inline"><mrow><msub><mi href="./10.61#E1">bei</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mi href="./10.1#p2.t1.r3">x</mi></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m10.png" altimg-height="25px" altimg-valign="-7px" altimg-width="176px" alttext="\displaystyle=M_{\nu}\left(x\right)\sin\theta_{\nu}\left(x\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><msub><mi href="./10.18#E1">M</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><msub><mi href="./10.18#E3">θ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.61#E1" title="(10.61.1) ‣ §10.61(i) Definitions ‣ §10.61 Definitions and Basic Properties ‣ Kelvin Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m62.png" altimg-height="23px" altimg-valign="-7px" altimg-width="70px" alttext="\operatorname{bei}_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.61#E1">bei</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: Kelvin function</a>,
<a href="./10.61#E1" title="(10.61.1) ‣ §10.61(i) Definitions ‣ §10.61 Definitions and Basic Properties ‣ Kelvin Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m64.png" altimg-height="23px" altimg-valign="-7px" altimg-width="73px" alttext="\operatorname{ber}_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.61#E1">ber</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: Kelvin function</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./10.18#E1" title="(10.18.1) ‣ §10.18(i) Definitions ‣ §10.18 Modulus and Phase Functions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="23px" altimg-valign="-7px" altimg-width="64px" alttext="M_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.18#E1">M</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: modulus of Bessel functions</a>,
<a href="./10.18#E3" title="(10.18.3) ‣ §10.18(i) Definitions ‣ §10.18 Modulus and Phase Functions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="\theta_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.18#E3">θ</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: phase of Bessel functions</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m76.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./10.1#p2.t1.r3" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./10.1#p2.t1.r3">x</mi></math>: real variable</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">9.10.9, 9.10.19</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E4" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex3" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">10.68.4</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m38.png" altimg-height="22px" altimg-valign="-5px" altimg-width="57px" alttext="\displaystyle\operatorname{ker}_{\nu}x" display="inline"><mrow><msub><mi href="./10.61#E2">ker</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mi href="./10.1#p2.t1.r3">x</mi></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m13.png" altimg-height="25px" altimg-valign="-7px" altimg-width="177px" alttext="\displaystyle=N_{\nu}\left(x\right)\cos\phi_{\nu}\left(x\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><msub><mi href="./10.18#E2">N</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><msub><mi href="./10.18#E3">ϕ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex4" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m37.png" altimg-height="22px" altimg-valign="-5px" altimg-width="55px" alttext="\displaystyle\operatorname{kei}_{\nu}x" display="inline"><mrow><msub><mi href="./10.61#E2">kei</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mi href="./10.1#p2.t1.r3">x</mi></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m14.png" altimg-height="25px" altimg-valign="-7px" altimg-width="175px" alttext="\displaystyle=N_{\nu}\left(x\right)\sin\phi_{\nu}\left(x\right)." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><msub><mi href="./10.18#E2">N</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><msub><mi href="./10.18#E3">ϕ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E4.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.61#E2" title="(10.61.2) ‣ §10.61(i) Definitions ‣ §10.61 Definitions and Basic Properties ‣ Kelvin Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="23px" altimg-valign="-7px" altimg-width="69px" alttext="\operatorname{kei}_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.61#E2">kei</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: Kelvin function</a>,
<a href="./10.61#E2" title="(10.61.2) ‣ §10.61(i) Definitions ‣ §10.61 Definitions and Basic Properties ‣ Kelvin Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m68.png" altimg-height="23px" altimg-valign="-7px" altimg-width="71px" alttext="\operatorname{ker}_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.61#E2">ker</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: Kelvin function</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./10.18#E2" title="(10.18.2) ‣ §10.18(i) Definitions ‣ §10.18 Modulus and Phase Functions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="N_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.18#E2">N</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: modulus of derivatives of Bessel functions</a>,
<a href="./10.18#E3" title="(10.18.3) ‣ §10.18(i) Definitions ‣ §10.18 Modulus and Phase Functions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m73.png" altimg-height="23px" altimg-valign="-7px" altimg-width="56px" alttext="\phi_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.18#E3">ϕ</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: phase of derivatives of Bessel functions</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m76.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./10.1#p2.t1.r3" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./10.1#p2.t1.r3">x</mi></math>: real variable</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E5" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex5" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">10.68.5</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m27.png" altimg-height="25px" altimg-valign="-7px" altimg-width="65px" alttext="\displaystyle M_{\nu}\left(x\right)" display="inline"><mrow><msub><mi href="./10.18#E1">M</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m2.png" altimg-height="29px" altimg-valign="-7px" altimg-width="199px" alttext="\displaystyle=({\operatorname{ber}_{\nu}^{2}}x+{\operatorname{bei}_{\nu}^{2}}x%
)^{\ifrac{1}{2}}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mrow><mrow><msubsup><mi href="./10.61#E1">ber</mi><mi href="./10.1#p2.t1.r5">ν</mi><mn>2</mn></msubsup><mo></mo><mi href="./10.1#p2.t1.r3">x</mi></mrow><mo>+</mo><mrow><msubsup><mi href="./10.61#E1">bei</mi><mi href="./10.1#p2.t1.r5">ν</mi><mn>2</mn></msubsup><mo></mo><mi href="./10.1#p2.t1.r3">x</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex6" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m29.png" altimg-height="25px" altimg-valign="-7px" altimg-width="62px" alttext="\displaystyle N_{\nu}\left(x\right)" display="inline"><mrow><msub><mi href="./10.18#E2">N</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m3.png" altimg-height="29px" altimg-valign="-7px" altimg-width="196px" alttext="\displaystyle=({\operatorname{ker}_{\nu}^{2}}x+{\operatorname{kei}_{\nu}^{2}}x%
)^{\ifrac{1}{2}}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mrow><mrow><msubsup><mi href="./10.61#E2">ker</mi><mi href="./10.1#p2.t1.r5">ν</mi><mn>2</mn></msubsup><mo></mo><mi href="./10.1#p2.t1.r3">x</mi></mrow><mo>+</mo><mrow><msubsup><mi href="./10.61#E2">kei</mi><mi href="./10.1#p2.t1.r5">ν</mi><mn>2</mn></msubsup><mo></mo><mi href="./10.1#p2.t1.r3">x</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E5.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.61#E1" title="(10.61.1) ‣ §10.61(i) Definitions ‣ §10.61 Definitions and Basic Properties ‣ Kelvin Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m62.png" altimg-height="23px" altimg-valign="-7px" altimg-width="70px" alttext="\operatorname{bei}_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.61#E1">bei</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: Kelvin function</a>,
<a href="./10.61#E1" title="(10.61.1) ‣ §10.61(i) Definitions ‣ §10.61 Definitions and Basic Properties ‣ Kelvin Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m64.png" altimg-height="23px" altimg-valign="-7px" altimg-width="73px" alttext="\operatorname{ber}_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.61#E1">ber</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: Kelvin function</a>,
<a href="./10.61#E2" title="(10.61.2) ‣ §10.61(i) Definitions ‣ §10.61 Definitions and Basic Properties ‣ Kelvin Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="23px" altimg-valign="-7px" altimg-width="69px" alttext="\operatorname{kei}_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.61#E2">kei</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: Kelvin function</a>,
<a href="./10.61#E2" title="(10.61.2) ‣ §10.61(i) Definitions ‣ §10.61 Definitions and Basic Properties ‣ Kelvin Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m68.png" altimg-height="23px" altimg-valign="-7px" altimg-width="71px" alttext="\operatorname{ker}_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.61#E2">ker</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: Kelvin function</a>,
<a href="./10.18#E1" title="(10.18.1) ‣ §10.18(i) Definitions ‣ §10.18 Modulus and Phase Functions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="23px" altimg-valign="-7px" altimg-width="64px" alttext="M_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.18#E1">M</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: modulus of Bessel functions</a>,
<a href="./10.18#E2" title="(10.18.2) ‣ §10.18(i) Definitions ‣ §10.18 Modulus and Phase Functions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="N_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.18#E2">N</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: modulus of derivatives of Bessel functions</a>,
<a href="./10.1#p2.t1.r3" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./10.1#p2.t1.r3">x</mi></math>: real variable</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">9.10.8, 9.10.18</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E6" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex7" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">10.68.6</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m43.png" altimg-height="25px" altimg-valign="-7px" altimg-width="56px" alttext="\displaystyle\theta_{\nu}\left(x\right)" display="inline"><mrow><msub><mi href="./10.18#E3">θ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m18.png" altimg-height="25px" altimg-valign="-7px" altimg-width="232px" alttext="\displaystyle=\operatorname{Arctan}\left(\operatorname{bei}_{\nu}x/%
\operatorname{ber}_{\nu}x\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./4.23#E3">Arctan</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><msub><mi href="./10.61#E1">bei</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mi href="./10.1#p2.t1.r3">x</mi></mrow><mo>/</mo><mrow><msub><mi href="./10.61#E1">ber</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mi href="./10.1#p2.t1.r3">x</mi></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex8" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m40.png" altimg-height="25px" altimg-valign="-7px" altimg-width="58px" alttext="\displaystyle\phi_{\nu}\left(x\right)" display="inline"><mrow><msub><mi href="./10.18#E3">ϕ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m19.png" altimg-height="25px" altimg-valign="-7px" altimg-width="229px" alttext="\displaystyle=\operatorname{Arctan}\left(\operatorname{kei}_{\nu}x/%
\operatorname{ker}_{\nu}x\right)." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./4.23#E3">Arctan</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><msub><mi href="./10.61#E2">kei</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mi href="./10.1#p2.t1.r3">x</mi></mrow><mo>/</mo><mrow><msub><mi href="./10.61#E2">ker</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mi href="./10.1#p2.t1.r3">x</mi></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E6.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.61#E1" title="(10.61.1) ‣ §10.61(i) Definitions ‣ §10.61 Definitions and Basic Properties ‣ Kelvin Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m62.png" altimg-height="23px" altimg-valign="-7px" altimg-width="70px" alttext="\operatorname{bei}_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.61#E1">bei</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: Kelvin function</a>,
<a href="./10.61#E1" title="(10.61.1) ‣ §10.61(i) Definitions ‣ §10.61 Definitions and Basic Properties ‣ Kelvin Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m64.png" altimg-height="23px" altimg-valign="-7px" altimg-width="73px" alttext="\operatorname{ber}_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.61#E1">ber</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: Kelvin function</a>,
<a href="./10.61#E2" title="(10.61.2) ‣ §10.61(i) Definitions ‣ §10.61 Definitions and Basic Properties ‣ Kelvin Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="23px" altimg-valign="-7px" altimg-width="69px" alttext="\operatorname{kei}_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.61#E2">kei</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: Kelvin function</a>,
<a href="./10.61#E2" title="(10.61.2) ‣ §10.61(i) Definitions ‣ §10.61 Definitions and Basic Properties ‣ Kelvin Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m68.png" altimg-height="23px" altimg-valign="-7px" altimg-width="71px" alttext="\operatorname{ker}_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.61#E2">ker</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: Kelvin function</a>,
<a href="./4.23#E3" title="(4.23.3) ‣ §4.23(i) General Definitions ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="18px" altimg-valign="-2px" altimg-width="78px" alttext="\operatorname{Arctan}\NVar{z}" display="inline"><mrow><mi href="./4.23#E3">Arctan</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: general arctangent function</a>,
<a href="./10.18#E3" title="(10.18.3) ‣ §10.18(i) Definitions ‣ §10.18 Modulus and Phase Functions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m73.png" altimg-height="23px" altimg-valign="-7px" altimg-width="56px" alttext="\phi_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.18#E3">ϕ</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: phase of derivatives of Bessel functions</a>,
<a href="./10.18#E3" title="(10.18.3) ‣ §10.18(i) Definitions ‣ §10.18 Modulus and Phase Functions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="\theta_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.18#E3">θ</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: phase of Bessel functions</a>,
<a href="./10.1#p2.t1.r3" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./10.1#p2.t1.r3">x</mi></math>: real variable</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">9.10.8, 9.10.18</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E7" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex9" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">10.68.7</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m25.png" altimg-height="25px" altimg-valign="-7px" altimg-width="79px" alttext="\displaystyle M_{-n}\left(x\right)" display="inline"><mrow><msub><mi href="./10.18#E1">M</mi><mrow><mo>-</mo><mi href="./10.1#p2.t1.r1">n</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m11.png" altimg-height="25px" altimg-valign="-7px" altimg-width="96px" alttext="\displaystyle=M_{n}\left(x\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msub><mi href="./10.18#E1">M</mi><mi href="./10.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex10" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m41.png" altimg-height="25px" altimg-valign="-7px" altimg-width="69px" alttext="\displaystyle\theta_{-n}\left(x\right)" display="inline"><mrow><msub><mi href="./10.18#E3">θ</mi><mrow><mo>-</mo><mi href="./10.1#p2.t1.r1">n</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m21.png" altimg-height="25px" altimg-valign="-7px" altimg-width="131px" alttext="\displaystyle=\theta_{n}\left(x\right)-n\pi." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><msub><mi href="./10.18#E3">θ</mi><mi href="./10.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow><mo>-</mo><mrow><mi href="./10.1#p2.t1.r1">n</mi><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E7.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m75.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./10.18#E1" title="(10.18.1) ‣ §10.18(i) Definitions ‣ §10.18 Modulus and Phase Functions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="23px" altimg-valign="-7px" altimg-width="64px" alttext="M_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.18#E1">M</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: modulus of Bessel functions</a>,
<a href="./10.18#E3" title="(10.18.3) ‣ §10.18(i) Definitions ‣ §10.18 Modulus and Phase Functions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="\theta_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.18#E3">θ</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: phase of Bessel functions</a>,
<a href="./10.1#p2.t1.r1" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./10.1#p2.t1.r1">n</mi></math>: integer</a> and
<a href="./10.1#p2.t1.r3" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./10.1#p2.t1.r3">x</mi></math>: real variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">9.10.10</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS2.p2" class="ltx_para">
<p class="ltx_p">With arguments <math class="ltx_Math" altimg="m45.png" altimg-height="23px" altimg-valign="-7px" altimg-width="31px" alttext="(x)" display="inline"><mrow><mo stretchy="false">(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo stretchy="false">)</mo></mrow></math> suppressed,
</p>
<table id="E8" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.68.8</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_Math ltx_eqn_cell ltx_align_center"><math class="ltx_Math" alttext="\operatorname{ber}_{\nu}'x=\tfrac{1}{2}M_{\nu+1}\cos\left(\theta_{\nu+1}-%
\tfrac{1}{4}\pi\right)-\tfrac{1}{2}M_{\nu-1}\cos\left(\theta_{\nu-1}-\tfrac{1}%
{4}\pi\right)=(\nu/x)M_{\nu}\cos\theta_{\nu}+M_{\nu+1}\cos\left(\theta_{\nu+1}%
-\tfrac{1}{4}\pi\right)=-(\nu/x)M_{\nu}\cos\theta_{\nu}-M_{\nu-1}\cos\left(%
\theta_{\nu-1}-\tfrac{1}{4}\pi\right)," display="block"><mrow><mtable align="baseline 1" columnalign="left" columnspacing="0.1em" displaystyle="true"><mtr><mtd><mrow><msubsup><mi href="./10.61#E1">ber</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′</mo></msubsup><mo></mo><mi href="./10.1#p2.t1.r3">x</mi></mrow></mtd><mtd><mo>=</mo><mrow><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><msub><mi href="./10.18#E1">M</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><msub><mi href="./10.18#E3">θ</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>-</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>4</mn></mfrac></mstyle><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>-</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><msub><mi href="./10.18#E1">M</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>-</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><msub><mi href="./10.18#E3">θ</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>-</mo><mn>1</mn></mrow></msub><mo>-</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>4</mn></mfrac></mstyle><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo><mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>/</mo><mi href="./10.1#p2.t1.r3">x</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msub><mi href="./10.18#E1">M</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><msub><mi href="./10.18#E3">θ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub></mrow></mrow><mo>+</mo><mrow><msub><mi href="./10.18#E1">M</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><msub><mi href="./10.18#E3">θ</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>-</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>4</mn></mfrac></mstyle><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo><mrow><mrow><mo>-</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>/</mo><mi href="./10.1#p2.t1.r3">x</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msub><mi href="./10.18#E1">M</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><msub><mi href="./10.18#E3">θ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub></mrow></mrow></mrow><mo>-</mo><mrow><msub><mi href="./10.18#E1">M</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>-</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><msub><mi href="./10.18#E3">θ</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>-</mo><mn>1</mn></mrow></msub><mo>-</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>4</mn></mfrac></mstyle><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mtd></mtr></mtable></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E8.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.61#E1" title="(10.61.1) ‣ §10.61(i) Definitions ‣ §10.61 Definitions and Basic Properties ‣ Kelvin Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m64.png" altimg-height="23px" altimg-valign="-7px" altimg-width="73px" alttext="\operatorname{ber}_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.61#E1">ber</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: Kelvin function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m75.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./10.18#E1" title="(10.18.1) ‣ §10.18(i) Definitions ‣ §10.18 Modulus and Phase Functions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="23px" altimg-valign="-7px" altimg-width="64px" alttext="M_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.18#E1">M</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: modulus of Bessel functions</a>,
<a href="./10.18#E3" title="(10.18.3) ‣ §10.18(i) Definitions ‣ §10.18 Modulus and Phase Functions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="\theta_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.18#E3">θ</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: phase of Bessel functions</a>,
<a href="./10.1#p2.t1.r3" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./10.1#p2.t1.r3">x</mi></math>: real variable</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">9.10.11</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E9" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.68.9</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_Math ltx_eqn_cell ltx_align_center"><math class="ltx_Math" alttext="\operatorname{bei}_{\nu}'x=\tfrac{1}{2}M_{\nu+1}\sin\left(\theta_{\nu+1}-%
\tfrac{1}{4}\pi\right)-\tfrac{1}{2}M_{\nu-1}\sin\left(\theta_{\nu-1}-\tfrac{1}%
{4}\pi\right)=(\nu/x)M_{\nu}\sin\theta_{\nu}+M_{\nu+1}\sin\left(\theta_{\nu+1}%
-\tfrac{1}{4}\pi\right)=-(\nu/x)M_{\nu}\sin\theta_{\nu}-M_{\nu-1}\sin\left(%
\theta_{\nu-1}-\tfrac{1}{4}\pi\right)." display="block"><mrow><mtable align="baseline 1" columnalign="left" columnspacing="0.1em" displaystyle="true"><mtr><mtd><mrow><msubsup><mi href="./10.61#E1">bei</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′</mo></msubsup><mo></mo><mi href="./10.1#p2.t1.r3">x</mi></mrow></mtd><mtd><mo>=</mo><mrow><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><msub><mi href="./10.18#E1">M</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><msub><mi href="./10.18#E3">θ</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>-</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>4</mn></mfrac></mstyle><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>-</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><msub><mi href="./10.18#E1">M</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>-</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><msub><mi href="./10.18#E3">θ</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>-</mo><mn>1</mn></mrow></msub><mo>-</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>4</mn></mfrac></mstyle><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo><mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>/</mo><mi href="./10.1#p2.t1.r3">x</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msub><mi href="./10.18#E1">M</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><msub><mi href="./10.18#E3">θ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub></mrow></mrow><mo>+</mo><mrow><msub><mi href="./10.18#E1">M</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><msub><mi href="./10.18#E3">θ</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>-</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>4</mn></mfrac></mstyle><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo><mrow><mrow><mo>-</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>/</mo><mi href="./10.1#p2.t1.r3">x</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msub><mi href="./10.18#E1">M</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><msub><mi href="./10.18#E3">θ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub></mrow></mrow></mrow><mo>-</mo><mrow><msub><mi href="./10.18#E1">M</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>-</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><msub><mi href="./10.18#E3">θ</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>-</mo><mn>1</mn></mrow></msub><mo>-</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>4</mn></mfrac></mstyle><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>.</mo></mtd></mtr></mtable></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E9.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.61#E1" title="(10.61.1) ‣ §10.61(i) Definitions ‣ §10.61 Definitions and Basic Properties ‣ Kelvin Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m62.png" altimg-height="23px" altimg-valign="-7px" altimg-width="70px" alttext="\operatorname{bei}_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.61#E1">bei</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: Kelvin function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m75.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./10.18#E1" title="(10.18.1) ‣ §10.18(i) Definitions ‣ §10.18 Modulus and Phase Functions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="23px" altimg-valign="-7px" altimg-width="64px" alttext="M_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.18#E1">M</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: modulus of Bessel functions</a>,
<a href="./10.18#E3" title="(10.18.3) ‣ §10.18(i) Definitions ‣ §10.18 Modulus and Phase Functions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="\theta_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.18#E3">θ</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: phase of Bessel functions</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m76.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./10.1#p2.t1.r3" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./10.1#p2.t1.r3">x</mi></math>: real variable</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">9.10.12</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E10" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex11" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">10.68.10</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m35.png" altimg-height="22px" altimg-valign="-2px" altimg-width="55px" alttext="\displaystyle\operatorname{ber}'x" display="inline"><mrow><msup><mi href="./10.61#E1">ber</mi><mo>′</mo></msup><mo></mo><mi href="./10.1#p2.t1.r3">x</mi></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m6.png" altimg-height="29px" altimg-valign="-9px" altimg-width="183px" alttext="\displaystyle=M_{1}\cos\left(\theta_{1}-\tfrac{1}{4}\pi\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msub><mi href="./10.18#E1">M</mi><mn>1</mn></msub><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><msub><mi href="./10.18#E3">θ</mi><mn>1</mn></msub><mo>-</mo><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex12" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m33.png" altimg-height="22px" altimg-valign="-2px" altimg-width="52px" alttext="\displaystyle\operatorname{bei}'x" display="inline"><mrow><msup><mi href="./10.61#E1">bei</mi><mo>′</mo></msup><mo></mo><mi href="./10.1#p2.t1.r3">x</mi></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m8.png" altimg-height="29px" altimg-valign="-9px" altimg-width="181px" alttext="\displaystyle=M_{1}\sin\left(\theta_{1}-\tfrac{1}{4}\pi\right)." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msub><mi href="./10.18#E1">M</mi><mn>1</mn></msub><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><msub><mi href="./10.18#E3">θ</mi><mn>1</mn></msub><mo>-</mo><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E10.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.61#E1" title="(10.61.1) ‣ §10.61(i) Definitions ‣ §10.61 Definitions and Basic Properties ‣ Kelvin Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m62.png" altimg-height="23px" altimg-valign="-7px" altimg-width="70px" alttext="\operatorname{bei}_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.61#E1">bei</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: Kelvin function</a>,
<a href="./10.61#E1" title="(10.61.1) ‣ §10.61(i) Definitions ‣ §10.61 Definitions and Basic Properties ‣ Kelvin Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m64.png" altimg-height="23px" altimg-valign="-7px" altimg-width="73px" alttext="\operatorname{ber}_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.61#E1">ber</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: Kelvin function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m75.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./10.18#E1" title="(10.18.1) ‣ §10.18(i) Definitions ‣ §10.18 Modulus and Phase Functions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="23px" altimg-valign="-7px" altimg-width="64px" alttext="M_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.18#E1">M</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: modulus of Bessel functions</a>,
<a href="./10.18#E3" title="(10.18.3) ‣ §10.18(i) Definitions ‣ §10.18 Modulus and Phase Functions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="\theta_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.18#E3">θ</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: phase of Bessel functions</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m76.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a> and
<a href="./10.1#p2.t1.r3" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./10.1#p2.t1.r3">x</mi></math>: real variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">9.10.13</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E11" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.68.11</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E11.png" altimg-height="29px" altimg-valign="-9px" altimg-width="783px" alttext="M_{\nu}'=(\nu/x)M_{\nu}+M_{\nu+1}\cos\left(\theta_{\nu+1}-\theta_{\nu}-\tfrac{%
1}{4}\pi\right)=-(\nu/x)M_{\nu}-M_{\nu-1}\cos\left(\theta_{\nu-1}-\theta_{\nu}%
-\tfrac{1}{4}\pi\right)," display="block"><mrow><mrow><msubsup><mi href="./10.18#E1">M</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′</mo></msubsup><mo>=</mo><mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>/</mo><mi href="./10.1#p2.t1.r3">x</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msub><mi href="./10.18#E1">M</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub></mrow><mo>+</mo><mrow><msub><mi href="./10.18#E1">M</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><msub><mi href="./10.18#E3">θ</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>-</mo><msub><mi href="./10.18#E3">θ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo>-</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>4</mn></mfrac></mstyle><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>=</mo><mrow><mrow><mo>-</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>/</mo><mi href="./10.1#p2.t1.r3">x</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msub><mi href="./10.18#E1">M</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub></mrow></mrow><mo>-</mo><mrow><msub><mi href="./10.18#E1">M</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>-</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><msub><mi href="./10.18#E3">θ</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>-</mo><mn>1</mn></mrow></msub><mo>-</mo><msub><mi href="./10.18#E3">θ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo>-</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>4</mn></mfrac></mstyle><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E11.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m75.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./10.18#E1" title="(10.18.1) ‣ §10.18(i) Definitions ‣ §10.18 Modulus and Phase Functions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="23px" altimg-valign="-7px" altimg-width="64px" alttext="M_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.18#E1">M</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: modulus of Bessel functions</a>,
<a href="./10.18#E3" title="(10.18.3) ‣ §10.18(i) Definitions ‣ §10.18 Modulus and Phase Functions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="\theta_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.18#E3">θ</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: phase of Bessel functions</a>,
<a href="./10.1#p2.t1.r3" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./10.1#p2.t1.r3">x</mi></math>: real variable</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">9.10.14</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E12" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.68.12</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E12.png" altimg-height="29px" altimg-valign="-9px" altimg-width="672px" alttext="\theta_{\nu}'=(M_{\nu+1}/M_{\nu})\sin\left(\theta_{\nu+1}-\theta_{\nu}-\tfrac{%
1}{4}\pi\right)=-(M_{\nu-1}/M_{\nu})\sin\left(\theta_{\nu-1}-\theta_{\nu}-%
\tfrac{1}{4}\pi\right)." display="block"><mrow><mrow><msubsup><mi href="./10.18#E3">θ</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′</mo></msubsup><mo>=</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><msub><mi href="./10.18#E1">M</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>/</mo><msub><mi href="./10.18#E1">M</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><msub><mi href="./10.18#E3">θ</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>-</mo><msub><mi href="./10.18#E3">θ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo>-</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>4</mn></mfrac></mstyle><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mrow><mo>-</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><msub><mi href="./10.18#E1">M</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>-</mo><mn>1</mn></mrow></msub><mo>/</mo><msub><mi href="./10.18#E1">M</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><msub><mi href="./10.18#E3">θ</mi><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo>-</mo><mn>1</mn></mrow></msub><mo>-</mo><msub><mi href="./10.18#E3">θ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo>-</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>4</mn></mfrac></mstyle><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E12.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m75.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./10.18#E1" title="(10.18.1) ‣ §10.18(i) Definitions ‣ §10.18 Modulus and Phase Functions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="23px" altimg-valign="-7px" altimg-width="64px" alttext="M_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.18#E1">M</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: modulus of Bessel functions</a>,
<a href="./10.18#E3" title="(10.18.3) ‣ §10.18(i) Definitions ‣ §10.18 Modulus and Phase Functions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="\theta_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.18#E3">θ</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: phase of Bessel functions</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m76.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">9.10.15</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E13" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex13" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">10.68.13</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m26.png" altimg-height="26px" altimg-valign="-7px" altimg-width="34px" alttext="\displaystyle M_{0}'" display="inline"><msubsup><mi href="./10.18#E1">M</mi><mn>0</mn><mo>′</mo></msubsup></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m7.png" altimg-height="29px" altimg-valign="-9px" altimg-width="226px" alttext="\displaystyle=M_{1}\cos\left(\theta_{1}-\theta_{0}-\tfrac{1}{4}\pi\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msub><mi href="./10.18#E1">M</mi><mn>1</mn></msub><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><msub><mi href="./10.18#E3">θ</mi><mn>1</mn></msub><mo>-</mo><msub><mi href="./10.18#E3">θ</mi><mn>0</mn></msub><mo>-</mo><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex14" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m42.png" altimg-height="26px" altimg-valign="-7px" altimg-width="24px" alttext="\displaystyle\theta_{0}'" display="inline"><msubsup><mi href="./10.18#E3">θ</mi><mn>0</mn><mo>′</mo></msubsup></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m1.png" altimg-height="29px" altimg-valign="-9px" altimg-width="276px" alttext="\displaystyle=(M_{1}/M_{0})\sin\left(\theta_{1}-\theta_{0}-\tfrac{1}{4}\pi%
\right)." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><msub><mi href="./10.18#E1">M</mi><mn>1</mn></msub><mo>/</mo><msub><mi href="./10.18#E1">M</mi><mn>0</mn></msub></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><msub><mi href="./10.18#E3">θ</mi><mn>1</mn></msub><mo>-</mo><msub><mi href="./10.18#E3">θ</mi><mn>0</mn></msub><mo>-</mo><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E13.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m75.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./10.18#E1" title="(10.18.1) ‣ §10.18(i) Definitions ‣ §10.18 Modulus and Phase Functions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="23px" altimg-valign="-7px" altimg-width="64px" alttext="M_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.18#E1">M</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: modulus of Bessel functions</a>,
<a href="./10.18#E3" title="(10.18.3) ‣ §10.18(i) Definitions ‣ §10.18 Modulus and Phase Functions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="\theta_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.18#E3">θ</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: phase of Bessel functions</a> and
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m76.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">9.10.16</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E14" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex15" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">10.68.14</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m30.png" altimg-height="30px" altimg-valign="-9px" altimg-width="132px" alttext="\displaystyle\ifrac{\mathrm{d}(x{M_{\nu}^{2}}\theta_{\nu}')}{\mathrm{d}x}" display="inline"><mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./10.1#p2.t1.r3">x</mi><mo></mo><msubsup><mi href="./10.18#E1">M</mi><mi href="./10.1#p2.t1.r5">ν</mi><mn>2</mn></msubsup><mo></mo><msubsup><mi href="./10.18#E3">θ</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′</mo></msubsup></mrow><mo stretchy="false">)</mo></mrow></mrow><mo href="./1.4#E4">/</mo><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi href="./10.1#p2.t1.r3">x</mi></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m24.png" altimg-height="28px" altimg-valign="-7px" altimg-width="74px" alttext="\displaystyle=x{M_{\nu}^{2}}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./10.1#p2.t1.r3">x</mi><mo></mo><msubsup><mi href="./10.18#E1">M</mi><mi href="./10.1#p2.t1.r5">ν</mi><mn>2</mn></msubsup></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex16" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m44.png" altimg-height="28px" altimg-valign="-7px" altimg-width="197px" alttext="\displaystyle x^{2}M_{\nu}''+xM_{\nu}'-\nu^{2}M_{\nu}" display="inline"><mrow><mrow><mrow><msup><mi href="./10.1#p2.t1.r3">x</mi><mn>2</mn></msup><mo></mo><msubsup><mi href="./10.18#E1">M</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′′</mo></msubsup></mrow><mo>+</mo><mrow><mi href="./10.1#p2.t1.r3">x</mi><mo></mo><msubsup><mi href="./10.18#E1">M</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′</mo></msubsup></mrow></mrow><mo>-</mo><mrow><msup><mi href="./10.1#p2.t1.r5">ν</mi><mn>2</mn></msup><mo></mo><msub><mi href="./10.18#E1">M</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m23.png" altimg-height="28px" altimg-valign="-7px" altimg-width="106px" alttext="\displaystyle=x^{2}M_{\nu}{\theta_{\nu}'^{2}}." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msup><mi href="./10.1#p2.t1.r3">x</mi><mn>2</mn></msup><mo></mo><msub><mi href="./10.18#E1">M</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mmultiscripts><mi href="./10.18#E3">θ</mi><mi href="./10.1#p2.t1.r5">ν</mi><mo>′</mo><none></none><mn>2</mn></mmultiscripts></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E14.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#E4" title="(1.4.4) ‣ §1.4(iii) Derivatives ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="29px" altimg-valign="-9px" altimg-width="27px" alttext="\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: derivative of <math class="ltx_Math" altimg="m80.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m82.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./10.18#E1" title="(10.18.1) ‣ §10.18(i) Definitions ‣ §10.18 Modulus and Phase Functions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="23px" altimg-valign="-7px" altimg-width="64px" alttext="M_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.18#E1">M</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: modulus of Bessel functions</a>,
<a href="./10.18#E3" title="(10.18.3) ‣ §10.18(i) Definitions ‣ §10.18 Modulus and Phase Functions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="\theta_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.18#E3">θ</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: phase of Bessel functions</a>,
<a href="./10.1#p2.t1.r3" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./10.1#p2.t1.r3">x</mi></math>: real variable</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">9.10.17</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS2.p3" class="ltx_para">
<p class="ltx_p">Equations () also hold with the
symbols <math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="33px" alttext="\operatorname{ber}" display="inline"><mi href="./10.61#E1">ber</mi></math>, <math class="ltx_Math" altimg="m61.png" altimg-height="18px" altimg-valign="-2px" altimg-width="30px" alttext="\operatorname{bei}" display="inline"><mi href="./10.61#E1">bei</mi></math>, <math class="ltx_Math" altimg="m46.png" altimg-height="18px" altimg-valign="-2px" altimg-width="26px" alttext="M" display="inline"><mi href="./10.18#E1">M</mi></math>, and <math class="ltx_Math" altimg="m77.png" altimg-height="18px" altimg-valign="-2px" altimg-width="14px" alttext="\theta" display="inline"><mi href="./10.18#E3">θ</mi></math>
replaced throughout by <math class="ltx_Math" altimg="m67.png" altimg-height="18px" altimg-valign="-2px" altimg-width="31px" alttext="\operatorname{ker}" display="inline"><mi href="./10.61#E2">ker</mi></math>, <math class="ltx_Math" altimg="m65.png" altimg-height="18px" altimg-valign="-2px" altimg-width="29px" alttext="\operatorname{kei}" display="inline"><mi href="./10.61#E2">kei</mi></math>, <math class="ltx_Math" altimg="m49.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="N" display="inline"><mi href="./10.18#E2">N</mi></math>, and
<math class="ltx_Math" altimg="m69.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="\phi" display="inline"><mi href="./10.18#E3">ϕ</mi></math>, respectively. In place of (),</p>
<table id="E15" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex17" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">10.68.15</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m28.png" altimg-height="25px" altimg-valign="-7px" altimg-width="75px" alttext="\displaystyle N_{-\nu}\left(x\right)" display="inline"><mrow><msub><mi href="./10.18#E2">N</mi><mrow><mo>-</mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m12.png" altimg-height="25px" altimg-valign="-7px" altimg-width="92px" alttext="\displaystyle=N_{\nu}\left(x\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msub><mi href="./10.18#E2">N</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex18" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m39.png" altimg-height="25px" altimg-valign="-7px" altimg-width="70px" alttext="\displaystyle\phi_{-\nu}\left(x\right)" display="inline"><mrow><msub><mi href="./10.18#E3">ϕ</mi><mrow><mo>-</mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m20.png" altimg-height="25px" altimg-valign="-7px" altimg-width="132px" alttext="\displaystyle=\phi_{\nu}\left(x\right)+\nu\pi." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><msub><mi href="./10.18#E3">ϕ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow><mo>+</mo><mrow><mi href="./10.1#p2.t1.r5">ν</mi><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E15.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m75.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./10.18#E2" title="(10.18.2) ‣ §10.18(i) Definitions ‣ §10.18 Modulus and Phase Functions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="N_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.18#E2">N</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: modulus of derivatives of Bessel functions</a>,
<a href="./10.18#E3" title="(10.18.3) ‣ §10.18(i) Definitions ‣ §10.18 Modulus and Phase Functions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m73.png" altimg-height="23px" altimg-valign="-7px" altimg-width="56px" alttext="\phi_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.18#E3">ϕ</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: phase of derivatives of Bessel functions</a>,
<a href="./10.1#p2.t1.r3" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./10.1#p2.t1.r3">x</mi></math>: real variable</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">9.10.20</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="iii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§10.68(iii) </span>Asymptotic Expansions for Large Argument</h2>
<div id="SS3.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="SS3.p1" class="ltx_para">
<p class="ltx_p">When <math class="ltx_Math" altimg="m59.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math> is fixed, <math class="ltx_Math" altimg="m58.png" altimg-height="24px" altimg-valign="-6px" altimg-width="73px" alttext="\mu=4\nu^{2}" display="inline"><mrow><mi>μ</mi><mo>=</mo><mrow><mn>4</mn><mo></mo><msup><mi href="./10.1#p2.t1.r5">ν</mi><mn>2</mn></msup></mrow></mrow></math>, and <math class="ltx_Math" altimg="m83.png" altimg-height="13px" altimg-valign="-2px" altimg-width="67px" alttext="x\to\infty" display="inline"><mrow><mi href="./10.1#p2.t1.r3">x</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></math></p>
</div>
<div id="SS3.p2" class="ltx_para">
<table id="EGx1" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E16">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.68.16</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m27.png" altimg-height="25px" altimg-valign="-7px" altimg-width="65px" alttext="\displaystyle M_{\nu}\left(x\right)" display="inline"><mrow><msub><mi href="./10.18#E1">M</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m15.png" altimg-height="61px" altimg-valign="-24px" altimg-width="722px" alttext="\displaystyle=\frac{e^{x/\sqrt{2}}}{(2\pi x)^{\frac{1}{2}}}\left(1-\frac{\mu-1%
}{8\sqrt{2}}\frac{1}{x}+\frac{(\mu-1)^{2}}{256}\frac{1}{x^{2}}-\frac{(\mu-1)(%
\mu^{2}+14\mu-399)}{6144\sqrt{2}}\frac{1}{x^{3}}+O\left(\frac{1}{x^{4}}\right)%
\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><mfrac><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mi href="./10.1#p2.t1.r3">x</mi><mo>/</mo><msqrt><mn>2</mn></msqrt></mrow></msup><msup><mrow><mo stretchy="false">(</mo><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi href="./10.1#p2.t1.r3">x</mi></mrow><mo stretchy="false">)</mo></mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></msup></mfrac></mstyle><mo></mo><mrow><mo>(</mo><mrow><mrow><mrow><mrow><mn>1</mn><mo>-</mo><mrow><mstyle displaystyle="true"><mfrac><mrow><mi>μ</mi><mo>-</mo><mn>1</mn></mrow><mrow><mn>8</mn><mo></mo><msqrt><mn>2</mn></msqrt></mrow></mfrac></mstyle><mo></mo><mstyle displaystyle="true"><mfrac><mn>1</mn><mi href="./10.1#p2.t1.r3">x</mi></mfrac></mstyle></mrow></mrow><mo>+</mo><mrow><mstyle displaystyle="true"><mfrac><msup><mrow><mo stretchy="false">(</mo><mrow><mi>μ</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mn>256</mn></mfrac></mstyle><mo></mo><mstyle displaystyle="true"><mfrac><mn>1</mn><msup><mi href="./10.1#p2.t1.r3">x</mi><mn>2</mn></msup></mfrac></mstyle></mrow></mrow><mo>-</mo><mrow><mstyle displaystyle="true"><mfrac><mrow><mrow><mo stretchy="false">(</mo><mrow><mi>μ</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><msup><mi>μ</mi><mn>2</mn></msup><mo>+</mo><mrow><mn>14</mn><mo></mo><mi>μ</mi></mrow></mrow><mo>-</mo><mn>399</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mrow><mn>6144</mn><mo></mo><msqrt><mn>2</mn></msqrt></mrow></mfrac></mstyle><mo></mo><mstyle displaystyle="true"><mfrac><mn>1</mn><msup><mi href="./10.1#p2.t1.r3">x</mi><mn>3</mn></msup></mfrac></mstyle></mrow></mrow><mo>+</mo><mrow><mi href="./2.1#E3">O</mi><mo></mo><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac><mn>1</mn><msup><mi href="./10.1#p2.t1.r3">x</mi><mn>4</mn></msup></mfrac></mstyle><mo>)</mo></mrow></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E16.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./2.1#E3" title="(2.1.3) ‣ §2.1(i) Asymptotic and Order Symbols ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="23px" altimg-valign="-7px" altimg-width="50px" alttext="O\left(\NVar{x}\right)" display="inline"><mrow><mi href="./2.1#E3">O</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>)</mo></mrow></mrow></math>: order not exceeding</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m75.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m57.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./10.18#E1" title="(10.18.1) ‣ §10.18(i) Definitions ‣ §10.18 Modulus and Phase Functions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="23px" altimg-valign="-7px" altimg-width="64px" alttext="M_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.18#E1">M</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: modulus of Bessel functions</a>,
<a href="./10.1#p2.t1.r3" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./10.1#p2.t1.r3">x</mi></math>: real variable</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">9.10.21</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E17">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.68.17</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m31.png" altimg-height="25px" altimg-valign="-7px" altimg-width="85px" alttext="\displaystyle\ln M_{\nu}\left(x\right)" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><msub><mi href="./10.18#E1">M</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m17.png" altimg-height="53px" altimg-valign="-21px" altimg-width="744px" alttext="\displaystyle=\frac{x}{\sqrt{2}}-\frac{1}{2}\ln\left(2\pi x\right)-\frac{\mu-1%
}{8\sqrt{2}}\frac{1}{x}-\frac{(\mu-1)(\mu-25)}{384\sqrt{2}}\frac{1}{x^{3}}-%
\frac{(\mu-1)(\mu-13)}{128}\frac{1}{x^{4}}+O\left(\frac{1}{x^{5}}\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mstyle displaystyle="true"><mfrac><mi href="./10.1#p2.t1.r3">x</mi><msqrt><mn>2</mn></msqrt></mfrac></mstyle><mo>-</mo><mrow><mstyle displaystyle="true"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi href="./10.1#p2.t1.r3">x</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>-</mo><mrow><mstyle displaystyle="true"><mfrac><mrow><mi>μ</mi><mo>-</mo><mn>1</mn></mrow><mrow><mn>8</mn><mo></mo><msqrt><mn>2</mn></msqrt></mrow></mfrac></mstyle><mo></mo><mstyle displaystyle="true"><mfrac><mn>1</mn><mi href="./10.1#p2.t1.r3">x</mi></mfrac></mstyle></mrow><mo>-</mo><mrow><mstyle displaystyle="true"><mfrac><mrow><mrow><mo stretchy="false">(</mo><mrow><mi>μ</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>μ</mi><mo>-</mo><mn>25</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mrow><mn>384</mn><mo></mo><msqrt><mn>2</mn></msqrt></mrow></mfrac></mstyle><mo></mo><mstyle displaystyle="true"><mfrac><mn>1</mn><msup><mi href="./10.1#p2.t1.r3">x</mi><mn>3</mn></msup></mfrac></mstyle></mrow><mo>-</mo><mrow><mstyle displaystyle="true"><mfrac><mrow><mrow><mo stretchy="false">(</mo><mrow><mi>μ</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>μ</mi><mo>-</mo><mn>13</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mn>128</mn></mfrac></mstyle><mo></mo><mstyle displaystyle="true"><mfrac><mn>1</mn><msup><mi href="./10.1#p2.t1.r3">x</mi><mn>4</mn></msup></mfrac></mstyle></mrow></mrow><mo>+</mo><mrow><mi href="./2.1#E3">O</mi><mo></mo><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac><mn>1</mn><msup><mi href="./10.1#p2.t1.r3">x</mi><mn>5</mn></msup></mfrac></mstyle><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E17.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./2.1#E3" title="(2.1.3) ‣ §2.1(i) Asymptotic and Order Symbols ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="23px" altimg-valign="-7px" altimg-width="50px" alttext="O\left(\NVar{x}\right)" display="inline"><mrow><mi href="./2.1#E3">O</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>)</mo></mrow></mrow></math>: order not exceeding</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m75.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./10.18#E1" title="(10.18.1) ‣ §10.18(i) Definitions ‣ §10.18 Modulus and Phase Functions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="23px" altimg-valign="-7px" altimg-width="64px" alttext="M_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.18#E1">M</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: modulus of Bessel functions</a>,
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a>,
<a href="./10.1#p2.t1.r3" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./10.1#p2.t1.r3">x</mi></math>: real variable</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">9.10.22</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E18">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.68.18</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m43.png" altimg-height="25px" altimg-valign="-7px" altimg-width="56px" alttext="\displaystyle\theta_{\nu}\left(x\right)" display="inline"><mrow><msub><mi href="./10.18#E3">θ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m16.png" altimg-height="53px" altimg-valign="-21px" altimg-width="679px" alttext="\displaystyle=\frac{x}{\sqrt{2}}+\left(\frac{1}{2}\nu-\frac{1}{8}\right)\pi+%
\frac{\mu-1}{8\sqrt{2}}\frac{1}{x}+\frac{\mu-1}{16}\frac{1}{x^{2}}-\frac{(\mu-%
1)(\mu-25)}{384\sqrt{2}}\frac{1}{x^{3}}+O\left(\frac{1}{x^{5}}\right)." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mrow><mstyle displaystyle="true"><mfrac><mi href="./10.1#p2.t1.r3">x</mi><msqrt><mn>2</mn></msqrt></mfrac></mstyle><mo>+</mo><mrow><mrow><mo>(</mo><mrow><mrow><mstyle displaystyle="true"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo>-</mo><mstyle displaystyle="true"><mfrac><mn>1</mn><mn>8</mn></mfrac></mstyle></mrow><mo>)</mo></mrow><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo>+</mo><mrow><mstyle displaystyle="true"><mfrac><mrow><mi>μ</mi><mo>-</mo><mn>1</mn></mrow><mrow><mn>8</mn><mo></mo><msqrt><mn>2</mn></msqrt></mrow></mfrac></mstyle><mo></mo><mstyle displaystyle="true"><mfrac><mn>1</mn><mi href="./10.1#p2.t1.r3">x</mi></mfrac></mstyle></mrow><mo>+</mo><mrow><mstyle displaystyle="true"><mfrac><mrow><mi>μ</mi><mo>-</mo><mn>1</mn></mrow><mn>16</mn></mfrac></mstyle><mo></mo><mstyle displaystyle="true"><mfrac><mn>1</mn><msup><mi href="./10.1#p2.t1.r3">x</mi><mn>2</mn></msup></mfrac></mstyle></mrow></mrow><mo>-</mo><mrow><mstyle displaystyle="true"><mfrac><mrow><mrow><mo stretchy="false">(</mo><mrow><mi>μ</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>μ</mi><mo>-</mo><mn>25</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mrow><mn>384</mn><mo></mo><msqrt><mn>2</mn></msqrt></mrow></mfrac></mstyle><mo></mo><mstyle displaystyle="true"><mfrac><mn>1</mn><msup><mi href="./10.1#p2.t1.r3">x</mi><mn>3</mn></msup></mfrac></mstyle></mrow></mrow><mo>+</mo><mrow><mi href="./2.1#E3">O</mi><mo></mo><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac><mn>1</mn><msup><mi href="./10.1#p2.t1.r3">x</mi><mn>5</mn></msup></mfrac></mstyle><mo>)</mo></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E18.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./2.1#E3" title="(2.1.3) ‣ §2.1(i) Asymptotic and Order Symbols ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="23px" altimg-valign="-7px" altimg-width="50px" alttext="O\left(\NVar{x}\right)" display="inline"><mrow><mi href="./2.1#E3">O</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>)</mo></mrow></mrow></math>: order not exceeding</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m75.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./10.18#E3" title="(10.18.3) ‣ §10.18(i) Definitions ‣ §10.18 Modulus and Phase Functions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="\theta_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.18#E3">θ</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: phase of Bessel functions</a>,
<a href="./10.1#p2.t1.r3" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./10.1#p2.t1.r3">x</mi></math>: real variable</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">9.10.23</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E19">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.68.19</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m29.png" altimg-height="25px" altimg-valign="-7px" altimg-width="62px" alttext="\displaystyle N_{\nu}\left(x\right)" display="inline"><mrow><msub><mi href="./10.18#E2">N</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m22.png" altimg-height="54px" altimg-valign="-21px" altimg-width="779px" alttext="\displaystyle=e^{-x/\sqrt{2}}\left(\frac{\pi}{2x}\right)^{\frac{1}{2}}\left(1+%
\frac{\mu-1}{8\sqrt{2}}\frac{1}{x}+\frac{(\mu-1)^{2}}{256}\frac{1}{x^{2}}+%
\frac{(\mu-1)(\mu^{2}+14\mu-399)}{6144\sqrt{2}}\frac{1}{x^{3}}+O\left(\frac{1}%
{x^{4}}\right)\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mi href="./10.1#p2.t1.r3">x</mi><mo>/</mo><msqrt><mn>2</mn></msqrt></mrow></mrow></msup><mo></mo><msup><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac><mi href="./3.12#E1">π</mi><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r3">x</mi></mrow></mfrac></mstyle><mo>)</mo></mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></msup><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mrow><mstyle displaystyle="true"><mfrac><mrow><mi>μ</mi><mo>-</mo><mn>1</mn></mrow><mrow><mn>8</mn><mo></mo><msqrt><mn>2</mn></msqrt></mrow></mfrac></mstyle><mo></mo><mstyle displaystyle="true"><mfrac><mn>1</mn><mi href="./10.1#p2.t1.r3">x</mi></mfrac></mstyle></mrow><mo>+</mo><mrow><mstyle displaystyle="true"><mfrac><msup><mrow><mo stretchy="false">(</mo><mrow><mi>μ</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mn>256</mn></mfrac></mstyle><mo></mo><mstyle displaystyle="true"><mfrac><mn>1</mn><msup><mi href="./10.1#p2.t1.r3">x</mi><mn>2</mn></msup></mfrac></mstyle></mrow><mo>+</mo><mrow><mstyle displaystyle="true"><mfrac><mrow><mrow><mo stretchy="false">(</mo><mrow><mi>μ</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><msup><mi>μ</mi><mn>2</mn></msup><mo>+</mo><mrow><mn>14</mn><mo></mo><mi>μ</mi></mrow></mrow><mo>-</mo><mn>399</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mrow><mn>6144</mn><mo></mo><msqrt><mn>2</mn></msqrt></mrow></mfrac></mstyle><mo></mo><mstyle displaystyle="true"><mfrac><mn>1</mn><msup><mi href="./10.1#p2.t1.r3">x</mi><mn>3</mn></msup></mfrac></mstyle></mrow><mo>+</mo><mrow><mi href="./2.1#E3">O</mi><mo></mo><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac><mn>1</mn><msup><mi href="./10.1#p2.t1.r3">x</mi><mn>4</mn></msup></mfrac></mstyle><mo>)</mo></mrow></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E19.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./2.1#E3" title="(2.1.3) ‣ §2.1(i) Asymptotic and Order Symbols ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="23px" altimg-valign="-7px" altimg-width="50px" alttext="O\left(\NVar{x}\right)" display="inline"><mrow><mi href="./2.1#E3">O</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>)</mo></mrow></mrow></math>: order not exceeding</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m75.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m57.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./10.18#E2" title="(10.18.2) ‣ §10.18(i) Definitions ‣ §10.18 Modulus and Phase Functions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="N_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.18#E2">N</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: modulus of derivatives of Bessel functions</a>,
<a href="./10.1#p2.t1.r3" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./10.1#p2.t1.r3">x</mi></math>: real variable</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">9.10.24</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E20">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.68.20</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m32.png" altimg-height="25px" altimg-valign="-7px" altimg-width="82px" alttext="\displaystyle\ln N_{\nu}\left(x\right)" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><msub><mi href="./10.18#E2">N</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m4.png" altimg-height="53px" altimg-valign="-21px" altimg-width="760px" alttext="\displaystyle=-\frac{x}{\sqrt{2}}+\frac{1}{2}\ln\left(\frac{\pi}{2x}\right)+%
\frac{\mu-1}{8\sqrt{2}}\frac{1}{x}+\frac{(\mu-1)(\mu-25)}{384\sqrt{2}}\frac{1}%
{x^{3}}-\frac{(\mu-1)(\mu-13)}{128}\frac{1}{x^{4}}+O\left(\frac{1}{x^{5}}%
\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mrow><mrow><mo>-</mo><mstyle displaystyle="true"><mfrac><mi href="./10.1#p2.t1.r3">x</mi><msqrt><mn>2</mn></msqrt></mfrac></mstyle></mrow><mo>+</mo><mrow><mstyle displaystyle="true"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac><mi href="./3.12#E1">π</mi><mrow><mn>2</mn><mo></mo><mi href="./10.1#p2.t1.r3">x</mi></mrow></mfrac></mstyle><mo>)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mstyle displaystyle="true"><mfrac><mrow><mi>μ</mi><mo>-</mo><mn>1</mn></mrow><mrow><mn>8</mn><mo></mo><msqrt><mn>2</mn></msqrt></mrow></mfrac></mstyle><mo></mo><mstyle displaystyle="true"><mfrac><mn>1</mn><mi href="./10.1#p2.t1.r3">x</mi></mfrac></mstyle></mrow><mo>+</mo><mrow><mstyle displaystyle="true"><mfrac><mrow><mrow><mo stretchy="false">(</mo><mrow><mi>μ</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>μ</mi><mo>-</mo><mn>25</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mrow><mn>384</mn><mo></mo><msqrt><mn>2</mn></msqrt></mrow></mfrac></mstyle><mo></mo><mstyle displaystyle="true"><mfrac><mn>1</mn><msup><mi href="./10.1#p2.t1.r3">x</mi><mn>3</mn></msup></mfrac></mstyle></mrow></mrow><mo>-</mo><mrow><mstyle displaystyle="true"><mfrac><mrow><mrow><mo stretchy="false">(</mo><mrow><mi>μ</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>μ</mi><mo>-</mo><mn>13</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mn>128</mn></mfrac></mstyle><mo></mo><mstyle displaystyle="true"><mfrac><mn>1</mn><msup><mi href="./10.1#p2.t1.r3">x</mi><mn>4</mn></msup></mfrac></mstyle></mrow></mrow><mo>+</mo><mrow><mi href="./2.1#E3">O</mi><mo></mo><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac><mn>1</mn><msup><mi href="./10.1#p2.t1.r3">x</mi><mn>5</mn></msup></mfrac></mstyle><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E20.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./2.1#E3" title="(2.1.3) ‣ §2.1(i) Asymptotic and Order Symbols ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="23px" altimg-valign="-7px" altimg-width="50px" alttext="O\left(\NVar{x}\right)" display="inline"><mrow><mi href="./2.1#E3">O</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>)</mo></mrow></mrow></math>: order not exceeding</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m75.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./10.18#E2" title="(10.18.2) ‣ §10.18(i) Definitions ‣ §10.18 Modulus and Phase Functions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="N_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.18#E2">N</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: modulus of derivatives of Bessel functions</a>,
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a>,
<a href="./10.1#p2.t1.r3" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./10.1#p2.t1.r3">x</mi></math>: real variable</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">9.10.25</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E21">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">10.68.21</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m40.png" altimg-height="25px" altimg-valign="-7px" altimg-width="58px" alttext="\displaystyle\phi_{\nu}\left(x\right)" display="inline"><mrow><msub><mi href="./10.18#E3">ϕ</mi><mi href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m5.png" altimg-height="53px" altimg-valign="-21px" altimg-width="695px" alttext="\displaystyle=-\frac{x}{\sqrt{2}}-\left(\frac{1}{2}\nu+\frac{1}{8}\right)\pi-%
\frac{\mu-1}{8\sqrt{2}}\frac{1}{x}+\frac{\mu-1}{16}\frac{1}{x^{2}}+\frac{(\mu-%
1)(\mu-25)}{384\sqrt{2}}\frac{1}{x^{3}}+O\left(\frac{1}{x^{5}}\right)." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mrow><mo>-</mo><mstyle displaystyle="true"><mfrac><mi href="./10.1#p2.t1.r3">x</mi><msqrt><mn>2</mn></msqrt></mfrac></mstyle></mrow><mo>-</mo><mrow><mrow><mo>(</mo><mrow><mrow><mstyle displaystyle="true"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./10.1#p2.t1.r5">ν</mi></mrow><mo>+</mo><mstyle displaystyle="true"><mfrac><mn>1</mn><mn>8</mn></mfrac></mstyle></mrow><mo>)</mo></mrow><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo>-</mo><mrow><mstyle displaystyle="true"><mfrac><mrow><mi>μ</mi><mo>-</mo><mn>1</mn></mrow><mrow><mn>8</mn><mo></mo><msqrt><mn>2</mn></msqrt></mrow></mfrac></mstyle><mo></mo><mstyle displaystyle="true"><mfrac><mn>1</mn><mi href="./10.1#p2.t1.r3">x</mi></mfrac></mstyle></mrow></mrow><mo>+</mo><mrow><mstyle displaystyle="true"><mfrac><mrow><mi>μ</mi><mo>-</mo><mn>1</mn></mrow><mn>16</mn></mfrac></mstyle><mo></mo><mstyle displaystyle="true"><mfrac><mn>1</mn><msup><mi href="./10.1#p2.t1.r3">x</mi><mn>2</mn></msup></mfrac></mstyle></mrow><mo>+</mo><mrow><mstyle displaystyle="true"><mfrac><mrow><mrow><mo stretchy="false">(</mo><mrow><mi>μ</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>μ</mi><mo>-</mo><mn>25</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mrow><mn>384</mn><mo></mo><msqrt><mn>2</mn></msqrt></mrow></mfrac></mstyle><mo></mo><mstyle displaystyle="true"><mfrac><mn>1</mn><msup><mi href="./10.1#p2.t1.r3">x</mi><mn>3</mn></msup></mfrac></mstyle></mrow><mo>+</mo><mrow><mi href="./2.1#E3">O</mi><mo></mo><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac><mn>1</mn><msup><mi href="./10.1#p2.t1.r3">x</mi><mn>5</mn></msup></mfrac></mstyle><mo>)</mo></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E21.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./2.1#E3" title="(2.1.3) ‣ §2.1(i) Asymptotic and Order Symbols ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="23px" altimg-valign="-7px" altimg-width="50px" alttext="O\left(\NVar{x}\right)" display="inline"><mrow><mi href="./2.1#E3">O</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>)</mo></mrow></mrow></math>: order not exceeding</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m75.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./10.18#E3" title="(10.18.3) ‣ §10.18(i) Definitions ‣ §10.18 Modulus and Phase Functions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m73.png" altimg-height="23px" altimg-valign="-7px" altimg-width="56px" alttext="\phi_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./10.18#E3">ϕ</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: phase of derivatives of Bessel functions</a>,
<a href="./10.1#p2.t1.r3" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./10.1#p2.t1.r3">x</mi></math>: real variable</a> and
<a href="./10.1#p2.t1.r5" title="§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./10.1#p2.t1.r5">ν</mi></math>: complex parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">9.10.26</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
</div>
</section>
<section id="iv" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§10.68(iv) </span>Further Properties</h2>
<div id="SS4.info" class="ltx_metadata ltx_info">
, pp. xi–xv)</cite>. However, care needs to be exercised with
the branches of the phases. Thus this reference gives
<math class="ltx_Math" altimg="m71.png" altimg-height="27px" altimg-valign="-9px" altimg-width="105px" alttext="\phi_{1}\left(0\right)=\tfrac{5}{4}\pi" display="inline"><mrow><mrow><msub><mi href="./10.18#E3">ϕ</mi><mn>1</mn></msub><mo></mo><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mfrac><mn>5</mn><mn>4</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow></math> (Eq. (6.10)), and
<math class="ltx_Math" altimg="m55.png" altimg-height="29px" altimg-valign="-9px" altimg-width="295px" alttext="\lim_{x\to\infty}(\phi_{1}\left(x\right)+(x/\sqrt{2}))=-\tfrac{5}{8}\pi" display="inline"><mrow><mrow><msub><mo>lim</mo><mrow><mi href="./10.1#p2.t1.r3">x</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></msub><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><msub><mi href="./10.18#E3">ϕ</mi><mn>1</mn></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow><mo>+</mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./10.1#p2.t1.r3">x</mi><mo>/</mo><msqrt><mn>2</mn></msqrt></mrow><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mo>-</mo><mrow><mfrac><mn>5</mn><mn>8</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow></mrow></math>
(Eqs. (10.20) and (Eqs. (10.26b)). However, numerical tabulations show that
if the second of these equations applies and <math class="ltx_Math" altimg="m72.png" altimg-height="23px" altimg-valign="-7px" altimg-width="55px" alttext="\phi_{1}\left(x\right)" display="inline"><mrow><msub><mi href="./10.18#E3">ϕ</mi><mn>1</mn></msub><mo></mo><mrow><mo>(</mo><mi href="./10.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math> is
continuous, then <math class="ltx_Math" altimg="m70.png" altimg-height="27px" altimg-valign="-9px" altimg-width="121px" alttext="\phi_{1}\left(0\right)=-\tfrac{3}{4}\pi" display="inline"><mrow><mrow><msub><mi href="./10.18#E3">ϕ</mi><mn>1</mn></msub><mo></mo><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mo>-</mo><mrow><mfrac><mn>3</mn><mn>4</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow></mrow></math>; compare
<cite class="ltx_cite ltx_citemacro_citet">Abramowitz and Stegun (</div>
</div>
</body></text>
</html>
</page>
<page>
<!DOCTYPE html><html>
<head>
<title>DLMF: 13.9 Zeros</title>
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<div class="ltx_page_navlogo"></dd>
</dl>
</div>
</div>
<h6>Contents</h6>
<ul class="ltx_toclist ltx_toclist_section">
<li class="ltx_tocentry"><a href="#i"><span class="ltx_tag ltx_tag_subsection">§13.9(i) </span>Zeros of <math class="ltx_Math" altimg="m13.png" altimg-height="23px" altimg-valign="-7px" altimg-width="92px" alttext="M\left(a,b,z\right)" display="inline"><mrow><mi href="./13.2#E2">M</mi><mo></mo><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></math></a></li>
<li class="ltx_tocentry"><a href="#ii"><span class="ltx_tag ltx_tag_subsection">§13.9(ii) </span>Zeros of <math class="ltx_Math" altimg="m24.png" altimg-height="23px" altimg-valign="-7px" altimg-width="86px" alttext="U\left(a,b,z\right)" display="inline"><mrow><mi href="./13.2#E6">U</mi><mo></mo><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></math></a></li>
</ul>
<section id="i" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§13.9(i) </span>Zeros of <math class="ltx_Math" altimg="m13.png" altimg-height="23px" altimg-valign="-7px" altimg-width="92px" alttext="M\left(a,b,z\right)" display="inline"><mrow><mi href="./13.2#E2">M</mi><mo></mo><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>
</h2>
<div id="SS1.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="SS1.p1" class="ltx_para">
<p class="ltx_p">If <math class="ltx_Math" altimg="m47.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi>a</mi></math> and <math class="ltx_Math" altimg="m51.png" altimg-height="21px" altimg-valign="-6px" altimg-width="185px" alttext="b-a\neq 0,-1,-2,\dots" display="inline"><mrow><mrow><mi>b</mi><mo>-</mo><mi>a</mi></mrow><mo>≠</mo><mrow><mn>0</mn><mo>,</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo>,</mo><mrow><mo>-</mo><mn>2</mn></mrow><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>, then <math class="ltx_Math" altimg="m13.png" altimg-height="23px" altimg-valign="-7px" altimg-width="92px" alttext="M\left(a,b,z\right)" display="inline"><mrow><mi href="./13.2#E2">M</mi><mo></mo><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></math> has infinitely
many <math class="ltx_Math" altimg="m68.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./13.1#p1.t1.r4">z</mi></math>-zeros in <math class="ltx_Math" altimg="m35.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\mathbb{C}" display="inline"><mi href="./front/introduction#Sx4.p1.t1.r1">ℂ</mi></math>. When <math class="ltx_Math" altimg="m43.png" altimg-height="21px" altimg-valign="-6px" altimg-width="71px" alttext="a,b\in\mathbb{R}" display="inline"><mrow><mrow><mi>a</mi><mo>,</mo><mi>b</mi></mrow><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mi href="./front/introduction#Sx4.p2.t1.r14">ℝ</mi></mrow></math> the number of real zeros is
finite. Let
<math class="ltx_Math" altimg="m63.png" altimg-height="23px" altimg-valign="-7px" altimg-width="58px" alttext="p(a,b)" display="inline"><mrow><mi href="./13.9#SS1.p1">p</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></mrow></math> be the number of positive zeros. Then</p>
<table id="EGx1" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E1">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">13.9.1</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m4.png" altimg-height="25px" altimg-valign="-7px" altimg-width="60px" alttext="\displaystyle p(a,b)" display="inline"><mrow><mi href="./13.9#SS1.p1">p</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m3.png" altimg-height="25px" altimg-valign="-7px" altimg-width="80px" alttext="\displaystyle=\left\lceil-a\right\rceil," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r17">⌈</mo><mrow><mo>-</mo><mi>a</mi></mrow><mo href="./front/introduction#Sx4.p1.t1.r17">⌉</mo></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m45.png" altimg-height="17px" altimg-valign="-3px" altimg-width="51px" alttext="a<0" display="inline"><mrow><mi>a</mi><mo><</mo><mn>0</mn></mrow></math>, <math class="ltx_Math" altimg="m55.png" altimg-height="20px" altimg-valign="-5px" altimg-width="49px" alttext="b\geq 0" display="inline"><mrow><mi>b</mi><mo>≥</mo><mn>0</mn></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./front/introduction#Sx4.p1.t1.r17" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m32.png" altimg-height="23px" altimg-valign="-7px" altimg-width="33px" alttext="\left\lceil\NVar{x}\right\rceil" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r17">⌈</mo><mi class="ltx_nvar">x</mi><mo href="./front/introduction#Sx4.p1.t1.r17">⌉</mo></mrow></math>: ceiling of <math class="ltx_Math" altimg="m66.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./13.9#SS1.p1" title="§13.9(i) Zeros of M ( a , b , z ) ‣ §13.9 Zeros ‣ Kummer Functions ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="23px" altimg-valign="-7px" altimg-width="58px" alttext="p(a,b)" display="inline"><mrow><mi href="./13.9#SS1.p1">p</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></mrow></math>: number of positive zeros</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E2">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">13.9.2</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m4.png" altimg-height="25px" altimg-valign="-7px" altimg-width="60px" alttext="\displaystyle p(a,b)" display="inline"><mrow><mi href="./13.9#SS1.p1">p</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m1.png" altimg-height="22px" altimg-valign="-6px" altimg-width="43px" alttext="\displaystyle=0," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mn>0</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m48.png" altimg-height="19px" altimg-valign="-5px" altimg-width="51px" alttext="a\geq 0" display="inline"><mrow><mi>a</mi><mo>≥</mo><mn>0</mn></mrow></math>, <math class="ltx_Math" altimg="m55.png" altimg-height="20px" altimg-valign="-5px" altimg-width="49px" alttext="b\geq 0" display="inline"><mrow><mi>b</mi><mo>≥</mo><mn>0</mn></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./13.9#SS1.p1" title="§13.9(i) Zeros of M ( a , b , z ) ‣ §13.9 Zeros ‣ Kummer Functions ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="23px" altimg-valign="-7px" altimg-width="58px" alttext="p(a,b)" display="inline"><mrow><mi href="./13.9#SS1.p1">p</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></mrow></math>: number of positive zeros</a></dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E3">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">13.9.3</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m4.png" altimg-height="25px" altimg-valign="-7px" altimg-width="60px" alttext="\displaystyle p(a,b)" display="inline"><mrow><mi href="./13.9#SS1.p1">p</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m2.png" altimg-height="22px" altimg-valign="-6px" altimg-width="43px" alttext="\displaystyle=1," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mn>1</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m48.png" altimg-height="19px" altimg-valign="-5px" altimg-width="51px" alttext="a\geq 0" display="inline"><mrow><mi>a</mi><mo>≥</mo><mn>0</mn></mrow></math>, <math class="ltx_Math" altimg="m5.png" altimg-height="19px" altimg-valign="-4px" altimg-width="101px" alttext="-1<b<0" display="inline"><mrow><mrow><mo>-</mo><mn>1</mn></mrow><mo><</mo><mi>b</mi><mo><</mo><mn>0</mn></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./13.9#SS1.p1" title="§13.9(i) Zeros of M ( a , b , z ) ‣ §13.9 Zeros ‣ Kummer Functions ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="23px" altimg-valign="-7px" altimg-width="58px" alttext="p(a,b)" display="inline"><mrow><mi href="./13.9#SS1.p1">p</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></mrow></math>: number of positive zeros</a></dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
<table id="E4" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">13.9.4</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E4.png" altimg-height="29px" altimg-valign="-9px" altimg-width="280px" alttext="p(a,b)=\left\lfloor-\tfrac{1}{2}b\right\rfloor-\left\lfloor-\tfrac{1}{2}(b+1)%
\right\rfloor," display="block"><mrow><mrow><mrow><mi href="./13.9#SS1.p1">p</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mo href="./front/introduction#Sx4.p1.t1.r16">⌊</mo><mrow><mo>-</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi>b</mi></mrow></mrow><mo href="./front/introduction#Sx4.p1.t1.r16">⌋</mo></mrow><mo>-</mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r16">⌊</mo><mrow><mo>-</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>b</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mo href="./front/introduction#Sx4.p1.t1.r16">⌋</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m48.png" altimg-height="19px" altimg-valign="-5px" altimg-width="51px" alttext="a\geq 0" display="inline"><mrow><mi>a</mi><mo>≥</mo><mn>0</mn></mrow></math>, <math class="ltx_Math" altimg="m58.png" altimg-height="20px" altimg-valign="-5px" altimg-width="65px" alttext="b\leq-1" display="inline"><mrow><mi>b</mi><mo>≤</mo><mrow><mo>-</mo><mn>1</mn></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E4.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./front/introduction#Sx4.p1.t1.r16" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m33.png" altimg-height="23px" altimg-valign="-7px" altimg-width="33px" alttext="\left\lfloor\NVar{x}\right\rfloor" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r16">⌊</mo><mi class="ltx_nvar">x</mi><mo href="./front/introduction#Sx4.p1.t1.r16">⌋</mo></mrow></math>: floor of <math class="ltx_Math" altimg="m66.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./13.9#SS1.p1" title="§13.9(i) Zeros of M ( a , b , z ) ‣ §13.9 Zeros ‣ Kummer Functions ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="23px" altimg-valign="-7px" altimg-width="58px" alttext="p(a,b)" display="inline"><mrow><mi href="./13.9#SS1.p1">p</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></mrow></math>: number of positive zeros</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E5" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">13.9.5</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E5.png" altimg-height="25px" altimg-valign="-7px" altimg-width="205px" alttext="p(a,b)=\left\lceil-a\right\rceil-\left\lceil-b\right\rceil," display="block"><mrow><mrow><mrow><mi href="./13.9#SS1.p1">p</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mo href="./front/introduction#Sx4.p1.t1.r17">⌈</mo><mrow><mo>-</mo><mi>a</mi></mrow><mo href="./front/introduction#Sx4.p1.t1.r17">⌉</mo></mrow><mo>-</mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r17">⌈</mo><mrow><mo>-</mo><mi>b</mi></mrow><mo href="./front/introduction#Sx4.p1.t1.r17">⌉</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m30.png" altimg-height="23px" altimg-valign="-7px" altimg-width="116px" alttext="\left\lceil-a\right\rceil\geq\left\lceil-b\right\rceil" display="inline"><mrow><mrow><mo href="./front/introduction#Sx4.p1.t1.r17">⌈</mo><mrow><mo>-</mo><mi>a</mi></mrow><mo href="./front/introduction#Sx4.p1.t1.r17">⌉</mo></mrow><mo>≥</mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r17">⌈</mo><mrow><mo>-</mo><mi>b</mi></mrow><mo href="./front/introduction#Sx4.p1.t1.r17">⌉</mo></mrow></mrow></math>, <math class="ltx_Math" altimg="m45.png" altimg-height="17px" altimg-valign="-3px" altimg-width="51px" alttext="a<0" display="inline"><mrow><mi>a</mi><mo><</mo><mn>0</mn></mrow></math>, <math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-3px" altimg-width="49px" alttext="b<0" display="inline"><mrow><mi>b</mi><mo><</mo><mn>0</mn></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E5.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./front/introduction#Sx4.p1.t1.r17" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m32.png" altimg-height="23px" altimg-valign="-7px" altimg-width="33px" alttext="\left\lceil\NVar{x}\right\rceil" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r17">⌈</mo><mi class="ltx_nvar">x</mi><mo href="./front/introduction#Sx4.p1.t1.r17">⌉</mo></mrow></math>: ceiling of <math class="ltx_Math" altimg="m66.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./13.9#SS1.p1" title="§13.9(i) Zeros of M ( a , b , z ) ‣ §13.9 Zeros ‣ Kummer Functions ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="23px" altimg-valign="-7px" altimg-width="58px" alttext="p(a,b)" display="inline"><mrow><mi href="./13.9#SS1.p1">p</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></mrow></math>: number of positive zeros</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E6" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">13.9.6</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E6.png" altimg-height="29px" altimg-valign="-9px" altimg-width="474px" alttext="p(a,b)=\left\lfloor\tfrac{1}{2}\left(\left\lceil-b\right\rceil-\left\lceil-a%
\right\rceil+1\right)\right\rfloor-\left\lfloor\tfrac{1}{2}\left(\left\lceil-b%
\right\rceil-\left\lceil-a\right\rceil\right)\right\rfloor," display="block"><mrow><mrow><mrow><mi href="./13.9#SS1.p1">p</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mo href="./front/introduction#Sx4.p1.t1.r16">⌊</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mrow><mo>(</mo><mrow><mrow><mrow><mo href="./front/introduction#Sx4.p1.t1.r17">⌈</mo><mrow><mo>-</mo><mi>b</mi></mrow><mo href="./front/introduction#Sx4.p1.t1.r17">⌉</mo></mrow><mo>-</mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r17">⌈</mo><mrow><mo>-</mo><mi>a</mi></mrow><mo href="./front/introduction#Sx4.p1.t1.r17">⌉</mo></mrow></mrow><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mo href="./front/introduction#Sx4.p1.t1.r16">⌋</mo></mrow><mo>-</mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r16">⌊</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mrow><mo>(</mo><mrow><mrow><mo href="./front/introduction#Sx4.p1.t1.r17">⌈</mo><mrow><mo>-</mo><mi>b</mi></mrow><mo href="./front/introduction#Sx4.p1.t1.r17">⌉</mo></mrow><mo>-</mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r17">⌈</mo><mrow><mo>-</mo><mi>a</mi></mrow><mo href="./front/introduction#Sx4.p1.t1.r17">⌉</mo></mrow></mrow><mo>)</mo></mrow></mrow><mo href="./front/introduction#Sx4.p1.t1.r16">⌋</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m31.png" altimg-height="23px" altimg-valign="-7px" altimg-width="153px" alttext="\left\lceil-b\right\rceil>\left\lceil-a\right\rceil>0" display="inline"><mrow><mrow><mo href="./front/introduction#Sx4.p1.t1.r17">⌈</mo><mrow><mo>-</mo><mi>b</mi></mrow><mo href="./front/introduction#Sx4.p1.t1.r17">⌉</mo></mrow><mo>></mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r17">⌈</mo><mrow><mo>-</mo><mi>a</mi></mrow><mo href="./front/introduction#Sx4.p1.t1.r17">⌉</mo></mrow><mo>></mo><mn>0</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E6.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./front/introduction#Sx4.p1.t1.r17" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m32.png" altimg-height="23px" altimg-valign="-7px" altimg-width="33px" alttext="\left\lceil\NVar{x}\right\rceil" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r17">⌈</mo><mi class="ltx_nvar">x</mi><mo href="./front/introduction#Sx4.p1.t1.r17">⌉</mo></mrow></math>: ceiling of <math class="ltx_Math" altimg="m66.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./front/introduction#Sx4.p1.t1.r16" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m33.png" altimg-height="23px" altimg-valign="-7px" altimg-width="33px" alttext="\left\lfloor\NVar{x}\right\rfloor" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r16">⌊</mo><mi class="ltx_nvar">x</mi><mo href="./front/introduction#Sx4.p1.t1.r16">⌋</mo></mrow></math>: floor of <math class="ltx_Math" altimg="m66.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./13.9#SS1.p1" title="§13.9(i) Zeros of M ( a , b , z ) ‣ §13.9 Zeros ‣ Kummer Functions ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="23px" altimg-valign="-7px" altimg-width="58px" alttext="p(a,b)" display="inline"><mrow><mi href="./13.9#SS1.p1">p</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></mrow></math>: number of positive zeros</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">The number of negative real zeros <math class="ltx_Math" altimg="m61.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="n(a,b)" display="inline"><mrow><mi href="./13.9#E7">n</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></mrow></math> is given by</p>
<table id="E7" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">13.9.7</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E7.png" altimg-height="25px" altimg-valign="-7px" altimg-width="180px" alttext="n(a,b)=p(b-a,b)." display="block"><mrow><mrow><mrow><mi href="./13.9#E7">n</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mi href="./13.9#SS1.p1">p</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>b</mi><mo>-</mo><mi>a</mi></mrow><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E7.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text"><math class="ltx_Math" altimg="m61.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="n(a,b)" display="inline"><mrow><mi href="./13.9#E7">n</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></mrow></math>: number of negative real zeros (locally)</span></dd>
<dt>Symbols:</dt>
<dd><a href="./13.9#SS1.p1" title="§13.9(i) Zeros of M ( a , b , z ) ‣ §13.9 Zeros ‣ Kummer Functions ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="23px" altimg-valign="-7px" altimg-width="58px" alttext="p(a,b)" display="inline"><mrow><mi href="./13.9#SS1.p1">p</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></mrow></math>: number of positive zeros</a></dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS1.p2" class="ltx_para">
<p class="ltx_p">When <math class="ltx_Math" altimg="m45.png" altimg-height="17px" altimg-valign="-3px" altimg-width="51px" alttext="a<0" display="inline"><mrow><mi>a</mi><mo><</mo><mn>0</mn></mrow></math> and <math class="ltx_Math" altimg="m53.png" altimg-height="18px" altimg-valign="-3px" altimg-width="49px" alttext="b>0" display="inline"><mrow><mi>b</mi><mo>></mo><mn>0</mn></mrow></math> let <math class="ltx_Math" altimg="m37.png" altimg-height="21px" altimg-valign="-6px" altimg-width="25px" alttext="\phi_{r}" display="inline"><msub><mi href="./13.9#SS1.p2">ϕ</mi><mi href="./13.9#i">r</mi></msub></math>, <math class="ltx_Math" altimg="m64.png" altimg-height="20px" altimg-valign="-6px" altimg-width="120px" alttext="r=1,2,3,\dots" display="inline"><mrow><mi href="./13.9#i">r</mi><mo>=</mo><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>, be the positive zeros of <math class="ltx_Math" altimg="m12.png" altimg-height="23px" altimg-valign="-7px" altimg-width="93px" alttext="M\left(a,b,x\right)" display="inline"><mrow><mi href="./13.2#E2">M</mi><mo></mo><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi href="./13.1#p1.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>
arranged in increasing order of magnitude, and let
<math class="ltx_Math" altimg="m60.png" altimg-height="22px" altimg-valign="-8px" altimg-width="53px" alttext="j_{b-1,r}" display="inline"><msub><mi href="./13.9#SS1.p2">j</mi><mrow><mrow><mi>b</mi><mo>-</mo><mn>1</mn></mrow><mo>,</mo><mi href="./13.9#i">r</mi></mrow></msub></math> be the <math class="ltx_Math" altimg="m65.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="r" display="inline"><mi href="./13.9#i">r</mi></math>th positive zero of the Bessel function
<math class="ltx_Math" altimg="m11.png" altimg-height="23px" altimg-valign="-7px" altimg-width="74px" alttext="J_{b-1}\left(x\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mrow><mi>b</mi><mo>-</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./13.1#p1.t1.r3">x</mi><mo>)</mo></mrow></mrow></math> (§). Then</p>
<table id="E8" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">13.9.8</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E8.png" altimg-height="65px" altimg-valign="-27px" altimg-width="452px" alttext="\phi_{r}=\frac{j_{b-1,r}^{2}}{2b-4a}\left(1+\frac{2b(b-2)+j_{b-1,r}^{2}}{3(2b-%
4a)^{2}}\right)+O\left(\frac{1}{a^{5}}\right)," display="block"><mrow><mrow><msub><mi href="./13.9#SS1.p2">ϕ</mi><mi href="./13.9#i">r</mi></msub><mo>=</mo><mrow><mrow><mfrac><msubsup><mi href="./13.9#SS1.p2">j</mi><mrow><mrow><mi>b</mi><mo>-</mo><mn>1</mn></mrow><mo>,</mo><mi href="./13.9#i">r</mi></mrow><mn>2</mn></msubsup><mrow><mrow><mn>2</mn><mo></mo><mi>b</mi></mrow><mo>-</mo><mrow><mn>4</mn><mo></mo><mi>a</mi></mrow></mrow></mfrac><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mfrac><mrow><mrow><mn>2</mn><mo></mo><mi>b</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>b</mi><mo>-</mo><mn>2</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>+</mo><msubsup><mi href="./13.9#SS1.p2">j</mi><mrow><mrow><mi>b</mi><mo>-</mo><mn>1</mn></mrow><mo>,</mo><mi href="./13.9#i">r</mi></mrow><mn>2</mn></msubsup></mrow><mrow><mn>3</mn><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>2</mn><mo></mo><mi>b</mi></mrow><mo>-</mo><mrow><mn>4</mn><mo></mo><mi>a</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mo>+</mo><mrow><mi href="./2.1#E3">O</mi><mo></mo><mrow><mo>(</mo><mfrac><mn>1</mn><msup><mi>a</mi><mn>5</mn></msup></mfrac><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E8.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./2.1#E3" title="(2.1.3) ‣ §2.1(i) Asymptotic and Order Symbols ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m15.png" altimg-height="23px" altimg-valign="-7px" altimg-width="50px" alttext="O\left(\NVar{x}\right)" display="inline"><mrow><mi href="./2.1#E3">O</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>)</mo></mrow></mrow></math>: order not exceeding</a>,
<a href="./13.9#i" title="§13.9(i) Zeros of M ( a , b , z ) ‣ §13.9 Zeros ‣ Kummer Functions ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m64.png" altimg-height="20px" altimg-valign="-6px" altimg-width="120px" alttext="r=1,2,3,\dots" display="inline"><mrow><mi href="./13.9#i">r</mi><mo>=</mo><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math></a>,
<a href="./13.9#SS1.p2" title="§13.9(i) Zeros of M ( a , b , z ) ‣ §13.9 Zeros ‣ Kummer Functions ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="21px" altimg-valign="-6px" altimg-width="25px" alttext="\phi_{r}" display="inline"><msub><mi href="./13.9#SS1.p2">ϕ</mi><mi href="./13.9#i">r</mi></msub></math>: positive zeros</a> and
<a href="./13.9#SS1.p2" title="§13.9(i) Zeros of M ( a , b , z ) ‣ §13.9 Zeros ‣ Kummer Functions ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="22px" altimg-valign="-8px" altimg-width="33px" alttext="j_{b,r}" display="inline"><msub><mi href="./13.9#SS1.p2">j</mi><mrow><mi>b</mi><mo>,</mo><mi href="./13.9#i">r</mi></mrow></msub></math>: positive zero of Bessel</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">as <math class="ltx_Math" altimg="m50.png" altimg-height="17px" altimg-valign="-4px" altimg-width="81px" alttext="a\to-\infty" display="inline"><mrow><mi>a</mi><mo>→</mo><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow></mrow></math> with <math class="ltx_Math" altimg="m65.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="r" display="inline"><mi href="./13.9#i">r</mi></math> fixed.
</p>
</div>
<div id="SS1.p3" class="ltx_para">
<p class="ltx_p">Inequalities for <math class="ltx_Math" altimg="m37.png" altimg-height="21px" altimg-valign="-6px" altimg-width="25px" alttext="\phi_{r}" display="inline"><msub><mi href="./13.9#SS1.p2">ϕ</mi><mi href="./13.9#i">r</mi></msub></math> are given in <cite class="ltx_cite ltx_citemacro_citet">Gatteschi ()</cite>, and
identities involving infinite series of all of the complex zeros of
<math class="ltx_Math" altimg="m12.png" altimg-height="23px" altimg-valign="-7px" altimg-width="93px" alttext="M\left(a,b,x\right)" display="inline"><mrow><mi href="./13.2#E2">M</mi><mo></mo><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi href="./13.1#p1.t1.r3">x</mi><mo>)</mo></mrow></mrow></math> are given in <cite class="ltx_cite ltx_citemacro_citet">Ahmed and Muldoon ()</cite>.
</p>
</div>
<div id="SS1.p4" class="ltx_para">
<p class="ltx_p">For fixed <math class="ltx_Math" altimg="m42.png" altimg-height="21px" altimg-valign="-6px" altimg-width="71px" alttext="a,b\in\mathbb{C}" display="inline"><mrow><mrow><mi>a</mi><mo>,</mo><mi>b</mi></mrow><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mi href="./front/introduction#Sx4.p1.t1.r1">ℂ</mi></mrow></math> the large <math class="ltx_Math" altimg="m68.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./13.1#p1.t1.r4">z</mi></math>-zeros of <math class="ltx_Math" altimg="m13.png" altimg-height="23px" altimg-valign="-7px" altimg-width="92px" alttext="M\left(a,b,z\right)" display="inline"><mrow><mi href="./13.2#E2">M</mi><mo></mo><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></math> satisfy</p>
<table id="E9" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">13.9.9</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E9.png" altimg-height="53px" altimg-valign="-21px" altimg-width="565px" alttext="z=\pm(2n+a)\pi\mathrm{i}+\ln\left(-\frac{\Gamma\left(a\right)}{\Gamma\left(b-a%
\right)}\left(\pm 2n\pi\mathrm{i}\right)^{b-2a}\right)+O\left(n^{-1}\ln n%
\right)," display="block"><mrow><mrow><mi href="./13.1#p1.t1.r4">z</mi><mo>=</mo><mrow><mrow><mo>±</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>2</mn><mo></mo><mi href="./13.1#p1.t1.r2">n</mi></mrow><mo>+</mo><mi>a</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi></mrow></mrow><mo>+</mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mrow><mfrac><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi>a</mi><mo>)</mo></mrow></mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi>b</mi><mo>-</mo><mi>a</mi></mrow><mo>)</mo></mrow></mrow></mfrac><mo></mo><msup><mrow><mo>(</mo><mrow><mo>±</mo><mrow><mn>2</mn><mo></mo><mi href="./13.1#p1.t1.r2">n</mi><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi></mrow></mrow><mo>)</mo></mrow><mrow><mi>b</mi><mo>-</mo><mrow><mn>2</mn><mo></mo><mi>a</mi></mrow></mrow></msup></mrow></mrow><mo>)</mo></mrow></mrow><mo>+</mo><mrow><mi href="./2.1#E3">O</mi><mo></mo><mrow><mo>(</mo><mrow><msup><mi href="./13.1#p1.t1.r2">n</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi href="./13.1#p1.t1.r2">n</mi></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E9.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./2.1#E3" title="(2.1.3) ‣ §2.1(i) Asymptotic and Order Symbols ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m15.png" altimg-height="23px" altimg-valign="-7px" altimg-width="50px" alttext="O\left(\NVar{x}\right)" display="inline"><mrow><mi href="./2.1#E3">O</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>)</mo></mrow></mrow></math>: order not exceeding</a>,
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m25.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m34.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a>,
<a href="./13.1#p1.t1.r2" title="§13.1 Special Notation ‣ Notation ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m62.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./13.1#p1.t1.r2">n</mi></math>: nonnegative integer</a> and
<a href="./13.1#p1.t1.r4" title="§13.1 Special Notation ‣ Notation ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m68.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./13.1#p1.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m62.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./13.1#p1.t1.r2">n</mi></math> is a large positive integer, and the logarithm takes its principal
value (§).</p>
</div>
<div id="SS1.p5" class="ltx_para">
<p class="ltx_p">Let <math class="ltx_Math" altimg="m19.png" altimg-height="21px" altimg-valign="-5px" altimg-width="28px" alttext="P_{\alpha}" display="inline"><msub><mi href="./13.9#SS1.p5">P</mi><mi>α</mi></msub></math> denote the closure of the domain that is bounded by the parabola
<math class="ltx_Math" altimg="m67.png" altimg-height="25px" altimg-valign="-7px" altimg-width="137px" alttext="y^{2}=4\alpha(x+\alpha)" display="inline"><mrow><msup><mi href="./13.1#p1.t1.r3">y</mi><mn>2</mn></msup><mo>=</mo><mrow><mn>4</mn><mo></mo><mi>α</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./13.1#p1.t1.r3">x</mi><mo>+</mo><mi>α</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></math> and contains the origin.
Then <math class="ltx_Math" altimg="m13.png" altimg-height="23px" altimg-valign="-7px" altimg-width="92px" alttext="M\left(a,b,z\right)" display="inline"><mrow><mi href="./13.2#E2">M</mi><mo></mo><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></math> has no zeros in the regions
<math class="ltx_Math" altimg="m20.png" altimg-height="25px" altimg-valign="-9px" altimg-width="46px" alttext="P_{\ifrac{b}{a}}" display="inline"><msub><mi href="./13.9#SS1.p5">P</mi><mrow><mi>b</mi><mo>/</mo><mi>a</mi></mrow></msub></math>, if <math class="ltx_Math" altimg="m7.png" altimg-height="20px" altimg-valign="-5px" altimg-width="86px" alttext="0<b\leq a" display="inline"><mrow><mn>0</mn><mo><</mo><mi>b</mi><mo>≤</mo><mi>a</mi></mrow></math>;
<math class="ltx_Math" altimg="m18.png" altimg-height="21px" altimg-valign="-5px" altimg-width="26px" alttext="P_{1}" display="inline"><msub><mi href="./13.9#SS1.p5">P</mi><mn>1</mn></msub></math>, if <math class="ltx_Math" altimg="m8.png" altimg-height="20px" altimg-valign="-5px" altimg-width="86px" alttext="1\leq a\leq b" display="inline"><mrow><mn>1</mn><mo>≤</mo><mi>a</mi><mo>≤</mo><mi>b</mi></mrow></math>;
<math class="ltx_Math" altimg="m19.png" altimg-height="21px" altimg-valign="-5px" altimg-width="28px" alttext="P_{\alpha}" display="inline"><msub><mi href="./13.9#SS1.p5">P</mi><mi>α</mi></msub></math>, where <math class="ltx_Math" altimg="m28.png" altimg-height="23px" altimg-valign="-7px" altimg-width="257px" alttext="\alpha=\ifrac{(2a-b+ab)}{(a(a+1))}" display="inline"><mrow><mi>α</mi><mo>=</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mrow><mn>2</mn><mo></mo><mi>a</mi></mrow><mo>-</mo><mi>b</mi></mrow><mo>+</mo><mrow><mi>a</mi><mo></mo><mi>b</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mo>/</mo><mrow><mo stretchy="false">(</mo><mrow><mi>a</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>a</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></math>, if <math class="ltx_Math" altimg="m6.png" altimg-height="17px" altimg-valign="-3px" altimg-width="88px" alttext="0<a<1" display="inline"><mrow><mn>0</mn><mo><</mo><mi>a</mi><mo><</mo><mn>1</mn></mrow></math> and
<math class="ltx_Math" altimg="m49.png" altimg-height="23px" altimg-valign="-7px" altimg-width="172px" alttext="a\leq b<\ifrac{2a}{(1-a)}" display="inline"><mrow><mi>a</mi><mo>≤</mo><mi>b</mi><mo><</mo><mrow><mrow><mn>2</mn><mo></mo><mi>a</mi></mrow><mo>/</mo><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><mi>a</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></math>. The same results apply for the <math class="ltx_Math" altimg="m62.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./13.1#p1.t1.r2">n</mi></math>th partial sums
of the Maclaurin series
() of <math class="ltx_Math" altimg="m13.png" altimg-height="23px" altimg-valign="-7px" altimg-width="92px" alttext="M\left(a,b,z\right)" display="inline"><mrow><mi href="./13.2#E2">M</mi><mo></mo><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>.</p>
</div>
<div id="SS1.p6" class="ltx_para">
<p class="ltx_p">More information on the location of real zeros can be found in
<cite class="ltx_cite ltx_citemacro_citet">Zarzo<span class="ltx_text ltx_bib_etal"> et al.</span> ()</cite>.</p>
</div>
<div id="SS1.p7" class="ltx_para">
<p class="ltx_p">For fixed <math class="ltx_Math" altimg="m54.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="b" display="inline"><mi>b</mi></math> and <math class="ltx_Math" altimg="m68.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./13.1#p1.t1.r4">z</mi></math> in <math class="ltx_Math" altimg="m35.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\mathbb{C}" display="inline"><mi href="./front/introduction#Sx4.p1.t1.r1">ℂ</mi></math> the large <math class="ltx_Math" altimg="m47.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi>a</mi></math>-zeros of <math class="ltx_Math" altimg="m13.png" altimg-height="23px" altimg-valign="-7px" altimg-width="92px" alttext="M\left(a,b,z\right)" display="inline"><mrow><mi href="./13.2#E2">M</mi><mo></mo><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></math> are
given by</p>
<table id="E10" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">13.9.10</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E10.png" altimg-height="50px" altimg-valign="-16px" altimg-width="745px" alttext="a=-\frac{\pi^{2}}{4z}\left(n^{2}+(b-\tfrac{3}{2})n\right)-\frac{1}{16z}\left((%
b-\tfrac{3}{2})^{2}\pi^{2}+\tfrac{4}{3}z^{2}-8b(z-1)-4b^{2}-3\right)+O\left(n^%
{-1}\right)," display="block"><mrow><mrow><mi>a</mi><mo>=</mo><mrow><mrow><mrow><mo>-</mo><mrow><mfrac><msup><mi href="./3.12#E1">π</mi><mn>2</mn></msup><mrow><mn>4</mn><mo></mo><mi href="./13.1#p1.t1.r4">z</mi></mrow></mfrac><mo></mo><mrow><mo>(</mo><mrow><msup><mi href="./13.1#p1.t1.r2">n</mi><mn>2</mn></msup><mo>+</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mi>b</mi><mo>-</mo><mstyle displaystyle="false"><mfrac><mn>3</mn><mn>2</mn></mfrac></mstyle></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi href="./13.1#p1.t1.r2">n</mi></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mrow><mn>16</mn><mo></mo><mi href="./13.1#p1.t1.r4">z</mi></mrow></mfrac><mo></mo><mrow><mo>(</mo><mrow><mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mi>b</mi><mo>-</mo><mstyle displaystyle="false"><mfrac><mn>3</mn><mn>2</mn></mfrac></mstyle></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo></mo><msup><mi href="./3.12#E1">π</mi><mn>2</mn></msup></mrow><mo>+</mo><mrow><mstyle displaystyle="false"><mfrac><mn>4</mn><mn>3</mn></mfrac></mstyle><mo></mo><msup><mi href="./13.1#p1.t1.r4">z</mi><mn>2</mn></msup></mrow></mrow><mo>-</mo><mrow><mn>8</mn><mo></mo><mi>b</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./13.1#p1.t1.r4">z</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>-</mo><mrow><mn>4</mn><mo></mo><msup><mi>b</mi><mn>2</mn></msup></mrow><mo>-</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mi href="./2.1#E3">O</mi><mo></mo><mrow><mo>(</mo><msup><mi href="./13.1#p1.t1.r2">n</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E10.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./2.1#E3" title="(2.1.3) ‣ §2.1(i) Asymptotic and Order Symbols ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m15.png" altimg-height="23px" altimg-valign="-7px" altimg-width="50px" alttext="O\left(\NVar{x}\right)" display="inline"><mrow><mi href="./2.1#E3">O</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>)</mo></mrow></mrow></math>: order not exceeding</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./13.1#p1.t1.r2" title="§13.1 Special Notation ‣ Notation ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m62.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./13.1#p1.t1.r2">n</mi></math>: nonnegative integer</a> and
<a href="./13.1#p1.t1.r4" title="§13.1 Special Notation ‣ Notation ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m68.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./13.1#p1.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m62.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./13.1#p1.t1.r2">n</mi></math> is a large positive integer.
</p>
</div>
<div id="SS1.p8" class="ltx_para">
<p class="ltx_p">For fixed <math class="ltx_Math" altimg="m47.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi>a</mi></math> and <math class="ltx_Math" altimg="m68.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./13.1#p1.t1.r4">z</mi></math> in <math class="ltx_Math" altimg="m35.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\mathbb{C}" display="inline"><mi href="./front/introduction#Sx4.p1.t1.r1">ℂ</mi></math> the function
<math class="ltx_Math" altimg="m13.png" altimg-height="23px" altimg-valign="-7px" altimg-width="92px" alttext="M\left(a,b,z\right)" display="inline"><mrow><mi href="./13.2#E2">M</mi><mo></mo><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></math> has only a finite number of <math class="ltx_Math" altimg="m54.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="b" display="inline"><mi>b</mi></math>-zeros.</p>
</div>
</section>
<section id="ii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§13.9(ii) </span>Zeros of <math class="ltx_Math" altimg="m24.png" altimg-height="23px" altimg-valign="-7px" altimg-width="86px" alttext="U\left(a,b,z\right)" display="inline"><mrow><mi href="./13.2#E6">U</mi><mo></mo><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>
</h2>
<div id="SS2.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="SS2.p1" class="ltx_para">
<p class="ltx_p">For fixed <math class="ltx_Math" altimg="m47.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi>a</mi></math> and <math class="ltx_Math" altimg="m54.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="b" display="inline"><mi>b</mi></math> in <math class="ltx_Math" altimg="m35.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\mathbb{C}" display="inline"><mi href="./front/introduction#Sx4.p1.t1.r1">ℂ</mi></math>, <math class="ltx_Math" altimg="m24.png" altimg-height="23px" altimg-valign="-7px" altimg-width="86px" alttext="U\left(a,b,z\right)" display="inline"><mrow><mi href="./13.2#E6">U</mi><mo></mo><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></math> has a finite number of
<math class="ltx_Math" altimg="m68.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./13.1#p1.t1.r4">z</mi></math>-zeros in the sector <math class="ltx_Math" altimg="m70.png" altimg-height="27px" altimg-valign="-9px" altimg-width="201px" alttext="|\operatorname{ph}z|\leq\tfrac{3}{2}\pi-\delta(<\tfrac{3}{2}\pi)" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mi href="./13.1#p1.t1.r4">z</mi></mrow><mo stretchy="false">|</mo></mrow><mo>≤</mo><mrow><mrow><mrow><mfrac><mn>3</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo>-</mo><mi href="./13.1#p1.t1.r5">δ</mi></mrow><mspace width="veryverythickmathspace"></mspace><mrow><mo stretchy="false">(</mo><mrow><mi></mi><mo><</mo><mrow><mfrac><mn>3</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></math>. Let <math class="ltx_Math" altimg="m22.png" altimg-height="23px" altimg-valign="-7px" altimg-width="62px" alttext="T(a,b)" display="inline"><mrow><mi href="./13.9#SS2.p1">T</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></mrow></math> be
the total number of zeros in the sector <math class="ltx_Math" altimg="m69.png" altimg-height="23px" altimg-valign="-7px" altimg-width="93px" alttext="|\operatorname{ph}z|<\pi" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mi href="./13.1#p1.t1.r4">z</mi></mrow><mo stretchy="false">|</mo></mrow><mo><</mo><mi href="./3.12#E1">π</mi></mrow></math>,
<math class="ltx_Math" altimg="m17.png" altimg-height="23px" altimg-valign="-7px" altimg-width="63px" alttext="P(a,b)" display="inline"><mrow><mi href="./13.9#SS2.p1">P</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></mrow></math> be the corresponding number of positive zeros, and
<math class="ltx_Math" altimg="m47.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi>a</mi></math>, <math class="ltx_Math" altimg="m54.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="b" display="inline"><mi>b</mi></math>, and <math class="ltx_Math" altimg="m44.png" altimg-height="19px" altimg-valign="-4px" altimg-width="82px" alttext="a-b+1" display="inline"><mrow><mrow><mi>a</mi><mo>-</mo><mi>b</mi></mrow><mo>+</mo><mn>1</mn></mrow></math> be nonintegers. For the case <math class="ltx_Math" altimg="m57.png" altimg-height="20px" altimg-valign="-5px" altimg-width="49px" alttext="b\leq 1" display="inline"><mrow><mi>b</mi><mo>≤</mo><mn>1</mn></mrow></math>
</p>
<table id="E11" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">13.9.11</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E11.png" altimg-height="25px" altimg-valign="-7px" altimg-width="174px" alttext="T(a,b)=\left\lfloor-a\right\rfloor+1," display="block"><mrow><mrow><mrow><mi href="./13.9#SS2.p1">T</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mo href="./front/introduction#Sx4.p1.t1.r16">⌊</mo><mrow><mo>-</mo><mi>a</mi></mrow><mo href="./front/introduction#Sx4.p1.t1.r16">⌋</mo></mrow><mo>+</mo><mn>1</mn></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m45.png" altimg-height="17px" altimg-valign="-3px" altimg-width="51px" alttext="a<0" display="inline"><mrow><mi>a</mi><mo><</mo><mn>0</mn></mrow></math>, <math class="ltx_Math" altimg="m27.png" altimg-height="23px" altimg-valign="-7px" altimg-width="195px" alttext="\Gamma\left(a\right)\Gamma\left(a-b+1\right)>0" display="inline"><mrow><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi>a</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mi>a</mi><mo>-</mo><mi>b</mi></mrow><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mrow><mo>></mo><mn>0</mn></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E11.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m25.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./front/introduction#Sx4.p1.t1.r16" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m33.png" altimg-height="23px" altimg-valign="-7px" altimg-width="33px" alttext="\left\lfloor\NVar{x}\right\rfloor" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r16">⌊</mo><mi class="ltx_nvar">x</mi><mo href="./front/introduction#Sx4.p1.t1.r16">⌋</mo></mrow></math>: floor of <math class="ltx_Math" altimg="m66.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./13.9#SS2.p1" title="§13.9(ii) Zeros of U ( a , b , z ) ‣ §13.9 Zeros ‣ Kummer Functions ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="23px" altimg-valign="-7px" altimg-width="62px" alttext="T(a,b)" display="inline"><mrow><mi href="./13.9#SS2.p1">T</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></mrow></math>: number of zeros</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E12" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">13.9.12</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E12.png" altimg-height="25px" altimg-valign="-7px" altimg-width="143px" alttext="T(a,b)=\left\lfloor-a\right\rfloor," display="block"><mrow><mrow><mrow><mi href="./13.9#SS2.p1">T</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r16">⌊</mo><mrow><mo>-</mo><mi>a</mi></mrow><mo href="./front/introduction#Sx4.p1.t1.r16">⌋</mo></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m45.png" altimg-height="17px" altimg-valign="-3px" altimg-width="51px" alttext="a<0" display="inline"><mrow><mi>a</mi><mo><</mo><mn>0</mn></mrow></math>, <math class="ltx_Math" altimg="m26.png" altimg-height="23px" altimg-valign="-7px" altimg-width="195px" alttext="\Gamma\left(a\right)\Gamma\left(a-b+1\right)<0" display="inline"><mrow><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi>a</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mi>a</mi><mo>-</mo><mi>b</mi></mrow><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mrow><mo><</mo><mn>0</mn></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E12.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m25.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./front/introduction#Sx4.p1.t1.r16" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m33.png" altimg-height="23px" altimg-valign="-7px" altimg-width="33px" alttext="\left\lfloor\NVar{x}\right\rfloor" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r16">⌊</mo><mi class="ltx_nvar">x</mi><mo href="./front/introduction#Sx4.p1.t1.r16">⌋</mo></mrow></math>: floor of <math class="ltx_Math" altimg="m66.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./13.9#SS2.p1" title="§13.9(ii) Zeros of U ( a , b , z ) ‣ §13.9 Zeros ‣ Kummer Functions ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="23px" altimg-valign="-7px" altimg-width="62px" alttext="T(a,b)" display="inline"><mrow><mi href="./13.9#SS2.p1">T</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></mrow></math>: number of zeros</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E13" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">13.9.13</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E13.png" altimg-height="25px" altimg-valign="-7px" altimg-width="106px" alttext="T(a,b)=0," display="block"><mrow><mrow><mrow><mi href="./13.9#SS2.p1">T</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mn>0</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m46.png" altimg-height="17px" altimg-valign="-3px" altimg-width="51px" alttext="a>0" display="inline"><mrow><mi>a</mi><mo>></mo><mn>0</mn></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E13.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./13.9#SS2.p1" title="§13.9(ii) Zeros of U ( a , b , z ) ‣ §13.9 Zeros ‣ Kummer Functions ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="23px" altimg-valign="-7px" altimg-width="62px" alttext="T(a,b)" display="inline"><mrow><mi href="./13.9#SS2.p1">T</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></mrow></math>: number of zeros</a></dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and
</p>
<table id="E14" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">13.9.14</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E14.png" altimg-height="25px" altimg-valign="-7px" altimg-width="196px" alttext="P(a,b)=\left\lceil b-a-1\right\rceil," display="block"><mrow><mrow><mrow><mi href="./13.9#SS2.p1">P</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r17">⌈</mo><mrow><mi>b</mi><mo>-</mo><mi>a</mi><mo>-</mo><mn>1</mn></mrow><mo href="./front/introduction#Sx4.p1.t1.r17">⌉</mo></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m40.png" altimg-height="19px" altimg-valign="-4px" altimg-width="84px" alttext="a+1<b" display="inline"><mrow><mrow><mi>a</mi><mo>+</mo><mn>1</mn></mrow><mo><</mo><mi>b</mi></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E14.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./front/introduction#Sx4.p1.t1.r17" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m32.png" altimg-height="23px" altimg-valign="-7px" altimg-width="33px" alttext="\left\lceil\NVar{x}\right\rceil" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r17">⌈</mo><mi class="ltx_nvar">x</mi><mo href="./front/introduction#Sx4.p1.t1.r17">⌉</mo></mrow></math>: ceiling of <math class="ltx_Math" altimg="m66.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./13.9#SS2.p1" title="§13.9(ii) Zeros of U ( a , b , z ) ‣ §13.9 Zeros ‣ Kummer Functions ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m17.png" altimg-height="23px" altimg-valign="-7px" altimg-width="63px" alttext="P(a,b)" display="inline"><mrow><mi href="./13.9#SS2.p1">P</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></mrow></math>: number of positive zeros</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E15" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">13.9.15</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E15.png" altimg-height="25px" altimg-valign="-7px" altimg-width="107px" alttext="P(a,b)=0," display="block"><mrow><mrow><mrow><mi href="./13.9#SS2.p1">P</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mn>0</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m41.png" altimg-height="20px" altimg-valign="-5px" altimg-width="84px" alttext="a+1\geq b" display="inline"><mrow><mrow><mi>a</mi><mo>+</mo><mn>1</mn></mrow><mo>≥</mo><mi>b</mi></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E15.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./13.9#SS2.p1" title="§13.9(ii) Zeros of U ( a , b , z ) ‣ §13.9 Zeros ‣ Kummer Functions ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m17.png" altimg-height="23px" altimg-valign="-7px" altimg-width="63px" alttext="P(a,b)" display="inline"><mrow><mi href="./13.9#SS2.p1">P</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></mrow></math>: number of positive zeros</a></dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">For the case <math class="ltx_Math" altimg="m56.png" altimg-height="20px" altimg-valign="-5px" altimg-width="49px" alttext="b\geq 1" display="inline"><mrow><mi>b</mi><mo>≥</mo><mn>1</mn></mrow></math> we can use <math class="ltx_Math" altimg="m21.png" altimg-height="23px" altimg-valign="-7px" altimg-width="248px" alttext="T(a,b)=T(a-b+1,2-b)" display="inline"><mrow><mrow><mi href="./13.9#SS2.p1">T</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mi href="./13.9#SS2.p1">T</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mi>a</mi><mo>-</mo><mi>b</mi></mrow><mo>+</mo><mn>1</mn></mrow><mo>,</mo><mrow><mn>2</mn><mo>-</mo><mi>b</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></math> and <math class="ltx_Math" altimg="m16.png" altimg-height="23px" altimg-valign="-7px" altimg-width="250px" alttext="P(a,b)=P(a-b+1,2-b)" display="inline"><mrow><mrow><mi href="./13.9#SS2.p1">P</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mi href="./13.9#SS2.p1">P</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mi>a</mi><mo>-</mo><mi>b</mi></mrow><mo>+</mo><mn>1</mn></mrow><mo>,</mo><mrow><mn>2</mn><mo>-</mo><mi>b</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></math>.</p>
</div>
<div id="SS2.p2" class="ltx_para">
<p class="ltx_p">In <cite class="ltx_cite ltx_citemacro_citet">Wimp ()</cite> it is shown that if <math class="ltx_Math" altimg="m43.png" altimg-height="21px" altimg-valign="-6px" altimg-width="71px" alttext="a,b\in\mathbb{R}" display="inline"><mrow><mrow><mi>a</mi><mo>,</mo><mi>b</mi></mrow><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mi href="./front/introduction#Sx4.p2.t1.r14">ℝ</mi></mrow></math> and <math class="ltx_Math" altimg="m9.png" altimg-height="19px" altimg-valign="-4px" altimg-width="110px" alttext="2a-b>-1" display="inline"><mrow><mrow><mrow><mn>2</mn><mo></mo><mi>a</mi></mrow><mo>-</mo><mi>b</mi></mrow><mo>></mo><mrow><mo>-</mo><mn>1</mn></mrow></mrow></math>,
then <math class="ltx_Math" altimg="m24.png" altimg-height="23px" altimg-valign="-7px" altimg-width="86px" alttext="U\left(a,b,z\right)" display="inline"><mrow><mi href="./13.2#E6">U</mi><mo></mo><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></math> has no zeros in the sector
<math class="ltx_Math" altimg="m71.png" altimg-height="27px" altimg-valign="-9px" altimg-width="106px" alttext="|\operatorname{ph}{z}|\leq\frac{1}{2}\pi" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mi href="./13.1#p1.t1.r4">z</mi></mrow><mo stretchy="false">|</mo></mrow><mo>≤</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow></math>.</p>
</div>
<div id="SS2.p3" class="ltx_para">
<p class="ltx_p">Inequalities for the zeros of <math class="ltx_Math" altimg="m23.png" altimg-height="23px" altimg-valign="-7px" altimg-width="87px" alttext="U\left(a,b,x\right)" display="inline"><mrow><mi href="./13.2#E6">U</mi><mo></mo><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi href="./13.1#p1.t1.r3">x</mi><mo>)</mo></mrow></mrow></math> are given in
<cite class="ltx_cite ltx_citemacro_citet">Gatteschi ()</cite>.</p>
</div>
<div id="SS2.p4" class="ltx_para">
<p class="ltx_p">For fixed <math class="ltx_Math" altimg="m54.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="b" display="inline"><mi>b</mi></math> and <math class="ltx_Math" altimg="m68.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./13.1#p1.t1.r4">z</mi></math> in <math class="ltx_Math" altimg="m35.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\mathbb{C}" display="inline"><mi href="./front/introduction#Sx4.p1.t1.r1">ℂ</mi></math> the large <math class="ltx_Math" altimg="m47.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi>a</mi></math>-zeros of <math class="ltx_Math" altimg="m24.png" altimg-height="23px" altimg-valign="-7px" altimg-width="86px" alttext="U\left(a,b,z\right)" display="inline"><mrow><mi href="./13.2#E6">U</mi><mo></mo><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></math> are
given by</p>
<table id="E16" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">13.9.16</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E16.png" altimg-height="56px" altimg-valign="-21px" altimg-width="695px" alttext="a=-n-\frac{2}{\pi}\sqrt{zn}-\frac{2z}{\pi^{2}}+\tfrac{1}{2}b+\tfrac{1}{4}+%
\frac{z^{2}\left(\frac{1}{3}-4\pi^{-2}\right)+z-(b-1)^{2}+\frac{1}{4}}{4\pi%
\sqrt{zn}}+O\left(\frac{1}{n}\right)," display="block"><mrow><mrow><mi>a</mi><mo>=</mo><mrow><mrow><mrow><mo>-</mo><mi href="./13.1#p1.t1.r2">n</mi></mrow><mo>-</mo><mrow><mfrac><mn>2</mn><mi href="./3.12#E1">π</mi></mfrac><mo></mo><msqrt><mrow><mi href="./13.1#p1.t1.r4">z</mi><mo></mo><mi href="./13.1#p1.t1.r2">n</mi></mrow></msqrt></mrow><mo>-</mo><mfrac><mrow><mn>2</mn><mo></mo><mi href="./13.1#p1.t1.r4">z</mi></mrow><msup><mi href="./3.12#E1">π</mi><mn>2</mn></msup></mfrac></mrow><mo>+</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi>b</mi></mrow><mo>+</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>4</mn></mfrac></mstyle><mo>+</mo><mfrac><mrow><mrow><mrow><mrow><msup><mi href="./13.1#p1.t1.r4">z</mi><mn>2</mn></msup><mo></mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac><mo>-</mo><mrow><mn>4</mn><mo></mo><msup><mi href="./3.12#E1">π</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></mrow></mrow><mo>)</mo></mrow></mrow><mo>+</mo><mi href="./13.1#p1.t1.r4">z</mi></mrow><mo>-</mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi>b</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow><mo>+</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow><mrow><mn>4</mn><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><msqrt><mrow><mi href="./13.1#p1.t1.r4">z</mi><mo></mo><mi href="./13.1#p1.t1.r2">n</mi></mrow></msqrt></mrow></mfrac><mo>+</mo><mrow><mi href="./2.1#E3">O</mi><mo></mo><mrow><mo>(</mo><mfrac><mn>1</mn><mi href="./13.1#p1.t1.r2">n</mi></mfrac><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E16.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./2.1#E3" title="(2.1.3) ‣ §2.1(i) Asymptotic and Order Symbols ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m15.png" altimg-height="23px" altimg-valign="-7px" altimg-width="50px" alttext="O\left(\NVar{x}\right)" display="inline"><mrow><mi href="./2.1#E3">O</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>)</mo></mrow></mrow></math>: order not exceeding</a>,
<a href="./2.1#SS3.p1" title="§2.1(iii) Asymptotic Expansions ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="12px" altimg-valign="-2px" altimg-width="20px" alttext="\sim" display="inline"><mo href="./2.1#SS3.p1">∼</mo></math>: Poincaré asymptotic expansion</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./13.1#p1.t1.r2" title="§13.1 Special Notation ‣ Notation ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m62.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./13.1#p1.t1.r2">n</mi></math>: nonnegative integer</a> and
<a href="./13.1#p1.t1.r4" title="§13.1 Special Notation ‣ Notation ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m68.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./13.1#p1.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>Errata (effective with 1.0.13):</dt>
<dd>
In applying changes in Version 1.0.12 to this equation, an editing error was made; it has been corrected.
<p><span class="ltx_font_italic">Reported 2016-09-12 by Adri Olde Daalhuis</span></p>
</dd>
<dt>Errata (effective with 1.0.12):</dt>
<dd>
Originally this equation was expressed in terms of the asymptotic symbol <math class="ltx_Math" altimg="m39.png" altimg-height="12px" altimg-valign="-2px" altimg-width="20px" alttext="\sim" display="inline"><mo href="./2.1#SS3.p1">∼</mo></math>. As a consequence of the use of
the <math class="ltx_Math" altimg="m14.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="O" display="inline"><mi href="./2.1#E3">O</mi></math> order symbol on the right hand side, <math class="ltx_Math" altimg="m39.png" altimg-height="12px" altimg-valign="-2px" altimg-width="20px" alttext="\sim" display="inline"><mo href="./2.1#SS3.p1">∼</mo></math> was replaced by <math class="ltx_Math" altimg="m10.png" altimg-height="12px" altimg-valign="-2px" altimg-width="20px" alttext="=" display="inline"><mo>=</mo></math>.
<p><span class="ltx_font_italic">Reported 2016-07-11 by Rudi Weikard</span></p>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m62.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./13.1#p1.t1.r2">n</mi></math> is a large positive integer.</p>
</div>
<div id="SS2.p5" class="ltx_para">
<p class="ltx_p">For fixed <math class="ltx_Math" altimg="m47.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi>a</mi></math> and <math class="ltx_Math" altimg="m68.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./13.1#p1.t1.r4">z</mi></math> in <math class="ltx_Math" altimg="m35.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\mathbb{C}" display="inline"><mi href="./front/introduction#Sx4.p1.t1.r1">ℂ</mi></math>,
<math class="ltx_Math" altimg="m24.png" altimg-height="23px" altimg-valign="-7px" altimg-width="86px" alttext="U\left(a,b,z\right)" display="inline"><mrow><mi href="./13.2#E6">U</mi><mo></mo><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></math> has two infinite strings of <math class="ltx_Math" altimg="m54.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="b" display="inline"><mi>b</mi></math>-zeros that are asymptotic to the
imaginary axis as <math class="ltx_Math" altimg="m72.png" altimg-height="23px" altimg-valign="-7px" altimg-width="75px" alttext="|b|\to\infty" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mi>b</mi><mo stretchy="false">|</mo></mrow><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></math>.</p>
</div>
</section>
</section>
</div>
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<span></div>
</div>
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<title>DLMF: 13.15 Recurrence Relations and Derivatives</title>
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<div id="SS1.p1" class="ltx_para">
<table id="EGx1" class="ltx_equationgroup ltx_eqn_table">
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<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">13.15.1</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m2.png" altimg-height="29px" altimg-valign="-9px" altimg-width="587px" alttext="\displaystyle(\kappa-\mu-\tfrac{1}{2})M_{\kappa-1,\mu}\left(z\right)+(z-2%
\kappa)M_{\kappa,\mu}\left(z\right)+(\kappa+\mu+\tfrac{1}{2})M_{\kappa+1,\mu}%
\left(z\right)" display="inline"><mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mi>κ</mi><mo>-</mo><mi>μ</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><msub><mi href="./13.14#E2">M</mi><mrow><mrow><mi>κ</mi><mo>-</mo><mn>1</mn></mrow><mo href="./13.14#E2">,</mo><mi>μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./13.1#p1.t1.r4">z</mi><mo>-</mo><mrow><mn>2</mn><mo></mo><mi>κ</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><msub><mi href="./13.14#E2">M</mi><mrow><mi>κ</mi><mo href="./13.14#E2">,</mo><mi>μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mi>κ</mi><mo>+</mo><mi>μ</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><msub><mi href="./13.14#E2">M</mi><mrow><mrow><mi>κ</mi><mo>+</mo><mn>1</mn></mrow><mo href="./13.14#E2">,</mo><mi>μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m20.png" altimg-height="22px" altimg-valign="-6px" altimg-width="43px" alttext="\displaystyle=0," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mn>0</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./13.14#E2" title="(13.14.2) ‣ Standard Solutions ‣ §13.14(i) Differential Equation ‣ §13.14 Definitions and Basic Properties ‣ Whittaker Functions ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m41.png" altimg-height="24px" altimg-valign="-8px" altimg-width="77px" alttext="M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./13.14#E2">M</mi><mrow><mi class="ltx_nvar">κ</mi><mo href="./13.14#E2">,</mo><mi class="ltx_nvar">μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Whittaker confluent hypergeometric function</a> and
<a href="./13.1#p1.t1.r4" title="§13.1 Special Notation ‣ Notation ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./13.1#p1.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">13.4.29</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E2">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">13.15.2</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m11.png" altimg-height="32px" altimg-valign="-12px" altimg-width="761px" alttext="\displaystyle 2\mu(1+2\mu)\sqrt{z}M_{\kappa-\frac{1}{2},\mu-\frac{1}{2}}\left(%
z\right)-(z+2\mu)(1+2\mu)M_{\kappa,\mu}\left(z\right)+(\kappa+\mu+\tfrac{1}{2}%
)\sqrt{z}M_{\kappa+\frac{1}{2},\mu+\frac{1}{2}}\left(z\right)" display="inline"><mrow><mrow><mrow><mn>2</mn><mo></mo><mi>μ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>+</mo><mrow><mn>2</mn><mo></mo><mi>μ</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msqrt><mi href="./13.1#p1.t1.r4">z</mi></msqrt><mo></mo><mrow><msub><mi href="./13.14#E2">M</mi><mrow><mrow><mi>κ</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo href="./13.14#E2">,</mo><mrow><mi>μ</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow><mo>-</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./13.1#p1.t1.r4">z</mi><mo>+</mo><mrow><mn>2</mn><mo></mo><mi>μ</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>+</mo><mrow><mn>2</mn><mo></mo><mi>μ</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><msub><mi href="./13.14#E2">M</mi><mrow><mi>κ</mi><mo href="./13.14#E2">,</mo><mi>μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>+</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mi>κ</mi><mo>+</mo><mi>μ</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msqrt><mi href="./13.1#p1.t1.r4">z</mi></msqrt><mo></mo><mrow><msub><mi href="./13.14#E2">M</mi><mrow><mrow><mi>κ</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo href="./13.14#E2">,</mo><mrow><mi>μ</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m20.png" altimg-height="22px" altimg-valign="-6px" altimg-width="43px" alttext="\displaystyle=0," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mn>0</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./13.14#E2" title="(13.14.2) ‣ Standard Solutions ‣ §13.14(i) Differential Equation ‣ §13.14 Definitions and Basic Properties ‣ Whittaker Functions ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m41.png" altimg-height="24px" altimg-valign="-8px" altimg-width="77px" alttext="M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./13.14#E2">M</mi><mrow><mi class="ltx_nvar">κ</mi><mo href="./13.14#E2">,</mo><mi class="ltx_nvar">μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Whittaker confluent hypergeometric function</a> and
<a href="./13.1#p1.t1.r4" title="§13.1 Special Notation ‣ Notation ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./13.1#p1.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E3">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">13.15.3</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m3.png" altimg-height="32px" altimg-valign="-12px" altimg-width="669px" alttext="\displaystyle(\kappa-\mu-\tfrac{1}{2})M_{\kappa-\frac{1}{2},\mu+\frac{1}{2}}%
\left(z\right)+(1+2\mu)\sqrt{z}M_{\kappa,\mu}\left(z\right)-(\kappa+\mu+\tfrac%
{1}{2})M_{\kappa+\frac{1}{2},\mu+\frac{1}{2}}\left(z\right)" display="inline"><mrow><mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mi>κ</mi><mo>-</mo><mi>μ</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><msub><mi href="./13.14#E2">M</mi><mrow><mrow><mi>κ</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo href="./13.14#E2">,</mo><mrow><mi>μ</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>+</mo><mrow><mn>2</mn><mo></mo><mi>μ</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msqrt><mi href="./13.1#p1.t1.r4">z</mi></msqrt><mo></mo><mrow><msub><mi href="./13.14#E2">M</mi><mrow><mi>κ</mi><mo href="./13.14#E2">,</mo><mi>μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>-</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mi>κ</mi><mo>+</mo><mi>μ</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><msub><mi href="./13.14#E2">M</mi><mrow><mrow><mi>κ</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo href="./13.14#E2">,</mo><mrow><mi>μ</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m20.png" altimg-height="22px" altimg-valign="-6px" altimg-width="43px" alttext="\displaystyle=0," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mn>0</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./13.14#E2" title="(13.14.2) ‣ Standard Solutions ‣ §13.14(i) Differential Equation ‣ §13.14 Definitions and Basic Properties ‣ Whittaker Functions ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m41.png" altimg-height="24px" altimg-valign="-8px" altimg-width="77px" alttext="M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./13.14#E2">M</mi><mrow><mi class="ltx_nvar">κ</mi><mo href="./13.14#E2">,</mo><mi class="ltx_nvar">μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Whittaker confluent hypergeometric function</a> and
<a href="./13.1#p1.t1.r4" title="§13.1 Special Notation ‣ Notation ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./13.1#p1.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E4">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">13.15.4</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m6.png" altimg-height="32px" altimg-valign="-12px" altimg-width="440px" alttext="\displaystyle 2\mu M_{\kappa-\frac{1}{2},\mu-\frac{1}{2}}\left(z\right)-2\mu M%
_{\kappa+\frac{1}{2},\mu-\frac{1}{2}}\left(z\right)-\sqrt{z}M_{\kappa,\mu}%
\left(z\right)" display="inline"><mrow><mrow><mn>2</mn><mo></mo><mi>μ</mi><mo></mo><mrow><msub><mi href="./13.14#E2">M</mi><mrow><mrow><mi>κ</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo href="./13.14#E2">,</mo><mrow><mi>μ</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow><mo>-</mo><mrow><mn>2</mn><mo></mo><mi>μ</mi><mo></mo><mrow><msub><mi href="./13.14#E2">M</mi><mrow><mrow><mi>κ</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo href="./13.14#E2">,</mo><mrow><mi>μ</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow><mo>-</mo><mrow><msqrt><mi href="./13.1#p1.t1.r4">z</mi></msqrt><mo></mo><mrow><msub><mi href="./13.14#E2">M</mi><mrow><mi>κ</mi><mo href="./13.14#E2">,</mo><mi>μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m20.png" altimg-height="22px" altimg-valign="-6px" altimg-width="43px" alttext="\displaystyle=0," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mn>0</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E4.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./13.14#E2" title="(13.14.2) ‣ Standard Solutions ‣ §13.14(i) Differential Equation ‣ §13.14 Definitions and Basic Properties ‣ Whittaker Functions ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m41.png" altimg-height="24px" altimg-valign="-8px" altimg-width="77px" alttext="M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./13.14#E2">M</mi><mrow><mi class="ltx_nvar">κ</mi><mo href="./13.14#E2">,</mo><mi class="ltx_nvar">μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Whittaker confluent hypergeometric function</a> and
<a href="./13.1#p1.t1.r4" title="§13.1 Special Notation ‣ Notation ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./13.1#p1.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">13.4.28</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E5">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">13.15.5</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m8.png" altimg-height="32px" altimg-valign="-12px" altimg-width="711px" alttext="\displaystyle 2\mu(1+2\mu)M_{\kappa,\mu}\left(z\right)-2\mu(1+2\mu)\sqrt{z}M_{%
\kappa-\frac{1}{2},\mu-\frac{1}{2}}\left(z\right)-(\kappa-\mu-\tfrac{1}{2})%
\sqrt{z}M_{\kappa-\frac{1}{2},\mu+\frac{1}{2}}\left(z\right)" display="inline"><mrow><mrow><mn>2</mn><mo></mo><mi>μ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>+</mo><mrow><mn>2</mn><mo></mo><mi>μ</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><msub><mi href="./13.14#E2">M</mi><mrow><mi>κ</mi><mo href="./13.14#E2">,</mo><mi>μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow><mo>-</mo><mrow><mn>2</mn><mo></mo><mi>μ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>+</mo><mrow><mn>2</mn><mo></mo><mi>μ</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msqrt><mi href="./13.1#p1.t1.r4">z</mi></msqrt><mo></mo><mrow><msub><mi href="./13.14#E2">M</mi><mrow><mrow><mi>κ</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo href="./13.14#E2">,</mo><mrow><mi>μ</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow><mo>-</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mi>κ</mi><mo>-</mo><mi>μ</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msqrt><mi href="./13.1#p1.t1.r4">z</mi></msqrt><mo></mo><mrow><msub><mi href="./13.14#E2">M</mi><mrow><mrow><mi>κ</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo href="./13.14#E2">,</mo><mrow><mi>μ</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m20.png" altimg-height="22px" altimg-valign="-6px" altimg-width="43px" alttext="\displaystyle=0," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mn>0</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E5.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./13.14#E2" title="(13.14.2) ‣ Standard Solutions ‣ §13.14(i) Differential Equation ‣ §13.14 Definitions and Basic Properties ‣ Whittaker Functions ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m41.png" altimg-height="24px" altimg-valign="-8px" altimg-width="77px" alttext="M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./13.14#E2">M</mi><mrow><mi class="ltx_nvar">κ</mi><mo href="./13.14#E2">,</mo><mi class="ltx_nvar">μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Whittaker confluent hypergeometric function</a> and
<a href="./13.1#p1.t1.r4" title="§13.1 Special Notation ‣ Notation ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./13.1#p1.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E6">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">13.15.6</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m9.png" altimg-height="32px" altimg-valign="-12px" altimg-width="761px" alttext="\displaystyle 2\mu(1+2\mu)\sqrt{z}M_{\kappa+\frac{1}{2},\mu-\frac{1}{2}}\left(%
z\right)+(z-2\mu)(1+2\mu)M_{\kappa,\mu}\left(z\right)+(\kappa-\mu-\tfrac{1}{2}%
)\sqrt{z}M_{\kappa-\frac{1}{2},\mu+\frac{1}{2}}\left(z\right)" display="inline"><mrow><mrow><mn>2</mn><mo></mo><mi>μ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>+</mo><mrow><mn>2</mn><mo></mo><mi>μ</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msqrt><mi href="./13.1#p1.t1.r4">z</mi></msqrt><mo></mo><mrow><msub><mi href="./13.14#E2">M</mi><mrow><mrow><mi>κ</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo href="./13.14#E2">,</mo><mrow><mi>μ</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./13.1#p1.t1.r4">z</mi><mo>-</mo><mrow><mn>2</mn><mo></mo><mi>μ</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>+</mo><mrow><mn>2</mn><mo></mo><mi>μ</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><msub><mi href="./13.14#E2">M</mi><mrow><mi>κ</mi><mo href="./13.14#E2">,</mo><mi>μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mi>κ</mi><mo>-</mo><mi>μ</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msqrt><mi href="./13.1#p1.t1.r4">z</mi></msqrt><mo></mo><mrow><msub><mi href="./13.14#E2">M</mi><mrow><mrow><mi>κ</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo href="./13.14#E2">,</mo><mrow><mi>μ</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m20.png" altimg-height="22px" altimg-valign="-6px" altimg-width="43px" alttext="\displaystyle=0," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mn>0</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E6.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./13.14#E2" title="(13.14.2) ‣ Standard Solutions ‣ §13.14(i) Differential Equation ‣ §13.14 Definitions and Basic Properties ‣ Whittaker Functions ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m41.png" altimg-height="24px" altimg-valign="-8px" altimg-width="77px" alttext="M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./13.14#E2">M</mi><mrow><mi class="ltx_nvar">κ</mi><mo href="./13.14#E2">,</mo><mi class="ltx_nvar">μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Whittaker confluent hypergeometric function</a> and
<a href="./13.1#p1.t1.r4" title="§13.1 Special Notation ‣ Notation ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./13.1#p1.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E7">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">13.15.7</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m10.png" altimg-height="32px" altimg-valign="-12px" altimg-width="711px" alttext="\displaystyle 2\mu(1+2\mu)\sqrt{z}M_{\kappa+\frac{1}{2},\mu-\frac{1}{2}}\left(%
z\right)-2\mu(1+2\mu)M_{\kappa,\mu}\left(z\right)+(\kappa+\mu+\tfrac{1}{2})%
\sqrt{z}M_{\kappa+\frac{1}{2},\mu+\frac{1}{2}}\left(z\right)" display="inline"><mrow><mrow><mrow><mn>2</mn><mo></mo><mi>μ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>+</mo><mrow><mn>2</mn><mo></mo><mi>μ</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msqrt><mi href="./13.1#p1.t1.r4">z</mi></msqrt><mo></mo><mrow><msub><mi href="./13.14#E2">M</mi><mrow><mrow><mi>κ</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo href="./13.14#E2">,</mo><mrow><mi>μ</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow><mo>-</mo><mrow><mn>2</mn><mo></mo><mi>μ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>+</mo><mrow><mn>2</mn><mo></mo><mi>μ</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><msub><mi href="./13.14#E2">M</mi><mrow><mi>κ</mi><mo href="./13.14#E2">,</mo><mi>μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>+</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mi>κ</mi><mo>+</mo><mi>μ</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msqrt><mi href="./13.1#p1.t1.r4">z</mi></msqrt><mo></mo><mrow><msub><mi href="./13.14#E2">M</mi><mrow><mrow><mi>κ</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo href="./13.14#E2">,</mo><mrow><mi>μ</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m21.png" altimg-height="18px" altimg-valign="-2px" altimg-width="43px" alttext="\displaystyle=0." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mn>0</mn></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E7.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./13.14#E2" title="(13.14.2) ‣ Standard Solutions ‣ §13.14(i) Differential Equation ‣ §13.14 Definitions and Basic Properties ‣ Whittaker Functions ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m41.png" altimg-height="24px" altimg-valign="-8px" altimg-width="77px" alttext="M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./13.14#E2">M</mi><mrow><mi class="ltx_nvar">κ</mi><mo href="./13.14#E2">,</mo><mi class="ltx_nvar">μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Whittaker confluent hypergeometric function</a> and
<a href="./13.1#p1.t1.r4" title="§13.1 Special Notation ‣ Notation ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./13.1#p1.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E8">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">13.15.8</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m27.png" altimg-height="32px" altimg-valign="-12px" altimg-width="495px" alttext="\displaystyle W_{\kappa+\frac{1}{2},\mu+\frac{1}{2}}\left(z\right)-\sqrt{z}W_{%
\kappa,\mu}\left(z\right)+(\kappa-\mu-\tfrac{1}{2})W_{\kappa-\frac{1}{2},\mu+%
\frac{1}{2}}\left(z\right)" display="inline"><mrow><mrow><mrow><msub><mi href="./13.14#E3">W</mi><mrow><mrow><mi>κ</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo href="./13.14#E3">,</mo><mrow><mi>μ</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow><mo>-</mo><mrow><msqrt><mi href="./13.1#p1.t1.r4">z</mi></msqrt><mo></mo><mrow><msub><mi href="./13.14#E3">W</mi><mrow><mi>κ</mi><mo href="./13.14#E3">,</mo><mi>μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>+</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mi>κ</mi><mo>-</mo><mi>μ</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><msub><mi href="./13.14#E3">W</mi><mrow><mrow><mi>κ</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo href="./13.14#E3">,</mo><mrow><mi>μ</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m20.png" altimg-height="22px" altimg-valign="-6px" altimg-width="43px" alttext="\displaystyle=0," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mn>0</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E8.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./13.14#E3" title="(13.14.3) ‣ Standard Solutions ‣ §13.14(i) Differential Equation ‣ §13.14 Definitions and Basic Properties ‣ Whittaker Functions ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m42.png" altimg-height="24px" altimg-valign="-8px" altimg-width="77px" alttext="W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./13.14#E3">W</mi><mrow><mi class="ltx_nvar">κ</mi><mo href="./13.14#E3">,</mo><mi class="ltx_nvar">μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Whittaker confluent hypergeometric function</a> and
<a href="./13.1#p1.t1.r4" title="§13.1 Special Notation ‣ Notation ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./13.1#p1.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E9">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">13.15.9</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m28.png" altimg-height="32px" altimg-valign="-12px" altimg-width="495px" alttext="\displaystyle W_{\kappa+\frac{1}{2},\mu-\frac{1}{2}}\left(z\right)-\sqrt{z}W_{%
\kappa,\mu}\left(z\right)+(\kappa+\mu-\tfrac{1}{2})W_{\kappa-\frac{1}{2},\mu-%
\frac{1}{2}}\left(z\right)" display="inline"><mrow><mrow><mrow><msub><mi href="./13.14#E3">W</mi><mrow><mrow><mi>κ</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo href="./13.14#E3">,</mo><mrow><mi>μ</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow><mo>-</mo><mrow><msqrt><mi href="./13.1#p1.t1.r4">z</mi></msqrt><mo></mo><mrow><msub><mi href="./13.14#E3">W</mi><mrow><mi>κ</mi><mo href="./13.14#E3">,</mo><mi>μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>+</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mi>κ</mi><mo>+</mo><mi>μ</mi></mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><msub><mi href="./13.14#E3">W</mi><mrow><mrow><mi>κ</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo href="./13.14#E3">,</mo><mrow><mi>μ</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m20.png" altimg-height="22px" altimg-valign="-6px" altimg-width="43px" alttext="\displaystyle=0," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mn>0</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E9.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./13.14#E3" title="(13.14.3) ‣ Standard Solutions ‣ §13.14(i) Differential Equation ‣ §13.14 Definitions and Basic Properties ‣ Whittaker Functions ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m42.png" altimg-height="24px" altimg-valign="-8px" altimg-width="77px" alttext="W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./13.14#E3">W</mi><mrow><mi class="ltx_nvar">κ</mi><mo href="./13.14#E3">,</mo><mi class="ltx_nvar">μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Whittaker confluent hypergeometric function</a> and
<a href="./13.1#p1.t1.r4" title="§13.1 Special Notation ‣ Notation ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./13.1#p1.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">13.4.30</span> (in different form)</span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E10">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">13.15.10</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m7.png" altimg-height="32px" altimg-valign="-12px" altimg-width="443px" alttext="\displaystyle 2\mu W_{\kappa,\mu}\left(z\right)-\sqrt{z}W_{\kappa+\frac{1}{2},%
\mu+\frac{1}{2}}\left(z\right)+\sqrt{z}W_{\kappa+\frac{1}{2},\mu-\frac{1}{2}}%
\left(z\right)" display="inline"><mrow><mrow><mrow><mn>2</mn><mo></mo><mi>μ</mi><mo></mo><mrow><msub><mi href="./13.14#E3">W</mi><mrow><mi>κ</mi><mo href="./13.14#E3">,</mo><mi>μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow><mo>-</mo><mrow><msqrt><mi href="./13.1#p1.t1.r4">z</mi></msqrt><mo></mo><mrow><msub><mi href="./13.14#E3">W</mi><mrow><mrow><mi>κ</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo href="./13.14#E3">,</mo><mrow><mi>μ</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>+</mo><mrow><msqrt><mi href="./13.1#p1.t1.r4">z</mi></msqrt><mo></mo><mrow><msub><mi href="./13.14#E3">W</mi><mrow><mrow><mi>κ</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo href="./13.14#E3">,</mo><mrow><mi>μ</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m20.png" altimg-height="22px" altimg-valign="-6px" altimg-width="43px" alttext="\displaystyle=0," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mn>0</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E10.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./13.14#E3" title="(13.14.3) ‣ Standard Solutions ‣ §13.14(i) Differential Equation ‣ §13.14 Definitions and Basic Properties ‣ Whittaker Functions ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m42.png" altimg-height="24px" altimg-valign="-8px" altimg-width="77px" alttext="W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./13.14#E3">W</mi><mrow><mi class="ltx_nvar">κ</mi><mo href="./13.14#E3">,</mo><mi class="ltx_nvar">μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Whittaker confluent hypergeometric function</a> and
<a href="./13.1#p1.t1.r4" title="§13.1 Special Notation ‣ Notation ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./13.1#p1.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E11">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">13.15.11</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m26.png" altimg-height="29px" altimg-valign="-9px" altimg-width="585px" alttext="\displaystyle W_{\kappa+1,\mu}\left(z\right)+(2\kappa-z)W_{\kappa,\mu}\left(z%
\right)+(\kappa-\mu-\tfrac{1}{2})(\kappa+\mu-\tfrac{1}{2})W_{\kappa-1,\mu}%
\left(z\right)" display="inline"><mrow><mrow><msub><mi href="./13.14#E3">W</mi><mrow><mrow><mi>κ</mi><mo>+</mo><mn>1</mn></mrow><mo href="./13.14#E3">,</mo><mi>μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow><mo>+</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>2</mn><mo></mo><mi>κ</mi></mrow><mo>-</mo><mi href="./13.1#p1.t1.r4">z</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><msub><mi href="./13.14#E3">W</mi><mrow><mi>κ</mi><mo href="./13.14#E3">,</mo><mi>μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mi>κ</mi><mo>-</mo><mi>μ</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mi>κ</mi><mo>+</mo><mi>μ</mi></mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><msub><mi href="./13.14#E3">W</mi><mrow><mrow><mi>κ</mi><mo>-</mo><mn>1</mn></mrow><mo href="./13.14#E3">,</mo><mi>μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m20.png" altimg-height="22px" altimg-valign="-6px" altimg-width="43px" alttext="\displaystyle=0," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mn>0</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E11.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./13.14#E3" title="(13.14.3) ‣ Standard Solutions ‣ §13.14(i) Differential Equation ‣ §13.14 Definitions and Basic Properties ‣ Whittaker Functions ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m42.png" altimg-height="24px" altimg-valign="-8px" altimg-width="77px" alttext="W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./13.14#E3">W</mi><mrow><mi class="ltx_nvar">κ</mi><mo href="./13.14#E3">,</mo><mi class="ltx_nvar">μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Whittaker confluent hypergeometric function</a> and
<a href="./13.1#p1.t1.r4" title="§13.1 Special Notation ‣ Notation ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./13.1#p1.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">13.4.31</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E12">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">13.15.12</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m4.png" altimg-height="32px" altimg-valign="-12px" altimg-width="644px" alttext="\displaystyle(\kappa-\mu-\tfrac{1}{2})\sqrt{z}W_{\kappa-\frac{1}{2},\mu+\frac{%
1}{2}}\left(z\right)+2\mu W_{\kappa,\mu}\left(z\right)-(\kappa+\mu-\tfrac{1}{2%
})\sqrt{z}W_{\kappa-\frac{1}{2},\mu-\frac{1}{2}}\left(z\right)" display="inline"><mrow><mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mi>κ</mi><mo>-</mo><mi>μ</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msqrt><mi href="./13.1#p1.t1.r4">z</mi></msqrt><mo></mo><mrow><msub><mi href="./13.14#E3">W</mi><mrow><mrow><mi>κ</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo href="./13.14#E3">,</mo><mrow><mi>μ</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mn>2</mn><mo></mo><mi>μ</mi><mo></mo><mrow><msub><mi href="./13.14#E3">W</mi><mrow><mi>κ</mi><mo href="./13.14#E3">,</mo><mi>μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>-</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mi>κ</mi><mo>+</mo><mi>μ</mi></mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msqrt><mi href="./13.1#p1.t1.r4">z</mi></msqrt><mo></mo><mrow><msub><mi href="./13.14#E3">W</mi><mrow><mrow><mi>κ</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo href="./13.14#E3">,</mo><mrow><mi>μ</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m20.png" altimg-height="22px" altimg-valign="-6px" altimg-width="43px" alttext="\displaystyle=0," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mn>0</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E12.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./13.14#E3" title="(13.14.3) ‣ Standard Solutions ‣ §13.14(i) Differential Equation ‣ §13.14 Definitions and Basic Properties ‣ Whittaker Functions ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m42.png" altimg-height="24px" altimg-valign="-8px" altimg-width="77px" alttext="W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./13.14#E3">W</mi><mrow><mi class="ltx_nvar">κ</mi><mo href="./13.14#E3">,</mo><mi class="ltx_nvar">μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Whittaker confluent hypergeometric function</a> and
<a href="./13.1#p1.t1.r4" title="§13.1 Special Notation ‣ Notation ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./13.1#p1.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E13">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">13.15.13</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m1.png" altimg-height="32px" altimg-valign="-12px" altimg-width="594px" alttext="\displaystyle(\kappa+\mu-\tfrac{1}{2})\sqrt{z}W_{\kappa-\frac{1}{2},\mu-\frac{%
1}{2}}\left(z\right)-(z+2\mu)W_{\kappa,\mu}\left(z\right)+\sqrt{z}W_{\kappa+%
\frac{1}{2},\mu+\frac{1}{2}}\left(z\right)" display="inline"><mrow><mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mi>κ</mi><mo>+</mo><mi>μ</mi></mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msqrt><mi href="./13.1#p1.t1.r4">z</mi></msqrt><mo></mo><mrow><msub><mi href="./13.14#E3">W</mi><mrow><mrow><mi>κ</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo href="./13.14#E3">,</mo><mrow><mi>μ</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow><mo>-</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./13.1#p1.t1.r4">z</mi><mo>+</mo><mrow><mn>2</mn><mo></mo><mi>μ</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><msub><mi href="./13.14#E3">W</mi><mrow><mi>κ</mi><mo href="./13.14#E3">,</mo><mi>μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>+</mo><mrow><msqrt><mi href="./13.1#p1.t1.r4">z</mi></msqrt><mo></mo><mrow><msub><mi href="./13.14#E3">W</mi><mrow><mrow><mi>κ</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo href="./13.14#E3">,</mo><mrow><mi>μ</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m20.png" altimg-height="22px" altimg-valign="-6px" altimg-width="43px" alttext="\displaystyle=0," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mn>0</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E13.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./13.14#E3" title="(13.14.3) ‣ Standard Solutions ‣ §13.14(i) Differential Equation ‣ §13.14 Definitions and Basic Properties ‣ Whittaker Functions ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m42.png" altimg-height="24px" altimg-valign="-8px" altimg-width="77px" alttext="W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./13.14#E3">W</mi><mrow><mi class="ltx_nvar">κ</mi><mo href="./13.14#E3">,</mo><mi class="ltx_nvar">μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Whittaker confluent hypergeometric function</a> and
<a href="./13.1#p1.t1.r4" title="§13.1 Special Notation ‣ Notation ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./13.1#p1.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E14">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">13.15.14</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m5.png" altimg-height="32px" altimg-valign="-12px" altimg-width="594px" alttext="\displaystyle(\kappa-\mu-\tfrac{1}{2})\sqrt{z}W_{\kappa-\frac{1}{2},\mu+\frac{%
1}{2}}\left(z\right)-(z-2\mu)W_{\kappa,\mu}\left(z\right)+\sqrt{z}W_{\kappa+%
\frac{1}{2},\mu-\frac{1}{2}}\left(z\right)" display="inline"><mrow><mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mi>κ</mi><mo>-</mo><mi>μ</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msqrt><mi href="./13.1#p1.t1.r4">z</mi></msqrt><mo></mo><mrow><msub><mi href="./13.14#E3">W</mi><mrow><mrow><mi>κ</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo href="./13.14#E3">,</mo><mrow><mi>μ</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow><mo>-</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./13.1#p1.t1.r4">z</mi><mo>-</mo><mrow><mn>2</mn><mo></mo><mi>μ</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><msub><mi href="./13.14#E3">W</mi><mrow><mi>κ</mi><mo href="./13.14#E3">,</mo><mi>μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>+</mo><mrow><msqrt><mi href="./13.1#p1.t1.r4">z</mi></msqrt><mo></mo><mrow><msub><mi href="./13.14#E3">W</mi><mrow><mrow><mi>κ</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo href="./13.14#E3">,</mo><mrow><mi>μ</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m21.png" altimg-height="18px" altimg-valign="-2px" altimg-width="43px" alttext="\displaystyle=0." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mn>0</mn></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E14.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./13.14#E3" title="(13.14.3) ‣ Standard Solutions ‣ §13.14(i) Differential Equation ‣ §13.14 Definitions and Basic Properties ‣ Whittaker Functions ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m42.png" altimg-height="24px" altimg-valign="-8px" altimg-width="77px" alttext="W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./13.14#E3">W</mi><mrow><mi class="ltx_nvar">κ</mi><mo href="./13.14#E3">,</mo><mi class="ltx_nvar">μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Whittaker confluent hypergeometric function</a> and
<a href="./13.1#p1.t1.r4" title="§13.1 Special Notation ‣ Notation ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./13.1#p1.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
</div>
</section>
<section id="ii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§13.15(ii) </span>Differentiation Formulas</h2>
<div id="SS2.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="SS2.p1" class="ltx_para">
<table id="EGx2" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E15">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">13.15.15</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m35.png" altimg-height="47px" altimg-valign="-16px" altimg-width="218px" alttext="\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(e^{\frac{1}{2}z}z%
^{\mu-\frac{1}{2}}M_{\kappa,\mu}\left(z\right)\right)" display="inline"><mrow><mstyle displaystyle="true"><mfrac><mpadded lspace="-1.7pt" width="-4.5pt"><msup><mo href="./1.4#E4">d</mo><mi href="./13.1#p1.t1.r2">n</mi></msup></mpadded><msup><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi href="./13.1#p1.t1.r4">z</mi></mrow><mi href="./1.4#E4">n</mi></msup></mfrac></mstyle><mo></mo><mrow><mo>(</mo><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./13.1#p1.t1.r4">z</mi></mrow></msup><mo></mo><msup><mi href="./13.1#p1.t1.r4">z</mi><mrow><mi>μ</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup><mo></mo><mrow><msub><mi href="./13.14#E2">M</mi><mrow><mi>κ</mi><mo href="./13.14#E2">,</mo><mi>μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m17.png" altimg-height="35px" altimg-valign="-12px" altimg-width="410px" alttext="\displaystyle=(-1)^{n}{\left(-2\mu\right)_{n}}e^{\frac{1}{2}z}z^{\mu-\frac{1}{%
2}(n+1)}M_{\kappa-\frac{1}{2}n,\mu-\frac{1}{2}n}\left(z\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./13.1#p1.t1.r2">n</mi></msup><mo></mo><msub><mrow><mo href="./5.2#iii">(</mo><mrow><mo>-</mo><mrow><mn>2</mn><mo></mo><mi>μ</mi></mrow></mrow><mo href="./5.2#iii">)</mo></mrow><mi href="./13.1#p1.t1.r2">n</mi></msub><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./13.1#p1.t1.r4">z</mi></mrow></msup><mo></mo><msup><mi href="./13.1#p1.t1.r4">z</mi><mrow><mi>μ</mi><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./13.1#p1.t1.r2">n</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></msup><mo></mo><mrow><msub><mi href="./13.14#E2">M</mi><mrow><mrow><mi>κ</mi><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./13.1#p1.t1.r2">n</mi></mrow></mrow><mo href="./13.14#E2">,</mo><mrow><mi>μ</mi><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./13.1#p1.t1.r2">n</mi></mrow></mrow></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E15.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#iii" title="§5.2(iii) Pochhammer’s Symbol ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="24px" altimg-valign="-8px" altimg-width="41px" alttext="{\left(\NVar{a}\right)_{\NVar{n}}}" display="inline"><msub><mrow><mo href="./5.2#iii">(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r5">a</mi><mo href="./5.2#iii">)</mo></mrow><mi class="ltx_nvar" href="./5.1#p2.t1.r1">n</mi></msub></math>: Pochhammer’s symbol (or shifted factorial)</a>,
<a href="./13.14#E2" title="(13.14.2) ‣ Standard Solutions ‣ §13.14(i) Differential Equation ‣ §13.14 Definitions and Basic Properties ‣ Whittaker Functions ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m41.png" altimg-height="24px" altimg-valign="-8px" altimg-width="77px" alttext="M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./13.14#E2">M</mi><mrow><mi class="ltx_nvar">κ</mi><mo href="./13.14#E2">,</mo><mi class="ltx_nvar">μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Whittaker confluent hypergeometric function</a>,
<a href="./1.4#E4" title="(1.4.4) ‣ §1.4(iii) Derivatives ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="29px" altimg-valign="-9px" altimg-width="27px" alttext="\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: derivative of <math class="ltx_Math" altimg="m47.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m45.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./13.1#p1.t1.r2" title="§13.1 Special Notation ‣ Notation ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./13.1#p1.t1.r2">n</mi></math>: nonnegative integer</a> and
<a href="./13.1#p1.t1.r4" title="§13.1 Special Notation ‣ Notation ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./13.1#p1.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E16">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">13.15.16</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m33.png" altimg-height="47px" altimg-valign="-16px" altimg-width="230px" alttext="\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(e^{\frac{1}{2}z}z%
^{-\mu-\frac{1}{2}}M_{\kappa,\mu}\left(z\right)\right)" display="inline"><mrow><mstyle displaystyle="true"><mfrac><mpadded lspace="-1.7pt" width="-4.5pt"><msup><mo href="./1.4#E4">d</mo><mi href="./13.1#p1.t1.r2">n</mi></msup></mpadded><msup><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi href="./13.1#p1.t1.r4">z</mi></mrow><mi href="./1.4#E4">n</mi></msup></mfrac></mstyle><mo></mo><mrow><mo>(</mo><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./13.1#p1.t1.r4">z</mi></mrow></msup><mo></mo><msup><mi href="./13.1#p1.t1.r4">z</mi><mrow><mrow><mo>-</mo><mi>μ</mi></mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup><mo></mo><mrow><msub><mi href="./13.14#E2">M</mi><mrow><mi>κ</mi><mo href="./13.14#E2">,</mo><mi>μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m22.png" altimg-height="59px" altimg-valign="-22px" altimg-width="425px" alttext="\displaystyle=\frac{{\left(\frac{1}{2}+\mu-\kappa\right)_{n}}}{{\left(1+2\mu%
\right)_{n}}}e^{\frac{1}{2}z}z^{-\mu-\frac{1}{2}(n+1)}M_{\kappa-\frac{1}{2}n,%
\mu+\frac{1}{2}n}\left(z\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><mfrac><msub><mrow><mo href="./5.2#iii">(</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>+</mo><mi>μ</mi></mrow><mo>-</mo><mi>κ</mi></mrow><mo href="./5.2#iii">)</mo></mrow><mi href="./13.1#p1.t1.r2">n</mi></msub><msub><mrow><mo href="./5.2#iii">(</mo><mrow><mn>1</mn><mo>+</mo><mrow><mn>2</mn><mo></mo><mi>μ</mi></mrow></mrow><mo href="./5.2#iii">)</mo></mrow><mi href="./13.1#p1.t1.r2">n</mi></msub></mfrac></mstyle><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./13.1#p1.t1.r4">z</mi></mrow></msup><mo></mo><msup><mi href="./13.1#p1.t1.r4">z</mi><mrow><mrow><mo>-</mo><mi>μ</mi></mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./13.1#p1.t1.r2">n</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></msup><mo></mo><mrow><msub><mi href="./13.14#E2">M</mi><mrow><mrow><mi>κ</mi><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./13.1#p1.t1.r2">n</mi></mrow></mrow><mo href="./13.14#E2">,</mo><mrow><mi>μ</mi><mo>+</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./13.1#p1.t1.r2">n</mi></mrow></mrow></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E16.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#iii" title="§5.2(iii) Pochhammer’s Symbol ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="24px" altimg-valign="-8px" altimg-width="41px" alttext="{\left(\NVar{a}\right)_{\NVar{n}}}" display="inline"><msub><mrow><mo href="./5.2#iii">(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r5">a</mi><mo href="./5.2#iii">)</mo></mrow><mi class="ltx_nvar" href="./5.1#p2.t1.r1">n</mi></msub></math>: Pochhammer’s symbol (or shifted factorial)</a>,
<a href="./13.14#E2" title="(13.14.2) ‣ Standard Solutions ‣ §13.14(i) Differential Equation ‣ §13.14 Definitions and Basic Properties ‣ Whittaker Functions ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m41.png" altimg-height="24px" altimg-valign="-8px" altimg-width="77px" alttext="M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./13.14#E2">M</mi><mrow><mi class="ltx_nvar">κ</mi><mo href="./13.14#E2">,</mo><mi class="ltx_nvar">μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Whittaker confluent hypergeometric function</a>,
<a href="./1.4#E4" title="(1.4.4) ‣ §1.4(iii) Derivatives ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="29px" altimg-valign="-9px" altimg-width="27px" alttext="\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: derivative of <math class="ltx_Math" altimg="m47.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m45.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./13.1#p1.t1.r2" title="§13.1 Special Notation ‣ Notation ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./13.1#p1.t1.r2">n</mi></math>: nonnegative integer</a> and
<a href="./13.1#p1.t1.r4" title="§13.1 Special Notation ‣ Notation ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./13.1#p1.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E17">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">13.15.17</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m39.png" altimg-height="54px" altimg-valign="-21px" altimg-width="270px" alttext="\displaystyle\left(z\frac{\mathrm{d}}{\mathrm{d}z}z\right)^{n}\left(e^{\frac{1%
}{2}z}z^{-\kappa-1}M_{\kappa,\mu}\left(z\right)\right)" display="inline"><mrow><msup><mrow><mo>(</mo><mrow><mi href="./13.1#p1.t1.r4">z</mi><mo></mo><mrow><mstyle displaystyle="true"><mfrac><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi href="./13.1#p1.t1.r4">z</mi></mrow></mfrac></mstyle><mo></mo><mi href="./13.1#p1.t1.r4">z</mi></mrow></mrow><mo>)</mo></mrow><mi href="./13.1#p1.t1.r2">n</mi></msup><mo></mo><mrow><mo>(</mo><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./13.1#p1.t1.r4">z</mi></mrow></msup><mo></mo><msup><mi href="./13.1#p1.t1.r4">z</mi><mrow><mrow><mo>-</mo><mi>κ</mi></mrow><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mrow><msub><mi href="./13.14#E2">M</mi><mrow><mi>κ</mi><mo href="./13.14#E2">,</mo><mi>μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m24.png" altimg-height="34px" altimg-valign="-10px" altimg-width="338px" alttext="\displaystyle={\left(\tfrac{1}{2}+\mu-\kappa\right)_{n}}e^{\frac{1}{2}z}z^{n-%
\kappa-1}M_{\kappa-n,\mu}\left(z\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msub><mrow><mo href="./5.2#iii">(</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>+</mo><mi>μ</mi></mrow><mo>-</mo><mi>κ</mi></mrow><mo href="./5.2#iii">)</mo></mrow><mi href="./13.1#p1.t1.r2">n</mi></msub><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./13.1#p1.t1.r4">z</mi></mrow></msup><mo></mo><msup><mi href="./13.1#p1.t1.r4">z</mi><mrow><mi href="./13.1#p1.t1.r2">n</mi><mo>-</mo><mi>κ</mi><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mrow><msub><mi href="./13.14#E2">M</mi><mrow><mrow><mi>κ</mi><mo>-</mo><mi href="./13.1#p1.t1.r2">n</mi></mrow><mo href="./13.14#E2">,</mo><mi>μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E17.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#iii" title="§5.2(iii) Pochhammer’s Symbol ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="24px" altimg-valign="-8px" altimg-width="41px" alttext="{\left(\NVar{a}\right)_{\NVar{n}}}" display="inline"><msub><mrow><mo href="./5.2#iii">(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r5">a</mi><mo href="./5.2#iii">)</mo></mrow><mi class="ltx_nvar" href="./5.1#p2.t1.r1">n</mi></msub></math>: Pochhammer’s symbol (or shifted factorial)</a>,
<a href="./13.14#E2" title="(13.14.2) ‣ Standard Solutions ‣ §13.14(i) Differential Equation ‣ §13.14 Definitions and Basic Properties ‣ Whittaker Functions ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m41.png" altimg-height="24px" altimg-valign="-8px" altimg-width="77px" alttext="M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./13.14#E2">M</mi><mrow><mi class="ltx_nvar">κ</mi><mo href="./13.14#E2">,</mo><mi class="ltx_nvar">μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Whittaker confluent hypergeometric function</a>,
<a href="./1.4#E4" title="(1.4.4) ‣ §1.4(iii) Derivatives ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="29px" altimg-valign="-9px" altimg-width="27px" alttext="\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: derivative of <math class="ltx_Math" altimg="m47.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m45.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./13.1#p1.t1.r2" title="§13.1 Special Notation ‣ Notation ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./13.1#p1.t1.r2">n</mi></math>: nonnegative integer</a> and
<a href="./13.1#p1.t1.r4" title="§13.1 Special Notation ‣ Notation ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./13.1#p1.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E18">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">13.15.18</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m31.png" altimg-height="47px" altimg-valign="-16px" altimg-width="230px" alttext="\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(e^{-\frac{1}{2}z}%
z^{\mu-\frac{1}{2}}M_{\kappa,\mu}\left(z\right)\right)" display="inline"><mrow><mstyle displaystyle="true"><mfrac><mpadded lspace="-1.7pt" width="-4.5pt"><msup><mo href="./1.4#E4">d</mo><mi href="./13.1#p1.t1.r2">n</mi></msup></mpadded><msup><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi href="./13.1#p1.t1.r4">z</mi></mrow><mi href="./1.4#E4">n</mi></msup></mfrac></mstyle><mo></mo><mrow><mo>(</mo><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./13.1#p1.t1.r4">z</mi></mrow></mrow></msup><mo></mo><msup><mi href="./13.1#p1.t1.r4">z</mi><mrow><mi>μ</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup><mo></mo><mrow><msub><mi href="./13.14#E2">M</mi><mrow><mi>κ</mi><mo href="./13.14#E2">,</mo><mi>μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m16.png" altimg-height="35px" altimg-valign="-12px" altimg-width="422px" alttext="\displaystyle=(-1)^{n}{\left(-2\mu\right)_{n}}e^{-\frac{1}{2}z}z^{\mu-\frac{1}%
{2}(n+1)}M_{\kappa+\frac{1}{2}n,\mu-\frac{1}{2}n}\left(z\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./13.1#p1.t1.r2">n</mi></msup><mo></mo><msub><mrow><mo href="./5.2#iii">(</mo><mrow><mo>-</mo><mrow><mn>2</mn><mo></mo><mi>μ</mi></mrow></mrow><mo href="./5.2#iii">)</mo></mrow><mi href="./13.1#p1.t1.r2">n</mi></msub><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./13.1#p1.t1.r4">z</mi></mrow></mrow></msup><mo></mo><msup><mi href="./13.1#p1.t1.r4">z</mi><mrow><mi>μ</mi><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./13.1#p1.t1.r2">n</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></msup><mo></mo><mrow><msub><mi href="./13.14#E2">M</mi><mrow><mrow><mi>κ</mi><mo>+</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./13.1#p1.t1.r2">n</mi></mrow></mrow><mo href="./13.14#E2">,</mo><mrow><mi>μ</mi><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./13.1#p1.t1.r2">n</mi></mrow></mrow></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E18.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#iii" title="§5.2(iii) Pochhammer’s Symbol ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="24px" altimg-valign="-8px" altimg-width="41px" alttext="{\left(\NVar{a}\right)_{\NVar{n}}}" display="inline"><msub><mrow><mo href="./5.2#iii">(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r5">a</mi><mo href="./5.2#iii">)</mo></mrow><mi class="ltx_nvar" href="./5.1#p2.t1.r1">n</mi></msub></math>: Pochhammer’s symbol (or shifted factorial)</a>,
<a href="./13.14#E2" title="(13.14.2) ‣ Standard Solutions ‣ §13.14(i) Differential Equation ‣ §13.14 Definitions and Basic Properties ‣ Whittaker Functions ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m41.png" altimg-height="24px" altimg-valign="-8px" altimg-width="77px" alttext="M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./13.14#E2">M</mi><mrow><mi class="ltx_nvar">κ</mi><mo href="./13.14#E2">,</mo><mi class="ltx_nvar">μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Whittaker confluent hypergeometric function</a>,
<a href="./1.4#E4" title="(1.4.4) ‣ §1.4(iii) Derivatives ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="29px" altimg-valign="-9px" altimg-width="27px" alttext="\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: derivative of <math class="ltx_Math" altimg="m47.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m45.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./13.1#p1.t1.r2" title="§13.1 Special Notation ‣ Notation ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./13.1#p1.t1.r2">n</mi></math>: nonnegative integer</a> and
<a href="./13.1#p1.t1.r4" title="§13.1 Special Notation ‣ Notation ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./13.1#p1.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E19">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">13.15.19</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m29.png" altimg-height="47px" altimg-valign="-16px" altimg-width="243px" alttext="\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(e^{-\frac{1}{2}z}%
z^{-\mu-\frac{1}{2}}M_{\kappa,\mu}\left(z\right)\right)" display="inline"><mrow><mstyle displaystyle="true"><mfrac><mpadded lspace="-1.7pt" width="-4.5pt"><msup><mo href="./1.4#E4">d</mo><mi href="./13.1#p1.t1.r2">n</mi></msup></mpadded><msup><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi href="./13.1#p1.t1.r4">z</mi></mrow><mi href="./1.4#E4">n</mi></msup></mfrac></mstyle><mo></mo><mrow><mo>(</mo><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./13.1#p1.t1.r4">z</mi></mrow></mrow></msup><mo></mo><msup><mi href="./13.1#p1.t1.r4">z</mi><mrow><mrow><mo>-</mo><mi>μ</mi></mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup><mo></mo><mrow><msub><mi href="./13.14#E2">M</mi><mrow><mi>κ</mi><mo href="./13.14#E2">,</mo><mi>μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m12.png" altimg-height="59px" altimg-valign="-22px" altimg-width="492px" alttext="\displaystyle=(-1)^{n}\frac{{\left(\frac{1}{2}+\mu+\kappa\right)_{n}}}{{\left(%
1+2\mu\right)_{n}}}e^{-\frac{1}{2}z}z^{-\mu-\frac{1}{2}(n+1)}\*M_{\kappa+\frac%
{1}{2}n,\mu+\frac{1}{2}n}\left(z\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./13.1#p1.t1.r2">n</mi></msup><mo></mo><mstyle displaystyle="true"><mfrac><msub><mrow><mo href="./5.2#iii">(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>+</mo><mi>μ</mi><mo>+</mo><mi>κ</mi></mrow><mo href="./5.2#iii">)</mo></mrow><mi href="./13.1#p1.t1.r2">n</mi></msub><msub><mrow><mo href="./5.2#iii">(</mo><mrow><mn>1</mn><mo>+</mo><mrow><mn>2</mn><mo></mo><mi>μ</mi></mrow></mrow><mo href="./5.2#iii">)</mo></mrow><mi href="./13.1#p1.t1.r2">n</mi></msub></mfrac></mstyle><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./13.1#p1.t1.r4">z</mi></mrow></mrow></msup><mo></mo><msup><mi href="./13.1#p1.t1.r4">z</mi><mrow><mrow><mo>-</mo><mi>μ</mi></mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./13.1#p1.t1.r2">n</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></msup><mo></mo><mrow><msub><mi href="./13.14#E2">M</mi><mrow><mrow><mi>κ</mi><mo>+</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./13.1#p1.t1.r2">n</mi></mrow></mrow><mo href="./13.14#E2">,</mo><mrow><mi>μ</mi><mo>+</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./13.1#p1.t1.r2">n</mi></mrow></mrow></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E19.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#iii" title="§5.2(iii) Pochhammer’s Symbol ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="24px" altimg-valign="-8px" altimg-width="41px" alttext="{\left(\NVar{a}\right)_{\NVar{n}}}" display="inline"><msub><mrow><mo href="./5.2#iii">(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r5">a</mi><mo href="./5.2#iii">)</mo></mrow><mi class="ltx_nvar" href="./5.1#p2.t1.r1">n</mi></msub></math>: Pochhammer’s symbol (or shifted factorial)</a>,
<a href="./13.14#E2" title="(13.14.2) ‣ Standard Solutions ‣ §13.14(i) Differential Equation ‣ §13.14 Definitions and Basic Properties ‣ Whittaker Functions ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m41.png" altimg-height="24px" altimg-valign="-8px" altimg-width="77px" alttext="M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./13.14#E2">M</mi><mrow><mi class="ltx_nvar">κ</mi><mo href="./13.14#E2">,</mo><mi class="ltx_nvar">μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Whittaker confluent hypergeometric function</a>,
<a href="./1.4#E4" title="(1.4.4) ‣ §1.4(iii) Derivatives ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="29px" altimg-valign="-9px" altimg-width="27px" alttext="\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: derivative of <math class="ltx_Math" altimg="m47.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m45.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./13.1#p1.t1.r2" title="§13.1 Special Notation ‣ Notation ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./13.1#p1.t1.r2">n</mi></math>: nonnegative integer</a> and
<a href="./13.1#p1.t1.r4" title="§13.1 Special Notation ‣ Notation ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./13.1#p1.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E20">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">13.15.20</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m37.png" altimg-height="54px" altimg-valign="-21px" altimg-width="270px" alttext="\displaystyle\left(z\frac{\mathrm{d}}{\mathrm{d}z}z\right)^{n}\left(e^{-\frac{%
1}{2}z}z^{\kappa-1}M_{\kappa,\mu}\left(z\right)\right)" display="inline"><mrow><msup><mrow><mo>(</mo><mrow><mi href="./13.1#p1.t1.r4">z</mi><mo></mo><mrow><mstyle displaystyle="true"><mfrac><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi href="./13.1#p1.t1.r4">z</mi></mrow></mfrac></mstyle><mo></mo><mi href="./13.1#p1.t1.r4">z</mi></mrow></mrow><mo>)</mo></mrow><mi href="./13.1#p1.t1.r2">n</mi></msup><mo></mo><mrow><mo>(</mo><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./13.1#p1.t1.r4">z</mi></mrow></mrow></msup><mo></mo><msup><mi href="./13.1#p1.t1.r4">z</mi><mrow><mi>κ</mi><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mrow><msub><mi href="./13.14#E2">M</mi><mrow><mi>κ</mi><mo href="./13.14#E2">,</mo><mi>μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m23.png" altimg-height="34px" altimg-valign="-10px" altimg-width="354px" alttext="\displaystyle={\left(\tfrac{1}{2}+\mu+\kappa\right)_{n}}e^{-\frac{1}{2}z}z^{%
\kappa+n-1}\*M_{\kappa+n,\mu}\left(z\right)." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msub><mrow><mo href="./5.2#iii">(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>+</mo><mi>μ</mi><mo>+</mo><mi>κ</mi></mrow><mo href="./5.2#iii">)</mo></mrow><mi href="./13.1#p1.t1.r2">n</mi></msub><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./13.1#p1.t1.r4">z</mi></mrow></mrow></msup><mo></mo><msup><mi href="./13.1#p1.t1.r4">z</mi><mrow><mrow><mi>κ</mi><mo>+</mo><mi href="./13.1#p1.t1.r2">n</mi></mrow><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mrow><msub><mi href="./13.14#E2">M</mi><mrow><mrow><mi>κ</mi><mo>+</mo><mi href="./13.1#p1.t1.r2">n</mi></mrow><mo href="./13.14#E2">,</mo><mi>μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E20.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#iii" title="§5.2(iii) Pochhammer’s Symbol ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="24px" altimg-valign="-8px" altimg-width="41px" alttext="{\left(\NVar{a}\right)_{\NVar{n}}}" display="inline"><msub><mrow><mo href="./5.2#iii">(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r5">a</mi><mo href="./5.2#iii">)</mo></mrow><mi class="ltx_nvar" href="./5.1#p2.t1.r1">n</mi></msub></math>: Pochhammer’s symbol (or shifted factorial)</a>,
<a href="./13.14#E2" title="(13.14.2) ‣ Standard Solutions ‣ §13.14(i) Differential Equation ‣ §13.14 Definitions and Basic Properties ‣ Whittaker Functions ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m41.png" altimg-height="24px" altimg-valign="-8px" altimg-width="77px" alttext="M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./13.14#E2">M</mi><mrow><mi class="ltx_nvar">κ</mi><mo href="./13.14#E2">,</mo><mi class="ltx_nvar">μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Whittaker confluent hypergeometric function</a>,
<a href="./1.4#E4" title="(1.4.4) ‣ §1.4(iii) Derivatives ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="29px" altimg-valign="-9px" altimg-width="27px" alttext="\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: derivative of <math class="ltx_Math" altimg="m47.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m45.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./13.1#p1.t1.r2" title="§13.1 Special Notation ‣ Notation ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./13.1#p1.t1.r2">n</mi></math>: nonnegative integer</a> and
<a href="./13.1#p1.t1.r4" title="§13.1 Special Notation ‣ Notation ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./13.1#p1.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E21">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">13.15.21</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m34.png" altimg-height="47px" altimg-valign="-16px" altimg-width="230px" alttext="\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(e^{\frac{1}{2}z}z%
^{-\mu-\frac{1}{2}}W_{\kappa,\mu}\left(z\right)\right)" display="inline"><mrow><mstyle displaystyle="true"><mfrac><mpadded lspace="-1.7pt" width="-4.5pt"><msup><mo href="./1.4#E4">d</mo><mi href="./13.1#p1.t1.r2">n</mi></msup></mpadded><msup><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi href="./13.1#p1.t1.r4">z</mi></mrow><mi href="./1.4#E4">n</mi></msup></mfrac></mstyle><mo></mo><mrow><mo>(</mo><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./13.1#p1.t1.r4">z</mi></mrow></msup><mo></mo><msup><mi href="./13.1#p1.t1.r4">z</mi><mrow><mrow><mo>-</mo><mi>μ</mi></mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup><mo></mo><mrow><msub><mi href="./13.14#E3">W</mi><mrow><mi>κ</mi><mo href="./13.14#E3">,</mo><mi>μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m18.png" altimg-height="35px" altimg-valign="-12px" altimg-width="475px" alttext="\displaystyle=(-1)^{n}{\left(\tfrac{1}{2}+\mu-\kappa\right)_{n}}e^{\frac{1}{2}%
z}z^{-\mu-\frac{1}{2}(n+1)}\*W_{\kappa-\frac{1}{2}n,\mu+\frac{1}{2}n}\left(z%
\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./13.1#p1.t1.r2">n</mi></msup><mo></mo><msub><mrow><mo href="./5.2#iii">(</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>+</mo><mi>μ</mi></mrow><mo>-</mo><mi>κ</mi></mrow><mo href="./5.2#iii">)</mo></mrow><mi href="./13.1#p1.t1.r2">n</mi></msub><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./13.1#p1.t1.r4">z</mi></mrow></msup><mo></mo><msup><mi href="./13.1#p1.t1.r4">z</mi><mrow><mrow><mo>-</mo><mi>μ</mi></mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./13.1#p1.t1.r2">n</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></msup><mo></mo><mrow><msub><mi href="./13.14#E3">W</mi><mrow><mrow><mi>κ</mi><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./13.1#p1.t1.r2">n</mi></mrow></mrow><mo href="./13.14#E3">,</mo><mrow><mi>μ</mi><mo>+</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./13.1#p1.t1.r2">n</mi></mrow></mrow></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E21.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#iii" title="§5.2(iii) Pochhammer’s Symbol ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="24px" altimg-valign="-8px" altimg-width="41px" alttext="{\left(\NVar{a}\right)_{\NVar{n}}}" display="inline"><msub><mrow><mo href="./5.2#iii">(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r5">a</mi><mo href="./5.2#iii">)</mo></mrow><mi class="ltx_nvar" href="./5.1#p2.t1.r1">n</mi></msub></math>: Pochhammer’s symbol (or shifted factorial)</a>,
<a href="./13.14#E3" title="(13.14.3) ‣ Standard Solutions ‣ §13.14(i) Differential Equation ‣ §13.14 Definitions and Basic Properties ‣ Whittaker Functions ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m42.png" altimg-height="24px" altimg-valign="-8px" altimg-width="77px" alttext="W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./13.14#E3">W</mi><mrow><mi class="ltx_nvar">κ</mi><mo href="./13.14#E3">,</mo><mi class="ltx_nvar">μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Whittaker confluent hypergeometric function</a>,
<a href="./1.4#E4" title="(1.4.4) ‣ §1.4(iii) Derivatives ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="29px" altimg-valign="-9px" altimg-width="27px" alttext="\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: derivative of <math class="ltx_Math" altimg="m47.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m45.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./13.1#p1.t1.r2" title="§13.1 Special Notation ‣ Notation ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./13.1#p1.t1.r2">n</mi></math>: nonnegative integer</a> and
<a href="./13.1#p1.t1.r4" title="§13.1 Special Notation ‣ Notation ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./13.1#p1.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E22">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">13.15.22</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m36.png" altimg-height="47px" altimg-valign="-16px" altimg-width="217px" alttext="\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(e^{\frac{1}{2}z}z%
^{\mu-\frac{1}{2}}W_{\kappa,\mu}\left(z\right)\right)" display="inline"><mrow><mstyle displaystyle="true"><mfrac><mpadded lspace="-1.7pt" width="-4.5pt"><msup><mo href="./1.4#E4">d</mo><mi href="./13.1#p1.t1.r2">n</mi></msup></mpadded><msup><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi href="./13.1#p1.t1.r4">z</mi></mrow><mi href="./1.4#E4">n</mi></msup></mfrac></mstyle><mo></mo><mrow><mo>(</mo><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./13.1#p1.t1.r4">z</mi></mrow></msup><mo></mo><msup><mi href="./13.1#p1.t1.r4">z</mi><mrow><mi>μ</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup><mo></mo><mrow><msub><mi href="./13.14#E3">W</mi><mrow><mi>κ</mi><mo href="./13.14#E3">,</mo><mi>μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m19.png" altimg-height="35px" altimg-valign="-12px" altimg-width="463px" alttext="\displaystyle=(-1)^{n}{\left(\tfrac{1}{2}-\mu-\kappa\right)_{n}}e^{\frac{1}{2}%
z}z^{\mu-\frac{1}{2}(n+1)}\*W_{\kappa-\frac{1}{2}n,\mu-\frac{1}{2}n}\left(z%
\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./13.1#p1.t1.r2">n</mi></msup><mo></mo><msub><mrow><mo href="./5.2#iii">(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>-</mo><mi>μ</mi><mo>-</mo><mi>κ</mi></mrow><mo href="./5.2#iii">)</mo></mrow><mi href="./13.1#p1.t1.r2">n</mi></msub><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./13.1#p1.t1.r4">z</mi></mrow></msup><mo></mo><msup><mi href="./13.1#p1.t1.r4">z</mi><mrow><mi>μ</mi><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./13.1#p1.t1.r2">n</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></msup><mo></mo><mrow><msub><mi href="./13.14#E3">W</mi><mrow><mrow><mi>κ</mi><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./13.1#p1.t1.r2">n</mi></mrow></mrow><mo href="./13.14#E3">,</mo><mrow><mi>μ</mi><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./13.1#p1.t1.r2">n</mi></mrow></mrow></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E22.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#iii" title="§5.2(iii) Pochhammer’s Symbol ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="24px" altimg-valign="-8px" altimg-width="41px" alttext="{\left(\NVar{a}\right)_{\NVar{n}}}" display="inline"><msub><mrow><mo href="./5.2#iii">(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r5">a</mi><mo href="./5.2#iii">)</mo></mrow><mi class="ltx_nvar" href="./5.1#p2.t1.r1">n</mi></msub></math>: Pochhammer’s symbol (or shifted factorial)</a>,
<a href="./13.14#E3" title="(13.14.3) ‣ Standard Solutions ‣ §13.14(i) Differential Equation ‣ §13.14 Definitions and Basic Properties ‣ Whittaker Functions ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m42.png" altimg-height="24px" altimg-valign="-8px" altimg-width="77px" alttext="W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./13.14#E3">W</mi><mrow><mi class="ltx_nvar">κ</mi><mo href="./13.14#E3">,</mo><mi class="ltx_nvar">μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Whittaker confluent hypergeometric function</a>,
<a href="./1.4#E4" title="(1.4.4) ‣ §1.4(iii) Derivatives ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="29px" altimg-valign="-9px" altimg-width="27px" alttext="\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: derivative of <math class="ltx_Math" altimg="m47.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m45.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./13.1#p1.t1.r2" title="§13.1 Special Notation ‣ Notation ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./13.1#p1.t1.r2">n</mi></math>: nonnegative integer</a> and
<a href="./13.1#p1.t1.r4" title="§13.1 Special Notation ‣ Notation ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./13.1#p1.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E23">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">13.15.23</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m40.png" altimg-height="54px" altimg-valign="-21px" altimg-width="269px" alttext="\displaystyle\left(z\frac{\mathrm{d}}{\mathrm{d}z}z\right)^{n}\left(e^{\frac{1%
}{2}z}z^{-\kappa-1}W_{\kappa,\mu}\left(z\right)\right)" display="inline"><mrow><msup><mrow><mo>(</mo><mrow><mi href="./13.1#p1.t1.r4">z</mi><mo></mo><mrow><mstyle displaystyle="true"><mfrac><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi href="./13.1#p1.t1.r4">z</mi></mrow></mfrac></mstyle><mo></mo><mi href="./13.1#p1.t1.r4">z</mi></mrow></mrow><mo>)</mo></mrow><mi href="./13.1#p1.t1.r2">n</mi></msup><mo></mo><mrow><mo>(</mo><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./13.1#p1.t1.r4">z</mi></mrow></msup><mo></mo><msup><mi href="./13.1#p1.t1.r4">z</mi><mrow><mrow><mo>-</mo><mi>κ</mi></mrow><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mrow><msub><mi href="./13.14#E3">W</mi><mrow><mi>κ</mi><mo href="./13.14#E3">,</mo><mi>μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m25.png" altimg-height="34px" altimg-valign="-10px" altimg-width="452px" alttext="\displaystyle={\left(\tfrac{1}{2}+\mu-\kappa\right)_{n}}{\left(\tfrac{1}{2}-%
\mu-\kappa\right)_{n}}e^{\frac{1}{2}z}z^{n-\kappa-1}W_{\kappa-n,\mu}\left(z%
\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msub><mrow><mo href="./5.2#iii">(</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>+</mo><mi>μ</mi></mrow><mo>-</mo><mi>κ</mi></mrow><mo href="./5.2#iii">)</mo></mrow><mi href="./13.1#p1.t1.r2">n</mi></msub><mo></mo><msub><mrow><mo href="./5.2#iii">(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>-</mo><mi>μ</mi><mo>-</mo><mi>κ</mi></mrow><mo href="./5.2#iii">)</mo></mrow><mi href="./13.1#p1.t1.r2">n</mi></msub><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./13.1#p1.t1.r4">z</mi></mrow></msup><mo></mo><msup><mi href="./13.1#p1.t1.r4">z</mi><mrow><mi href="./13.1#p1.t1.r2">n</mi><mo>-</mo><mi>κ</mi><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mrow><msub><mi href="./13.14#E3">W</mi><mrow><mrow><mi>κ</mi><mo>-</mo><mi href="./13.1#p1.t1.r2">n</mi></mrow><mo href="./13.14#E3">,</mo><mi>μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E23.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#iii" title="§5.2(iii) Pochhammer’s Symbol ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="24px" altimg-valign="-8px" altimg-width="41px" alttext="{\left(\NVar{a}\right)_{\NVar{n}}}" display="inline"><msub><mrow><mo href="./5.2#iii">(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r5">a</mi><mo href="./5.2#iii">)</mo></mrow><mi class="ltx_nvar" href="./5.1#p2.t1.r1">n</mi></msub></math>: Pochhammer’s symbol (or shifted factorial)</a>,
<a href="./13.14#E3" title="(13.14.3) ‣ Standard Solutions ‣ §13.14(i) Differential Equation ‣ §13.14 Definitions and Basic Properties ‣ Whittaker Functions ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m42.png" altimg-height="24px" altimg-valign="-8px" altimg-width="77px" alttext="W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./13.14#E3">W</mi><mrow><mi class="ltx_nvar">κ</mi><mo href="./13.14#E3">,</mo><mi class="ltx_nvar">μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Whittaker confluent hypergeometric function</a>,
<a href="./1.4#E4" title="(1.4.4) ‣ §1.4(iii) Derivatives ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="29px" altimg-valign="-9px" altimg-width="27px" alttext="\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: derivative of <math class="ltx_Math" altimg="m47.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m45.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./13.1#p1.t1.r2" title="§13.1 Special Notation ‣ Notation ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./13.1#p1.t1.r2">n</mi></math>: nonnegative integer</a> and
<a href="./13.1#p1.t1.r4" title="§13.1 Special Notation ‣ Notation ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./13.1#p1.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E24">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">13.15.24</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m30.png" altimg-height="47px" altimg-valign="-16px" altimg-width="242px" alttext="\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(e^{-\frac{1}{2}z}%
z^{-\mu-\frac{1}{2}}W_{\kappa,\mu}\left(z\right)\right)" display="inline"><mrow><mstyle displaystyle="true"><mfrac><mpadded lspace="-1.7pt" width="-4.5pt"><msup><mo href="./1.4#E4">d</mo><mi href="./13.1#p1.t1.r2">n</mi></msup></mpadded><msup><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi href="./13.1#p1.t1.r4">z</mi></mrow><mi href="./1.4#E4">n</mi></msup></mfrac></mstyle><mo></mo><mrow><mo>(</mo><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./13.1#p1.t1.r4">z</mi></mrow></mrow></msup><mo></mo><msup><mi href="./13.1#p1.t1.r4">z</mi><mrow><mrow><mo>-</mo><mi>μ</mi></mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup><mo></mo><mrow><msub><mi href="./13.14#E3">W</mi><mrow><mi>κ</mi><mo href="./13.14#E3">,</mo><mi>μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m13.png" altimg-height="35px" altimg-valign="-12px" altimg-width="370px" alttext="\displaystyle=(-1)^{n}e^{-\frac{1}{2}z}z^{-\mu-\frac{1}{2}(n+1)}W_{\kappa+%
\frac{1}{2}n,\mu+\frac{1}{2}n}\left(z\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./13.1#p1.t1.r2">n</mi></msup><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./13.1#p1.t1.r4">z</mi></mrow></mrow></msup><mo></mo><msup><mi href="./13.1#p1.t1.r4">z</mi><mrow><mrow><mo>-</mo><mi>μ</mi></mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./13.1#p1.t1.r2">n</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></msup><mo></mo><mrow><msub><mi href="./13.14#E3">W</mi><mrow><mrow><mi>κ</mi><mo>+</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./13.1#p1.t1.r2">n</mi></mrow></mrow><mo href="./13.14#E3">,</mo><mrow><mi>μ</mi><mo>+</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./13.1#p1.t1.r2">n</mi></mrow></mrow></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E24.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./13.14#E3" title="(13.14.3) ‣ Standard Solutions ‣ §13.14(i) Differential Equation ‣ §13.14 Definitions and Basic Properties ‣ Whittaker Functions ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m42.png" altimg-height="24px" altimg-valign="-8px" altimg-width="77px" alttext="W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./13.14#E3">W</mi><mrow><mi class="ltx_nvar">κ</mi><mo href="./13.14#E3">,</mo><mi class="ltx_nvar">μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Whittaker confluent hypergeometric function</a>,
<a href="./1.4#E4" title="(1.4.4) ‣ §1.4(iii) Derivatives ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="29px" altimg-valign="-9px" altimg-width="27px" alttext="\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: derivative of <math class="ltx_Math" altimg="m47.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m45.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./13.1#p1.t1.r2" title="§13.1 Special Notation ‣ Notation ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./13.1#p1.t1.r2">n</mi></math>: nonnegative integer</a> and
<a href="./13.1#p1.t1.r4" title="§13.1 Special Notation ‣ Notation ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./13.1#p1.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E25">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">13.15.25</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m32.png" altimg-height="47px" altimg-valign="-16px" altimg-width="230px" alttext="\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(e^{-\frac{1}{2}z}%
z^{\mu-\frac{1}{2}}W_{\kappa,\mu}\left(z\right)\right)" display="inline"><mrow><mstyle displaystyle="true"><mfrac><mpadded lspace="-1.7pt" width="-4.5pt"><msup><mo href="./1.4#E4">d</mo><mi href="./13.1#p1.t1.r2">n</mi></msup></mpadded><msup><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi href="./13.1#p1.t1.r4">z</mi></mrow><mi href="./1.4#E4">n</mi></msup></mfrac></mstyle><mo></mo><mrow><mo>(</mo><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./13.1#p1.t1.r4">z</mi></mrow></mrow></msup><mo></mo><msup><mi href="./13.1#p1.t1.r4">z</mi><mrow><mi>μ</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup><mo></mo><mrow><msub><mi href="./13.14#E3">W</mi><mrow><mi>κ</mi><mo href="./13.14#E3">,</mo><mi>μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m15.png" altimg-height="35px" altimg-valign="-12px" altimg-width="358px" alttext="\displaystyle=(-1)^{n}e^{-\frac{1}{2}z}z^{\mu-\frac{1}{2}(n+1)}W_{\kappa+\frac%
{1}{2}n,\mu-\frac{1}{2}n}\left(z\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./13.1#p1.t1.r2">n</mi></msup><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./13.1#p1.t1.r4">z</mi></mrow></mrow></msup><mo></mo><msup><mi href="./13.1#p1.t1.r4">z</mi><mrow><mi>μ</mi><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./13.1#p1.t1.r2">n</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></msup><mo></mo><mrow><msub><mi href="./13.14#E3">W</mi><mrow><mrow><mi>κ</mi><mo>+</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./13.1#p1.t1.r2">n</mi></mrow></mrow><mo href="./13.14#E3">,</mo><mrow><mi>μ</mi><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./13.1#p1.t1.r2">n</mi></mrow></mrow></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E25.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./13.14#E3" title="(13.14.3) ‣ Standard Solutions ‣ §13.14(i) Differential Equation ‣ §13.14 Definitions and Basic Properties ‣ Whittaker Functions ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m42.png" altimg-height="24px" altimg-valign="-8px" altimg-width="77px" alttext="W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./13.14#E3">W</mi><mrow><mi class="ltx_nvar">κ</mi><mo href="./13.14#E3">,</mo><mi class="ltx_nvar">μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Whittaker confluent hypergeometric function</a>,
<a href="./1.4#E4" title="(1.4.4) ‣ §1.4(iii) Derivatives ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="29px" altimg-valign="-9px" altimg-width="27px" alttext="\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: derivative of <math class="ltx_Math" altimg="m47.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m45.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./13.1#p1.t1.r2" title="§13.1 Special Notation ‣ Notation ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./13.1#p1.t1.r2">n</mi></math>: nonnegative integer</a> and
<a href="./13.1#p1.t1.r4" title="§13.1 Special Notation ‣ Notation ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./13.1#p1.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E26">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">13.15.26</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m38.png" altimg-height="54px" altimg-valign="-21px" altimg-width="269px" alttext="\displaystyle\left(z\frac{\mathrm{d}}{\mathrm{d}z}z\right)^{n}\left(e^{-\frac{%
1}{2}z}z^{\kappa-1}W_{\kappa,\mu}\left(z\right)\right)" display="inline"><mrow><msup><mrow><mo>(</mo><mrow><mi href="./13.1#p1.t1.r4">z</mi><mo></mo><mrow><mstyle displaystyle="true"><mfrac><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi href="./13.1#p1.t1.r4">z</mi></mrow></mfrac></mstyle><mo></mo><mi href="./13.1#p1.t1.r4">z</mi></mrow></mrow><mo>)</mo></mrow><mi href="./13.1#p1.t1.r2">n</mi></msup><mo></mo><mrow><mo>(</mo><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./13.1#p1.t1.r4">z</mi></mrow></mrow></msup><mo></mo><msup><mi href="./13.1#p1.t1.r4">z</mi><mrow><mi>κ</mi><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mrow><msub><mi href="./13.14#E3">W</mi><mrow><mi>κ</mi><mo href="./13.14#E3">,</mo><mi>μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m14.png" altimg-height="31px" altimg-valign="-8px" altimg-width="288px" alttext="\displaystyle=(-1)^{n}e^{-\frac{1}{2}z}z^{\kappa+n-1}W_{\kappa+n,\mu}\left(z%
\right)." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./13.1#p1.t1.r2">n</mi></msup><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./13.1#p1.t1.r4">z</mi></mrow></mrow></msup><mo></mo><msup><mi href="./13.1#p1.t1.r4">z</mi><mrow><mrow><mi>κ</mi><mo>+</mo><mi href="./13.1#p1.t1.r2">n</mi></mrow><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mrow><msub><mi href="./13.14#E3">W</mi><mrow><mrow><mi>κ</mi><mo>+</mo><mi href="./13.1#p1.t1.r2">n</mi></mrow><mo href="./13.14#E3">,</mo><mi>μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E26.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./13.14#E3" title="(13.14.3) ‣ Standard Solutions ‣ §13.14(i) Differential Equation ‣ §13.14 Definitions and Basic Properties ‣ Whittaker Functions ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m42.png" altimg-height="24px" altimg-valign="-8px" altimg-width="77px" alttext="W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./13.14#E3">W</mi><mrow><mi class="ltx_nvar">κ</mi><mo href="./13.14#E3">,</mo><mi class="ltx_nvar">μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Whittaker confluent hypergeometric function</a>,
<a href="./1.4#E4" title="(1.4.4) ‣ §1.4(iii) Derivatives ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="29px" altimg-valign="-9px" altimg-width="27px" alttext="\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: derivative of <math class="ltx_Math" altimg="m47.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m45.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./13.1#p1.t1.r2" title="§13.1 Special Notation ‣ Notation ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./13.1#p1.t1.r2">n</mi></math>: nonnegative integer</a> and
<a href="./13.1#p1.t1.r4" title="§13.1 Special Notation ‣ Notation ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./13.1#p1.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
<p class="ltx_p">Other versions of several of the identities in this subsection can be constructed by use of
(</div>
</div>
</body></text>
</html>
</page>
<page>
<!DOCTYPE html><html>
<head>
<title>DLMF: 13.23 Integrals</title>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
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<div class="ltx_page_navlogo">.</p>
<table id="E1" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">13.23.1</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E1.png" altimg-height="66px" altimg-valign="-31px" altimg-width="684px" alttext="\int_{0}^{\infty}e^{-zt}t^{\nu-1}M_{\kappa,\mu}\left(t\right)\mathrm{d}t=\frac%
{\Gamma\left(\mu+\nu+\tfrac{1}{2}\right)}{\left(z+\frac{1}{2}\right)^{\mu+\nu+%
\frac{1}{2}}}\*{{}_{2}F_{1}}\left({\tfrac{1}{2}+\mu-\kappa,\tfrac{1}{2}+\mu+%
\nu\atop 1+2\mu};\frac{1}{z+\frac{1}{2}}\right)," display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mi href="./13.1#p1.t1.r4">z</mi><mo></mo><mi>t</mi></mrow></mrow></msup><mo></mo><msup><mi>t</mi><mrow><mi>ν</mi><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mrow><msub><mi href="./13.14#E2">M</mi><mrow><mi>κ</mi><mo href="./13.14#E2">,</mo><mi>μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow><mo>=</mo><mrow><mfrac><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi>μ</mi><mo>+</mo><mi>ν</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><msup><mrow><mo>(</mo><mrow><mi href="./13.1#p1.t1.r4">z</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow><mrow><mi>μ</mi><mo>+</mo><mi>ν</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup></mfrac><mo></mo><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>+</mo><mi>μ</mi></mrow><mo>-</mo><mi>κ</mi></mrow><mo>,</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>+</mo><mi>μ</mi><mo>+</mo><mi>ν</mi></mrow></mrow><mrow><mn>1</mn><mo>+</mo><mrow><mn>2</mn><mo></mo><mi>μ</mi></mrow></mrow></mfrac><mo>;</mo><mfrac><mn>1</mn><mrow><mi href="./13.1#p1.t1.r4">z</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mfrac><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m20.png" altimg-height="27px" altimg-valign="-9px" altimg-width="155px" alttext="\Re(\mu+\nu+\tfrac{1}{2})>0" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>μ</mi><mo>+</mo><mi>ν</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>></mo><mn>0</mn></mrow></math>,
<math class="ltx_Math" altimg="m18.png" altimg-height="27px" altimg-valign="-9px" altimg-width="68px" alttext="\Re z>\tfrac{1}{2}" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./13.1#p1.t1.r4">z</mi></mrow><mo>></mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m15.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./16.2" title="§16.2 Definition and Analytic Properties ‣ Generalized Hypergeometric Functions ‣ Chapter 16 Generalized Hypergeometric Functions and Meijer G -Function" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="23px" altimg-valign="-7px" altimg-width="118px" alttext="{{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mrow><mi class="ltx_nvar" href="./16.1#p2.t1.r5">a</mi><mo>,</mo><mi class="ltx_nvar" href="./16.1#p2.t1.r5">b</mi></mrow><mo>;</mo><mi class="ltx_nvar">c</mi><mo>;</mo><mi class="ltx_nvar" href="./16.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: <math class="ltx_Math" altimg="m8.png" altimg-height="23px" altimg-valign="-7px" altimg-width="124px" alttext="=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./16.1#p2.t1.r5">a</mi><mo>,</mo><mi class="ltx_nvar" href="./16.1#p2.t1.r5">b</mi><mo>;</mo><mi class="ltx_nvar">c</mi><mo>;</mo><mi class="ltx_nvar" href="./16.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></math> notation for Gauss’ hypergeometric function</a>,
<a href="./13.14#E2" title="(13.14.2) ‣ Standard Solutions ‣ §13.14(i) Differential Equation ‣ §13.14 Definitions and Basic Properties ‣ Whittaker Functions ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m12.png" altimg-height="24px" altimg-valign="-8px" altimg-width="77px" alttext="M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./13.14#E2">M</mi><mrow><mi class="ltx_nvar">κ</mi><mo href="./13.14#E2">,</mo><mi class="ltx_nvar">μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Whittaker confluent hypergeometric function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m28.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m40.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m26.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a> and
<a href="./13.1#p1.t1.r4" title="§13.1 Special Notation ‣ Notation ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./13.1#p1.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E2" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">13.23.2</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E2.png" altimg-height="53px" altimg-valign="-20px" altimg-width="601px" alttext="\int_{0}^{\infty}e^{-zt}t^{\mu-\frac{1}{2}}M_{\kappa,\mu}\left(t\right)\mathrm%
{d}t=\Gamma\left(2\mu+1\right)\left(z+\tfrac{1}{2}\right)^{-\kappa-\mu-\frac{1%
}{2}}\*\left(z-\tfrac{1}{2}\right)^{\kappa-\mu-\frac{1}{2}}," display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mi href="./13.1#p1.t1.r4">z</mi><mo></mo><mi>t</mi></mrow></mrow></msup><mo></mo><msup><mi>t</mi><mrow><mi>μ</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup><mo></mo><mrow><msub><mi href="./13.14#E2">M</mi><mrow><mi>κ</mi><mo href="./13.14#E2">,</mo><mi>μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow><mo>=</mo><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mn>2</mn><mo></mo><mi>μ</mi></mrow><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mo></mo><msup><mrow><mo>(</mo><mrow><mi href="./13.1#p1.t1.r4">z</mi><mo>+</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle></mrow><mo>)</mo></mrow><mrow><mrow><mo>-</mo><mi>κ</mi></mrow><mo>-</mo><mi>μ</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup><mo></mo><msup><mrow><mo>(</mo><mrow><mi href="./13.1#p1.t1.r4">z</mi><mo>-</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle></mrow><mo>)</mo></mrow><mrow><mi>κ</mi><mo>-</mo><mi>μ</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m23.png" altimg-height="27px" altimg-valign="-9px" altimg-width="85px" alttext="\Re\mu>-\tfrac{1}{2}" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>μ</mi></mrow><mo>></mo><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow></math>,
<math class="ltx_Math" altimg="m18.png" altimg-height="27px" altimg-valign="-9px" altimg-width="68px" alttext="\Re z>\tfrac{1}{2}" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./13.1#p1.t1.r4">z</mi></mrow><mo>></mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m15.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./13.14#E2" title="(13.14.2) ‣ Standard Solutions ‣ §13.14(i) Differential Equation ‣ §13.14 Definitions and Basic Properties ‣ Whittaker Functions ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m12.png" altimg-height="24px" altimg-valign="-8px" altimg-width="77px" alttext="M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./13.14#E2">M</mi><mrow><mi class="ltx_nvar">κ</mi><mo href="./13.14#E2">,</mo><mi class="ltx_nvar">μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Whittaker confluent hypergeometric function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m28.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m40.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m26.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a> and
<a href="./13.1#p1.t1.r4" title="§13.1 Special Notation ‣ Notation ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./13.1#p1.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E3" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">13.23.3</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E3.png" altimg-height="59px" altimg-valign="-24px" altimg-width="589px" alttext="\frac{1}{\Gamma\left(1+2\mu\right)}\int_{0}^{\infty}e^{-\frac{1}{2}t}t^{\nu-1}%
M_{\kappa,\mu}\left(t\right)\mathrm{d}t=\frac{\Gamma\left(\mu+\nu+\frac{1}{2}%
\right)\Gamma\left(\kappa-\nu\right)}{\Gamma\left(\frac{1}{2}+\mu+\kappa\right%
)\Gamma\left(\frac{1}{2}+\mu-\nu\right)}," display="block"><mrow><mrow><mrow><mfrac><mn>1</mn><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mrow><mn>2</mn><mo></mo><mi>μ</mi></mrow></mrow><mo>)</mo></mrow></mrow></mfrac><mo></mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi>t</mi></mrow></mrow></msup><mo></mo><msup><mi>t</mi><mrow><mi>ν</mi><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mrow><msub><mi href="./13.14#E2">M</mi><mrow><mi>κ</mi><mo href="./13.14#E2">,</mo><mi>μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow><mo>=</mo><mfrac><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi>μ</mi><mo>+</mo><mi>ν</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi>κ</mi><mo>-</mo><mi>ν</mi></mrow><mo>)</mo></mrow></mrow></mrow><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>+</mo><mi>μ</mi><mo>+</mo><mi>κ</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>+</mo><mi>μ</mi></mrow><mo>-</mo><mi>ν</mi></mrow><mo>)</mo></mrow></mrow></mrow></mfrac></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m6.png" altimg-height="27px" altimg-valign="-9px" altimg-width="188px" alttext="-\tfrac{1}{2}-\Re\mu<\Re\nu<\Re\kappa" display="inline"><mrow><mrow><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>-</mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>μ</mi></mrow></mrow><mo><</mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>ν</mi></mrow><mo><</mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>κ</mi></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m15.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./13.14#E2" title="(13.14.2) ‣ Standard Solutions ‣ §13.14(i) Differential Equation ‣ §13.14 Definitions and Basic Properties ‣ Whittaker Functions ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m12.png" altimg-height="24px" altimg-valign="-8px" altimg-width="77px" alttext="M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./13.14#E2">M</mi><mrow><mi class="ltx_nvar">κ</mi><mo href="./13.14#E2">,</mo><mi class="ltx_nvar">μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Whittaker confluent hypergeometric function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m28.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m40.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m26.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a> and
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E4" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">13.23.4</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E4.png" altimg-height="55px" altimg-valign="-21px" altimg-width="797px" alttext="\int_{0}^{\infty}e^{-zt}t^{\nu-1}W_{\kappa,\mu}\left(t\right)\mathrm{d}t=%
\Gamma\left(\tfrac{1}{2}+\mu+\nu\right)\Gamma\left(\tfrac{1}{2}-\mu+\nu\right)%
\*{{}_{2}{\mathbf{F}}_{1}}\left({\tfrac{1}{2}-\mu+\nu,\tfrac{1}{2}+\mu+\nu%
\atop\nu-\kappa+1};\tfrac{1}{2}-z\right)," display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mi href="./13.1#p1.t1.r4">z</mi><mo></mo><mi>t</mi></mrow></mrow></msup><mo></mo><msup><mi>t</mi><mrow><mi>ν</mi><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mrow><msub><mi href="./13.14#E3">W</mi><mrow><mi>κ</mi><mo href="./13.14#E3">,</mo><mi>μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow><mo>=</mo><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo>+</mo><mi>μ</mi><mo>+</mo><mi>ν</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo>-</mo><mi>μ</mi></mrow><mo>+</mo><mi>ν</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mmultiscripts><mi href="./16.2#E5" mathvariant="bold">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>-</mo><mi>μ</mi></mrow><mo>+</mo><mi>ν</mi></mrow><mo>,</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>+</mo><mi>μ</mi><mo>+</mo><mi>ν</mi></mrow></mrow><mrow><mrow><mi>ν</mi><mo>-</mo><mi>κ</mi></mrow><mo>+</mo><mn>1</mn></mrow></mfrac><mo>;</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo>-</mo><mi href="./13.1#p1.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m21.png" altimg-height="27px" altimg-valign="-9px" altimg-width="146px" alttext="\Re(\nu+\tfrac{1}{2})>|\Re\mu|" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>ν</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>></mo><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>μ</mi></mrow><mo stretchy="false">|</mo></mrow></mrow></math>,
<math class="ltx_Math" altimg="m16.png" altimg-height="27px" altimg-valign="-9px" altimg-width="84px" alttext="\Re z>-\tfrac{1}{2}" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./13.1#p1.t1.r4">z</mi></mrow><mo>></mo><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E4.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m15.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./13.14#E3" title="(13.14.3) ‣ Standard Solutions ‣ §13.14(i) Differential Equation ‣ §13.14 Definitions and Basic Properties ‣ Whittaker Functions ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m13.png" altimg-height="24px" altimg-valign="-8px" altimg-width="77px" alttext="W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./13.14#E3">W</mi><mrow><mi class="ltx_nvar">κ</mi><mo href="./13.14#E3">,</mo><mi class="ltx_nvar">μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Whittaker confluent hypergeometric function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m28.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m40.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./16.2#E5" title="(16.2.5) ‣ §16.2(v) Behavior with Respect to Parameters ‣ §16.2 Definition and Analytic Properties ‣ Generalized Hypergeometric Functions ‣ Chapter 16 Generalized Hypergeometric Functions and Meijer G -Function" class="ltx_ref"><math class="ltx_Math" altimg="m45.png" altimg-height="24px" altimg-valign="-8px" altimg-width="107px" alttext="{{}_{\NVar{p}}{\mathbf{F}}_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}%
};\NVar{z}\right)" display="inline"><mrow><mmultiscripts><mi href="./16.2#E5" mathvariant="bold">F</mi><mi class="ltx_nvar" href="./16.1#p2.t1.r1">q</mi><none></none><mprescripts></mprescripts><mi class="ltx_nvar" href="./16.1#p2.t1.r1">p</mi><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" mathvariant="bold">a</mi><mo>;</mo><mi class="ltx_nvar" mathvariant="bold">b</mi><mo>;</mo><mi class="ltx_nvar" href="./16.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m46.png" altimg-height="27px" altimg-valign="-9px" altimg-width="92px" alttext="{{}_{\NVar{p}}{\mathbf{F}}_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{%
\mathbf{b}}};\NVar{z}\right)" display="inline"><mrow><mmultiscripts><mi href="./16.2#E5" mathvariant="bold">F</mi><mi class="ltx_nvar" href="./16.1#p2.t1.r1">q</mi><none></none><mprescripts></mprescripts><mi class="ltx_nvar" href="./16.1#p2.t1.r1">p</mi><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mi class="ltx_nvar" mathvariant="bold">a</mi><mi class="ltx_nvar" mathvariant="bold">b</mi></mfrac><mo>;</mo><mi class="ltx_nvar" href="./16.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: scaled (or Olver’s) generalized hypergeometric function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m26.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a> and
<a href="./13.1#p1.t1.r4" title="§13.1 Special Notation ‣ Notation ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./13.1#p1.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E5" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">13.23.5</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E5.png" altimg-height="59px" altimg-valign="-24px" altimg-width="577px" alttext="\int_{0}^{\infty}e^{\frac{1}{2}t}t^{\nu-1}W_{\kappa,\mu}\left(t\right)\mathrm{%
d}t=\frac{\Gamma\left(\frac{1}{2}+\mu+\nu\right)\Gamma\left(\frac{1}{2}-\mu+%
\nu\right)\Gamma\left(-\kappa-\nu\right)}{\Gamma\left(\frac{1}{2}+\mu-\kappa%
\right)\Gamma\left(\frac{1}{2}-\mu-\kappa\right)}," display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi>t</mi></mrow></msup><mo></mo><msup><mi>t</mi><mrow><mi>ν</mi><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mrow><msub><mi href="./13.14#E3">W</mi><mrow><mi>κ</mi><mo href="./13.14#E3">,</mo><mi>μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow><mo>=</mo><mfrac><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>+</mo><mi>μ</mi><mo>+</mo><mi>ν</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>-</mo><mi>μ</mi></mrow><mo>+</mo><mi>ν</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mo>-</mo><mi>κ</mi></mrow><mo>-</mo><mi>ν</mi></mrow><mo>)</mo></mrow></mrow></mrow><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>+</mo><mi>μ</mi></mrow><mo>-</mo><mi>κ</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>-</mo><mi>μ</mi><mo>-</mo><mi>κ</mi></mrow><mo>)</mo></mrow></mrow></mrow></mfrac></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m47.png" altimg-height="27px" altimg-valign="-9px" altimg-width="199px" alttext="|\Re\mu|-\tfrac{1}{2}<\Re\nu<-\Re\kappa" display="inline"><mrow><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>μ</mi></mrow><mo stretchy="false">|</mo></mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo><</mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>ν</mi></mrow><mo><</mo><mrow><mo>-</mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>κ</mi></mrow></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E5.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m15.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./13.14#E3" title="(13.14.3) ‣ Standard Solutions ‣ §13.14(i) Differential Equation ‣ §13.14 Definitions and Basic Properties ‣ Whittaker Functions ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m13.png" altimg-height="24px" altimg-valign="-8px" altimg-width="77px" alttext="W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./13.14#E3">W</mi><mrow><mi class="ltx_nvar">κ</mi><mo href="./13.14#E3">,</mo><mi class="ltx_nvar">μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Whittaker confluent hypergeometric function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m28.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m40.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m26.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a> and
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E6" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">13.23.6</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E6.png" altimg-height="60px" altimg-valign="-24px" altimg-width="641px" alttext="\frac{1}{\Gamma\left(1+2\mu\right)2\pi\mathrm{i}}\int_{-\infty}^{(0+)}e^{zt+%
\frac{1}{2}t^{-1}}t^{\kappa}M_{\kappa,\mu}\left(t^{-1}\right)\mathrm{d}t=\frac%
{z^{-\kappa-\frac{1}{2}}}{\Gamma\left(\frac{1}{2}+\mu-\kappa\right)}I_{2\mu}%
\left(2\sqrt{z}\right)," display="block"><mrow><mrow><mrow><mfrac><mn>1</mn><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mrow><mn>2</mn><mo></mo><mi>μ</mi></mrow></mrow><mo>)</mo></mrow></mrow><mo></mo><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi></mrow></mfrac><mo></mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mo stretchy="false">(</mo><mrow><mn>0</mn><mo>+</mo></mrow><mo stretchy="false">)</mo></mrow></msubsup><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mrow><mi href="./13.1#p1.t1.r4">z</mi><mo></mo><mi>t</mi></mrow><mo>+</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><msup><mi>t</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></mrow></mrow></msup><mo></mo><msup><mi>t</mi><mi>κ</mi></msup><mo></mo><mrow><msub><mi href="./13.14#E2">M</mi><mrow><mi>κ</mi><mo href="./13.14#E2">,</mo><mi>μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><msup><mi>t</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow><mo>=</mo><mrow><mfrac><msup><mi href="./13.1#p1.t1.r4">z</mi><mrow><mrow><mo>-</mo><mi>κ</mi></mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>+</mo><mi>μ</mi></mrow><mo>-</mo><mi>κ</mi></mrow><mo>)</mo></mrow></mrow></mfrac><mo></mo><mrow><msub><mi href="./10.25#E2">I</mi><mrow><mn>2</mn><mo></mo><mi>μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><msqrt><mi href="./13.1#p1.t1.r4">z</mi></msqrt></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m17.png" altimg-height="18px" altimg-valign="-3px" altimg-width="65px" alttext="\Re z>0" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./13.1#p1.t1.r4">z</mi></mrow><mo>></mo><mn>0</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E6.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m15.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./13.14#E2" title="(13.14.2) ‣ Standard Solutions ‣ §13.14(i) Differential Equation ‣ §13.14 Definitions and Basic Properties ‣ Whittaker Functions ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m12.png" altimg-height="24px" altimg-valign="-8px" altimg-width="77px" alttext="M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./13.14#E2">M</mi><mrow><mi class="ltx_nvar">κ</mi><mo href="./13.14#E2">,</mo><mi class="ltx_nvar">μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Whittaker confluent hypergeometric function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m33.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m28.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m40.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m26.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./10.25#E2" title="(10.25.2) ‣ §10.25(ii) Standard Solutions ‣ §10.25 Definitions ‣ Modified Bessel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m9.png" altimg-height="23px" altimg-valign="-7px" altimg-width="52px" alttext="I_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.25#E2">I</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: modified Bessel function of the first kind</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a> and
<a href="./13.1#p1.t1.r4" title="§13.1 Special Notation ‣ Notation ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./13.1#p1.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E7" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">13.23.7</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E7.png" altimg-height="60px" altimg-valign="-24px" altimg-width="680px" alttext="\frac{1}{2\pi\mathrm{i}}\int_{-\infty}^{(0+)}e^{zt+\frac{1}{2}t^{-1}}t^{\kappa%
}W_{\kappa,\mu}\left(t^{-1}\right)\mathrm{d}t=\frac{2z^{-\kappa-\frac{1}{2}}}{%
\Gamma\left(\frac{1}{2}+\mu-\kappa\right)\Gamma\left(\frac{1}{2}-\mu-\kappa%
\right)}K_{2\mu}\left(2\sqrt{z}\right)," display="block"><mrow><mrow><mrow><mfrac><mn>1</mn><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi></mrow></mfrac><mo></mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mo stretchy="false">(</mo><mrow><mn>0</mn><mo>+</mo></mrow><mo stretchy="false">)</mo></mrow></msubsup><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mrow><mi href="./13.1#p1.t1.r4">z</mi><mo></mo><mi>t</mi></mrow><mo>+</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><msup><mi>t</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></mrow></mrow></msup><mo></mo><msup><mi>t</mi><mi>κ</mi></msup><mo></mo><mrow><msub><mi href="./13.14#E3">W</mi><mrow><mi>κ</mi><mo href="./13.14#E3">,</mo><mi>μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><msup><mi>t</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow><mo>=</mo><mrow><mfrac><mrow><mn>2</mn><mo></mo><msup><mi href="./13.1#p1.t1.r4">z</mi><mrow><mrow><mo>-</mo><mi>κ</mi></mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup></mrow><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>+</mo><mi>μ</mi></mrow><mo>-</mo><mi>κ</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>-</mo><mi>μ</mi><mo>-</mo><mi>κ</mi></mrow><mo>)</mo></mrow></mrow></mrow></mfrac><mo></mo><mrow><msub><mi href="./10.25#E3">K</mi><mrow><mn>2</mn><mo></mo><mi>μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><msqrt><mi href="./13.1#p1.t1.r4">z</mi></msqrt></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m17.png" altimg-height="18px" altimg-valign="-3px" altimg-width="65px" alttext="\Re z>0" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./13.1#p1.t1.r4">z</mi></mrow><mo>></mo><mn>0</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E7.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m15.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./13.14#E3" title="(13.14.3) ‣ Standard Solutions ‣ §13.14(i) Differential Equation ‣ §13.14 Definitions and Basic Properties ‣ Whittaker Functions ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m13.png" altimg-height="24px" altimg-valign="-8px" altimg-width="77px" alttext="W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./13.14#E3">W</mi><mrow><mi class="ltx_nvar">κ</mi><mo href="./13.14#E3">,</mo><mi class="ltx_nvar">μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Whittaker confluent hypergeometric function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m33.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m28.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m40.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m26.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./10.25#E3" title="(10.25.3) ‣ §10.25(ii) Standard Solutions ‣ §10.25 Definitions ‣ Modified Bessel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m11.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="K_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.25#E3">K</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: modified Bessel function of the second kind</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a> and
<a href="./13.1#p1.t1.r4" title="§13.1 Special Notation ‣ Notation ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./13.1#p1.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS1.p2" class="ltx_para">
<p class="ltx_p">For additional Laplace and Mellin transforms see
<cite class="ltx_cite ltx_citemacro_citet">Erdélyi<span class="ltx_text ltx_bib_etal"> et al.</span> (</dd>
</dl>
</div>
</div>
<div id="SS2.p1" class="ltx_para">
<table id="E8" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">13.23.8</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_Math ltx_eqn_cell ltx_align_center"><math class="ltx_Math" alttext="\frac{1}{\Gamma\left(1+2\mu\right)}\int_{0}^{\infty}\cos\left(2xt\right)e^{-%
\frac{1}{2}t^{2}}t^{-2\mu-1}M_{\kappa,\mu}\left(t^{2}\right)\mathrm{d}t=\frac{%
\sqrt{\pi}e^{-\frac{1}{2}x^{2}}x^{\mu+\kappa-1}}{2\Gamma\left(\frac{1}{2}+\mu+%
\kappa\right)}W_{\frac{1}{2}\kappa-\frac{3}{2}\mu,\frac{1}{2}\kappa+\frac{1}{2%
}\mu}\left(x^{2}\right)," display="block"><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mrow><mfrac><mn>1</mn><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mrow><mn>2</mn><mo></mo><mi>μ</mi></mrow></mrow><mo>)</mo></mrow></mrow></mfrac><mo></mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><mrow><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><mi href="./13.1#p1.t1.r3">x</mi><mo></mo><mi>t</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mrow></msup><mo></mo><msup><mi>t</mi><mrow><mrow><mo>-</mo><mrow><mn>2</mn><mo></mo><mi>μ</mi></mrow></mrow><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mrow><msub><mi href="./13.14#E2">M</mi><mrow><mi>κ</mi><mo href="./13.14#E2">,</mo><mi>μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><msup><mi>t</mi><mn>2</mn></msup><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>=</mo><mrow><mfrac><mrow><msqrt><mi href="./3.12#E1">π</mi></msqrt><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><msup><mi href="./13.1#p1.t1.r3">x</mi><mn>2</mn></msup></mrow></mrow></msup><mo></mo><msup><mi href="./13.1#p1.t1.r3">x</mi><mrow><mrow><mi>μ</mi><mo>+</mo><mi>κ</mi></mrow><mo>-</mo><mn>1</mn></mrow></msup></mrow><mrow><mn>2</mn><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>+</mo><mi>μ</mi><mo>+</mo><mi>κ</mi></mrow><mo>)</mo></mrow></mrow></mrow></mfrac><mo></mo><mrow><msub><mi href="./13.14#E3">W</mi><mrow><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi>κ</mi></mrow><mo>-</mo><mrow><mfrac><mn>3</mn><mn>2</mn></mfrac><mo></mo><mi>μ</mi></mrow></mrow><mo href="./13.14#E3">,</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi>κ</mi></mrow><mo>+</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi>μ</mi></mrow></mrow></mrow></msub><mo></mo><mrow><mo>(</mo><msup><mi href="./13.1#p1.t1.r3">x</mi><mn>2</mn></msup><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mtd></mtr></mtable></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m19.png" altimg-height="27px" altimg-valign="-9px" altimg-width="137px" alttext="\Re(\kappa+\mu)>-\tfrac{1}{2}" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>κ</mi><mo>+</mo><mi>μ</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>></mo><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E8.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m15.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./13.14#E2" title="(13.14.2) ‣ Standard Solutions ‣ §13.14(i) Differential Equation ‣ §13.14 Definitions and Basic Properties ‣ Whittaker Functions ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m12.png" altimg-height="24px" altimg-valign="-8px" altimg-width="77px" alttext="M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./13.14#E2">M</mi><mrow><mi class="ltx_nvar">κ</mi><mo href="./13.14#E2">,</mo><mi class="ltx_nvar">μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Whittaker confluent hypergeometric function</a>,
<a href="./13.14#E3" title="(13.14.3) ‣ Standard Solutions ‣ §13.14(i) Differential Equation ‣ §13.14 Definitions and Basic Properties ‣ Whittaker Functions ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m13.png" altimg-height="24px" altimg-valign="-8px" altimg-width="77px" alttext="W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./13.14#E3">W</mi><mrow><mi class="ltx_nvar">κ</mi><mo href="./13.14#E3">,</mo><mi class="ltx_nvar">μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Whittaker confluent hypergeometric function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m33.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m25.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m28.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m40.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m26.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a> and
<a href="./13.1#p1.t1.r3" title="§13.1 Special Notation ‣ Notation ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m40.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./13.1#p1.t1.r3">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS2.p2" class="ltx_para">
<p class="ltx_p">For additional Fourier transforms see <cite class="ltx_cite ltx_citemacro_citet">Erdélyi<span class="ltx_text ltx_bib_etal"> et al.</span> (.</p>
<table id="E9" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">13.23.9</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_Math ltx_eqn_cell ltx_align_center"><math class="ltx_Math" alttext="\int_{0}^{\infty}e^{-\frac{1}{2}t}t^{\mu-\frac{1}{2}(\nu+1)}M_{\kappa,\mu}%
\left(t\right)J_{\nu}\left(2\sqrt{xt}\right)\mathrm{d}t=\frac{\Gamma\left(1+2%
\mu\right)}{\Gamma\left(\frac{1}{2}-\mu+\kappa+\nu\right)}\*e^{-\frac{1}{2}x}x%
^{\frac{1}{2}(\kappa-\mu-\frac{3}{2})}\*M_{\frac{1}{2}(\kappa+3\mu-\nu+\frac{1%
}{2}),\frac{1}{2}(\kappa-\mu+\nu-\frac{1}{2})}\left(x\right)," display="block"><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi>t</mi></mrow></mrow></msup><mo></mo><msup><mi>t</mi><mrow><mi>μ</mi><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>ν</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></msup><mo></mo><mrow><msub><mi href="./13.14#E2">M</mi><mrow><mi>κ</mi><mo href="./13.14#E2">,</mo><mi>μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mi>ν</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><msqrt><mrow><mi href="./13.1#p1.t1.r3">x</mi><mo></mo><mi>t</mi></mrow></msqrt></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>=</mo><mrow><mfrac><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mrow><mn>2</mn><mo></mo><mi>μ</mi></mrow></mrow><mo>)</mo></mrow></mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>-</mo><mi>μ</mi></mrow><mo>+</mo><mi>κ</mi><mo>+</mo><mi>ν</mi></mrow><mo>)</mo></mrow></mrow></mfrac><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./13.1#p1.t1.r3">x</mi></mrow></mrow></msup><mo></mo><msup><mi href="./13.1#p1.t1.r3">x</mi><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>κ</mi><mo>-</mo><mi>μ</mi><mo>-</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow><mo stretchy="false">)</mo></mrow></mrow></msup><mo></mo><mrow><msub><mi href="./13.14#E2">M</mi><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mrow><mi>κ</mi><mo>+</mo><mrow><mn>3</mn><mo></mo><mi>μ</mi></mrow></mrow><mo>-</mo><mi>ν</mi></mrow><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo stretchy="false">)</mo></mrow></mrow><mo href="./13.14#E2">,</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mrow><mi>κ</mi><mo>-</mo><mi>μ</mi></mrow><mo>+</mo><mi>ν</mi></mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./13.1#p1.t1.r3">x</mi><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mtd></mtr></mtable></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m39.png" altimg-height="17px" altimg-valign="-3px" altimg-width="52px" alttext="x>0" display="inline"><mrow><mi href="./13.1#p1.t1.r3">x</mi><mo>></mo><mn>0</mn></mrow></math>, <math class="ltx_Math" altimg="m7.png" altimg-height="27px" altimg-valign="-9px" altimg-width="239px" alttext="-\tfrac{1}{2}<\Re\mu<\Re(\kappa+\tfrac{1}{2}\nu)+\tfrac{3}{4}" display="inline"><mrow><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo><</mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>μ</mi></mrow><mo><</mo><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>κ</mi><mo>+</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi>ν</mi></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>+</mo><mfrac><mn>3</mn><mn>4</mn></mfrac></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E9.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m10.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m15.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./13.14#E2" title="(13.14.2) ‣ Standard Solutions ‣ §13.14(i) Differential Equation ‣ §13.14 Definitions and Basic Properties ‣ Whittaker Functions ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m12.png" altimg-height="24px" altimg-valign="-8px" altimg-width="77px" alttext="M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./13.14#E2">M</mi><mrow><mi class="ltx_nvar">κ</mi><mo href="./13.14#E2">,</mo><mi class="ltx_nvar">μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Whittaker confluent hypergeometric function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m28.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m40.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m26.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a> and
<a href="./13.1#p1.t1.r3" title="§13.1 Special Notation ‣ Notation ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m40.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./13.1#p1.t1.r3">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E10" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">13.23.10</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_Math ltx_eqn_cell ltx_align_center"><math class="ltx_Math" alttext="\frac{1}{\Gamma\left(1+2\mu\right)}\int_{0}^{\infty}e^{-\frac{1}{2}t}t^{\frac{%
1}{2}(\nu-1)-\mu}M_{\kappa,\mu}\left(t\right)J_{\nu}\left(2\sqrt{xt}\right)%
\mathrm{d}t=\frac{e^{-\frac{1}{2}x}x^{\frac{1}{2}(\kappa+\mu-\frac{3}{2})}}{%
\Gamma\left(\frac{1}{2}+\mu+\kappa\right)}\*W_{\frac{1}{2}(\kappa-3\mu+\nu+%
\frac{1}{2}),\frac{1}{2}(\kappa+\mu-\nu-\frac{1}{2})}\left(x\right)," display="block"><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mrow><mfrac><mn>1</mn><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mrow><mn>2</mn><mo></mo><mi>μ</mi></mrow></mrow><mo>)</mo></mrow></mrow></mfrac><mo></mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi>t</mi></mrow></mrow></msup><mo></mo><msup><mi>t</mi><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>ν</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>-</mo><mi>μ</mi></mrow></msup><mo></mo><mrow><msub><mi href="./13.14#E2">M</mi><mrow><mi>κ</mi><mo href="./13.14#E2">,</mo><mi>μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mi>ν</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><msqrt><mrow><mi href="./13.1#p1.t1.r3">x</mi><mo></mo><mi>t</mi></mrow></msqrt></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>=</mo><mrow><mfrac><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./13.1#p1.t1.r3">x</mi></mrow></mrow></msup><mo></mo><msup><mi href="./13.1#p1.t1.r3">x</mi><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mi>κ</mi><mo>+</mo><mi>μ</mi></mrow><mo>-</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow><mo stretchy="false">)</mo></mrow></mrow></msup></mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>+</mo><mi>μ</mi><mo>+</mo><mi>κ</mi></mrow><mo>)</mo></mrow></mrow></mfrac><mo></mo><mrow><msub><mi href="./13.14#E3">W</mi><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mi>κ</mi><mo>-</mo><mrow><mn>3</mn><mo></mo><mi>μ</mi></mrow></mrow><mo>+</mo><mi>ν</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo stretchy="false">)</mo></mrow></mrow><mo href="./13.14#E3">,</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mi>κ</mi><mo>+</mo><mi>μ</mi></mrow><mo>-</mo><mi>ν</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./13.1#p1.t1.r3">x</mi><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mtd></mtr></mtable></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m39.png" altimg-height="17px" altimg-valign="-3px" altimg-width="52px" alttext="x>0" display="inline"><mrow><mi href="./13.1#p1.t1.r3">x</mi><mo>></mo><mn>0</mn></mrow></math>, <math class="ltx_Math" altimg="m5.png" altimg-height="27px" altimg-valign="-9px" altimg-width="233px" alttext="-1<\Re\nu<2\Re(\mu+\kappa)+\tfrac{1}{2}" display="inline"><mrow><mrow><mo>-</mo><mn>1</mn></mrow><mo><</mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>ν</mi></mrow><mo><</mo><mrow><mrow><mn>2</mn><mo></mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>μ</mi><mo>+</mo><mi>κ</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E10.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m10.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m15.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./13.14#E2" title="(13.14.2) ‣ Standard Solutions ‣ §13.14(i) Differential Equation ‣ §13.14 Definitions and Basic Properties ‣ Whittaker Functions ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m12.png" altimg-height="24px" altimg-valign="-8px" altimg-width="77px" alttext="M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./13.14#E2">M</mi><mrow><mi class="ltx_nvar">κ</mi><mo href="./13.14#E2">,</mo><mi class="ltx_nvar">μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Whittaker confluent hypergeometric function</a>,
<a href="./13.14#E3" title="(13.14.3) ‣ Standard Solutions ‣ §13.14(i) Differential Equation ‣ §13.14 Definitions and Basic Properties ‣ Whittaker Functions ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m13.png" altimg-height="24px" altimg-valign="-8px" altimg-width="77px" alttext="W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./13.14#E3">W</mi><mrow><mi class="ltx_nvar">κ</mi><mo href="./13.14#E3">,</mo><mi class="ltx_nvar">μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Whittaker confluent hypergeometric function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m28.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m40.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m26.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a> and
<a href="./13.1#p1.t1.r3" title="§13.1 Special Notation ‣ Notation ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m40.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./13.1#p1.t1.r3">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E11" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">13.23.11</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_Math ltx_eqn_cell ltx_align_center"><math class="ltx_Math" alttext="\int_{0}^{\infty}e^{\frac{1}{2}t}t^{\frac{1}{2}(\nu-1)-\mu}W_{\kappa,\mu}\left%
(t\right)J_{\nu}\left(2\sqrt{xt}\right)\mathrm{d}t=\frac{\Gamma\left(\nu-2\mu+%
1\right)}{\Gamma\left(\frac{1}{2}+\mu-\kappa\right)}\*e^{\frac{1}{2}x}x^{\frac%
{1}{2}(\mu-\kappa-\frac{3}{2})}\*W_{\frac{1}{2}(\kappa+3\mu-\nu-\frac{1}{2}),%
\frac{1}{2}(\kappa-\mu+\nu+\frac{1}{2})}\left(x\right)," display="block"><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi>t</mi></mrow></msup><mo></mo><msup><mi>t</mi><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>ν</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>-</mo><mi>μ</mi></mrow></msup><mo></mo><mrow><msub><mi href="./13.14#E3">W</mi><mrow><mi>κ</mi><mo href="./13.14#E3">,</mo><mi>μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mi>ν</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><msqrt><mrow><mi href="./13.1#p1.t1.r3">x</mi><mo></mo><mi>t</mi></mrow></msqrt></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>=</mo><mrow><mfrac><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mi>ν</mi><mo>-</mo><mrow><mn>2</mn><mo></mo><mi>μ</mi></mrow></mrow><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>+</mo><mi>μ</mi></mrow><mo>-</mo><mi>κ</mi></mrow><mo>)</mo></mrow></mrow></mfrac><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./13.1#p1.t1.r3">x</mi></mrow></msup><mo></mo><msup><mi href="./13.1#p1.t1.r3">x</mi><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>μ</mi><mo>-</mo><mi>κ</mi><mo>-</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow><mo stretchy="false">)</mo></mrow></mrow></msup><mo></mo><mrow><msub><mi href="./13.14#E3">W</mi><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mi>κ</mi><mo>+</mo><mrow><mn>3</mn><mo></mo><mi>μ</mi></mrow></mrow><mo>-</mo><mi>ν</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo stretchy="false">)</mo></mrow></mrow><mo href="./13.14#E3">,</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mi>κ</mi><mo>-</mo><mi>μ</mi></mrow><mo>+</mo><mi>ν</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./13.1#p1.t1.r3">x</mi><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mtd></mtr></mtable></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m39.png" altimg-height="17px" altimg-valign="-3px" altimg-width="52px" alttext="x>0" display="inline"><mrow><mi href="./13.1#p1.t1.r3">x</mi><mo>></mo><mn>0</mn></mrow></math>, <math class="ltx_Math" altimg="m30.png" altimg-height="27px" altimg-valign="-9px" altimg-width="365px" alttext="\max(2\Re\mu-1,-1)<\Re\nu<2\Re(\mu-\kappa)+\tfrac{3}{2}" display="inline"><mrow><mrow><mi>max</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>2</mn><mo></mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>μ</mi></mrow></mrow><mo>-</mo><mn>1</mn></mrow><mo>,</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mo><</mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>ν</mi></mrow><mo><</mo><mrow><mrow><mn>2</mn><mo></mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>μ</mi><mo>-</mo><mi>κ</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>+</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E11.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m10.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m15.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./13.14#E3" title="(13.14.3) ‣ Standard Solutions ‣ §13.14(i) Differential Equation ‣ §13.14 Definitions and Basic Properties ‣ Whittaker Functions ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m13.png" altimg-height="24px" altimg-valign="-8px" altimg-width="77px" alttext="W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./13.14#E3">W</mi><mrow><mi class="ltx_nvar">κ</mi><mo href="./13.14#E3">,</mo><mi class="ltx_nvar">μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Whittaker confluent hypergeometric function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m28.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m40.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m26.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a> and
<a href="./13.1#p1.t1.r3" title="§13.1 Special Notation ‣ Notation ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m40.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./13.1#p1.t1.r3">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E12" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">13.23.12</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_Math ltx_eqn_cell ltx_align_center"><math class="ltx_Math" alttext="\int_{0}^{\infty}e^{-\frac{1}{2}t}t^{\frac{1}{2}(\nu-1)-\mu}W_{\kappa,\mu}%
\left(t\right)J_{\nu}\left(2\sqrt{xt}\right)\mathrm{d}t=\frac{\Gamma\left(\nu-%
2\mu+1\right)}{\Gamma\left(\frac{3}{2}-\mu-\kappa+\nu\right)}\*e^{-\frac{1}{2}%
x}x^{\frac{1}{2}(\mu+\kappa-\frac{3}{2})}\*M_{\frac{1}{2}(\kappa-3\mu+\nu+%
\frac{1}{2}),\frac{1}{2}(\nu-\mu-\kappa+\frac{1}{2})}\left(x\right)," display="block"><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi>t</mi></mrow></mrow></msup><mo></mo><msup><mi>t</mi><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>ν</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>-</mo><mi>μ</mi></mrow></msup><mo></mo><mrow><msub><mi href="./13.14#E3">W</mi><mrow><mi>κ</mi><mo href="./13.14#E3">,</mo><mi>μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mi>ν</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><msqrt><mrow><mi href="./13.1#p1.t1.r3">x</mi><mo></mo><mi>t</mi></mrow></msqrt></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>=</mo><mrow><mfrac><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mi>ν</mi><mo>-</mo><mrow><mn>2</mn><mo></mo><mi>μ</mi></mrow></mrow><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mfrac><mn>3</mn><mn>2</mn></mfrac><mo>-</mo><mi>μ</mi><mo>-</mo><mi>κ</mi></mrow><mo>+</mo><mi>ν</mi></mrow><mo>)</mo></mrow></mrow></mfrac><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./13.1#p1.t1.r3">x</mi></mrow></mrow></msup><mo></mo><msup><mi href="./13.1#p1.t1.r3">x</mi><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mi>μ</mi><mo>+</mo><mi>κ</mi></mrow><mo>-</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow><mo stretchy="false">)</mo></mrow></mrow></msup><mo></mo><mrow><msub><mi href="./13.14#E2">M</mi><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mi>κ</mi><mo>-</mo><mrow><mn>3</mn><mo></mo><mi>μ</mi></mrow></mrow><mo>+</mo><mi>ν</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo stretchy="false">)</mo></mrow></mrow><mo href="./13.14#E2">,</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mi>ν</mi><mo>-</mo><mi>μ</mi><mo>-</mo><mi>κ</mi></mrow><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./13.1#p1.t1.r3">x</mi><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mtd></mtr></mtable></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m39.png" altimg-height="17px" altimg-valign="-3px" altimg-width="52px" alttext="x>0" display="inline"><mrow><mi href="./13.1#p1.t1.r3">x</mi><mo>></mo><mn>0</mn></mrow></math>, <math class="ltx_Math" altimg="m31.png" altimg-height="23px" altimg-valign="-7px" altimg-width="214px" alttext="\max(2\Re\mu-1,-1)<\Re\nu" display="inline"><mrow><mrow><mi>max</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>2</mn><mo></mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>μ</mi></mrow></mrow><mo>-</mo><mn>1</mn></mrow><mo>,</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mo><</mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>ν</mi></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E12.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m10.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m15.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./13.14#E2" title="(13.14.2) ‣ Standard Solutions ‣ §13.14(i) Differential Equation ‣ §13.14 Definitions and Basic Properties ‣ Whittaker Functions ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m12.png" altimg-height="24px" altimg-valign="-8px" altimg-width="77px" alttext="M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./13.14#E2">M</mi><mrow><mi class="ltx_nvar">κ</mi><mo href="./13.14#E2">,</mo><mi class="ltx_nvar">μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Whittaker confluent hypergeometric function</a>,
<a href="./13.14#E3" title="(13.14.3) ‣ Standard Solutions ‣ §13.14(i) Differential Equation ‣ §13.14 Definitions and Basic Properties ‣ Whittaker Functions ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m13.png" altimg-height="24px" altimg-valign="-8px" altimg-width="77px" alttext="W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./13.14#E3">W</mi><mrow><mi class="ltx_nvar">κ</mi><mo href="./13.14#E3">,</mo><mi class="ltx_nvar">μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Whittaker confluent hypergeometric function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m28.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m40.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m26.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a> and
<a href="./13.1#p1.t1.r3" title="§13.1 Special Notation ‣ Notation ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m40.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./13.1#p1.t1.r3">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS3.p2" class="ltx_para">
<p class="ltx_p">For additional Hankel transforms and also other Bessel transforms
see <cite class="ltx_cite ltx_citemacro_citet">Erdélyi<span class="ltx_text ltx_bib_etal"> et al.</span> (</dd>
</dl>
</div>
</div>
<div id="SS4.p1" class="ltx_para">
<p class="ltx_p">Let <math class="ltx_Math" altimg="m37.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="f(x)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./13.1#p1.t1.r3">x</mi><mo stretchy="false">)</mo></mrow></mrow></math> be absolutely integrable on the interval <math class="ltx_Math" altimg="m14.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="[r,R]" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">[</mo><mi>r</mi><mo href="./front/introduction#Sx4.p1.t1.r29">,</mo><mi>R</mi><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">]</mo></mrow></math> for all positive <math class="ltx_Math" altimg="m38.png" altimg-height="18px" altimg-valign="-3px" altimg-width="56px" alttext="r<R" display="inline"><mrow><mi>r</mi><mo><</mo><mi>R</mi></mrow></math>,
<math class="ltx_Math" altimg="m36.png" altimg-height="23px" altimg-valign="-7px" altimg-width="133px" alttext="f(x)=O\left(x^{\rho_{0}}\right)" display="inline"><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./13.1#p1.t1.r3">x</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mi href="./2.1#E3">O</mi><mo></mo><mrow><mo>(</mo><msup><mi href="./13.1#p1.t1.r3">x</mi><msub><mi>ρ</mi><mn>0</mn></msub></msup><mo>)</mo></mrow></mrow></mrow></math> as <math class="ltx_Math" altimg="m41.png" altimg-height="18px" altimg-valign="-4px" altimg-width="72px" alttext="x\to 0+" display="inline"><mrow><mi href="./13.1#p1.t1.r3">x</mi><mo>→</mo><mrow><mn>0</mn><mo>+</mo></mrow></mrow></math>, and
<math class="ltx_Math" altimg="m35.png" altimg-height="24px" altimg-valign="-7px" altimg-width="152px" alttext="f(x)=O\left(e^{-\rho_{1}x}\right)" display="inline"><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./13.1#p1.t1.r3">x</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mi href="./2.1#E3">O</mi><mo></mo><mrow><mo>(</mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><msub><mi>ρ</mi><mn>1</mn></msub><mo></mo><mi href="./13.1#p1.t1.r3">x</mi></mrow></mrow></msup><mo>)</mo></mrow></mrow></mrow></math> as <math class="ltx_Math" altimg="m42.png" altimg-height="17px" altimg-valign="-4px" altimg-width="82px" alttext="x\to+\infty" display="inline"><mrow><mi href="./13.1#p1.t1.r3">x</mi><mo>→</mo><mrow><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></mrow></math>, where
<math class="ltx_Math" altimg="m34.png" altimg-height="27px" altimg-valign="-9px" altimg-width="63px" alttext="\rho_{1}>\frac{1}{2}" display="inline"><mrow><msub><mi>ρ</mi><mn>1</mn></msub><mo>></mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math>. Then for <math class="ltx_Math" altimg="m32.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\mu" display="inline"><mi>μ</mi></math> in the half-plane <math class="ltx_Math" altimg="m24.png" altimg-height="27px" altimg-valign="-9px" altimg-width="270px" alttext="\Re\mu\geq\mu_{1}>\max\left(-\rho_{0},\Re\kappa-\frac{1}{2}\right)" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>μ</mi></mrow><mo>≥</mo><msub><mi>μ</mi><mn>1</mn></msub><mo>></mo><mrow><mi>max</mi><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><msub><mi>ρ</mi><mn>0</mn></msub></mrow><mo>,</mo><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>κ</mi></mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow></mrow></math></p>
<table id="EGx1" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E13">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">13.23.13</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m4.png" altimg-height="25px" altimg-valign="-7px" altimg-width="44px" alttext="\displaystyle g(\mu)" display="inline"><mrow><mi>g</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>μ</mi><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m1.png" altimg-height="52px" altimg-valign="-21px" altimg-width="344px" alttext="\displaystyle=\frac{1}{\Gamma\left(1+2\mu\right)}\int_{0}^{\infty}f(x)x^{-%
\frac{3}{2}}M_{\kappa,\mu}\left(x\right)\mathrm{d}x," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><mfrac><mn>1</mn><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mrow><mn>2</mn><mo></mo><mi>μ</mi></mrow></mrow><mo>)</mo></mrow></mrow></mfrac></mstyle><mo></mo><mrow><mstyle displaystyle="true"><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup></mstyle><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./13.1#p1.t1.r3">x</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><msup><mi href="./13.1#p1.t1.r3">x</mi><mrow><mo>-</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow></msup><mo></mo><mrow><msub><mi href="./13.14#E2">M</mi><mrow><mi>κ</mi><mo href="./13.14#E2">,</mo><mi>μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./13.1#p1.t1.r3">x</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./13.1#p1.t1.r3">x</mi></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E13.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m15.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./13.14#E2" title="(13.14.2) ‣ Standard Solutions ‣ §13.14(i) Differential Equation ‣ §13.14 Definitions and Basic Properties ‣ Whittaker Functions ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m12.png" altimg-height="24px" altimg-valign="-8px" altimg-width="77px" alttext="M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./13.14#E2">M</mi><mrow><mi class="ltx_nvar">κ</mi><mo href="./13.14#E2">,</mo><mi class="ltx_nvar">μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Whittaker confluent hypergeometric function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m28.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m40.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m26.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a> and
<a href="./13.1#p1.t1.r3" title="§13.1 Special Notation ‣ Notation ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m40.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./13.1#p1.t1.r3">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E14">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">13.23.14</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m3.png" altimg-height="25px" altimg-valign="-7px" altimg-width="45px" alttext="\displaystyle f(x)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./13.1#p1.t1.r3">x</mi><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m2.png" altimg-height="58px" altimg-valign="-23px" altimg-width="433px" alttext="\displaystyle=\frac{1}{\pi\mathrm{i}\sqrt{x}}\int_{\mu_{1}-\mathrm{i}\infty}^{%
\mu_{1}+\mathrm{i}\infty}\mu g(\mu)\Gamma\left(\tfrac{1}{2}+\mu-\kappa\right)W%
_{\kappa,\mu}\left(x\right)\mathrm{d}\mu." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><mfrac><mn>1</mn><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi><mo></mo><msqrt><mi href="./13.1#p1.t1.r3">x</mi></msqrt></mrow></mfrac></mstyle><mo></mo><mrow><mstyle displaystyle="true"><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><msub><mi>μ</mi><mn>1</mn></msub><mo>-</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi mathvariant="normal">∞</mi></mrow></mrow><mrow><msub><mi>μ</mi><mn>1</mn></msub><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi mathvariant="normal">∞</mi></mrow></mrow></msubsup></mstyle><mrow><mi>μ</mi><mo></mo><mrow><mi>g</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>μ</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>+</mo><mi>μ</mi></mrow><mo>-</mo><mi>κ</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./13.14#E3">W</mi><mrow><mi>κ</mi><mo href="./13.14#E3">,</mo><mi>μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./13.1#p1.t1.r3">x</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>μ</mi></mrow></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E14.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m15.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./13.14#E3" title="(13.14.3) ‣ Standard Solutions ‣ §13.14(i) Differential Equation ‣ §13.14 Definitions and Basic Properties ‣ Whittaker Functions ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m13.png" altimg-height="24px" altimg-valign="-8px" altimg-width="77px" alttext="W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./13.14#E3">W</mi><mrow><mi class="ltx_nvar">κ</mi><mo href="./13.14#E3">,</mo><mi class="ltx_nvar">μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Whittaker confluent hypergeometric function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m33.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m28.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m40.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m26.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a> and
<a href="./13.1#p1.t1.r3" title="§13.1 Special Notation ‣ Notation ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m40.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./13.1#p1.t1.r3">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
</div>
<div id="SS4.p2" class="ltx_para">
<p class="ltx_p">For additional integral transforms see <cite class="ltx_cite ltx_citemacro_citet">Magnus<span class="ltx_text ltx_bib_etal"> et al.</span> (</div>
</div>
</body></text>
</html>
</page>
<page>
<!DOCTYPE html><html>
<head>
<title>DLMF: 33.16 Connection Formulas</title>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<!--GOOGLE BOOTSTRAP--></head>
<text><body class="color_default textfont_default titlefont_default fontsize_default navbar_default">
<div class="ltx_page_navbar">
<div class="ltx_page_navlogo"><a href="./33#PT3" title="Variables r , ϵ ‣ Chapter 33 Coulomb Functions" class="ltx_ref" rel="up"><span class="ltx_text ltx_ref_title">Variables <math class="ltx_Math" altimg="m57.png" altimg-height="16px" altimg-valign="-6px" altimg-width="30px" alttext="r,\epsilon" display="inline"><mrow><mi href="./33.1#p1.t1.r2">r</mi><mo>,</mo><mi href="./33.1#p1.t1.r4">ϵ</mi></mrow></math></span></a></dd>
</dl>
</div>
</div>
<h6>Contents</h6>
<ul class="ltx_toclist ltx_toclist_section">
<li class="ltx_tocentry"><a href="#i"><span class="ltx_tag ltx_tag_subsection">§33.16(i) </span><math class="ltx_Math" altimg="m26.png" altimg-height="21px" altimg-valign="-5px" altimg-width="25px" alttext="F_{\ell}" display="inline"><msub><mi href="./33.2#E3">F</mi><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></msub></math> and <math class="ltx_Math" altimg="m28.png" altimg-height="21px" altimg-valign="-5px" altimg-width="28px" alttext="G_{\ell}" display="inline"><msub><mi href="./33.2#E11">G</mi><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></msub></math> in Terms
of <math class="ltx_Math" altimg="m52.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi href="./33.14#E4">f</mi></math> and <math class="ltx_Math" altimg="m54.png" altimg-height="18px" altimg-valign="-2px" altimg-width="16px" alttext="h" display="inline"><mi href="./33.14#E7">h</mi></math></a></li>
<li class="ltx_tocentry"><a href="#ii"><span class="ltx_tag ltx_tag_subsection">§33.16(ii) </span><math class="ltx_Math" altimg="m52.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi href="./33.14#E4">f</mi></math> and <math class="ltx_Math" altimg="m54.png" altimg-height="18px" altimg-valign="-2px" altimg-width="16px" alttext="h" display="inline"><mi href="./33.14#E7">h</mi></math> in Terms of
<math class="ltx_Math" altimg="m26.png" altimg-height="21px" altimg-valign="-5px" altimg-width="25px" alttext="F_{\ell}" display="inline"><msub><mi href="./33.2#E3">F</mi><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></msub></math> and <math class="ltx_Math" altimg="m28.png" altimg-height="21px" altimg-valign="-5px" altimg-width="28px" alttext="G_{\ell}" display="inline"><msub><mi href="./33.2#E11">G</mi><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></msub></math> when <math class="ltx_Math" altimg="m37.png" altimg-height="17px" altimg-valign="-3px" altimg-width="49px" alttext="\epsilon>0" display="inline"><mrow><mi href="./33.1#p1.t1.r4">ϵ</mi><mo>></mo><mn>0</mn></mrow></math></a></li>
<li class="ltx_tocentry"><a href="#iii"><span class="ltx_tag ltx_tag_subsection">§33.16(iii) </span><math class="ltx_Math" altimg="m52.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi href="./33.14#E4">f</mi></math> and <math class="ltx_Math" altimg="m54.png" altimg-height="18px" altimg-valign="-2px" altimg-width="16px" alttext="h" display="inline"><mi href="./33.14#E7">h</mi></math> in Terms of
<math class="ltx_Math" altimg="m31.png" altimg-height="24px" altimg-valign="-8px" altimg-width="77px" alttext="W_{\kappa,\mu}\left(z\right)" display="inline"><mrow><msub><mi href="./13.14#E3">W</mi><mrow><mi>κ</mi><mo href="./13.14#E3">,</mo><mi>μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></math> when <math class="ltx_Math" altimg="m36.png" altimg-height="17px" altimg-valign="-3px" altimg-width="49px" alttext="\epsilon<0" display="inline"><mrow><mi href="./33.1#p1.t1.r4">ϵ</mi><mo><</mo><mn>0</mn></mrow></math></a></li>
<li class="ltx_tocentry"><a href="#iv"><span class="ltx_tag ltx_tag_subsection">§33.16(iv) </span><math class="ltx_Math" altimg="m61.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="s" display="inline"><mi href="./33.14#E9">s</mi></math> and <math class="ltx_Math" altimg="m50.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="c" display="inline"><mi href="./33.14#E9">c</mi></math> in Terms of
<math class="ltx_Math" altimg="m26.png" altimg-height="21px" altimg-valign="-5px" altimg-width="25px" alttext="F_{\ell}" display="inline"><msub><mi href="./33.2#E3">F</mi><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></msub></math> and <math class="ltx_Math" altimg="m28.png" altimg-height="21px" altimg-valign="-5px" altimg-width="28px" alttext="G_{\ell}" display="inline"><msub><mi href="./33.2#E11">G</mi><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></msub></math> when <math class="ltx_Math" altimg="m37.png" altimg-height="17px" altimg-valign="-3px" altimg-width="49px" alttext="\epsilon>0" display="inline"><mrow><mi href="./33.1#p1.t1.r4">ϵ</mi><mo>></mo><mn>0</mn></mrow></math></a></li>
<li class="ltx_tocentry"><a href="#v"><span class="ltx_tag ltx_tag_subsection">§33.16(v) </span><math class="ltx_Math" altimg="m61.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="s" display="inline"><mi href="./33.14#E9">s</mi></math> and <math class="ltx_Math" altimg="m50.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="c" display="inline"><mi href="./33.14#E9">c</mi></math> in Terms of
<math class="ltx_Math" altimg="m31.png" altimg-height="24px" altimg-valign="-8px" altimg-width="77px" alttext="W_{\kappa,\mu}\left(z\right)" display="inline"><mrow><msub><mi href="./13.14#E3">W</mi><mrow><mi>κ</mi><mo href="./13.14#E3">,</mo><mi>μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></math> when <math class="ltx_Math" altimg="m36.png" altimg-height="17px" altimg-valign="-3px" altimg-width="49px" alttext="\epsilon<0" display="inline"><mrow><mi href="./33.1#p1.t1.r4">ϵ</mi><mo><</mo><mn>0</mn></mrow></math></a></li>
</ul>
<section id="i" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§33.16(i) </span><math class="ltx_Math" altimg="m26.png" altimg-height="21px" altimg-valign="-5px" altimg-width="25px" alttext="F_{\ell}" display="inline"><msub><mi href="./33.2#E3">F</mi><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></msub></math> and <math class="ltx_Math" altimg="m28.png" altimg-height="21px" altimg-valign="-5px" altimg-width="28px" alttext="G_{\ell}" display="inline"><msub><mi href="./33.2#E11">G</mi><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></msub></math> in Terms
of <math class="ltx_Math" altimg="m52.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi href="./33.14#E4">f</mi></math> and <math class="ltx_Math" altimg="m54.png" altimg-height="18px" altimg-valign="-2px" altimg-width="16px" alttext="h" display="inline"><mi href="./33.14#E7">h</mi></math>
</h2>
<div id="SS1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Keywords:</dt>
<dd><a class="ltx_keyword" href="./idx/C#Coulombfunctionsvariablesre">Coulomb functions: variables <math class="ltx_Math" altimg="m44.png" altimg-height="16px" altimg-valign="-6px" altimg-width="34px" alttext="\rho,\eta" display="inline"><mrow><mi href="./33.1#p1.t1.r3">ρ</mi><mo>,</mo><mi href="./33.1#p1.t1.r4">η</mi></mrow></math></a></dd>
</dl>
</div>
</div>
<div id="SS1.p1" class="ltx_para">
<table id="E1" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">33.16.1</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E1.png" altimg-height="53px" altimg-valign="-21px" altimg-width="376px" alttext="F_{\ell}\left(\eta,\rho\right)=\dfrac{(2\ell+1)!C_{\ell}\left(\eta\right)}{(-2%
\eta)^{\ell+1}}f\left(1/\eta^{2},\ell;-\eta\rho\right)," display="block"><mrow><mrow><mrow><msub><mi href="./33.2#E3">F</mi><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./33.1#p1.t1.r4">η</mi><mo>,</mo><mi href="./33.1#p1.t1.r3">ρ</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mfrac><mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>2</mn><mo></mo><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></mrow><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow><mo></mo><mrow><msub><mi href="./33.2#E5">C</mi><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./33.1#p1.t1.r4">η</mi><mo>)</mo></mrow></mrow></mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mrow><mn>2</mn><mo></mo><mi href="./33.1#p1.t1.r4">η</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mrow><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi><mo>+</mo><mn>1</mn></mrow></msup></mfrac><mo></mo><mrow><mi href="./33.14#E4">f</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>/</mo><msup><mi href="./33.1#p1.t1.r4">η</mi><mn>2</mn></msup></mrow><mo>,</mo><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi><mo>;</mo><mrow><mo>-</mo><mrow><mi href="./33.1#p1.t1.r4">η</mi><mo></mo><mi href="./33.1#p1.t1.r3">ρ</mi></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./33.2#E5" title="(33.2.5) ‣ §33.2(ii) Regular Solution F ℓ ( η , ρ ) ‣ §33.2 Definitions and Basic Properties ‣ Variables ρ , η ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m23.png" altimg-height="23px" altimg-valign="-7px" altimg-width="56px" alttext="C_{\NVar{\ell}}\left(\NVar{\eta}\right)" display="inline"><mrow><msub><mi href="./33.2#E5">C</mi><mi class="ltx_nvar" href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./33.1#p1.t1.r4">η</mi><mo>)</mo></mrow></mrow></math>: normalizing constant for Coulomb radial functions</a>,
<a href="./33.14#E4" title="(33.14.4) ‣ §33.14(ii) Regular Solution f ( ϵ , ℓ ; r ) ‣ §33.14 Definitions and Basic Properties ‣ Variables r , ϵ ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="23px" altimg-valign="-7px" altimg-width="79px" alttext="f\left(\NVar{\epsilon},\NVar{\ell};\NVar{r}\right)" display="inline"><mrow><mi href="./33.14#E4">f</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./33.1#p1.t1.r4">ϵ</mi><mo>,</mo><mi class="ltx_nvar" href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi><mo>;</mo><mi class="ltx_nvar" href="./33.1#p1.t1.r2">r</mi><mo>)</mo></mrow></mrow></math>: regular Coulomb function</a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m21.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m56.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./33.2#E3" title="(33.2.3) ‣ §33.2(ii) Regular Solution F ℓ ( η , ρ ) ‣ §33.2 Definitions and Basic Properties ‣ Variables ρ , η ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m25.png" altimg-height="23px" altimg-valign="-7px" altimg-width="73px" alttext="F_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}\right)" display="inline"><mrow><msub><mi href="./33.2#E3">F</mi><mi class="ltx_nvar" href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./33.1#p1.t1.r4">η</mi><mo>,</mo><mi class="ltx_nvar" href="./33.1#p1.t1.r3">ρ</mi><mo>)</mo></mrow></mrow></math>: regular Coulomb radial function</a>,
<a href="./33.1#p1.t1.r1" title="§33.1 Special Notation ‣ Notation ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m35.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="\ell" display="inline"><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></math>: nonnegative integer</a>,
<a href="./33.1#p1.t1.r3" title="§33.1 Special Notation ‣ Notation ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m45.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="\rho" display="inline"><mi href="./33.1#p1.t1.r3">ρ</mi></math>: nonnegative real variable</a> and
<a href="./33.1#p1.t1.r4" title="§33.1 Special Notation ‣ Notation ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="\eta" display="inline"><mi href="./33.1#p1.t1.r4">η</mi></math>: real parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E2" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">33.16.2</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E2.png" altimg-height="55px" altimg-valign="-21px" altimg-width="378px" alttext="G_{\ell}\left(\eta,\rho\right)=\dfrac{\pi(-2\eta)^{\ell}}{(2\ell+1)!C_{\ell}%
\left(\eta\right)}h\left(1/\eta^{2},\ell;-\eta\rho\right)," display="block"><mrow><mrow><mrow><msub><mi href="./33.2#E11">G</mi><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./33.1#p1.t1.r4">η</mi><mo>,</mo><mi href="./33.1#p1.t1.r3">ρ</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mfrac><mrow><mi href="./3.12#E1">π</mi><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mrow><mn>2</mn><mo></mo><mi href="./33.1#p1.t1.r4">η</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></msup></mrow><mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>2</mn><mo></mo><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></mrow><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow><mo></mo><mrow><msub><mi href="./33.2#E5">C</mi><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./33.1#p1.t1.r4">η</mi><mo>)</mo></mrow></mrow></mrow></mfrac><mo></mo><mrow><mi href="./33.14#E7">h</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>/</mo><msup><mi href="./33.1#p1.t1.r4">η</mi><mn>2</mn></msup></mrow><mo>,</mo><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi><mo>;</mo><mrow><mo>-</mo><mrow><mi href="./33.1#p1.t1.r4">η</mi><mo></mo><mi href="./33.1#p1.t1.r3">ρ</mi></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./33.2#E5" title="(33.2.5) ‣ §33.2(ii) Regular Solution F ℓ ( η , ρ ) ‣ §33.2 Definitions and Basic Properties ‣ Variables ρ , η ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m23.png" altimg-height="23px" altimg-valign="-7px" altimg-width="56px" alttext="C_{\NVar{\ell}}\left(\NVar{\eta}\right)" display="inline"><mrow><msub><mi href="./33.2#E5">C</mi><mi class="ltx_nvar" href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./33.1#p1.t1.r4">η</mi><mo>)</mo></mrow></mrow></math>: normalizing constant for Coulomb radial functions</a>,
<a href="./33.14#E7" title="(33.14.7) ‣ §33.14(iii) Irregular Solution h ( ϵ , ℓ ; r ) ‣ §33.14 Definitions and Basic Properties ‣ Variables r , ϵ ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m55.png" altimg-height="23px" altimg-valign="-7px" altimg-width="78px" alttext="h\left(\NVar{\epsilon},\NVar{\ell};\NVar{r}\right)" display="inline"><mrow><mi href="./33.14#E7">h</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./33.1#p1.t1.r4">ϵ</mi><mo>,</mo><mi class="ltx_nvar" href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi><mo>;</mo><mi class="ltx_nvar" href="./33.1#p1.t1.r2">r</mi><mo>)</mo></mrow></mrow></math>: irregular Coulomb function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m21.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m56.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./33.2#E11" title="(33.2.11) ‣ §33.2(iii) Irregular Solutions G ℓ ( η , ρ ) , H ± ℓ ( η , ρ ) ‣ §33.2 Definitions and Basic Properties ‣ Variables ρ , η ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m27.png" altimg-height="23px" altimg-valign="-7px" altimg-width="76px" alttext="G_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}\right)" display="inline"><mrow><msub><mi href="./33.2#E11">G</mi><mi class="ltx_nvar" href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./33.1#p1.t1.r4">η</mi><mo>,</mo><mi class="ltx_nvar" href="./33.1#p1.t1.r3">ρ</mi><mo>)</mo></mrow></mrow></math>: irregular Coulomb radial function</a>,
<a href="./33.1#p1.t1.r1" title="§33.1 Special Notation ‣ Notation ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m35.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="\ell" display="inline"><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></math>: nonnegative integer</a>,
<a href="./33.1#p1.t1.r3" title="§33.1 Special Notation ‣ Notation ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m45.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="\rho" display="inline"><mi href="./33.1#p1.t1.r3">ρ</mi></math>: nonnegative real variable</a> and
<a href="./33.1#p1.t1.r4" title="§33.1 Special Notation ‣ Notation ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="\eta" display="inline"><mi href="./33.1#p1.t1.r4">η</mi></math>: real parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m24.png" altimg-height="23px" altimg-valign="-7px" altimg-width="56px" alttext="C_{\ell}\left(\eta\right)" display="inline"><mrow><msub><mi href="./33.2#E5">C</mi><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./33.1#p1.t1.r4">η</mi><mo>)</mo></mrow></mrow></math> is given by ().</p>
</div>
</section>
<section id="ii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§33.16(ii) </span><math class="ltx_Math" altimg="m52.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi href="./33.14#E4">f</mi></math> and <math class="ltx_Math" altimg="m54.png" altimg-height="18px" altimg-valign="-2px" altimg-width="16px" alttext="h" display="inline"><mi href="./33.14#E7">h</mi></math> in Terms of
<math class="ltx_Math" altimg="m26.png" altimg-height="21px" altimg-valign="-5px" altimg-width="25px" alttext="F_{\ell}" display="inline"><msub><mi href="./33.2#E3">F</mi><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></msub></math> and <math class="ltx_Math" altimg="m28.png" altimg-height="21px" altimg-valign="-5px" altimg-width="28px" alttext="G_{\ell}" display="inline"><msub><mi href="./33.2#E11">G</mi><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></msub></math> when <math class="ltx_Math" altimg="m37.png" altimg-height="17px" altimg-valign="-3px" altimg-width="49px" alttext="\epsilon>0" display="inline"><mrow><mi href="./33.1#p1.t1.r4">ϵ</mi><mo>></mo><mn>0</mn></mrow></math>
</h2>
<div id="SS2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Keywords:</dt>
<dd><a class="ltx_keyword" href="./idx/C#Coulombfunctionsvariablesr">Coulomb functions: variables <math class="ltx_Math" altimg="m57.png" altimg-height="16px" altimg-valign="-6px" altimg-width="30px" alttext="r,\epsilon" display="inline"><mrow><mi href="./33.1#p1.t1.r2">r</mi><mo>,</mo><mi href="./33.1#p1.t1.r4">ϵ</mi></mrow></math></a></dd>
</dl>
</div>
</div>
<div id="SS2.p1" class="ltx_para">
<p class="ltx_p">When <math class="ltx_Math" altimg="m37.png" altimg-height="17px" altimg-valign="-3px" altimg-width="49px" alttext="\epsilon>0" display="inline"><mrow><mi href="./33.1#p1.t1.r4">ϵ</mi><mo>></mo><mn>0</mn></mrow></math> denote</p>
<table id="E3" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">33.16.3</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E3.png" altimg-height="29px" altimg-valign="-7px" altimg-width="129px" alttext="\tau=\epsilon^{1/2}(>0)," display="block"><mrow><mrow><mi href="./33.16#E3">τ</mi><mo>=</mo><mrow><msup><mi href="./33.1#p1.t1.r4">ϵ</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mspace width="veryverythickmathspace"></mspace><mrow><mo stretchy="false">(</mo><mrow><mi></mi><mo>></mo><mn>0</mn></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text"><math class="ltx_Math" altimg="m47.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\tau" display="inline"><mi href="./33.16#E3">τ</mi></math>: parameter (locally)</span></dd>
<dt>Symbols:</dt>
<dd><a href="./33.1#p1.t1.r4" title="§33.1 Special Notation ‣ Notation ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="13px" altimg-valign="-2px" altimg-width="12px" alttext="\epsilon" display="inline"><mi href="./33.1#p1.t1.r4">ϵ</mi></math>: real parameter</a></dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and again define <math class="ltx_Math" altimg="m22.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="A(\epsilon,\ell)" display="inline"><mrow><mi href="./33.14#E11">A</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./33.1#p1.t1.r4">ϵ</mi><mo>,</mo><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi><mo stretchy="false">)</mo></mrow></mrow></math> by (). Then for <math class="ltx_Math" altimg="m59.png" altimg-height="17px" altimg-valign="-3px" altimg-width="50px" alttext="r>0" display="inline"><mrow><mi href="./33.1#p1.t1.r2">r</mi><mo>></mo><mn>0</mn></mrow></math></p>
<table id="E4" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">33.16.4</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E4.png" altimg-height="61px" altimg-valign="-21px" altimg-width="410px" alttext="f\left(\epsilon,\ell;r\right)=\left(\frac{2}{\pi\tau}\frac{1-e^{-2\pi/\tau}}{A%
(\epsilon,\ell)}\right)^{\ifrac{1}{2}}F_{\ell}\left(-1/\tau,\tau r\right)," display="block"><mrow><mrow><mrow><mi href="./33.14#E4">f</mi><mo></mo><mrow><mo>(</mo><mi href="./33.1#p1.t1.r4">ϵ</mi><mo>,</mo><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi><mo>;</mo><mi href="./33.1#p1.t1.r2">r</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msup><mrow><mo>(</mo><mrow><mfrac><mn>2</mn><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi href="./33.16#E3">τ</mi></mrow></mfrac><mo></mo><mfrac><mrow><mn>1</mn><mo>-</mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo>/</mo><mi href="./33.16#E3">τ</mi></mrow></mrow></msup></mrow><mrow><mi href="./33.14#E11">A</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./33.1#p1.t1.r4">ϵ</mi><mo>,</mo><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi><mo stretchy="false">)</mo></mrow></mrow></mfrac></mrow><mo>)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo></mo><mrow><msub><mi href="./33.2#E3">F</mi><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mrow><mn>1</mn><mo>/</mo><mi href="./33.16#E3">τ</mi></mrow></mrow><mo>,</mo><mrow><mi href="./33.16#E3">τ</mi><mo></mo><mi href="./33.1#p1.t1.r2">r</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E4.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./33.14#E4" title="(33.14.4) ‣ §33.14(ii) Regular Solution f ( ϵ , ℓ ; r ) ‣ §33.14 Definitions and Basic Properties ‣ Variables r , ϵ ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="23px" altimg-valign="-7px" altimg-width="79px" alttext="f\left(\NVar{\epsilon},\NVar{\ell};\NVar{r}\right)" display="inline"><mrow><mi href="./33.14#E4">f</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./33.1#p1.t1.r4">ϵ</mi><mo>,</mo><mi class="ltx_nvar" href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi><mo>;</mo><mi class="ltx_nvar" href="./33.1#p1.t1.r2">r</mi><mo>)</mo></mrow></mrow></math>: regular Coulomb function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m41.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./33.2#E3" title="(33.2.3) ‣ §33.2(ii) Regular Solution F ℓ ( η , ρ ) ‣ §33.2 Definitions and Basic Properties ‣ Variables ρ , η ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m25.png" altimg-height="23px" altimg-valign="-7px" altimg-width="73px" alttext="F_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}\right)" display="inline"><mrow><msub><mi href="./33.2#E3">F</mi><mi class="ltx_nvar" href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./33.1#p1.t1.r4">η</mi><mo>,</mo><mi class="ltx_nvar" href="./33.1#p1.t1.r3">ρ</mi><mo>)</mo></mrow></mrow></math>: regular Coulomb radial function</a>,
<a href="./33.1#p1.t1.r1" title="§33.1 Special Notation ‣ Notation ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m35.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="\ell" display="inline"><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></math>: nonnegative integer</a>,
<a href="./33.1#p1.t1.r2" title="§33.1 Special Notation ‣ Notation ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="r" display="inline"><mi href="./33.1#p1.t1.r2">r</mi></math>: real variable</a>,
<a href="./33.1#p1.t1.r4" title="§33.1 Special Notation ‣ Notation ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="13px" altimg-valign="-2px" altimg-width="12px" alttext="\epsilon" display="inline"><mi href="./33.1#p1.t1.r4">ϵ</mi></math>: real parameter</a>,
<a href="./33.14#E11" title="(33.14.11) ‣ §33.14(iv) Solutions s ( ϵ , ℓ ; r ) and c ( ϵ , ℓ ; r ) ‣ §33.14 Definitions and Basic Properties ‣ Variables r , ϵ ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="A(\epsilon,\ell)" display="inline"><mrow><mi href="./33.14#E11">A</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./33.1#p1.t1.r4">ϵ</mi><mo>,</mo><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function</a> and
<a href="./33.16#E3" title="(33.16.3) ‣ §33.16(ii) f and h in Terms of F ℓ and G ℓ when > ϵ 0 ‣ §33.16 Connection Formulas ‣ Variables r , ϵ ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\tau" display="inline"><mi href="./33.16#E3">τ</mi></math>: parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E5" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">33.16.5</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E5.png" altimg-height="60px" altimg-valign="-21px" altimg-width="413px" alttext="h\left(\epsilon,\ell;r\right)=\left(\frac{2}{\pi\tau}\frac{A(\epsilon,\ell)}{1%
-e^{-2\pi/\tau}}\right)^{\ifrac{1}{2}}G_{\ell}\left(-1/\tau,\tau r\right)." display="block"><mrow><mrow><mrow><mi href="./33.14#E7">h</mi><mo></mo><mrow><mo>(</mo><mi href="./33.1#p1.t1.r4">ϵ</mi><mo>,</mo><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi><mo>;</mo><mi href="./33.1#p1.t1.r2">r</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msup><mrow><mo>(</mo><mrow><mfrac><mn>2</mn><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi href="./33.16#E3">τ</mi></mrow></mfrac><mo></mo><mfrac><mrow><mi href="./33.14#E11">A</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./33.1#p1.t1.r4">ϵ</mi><mo>,</mo><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi><mo stretchy="false">)</mo></mrow></mrow><mrow><mn>1</mn><mo>-</mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo>/</mo><mi href="./33.16#E3">τ</mi></mrow></mrow></msup></mrow></mfrac></mrow><mo>)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo></mo><mrow><msub><mi href="./33.2#E11">G</mi><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mrow><mn>1</mn><mo>/</mo><mi href="./33.16#E3">τ</mi></mrow></mrow><mo>,</mo><mrow><mi href="./33.16#E3">τ</mi><mo></mo><mi href="./33.1#p1.t1.r2">r</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E5.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./33.14#E7" title="(33.14.7) ‣ §33.14(iii) Irregular Solution h ( ϵ , ℓ ; r ) ‣ §33.14 Definitions and Basic Properties ‣ Variables r , ϵ ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m55.png" altimg-height="23px" altimg-valign="-7px" altimg-width="78px" alttext="h\left(\NVar{\epsilon},\NVar{\ell};\NVar{r}\right)" display="inline"><mrow><mi href="./33.14#E7">h</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./33.1#p1.t1.r4">ϵ</mi><mo>,</mo><mi class="ltx_nvar" href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi><mo>;</mo><mi class="ltx_nvar" href="./33.1#p1.t1.r2">r</mi><mo>)</mo></mrow></mrow></math>: irregular Coulomb function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m41.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./33.2#E11" title="(33.2.11) ‣ §33.2(iii) Irregular Solutions G ℓ ( η , ρ ) , H ± ℓ ( η , ρ ) ‣ §33.2 Definitions and Basic Properties ‣ Variables ρ , η ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m27.png" altimg-height="23px" altimg-valign="-7px" altimg-width="76px" alttext="G_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}\right)" display="inline"><mrow><msub><mi href="./33.2#E11">G</mi><mi class="ltx_nvar" href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./33.1#p1.t1.r4">η</mi><mo>,</mo><mi class="ltx_nvar" href="./33.1#p1.t1.r3">ρ</mi><mo>)</mo></mrow></mrow></math>: irregular Coulomb radial function</a>,
<a href="./33.1#p1.t1.r1" title="§33.1 Special Notation ‣ Notation ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m35.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="\ell" display="inline"><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></math>: nonnegative integer</a>,
<a href="./33.1#p1.t1.r2" title="§33.1 Special Notation ‣ Notation ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="r" display="inline"><mi href="./33.1#p1.t1.r2">r</mi></math>: real variable</a>,
<a href="./33.1#p1.t1.r4" title="§33.1 Special Notation ‣ Notation ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="13px" altimg-valign="-2px" altimg-width="12px" alttext="\epsilon" display="inline"><mi href="./33.1#p1.t1.r4">ϵ</mi></math>: real parameter</a>,
<a href="./33.14#E11" title="(33.14.11) ‣ §33.14(iv) Solutions s ( ϵ , ℓ ; r ) and c ( ϵ , ℓ ; r ) ‣ §33.14 Definitions and Basic Properties ‣ Variables r , ϵ ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="A(\epsilon,\ell)" display="inline"><mrow><mi href="./33.14#E11">A</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./33.1#p1.t1.r4">ϵ</mi><mo>,</mo><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function</a> and
<a href="./33.16#E3" title="(33.16.3) ‣ §33.16(ii) f and h in Terms of F ℓ and G ℓ when > ϵ 0 ‣ §33.16 Connection Formulas ‣ Variables r , ϵ ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\tau" display="inline"><mi href="./33.16#E3">τ</mi></math>: parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Alternatively, for <math class="ltx_Math" altimg="m58.png" altimg-height="17px" altimg-valign="-3px" altimg-width="50px" alttext="r<0" display="inline"><mrow><mi href="./33.1#p1.t1.r2">r</mi><mo><</mo><mn>0</mn></mrow></math></p>
<table id="EGx1" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E6">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">33.16.6</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m18.png" altimg-height="25px" altimg-valign="-7px" altimg-width="80px" alttext="\displaystyle f\left(\epsilon,\ell;r\right)" display="inline"><mrow><mi href="./33.14#E4">f</mi><mo></mo><mrow><mo>(</mo><mi href="./33.1#p1.t1.r4">ϵ</mi><mo>,</mo><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi><mo>;</mo><mi href="./33.1#p1.t1.r2">r</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m1.png" altimg-height="61px" altimg-valign="-21px" altimg-width="390px" alttext="\displaystyle=(-1)^{\ell+1}\left(\frac{2}{\pi\tau}\frac{e^{2\pi/\tau}-1}{A(%
\epsilon,\ell)}\right)^{\ifrac{1}{2}}F_{\ell}\left(1/\tau,-\tau r\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mrow><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi><mo>+</mo><mn>1</mn></mrow></msup><mo></mo><msup><mrow><mo>(</mo><mrow><mstyle displaystyle="true"><mfrac><mn>2</mn><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi href="./33.16#E3">τ</mi></mrow></mfrac></mstyle><mo></mo><mstyle displaystyle="true"><mfrac><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo>/</mo><mi href="./33.16#E3">τ</mi></mrow></msup><mo>-</mo><mn>1</mn></mrow><mrow><mi href="./33.14#E11">A</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./33.1#p1.t1.r4">ϵ</mi><mo>,</mo><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi><mo stretchy="false">)</mo></mrow></mrow></mfrac></mstyle></mrow><mo>)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo></mo><mrow><msub><mi href="./33.2#E3">F</mi><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>/</mo><mi href="./33.16#E3">τ</mi></mrow><mo>,</mo><mrow><mo>-</mo><mrow><mi href="./33.16#E3">τ</mi><mo></mo><mi href="./33.1#p1.t1.r2">r</mi></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E6.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./33.14#E4" title="(33.14.4) ‣ §33.14(ii) Regular Solution f ( ϵ , ℓ ; r ) ‣ §33.14 Definitions and Basic Properties ‣ Variables r , ϵ ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="23px" altimg-valign="-7px" altimg-width="79px" alttext="f\left(\NVar{\epsilon},\NVar{\ell};\NVar{r}\right)" display="inline"><mrow><mi href="./33.14#E4">f</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./33.1#p1.t1.r4">ϵ</mi><mo>,</mo><mi class="ltx_nvar" href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi><mo>;</mo><mi class="ltx_nvar" href="./33.1#p1.t1.r2">r</mi><mo>)</mo></mrow></mrow></math>: regular Coulomb function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m41.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./33.2#E3" title="(33.2.3) ‣ §33.2(ii) Regular Solution F ℓ ( η , ρ ) ‣ §33.2 Definitions and Basic Properties ‣ Variables ρ , η ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m25.png" altimg-height="23px" altimg-valign="-7px" altimg-width="73px" alttext="F_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}\right)" display="inline"><mrow><msub><mi href="./33.2#E3">F</mi><mi class="ltx_nvar" href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./33.1#p1.t1.r4">η</mi><mo>,</mo><mi class="ltx_nvar" href="./33.1#p1.t1.r3">ρ</mi><mo>)</mo></mrow></mrow></math>: regular Coulomb radial function</a>,
<a href="./33.1#p1.t1.r1" title="§33.1 Special Notation ‣ Notation ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m35.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="\ell" display="inline"><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></math>: nonnegative integer</a>,
<a href="./33.1#p1.t1.r2" title="§33.1 Special Notation ‣ Notation ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="r" display="inline"><mi href="./33.1#p1.t1.r2">r</mi></math>: real variable</a>,
<a href="./33.1#p1.t1.r4" title="§33.1 Special Notation ‣ Notation ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="13px" altimg-valign="-2px" altimg-width="12px" alttext="\epsilon" display="inline"><mi href="./33.1#p1.t1.r4">ϵ</mi></math>: real parameter</a>,
<a href="./33.14#E11" title="(33.14.11) ‣ §33.14(iv) Solutions s ( ϵ , ℓ ; r ) and c ( ϵ , ℓ ; r ) ‣ §33.14 Definitions and Basic Properties ‣ Variables r , ϵ ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="A(\epsilon,\ell)" display="inline"><mrow><mi href="./33.14#E11">A</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./33.1#p1.t1.r4">ϵ</mi><mo>,</mo><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function</a> and
<a href="./33.16#E3" title="(33.16.3) ‣ §33.16(ii) f and h in Terms of F ℓ and G ℓ when > ϵ 0 ‣ §33.16 Connection Formulas ‣ Variables r , ϵ ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\tau" display="inline"><mi href="./33.16#E3">τ</mi></math>: parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E7">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">33.16.7</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m19.png" altimg-height="25px" altimg-valign="-7px" altimg-width="80px" alttext="\displaystyle h\left(\epsilon,\ell;r\right)" display="inline"><mrow><mi href="./33.14#E7">h</mi><mo></mo><mrow><mo>(</mo><mi href="./33.1#p1.t1.r4">ϵ</mi><mo>,</mo><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi><mo>;</mo><mi href="./33.1#p1.t1.r2">r</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m2.png" altimg-height="60px" altimg-valign="-21px" altimg-width="373px" alttext="\displaystyle=(-1)^{\ell}\left(\frac{2}{\pi\tau}\frac{A(\epsilon,\ell)}{e^{2%
\pi/\tau}-1}\right)^{\ifrac{1}{2}}G_{\ell}\left(1/\tau,-\tau r\right)." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></msup><mo></mo><msup><mrow><mo>(</mo><mrow><mstyle displaystyle="true"><mfrac><mn>2</mn><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi href="./33.16#E3">τ</mi></mrow></mfrac></mstyle><mo></mo><mstyle displaystyle="true"><mfrac><mrow><mi href="./33.14#E11">A</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./33.1#p1.t1.r4">ϵ</mi><mo>,</mo><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi><mo stretchy="false">)</mo></mrow></mrow><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo>/</mo><mi href="./33.16#E3">τ</mi></mrow></msup><mo>-</mo><mn>1</mn></mrow></mfrac></mstyle></mrow><mo>)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo></mo><mrow><msub><mi href="./33.2#E11">G</mi><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>/</mo><mi href="./33.16#E3">τ</mi></mrow><mo>,</mo><mrow><mo>-</mo><mrow><mi href="./33.16#E3">τ</mi><mo></mo><mi href="./33.1#p1.t1.r2">r</mi></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E7.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./33.14#E7" title="(33.14.7) ‣ §33.14(iii) Irregular Solution h ( ϵ , ℓ ; r ) ‣ §33.14 Definitions and Basic Properties ‣ Variables r , ϵ ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m55.png" altimg-height="23px" altimg-valign="-7px" altimg-width="78px" alttext="h\left(\NVar{\epsilon},\NVar{\ell};\NVar{r}\right)" display="inline"><mrow><mi href="./33.14#E7">h</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./33.1#p1.t1.r4">ϵ</mi><mo>,</mo><mi class="ltx_nvar" href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi><mo>;</mo><mi class="ltx_nvar" href="./33.1#p1.t1.r2">r</mi><mo>)</mo></mrow></mrow></math>: irregular Coulomb function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m41.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./33.2#E11" title="(33.2.11) ‣ §33.2(iii) Irregular Solutions G ℓ ( η , ρ ) , H ± ℓ ( η , ρ ) ‣ §33.2 Definitions and Basic Properties ‣ Variables ρ , η ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m27.png" altimg-height="23px" altimg-valign="-7px" altimg-width="76px" alttext="G_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}\right)" display="inline"><mrow><msub><mi href="./33.2#E11">G</mi><mi class="ltx_nvar" href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./33.1#p1.t1.r4">η</mi><mo>,</mo><mi class="ltx_nvar" href="./33.1#p1.t1.r3">ρ</mi><mo>)</mo></mrow></mrow></math>: irregular Coulomb radial function</a>,
<a href="./33.1#p1.t1.r1" title="§33.1 Special Notation ‣ Notation ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m35.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="\ell" display="inline"><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></math>: nonnegative integer</a>,
<a href="./33.1#p1.t1.r2" title="§33.1 Special Notation ‣ Notation ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="r" display="inline"><mi href="./33.1#p1.t1.r2">r</mi></math>: real variable</a>,
<a href="./33.1#p1.t1.r4" title="§33.1 Special Notation ‣ Notation ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="13px" altimg-valign="-2px" altimg-width="12px" alttext="\epsilon" display="inline"><mi href="./33.1#p1.t1.r4">ϵ</mi></math>: real parameter</a>,
<a href="./33.14#E11" title="(33.14.11) ‣ §33.14(iv) Solutions s ( ϵ , ℓ ; r ) and c ( ϵ , ℓ ; r ) ‣ §33.14 Definitions and Basic Properties ‣ Variables r , ϵ ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="A(\epsilon,\ell)" display="inline"><mrow><mi href="./33.14#E11">A</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./33.1#p1.t1.r4">ϵ</mi><mo>,</mo><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function</a> and
<a href="./33.16#E3" title="(33.16.3) ‣ §33.16(ii) f and h in Terms of F ℓ and G ℓ when > ϵ 0 ‣ §33.16 Connection Formulas ‣ Variables r , ϵ ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\tau" display="inline"><mi href="./33.16#E3">τ</mi></math>: parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
</div>
</section>
<section id="iii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§33.16(iii) </span><math class="ltx_Math" altimg="m52.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi href="./33.14#E4">f</mi></math> and <math class="ltx_Math" altimg="m54.png" altimg-height="18px" altimg-valign="-2px" altimg-width="16px" alttext="h" display="inline"><mi href="./33.14#E7">h</mi></math> in Terms of
<math class="ltx_Math" altimg="m31.png" altimg-height="24px" altimg-valign="-8px" altimg-width="77px" alttext="W_{\kappa,\mu}\left(z\right)" display="inline"><mrow><msub><mi href="./13.14#E3">W</mi><mrow><mi>κ</mi><mo href="./13.14#E3">,</mo><mi>μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></math> when <math class="ltx_Math" altimg="m36.png" altimg-height="17px" altimg-valign="-3px" altimg-width="49px" alttext="\epsilon<0" display="inline"><mrow><mi href="./33.1#p1.t1.r4">ϵ</mi><mo><</mo><mn>0</mn></mrow></math>
</h2>
<div id="SS3.info" class="ltx_metadata ltx_info">
, <a class="ltx_keyword" href="./idx/C#Coulombfunctionsvariablesr">Coulomb functions: variables <math class="ltx_Math" altimg="m57.png" altimg-height="16px" altimg-valign="-6px" altimg-width="30px" alttext="r,\epsilon" display="inline"><mrow><mi href="./33.1#p1.t1.r2">r</mi><mo>,</mo><mi href="./33.1#p1.t1.r4">ϵ</mi></mrow></math></a>, </dd>
</dl>
</div>
</div>
<div id="SS3.p1" class="ltx_para">
<p class="ltx_p">When <math class="ltx_Math" altimg="m36.png" altimg-height="17px" altimg-valign="-3px" altimg-width="49px" alttext="\epsilon<0" display="inline"><mrow><mi href="./33.1#p1.t1.r4">ϵ</mi><mo><</mo><mn>0</mn></mrow></math> denote</p>
<table id="E8" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">33.16.8</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E8.png" altimg-height="29px" altimg-valign="-7px" altimg-width="180px" alttext="\nu=1/(-\epsilon)^{1/2}(>0)," display="block"><mrow><mrow><mi href="./33.16#E8">ν</mi><mo>=</mo><mrow><mrow><mn>1</mn><mo>/</mo><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mi href="./33.1#p1.t1.r4">ϵ</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow><mspace width="veryverythickmathspace"></mspace><mrow><mo stretchy="false">(</mo><mrow><mi></mi><mo>></mo><mn>0</mn></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E8.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text"><math class="ltx_Math" altimg="m42.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./33.16#E8">ν</mi></math>: parameter (locally)</span></dd>
<dt>Symbols:</dt>
<dd><a href="./33.1#p1.t1.r4" title="§33.1 Special Notation ‣ Notation ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="13px" altimg-valign="-2px" altimg-width="12px" alttext="\epsilon" display="inline"><mi href="./33.1#p1.t1.r4">ϵ</mi></math>: real parameter</a></dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E9" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex1" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">33.16.9</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m16.png" altimg-height="25px" altimg-valign="-7px" altimg-width="66px" alttext="\displaystyle\zeta_{\ell}(\nu,r)" display="inline"><mrow><msub><mi href="./33.16#E9">ζ</mi><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./33.16#E8">ν</mi><mo>,</mo><mi href="./33.1#p1.t1.r2">r</mi><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m9.png" altimg-height="30px" altimg-valign="-12px" altimg-width="158px" alttext="\displaystyle=W_{\nu,\ell+\frac{1}{2}}\left(2r/\nu\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msub><mi href="./13.14#E3">W</mi><mrow><mi href="./33.16#E8">ν</mi><mo href="./13.14#E3">,</mo><mrow><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow></msub><mo></mo><mrow><mo>(</mo><mrow><mrow><mn>2</mn><mo></mo><mi href="./33.1#p1.t1.r2">r</mi></mrow><mo>/</mo><mi href="./33.16#E8">ν</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex2" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m15.png" altimg-height="25px" altimg-valign="-7px" altimg-width="66px" alttext="\displaystyle\xi_{\ell}(\nu,r)" display="inline"><mrow><msub><mi href="./33.16#E9">ξ</mi><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./33.16#E8">ν</mi><mo>,</mo><mi href="./33.1#p1.t1.r2">r</mi><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m10.png" altimg-height="41px" altimg-valign="-15px" altimg-width="273px" alttext="\displaystyle=\Re\left(e^{\mathrm{i}\pi\nu}W_{-\nu,\ell+\frac{1}{2}}\left(e^{%
\mathrm{i}\pi}2r/\nu\right)\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mrow><mo>(</mo><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi href="./33.16#E8">ν</mi></mrow></msup><mo></mo><mrow><msub><mi href="./13.14#E3">W</mi><mrow><mrow><mo>-</mo><mi href="./33.16#E8">ν</mi></mrow><mo href="./13.14#E3">,</mo><mrow><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow></msub><mo></mo><mrow><mo>(</mo><mrow><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./3.12#E1">π</mi></mrow></msup><mo></mo><mn>2</mn><mo></mo><mi href="./33.1#p1.t1.r2">r</mi></mrow><mo>/</mo><mi href="./33.16#E8">ν</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E9.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd>
<span class="ltx_text"><math class="ltx_Math" altimg="m49.png" altimg-height="23px" altimg-valign="-7px" altimg-width="65px" alttext="\zeta_{\ell}(\nu,r)" display="inline"><mrow><msub><mi href="./33.16#E9">ζ</mi><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./33.16#E8">ν</mi><mo>,</mo><mi href="./33.1#p1.t1.r2">r</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function (locally)</span> and
<span class="ltx_text"><math class="ltx_Math" altimg="m48.png" altimg-height="23px" altimg-valign="-7px" altimg-width="65px" alttext="\xi_{\ell}(\nu,r)" display="inline"><mrow><msub><mi href="./33.16#E9">ξ</mi><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./33.16#E8">ν</mi><mo>,</mo><mi href="./33.1#p1.t1.r2">r</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function (locally)</span>
</dd>
<dt>Symbols:</dt>
<dd>
<a href="./13.14#E3" title="(13.14.3) ‣ Standard Solutions ‣ §13.14(i) Differential Equation ‣ §13.14 Definitions and Basic Properties ‣ Whittaker Functions ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m30.png" altimg-height="24px" altimg-valign="-8px" altimg-width="77px" alttext="W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./13.14#E3">W</mi><mrow><mi class="ltx_nvar">κ</mi><mo href="./13.14#E3">,</mo><mi class="ltx_nvar">μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Whittaker confluent hypergeometric function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m41.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m33.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./33.1#p1.t1.r1" title="§33.1 Special Notation ‣ Notation ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m35.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="\ell" display="inline"><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></math>: nonnegative integer</a>,
<a href="./33.1#p1.t1.r2" title="§33.1 Special Notation ‣ Notation ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="r" display="inline"><mi href="./33.1#p1.t1.r2">r</mi></math>: real variable</a> and
<a href="./33.16#E8" title="(33.16.8) ‣ §33.16(iii) f and h in Terms of W κ , μ ( z ) when < ϵ 0 ‣ §33.16 Connection Formulas ‣ Variables r , ϵ ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m42.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./33.16#E8">ν</mi></math>: parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and again define <math class="ltx_Math" altimg="m22.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="A(\epsilon,\ell)" display="inline"><mrow><mi href="./33.14#E11">A</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./33.1#p1.t1.r4">ϵ</mi><mo>,</mo><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi><mo stretchy="false">)</mo></mrow></mrow></math> by (). Then for <math class="ltx_Math" altimg="m59.png" altimg-height="17px" altimg-valign="-3px" altimg-width="50px" alttext="r>0" display="inline"><mrow><mi href="./33.1#p1.t1.r2">r</mi><mo>></mo><mn>0</mn></mrow></math></p>
<table id="EGx2" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E10">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">33.16.10</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m18.png" altimg-height="25px" altimg-valign="-7px" altimg-width="80px" alttext="\displaystyle f\left(\epsilon,\ell;r\right)" display="inline"><mrow><mi href="./33.14#E4">f</mi><mo></mo><mrow><mo>(</mo><mi href="./33.1#p1.t1.r4">ϵ</mi><mo>,</mo><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi><mo>;</mo><mi href="./33.1#p1.t1.r2">r</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m4.png" altimg-height="53px" altimg-valign="-21px" altimg-width="546px" alttext="\displaystyle=(-1)^{\ell}\nu^{\ell+1}\left(-\frac{\cos\left(\pi\nu\right)\zeta%
_{\ell}(\nu,r)}{\Gamma\left(\ell+1+\nu\right)}+\frac{\sin\left(\pi\nu\right)%
\Gamma\left(\nu-\ell\right)\xi_{\ell}(\nu,r)}{\pi}\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></msup><mo></mo><msup><mi href="./33.16#E8">ν</mi><mrow><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi><mo>+</mo><mn>1</mn></mrow></msup><mo></mo><mrow><mo>(</mo><mrow><mrow><mo>-</mo><mstyle displaystyle="true"><mfrac><mrow><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi href="./33.16#E8">ν</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./33.16#E9">ζ</mi><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./33.16#E8">ν</mi><mo>,</mo><mi href="./33.1#p1.t1.r2">r</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi><mo>+</mo><mn>1</mn><mo>+</mo><mi href="./33.16#E8">ν</mi></mrow><mo>)</mo></mrow></mrow></mfrac></mstyle></mrow><mo>+</mo><mstyle displaystyle="true"><mfrac><mrow><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi href="./33.16#E8">ν</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./33.16#E8">ν</mi><mo>-</mo><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./33.16#E9">ξ</mi><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./33.16#E8">ν</mi><mo>,</mo><mi href="./33.1#p1.t1.r2">r</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mi href="./3.12#E1">π</mi></mfrac></mstyle></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E10.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./33.14#E4" title="(33.14.4) ‣ §33.14(ii) Regular Solution f ( ϵ , ℓ ; r ) ‣ §33.14 Definitions and Basic Properties ‣ Variables r , ϵ ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="23px" altimg-valign="-7px" altimg-width="79px" alttext="f\left(\NVar{\epsilon},\NVar{\ell};\NVar{r}\right)" display="inline"><mrow><mi href="./33.14#E4">f</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./33.1#p1.t1.r4">ϵ</mi><mo>,</mo><mi class="ltx_nvar" href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi><mo>;</mo><mi class="ltx_nvar" href="./33.1#p1.t1.r2">r</mi><mo>)</mo></mrow></mrow></math>: regular Coulomb function</a>,
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m32.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m34.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m46.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./33.1#p1.t1.r1" title="§33.1 Special Notation ‣ Notation ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m35.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="\ell" display="inline"><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></math>: nonnegative integer</a>,
<a href="./33.1#p1.t1.r2" title="§33.1 Special Notation ‣ Notation ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="r" display="inline"><mi href="./33.1#p1.t1.r2">r</mi></math>: real variable</a>,
<a href="./33.1#p1.t1.r4" title="§33.1 Special Notation ‣ Notation ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="13px" altimg-valign="-2px" altimg-width="12px" alttext="\epsilon" display="inline"><mi href="./33.1#p1.t1.r4">ϵ</mi></math>: real parameter</a>,
<a href="./33.16#E8" title="(33.16.8) ‣ §33.16(iii) f and h in Terms of W κ , μ ( z ) when < ϵ 0 ‣ §33.16 Connection Formulas ‣ Variables r , ϵ ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m42.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./33.16#E8">ν</mi></math>: parameter</a>,
<a href="./33.16#E9" title="(33.16.9) ‣ §33.16(iii) f and h in Terms of W κ , μ ( z ) when < ϵ 0 ‣ §33.16 Connection Formulas ‣ Variables r , ϵ ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="23px" altimg-valign="-7px" altimg-width="65px" alttext="\zeta_{\ell}(\nu,r)" display="inline"><mrow><msub><mi href="./33.16#E9">ζ</mi><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./33.16#E8">ν</mi><mo>,</mo><mi href="./33.1#p1.t1.r2">r</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function</a> and
<a href="./33.16#E9" title="(33.16.9) ‣ §33.16(iii) f and h in Terms of W κ , μ ( z ) when < ϵ 0 ‣ §33.16 Connection Formulas ‣ Variables r , ϵ ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="23px" altimg-valign="-7px" altimg-width="65px" alttext="\xi_{\ell}(\nu,r)" display="inline"><mrow><msub><mi href="./33.16#E9">ξ</mi><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./33.16#E8">ν</mi><mo>,</mo><mi href="./33.1#p1.t1.r2">r</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E11">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">33.16.11</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m19.png" altimg-height="25px" altimg-valign="-7px" altimg-width="80px" alttext="\displaystyle h\left(\epsilon,\ell;r\right)" display="inline"><mrow><mi href="./33.14#E7">h</mi><mo></mo><mrow><mo>(</mo><mi href="./33.1#p1.t1.r4">ϵ</mi><mo>,</mo><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi><mo>;</mo><mi href="./33.1#p1.t1.r2">r</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m3.png" altimg-height="53px" altimg-valign="-21px" altimg-width="586px" alttext="\displaystyle=(-1)^{\ell}\nu^{\ell+1}A(\epsilon,\ell)\left(\frac{\sin\left(\pi%
\nu\right)\zeta_{\ell}(\nu,r)}{\Gamma\left(\ell+1+\nu\right)}+\frac{\cos\left(%
\pi\nu\right)\Gamma\left(\nu-\ell\right)\xi_{\ell}(\nu,r)}{\pi}\right)." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></msup><mo></mo><msup><mi href="./33.16#E8">ν</mi><mrow><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi><mo>+</mo><mn>1</mn></mrow></msup><mo></mo><mrow><mi href="./33.14#E11">A</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./33.1#p1.t1.r4">ϵ</mi><mo>,</mo><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo>(</mo><mrow><mstyle displaystyle="true"><mfrac><mrow><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi href="./33.16#E8">ν</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./33.16#E9">ζ</mi><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./33.16#E8">ν</mi><mo>,</mo><mi href="./33.1#p1.t1.r2">r</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi><mo>+</mo><mn>1</mn><mo>+</mo><mi href="./33.16#E8">ν</mi></mrow><mo>)</mo></mrow></mrow></mfrac></mstyle><mo>+</mo><mstyle displaystyle="true"><mfrac><mrow><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi href="./33.16#E8">ν</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./33.16#E8">ν</mi><mo>-</mo><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./33.16#E9">ξ</mi><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./33.16#E8">ν</mi><mo>,</mo><mi href="./33.1#p1.t1.r2">r</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mi href="./3.12#E1">π</mi></mfrac></mstyle></mrow><mo>)</mo></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E11.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./33.14#E7" title="(33.14.7) ‣ §33.14(iii) Irregular Solution h ( ϵ , ℓ ; r ) ‣ §33.14 Definitions and Basic Properties ‣ Variables r , ϵ ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m55.png" altimg-height="23px" altimg-valign="-7px" altimg-width="78px" alttext="h\left(\NVar{\epsilon},\NVar{\ell};\NVar{r}\right)" display="inline"><mrow><mi href="./33.14#E7">h</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./33.1#p1.t1.r4">ϵ</mi><mo>,</mo><mi class="ltx_nvar" href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi><mo>;</mo><mi class="ltx_nvar" href="./33.1#p1.t1.r2">r</mi><mo>)</mo></mrow></mrow></math>: irregular Coulomb function</a>,
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m32.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m34.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m46.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./33.1#p1.t1.r1" title="§33.1 Special Notation ‣ Notation ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m35.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="\ell" display="inline"><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></math>: nonnegative integer</a>,
<a href="./33.1#p1.t1.r2" title="§33.1 Special Notation ‣ Notation ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="r" display="inline"><mi href="./33.1#p1.t1.r2">r</mi></math>: real variable</a>,
<a href="./33.1#p1.t1.r4" title="§33.1 Special Notation ‣ Notation ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="13px" altimg-valign="-2px" altimg-width="12px" alttext="\epsilon" display="inline"><mi href="./33.1#p1.t1.r4">ϵ</mi></math>: real parameter</a>,
<a href="./33.14#E11" title="(33.14.11) ‣ §33.14(iv) Solutions s ( ϵ , ℓ ; r ) and c ( ϵ , ℓ ; r ) ‣ §33.14 Definitions and Basic Properties ‣ Variables r , ϵ ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="A(\epsilon,\ell)" display="inline"><mrow><mi href="./33.14#E11">A</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./33.1#p1.t1.r4">ϵ</mi><mo>,</mo><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function</a>,
<a href="./33.16#E8" title="(33.16.8) ‣ §33.16(iii) f and h in Terms of W κ , μ ( z ) when < ϵ 0 ‣ §33.16 Connection Formulas ‣ Variables r , ϵ ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m42.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./33.16#E8">ν</mi></math>: parameter</a>,
<a href="./33.16#E9" title="(33.16.9) ‣ §33.16(iii) f and h in Terms of W κ , μ ( z ) when < ϵ 0 ‣ §33.16 Connection Formulas ‣ Variables r , ϵ ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="23px" altimg-valign="-7px" altimg-width="65px" alttext="\zeta_{\ell}(\nu,r)" display="inline"><mrow><msub><mi href="./33.16#E9">ζ</mi><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./33.16#E8">ν</mi><mo>,</mo><mi href="./33.1#p1.t1.r2">r</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function</a> and
<a href="./33.16#E9" title="(33.16.9) ‣ §33.16(iii) f and h in Terms of W κ , μ ( z ) when < ϵ 0 ‣ §33.16 Connection Formulas ‣ Variables r , ϵ ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="23px" altimg-valign="-7px" altimg-width="65px" alttext="\xi_{\ell}(\nu,r)" display="inline"><mrow><msub><mi href="./33.16#E9">ξ</mi><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./33.16#E8">ν</mi><mo>,</mo><mi href="./33.1#p1.t1.r2">r</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
<p class="ltx_p">Alternatively, for <math class="ltx_Math" altimg="m58.png" altimg-height="17px" altimg-valign="-3px" altimg-width="50px" alttext="r<0" display="inline"><mrow><mi href="./33.1#p1.t1.r2">r</mi><mo><</mo><mn>0</mn></mrow></math></p>
<table id="E12" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">33.16.12</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E12.png" altimg-height="56px" altimg-valign="-21px" altimg-width="691px" alttext="f\left(\epsilon,\ell;r\right)=\frac{(-1)^{\ell}\nu^{\ell+1}}{\pi}\left(-\frac{%
\pi\xi_{\ell}(-\nu,r)}{\Gamma\left(\ell+1+\nu\right)}+\sin\left(\pi\nu\right)%
\cos\left(\pi\nu\right)\Gamma\left(\nu-\ell\right)\zeta_{\ell}(-\nu,r)\right)," display="block"><mrow><mrow><mrow><mi href="./33.14#E4">f</mi><mo></mo><mrow><mo>(</mo><mi href="./33.1#p1.t1.r4">ϵ</mi><mo>,</mo><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi><mo>;</mo><mi href="./33.1#p1.t1.r2">r</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mfrac><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></msup><mo></mo><msup><mi href="./33.16#E8">ν</mi><mrow><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><mi href="./3.12#E1">π</mi></mfrac><mo></mo><mrow><mo>(</mo><mrow><mrow><mo>-</mo><mfrac><mrow><mi href="./3.12#E1">π</mi><mo></mo><mrow><msub><mi href="./33.16#E9">ξ</mi><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mi href="./33.16#E8">ν</mi></mrow><mo>,</mo><mi href="./33.1#p1.t1.r2">r</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi><mo>+</mo><mn>1</mn><mo>+</mo><mi href="./33.16#E8">ν</mi></mrow><mo>)</mo></mrow></mrow></mfrac></mrow><mo>+</mo><mrow><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi href="./33.16#E8">ν</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi href="./33.16#E8">ν</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./33.16#E8">ν</mi><mo>-</mo><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./33.16#E9">ζ</mi><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mi href="./33.16#E8">ν</mi></mrow><mo>,</mo><mi href="./33.1#p1.t1.r2">r</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E12.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./33.14#E4" title="(33.14.4) ‣ §33.14(ii) Regular Solution f ( ϵ , ℓ ; r ) ‣ §33.14 Definitions and Basic Properties ‣ Variables r , ϵ ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="23px" altimg-valign="-7px" altimg-width="79px" alttext="f\left(\NVar{\epsilon},\NVar{\ell};\NVar{r}\right)" display="inline"><mrow><mi href="./33.14#E4">f</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./33.1#p1.t1.r4">ϵ</mi><mo>,</mo><mi class="ltx_nvar" href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi><mo>;</mo><mi class="ltx_nvar" href="./33.1#p1.t1.r2">r</mi><mo>)</mo></mrow></mrow></math>: regular Coulomb function</a>,
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m32.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m34.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m46.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./33.1#p1.t1.r1" title="§33.1 Special Notation ‣ Notation ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m35.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="\ell" display="inline"><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></math>: nonnegative integer</a>,
<a href="./33.1#p1.t1.r2" title="§33.1 Special Notation ‣ Notation ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="r" display="inline"><mi href="./33.1#p1.t1.r2">r</mi></math>: real variable</a>,
<a href="./33.1#p1.t1.r4" title="§33.1 Special Notation ‣ Notation ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="13px" altimg-valign="-2px" altimg-width="12px" alttext="\epsilon" display="inline"><mi href="./33.1#p1.t1.r4">ϵ</mi></math>: real parameter</a>,
<a href="./33.16#E8" title="(33.16.8) ‣ §33.16(iii) f and h in Terms of W κ , μ ( z ) when < ϵ 0 ‣ §33.16 Connection Formulas ‣ Variables r , ϵ ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m42.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./33.16#E8">ν</mi></math>: parameter</a>,
<a href="./33.16#E9" title="(33.16.9) ‣ §33.16(iii) f and h in Terms of W κ , μ ( z ) when < ϵ 0 ‣ §33.16 Connection Formulas ‣ Variables r , ϵ ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="23px" altimg-valign="-7px" altimg-width="65px" alttext="\zeta_{\ell}(\nu,r)" display="inline"><mrow><msub><mi href="./33.16#E9">ζ</mi><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./33.16#E8">ν</mi><mo>,</mo><mi href="./33.1#p1.t1.r2">r</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function</a> and
<a href="./33.16#E9" title="(33.16.9) ‣ §33.16(iii) f and h in Terms of W κ , μ ( z ) when < ϵ 0 ‣ §33.16 Connection Formulas ‣ Variables r , ϵ ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="23px" altimg-valign="-7px" altimg-width="65px" alttext="\xi_{\ell}(\nu,r)" display="inline"><mrow><msub><mi href="./33.16#E9">ξ</mi><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./33.16#E8">ν</mi><mo>,</mo><mi href="./33.1#p1.t1.r2">r</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E13" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">33.16.13</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E13.png" altimg-height="28px" altimg-valign="-7px" altimg-width="432px" alttext="h\left(\epsilon,\ell;r\right)=(-1)^{\ell}\nu^{\ell+1}A(\epsilon,\ell)\Gamma%
\left(\nu-\ell\right)\zeta_{\ell}(-\nu,r)/\pi." display="block"><mrow><mrow><mrow><mi href="./33.14#E7">h</mi><mo></mo><mrow><mo>(</mo><mi href="./33.1#p1.t1.r4">ϵ</mi><mo>,</mo><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi><mo>;</mo><mi href="./33.1#p1.t1.r2">r</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></msup><mo></mo><msup><mi href="./33.16#E8">ν</mi><mrow><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi><mo>+</mo><mn>1</mn></mrow></msup><mo></mo><mrow><mi href="./33.14#E11">A</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./33.1#p1.t1.r4">ϵ</mi><mo>,</mo><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./33.16#E8">ν</mi><mo>-</mo><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./33.16#E9">ζ</mi><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mi href="./33.16#E8">ν</mi></mrow><mo>,</mo><mi href="./33.1#p1.t1.r2">r</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>/</mo><mi href="./3.12#E1">π</mi></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E13.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./33.14#E7" title="(33.14.7) ‣ §33.14(iii) Irregular Solution h ( ϵ , ℓ ; r ) ‣ §33.14 Definitions and Basic Properties ‣ Variables r , ϵ ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m55.png" altimg-height="23px" altimg-valign="-7px" altimg-width="78px" alttext="h\left(\NVar{\epsilon},\NVar{\ell};\NVar{r}\right)" display="inline"><mrow><mi href="./33.14#E7">h</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./33.1#p1.t1.r4">ϵ</mi><mo>,</mo><mi class="ltx_nvar" href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi><mo>;</mo><mi class="ltx_nvar" href="./33.1#p1.t1.r2">r</mi><mo>)</mo></mrow></mrow></math>: irregular Coulomb function</a>,
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m32.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./33.1#p1.t1.r1" title="§33.1 Special Notation ‣ Notation ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m35.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="\ell" display="inline"><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></math>: nonnegative integer</a>,
<a href="./33.1#p1.t1.r2" title="§33.1 Special Notation ‣ Notation ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="r" display="inline"><mi href="./33.1#p1.t1.r2">r</mi></math>: real variable</a>,
<a href="./33.1#p1.t1.r4" title="§33.1 Special Notation ‣ Notation ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="13px" altimg-valign="-2px" altimg-width="12px" alttext="\epsilon" display="inline"><mi href="./33.1#p1.t1.r4">ϵ</mi></math>: real parameter</a>,
<a href="./33.14#E11" title="(33.14.11) ‣ §33.14(iv) Solutions s ( ϵ , ℓ ; r ) and c ( ϵ , ℓ ; r ) ‣ §33.14 Definitions and Basic Properties ‣ Variables r , ϵ ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="A(\epsilon,\ell)" display="inline"><mrow><mi href="./33.14#E11">A</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./33.1#p1.t1.r4">ϵ</mi><mo>,</mo><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function</a>,
<a href="./33.16#E8" title="(33.16.8) ‣ §33.16(iii) f and h in Terms of W κ , μ ( z ) when < ϵ 0 ‣ §33.16 Connection Formulas ‣ Variables r , ϵ ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m42.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./33.16#E8">ν</mi></math>: parameter</a> and
<a href="./33.16#E9" title="(33.16.9) ‣ §33.16(iii) f and h in Terms of W κ , μ ( z ) when < ϵ 0 ‣ §33.16 Connection Formulas ‣ Variables r , ϵ ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="23px" altimg-valign="-7px" altimg-width="65px" alttext="\zeta_{\ell}(\nu,r)" display="inline"><mrow><msub><mi href="./33.16#E9">ζ</mi><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./33.16#E8">ν</mi><mo>,</mo><mi href="./33.1#p1.t1.r2">r</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="iv" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§33.16(iv) </span><math class="ltx_Math" altimg="m61.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="s" display="inline"><mi href="./33.14#E9">s</mi></math> and <math class="ltx_Math" altimg="m50.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="c" display="inline"><mi href="./33.14#E9">c</mi></math> in Terms of
<math class="ltx_Math" altimg="m26.png" altimg-height="21px" altimg-valign="-5px" altimg-width="25px" alttext="F_{\ell}" display="inline"><msub><mi href="./33.2#E3">F</mi><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></msub></math> and <math class="ltx_Math" altimg="m28.png" altimg-height="21px" altimg-valign="-5px" altimg-width="28px" alttext="G_{\ell}" display="inline"><msub><mi href="./33.2#E11">G</mi><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></msub></math> when <math class="ltx_Math" altimg="m37.png" altimg-height="17px" altimg-valign="-3px" altimg-width="49px" alttext="\epsilon>0" display="inline"><mrow><mi href="./33.1#p1.t1.r4">ϵ</mi><mo>></mo><mn>0</mn></mrow></math>
</h2>
<div id="SS4.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Keywords:</dt>
<dd><a class="ltx_keyword" href="./idx/C#Coulombfunctionsvariablesr">Coulomb functions: variables <math class="ltx_Math" altimg="m57.png" altimg-height="16px" altimg-valign="-6px" altimg-width="30px" alttext="r,\epsilon" display="inline"><mrow><mi href="./33.1#p1.t1.r2">r</mi><mo>,</mo><mi href="./33.1#p1.t1.r4">ϵ</mi></mrow></math></a></dd>
</dl>
</div>
</div>
<div id="SS4.p1" class="ltx_para">
<p class="ltx_p">When <math class="ltx_Math" altimg="m37.png" altimg-height="17px" altimg-valign="-3px" altimg-width="49px" alttext="\epsilon>0" display="inline"><mrow><mi href="./33.1#p1.t1.r4">ϵ</mi><mo>></mo><mn>0</mn></mrow></math>, again denote <math class="ltx_Math" altimg="m47.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\tau" display="inline"><mi href="./33.16#E3">τ</mi></math> by (). Then for
<math class="ltx_Math" altimg="m59.png" altimg-height="17px" altimg-valign="-3px" altimg-width="50px" alttext="r>0" display="inline"><mrow><mi href="./33.1#p1.t1.r2">r</mi><mo>></mo><mn>0</mn></mrow></math></p>
<table id="E14" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex3" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">33.16.14</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m20.png" altimg-height="25px" altimg-valign="-7px" altimg-width="78px" alttext="\displaystyle s\left(\epsilon,\ell;r\right)" display="inline"><mrow><mi href="./33.14#E9">s</mi><mo></mo><mrow><mo>(</mo><mi href="./33.1#p1.t1.r4">ϵ</mi><mo>,</mo><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi><mo>;</mo><mi href="./33.1#p1.t1.r2">r</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m5.png" altimg-height="29px" altimg-valign="-7px" altimg-width="226px" alttext="\displaystyle=(\pi\tau)^{-1/2}F_{\ell}\left(-1/\tau,\tau r\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi href="./33.16#E3">τ</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></mrow></msup><mo></mo><mrow><msub><mi href="./33.2#E3">F</mi><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mrow><mn>1</mn><mo>/</mo><mi href="./33.16#E3">τ</mi></mrow></mrow><mo>,</mo><mrow><mi href="./33.16#E3">τ</mi><mo></mo><mi href="./33.1#p1.t1.r2">r</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex4" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m17.png" altimg-height="25px" altimg-valign="-7px" altimg-width="77px" alttext="\displaystyle c\left(\epsilon,\ell;r\right)" display="inline"><mrow><mi href="./33.14#E9">c</mi><mo></mo><mrow><mo>(</mo><mi href="./33.1#p1.t1.r4">ϵ</mi><mo>,</mo><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi><mo>;</mo><mi href="./33.1#p1.t1.r2">r</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m7.png" altimg-height="29px" altimg-valign="-7px" altimg-width="229px" alttext="\displaystyle=(\pi\tau)^{-1/2}G_{\ell}\left(-1/\tau,\tau r\right)." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi href="./33.16#E3">τ</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></mrow></msup><mo></mo><mrow><msub><mi href="./33.2#E11">G</mi><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mrow><mn>1</mn><mo>/</mo><mi href="./33.16#E3">τ</mi></mrow></mrow><mo>,</mo><mrow><mi href="./33.16#E3">τ</mi><mo></mo><mi href="./33.1#p1.t1.r2">r</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E14.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./33.14#E9" title="(33.14.9) ‣ §33.14(iv) Solutions s ( ϵ , ℓ ; r ) and c ( ϵ , ℓ ; r ) ‣ §33.14 Definitions and Basic Properties ‣ Variables r , ϵ ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="23px" altimg-valign="-7px" altimg-width="75px" alttext="c\left(\NVar{\epsilon},\NVar{\ell};\NVar{r}\right)" display="inline"><mrow><mi href="./33.14#E9">c</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./33.1#p1.t1.r4">ϵ</mi><mo>,</mo><mi class="ltx_nvar" href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi><mo>;</mo><mi class="ltx_nvar" href="./33.1#p1.t1.r2">r</mi><mo>)</mo></mrow></mrow></math>: irregular Coulomb function</a>,
<a href="./33.14#E9" title="(33.14.9) ‣ §33.14(iv) Solutions s ( ϵ , ℓ ; r ) and c ( ϵ , ℓ ; r ) ‣ §33.14 Definitions and Basic Properties ‣ Variables r , ϵ ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m62.png" altimg-height="23px" altimg-valign="-7px" altimg-width="76px" alttext="s\left(\NVar{\epsilon},\NVar{\ell};\NVar{r}\right)" display="inline"><mrow><mi href="./33.14#E9">s</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./33.1#p1.t1.r4">ϵ</mi><mo>,</mo><mi class="ltx_nvar" href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi><mo>;</mo><mi class="ltx_nvar" href="./33.1#p1.t1.r2">r</mi><mo>)</mo></mrow></mrow></math>: regular Coulomb function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./33.2#E11" title="(33.2.11) ‣ §33.2(iii) Irregular Solutions G ℓ ( η , ρ ) , H ± ℓ ( η , ρ ) ‣ §33.2 Definitions and Basic Properties ‣ Variables ρ , η ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m27.png" altimg-height="23px" altimg-valign="-7px" altimg-width="76px" alttext="G_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}\right)" display="inline"><mrow><msub><mi href="./33.2#E11">G</mi><mi class="ltx_nvar" href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./33.1#p1.t1.r4">η</mi><mo>,</mo><mi class="ltx_nvar" href="./33.1#p1.t1.r3">ρ</mi><mo>)</mo></mrow></mrow></math>: irregular Coulomb radial function</a>,
<a href="./33.2#E3" title="(33.2.3) ‣ §33.2(ii) Regular Solution F ℓ ( η , ρ ) ‣ §33.2 Definitions and Basic Properties ‣ Variables ρ , η ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m25.png" altimg-height="23px" altimg-valign="-7px" altimg-width="73px" alttext="F_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}\right)" display="inline"><mrow><msub><mi href="./33.2#E3">F</mi><mi class="ltx_nvar" href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./33.1#p1.t1.r4">η</mi><mo>,</mo><mi class="ltx_nvar" href="./33.1#p1.t1.r3">ρ</mi><mo>)</mo></mrow></mrow></math>: regular Coulomb radial function</a>,
<a href="./33.1#p1.t1.r1" title="§33.1 Special Notation ‣ Notation ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m35.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="\ell" display="inline"><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></math>: nonnegative integer</a>,
<a href="./33.1#p1.t1.r2" title="§33.1 Special Notation ‣ Notation ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="r" display="inline"><mi href="./33.1#p1.t1.r2">r</mi></math>: real variable</a>,
<a href="./33.1#p1.t1.r4" title="§33.1 Special Notation ‣ Notation ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="13px" altimg-valign="-2px" altimg-width="12px" alttext="\epsilon" display="inline"><mi href="./33.1#p1.t1.r4">ϵ</mi></math>: real parameter</a> and
<a href="./33.16#E3" title="(33.16.3) ‣ §33.16(ii) f and h in Terms of F ℓ and G ℓ when > ϵ 0 ‣ §33.16 Connection Formulas ‣ Variables r , ϵ ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\tau" display="inline"><mi href="./33.16#E3">τ</mi></math>: parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Alternatively, for <math class="ltx_Math" altimg="m58.png" altimg-height="17px" altimg-valign="-3px" altimg-width="50px" alttext="r<0" display="inline"><mrow><mi href="./33.1#p1.t1.r2">r</mi><mo><</mo><mn>0</mn></mrow></math></p>
<table id="E15" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex5" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">33.16.15</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m20.png" altimg-height="25px" altimg-valign="-7px" altimg-width="78px" alttext="\displaystyle s\left(\epsilon,\ell;r\right)" display="inline"><mrow><mi href="./33.14#E9">s</mi><mo></mo><mrow><mo>(</mo><mi href="./33.1#p1.t1.r4">ϵ</mi><mo>,</mo><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi><mo>;</mo><mi href="./33.1#p1.t1.r2">r</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m6.png" altimg-height="29px" altimg-valign="-7px" altimg-width="226px" alttext="\displaystyle=(\pi\tau)^{-1/2}F_{\ell}\left(1/\tau,-\tau r\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi href="./33.16#E3">τ</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></mrow></msup><mo></mo><mrow><msub><mi href="./33.2#E3">F</mi><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>/</mo><mi href="./33.16#E3">τ</mi></mrow><mo>,</mo><mrow><mo>-</mo><mrow><mi href="./33.16#E3">τ</mi><mo></mo><mi href="./33.1#p1.t1.r2">r</mi></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex6" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m17.png" altimg-height="25px" altimg-valign="-7px" altimg-width="77px" alttext="\displaystyle c\left(\epsilon,\ell;r\right)" display="inline"><mrow><mi href="./33.14#E9">c</mi><mo></mo><mrow><mo>(</mo><mi href="./33.1#p1.t1.r4">ϵ</mi><mo>,</mo><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi><mo>;</mo><mi href="./33.1#p1.t1.r2">r</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m8.png" altimg-height="29px" altimg-valign="-7px" altimg-width="229px" alttext="\displaystyle=(\pi\tau)^{-1/2}G_{\ell}\left(1/\tau,-\tau r\right)." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi href="./33.16#E3">τ</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></mrow></msup><mo></mo><mrow><msub><mi href="./33.2#E11">G</mi><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>/</mo><mi href="./33.16#E3">τ</mi></mrow><mo>,</mo><mrow><mo>-</mo><mrow><mi href="./33.16#E3">τ</mi><mo></mo><mi href="./33.1#p1.t1.r2">r</mi></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E15.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./33.14#E9" title="(33.14.9) ‣ §33.14(iv) Solutions s ( ϵ , ℓ ; r ) and c ( ϵ , ℓ ; r ) ‣ §33.14 Definitions and Basic Properties ‣ Variables r , ϵ ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="23px" altimg-valign="-7px" altimg-width="75px" alttext="c\left(\NVar{\epsilon},\NVar{\ell};\NVar{r}\right)" display="inline"><mrow><mi href="./33.14#E9">c</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./33.1#p1.t1.r4">ϵ</mi><mo>,</mo><mi class="ltx_nvar" href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi><mo>;</mo><mi class="ltx_nvar" href="./33.1#p1.t1.r2">r</mi><mo>)</mo></mrow></mrow></math>: irregular Coulomb function</a>,
<a href="./33.14#E9" title="(33.14.9) ‣ §33.14(iv) Solutions s ( ϵ , ℓ ; r ) and c ( ϵ , ℓ ; r ) ‣ §33.14 Definitions and Basic Properties ‣ Variables r , ϵ ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m62.png" altimg-height="23px" altimg-valign="-7px" altimg-width="76px" alttext="s\left(\NVar{\epsilon},\NVar{\ell};\NVar{r}\right)" display="inline"><mrow><mi href="./33.14#E9">s</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./33.1#p1.t1.r4">ϵ</mi><mo>,</mo><mi class="ltx_nvar" href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi><mo>;</mo><mi class="ltx_nvar" href="./33.1#p1.t1.r2">r</mi><mo>)</mo></mrow></mrow></math>: regular Coulomb function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./33.2#E11" title="(33.2.11) ‣ §33.2(iii) Irregular Solutions G ℓ ( η , ρ ) , H ± ℓ ( η , ρ ) ‣ §33.2 Definitions and Basic Properties ‣ Variables ρ , η ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m27.png" altimg-height="23px" altimg-valign="-7px" altimg-width="76px" alttext="G_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}\right)" display="inline"><mrow><msub><mi href="./33.2#E11">G</mi><mi class="ltx_nvar" href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./33.1#p1.t1.r4">η</mi><mo>,</mo><mi class="ltx_nvar" href="./33.1#p1.t1.r3">ρ</mi><mo>)</mo></mrow></mrow></math>: irregular Coulomb radial function</a>,
<a href="./33.2#E3" title="(33.2.3) ‣ §33.2(ii) Regular Solution F ℓ ( η , ρ ) ‣ §33.2 Definitions and Basic Properties ‣ Variables ρ , η ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m25.png" altimg-height="23px" altimg-valign="-7px" altimg-width="73px" alttext="F_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}\right)" display="inline"><mrow><msub><mi href="./33.2#E3">F</mi><mi class="ltx_nvar" href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./33.1#p1.t1.r4">η</mi><mo>,</mo><mi class="ltx_nvar" href="./33.1#p1.t1.r3">ρ</mi><mo>)</mo></mrow></mrow></math>: regular Coulomb radial function</a>,
<a href="./33.1#p1.t1.r1" title="§33.1 Special Notation ‣ Notation ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m35.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="\ell" display="inline"><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></math>: nonnegative integer</a>,
<a href="./33.1#p1.t1.r2" title="§33.1 Special Notation ‣ Notation ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="r" display="inline"><mi href="./33.1#p1.t1.r2">r</mi></math>: real variable</a>,
<a href="./33.1#p1.t1.r4" title="§33.1 Special Notation ‣ Notation ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="13px" altimg-valign="-2px" altimg-width="12px" alttext="\epsilon" display="inline"><mi href="./33.1#p1.t1.r4">ϵ</mi></math>: real parameter</a> and
<a href="./33.16#E3" title="(33.16.3) ‣ §33.16(ii) f and h in Terms of F ℓ and G ℓ when > ϵ 0 ‣ §33.16 Connection Formulas ‣ Variables r , ϵ ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\tau" display="inline"><mi href="./33.16#E3">τ</mi></math>: parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="v" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§33.16(v) </span><math class="ltx_Math" altimg="m61.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="s" display="inline"><mi href="./33.14#E9">s</mi></math> and <math class="ltx_Math" altimg="m50.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="c" display="inline"><mi href="./33.14#E9">c</mi></math> in Terms of
<math class="ltx_Math" altimg="m31.png" altimg-height="24px" altimg-valign="-8px" altimg-width="77px" alttext="W_{\kappa,\mu}\left(z\right)" display="inline"><mrow><msub><mi href="./13.14#E3">W</mi><mrow><mi>κ</mi><mo href="./13.14#E3">,</mo><mi>μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></math> when <math class="ltx_Math" altimg="m36.png" altimg-height="17px" altimg-valign="-3px" altimg-width="49px" alttext="\epsilon<0" display="inline"><mrow><mi href="./33.1#p1.t1.r4">ϵ</mi><mo><</mo><mn>0</mn></mrow></math>
</h2>
<div id="SS5.info" class="ltx_metadata ltx_info">
, <a class="ltx_keyword" href="./idx/C#Coulombfunctionsvariablesr">Coulomb functions: variables <math class="ltx_Math" altimg="m57.png" altimg-height="16px" altimg-valign="-6px" altimg-width="30px" alttext="r,\epsilon" display="inline"><mrow><mi href="./33.1#p1.t1.r2">r</mi><mo>,</mo><mi href="./33.1#p1.t1.r4">ϵ</mi></mrow></math></a>, </dd>
</dl>
</div>
</div>
<div id="SS5.p1" class="ltx_para">
<p class="ltx_p">When <math class="ltx_Math" altimg="m36.png" altimg-height="17px" altimg-valign="-3px" altimg-width="49px" alttext="\epsilon<0" display="inline"><mrow><mi href="./33.1#p1.t1.r4">ϵ</mi><mo><</mo><mn>0</mn></mrow></math> denote <math class="ltx_Math" altimg="m42.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./33.16#E8">ν</mi></math>, <math class="ltx_Math" altimg="m49.png" altimg-height="23px" altimg-valign="-7px" altimg-width="65px" alttext="\zeta_{\ell}(\nu,r)" display="inline"><mrow><msub><mi href="./33.16#E9">ζ</mi><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./33.16#E8">ν</mi><mo>,</mo><mi href="./33.1#p1.t1.r2">r</mi><mo stretchy="false">)</mo></mrow></mrow></math>, and <math class="ltx_Math" altimg="m48.png" altimg-height="23px" altimg-valign="-7px" altimg-width="65px" alttext="\xi_{\ell}(\nu,r)" display="inline"><mrow><msub><mi href="./33.16#E9">ξ</mi><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./33.16#E8">ν</mi><mo>,</mo><mi href="./33.1#p1.t1.r2">r</mi><mo stretchy="false">)</mo></mrow></mrow></math> by
(). Also denote</p>
<table id="E16" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">33.16.16</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E16.png" altimg-height="35px" altimg-valign="-9px" altimg-width="364px" alttext="K(\nu,\ell)=\left(\nu^{2}\Gamma\left(\nu+\ell+1\right)\Gamma\left(\nu-\ell%
\right)\right)^{-1/2}." display="block"><mrow><mrow><mrow><mi href="./33.16#E16">K</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./33.16#E8">ν</mi><mo>,</mo><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><msup><mrow><mo>(</mo><mrow><msup><mi href="./33.16#E8">ν</mi><mn>2</mn></msup><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./33.16#E8">ν</mi><mo>+</mo><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./33.16#E8">ν</mi><mo>-</mo><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>)</mo></mrow><mrow><mo>-</mo><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></mrow></msup></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E16.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text"><math class="ltx_Math" altimg="m29.png" altimg-height="23px" altimg-valign="-7px" altimg-width="65px" alttext="K(\nu,\ell)" display="inline"><mrow><mi href="./33.16#E16">K</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./33.16#E8">ν</mi><mo>,</mo><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi><mo stretchy="false">)</mo></mrow></mrow></math>: factor (locally)</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m32.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./33.1#p1.t1.r1" title="§33.1 Special Notation ‣ Notation ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m35.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="\ell" display="inline"><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></math>: nonnegative integer</a> and
<a href="./33.16#E8" title="(33.16.8) ‣ §33.16(iii) f and h in Terms of W κ , μ ( z ) when < ϵ 0 ‣ §33.16 Connection Formulas ‣ Variables r , ϵ ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m42.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./33.16#E8">ν</mi></math>: parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Then for <math class="ltx_Math" altimg="m59.png" altimg-height="17px" altimg-valign="-3px" altimg-width="50px" alttext="r>0" display="inline"><mrow><mi href="./33.1#p1.t1.r2">r</mi><mo>></mo><mn>0</mn></mrow></math></p>
<table id="E17" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex7" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">33.16.17</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m20.png" altimg-height="25px" altimg-valign="-7px" altimg-width="78px" alttext="\displaystyle s\left(\epsilon,\ell;r\right)" display="inline"><mrow><mi href="./33.14#E9">s</mi><mo></mo><mrow><mo>(</mo><mi href="./33.1#p1.t1.r4">ϵ</mi><mo>,</mo><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi><mo>;</mo><mi href="./33.1#p1.t1.r2">r</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m13.png" altimg-height="56px" altimg-valign="-21px" altimg-width="498px" alttext="\displaystyle=\frac{(-1)^{\ell}}{2\nu^{1/2}}\left(\frac{\sin\left(\pi\nu\right%
)}{\pi K(\nu,\ell)}\xi_{\ell}(\nu,r)-\cos\left(\pi\nu\right)\nu^{2}K(\nu,\ell)%
\zeta_{\ell}(\nu,r)\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><mfrac><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></msup><mrow><mn>2</mn><mo></mo><msup><mi href="./33.16#E8">ν</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></mfrac></mstyle><mo></mo><mrow><mo>(</mo><mrow><mrow><mstyle displaystyle="true"><mfrac><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi href="./33.16#E8">ν</mi></mrow><mo>)</mo></mrow></mrow><mrow><mi href="./3.12#E1">π</mi><mo></mo><mrow><mi href="./33.16#E16">K</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./33.16#E8">ν</mi><mo>,</mo><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></mfrac></mstyle><mo></mo><mrow><msub><mi href="./33.16#E9">ξ</mi><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./33.16#E8">ν</mi><mo>,</mo><mi href="./33.1#p1.t1.r2">r</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>-</mo><mrow><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi href="./33.16#E8">ν</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><msup><mi href="./33.16#E8">ν</mi><mn>2</mn></msup><mo></mo><mrow><mi href="./33.16#E16">K</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./33.16#E8">ν</mi><mo>,</mo><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./33.16#E9">ζ</mi><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./33.16#E8">ν</mi><mo>,</mo><mi href="./33.1#p1.t1.r2">r</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex8" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m17.png" altimg-height="25px" altimg-valign="-7px" altimg-width="77px" alttext="\displaystyle c\left(\epsilon,\ell;r\right)" display="inline"><mrow><mi href="./33.14#E9">c</mi><mo></mo><mrow><mo>(</mo><mi href="./33.1#p1.t1.r4">ϵ</mi><mo>,</mo><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi><mo>;</mo><mi href="./33.1#p1.t1.r2">r</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m12.png" altimg-height="56px" altimg-valign="-21px" altimg-width="496px" alttext="\displaystyle=\frac{(-1)^{\ell}}{2\nu^{1/2}}\left(\frac{\cos\left(\pi\nu\right%
)}{\pi K(\nu,\ell)}\xi_{\ell}(\nu,r)+\sin\left(\pi\nu\right)\nu^{2}K(\nu,\ell)%
\zeta_{\ell}(\nu,r)\right)." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><mfrac><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></msup><mrow><mn>2</mn><mo></mo><msup><mi href="./33.16#E8">ν</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></mfrac></mstyle><mo></mo><mrow><mo>(</mo><mrow><mrow><mstyle displaystyle="true"><mfrac><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi href="./33.16#E8">ν</mi></mrow><mo>)</mo></mrow></mrow><mrow><mi href="./3.12#E1">π</mi><mo></mo><mrow><mi href="./33.16#E16">K</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./33.16#E8">ν</mi><mo>,</mo><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></mfrac></mstyle><mo></mo><mrow><msub><mi href="./33.16#E9">ξ</mi><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./33.16#E8">ν</mi><mo>,</mo><mi href="./33.1#p1.t1.r2">r</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi href="./33.16#E8">ν</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><msup><mi href="./33.16#E8">ν</mi><mn>2</mn></msup><mo></mo><mrow><mi href="./33.16#E16">K</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./33.16#E8">ν</mi><mo>,</mo><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./33.16#E9">ζ</mi><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./33.16#E8">ν</mi><mo>,</mo><mi href="./33.1#p1.t1.r2">r</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E17.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./33.14#E9" title="(33.14.9) ‣ §33.14(iv) Solutions s ( ϵ , ℓ ; r ) and c ( ϵ , ℓ ; r ) ‣ §33.14 Definitions and Basic Properties ‣ Variables r , ϵ ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="23px" altimg-valign="-7px" altimg-width="75px" alttext="c\left(\NVar{\epsilon},\NVar{\ell};\NVar{r}\right)" display="inline"><mrow><mi href="./33.14#E9">c</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./33.1#p1.t1.r4">ϵ</mi><mo>,</mo><mi class="ltx_nvar" href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi><mo>;</mo><mi class="ltx_nvar" href="./33.1#p1.t1.r2">r</mi><mo>)</mo></mrow></mrow></math>: irregular Coulomb function</a>,
<a href="./33.14#E9" title="(33.14.9) ‣ §33.14(iv) Solutions s ( ϵ , ℓ ; r ) and c ( ϵ , ℓ ; r ) ‣ §33.14 Definitions and Basic Properties ‣ Variables r , ϵ ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m62.png" altimg-height="23px" altimg-valign="-7px" altimg-width="76px" alttext="s\left(\NVar{\epsilon},\NVar{\ell};\NVar{r}\right)" display="inline"><mrow><mi href="./33.14#E9">s</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./33.1#p1.t1.r4">ϵ</mi><mo>,</mo><mi class="ltx_nvar" href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi><mo>;</mo><mi class="ltx_nvar" href="./33.1#p1.t1.r2">r</mi><mo>)</mo></mrow></mrow></math>: regular Coulomb function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m34.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m46.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./33.1#p1.t1.r1" title="§33.1 Special Notation ‣ Notation ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m35.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="\ell" display="inline"><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></math>: nonnegative integer</a>,
<a href="./33.1#p1.t1.r2" title="§33.1 Special Notation ‣ Notation ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="r" display="inline"><mi href="./33.1#p1.t1.r2">r</mi></math>: real variable</a>,
<a href="./33.1#p1.t1.r4" title="§33.1 Special Notation ‣ Notation ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="13px" altimg-valign="-2px" altimg-width="12px" alttext="\epsilon" display="inline"><mi href="./33.1#p1.t1.r4">ϵ</mi></math>: real parameter</a>,
<a href="./33.16#E8" title="(33.16.8) ‣ §33.16(iii) f and h in Terms of W κ , μ ( z ) when < ϵ 0 ‣ §33.16 Connection Formulas ‣ Variables r , ϵ ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m42.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./33.16#E8">ν</mi></math>: parameter</a>,
<a href="./33.16#E9" title="(33.16.9) ‣ §33.16(iii) f and h in Terms of W κ , μ ( z ) when < ϵ 0 ‣ §33.16 Connection Formulas ‣ Variables r , ϵ ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="23px" altimg-valign="-7px" altimg-width="65px" alttext="\zeta_{\ell}(\nu,r)" display="inline"><mrow><msub><mi href="./33.16#E9">ζ</mi><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./33.16#E8">ν</mi><mo>,</mo><mi href="./33.1#p1.t1.r2">r</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function</a>,
<a href="./33.16#E9" title="(33.16.9) ‣ §33.16(iii) f and h in Terms of W κ , μ ( z ) when < ϵ 0 ‣ §33.16 Connection Formulas ‣ Variables r , ϵ ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="23px" altimg-valign="-7px" altimg-width="65px" alttext="\xi_{\ell}(\nu,r)" display="inline"><mrow><msub><mi href="./33.16#E9">ξ</mi><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./33.16#E8">ν</mi><mo>,</mo><mi href="./33.1#p1.t1.r2">r</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function</a> and
<a href="./33.16#E16" title="(33.16.16) ‣ §33.16(v) s and c in Terms of W κ , μ ( z ) when < ϵ 0 ‣ §33.16 Connection Formulas ‣ Variables r , ϵ ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="23px" altimg-valign="-7px" altimg-width="65px" alttext="K(\nu,\ell)" display="inline"><mrow><mi href="./33.16#E16">K</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./33.16#E8">ν</mi><mo>,</mo><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi><mo stretchy="false">)</mo></mrow></mrow></math>: factor</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Alternatively, for <math class="ltx_Math" altimg="m58.png" altimg-height="17px" altimg-valign="-3px" altimg-width="50px" alttext="r<0" display="inline"><mrow><mi href="./33.1#p1.t1.r2">r</mi><mo><</mo><mn>0</mn></mrow></math></p>
<table id="E18" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex9" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">33.16.18</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m20.png" altimg-height="25px" altimg-valign="-7px" altimg-width="78px" alttext="\displaystyle s\left(\epsilon,\ell;r\right)" display="inline"><mrow><mi href="./33.14#E9">s</mi><mo></mo><mrow><mo>(</mo><mi href="./33.1#p1.t1.r4">ϵ</mi><mo>,</mo><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi><mo>;</mo><mi href="./33.1#p1.t1.r2">r</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m11.png" altimg-height="56px" altimg-valign="-21px" altimg-width="588px" alttext="\displaystyle=\frac{(-1)^{\ell+1}}{2^{1/2}}\left(\frac{\nu^{3/2}}{K(\nu,\ell)}%
\xi_{\ell}(-\nu,r)-\frac{\sin\left(\pi\nu\right)\cos\left(\pi\nu\right)}{\pi%
\nu^{1/2}}K(\nu,\ell)\zeta_{\ell}(-\nu,r)\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><mfrac><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mrow><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi><mo>+</mo><mn>1</mn></mrow></msup><msup><mn>2</mn><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mfrac></mstyle><mo></mo><mrow><mo>(</mo><mrow><mrow><mstyle displaystyle="true"><mfrac><msup><mi href="./33.16#E8">ν</mi><mrow><mn>3</mn><mo>/</mo><mn>2</mn></mrow></msup><mrow><mi href="./33.16#E16">K</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./33.16#E8">ν</mi><mo>,</mo><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi><mo stretchy="false">)</mo></mrow></mrow></mfrac></mstyle><mo></mo><mrow><msub><mi href="./33.16#E9">ξ</mi><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mi href="./33.16#E8">ν</mi></mrow><mo>,</mo><mi href="./33.1#p1.t1.r2">r</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>-</mo><mrow><mstyle displaystyle="true"><mfrac><mrow><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi href="./33.16#E8">ν</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi href="./33.16#E8">ν</mi></mrow><mo>)</mo></mrow></mrow></mrow><mrow><mi href="./3.12#E1">π</mi><mo></mo><msup><mi href="./33.16#E8">ν</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></mfrac></mstyle><mo></mo><mrow><mi href="./33.16#E16">K</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./33.16#E8">ν</mi><mo>,</mo><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./33.16#E9">ζ</mi><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mi href="./33.16#E8">ν</mi></mrow><mo>,</mo><mi href="./33.1#p1.t1.r2">r</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex10" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m17.png" altimg-height="25px" altimg-valign="-7px" altimg-width="77px" alttext="\displaystyle c\left(\epsilon,\ell;r\right)" display="inline"><mrow><mi href="./33.14#E9">c</mi><mo></mo><mrow><mo>(</mo><mi href="./33.1#p1.t1.r4">ϵ</mi><mo>,</mo><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi><mo>;</mo><mi href="./33.1#p1.t1.r2">r</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m14.png" altimg-height="55px" altimg-valign="-21px" altimg-width="248px" alttext="\displaystyle=\frac{(-1)^{\ell}}{\pi(2\nu)^{1/2}}K(\nu,\ell)\zeta_{\ell}(-\nu,%
r)." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><mfrac><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></msup><mrow><mi href="./3.12#E1">π</mi><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mn>2</mn><mo></mo><mi href="./33.16#E8">ν</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></mfrac></mstyle><mo></mo><mrow><mi href="./33.16#E16">K</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./33.16#E8">ν</mi><mo>,</mo><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./33.16#E9">ζ</mi><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mi href="./33.16#E8">ν</mi></mrow><mo>,</mo><mi href="./33.1#p1.t1.r2">r</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E18.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./33.14#E9" title="(33.14.9) ‣ §33.14(iv) Solutions s ( ϵ , ℓ ; r ) and c ( ϵ , ℓ ; r ) ‣ §33.14 Definitions and Basic Properties ‣ Variables r , ϵ ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="23px" altimg-valign="-7px" altimg-width="75px" alttext="c\left(\NVar{\epsilon},\NVar{\ell};\NVar{r}\right)" display="inline"><mrow><mi href="./33.14#E9">c</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./33.1#p1.t1.r4">ϵ</mi><mo>,</mo><mi class="ltx_nvar" href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi><mo>;</mo><mi class="ltx_nvar" href="./33.1#p1.t1.r2">r</mi><mo>)</mo></mrow></mrow></math>: irregular Coulomb function</a>,
<a href="./33.14#E9" title="(33.14.9) ‣ §33.14(iv) Solutions s ( ϵ , ℓ ; r ) and c ( ϵ , ℓ ; r ) ‣ §33.14 Definitions and Basic Properties ‣ Variables r , ϵ ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m62.png" altimg-height="23px" altimg-valign="-7px" altimg-width="76px" alttext="s\left(\NVar{\epsilon},\NVar{\ell};\NVar{r}\right)" display="inline"><mrow><mi href="./33.14#E9">s</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./33.1#p1.t1.r4">ϵ</mi><mo>,</mo><mi class="ltx_nvar" href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi><mo>;</mo><mi class="ltx_nvar" href="./33.1#p1.t1.r2">r</mi><mo>)</mo></mrow></mrow></math>: regular Coulomb function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m34.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m46.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./33.1#p1.t1.r1" title="§33.1 Special Notation ‣ Notation ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m35.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="\ell" display="inline"><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></math>: nonnegative integer</a>,
<a href="./33.1#p1.t1.r2" title="§33.1 Special Notation ‣ Notation ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="r" display="inline"><mi href="./33.1#p1.t1.r2">r</mi></math>: real variable</a>,
<a href="./33.1#p1.t1.r4" title="§33.1 Special Notation ‣ Notation ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="13px" altimg-valign="-2px" altimg-width="12px" alttext="\epsilon" display="inline"><mi href="./33.1#p1.t1.r4">ϵ</mi></math>: real parameter</a>,
<a href="./33.16#E8" title="(33.16.8) ‣ §33.16(iii) f and h in Terms of W κ , μ ( z ) when < ϵ 0 ‣ §33.16 Connection Formulas ‣ Variables r , ϵ ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m42.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./33.16#E8">ν</mi></math>: parameter</a>,
<a href="./33.16#E9" title="(33.16.9) ‣ §33.16(iii) f and h in Terms of W κ , μ ( z ) when < ϵ 0 ‣ §33.16 Connection Formulas ‣ Variables r , ϵ ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="23px" altimg-valign="-7px" altimg-width="65px" alttext="\zeta_{\ell}(\nu,r)" display="inline"><mrow><msub><mi href="./33.16#E9">ζ</mi><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./33.16#E8">ν</mi><mo>,</mo><mi href="./33.1#p1.t1.r2">r</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function</a>,
<a href="./33.16#E9" title="(33.16.9) ‣ §33.16(iii) f and h in Terms of W κ , μ ( z ) when < ϵ 0 ‣ §33.16 Connection Formulas ‣ Variables r , ϵ ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="23px" altimg-valign="-7px" altimg-width="65px" alttext="\xi_{\ell}(\nu,r)" display="inline"><mrow><msub><mi href="./33.16#E9">ξ</mi><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./33.16#E8">ν</mi><mo>,</mo><mi href="./33.1#p1.t1.r2">r</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function</a> and
<a href="./33.16#E16" title="(33.16.16) ‣ §33.16(v) s and c in Terms of W κ , μ ( z ) when < ϵ 0 ‣ §33.16 Connection Formulas ‣ Variables r , ϵ ‣ Chapter 33 Coulomb Functions" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="23px" altimg-valign="-7px" altimg-width="65px" alttext="K(\nu,\ell)" display="inline"><mrow><mi href="./33.16#E16">K</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./33.16#E8">ν</mi><mo>,</mo><mi href="./33.1#p1.t1.r1" mathvariant="normal">ℓ</mi><mo stretchy="false">)</mo></mrow></mrow></math>: factor</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
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</section>
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<title>DLMF: 26.6 Other Lattice Path Numbers</title>
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<section id="Px1" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Delannoy Number <math class="ltx_Math" altimg="m52.png" altimg-height="23px" altimg-valign="-7px" altimg-width="75px" alttext="D(m,n)" display="inline"><mrow><mi href="./26.6#E1">D</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./26.1#p2.t1.r2">m</mi><mo>,</mo><mi href="./26.1#p2.t1.r2">n</mi><mo stretchy="false">)</mo></mrow></mrow></math>
</h3>
<div id="Px1.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px1.p1" class="ltx_para">
<p class="ltx_p"><math class="ltx_Math" altimg="m52.png" altimg-height="23px" altimg-valign="-7px" altimg-width="75px" alttext="D(m,n)" display="inline"><mrow><mi href="./26.6#E1">D</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./26.1#p2.t1.r2">m</mi><mo>,</mo><mi href="./26.1#p2.t1.r2">n</mi><mo stretchy="false">)</mo></mrow></mrow></math> is the number of paths from <math class="ltx_Math" altimg="m5.png" altimg-height="23px" altimg-valign="-7px" altimg-width="49px" alttext="(0,0)" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mn>0</mn><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mn>0</mn><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math> to <math class="ltx_Math" altimg="m11.png" altimg-height="23px" altimg-valign="-7px" altimg-width="58px" alttext="(m,n)" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi href="./26.1#p2.t1.r2">m</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi href="./26.1#p2.t1.r2">n</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math> that are composed of
directed line segments of the form <math class="ltx_Math" altimg="m8.png" altimg-height="23px" altimg-valign="-7px" altimg-width="49px" alttext="(1,0)" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mn>1</mn><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mn>0</mn><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math>, <math class="ltx_Math" altimg="m6.png" altimg-height="23px" altimg-valign="-7px" altimg-width="49px" alttext="(0,1)" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mn>0</mn><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mn>1</mn><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math>, or <math class="ltx_Math" altimg="m9.png" altimg-height="23px" altimg-valign="-7px" altimg-width="49px" alttext="(1,1)" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mn>1</mn><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mn>1</mn><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math>.</p>
<table id="E1" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">26.6.1</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E1.png" altimg-height="64px" altimg-valign="-28px" altimg-width="468px" alttext="D(m,n)=\sum_{k=0}^{n}\genfrac{(}{)}{0.0pt}{}{n}{k}\genfrac{(}{)}{0.0pt}{}{m+n-%
k}{n}=\sum_{k=0}^{n}2^{k}\genfrac{(}{)}{0.0pt}{}{m}{k}\genfrac{(}{)}{0.0pt}{}{%
n}{k}." display="block"><mrow><mrow><mrow><mi href="./26.6#E1">D</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./26.1#p2.t1.r2">m</mi><mo>,</mo><mi href="./26.1#p2.t1.r2">n</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./26.1#p2.t1.r2">k</mi><mo>=</mo><mn>0</mn></mrow><mi href="./26.1#p2.t1.r2">n</mi></munderover><mrow><mrow><mo href="./1.2#i">(</mo><mfrac linethickness="0.0pt"><mi href="./26.1#p2.t1.r2">n</mi><mi href="./26.1#p2.t1.r2">k</mi></mfrac><mo href="./1.2#i">)</mo></mrow><mo></mo><mrow><mo href="./1.2#i">(</mo><mfrac linethickness="0.0pt"><mrow><mrow><mi href="./26.1#p2.t1.r2">m</mi><mo>+</mo><mi href="./26.1#p2.t1.r2">n</mi></mrow><mo>-</mo><mi href="./26.1#p2.t1.r2">k</mi></mrow><mi href="./26.1#p2.t1.r2">n</mi></mfrac><mo href="./1.2#i">)</mo></mrow></mrow></mrow><mo>=</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./26.1#p2.t1.r2">k</mi><mo>=</mo><mn>0</mn></mrow><mi href="./26.1#p2.t1.r2">n</mi></munderover><mrow><msup><mn>2</mn><mi href="./26.1#p2.t1.r2">k</mi></msup><mo></mo><mrow><mo href="./1.2#i">(</mo><mfrac linethickness="0.0pt"><mi href="./26.1#p2.t1.r2">m</mi><mi href="./26.1#p2.t1.r2">k</mi></mfrac><mo href="./1.2#i">)</mo></mrow><mo></mo><mrow><mo href="./1.2#i">(</mo><mfrac linethickness="0.0pt"><mi href="./26.1#p2.t1.r2">n</mi><mi href="./26.1#p2.t1.r2">k</mi></mfrac><mo href="./1.2#i">)</mo></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text"><math class="ltx_Math" altimg="m52.png" altimg-height="23px" altimg-valign="-7px" altimg-width="75px" alttext="D(m,n)" display="inline"><mrow><mi href="./26.6#E1">D</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./26.1#p2.t1.r2">m</mi><mo>,</mo><mi href="./26.1#p2.t1.r2">n</mi><mo stretchy="false">)</mo></mrow></mrow></math>: Delannoy number (locally)</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./1.2#i" title="§1.2(i) Binomial Coefficients ‣ §1.2 Elementary Algebra ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m57.png" altimg-height="27px" altimg-valign="-9px" altimg-width="37px" alttext="\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}" display="inline"><mrow><mo href="./1.2#i">(</mo><mfrac linethickness="0.0pt"><mi class="ltx_nvar" href="./1.1#p2.t1.r5">m</mi><mi class="ltx_nvar" href="./1.1#p2.t1.r5">n</mi></mfrac><mo href="./1.2#i">)</mo></mrow></math>: binomial coefficient</a>,
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./26.1#p2.t1.r2">k</mi></math>: nonnegative integer</a>,
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./26.1#p2.t1.r2">m</mi></math>: nonnegative integer</a> and
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m61.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math>: nonnegative integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">See Table . </p>
</div>
<figure id="T1" class="ltx_table">
<figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_table">Table 26.6.1: </span>Delannoy numbers <math class="ltx_Math" altimg="m52.png" altimg-height="23px" altimg-valign="-7px" altimg-width="75px" alttext="D(m,n)" display="inline"><mrow><mi href="./26.6#E1">D</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./26.1#p2.t1.r2">m</mi><mo>,</mo><mi href="./26.1#p2.t1.r2">n</mi><mo stretchy="false">)</mo></mrow></mrow></math>.
</figcaption>
<table id="T1.t1" class="ltx_tabular ltx_centering ltx_guessed_headers ltx_align_middle">
<thead class="ltx_thead">
<tr id="T1.t1.r1" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_th_row ltx_border_r ltx_border_tt" rowspan="2"><math class="ltx_Math" altimg="m60.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./26.1#p2.t1.r2">m</mi></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" colspan="11"><math class="ltx_Math" altimg="m61.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math></th>
</tr>
<tr id="T1.t1.r2" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_column">0</th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column">1</th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column">2</th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column">3</th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column">4</th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column">5</th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column">6</th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column">7</th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column">8</th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column">9</th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column">10</th>
</tr>
</thead>
<tbody class="ltx_tbody">
<tr id="T1.t1.r3" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_r">0</th>
<td class="ltx_td ltx_align_right ltx_border_t">1</td>
<td class="ltx_td ltx_align_right ltx_border_t">1</td>
<td class="ltx_td ltx_align_right ltx_border_t">1</td>
<td class="ltx_td ltx_align_right ltx_border_t">1</td>
<td class="ltx_td ltx_align_right ltx_border_t">1</td>
<td class="ltx_td ltx_align_right ltx_border_t">1</td>
<td class="ltx_td ltx_align_right ltx_border_t">1</td>
<td class="ltx_td ltx_align_right ltx_border_t">1</td>
<td class="ltx_td ltx_align_right ltx_border_t">1</td>
<td class="ltx_td ltx_align_right ltx_border_t">1</td>
<td class="ltx_td ltx_align_right ltx_border_t">1</td>
</tr>
<tr id="T1.t1.r4" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_r">1</th>
<td class="ltx_td ltx_align_right">1</td>
<td class="ltx_td ltx_align_right">3</td>
<td class="ltx_td ltx_align_right">5</td>
<td class="ltx_td ltx_align_right">7</td>
<td class="ltx_td ltx_align_right">9</td>
<td class="ltx_td ltx_align_right">11</td>
<td class="ltx_td ltx_align_right">13</td>
<td class="ltx_td ltx_align_right">15</td>
<td class="ltx_td ltx_align_right">17</td>
<td class="ltx_td ltx_align_right">19</td>
<td class="ltx_td ltx_align_right">21</td>
</tr>
<tr id="T1.t1.r5" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_r">2</th>
<td class="ltx_td ltx_align_right">1</td>
<td class="ltx_td ltx_align_right">5</td>
<td class="ltx_td ltx_align_right">13</td>
<td class="ltx_td ltx_align_right">25</td>
<td class="ltx_td ltx_align_right">41</td>
<td class="ltx_td ltx_align_right">61</td>
<td class="ltx_td ltx_align_right">85</td>
<td class="ltx_td ltx_align_right">113</td>
<td class="ltx_td ltx_align_right">145</td>
<td class="ltx_td ltx_align_right">181</td>
<td class="ltx_td ltx_align_right">221</td>
</tr>
<tr id="T1.t1.r6" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_r">3</th>
<td class="ltx_td ltx_align_right">1</td>
<td class="ltx_td ltx_align_right">7</td>
<td class="ltx_td ltx_align_right">25</td>
<td class="ltx_td ltx_align_right">63</td>
<td class="ltx_td ltx_align_right">129</td>
<td class="ltx_td ltx_align_right">231</td>
<td class="ltx_td ltx_align_right">377</td>
<td class="ltx_td ltx_align_right">575</td>
<td class="ltx_td ltx_align_right">833</td>
<td class="ltx_td ltx_align_right">1159</td>
<td class="ltx_td ltx_align_right">1561</td>
</tr>
<tr id="T1.t1.r7" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_T ltx_border_r">4</th>
<td class="ltx_td ltx_align_right ltx_border_T">1</td>
<td class="ltx_td ltx_align_right ltx_border_T">9</td>
<td class="ltx_td ltx_align_right ltx_border_T">41</td>
<td class="ltx_td ltx_align_right ltx_border_T">129</td>
<td class="ltx_td ltx_align_right ltx_border_T">321</td>
<td class="ltx_td ltx_align_right ltx_border_T">681</td>
<td class="ltx_td ltx_align_right ltx_border_T">1289</td>
<td class="ltx_td ltx_align_right ltx_border_T">2241</td>
<td class="ltx_td ltx_align_right ltx_border_T">3649</td>
<td class="ltx_td ltx_align_right ltx_border_T">5641</td>
<td class="ltx_td ltx_align_right ltx_border_T">8361</td>
</tr>
<tr id="T1.t1.r8" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_r">5</th>
<td class="ltx_td ltx_align_right">1</td>
<td class="ltx_td ltx_align_right">11</td>
<td class="ltx_td ltx_align_right">61</td>
<td class="ltx_td ltx_align_right">231</td>
<td class="ltx_td ltx_align_right">681</td>
<td class="ltx_td ltx_align_right">1683</td>
<td class="ltx_td ltx_align_right">3653</td>
<td class="ltx_td ltx_align_right">7183</td>
<td class="ltx_td ltx_align_right">13073</td>
<td class="ltx_td ltx_align_right">22363</td>
<td class="ltx_td ltx_align_right">36365</td>
</tr>
<tr id="T1.t1.r9" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_r">6</th>
<td class="ltx_td ltx_align_right">1</td>
<td class="ltx_td ltx_align_right">13</td>
<td class="ltx_td ltx_align_right">85</td>
<td class="ltx_td ltx_align_right">377</td>
<td class="ltx_td ltx_align_right">1289</td>
<td class="ltx_td ltx_align_right">3653</td>
<td class="ltx_td ltx_align_right">8989</td>
<td class="ltx_td ltx_align_right">19825</td>
<td class="ltx_td ltx_align_right">40081</td>
<td class="ltx_td ltx_align_right">75517</td>
<td class="ltx_td ltx_align_right">1 34245</td>
</tr>
<tr id="T1.t1.r10" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_T ltx_border_r">7</th>
<td class="ltx_td ltx_align_right ltx_border_T">1</td>
<td class="ltx_td ltx_align_right ltx_border_T">15</td>
<td class="ltx_td ltx_align_right ltx_border_T">113</td>
<td class="ltx_td ltx_align_right ltx_border_T">575</td>
<td class="ltx_td ltx_align_right ltx_border_T">2241</td>
<td class="ltx_td ltx_align_right ltx_border_T">7183</td>
<td class="ltx_td ltx_align_right ltx_border_T">19825</td>
<td class="ltx_td ltx_align_right ltx_border_T">48639</td>
<td class="ltx_td ltx_align_right ltx_border_T">1 08545</td>
<td class="ltx_td ltx_align_right ltx_border_T">2 24143</td>
<td class="ltx_td ltx_align_right ltx_border_T">4 33905</td>
</tr>
<tr id="T1.t1.r11" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_r">8</th>
<td class="ltx_td ltx_align_right">1</td>
<td class="ltx_td ltx_align_right">17</td>
<td class="ltx_td ltx_align_right">145</td>
<td class="ltx_td ltx_align_right">833</td>
<td class="ltx_td ltx_align_right">3649</td>
<td class="ltx_td ltx_align_right">13073</td>
<td class="ltx_td ltx_align_right">40081</td>
<td class="ltx_td ltx_align_right">1 08545</td>
<td class="ltx_td ltx_align_right">2 65729</td>
<td class="ltx_td ltx_align_right">5 98417</td>
<td class="ltx_td ltx_align_right">12 56465</td>
</tr>
<tr id="T1.t1.r12" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_r">9</th>
<td class="ltx_td ltx_align_right">1</td>
<td class="ltx_td ltx_align_right">19</td>
<td class="ltx_td ltx_align_right">181</td>
<td class="ltx_td ltx_align_right">1159</td>
<td class="ltx_td ltx_align_right">5641</td>
<td class="ltx_td ltx_align_right">22363</td>
<td class="ltx_td ltx_align_right">75517</td>
<td class="ltx_td ltx_align_right">2 24143</td>
<td class="ltx_td ltx_align_right">5 98417</td>
<td class="ltx_td ltx_align_right">14 62563</td>
<td class="ltx_td ltx_align_right">33 17445</td>
</tr>
<tr id="T1.t1.r13" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_b ltx_border_r">10</th>
<td class="ltx_td ltx_align_right ltx_border_b">1</td>
<td class="ltx_td ltx_align_right ltx_border_b">21</td>
<td class="ltx_td ltx_align_right ltx_border_b">221</td>
<td class="ltx_td ltx_align_right ltx_border_b">1561</td>
<td class="ltx_td ltx_align_right ltx_border_b">8361</td>
<td class="ltx_td ltx_align_right ltx_border_b">36365</td>
<td class="ltx_td ltx_align_right ltx_border_b">1 34245</td>
<td class="ltx_td ltx_align_right ltx_border_b">4 33905</td>
<td class="ltx_td ltx_align_right ltx_border_b">12 56465</td>
<td class="ltx_td ltx_align_right ltx_border_b">33 17445</td>
<td class="ltx_td ltx_align_right ltx_border_b">80 97453</td>
</tr>
</tbody>
</table>
<div id="T1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./26.1#p2.t1.r2">m</mi></math>: nonnegative integer</a>,
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m61.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math>: nonnegative integer</a> and
<a href="./26.6#E1" title="(26.6.1) ‣ Delannoy Number D ( m , n ) ‣ §26.6(i) Definitions ‣ §26.6 Other Lattice Path Numbers ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="23px" altimg-valign="-7px" altimg-width="75px" alttext="D(m,n)" display="inline"><mrow><mi href="./26.6#E1">D</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./26.1#p2.t1.r2">m</mi><mo>,</mo><mi href="./26.1#p2.t1.r2">n</mi><mo stretchy="false">)</mo></mrow></mrow></math>: Delannoy number</a>
</dd>
<dt>Keywords:</dt>
<dd></dd>
</dl>
</div>
</div>
</figure>
</section>
<section id="Px2" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Motzkin Number <math class="ltx_Math" altimg="m54.png" altimg-height="23px" altimg-valign="-7px" altimg-width="53px" alttext="M(n)" display="inline"><mrow><mi href="./26.6#E2">M</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./26.1#p2.t1.r2">n</mi><mo stretchy="false">)</mo></mrow></mrow></math>
</h3>
<div id="Px2.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px2.p1" class="ltx_para">
<p class="ltx_p"><math class="ltx_Math" altimg="m54.png" altimg-height="23px" altimg-valign="-7px" altimg-width="53px" alttext="M(n)" display="inline"><mrow><mi href="./26.6#E2">M</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./26.1#p2.t1.r2">n</mi><mo stretchy="false">)</mo></mrow></mrow></math> is the number of lattice paths from <math class="ltx_Math" altimg="m5.png" altimg-height="23px" altimg-valign="-7px" altimg-width="49px" alttext="(0,0)" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mn>0</mn><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mn>0</mn><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math> to <math class="ltx_Math" altimg="m12.png" altimg-height="23px" altimg-valign="-7px" altimg-width="53px" alttext="(n,n)" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi href="./26.1#p2.t1.r2">n</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi href="./26.1#p2.t1.r2">n</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math> that stay on or
above the line <math class="ltx_Math" altimg="m67.png" altimg-height="16px" altimg-valign="-6px" altimg-width="53px" alttext="y=x" display="inline"><mrow><mi>y</mi><mo>=</mo><mi href="./26.1#p2.t1.r1">x</mi></mrow></math> and are composed of directed line segments of the form
<math class="ltx_Math" altimg="m10.png" altimg-height="23px" altimg-valign="-7px" altimg-width="49px" alttext="(2,0)" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mn>2</mn><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mn>0</mn><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math>, <math class="ltx_Math" altimg="m7.png" altimg-height="23px" altimg-valign="-7px" altimg-width="49px" alttext="(0,2)" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mn>0</mn><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mn>2</mn><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math>, or <math class="ltx_Math" altimg="m9.png" altimg-height="23px" altimg-valign="-7px" altimg-width="49px" alttext="(1,1)" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mn>1</mn><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mn>1</mn><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math>.</p>
<table id="E2" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">26.6.2</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E2.png" altimg-height="64px" altimg-valign="-28px" altimg-width="378px" alttext="M(n)=\sum_{k=0}^{n}\frac{(-1)^{k}}{n+2-k}\genfrac{(}{)}{0.0pt}{}{n}{k}\genfrac%
{(}{)}{0.0pt}{}{2n+2-2k}{n+1-k}." display="block"><mrow><mrow><mrow><mi href="./26.6#E2">M</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./26.1#p2.t1.r2">n</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./26.1#p2.t1.r2">k</mi><mo>=</mo><mn>0</mn></mrow><mi href="./26.1#p2.t1.r2">n</mi></munderover><mrow><mfrac><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./26.1#p2.t1.r2">k</mi></msup><mrow><mrow><mi href="./26.1#p2.t1.r2">n</mi><mo>+</mo><mn>2</mn></mrow><mo>-</mo><mi href="./26.1#p2.t1.r2">k</mi></mrow></mfrac><mo></mo><mrow><mo href="./1.2#i">(</mo><mfrac linethickness="0.0pt"><mi href="./26.1#p2.t1.r2">n</mi><mi href="./26.1#p2.t1.r2">k</mi></mfrac><mo href="./1.2#i">)</mo></mrow><mo></mo><mrow><mo href="./1.2#i">(</mo><mfrac linethickness="0.0pt"><mrow><mrow><mrow><mn>2</mn><mo></mo><mi href="./26.1#p2.t1.r2">n</mi></mrow><mo>+</mo><mn>2</mn></mrow><mo>-</mo><mrow><mn>2</mn><mo></mo><mi href="./26.1#p2.t1.r2">k</mi></mrow></mrow><mrow><mrow><mi href="./26.1#p2.t1.r2">n</mi><mo>+</mo><mn>1</mn></mrow><mo>-</mo><mi href="./26.1#p2.t1.r2">k</mi></mrow></mfrac><mo href="./1.2#i">)</mo></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text"><math class="ltx_Math" altimg="m54.png" altimg-height="23px" altimg-valign="-7px" altimg-width="53px" alttext="M(n)" display="inline"><mrow><mi href="./26.6#E2">M</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./26.1#p2.t1.r2">n</mi><mo stretchy="false">)</mo></mrow></mrow></math>: Motzkin number (locally)</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./1.2#i" title="§1.2(i) Binomial Coefficients ‣ §1.2 Elementary Algebra ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m57.png" altimg-height="27px" altimg-valign="-9px" altimg-width="37px" alttext="\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}" display="inline"><mrow><mo href="./1.2#i">(</mo><mfrac linethickness="0.0pt"><mi class="ltx_nvar" href="./1.1#p2.t1.r5">m</mi><mi class="ltx_nvar" href="./1.1#p2.t1.r5">n</mi></mfrac><mo href="./1.2#i">)</mo></mrow></math>: binomial coefficient</a>,
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./26.1#p2.t1.r2">k</mi></math>: nonnegative integer</a> and
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m61.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math>: nonnegative integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">See Table . </p>
</div>
<figure id="T2" class="ltx_table">
<figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_table">Table 26.6.2: </span>Motzkin numbers <math class="ltx_Math" altimg="m54.png" altimg-height="23px" altimg-valign="-7px" altimg-width="53px" alttext="M(n)" display="inline"><mrow><mi href="./26.6#E2">M</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./26.1#p2.t1.r2">n</mi><mo stretchy="false">)</mo></mrow></mrow></math>.
</figcaption>
<table id="T2.t1" class="ltx_tabular ltx_centering ltx_guessed_headers ltx_align_middle">
<thead class="ltx_thead">
<tr id="T2.t1.r1" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_th_row ltx_border_tt"><math class="ltx_Math" altimg="m61.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math></th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_border_r ltx_border_tt"><math class="ltx_Math" altimg="m54.png" altimg-height="23px" altimg-valign="-7px" altimg-width="53px" alttext="M(n)" display="inline"><mrow><mi href="./26.6#E2">M</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./26.1#p2.t1.r2">n</mi><mo stretchy="false">)</mo></mrow></mrow></math></th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_th_row ltx_border_tt"><math class="ltx_Math" altimg="m61.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math></th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_border_r ltx_border_tt"><math class="ltx_Math" altimg="m54.png" altimg-height="23px" altimg-valign="-7px" altimg-width="53px" alttext="M(n)" display="inline"><mrow><mi href="./26.6#E2">M</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./26.1#p2.t1.r2">n</mi><mo stretchy="false">)</mo></mrow></mrow></math></th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_th_row ltx_border_tt"><math class="ltx_Math" altimg="m61.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math></th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_border_r ltx_border_tt"><math class="ltx_Math" altimg="m54.png" altimg-height="23px" altimg-valign="-7px" altimg-width="53px" alttext="M(n)" display="inline"><mrow><mi href="./26.6#E2">M</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./26.1#p2.t1.r2">n</mi><mo stretchy="false">)</mo></mrow></mrow></math></th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_th_row ltx_border_tt"><math class="ltx_Math" altimg="m61.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math></th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_border_r ltx_border_tt"><math class="ltx_Math" altimg="m54.png" altimg-height="23px" altimg-valign="-7px" altimg-width="53px" alttext="M(n)" display="inline"><mrow><mi href="./26.6#E2">M</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./26.1#p2.t1.r2">n</mi><mo stretchy="false">)</mo></mrow></mrow></math></th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_th_row ltx_border_tt"><math class="ltx_Math" altimg="m61.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math></th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_border_tt"><math class="ltx_Math" altimg="m54.png" altimg-height="23px" altimg-valign="-7px" altimg-width="53px" alttext="M(n)" display="inline"><mrow><mi href="./26.6#E2">M</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./26.1#p2.t1.r2">n</mi><mo stretchy="false">)</mo></mrow></mrow></math></th>
</tr>
</thead>
<tbody class="ltx_tbody">
<tr id="T2.t1.r2" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_t">0</th>
<td class="ltx_td ltx_align_right ltx_border_r ltx_border_t">1</td>
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_t">4</th>
<td class="ltx_td ltx_align_right ltx_border_r ltx_border_t">9</td>
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_t">8</th>
<td class="ltx_td ltx_align_right ltx_border_r ltx_border_t">323</td>
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_t">12</th>
<td class="ltx_td ltx_align_right ltx_border_r ltx_border_t">15511</td>
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_t">16</th>
<td class="ltx_td ltx_align_right ltx_border_t">8 53467</td>
</tr>
<tr id="T2.t1.r3" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row">1</th>
<td class="ltx_td ltx_align_right ltx_border_r">1</td>
<th class="ltx_td ltx_align_right ltx_th ltx_th_row">5</th>
<td class="ltx_td ltx_align_right ltx_border_r">21</td>
<th class="ltx_td ltx_align_right ltx_th ltx_th_row">9</th>
<td class="ltx_td ltx_align_right ltx_border_r">835</td>
<th class="ltx_td ltx_align_right ltx_th ltx_th_row">13</th>
<td class="ltx_td ltx_align_right ltx_border_r">41835</td>
<th class="ltx_td ltx_align_right ltx_th ltx_th_row">17</th>
<td class="ltx_td ltx_align_right">23 56779</td>
</tr>
<tr id="T2.t1.r4" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row">2</th>
<td class="ltx_td ltx_align_right ltx_border_r">2</td>
<th class="ltx_td ltx_align_right ltx_th ltx_th_row">6</th>
<td class="ltx_td ltx_align_right ltx_border_r">51</td>
<th class="ltx_td ltx_align_right ltx_th ltx_th_row">10</th>
<td class="ltx_td ltx_align_right ltx_border_r">2188</td>
<th class="ltx_td ltx_align_right ltx_th ltx_th_row">14</th>
<td class="ltx_td ltx_align_right ltx_border_r">1 13634</td>
<th class="ltx_td ltx_align_right ltx_th ltx_th_row">18</th>
<td class="ltx_td ltx_align_right">65 36382</td>
</tr>
<tr id="T2.t1.r5" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_b">3</th>
<td class="ltx_td ltx_align_right ltx_border_b ltx_border_r">4</td>
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_b">7</th>
<td class="ltx_td ltx_align_right ltx_border_b ltx_border_r">127</td>
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_b">11</th>
<td class="ltx_td ltx_align_right ltx_border_b ltx_border_r">5798</td>
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_b">15</th>
<td class="ltx_td ltx_align_right ltx_border_b ltx_border_r">3 10572</td>
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_b">19</th>
<td class="ltx_td ltx_align_right ltx_border_b">181 99284</td>
</tr>
</tbody>
</table>
<div id="T2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m61.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math>: nonnegative integer</a> and
<a href="./26.6#E2" title="(26.6.2) ‣ Motzkin Number M ( n ) ‣ §26.6(i) Definitions ‣ §26.6 Other Lattice Path Numbers ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="23px" altimg-valign="-7px" altimg-width="53px" alttext="M(n)" display="inline"><mrow><mi href="./26.6#E2">M</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./26.1#p2.t1.r2">n</mi><mo stretchy="false">)</mo></mrow></mrow></math>: Motzkin number</a>
</dd>
<dt>Keywords:</dt>
<dd></dd>
</dl>
</div>
</div>
</figure>
</section>
<section id="Px3" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Narayana Number <math class="ltx_Math" altimg="m56.png" altimg-height="23px" altimg-valign="-7px" altimg-width="70px" alttext="N(n,k)" display="inline"><mrow><mi href="./26.6#E3">N</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./26.1#p2.t1.r2">n</mi><mo>,</mo><mi href="./26.1#p2.t1.r2">k</mi><mo stretchy="false">)</mo></mrow></mrow></math>
</h3>
<div id="Px3.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px3.p1" class="ltx_para">
<p class="ltx_p"><math class="ltx_Math" altimg="m56.png" altimg-height="23px" altimg-valign="-7px" altimg-width="70px" alttext="N(n,k)" display="inline"><mrow><mi href="./26.6#E3">N</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./26.1#p2.t1.r2">n</mi><mo>,</mo><mi href="./26.1#p2.t1.r2">k</mi><mo stretchy="false">)</mo></mrow></mrow></math> is the number of lattice paths from <math class="ltx_Math" altimg="m5.png" altimg-height="23px" altimg-valign="-7px" altimg-width="49px" alttext="(0,0)" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mn>0</mn><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mn>0</mn><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math> to <math class="ltx_Math" altimg="m12.png" altimg-height="23px" altimg-valign="-7px" altimg-width="53px" alttext="(n,n)" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi href="./26.1#p2.t1.r2">n</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi href="./26.1#p2.t1.r2">n</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math> that stay on or
above the line <math class="ltx_Math" altimg="m67.png" altimg-height="16px" altimg-valign="-6px" altimg-width="53px" alttext="y=x" display="inline"><mrow><mi>y</mi><mo>=</mo><mi href="./26.1#p2.t1.r1">x</mi></mrow></math>, are composed of directed line segments of the form
<math class="ltx_Math" altimg="m8.png" altimg-height="23px" altimg-valign="-7px" altimg-width="49px" alttext="(1,0)" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mn>1</mn><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mn>0</mn><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math> or <math class="ltx_Math" altimg="m6.png" altimg-height="23px" altimg-valign="-7px" altimg-width="49px" alttext="(0,1)" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mn>0</mn><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mn>1</mn><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math>, and for which there are exactly <math class="ltx_Math" altimg="m58.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./26.1#p2.t1.r2">k</mi></math> occurrences at which a
segment of the form <math class="ltx_Math" altimg="m6.png" altimg-height="23px" altimg-valign="-7px" altimg-width="49px" alttext="(0,1)" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mn>0</mn><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mn>1</mn><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math> is followed by a segment of the form <math class="ltx_Math" altimg="m8.png" altimg-height="23px" altimg-valign="-7px" altimg-width="49px" alttext="(1,0)" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mn>1</mn><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mn>0</mn><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math>.</p>
<table id="E3" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">26.6.3</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E3.png" altimg-height="53px" altimg-valign="-21px" altimg-width="236px" alttext="N(n,k)=\frac{1}{n}\genfrac{(}{)}{0.0pt}{}{n}{k}\genfrac{(}{)}{0.0pt}{}{n}{k-1}." display="block"><mrow><mrow><mrow><mi href="./26.6#E3">N</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./26.1#p2.t1.r2">n</mi><mo>,</mo><mi href="./26.1#p2.t1.r2">k</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mfrac><mn>1</mn><mi href="./26.1#p2.t1.r2">n</mi></mfrac><mo></mo><mrow><mo href="./1.2#i">(</mo><mfrac linethickness="0.0pt"><mi href="./26.1#p2.t1.r2">n</mi><mi href="./26.1#p2.t1.r2">k</mi></mfrac><mo href="./1.2#i">)</mo></mrow><mo></mo><mrow><mo href="./1.2#i">(</mo><mfrac linethickness="0.0pt"><mi href="./26.1#p2.t1.r2">n</mi><mrow><mi href="./26.1#p2.t1.r2">k</mi><mo>-</mo><mn>1</mn></mrow></mfrac><mo href="./1.2#i">)</mo></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text"><math class="ltx_Math" altimg="m56.png" altimg-height="23px" altimg-valign="-7px" altimg-width="70px" alttext="N(n,k)" display="inline"><mrow><mi href="./26.6#E3">N</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./26.1#p2.t1.r2">n</mi><mo>,</mo><mi href="./26.1#p2.t1.r2">k</mi><mo stretchy="false">)</mo></mrow></mrow></math>: Narayana number (locally)</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./1.2#i" title="§1.2(i) Binomial Coefficients ‣ §1.2 Elementary Algebra ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m57.png" altimg-height="27px" altimg-valign="-9px" altimg-width="37px" alttext="\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}" display="inline"><mrow><mo href="./1.2#i">(</mo><mfrac linethickness="0.0pt"><mi class="ltx_nvar" href="./1.1#p2.t1.r5">m</mi><mi class="ltx_nvar" href="./1.1#p2.t1.r5">n</mi></mfrac><mo href="./1.2#i">)</mo></mrow></math>: binomial coefficient</a>,
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./26.1#p2.t1.r2">k</mi></math>: nonnegative integer</a> and
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m61.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math>: nonnegative integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">See Table . </p>
</div>
<figure id="T3" class="ltx_table">
<figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_table">Table 26.6.3: </span>Narayana numbers <math class="ltx_Math" altimg="m56.png" altimg-height="23px" altimg-valign="-7px" altimg-width="70px" alttext="N(n,k)" display="inline"><mrow><mi href="./26.6#E3">N</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./26.1#p2.t1.r2">n</mi><mo>,</mo><mi href="./26.1#p2.t1.r2">k</mi><mo stretchy="false">)</mo></mrow></mrow></math>.
</figcaption>
<table id="T3.t1" class="ltx_tabular ltx_centering ltx_guessed_headers ltx_align_middle">
<thead class="ltx_thead">
<tr id="T3.t1.r1" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_th_row ltx_border_r ltx_border_tt" rowspan="2"><math class="ltx_Math" altimg="m61.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" colspan="11"><math class="ltx_Math" altimg="m58.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./26.1#p2.t1.r2">k</mi></math></th>
</tr>
<tr id="T3.t1.r2" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_column">0</th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column">1</th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column">2</th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column">3</th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column">4</th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column">5</th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column">6</th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column">7</th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column">8</th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column">9</th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column">10</th>
</tr>
</thead>
<tbody class="ltx_tbody">
<tr id="T3.t1.r3" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_r">0</th>
<td class="ltx_td ltx_align_right ltx_border_t">1</td>
<td class="ltx_td ltx_border_t"></td>
<td class="ltx_td ltx_border_t"></td>
<td class="ltx_td ltx_border_t"></td>
<td class="ltx_td ltx_border_t"></td>
<td class="ltx_td ltx_border_t"></td>
<td class="ltx_td ltx_border_t"></td>
<td class="ltx_td ltx_border_t"></td>
<td class="ltx_td ltx_border_t"></td>
<td class="ltx_td ltx_border_t"></td>
<td class="ltx_td ltx_border_t"></td>
</tr>
<tr id="T3.t1.r4" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_r">1</th>
<td class="ltx_td ltx_align_right">0</td>
<td class="ltx_td ltx_align_right">1</td>
<td class="ltx_td"></td>
<td class="ltx_td"></td>
<td class="ltx_td"></td>
<td class="ltx_td"></td>
<td class="ltx_td"></td>
<td class="ltx_td"></td>
<td class="ltx_td"></td>
<td class="ltx_td"></td>
<td class="ltx_td"></td>
</tr>
<tr id="T3.t1.r5" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_r">2</th>
<td class="ltx_td ltx_align_right">0</td>
<td class="ltx_td ltx_align_right">1</td>
<td class="ltx_td ltx_align_right">1</td>
<td class="ltx_td"></td>
<td class="ltx_td"></td>
<td class="ltx_td"></td>
<td class="ltx_td"></td>
<td class="ltx_td"></td>
<td class="ltx_td"></td>
<td class="ltx_td"></td>
<td class="ltx_td"></td>
</tr>
<tr id="T3.t1.r6" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_r">3</th>
<td class="ltx_td ltx_align_right">0</td>
<td class="ltx_td ltx_align_right">1</td>
<td class="ltx_td ltx_align_right">3</td>
<td class="ltx_td ltx_align_right">1</td>
<td class="ltx_td"></td>
<td class="ltx_td"></td>
<td class="ltx_td"></td>
<td class="ltx_td"></td>
<td class="ltx_td"></td>
<td class="ltx_td"></td>
<td class="ltx_td"></td>
</tr>
<tr id="T3.t1.r7" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_T ltx_border_r">4</th>
<td class="ltx_td ltx_align_right ltx_border_T">0</td>
<td class="ltx_td ltx_align_right ltx_border_T">1</td>
<td class="ltx_td ltx_align_right ltx_border_T">6</td>
<td class="ltx_td ltx_align_right ltx_border_T">6</td>
<td class="ltx_td ltx_align_right ltx_border_T">1</td>
<td class="ltx_td ltx_border_T"></td>
<td class="ltx_td ltx_border_T"></td>
<td class="ltx_td ltx_border_T"></td>
<td class="ltx_td ltx_border_T"></td>
<td class="ltx_td ltx_border_T"></td>
<td class="ltx_td ltx_border_T"></td>
</tr>
<tr id="T3.t1.r8" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_r">5</th>
<td class="ltx_td ltx_align_right">0</td>
<td class="ltx_td ltx_align_right">1</td>
<td class="ltx_td ltx_align_right">10</td>
<td class="ltx_td ltx_align_right">20</td>
<td class="ltx_td ltx_align_right">10</td>
<td class="ltx_td ltx_align_right">1</td>
<td class="ltx_td"></td>
<td class="ltx_td"></td>
<td class="ltx_td"></td>
<td class="ltx_td"></td>
<td class="ltx_td"></td>
</tr>
<tr id="T3.t1.r9" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_r">6</th>
<td class="ltx_td ltx_align_right">0</td>
<td class="ltx_td ltx_align_right">1</td>
<td class="ltx_td ltx_align_right">15</td>
<td class="ltx_td ltx_align_right">50</td>
<td class="ltx_td ltx_align_right">50</td>
<td class="ltx_td ltx_align_right">15</td>
<td class="ltx_td ltx_align_right">1</td>
<td class="ltx_td"></td>
<td class="ltx_td"></td>
<td class="ltx_td"></td>
<td class="ltx_td"></td>
</tr>
<tr id="T3.t1.r10" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_T ltx_border_r">7</th>
<td class="ltx_td ltx_align_right ltx_border_T">0</td>
<td class="ltx_td ltx_align_right ltx_border_T">1</td>
<td class="ltx_td ltx_align_right ltx_border_T">21</td>
<td class="ltx_td ltx_align_right ltx_border_T">105</td>
<td class="ltx_td ltx_align_right ltx_border_T">175</td>
<td class="ltx_td ltx_align_right ltx_border_T">105</td>
<td class="ltx_td ltx_align_right ltx_border_T">21</td>
<td class="ltx_td ltx_align_right ltx_border_T">1</td>
<td class="ltx_td ltx_border_T"></td>
<td class="ltx_td ltx_border_T"></td>
<td class="ltx_td ltx_border_T"></td>
</tr>
<tr id="T3.t1.r11" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_r">8</th>
<td class="ltx_td ltx_align_right">0</td>
<td class="ltx_td ltx_align_right">1</td>
<td class="ltx_td ltx_align_right">28</td>
<td class="ltx_td ltx_align_right">196</td>
<td class="ltx_td ltx_align_right">490</td>
<td class="ltx_td ltx_align_right">490</td>
<td class="ltx_td ltx_align_right">196</td>
<td class="ltx_td ltx_align_right">28</td>
<td class="ltx_td ltx_align_right">1</td>
<td class="ltx_td"></td>
<td class="ltx_td"></td>
</tr>
<tr id="T3.t1.r12" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_r">9</th>
<td class="ltx_td ltx_align_right">0</td>
<td class="ltx_td ltx_align_right">1</td>
<td class="ltx_td ltx_align_right">36</td>
<td class="ltx_td ltx_align_right">336</td>
<td class="ltx_td ltx_align_right">1176</td>
<td class="ltx_td ltx_align_right">1764</td>
<td class="ltx_td ltx_align_right">1176</td>
<td class="ltx_td ltx_align_right">336</td>
<td class="ltx_td ltx_align_right">36</td>
<td class="ltx_td ltx_align_right">1</td>
<td class="ltx_td"></td>
</tr>
<tr id="T3.t1.r13" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_b ltx_border_r">10</th>
<td class="ltx_td ltx_align_right ltx_border_b">0</td>
<td class="ltx_td ltx_align_right ltx_border_b">1</td>
<td class="ltx_td ltx_align_right ltx_border_b">45</td>
<td class="ltx_td ltx_align_right ltx_border_b">540</td>
<td class="ltx_td ltx_align_right ltx_border_b">2520</td>
<td class="ltx_td ltx_align_right ltx_border_b">5292</td>
<td class="ltx_td ltx_align_right ltx_border_b">5292</td>
<td class="ltx_td ltx_align_right ltx_border_b">2520</td>
<td class="ltx_td ltx_align_right ltx_border_b">540</td>
<td class="ltx_td ltx_align_right ltx_border_b">45</td>
<td class="ltx_td ltx_align_right ltx_border_b">1</td>
</tr>
</tbody>
</table>
<div id="T3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./26.1#p2.t1.r2">k</mi></math>: nonnegative integer</a>,
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m61.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math>: nonnegative integer</a> and
<a href="./26.6#E3" title="(26.6.3) ‣ Narayana Number N ( n , k ) ‣ §26.6(i) Definitions ‣ §26.6 Other Lattice Path Numbers ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="23px" altimg-valign="-7px" altimg-width="70px" alttext="N(n,k)" display="inline"><mrow><mi href="./26.6#E3">N</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./26.1#p2.t1.r2">n</mi><mo>,</mo><mi href="./26.1#p2.t1.r2">k</mi><mo stretchy="false">)</mo></mrow></mrow></math>: Narayana number</a>
</dd>
<dt>Keywords:</dt>
<dd></dd>
</dl>
</div>
</div>
</figure>
</section>
<section id="Px4" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Schröder Number <math class="ltx_Math" altimg="m65.png" altimg-height="23px" altimg-valign="-7px" altimg-width="41px" alttext="r(n)" display="inline"><mrow><mi href="./26.6#E4">r</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./26.1#p2.t1.r2">n</mi><mo stretchy="false">)</mo></mrow></mrow></math>
</h3>
<div id="Px4.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px4.p1" class="ltx_para">
<p class="ltx_p"><math class="ltx_Math" altimg="m65.png" altimg-height="23px" altimg-valign="-7px" altimg-width="41px" alttext="r(n)" display="inline"><mrow><mi href="./26.6#E4">r</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./26.1#p2.t1.r2">n</mi><mo stretchy="false">)</mo></mrow></mrow></math> is the number of paths from <math class="ltx_Math" altimg="m5.png" altimg-height="23px" altimg-valign="-7px" altimg-width="49px" alttext="(0,0)" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mn>0</mn><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mn>0</mn><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math> to <math class="ltx_Math" altimg="m12.png" altimg-height="23px" altimg-valign="-7px" altimg-width="53px" alttext="(n,n)" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi href="./26.1#p2.t1.r2">n</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi href="./26.1#p2.t1.r2">n</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math> that stay on or above the
diagonal <math class="ltx_Math" altimg="m67.png" altimg-height="16px" altimg-valign="-6px" altimg-width="53px" alttext="y=x" display="inline"><mrow><mi>y</mi><mo>=</mo><mi href="./26.1#p2.t1.r1">x</mi></mrow></math> and are composed of directed line segments of the form <math class="ltx_Math" altimg="m8.png" altimg-height="23px" altimg-valign="-7px" altimg-width="49px" alttext="(1,0)" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mn>1</mn><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mn>0</mn><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math>,
<math class="ltx_Math" altimg="m6.png" altimg-height="23px" altimg-valign="-7px" altimg-width="49px" alttext="(0,1)" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mn>0</mn><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mn>1</mn><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math>, or <math class="ltx_Math" altimg="m9.png" altimg-height="23px" altimg-valign="-7px" altimg-width="49px" alttext="(1,1)" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mn>1</mn><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mn>1</mn><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math>.</p>
<table id="E4" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">26.6.4</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E4.png" altimg-height="25px" altimg-valign="-7px" altimg-width="299px" alttext="r(n)=D(n,n)-D(n+1,n-1)," display="block"><mrow><mrow><mrow><mi href="./26.6#E4">r</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./26.1#p2.t1.r2">n</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mi href="./26.6#E1">D</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./26.1#p2.t1.r2">n</mi><mo>,</mo><mi href="./26.1#p2.t1.r2">n</mi><mo stretchy="false">)</mo></mrow></mrow><mo>-</mo><mrow><mi href="./26.6#E1">D</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./26.1#p2.t1.r2">n</mi><mo>+</mo><mn>1</mn></mrow><mo>,</mo><mrow><mi href="./26.1#p2.t1.r2">n</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m62.png" altimg-height="19px" altimg-valign="-5px" altimg-width="53px" alttext="n\geq 1" display="inline"><mrow><mi href="./26.1#p2.t1.r2">n</mi><mo>≥</mo><mn>1</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E4.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text"><math class="ltx_Math" altimg="m65.png" altimg-height="23px" altimg-valign="-7px" altimg-width="41px" alttext="r(n)" display="inline"><mrow><mi href="./26.6#E4">r</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./26.1#p2.t1.r2">n</mi><mo stretchy="false">)</mo></mrow></mrow></math>: Schröder number (locally)</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m61.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math>: nonnegative integer</a> and
<a href="./26.6#E1" title="(26.6.1) ‣ Delannoy Number D ( m , n ) ‣ §26.6(i) Definitions ‣ §26.6 Other Lattice Path Numbers ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="23px" altimg-valign="-7px" altimg-width="75px" alttext="D(m,n)" display="inline"><mrow><mi href="./26.6#E1">D</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./26.1#p2.t1.r2">m</mi><mo>,</mo><mi href="./26.1#p2.t1.r2">n</mi><mo stretchy="false">)</mo></mrow></mrow></math>: Delannoy number</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">See Table . </p>
</div>
<figure id="T4" class="ltx_table">
<figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_table">Table 26.6.4: </span>Schröder numbers <math class="ltx_Math" altimg="m65.png" altimg-height="23px" altimg-valign="-7px" altimg-width="41px" alttext="r(n)" display="inline"><mrow><mi href="./26.6#E4">r</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./26.1#p2.t1.r2">n</mi><mo stretchy="false">)</mo></mrow></mrow></math>.
</figcaption>
<table id="T4.t1" class="ltx_tabular ltx_centering ltx_guessed_headers ltx_align_middle">
<thead class="ltx_thead">
<tr id="T4.t1.r1" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_th_row ltx_border_tt"><math class="ltx_Math" altimg="m61.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math></th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_border_r ltx_border_tt"><math class="ltx_Math" altimg="m65.png" altimg-height="23px" altimg-valign="-7px" altimg-width="41px" alttext="r(n)" display="inline"><mrow><mi href="./26.6#E4">r</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./26.1#p2.t1.r2">n</mi><mo stretchy="false">)</mo></mrow></mrow></math></th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_th_row ltx_border_tt"><math class="ltx_Math" altimg="m61.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math></th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_border_r ltx_border_tt"><math class="ltx_Math" altimg="m65.png" altimg-height="23px" altimg-valign="-7px" altimg-width="41px" alttext="r(n)" display="inline"><mrow><mi href="./26.6#E4">r</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./26.1#p2.t1.r2">n</mi><mo stretchy="false">)</mo></mrow></mrow></math></th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_th_row ltx_border_tt"><math class="ltx_Math" altimg="m61.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math></th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_border_r ltx_border_tt"><math class="ltx_Math" altimg="m65.png" altimg-height="23px" altimg-valign="-7px" altimg-width="41px" alttext="r(n)" display="inline"><mrow><mi href="./26.6#E4">r</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./26.1#p2.t1.r2">n</mi><mo stretchy="false">)</mo></mrow></mrow></math></th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_th_row ltx_border_tt"><math class="ltx_Math" altimg="m61.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math></th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_border_r ltx_border_tt"><math class="ltx_Math" altimg="m65.png" altimg-height="23px" altimg-valign="-7px" altimg-width="41px" alttext="r(n)" display="inline"><mrow><mi href="./26.6#E4">r</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./26.1#p2.t1.r2">n</mi><mo stretchy="false">)</mo></mrow></mrow></math></th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_th_row ltx_border_tt"><math class="ltx_Math" altimg="m61.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math></th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_border_tt"><math class="ltx_Math" altimg="m65.png" altimg-height="23px" altimg-valign="-7px" altimg-width="41px" alttext="r(n)" display="inline"><mrow><mi href="./26.6#E4">r</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./26.1#p2.t1.r2">n</mi><mo stretchy="false">)</mo></mrow></mrow></math></th>
</tr>
</thead>
<tbody class="ltx_tbody">
<tr id="T4.t1.r2" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_t"><math class="ltx_Math" altimg="m13.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></th>
<td class="ltx_td ltx_align_right ltx_border_r ltx_border_t"><math class="ltx_Math" altimg="m28.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="1" display="inline"><mn>1</mn></math></td>
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_t"><math class="ltx_Math" altimg="m39.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="4" display="inline"><mn>4</mn></math></th>
<td class="ltx_td ltx_align_right ltx_border_r ltx_border_t"><math class="ltx_Math" altimg="m48.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="90" display="inline"><mn>90</mn></math></td>
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_t"><math class="ltx_Math" altimg="m47.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="8" display="inline"><mn>8</mn></math></th>
<td class="ltx_td ltx_align_right ltx_border_r ltx_border_t"><math class="ltx_Math" altimg="m38.png" altimg-height="17px" altimg-valign="-2px" altimg-width="54px" alttext="41586" display="inline"><mn>41586</mn></math></td>
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_t"><math class="ltx_Math" altimg="m18.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="12" display="inline"><mn>12</mn></math></th>
<td class="ltx_td ltx_align_right ltx_border_r ltx_border_t"><math class="ltx_Math" altimg="m30.png" altimg-height="17px" altimg-valign="-2px" altimg-width="90px" alttext="272\;97738" display="inline"><mn>272 97738</mn></math></td>
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_t"><math class="ltx_Math" altimg="m23.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="16" display="inline"><mn>16</mn></math></th>
<td class="ltx_td ltx_align_right ltx_border_t"><math class="ltx_Math" altimg="m32.png" altimg-height="17px" altimg-valign="-2px" altimg-width="125px" alttext="2\;09271\;56706" display="inline"><mn>2 09271 56706</mn></math></td>
</tr>
<tr id="T4.t1.r3" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row"><math class="ltx_Math" altimg="m28.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="1" display="inline"><mn>1</mn></math></th>
<td class="ltx_td ltx_align_right ltx_border_r"><math class="ltx_Math" altimg="m33.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="2" display="inline"><mn>2</mn></math></td>
<th class="ltx_td ltx_align_right ltx_th ltx_th_row"><math class="ltx_Math" altimg="m41.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="5" display="inline"><mn>5</mn></math></th>
<td class="ltx_td ltx_align_right ltx_border_r"><math class="ltx_Math" altimg="m36.png" altimg-height="17px" altimg-valign="-2px" altimg-width="34px" alttext="394" display="inline"><mn>394</mn></math></td>
<th class="ltx_td ltx_align_right ltx_th ltx_th_row"><math class="ltx_Math" altimg="m49.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="9" display="inline"><mn>9</mn></math></th>
<td class="ltx_td ltx_align_right ltx_border_r"><math class="ltx_Math" altimg="m31.png" altimg-height="17px" altimg-valign="-2px" altimg-width="70px" alttext="2\;06098" display="inline"><mn>2 06098</mn></math></td>
<th class="ltx_td ltx_align_right ltx_th ltx_th_row"><math class="ltx_Math" altimg="m19.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="13" display="inline"><mn>13</mn></math></th>
<td class="ltx_td ltx_align_right ltx_border_r"><math class="ltx_Math" altimg="m20.png" altimg-height="17px" altimg-valign="-2px" altimg-width="99px" alttext="1420\;78746" display="inline"><mn>1420 78746</mn></math></td>
<th class="ltx_td ltx_align_right ltx_th ltx_th_row"><math class="ltx_Math" altimg="m24.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="17" display="inline"><mn>17</mn></math></th>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m16.png" altimg-height="17px" altimg-valign="-2px" altimg-width="135px" alttext="11\;18180\;26018" display="inline"><mn>11 18180 26018</mn></math></td>
</tr>
<tr id="T4.t1.r4" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row"><math class="ltx_Math" altimg="m33.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="2" display="inline"><mn>2</mn></math></th>
<td class="ltx_td ltx_align_right ltx_border_r"><math class="ltx_Math" altimg="m43.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="6" display="inline"><mn>6</mn></math></td>
<th class="ltx_td ltx_align_right ltx_th ltx_th_row"><math class="ltx_Math" altimg="m43.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="6" display="inline"><mn>6</mn></math></th>
<td class="ltx_td ltx_align_right ltx_border_r"><math class="ltx_Math" altimg="m25.png" altimg-height="17px" altimg-valign="-2px" altimg-width="44px" alttext="1806" display="inline"><mn>1806</mn></math></td>
<th class="ltx_td ltx_align_right ltx_th ltx_th_row"><math class="ltx_Math" altimg="m15.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="10" display="inline"><mn>10</mn></math></th>
<td class="ltx_td ltx_align_right ltx_border_r"><math class="ltx_Math" altimg="m14.png" altimg-height="17px" altimg-valign="-2px" altimg-width="80px" alttext="10\;37718" display="inline"><mn>10 37718</mn></math></td>
<th class="ltx_td ltx_align_right ltx_th ltx_th_row"><math class="ltx_Math" altimg="m21.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="14" display="inline"><mn>14</mn></math></th>
<td class="ltx_td ltx_align_right ltx_border_r"><math class="ltx_Math" altimg="m44.png" altimg-height="17px" altimg-valign="-2px" altimg-width="99px" alttext="7453\;87038" display="inline"><mn>7453 87038</mn></math></td>
<th class="ltx_td ltx_align_right ltx_th ltx_th_row"><math class="ltx_Math" altimg="m26.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="18" display="inline"><mn>18</mn></math></th>
<td class="ltx_td ltx_align_right"><math class="ltx_Math" altimg="m42.png" altimg-height="17px" altimg-valign="-2px" altimg-width="135px" alttext="60\;03188\;53926" display="inline"><mn>60 03188 53926</mn></math></td>
</tr>
<tr id="T4.t1.r5" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_b"><math class="ltx_Math" altimg="m37.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="3" display="inline"><mn>3</mn></math></th>
<td class="ltx_td ltx_align_right ltx_border_b ltx_border_r"><math class="ltx_Math" altimg="m29.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="22" display="inline"><mn>22</mn></math></td>
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_b"><math class="ltx_Math" altimg="m45.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="7" display="inline"><mn>7</mn></math></th>
<td class="ltx_td ltx_align_right ltx_border_b ltx_border_r"><math class="ltx_Math" altimg="m46.png" altimg-height="17px" altimg-valign="-2px" altimg-width="44px" alttext="8558" display="inline"><mn>8558</mn></math></td>
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_b"><math class="ltx_Math" altimg="m17.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="11" display="inline"><mn>11</mn></math></th>
<td class="ltx_td ltx_align_right ltx_border_b ltx_border_r"><math class="ltx_Math" altimg="m40.png" altimg-height="17px" altimg-valign="-2px" altimg-width="80px" alttext="52\;93446" display="inline"><mn>52 93446</mn></math></td>
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_b"><math class="ltx_Math" altimg="m22.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="15" display="inline"><mn>15</mn></math></th>
<td class="ltx_td ltx_align_right ltx_border_b ltx_border_r"><math class="ltx_Math" altimg="m35.png" altimg-height="17px" altimg-valign="-2px" altimg-width="109px" alttext="39376\;03038" display="inline"><mn>39376 03038</mn></math></td>
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_b"><math class="ltx_Math" altimg="m27.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="19" display="inline"><mn>19</mn></math></th>
<td class="ltx_td ltx_align_right ltx_border_b"><math class="ltx_Math" altimg="m34.png" altimg-height="17px" altimg-valign="-2px" altimg-width="145px" alttext="323\;67243\;17174" display="inline"><mn>323 67243 17174</mn></math></td>
</tr>
</tbody>
</table>
<div id="T4.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m61.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math>: nonnegative integer</a> and
<a href="./26.6#E4" title="(26.6.4) ‣ Schröder Number r ( n ) ‣ §26.6(i) Definitions ‣ §26.6 Other Lattice Path Numbers ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m65.png" altimg-height="23px" altimg-valign="-7px" altimg-width="41px" alttext="r(n)" display="inline"><mrow><mi href="./26.6#E4">r</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./26.1#p2.t1.r2">n</mi><mo stretchy="false">)</mo></mrow></mrow></math>: Schröder number</a>
</dd>
<dt>Keywords:</dt>
<dd></dd>
</dl>
</div>
</div>
<div id="SS2.p1" class="ltx_para">
<p class="ltx_p">For sufficiently small <math class="ltx_Math" altimg="m69.png" altimg-height="23px" altimg-valign="-7px" altimg-width="27px" alttext="|x|" display="inline"><mrow><mo stretchy="false">|</mo><mi href="./26.6#SS2.p1">x</mi><mo stretchy="false">|</mo></mrow></math> and <math class="ltx_Math" altimg="m70.png" altimg-height="23px" altimg-valign="-7px" altimg-width="26px" alttext="|y|" display="inline"><mrow><mo stretchy="false">|</mo><mi href="./26.6#SS2.p1">y</mi><mo stretchy="false">|</mo></mrow></math>,
</p>
<table id="E5" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">26.6.5</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E5.png" altimg-height="66px" altimg-valign="-30px" altimg-width="345px" alttext="\sum_{m,n=0}^{\infty}D(m,n)x^{m}y^{n}=\frac{1}{1-x-y-xy}," display="block"><mrow><mrow><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mrow><mi href="./26.1#p2.t1.r2">m</mi><mo>,</mo><mi href="./26.1#p2.t1.r2">n</mi></mrow><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><mrow><mi href="./26.6#E1">D</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./26.1#p2.t1.r2">m</mi><mo>,</mo><mi href="./26.1#p2.t1.r2">n</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><msup><mi href="./26.6#SS2.p1">x</mi><mi href="./26.1#p2.t1.r2">m</mi></msup><mo></mo><msup><mi href="./26.6#SS2.p1">y</mi><mi href="./26.1#p2.t1.r2">n</mi></msup></mrow></mrow><mo>=</mo><mfrac><mn>1</mn><mrow><mn>1</mn><mo>-</mo><mi href="./26.6#SS2.p1">x</mi><mo>-</mo><mi href="./26.6#SS2.p1">y</mi><mo>-</mo><mrow><mi href="./26.6#SS2.p1">x</mi><mo></mo><mi href="./26.6#SS2.p1">y</mi></mrow></mrow></mfrac></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E5.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./26.1#p2.t1.r2">m</mi></math>: nonnegative integer</a>,
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m61.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math>: nonnegative integer</a>,
<a href="./26.6#E1" title="(26.6.1) ‣ Delannoy Number D ( m , n ) ‣ §26.6(i) Definitions ‣ §26.6 Other Lattice Path Numbers ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="23px" altimg-valign="-7px" altimg-width="75px" alttext="D(m,n)" display="inline"><mrow><mi href="./26.6#E1">D</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./26.1#p2.t1.r2">m</mi><mo>,</mo><mi href="./26.1#p2.t1.r2">n</mi><mo stretchy="false">)</mo></mrow></mrow></math>: Delannoy number</a>,
<a href="./26.6#SS2.p1" title="§26.6(ii) Generating Functions ‣ §26.6 Other Lattice Path Numbers ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./26.6#SS2.p1">x</mi></math>: small</a> and
<a href="./26.6#SS2.p1" title="§26.6(ii) Generating Functions ‣ §26.6 Other Lattice Path Numbers ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m68.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./26.6#SS2.p1">y</mi></math>: small</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS2.p2" class="ltx_para">
<table id="E6" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">26.6.6</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E6.png" altimg-height="64px" altimg-valign="-27px" altimg-width="285px" alttext="\sum_{n=0}^{\infty}D(n,n)x^{n}=\frac{1}{\sqrt{1-6x+x^{2}}}," display="block"><mrow><mrow><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./26.1#p2.t1.r2">n</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><mrow><mi href="./26.6#E1">D</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./26.1#p2.t1.r2">n</mi><mo>,</mo><mi href="./26.1#p2.t1.r2">n</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><msup><mi href="./26.6#SS2.p1">x</mi><mi href="./26.1#p2.t1.r2">n</mi></msup></mrow></mrow><mo>=</mo><mfrac><mn>1</mn><msqrt><mrow><mrow><mn>1</mn><mo>-</mo><mrow><mn>6</mn><mo></mo><mi href="./26.6#SS2.p1">x</mi></mrow></mrow><mo>+</mo><msup><mi href="./26.6#SS2.p1">x</mi><mn>2</mn></msup></mrow></msqrt></mfrac></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E6.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m61.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math>: nonnegative integer</a>,
<a href="./26.6#E1" title="(26.6.1) ‣ Delannoy Number D ( m , n ) ‣ §26.6(i) Definitions ‣ §26.6 Other Lattice Path Numbers ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="23px" altimg-valign="-7px" altimg-width="75px" alttext="D(m,n)" display="inline"><mrow><mi href="./26.6#E1">D</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./26.1#p2.t1.r2">m</mi><mo>,</mo><mi href="./26.1#p2.t1.r2">n</mi><mo stretchy="false">)</mo></mrow></mrow></math>: Delannoy number</a> and
<a href="./26.6#SS2.p1" title="§26.6(ii) Generating Functions ‣ §26.6 Other Lattice Path Numbers ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./26.6#SS2.p1">x</mi></math>: small</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS2.p3" class="ltx_para">
<table id="E7" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">26.6.7</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E7.png" altimg-height="64px" altimg-valign="-27px" altimg-width="349px" alttext="\sum_{n=0}^{\infty}M(n)x^{n}=\frac{1-x-\sqrt{1-2x-3x^{2}}}{2x^{2}}," display="block"><mrow><mrow><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./26.1#p2.t1.r2">n</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><mrow><mi href="./26.6#E2">M</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./26.1#p2.t1.r2">n</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><msup><mi href="./26.6#SS2.p1">x</mi><mi href="./26.1#p2.t1.r2">n</mi></msup></mrow></mrow><mo>=</mo><mfrac><mrow><mn>1</mn><mo>-</mo><mi href="./26.6#SS2.p1">x</mi><mo>-</mo><msqrt><mrow><mn>1</mn><mo>-</mo><mrow><mn>2</mn><mo></mo><mi href="./26.6#SS2.p1">x</mi></mrow><mo>-</mo><mrow><mn>3</mn><mo></mo><msup><mi href="./26.6#SS2.p1">x</mi><mn>2</mn></msup></mrow></mrow></msqrt></mrow><mrow><mn>2</mn><mo></mo><msup><mi href="./26.6#SS2.p1">x</mi><mn>2</mn></msup></mrow></mfrac></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E7.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m61.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math>: nonnegative integer</a>,
<a href="./26.6#E2" title="(26.6.2) ‣ Motzkin Number M ( n ) ‣ §26.6(i) Definitions ‣ §26.6 Other Lattice Path Numbers ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="23px" altimg-valign="-7px" altimg-width="53px" alttext="M(n)" display="inline"><mrow><mi href="./26.6#E2">M</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./26.1#p2.t1.r2">n</mi><mo stretchy="false">)</mo></mrow></mrow></math>: Motzkin number</a> and
<a href="./26.6#SS2.p1" title="§26.6(ii) Generating Functions ‣ §26.6 Other Lattice Path Numbers ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./26.6#SS2.p1">x</mi></math>: small</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS2.p4" class="ltx_para">
<table id="E8" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">26.6.8</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E8.png" altimg-height="67px" altimg-valign="-31px" altimg-width="520px" alttext="\sum_{n,k=1}^{\infty}N(n,k)x^{n}y^{k}=\frac{1-x-xy-\sqrt{(1-x-xy)^{2}-4x^{2}y}%
}{2x}," display="block"><mrow><mrow><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mrow><mi href="./26.1#p2.t1.r2">n</mi><mo>,</mo><mi href="./26.1#p2.t1.r2">k</mi></mrow><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><mrow><mi href="./26.6#E3">N</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./26.1#p2.t1.r2">n</mi><mo>,</mo><mi href="./26.1#p2.t1.r2">k</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><msup><mi href="./26.6#SS2.p1">x</mi><mi href="./26.1#p2.t1.r2">n</mi></msup><mo></mo><msup><mi href="./26.6#SS2.p1">y</mi><mi href="./26.1#p2.t1.r2">k</mi></msup></mrow></mrow><mo>=</mo><mfrac><mrow><mn>1</mn><mo>-</mo><mi href="./26.6#SS2.p1">x</mi><mo>-</mo><mrow><mi href="./26.6#SS2.p1">x</mi><mo></mo><mi href="./26.6#SS2.p1">y</mi></mrow><mo>-</mo><msqrt><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><mi href="./26.6#SS2.p1">x</mi><mo>-</mo><mrow><mi href="./26.6#SS2.p1">x</mi><mo></mo><mi href="./26.6#SS2.p1">y</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>-</mo><mrow><mn>4</mn><mo></mo><msup><mi href="./26.6#SS2.p1">x</mi><mn>2</mn></msup><mo></mo><mi href="./26.6#SS2.p1">y</mi></mrow></mrow></msqrt></mrow><mrow><mn>2</mn><mo></mo><mi href="./26.6#SS2.p1">x</mi></mrow></mfrac></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E8.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./26.1#p2.t1.r2">k</mi></math>: nonnegative integer</a>,
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m61.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math>: nonnegative integer</a>,
<a href="./26.6#E3" title="(26.6.3) ‣ Narayana Number N ( n , k ) ‣ §26.6(i) Definitions ‣ §26.6 Other Lattice Path Numbers ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="23px" altimg-valign="-7px" altimg-width="70px" alttext="N(n,k)" display="inline"><mrow><mi href="./26.6#E3">N</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./26.1#p2.t1.r2">n</mi><mo>,</mo><mi href="./26.1#p2.t1.r2">k</mi><mo stretchy="false">)</mo></mrow></mrow></math>: Narayana number</a>,
<a href="./26.6#SS2.p1" title="§26.6(ii) Generating Functions ‣ §26.6 Other Lattice Path Numbers ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./26.6#SS2.p1">x</mi></math>: small</a> and
<a href="./26.6#SS2.p1" title="§26.6(ii) Generating Functions ‣ §26.6 Other Lattice Path Numbers ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m68.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./26.6#SS2.p1">y</mi></math>: small</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS2.p5" class="ltx_para">
<table id="E9" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">26.6.9</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E9.png" altimg-height="64px" altimg-valign="-27px" altimg-width="327px" alttext="\sum_{n=0}^{\infty}r(n)x^{n}=\frac{1-x-\sqrt{1-6x+x^{2}}}{2x}." display="block"><mrow><mrow><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./26.1#p2.t1.r2">n</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><mrow><mi href="./26.6#E4">r</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./26.1#p2.t1.r2">n</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><msup><mi href="./26.6#SS2.p1">x</mi><mi href="./26.1#p2.t1.r2">n</mi></msup></mrow></mrow><mo>=</mo><mfrac><mrow><mn>1</mn><mo>-</mo><mi href="./26.6#SS2.p1">x</mi><mo>-</mo><msqrt><mrow><mrow><mn>1</mn><mo>-</mo><mrow><mn>6</mn><mo></mo><mi href="./26.6#SS2.p1">x</mi></mrow></mrow><mo>+</mo><msup><mi href="./26.6#SS2.p1">x</mi><mn>2</mn></msup></mrow></msqrt></mrow><mrow><mn>2</mn><mo></mo><mi href="./26.6#SS2.p1">x</mi></mrow></mfrac></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E9.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m61.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math>: nonnegative integer</a>,
<a href="./26.6#E4" title="(26.6.4) ‣ Schröder Number r ( n ) ‣ §26.6(i) Definitions ‣ §26.6 Other Lattice Path Numbers ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m65.png" altimg-height="23px" altimg-valign="-7px" altimg-width="41px" alttext="r(n)" display="inline"><mrow><mi href="./26.6#E4">r</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./26.1#p2.t1.r2">n</mi><mo stretchy="false">)</mo></mrow></mrow></math>: Schröder number</a> and
<a href="./26.6#SS2.p1" title="§26.6(ii) Generating Functions ‣ §26.6 Other Lattice Path Numbers ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./26.6#SS2.p1">x</mi></math>: small</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="iii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§26.6(iii) </span>Recurrence Relations</h2>
<div id="SS3.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="SS3.p1" class="ltx_para">
<table id="EGx1" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E10">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">26.6.10</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m3.png" altimg-height="25px" altimg-valign="-7px" altimg-width="77px" alttext="\displaystyle D(m,n)" display="inline"><mrow><mi href="./26.6#E1">D</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./26.1#p2.t1.r2">m</mi><mo>,</mo><mi href="./26.1#p2.t1.r2">n</mi><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m1.png" altimg-height="25px" altimg-valign="-7px" altimg-width="431px" alttext="\displaystyle=D(m,n-1)+D(m-1,n)+D(m-1,n-1)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mi href="./26.6#E1">D</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./26.1#p2.t1.r2">m</mi><mo>,</mo><mrow><mi href="./26.1#p2.t1.r2">n</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>+</mo><mrow><mi href="./26.6#E1">D</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./26.1#p2.t1.r2">m</mi><mo>-</mo><mn>1</mn></mrow><mo>,</mo><mi href="./26.1#p2.t1.r2">n</mi><mo stretchy="false">)</mo></mrow></mrow><mo>+</mo><mrow><mi href="./26.6#E1">D</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./26.1#p2.t1.r2">m</mi><mo>-</mo><mn>1</mn></mrow><mo>,</mo><mrow><mi href="./26.1#p2.t1.r2">n</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m59.png" altimg-height="20px" altimg-valign="-6px" altimg-width="79px" alttext="m,n\geq 1" display="inline"><mrow><mrow><mi href="./26.1#p2.t1.r2">m</mi><mo>,</mo><mi href="./26.1#p2.t1.r2">n</mi></mrow><mo>≥</mo><mn>1</mn></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E10.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./26.1#p2.t1.r2">m</mi></math>: nonnegative integer</a>,
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m61.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math>: nonnegative integer</a> and
<a href="./26.6#E1" title="(26.6.1) ‣ Delannoy Number D ( m , n ) ‣ §26.6(i) Definitions ‣ §26.6 Other Lattice Path Numbers ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="23px" altimg-valign="-7px" altimg-width="75px" alttext="D(m,n)" display="inline"><mrow><mi href="./26.6#E1">D</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./26.1#p2.t1.r2">m</mi><mo>,</mo><mi href="./26.1#p2.t1.r2">n</mi><mo stretchy="false">)</mo></mrow></mrow></math>: Delannoy number</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E11">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">26.6.11</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m4.png" altimg-height="25px" altimg-valign="-7px" altimg-width="55px" alttext="\displaystyle M(n)" display="inline"><mrow><mi href="./26.6#E2">M</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./26.1#p2.t1.r2">n</mi><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m2.png" altimg-height="64px" altimg-valign="-28px" altimg-width="342px" alttext="\displaystyle=M(n-1)+\sum_{k=2}^{n}M(k-2)\,M(n-k)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mi href="./26.6#E2">M</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./26.1#p2.t1.r2">n</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>+</mo><mrow><mstyle displaystyle="true"><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./26.1#p2.t1.r2">k</mi><mo>=</mo><mn>2</mn></mrow><mi href="./26.1#p2.t1.r2">n</mi></munderover></mstyle><mrow><mrow><mi href="./26.6#E2">M</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./26.1#p2.t1.r2">k</mi><mo>-</mo><mn>2</mn></mrow><mo rspace="4.2pt" stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mi href="./26.6#E2">M</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./26.1#p2.t1.r2">n</mi><mo>-</mo><mi href="./26.1#p2.t1.r2">k</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m63.png" altimg-height="19px" altimg-valign="-5px" altimg-width="53px" alttext="n\geq 2" display="inline"><mrow><mi href="./26.1#p2.t1.r2">n</mi><mo>≥</mo><mn>2</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E11.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./26.1#p2.t1.r2">k</mi></math>: nonnegative integer</a>,
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m61.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math>: nonnegative integer</a> and
<a href="./26.6#E2" title="(26.6.2) ‣ Motzkin Number M ( n ) ‣ §26.6(i) Definitions ‣ §26.6 Other Lattice Path Numbers ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="23px" altimg-valign="-7px" altimg-width="53px" alttext="M(n)" display="inline"><mrow><mi href="./26.6#E2">M</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./26.1#p2.t1.r2">n</mi><mo stretchy="false">)</mo></mrow></mrow></math>: Motzkin number</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
</div>
</section>
<section id="iv" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§26.6(iv) </span>Identities</h2>
<div id="SS4.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="SS4.p1" class="ltx_para">
<table id="E12" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">26.6.12</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E12.png" altimg-height="64px" altimg-valign="-28px" altimg-width="182px" alttext="C\left(n\right)=\sum_{k=1}^{n}N(n,k)," display="block"><mrow><mrow><mrow><mi href="./26.5#E1">C</mi><mo></mo><mrow><mo>(</mo><mi href="./26.1#p2.t1.r2">n</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./26.1#p2.t1.r2">k</mi><mo>=</mo><mn>1</mn></mrow><mi href="./26.1#p2.t1.r2">n</mi></munderover><mrow><mi href="./26.6#E3">N</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./26.1#p2.t1.r2">n</mi><mo>,</mo><mi href="./26.1#p2.t1.r2">k</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E12.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./26.5#E1" title="(26.5.1) ‣ §26.5(i) Definitions ‣ §26.5 Lattice Paths: Catalan Numbers ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="23px" altimg-valign="-7px" altimg-width="51px" alttext="C\left(\NVar{n}\right)" display="inline"><mrow><mi href="./26.5#E1">C</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./26.1#p2.t1.r2">n</mi><mo>)</mo></mrow></mrow></math>: Catalan number</a>,
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./26.1#p2.t1.r2">k</mi></math>: nonnegative integer</a>,
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m61.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math>: nonnegative integer</a> and
<a href="./26.6#E3" title="(26.6.3) ‣ Narayana Number N ( n , k ) ‣ §26.6(i) Definitions ‣ §26.6 Other Lattice Path Numbers ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="23px" altimg-valign="-7px" altimg-width="70px" alttext="N(n,k)" display="inline"><mrow><mi href="./26.6#E3">N</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./26.1#p2.t1.r2">n</mi><mo>,</mo><mi href="./26.1#p2.t1.r2">k</mi><mo stretchy="false">)</mo></mrow></mrow></math>: Narayana number</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS4.p2" class="ltx_para">
<table id="E13" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">26.6.13</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E13.png" altimg-height="64px" altimg-valign="-28px" altimg-width="328px" alttext="M(n)=\sum_{k=0}^{n}(-1)^{k}\genfrac{(}{)}{0.0pt}{}{n}{k}C\left(n+1-k\right)," display="block"><mrow><mrow><mrow><mi href="./26.6#E2">M</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./26.1#p2.t1.r2">n</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./26.1#p2.t1.r2">k</mi><mo>=</mo><mn>0</mn></mrow><mi href="./26.1#p2.t1.r2">n</mi></munderover><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./26.1#p2.t1.r2">k</mi></msup><mo></mo><mrow><mo href="./1.2#i">(</mo><mfrac linethickness="0.0pt"><mi href="./26.1#p2.t1.r2">n</mi><mi href="./26.1#p2.t1.r2">k</mi></mfrac><mo href="./1.2#i">)</mo></mrow><mo></mo><mrow><mi href="./26.5#E1">C</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mi href="./26.1#p2.t1.r2">n</mi><mo>+</mo><mn>1</mn></mrow><mo>-</mo><mi href="./26.1#p2.t1.r2">k</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E13.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./26.5#E1" title="(26.5.1) ‣ §26.5(i) Definitions ‣ §26.5 Lattice Paths: Catalan Numbers ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="23px" altimg-valign="-7px" altimg-width="51px" alttext="C\left(\NVar{n}\right)" display="inline"><mrow><mi href="./26.5#E1">C</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./26.1#p2.t1.r2">n</mi><mo>)</mo></mrow></mrow></math>: Catalan number</a>,
<a href="./1.2#i" title="§1.2(i) Binomial Coefficients ‣ §1.2 Elementary Algebra ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m57.png" altimg-height="27px" altimg-valign="-9px" altimg-width="37px" alttext="\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}" display="inline"><mrow><mo href="./1.2#i">(</mo><mfrac linethickness="0.0pt"><mi class="ltx_nvar" href="./1.1#p2.t1.r5">m</mi><mi class="ltx_nvar" href="./1.1#p2.t1.r5">n</mi></mfrac><mo href="./1.2#i">)</mo></mrow></math>: binomial coefficient</a>,
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./26.1#p2.t1.r2">k</mi></math>: nonnegative integer</a>,
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m61.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math>: nonnegative integer</a> and
<a href="./26.6#E2" title="(26.6.2) ‣ Motzkin Number M ( n ) ‣ §26.6(i) Definitions ‣ §26.6 Other Lattice Path Numbers ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="23px" altimg-valign="-7px" altimg-width="53px" alttext="M(n)" display="inline"><mrow><mi href="./26.6#E2">M</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./26.1#p2.t1.r2">n</mi><mo stretchy="false">)</mo></mrow></mrow></math>: Motzkin number</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS4.p3" class="ltx_para">
<table id="E14" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">26.6.14</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E14.png" altimg-height="67px" altimg-valign="-28px" altimg-width="310px" alttext="C\left(n\right)=\sum_{k=0}^{2n}(-1)^{k}\genfrac{(}{)}{0.0pt}{}{2n}{k}M(2n-k)." display="block"><mrow><mrow><mrow><mi href="./26.5#E1">C</mi><mo></mo><mrow><mo>(</mo><mi href="./26.1#p2.t1.r2">n</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./26.1#p2.t1.r2">k</mi><mo>=</mo><mn>0</mn></mrow><mrow><mn>2</mn><mo></mo><mi href="./26.1#p2.t1.r2">n</mi></mrow></munderover><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./26.1#p2.t1.r2">k</mi></msup><mo></mo><mrow><mo href="./1.2#i">(</mo><mfrac linethickness="0.0pt"><mrow><mn>2</mn><mo></mo><mi href="./26.1#p2.t1.r2">n</mi></mrow><mi href="./26.1#p2.t1.r2">k</mi></mfrac><mo href="./1.2#i">)</mo></mrow><mo></mo><mrow><mi href="./26.6#E2">M</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>2</mn><mo></mo><mi href="./26.1#p2.t1.r2">n</mi></mrow><mo>-</mo><mi href="./26.1#p2.t1.r2">k</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E14.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./26.5#E1" title="(26.5.1) ‣ §26.5(i) Definitions ‣ §26.5 Lattice Paths: Catalan Numbers ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="23px" altimg-valign="-7px" altimg-width="51px" alttext="C\left(\NVar{n}\right)" display="inline"><mrow><mi href="./26.5#E1">C</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./26.1#p2.t1.r2">n</mi><mo>)</mo></mrow></mrow></math>: Catalan number</a>,
<a href="./1.2#i" title="§1.2(i) Binomial Coefficients ‣ §1.2 Elementary Algebra ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m57.png" altimg-height="27px" altimg-valign="-9px" altimg-width="37px" alttext="\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}" display="inline"><mrow><mo href="./1.2#i">(</mo><mfrac linethickness="0.0pt"><mi class="ltx_nvar" href="./1.1#p2.t1.r5">m</mi><mi class="ltx_nvar" href="./1.1#p2.t1.r5">n</mi></mfrac><mo href="./1.2#i">)</mo></mrow></math>: binomial coefficient</a>,
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./26.1#p2.t1.r2">k</mi></math>: nonnegative integer</a>,
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m61.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math>: nonnegative integer</a> and
<a href="./26.6#E2" title="(26.6.2) ‣ Motzkin Number M ( n ) ‣ §26.6(i) Definitions ‣ §26.6 Other Lattice Path Numbers ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="23px" altimg-valign="-7px" altimg-width="53px" alttext="M(n)" display="inline"><mrow><mi href="./26.6#E2">M</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./26.1#p2.t1.r2">n</mi><mo stretchy="false">)</mo></mrow></mrow></math>: Motzkin number</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
</section>
</div>
<div class="ltx_page_footer">
<span></div>
</div>
</body></text>
</html>
</page>
<page>
<!DOCTYPE html><html>
<head>
<title>DLMF: 26.10 Integer Partitions: Other Restrictions</title>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<!--GOOGLE BOOTSTRAP--></head>
<text><body class="color_default textfont_default titlefont_default fontsize_default navbar_default">
<div class="ltx_page_navbar">
<div class="ltx_page_navlogo"><div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m69.png" altimg-height="23px" altimg-valign="-7px" altimg-width="135px" alttext="p\left(\NVar{\mathrm{condition}},\NVar{n}\right)" display="inline"><mrow><mi href="./26.10#i">p</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">condition</mi><mo>,</mo><mi class="ltx_nvar" href="./26.1#p2.t1.r2">n</mi><mo>)</mo></mrow></mrow></math>: restricted number of partions of <math class="ltx_Math" altimg="m66.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math></span></dd>
<dt>Keywords:</dt>
<dd></dd>
</dl>
</div>
</div>
<div id="SS1.p1" class="ltx_para">
<p class="ltx_p"><math class="ltx_Math" altimg="m74.png" altimg-height="23px" altimg-valign="-7px" altimg-width="70px" alttext="p\left(\mathcal{D},n\right)" display="inline"><mrow><mi href="./26.10#i">p</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_font_mathcaligraphic">𝒟</mi><mo>,</mo><mi href="./26.1#p2.t1.r2">n</mi><mo>)</mo></mrow></mrow></math> denotes the number of partitions
of <math class="ltx_Math" altimg="m66.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math> into distinct parts.
<math class="ltx_Math" altimg="m85.png" altimg-height="23px" altimg-valign="-7px" altimg-width="85px" alttext="p_{m}\left(\mathcal{D},n\right)" display="inline"><mrow><msub><mi href="./26.9#i">p</mi><mi href="./26.1#p2.t1.r2">m</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_font_mathcaligraphic">𝒟</mi><mo>,</mo><mi href="./26.1#p2.t1.r2">n</mi><mo>)</mo></mrow></mrow></math> denotes the number of partitions
of <math class="ltx_Math" altimg="m66.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math> into at most <math class="ltx_Math" altimg="m61.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./26.1#p2.t1.r2">m</mi></math> distinct parts.
<math class="ltx_Math" altimg="m79.png" altimg-height="23px" altimg-valign="-7px" altimg-width="81px" alttext="p\left(\mathcal{D}k,n\right)" display="inline"><mrow><mi href="./26.10#i">p</mi><mo></mo><mrow><mo>(</mo><mrow><mi class="ltx_font_mathcaligraphic">𝒟</mi><mo></mo><mi href="./26.1#p2.t1.r2">k</mi></mrow><mo>,</mo><mi href="./26.1#p2.t1.r2">n</mi><mo>)</mo></mrow></mrow></math> denotes the number of partitions of
<math class="ltx_Math" altimg="m66.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math> into parts with difference at least <math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./26.1#p2.t1.r2">k</mi></math>.
<math class="ltx_Math" altimg="m77.png" altimg-height="24px" altimg-valign="-7px" altimg-width="85px" alttext="p\left(\mathcal{D}^{\prime}3,n\right)" display="inline"><mrow><mi href="./26.10#i">p</mi><mo></mo><mrow><mo>(</mo><mrow><msup><mi class="ltx_font_mathcaligraphic">𝒟</mi><mo>′</mo></msup><mo></mo><mn>3</mn></mrow><mo>,</mo><mi href="./26.1#p2.t1.r2">n</mi><mo>)</mo></mrow></mrow></math> denotes the
number of partitions of <math class="ltx_Math" altimg="m66.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math> into parts with difference at least 3, except that
multiples of 3 must differ by at least 6.
<math class="ltx_Math" altimg="m81.png" altimg-height="23px" altimg-valign="-7px" altimg-width="70px" alttext="p\left(\mathcal{O},n\right)" display="inline"><mrow><mi href="./26.10#i">p</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_font_mathcaligraphic">𝒪</mi><mo>,</mo><mi href="./26.1#p2.t1.r2">n</mi><mo>)</mo></mrow></mrow></math> denotes the number
of partitions of <math class="ltx_Math" altimg="m66.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math> into odd parts.
<math class="ltx_Math" altimg="m73.png" altimg-height="23px" altimg-valign="-7px" altimg-width="82px" alttext="p\left(\in\!S,n\right)" display="inline"><mrow><mi href="./26.10#i">p</mi><mo></mo><mrow><mo>(</mo><mrow><mi></mi><mo href="./front/introduction#Sx4.p1.t1.r9" rspace="0.8pt">∈</mo><mi href="./26.10#SS1.p1">S</mi></mrow><mo>,</mo><mi href="./26.1#p2.t1.r2">n</mi><mo>)</mo></mrow></mrow></math> denotes the number of
partitions of <math class="ltx_Math" altimg="m66.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math> into parts taken from the set <math class="ltx_Math" altimg="m40.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi href="./26.10#SS1.p1">S</mi></math>.
The set <math class="ltx_Math" altimg="m53.png" altimg-height="23px" altimg-valign="-7px" altimg-width="237px" alttext="\{n\geq 1\>|\>n\equiv\pm j\ \pmod{k}\}" display="inline"><mrow><mo stretchy="false">{</mo><mrow><mi href="./26.1#p2.t1.r2">n</mi><mo>≥</mo><mpadded width="+2.2pt"><mn>1</mn></mpadded></mrow><mo rspace="4.7pt" stretchy="false">|</mo><mrow><mi href="./26.1#p2.t1.r2">n</mi><mo>≡</mo><mrow><mrow><mo>±</mo><mpadded width="+5pt"><mi href="./26.1#p2.t1.r2">j</mi></mpadded></mrow><mspace width="veryverythickmathspace"></mspace><mrow><mo lspace="8.1pt" stretchy="false">(</mo><mrow><mo>mod</mo><mi href="./26.1#p2.t1.r2">k</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mo stretchy="false">}</mo></mrow></math> is denoted by <math class="ltx_Math" altimg="m36.png" altimg-height="23px" altimg-valign="-8px" altimg-width="40px" alttext="A_{j,k}" display="inline"><msub><mi href="./26.10#SS1.p1">A</mi><mrow><mi href="./26.1#p2.t1.r2">j</mi><mo>,</mo><mi href="./26.1#p2.t1.r2">k</mi></mrow></msub></math>. The
set <math class="ltx_Math" altimg="m52.png" altimg-height="23px" altimg-valign="-7px" altimg-width="104px" alttext="\{2,3,4,\ldots\}" display="inline"><mrow><mo stretchy="false">{</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>,</mo><mi mathvariant="normal">…</mi><mo stretchy="false">}</mo></mrow></math> is denoted by <math class="ltx_Math" altimg="m42.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="T" display="inline"><mi href="./26.10#SS1.p1">T</mi></math>.
If more than one restriction applies, then the restrictions are separated by
commas, for example, <math class="ltx_Math" altimg="m75.png" altimg-height="23px" altimg-valign="-7px" altimg-width="116px" alttext="p\left(\mathcal{D}2,\hbox{}\!\!\in\!T,n\right)" display="inline"><mrow><mi href="./26.10#i">p</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mrow><mi class="ltx_font_mathcaligraphic">𝒟</mi><mo></mo><mn>2</mn></mrow><mo>,</mo><mpadded width="-3.3pt"><mrow></mrow></mpadded></mrow><mo href="./front/introduction#Sx4.p1.t1.r9" rspace="0.8pt">∈</mo><mi href="./26.10#SS1.p1">T</mi></mrow><mo>,</mo><mi href="./26.1#p2.t1.r2">n</mi><mo>)</mo></mrow></mrow></math>.
See Table .</p>
</div>
<div id="SS1.p2" class="ltx_para">
<table id="E1" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">26.10.1</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E1.png" altimg-height="25px" altimg-valign="-7px" altimg-width="315px" alttext="p\left(\mathcal{D},0\right)=p\left(\mathcal{D}k,0\right)=p\left(\in\!S,0\right%
)=1." display="block"><mrow><mrow><mrow><mi href="./26.10#i">p</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_font_mathcaligraphic">𝒟</mi><mo>,</mo><mn>0</mn><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mi href="./26.10#i">p</mi><mo></mo><mrow><mo>(</mo><mrow><mi class="ltx_font_mathcaligraphic">𝒟</mi><mo></mo><mi href="./26.1#p2.t1.r2">k</mi></mrow><mo>,</mo><mn>0</mn><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mi href="./26.10#i">p</mi><mo></mo><mrow><mo>(</mo><mrow><mi></mi><mo href="./front/introduction#Sx4.p1.t1.r9" rspace="0.8pt">∈</mo><mi href="./26.10#SS1.p1">S</mi></mrow><mo>,</mo><mn>0</mn><mo>)</mo></mrow></mrow><mo>=</mo><mn>1</mn></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./front/introduction#Sx4.p1.t1.r9" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="15px" altimg-valign="-3px" altimg-width="18px" alttext="\in" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo></math>: element of</a>,
<a href="./26.10#i" title="§26.10(i) Definitions ‣ §26.10 Integer Partitions: Other Restrictions ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="23px" altimg-valign="-7px" altimg-width="135px" alttext="p\left(\NVar{\mathrm{condition}},\NVar{n}\right)" display="inline"><mrow><mi href="./26.10#i">p</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">condition</mi><mo>,</mo><mi class="ltx_nvar" href="./26.1#p2.t1.r2">n</mi><mo>)</mo></mrow></mrow></math>: restricted number of partions of <math class="ltx_Math" altimg="m66.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math></a>,
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./26.1#p2.t1.r2">k</mi></math>: nonnegative integer</a> and
<a href="./26.10#SS1.p1" title="§26.10(i) Definitions ‣ §26.10 Integer Partitions: Other Restrictions ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m40.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi href="./26.10#SS1.p1">S</mi></math>: set</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<figure id="T1" class="ltx_table">
<figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_table">Table 26.10.1: </span>Partitions restricted by difference conditions, or equivalently with parts from
<math class="ltx_Math" altimg="m36.png" altimg-height="23px" altimg-valign="-8px" altimg-width="40px" alttext="A_{j,k}" display="inline"><msub><mi href="./26.10#SS1.p1">A</mi><mrow><mi href="./26.1#p2.t1.r2">j</mi><mo>,</mo><mi href="./26.1#p2.t1.r2">k</mi></mrow></msub></math>.
</figcaption>
<table id="T1.t1" class="ltx_tabular ltx_centering ltx_guessed_headers ltx_align_middle">
<thead class="ltx_thead">
<tr id="T1.t1.r1" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_th_row ltx_border_r ltx_border_tt" style="padding-left:4.4pt;padding-right:4.4pt;"></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r ltx_border_tt" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m74.png" altimg-height="23px" altimg-valign="-7px" altimg-width="70px" alttext="p\left(\mathcal{D},n\right)" display="inline"><mrow><mi href="./26.10#i">p</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_font_mathcaligraphic">𝒟</mi><mo>,</mo><mi href="./26.1#p2.t1.r2">n</mi><mo>)</mo></mrow></mrow></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r ltx_border_tt" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m76.png" altimg-height="23px" altimg-valign="-7px" altimg-width="80px" alttext="p\left(\mathcal{D}2,n\right)" display="inline"><mrow><mi href="./26.10#i">p</mi><mo></mo><mrow><mo>(</mo><mrow><mi class="ltx_font_mathcaligraphic">𝒟</mi><mo></mo><mn>2</mn></mrow><mo>,</mo><mi href="./26.1#p2.t1.r2">n</mi><mo>)</mo></mrow></mrow></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r ltx_border_tt" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m75.png" altimg-height="23px" altimg-valign="-7px" altimg-width="116px" alttext="p\left(\mathcal{D}2,\hbox{}\!\!\in\!T,n\right)" display="inline"><mrow><mi href="./26.10#i">p</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mrow><mi class="ltx_font_mathcaligraphic">𝒟</mi><mo></mo><mn>2</mn></mrow><mo>,</mo><mpadded width="-3.3pt"><mrow></mrow></mpadded></mrow><mo href="./front/introduction#Sx4.p1.t1.r9" rspace="0.8pt">∈</mo><mi href="./26.10#SS1.p1">T</mi></mrow><mo>,</mo><mi href="./26.1#p2.t1.r2">n</mi><mo>)</mo></mrow></mrow></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m77.png" altimg-height="24px" altimg-valign="-7px" altimg-width="85px" alttext="p\left(\mathcal{D}^{\prime}3,n\right)" display="inline"><mrow><mi href="./26.10#i">p</mi><mo></mo><mrow><mo>(</mo><mrow><msup><mi class="ltx_font_mathcaligraphic">𝒟</mi><mo>′</mo></msup><mo></mo><mn>3</mn></mrow><mo>,</mo><mi href="./26.1#p2.t1.r2">n</mi><mo>)</mo></mrow></mrow></math></th>
</tr>
<tr id="T1.t1.r2" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_th_row ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m66.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;">and</th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;">and</th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;">and</th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column" style="padding-left:4.4pt;padding-right:4.4pt;">and</th>
</tr>
<tr id="T1.t1.r3" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_th_row ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m81.png" altimg-height="23px" altimg-valign="-7px" altimg-width="70px" alttext="p\left(\mathcal{O},n\right)" display="inline"><mrow><mi href="./26.10#i">p</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_font_mathcaligraphic">𝒪</mi><mo>,</mo><mi href="./26.1#p2.t1.r2">n</mi><mo>)</mo></mrow></mrow></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m70.png" altimg-height="24px" altimg-valign="-8px" altimg-width="106px" alttext="p\left(\in\!A_{1,5},n\right)" display="inline"><mrow><mi href="./26.10#i">p</mi><mo></mo><mrow><mo>(</mo><mrow><mi></mi><mo href="./front/introduction#Sx4.p1.t1.r9" rspace="0.8pt">∈</mo><msub><mi href="./26.10#SS1.p1">A</mi><mrow><mn>1</mn><mo>,</mo><mn>5</mn></mrow></msub></mrow><mo>,</mo><mi href="./26.1#p2.t1.r2">n</mi><mo>)</mo></mrow></mrow></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m72.png" altimg-height="24px" altimg-valign="-8px" altimg-width="106px" alttext="p\left(\in\!A_{2,5},n\right)" display="inline"><mrow><mi href="./26.10#i">p</mi><mo></mo><mrow><mo>(</mo><mrow><mi></mi><mo href="./front/introduction#Sx4.p1.t1.r9" rspace="0.8pt">∈</mo><msub><mi href="./26.10#SS1.p1">A</mi><mrow><mn>2</mn><mo>,</mo><mn>5</mn></mrow></msub></mrow><mo>,</mo><mi href="./26.1#p2.t1.r2">n</mi><mo>)</mo></mrow></mrow></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m71.png" altimg-height="24px" altimg-valign="-8px" altimg-width="106px" alttext="p\left(\in\!A_{1,6},n\right)" display="inline"><mrow><mi href="./26.10#i">p</mi><mo></mo><mrow><mo>(</mo><mrow><mi></mi><mo href="./front/introduction#Sx4.p1.t1.r9" rspace="0.8pt">∈</mo><msub><mi href="./26.10#SS1.p1">A</mi><mrow><mn>1</mn><mo>,</mo><mn>6</mn></mrow></msub></mrow><mo>,</mo><mi href="./26.1#p2.t1.r2">n</mi><mo>)</mo></mrow></mrow></math></th>
</tr>
</thead>
<tbody class="ltx_tbody">
<tr id="T1.t1.r4" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m5.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></th>
<td class="ltx_td ltx_align_right ltx_border_r ltx_border_t" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m16.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="1" display="inline"><mn>1</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_r ltx_border_t" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m16.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="1" display="inline"><mn>1</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_r ltx_border_t" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m16.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="1" display="inline"><mn>1</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_t" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m16.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="1" display="inline"><mn>1</mn></math></td>
</tr>
<tr id="T1.t1.r5" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m16.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="1" display="inline"><mn>1</mn></math></th>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m16.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="1" display="inline"><mn>1</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m16.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="1" display="inline"><mn>1</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m5.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m16.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="1" display="inline"><mn>1</mn></math></td>
</tr>
<tr id="T1.t1.r6" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m22.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="2" display="inline"><mn>2</mn></math></th>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m16.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="1" display="inline"><mn>1</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m16.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="1" display="inline"><mn>1</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m16.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="1" display="inline"><mn>1</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m16.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="1" display="inline"><mn>1</mn></math></td>
</tr>
<tr id="T1.t1.r7" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_T ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m26.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="3" display="inline"><mn>3</mn></math></th>
<td class="ltx_td ltx_align_right ltx_border_T ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m22.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="2" display="inline"><mn>2</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_T ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m16.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="1" display="inline"><mn>1</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_T ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m16.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="1" display="inline"><mn>1</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m16.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="1" display="inline"><mn>1</mn></math></td>
</tr>
<tr id="T1.t1.r8" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m28.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="4" display="inline"><mn>4</mn></math></th>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m22.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="2" display="inline"><mn>2</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m22.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="2" display="inline"><mn>2</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m16.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="1" display="inline"><mn>1</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m16.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="1" display="inline"><mn>1</mn></math></td>
</tr>
<tr id="T1.t1.r9" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m30.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="5" display="inline"><mn>5</mn></math></th>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m26.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="3" display="inline"><mn>3</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m22.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="2" display="inline"><mn>2</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m16.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="1" display="inline"><mn>1</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m22.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="2" display="inline"><mn>2</mn></math></td>
</tr>
<tr id="T1.t1.r10" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_T ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m32.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="6" display="inline"><mn>6</mn></math></th>
<td class="ltx_td ltx_align_right ltx_border_T ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m28.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="4" display="inline"><mn>4</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_T ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m26.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="3" display="inline"><mn>3</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_T ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m22.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="2" display="inline"><mn>2</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m22.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="2" display="inline"><mn>2</mn></math></td>
</tr>
<tr id="T1.t1.r11" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m33.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="7" display="inline"><mn>7</mn></math></th>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m30.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="5" display="inline"><mn>5</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m26.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="3" display="inline"><mn>3</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m22.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="2" display="inline"><mn>2</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m26.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="3" display="inline"><mn>3</mn></math></td>
</tr>
<tr id="T1.t1.r12" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m34.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="8" display="inline"><mn>8</mn></math></th>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m32.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="6" display="inline"><mn>6</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m28.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="4" display="inline"><mn>4</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m26.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="3" display="inline"><mn>3</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m26.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="3" display="inline"><mn>3</mn></math></td>
</tr>
<tr id="T1.t1.r13" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_T ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m35.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="9" display="inline"><mn>9</mn></math></th>
<td class="ltx_td ltx_align_right ltx_border_T ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m34.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="8" display="inline"><mn>8</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_T ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m30.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="5" display="inline"><mn>5</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_T ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m26.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="3" display="inline"><mn>3</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m26.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="3" display="inline"><mn>3</mn></math></td>
</tr>
<tr id="T1.t1.r14" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m6.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="10" display="inline"><mn>10</mn></math></th>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m6.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="10" display="inline"><mn>10</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m32.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="6" display="inline"><mn>6</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m28.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="4" display="inline"><mn>4</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m28.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="4" display="inline"><mn>4</mn></math></td>
</tr>
<tr id="T1.t1.r15" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m7.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="11" display="inline"><mn>11</mn></math></th>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m8.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="12" display="inline"><mn>12</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m33.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="7" display="inline"><mn>7</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m28.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="4" display="inline"><mn>4</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m30.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="5" display="inline"><mn>5</mn></math></td>
</tr>
<tr id="T1.t1.r16" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_T ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m8.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="12" display="inline"><mn>12</mn></math></th>
<td class="ltx_td ltx_align_right ltx_border_T ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m11.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="15" display="inline"><mn>15</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_T ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m35.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="9" display="inline"><mn>9</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_T ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m32.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="6" display="inline"><mn>6</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m32.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="6" display="inline"><mn>6</mn></math></td>
</tr>
<tr id="T1.t1.r17" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m9.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="13" display="inline"><mn>13</mn></math></th>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m14.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="18" display="inline"><mn>18</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m6.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="10" display="inline"><mn>10</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m32.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="6" display="inline"><mn>6</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m33.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="7" display="inline"><mn>7</mn></math></td>
</tr>
<tr id="T1.t1.r18" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m10.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="14" display="inline"><mn>14</mn></math></th>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m18.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="22" display="inline"><mn>22</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m8.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="12" display="inline"><mn>12</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m34.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="8" display="inline"><mn>8</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m34.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="8" display="inline"><mn>8</mn></math></td>
</tr>
<tr id="T1.t1.r19" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_T ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m11.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="15" display="inline"><mn>15</mn></math></th>
<td class="ltx_td ltx_align_right ltx_border_T ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m21.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="27" display="inline"><mn>27</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_T ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m10.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="14" display="inline"><mn>14</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_T ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m35.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="9" display="inline"><mn>9</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m35.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="9" display="inline"><mn>9</mn></math></td>
</tr>
<tr id="T1.t1.r20" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m12.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="16" display="inline"><mn>16</mn></math></th>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m24.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="32" display="inline"><mn>32</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m13.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="17" display="inline"><mn>17</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m7.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="11" display="inline"><mn>11</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m6.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="10" display="inline"><mn>10</mn></math></td>
</tr>
<tr id="T1.t1.r21" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m13.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="17" display="inline"><mn>17</mn></math></th>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m25.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="38" display="inline"><mn>38</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m15.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="19" display="inline"><mn>19</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m8.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="12" display="inline"><mn>12</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m8.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="12" display="inline"><mn>12</mn></math></td>
</tr>
<tr id="T1.t1.r22" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_T ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m14.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="18" display="inline"><mn>18</mn></math></th>
<td class="ltx_td ltx_align_right ltx_border_T ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m27.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="46" display="inline"><mn>46</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_T ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m19.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="23" display="inline"><mn>23</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_T ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m11.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="15" display="inline"><mn>15</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m10.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="14" display="inline"><mn>14</mn></math></td>
</tr>
<tr id="T1.t1.r23" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m15.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="19" display="inline"><mn>19</mn></math></th>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m29.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="54" display="inline"><mn>54</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m20.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="26" display="inline"><mn>26</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m12.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="16" display="inline"><mn>16</mn></math></td>
<td class="ltx_td ltx_align_right" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m12.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="16" display="inline"><mn>16</mn></math></td>
</tr>
<tr id="T1.t1.r24" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_b ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m17.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="20" display="inline"><mn>20</mn></math></th>
<td class="ltx_td ltx_align_right ltx_border_b ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m31.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="64" display="inline"><mn>64</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_b ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m23.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="31" display="inline"><mn>31</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_b ltx_border_r" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m17.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="20" display="inline"><mn>20</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_b" style="padding-left:4.4pt;padding-right:4.4pt;"><math class="ltx_Math" altimg="m14.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="18" display="inline"><mn>18</mn></math></td>
</tr>
</tbody>
</table>
<div id="T1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./front/introduction#Sx4.p1.t1.r9" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="15px" altimg-valign="-3px" altimg-width="18px" alttext="\in" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo></math>: element of</a>,
<a href="./26.10#i" title="§26.10(i) Definitions ‣ §26.10 Integer Partitions: Other Restrictions ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="23px" altimg-valign="-7px" altimg-width="135px" alttext="p\left(\NVar{\mathrm{condition}},\NVar{n}\right)" display="inline"><mrow><mi href="./26.10#i">p</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">condition</mi><mo>,</mo><mi class="ltx_nvar" href="./26.1#p2.t1.r2">n</mi><mo>)</mo></mrow></mrow></math>: restricted number of partions of <math class="ltx_Math" altimg="m66.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math></a>,
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="20px" altimg-valign="-6px" altimg-width="14px" alttext="j" display="inline"><mi href="./26.1#p2.t1.r2">j</mi></math>: nonnegative integer</a>,
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./26.1#p2.t1.r2">k</mi></math>: nonnegative integer</a>,
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math>: nonnegative integer</a>,
<a href="./26.10#SS1.p1" title="§26.10(i) Definitions ‣ §26.10 Integer Partitions: Other Restrictions ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m36.png" altimg-height="23px" altimg-valign="-8px" altimg-width="40px" alttext="A_{j,k}" display="inline"><msub><mi href="./26.10#SS1.p1">A</mi><mrow><mi href="./26.1#p2.t1.r2">j</mi><mo>,</mo><mi href="./26.1#p2.t1.r2">k</mi></mrow></msub></math>: set</a> and
<a href="./26.10#SS1.p1" title="§26.10(i) Definitions ‣ §26.10 Integer Partitions: Other Restrictions ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m42.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="T" display="inline"><mi href="./26.10#SS1.p1">T</mi></math>: set</a>
</dd>
<dt>Keywords:</dt>
<dd></dd>
</dl>
</div>
</div>
<div id="SS2.p1" class="ltx_para">
<p class="ltx_p">Throughout this subsection it is assumed that <math class="ltx_Math" altimg="m89.png" altimg-height="23px" altimg-valign="-7px" altimg-width="62px" alttext="|q|<1" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mi href="./26.9#SS2.p1">q</mi><mo stretchy="false">|</mo></mrow><mo><</mo><mn>1</mn></mrow></math>.</p>
<table id="E2" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">26.10.2</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_Math ltx_eqn_cell ltx_align_center"><math class="ltx_Math" alttext="\sum_{n=0}^{\infty}p\left(\mathcal{D},n\right)q^{n}=\prod_{j=1}^{\infty}(1+q^{%
j})=\prod_{j=1}^{\infty}\frac{1}{1-q^{2j-1}}=1+\sum_{m=1}^{\infty}\frac{q^{m(m%
+1)/2}}{(1-q)(1-q^{2})\cdots(1-q^{m})}=1+\sum_{m=1}^{\infty}q^{m}(1+q)(1+q^{2}%
)\cdots\*(1+q^{m-1})," display="block"><mrow><mtable align="baseline 1" columnalign="left" columnspacing="0.1em" displaystyle="true"><mtr><mtd><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./26.1#p2.t1.r2">n</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><mrow><mi href="./26.10#i">p</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_font_mathcaligraphic">𝒟</mi><mo>,</mo><mi href="./26.1#p2.t1.r2">n</mi><mo>)</mo></mrow></mrow><mo></mo><msup><mi href="./26.9#SS2.p1">q</mi><mi href="./26.1#p2.t1.r2">n</mi></msup></mrow></mrow></mtd><mtd><mo>=</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∏</mo><mrow><mi href="./26.1#p2.t1.r2">j</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>+</mo><msup><mi href="./26.9#SS2.p1">q</mi><mi href="./26.1#p2.t1.r2">j</mi></msup></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∏</mo><mrow><mi href="./26.1#p2.t1.r2">j</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mfrac><mn>1</mn><mrow><mn>1</mn><mo>-</mo><msup><mi href="./26.9#SS2.p1">q</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./26.1#p2.t1.r2">j</mi></mrow><mo>-</mo><mn>1</mn></mrow></msup></mrow></mfrac></mrow><mo>=</mo><mrow><mn>1</mn><mo>+</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./26.1#p2.t1.r2">m</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mfrac><msup><mi href="./26.9#SS2.p1">q</mi><mrow><mrow><mi href="./26.1#p2.t1.r2">m</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./26.1#p2.t1.r2">m</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>/</mo><mn>2</mn></mrow></msup><mrow><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><mi href="./26.9#SS2.p1">q</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><msup><mi href="./26.9#SS2.p1">q</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi mathvariant="normal">⋯</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><msup><mi href="./26.9#SS2.p1">q</mi><mi href="./26.1#p2.t1.r2">m</mi></msup></mrow><mo stretchy="false">)</mo></mrow></mrow></mfrac></mrow></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo><mrow><mn>1</mn><mo>+</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./26.1#p2.t1.r2">m</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><msup><mi href="./26.9#SS2.p1">q</mi><mi href="./26.1#p2.t1.r2">m</mi></msup><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>+</mo><mi href="./26.9#SS2.p1">q</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>+</mo><msup><mi href="./26.9#SS2.p1">q</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi mathvariant="normal">⋯</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>+</mo><msup><mi href="./26.9#SS2.p1">q</mi><mrow><mi href="./26.1#p2.t1.r2">m</mi><mo>-</mo><mn>1</mn></mrow></msup></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mtd></mtr></mtable></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./26.10#i" title="§26.10(i) Definitions ‣ §26.10 Integer Partitions: Other Restrictions ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="23px" altimg-valign="-7px" altimg-width="135px" alttext="p\left(\NVar{\mathrm{condition}},\NVar{n}\right)" display="inline"><mrow><mi href="./26.10#i">p</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">condition</mi><mo>,</mo><mi class="ltx_nvar" href="./26.1#p2.t1.r2">n</mi><mo>)</mo></mrow></mrow></math>: restricted number of partions of <math class="ltx_Math" altimg="m66.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math></a>,
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="20px" altimg-valign="-6px" altimg-width="14px" alttext="j" display="inline"><mi href="./26.1#p2.t1.r2">j</mi></math>: nonnegative integer</a>,
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m61.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./26.1#p2.t1.r2">m</mi></math>: nonnegative integer</a>,
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math>: nonnegative integer</a> and
<a href="./26.9#SS2.p1" title="§26.9(ii) Generating Functions ‣ §26.9 Integer Partitions: Restricted Number and Part Size ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m86.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="q" display="inline"><mi href="./26.9#SS2.p1">q</mi></math>: parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">24.2.2</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where the last right-hand side is the sum over <math class="ltx_Math" altimg="m62.png" altimg-height="19px" altimg-valign="-5px" altimg-width="58px" alttext="m\geq 0" display="inline"><mrow><mi href="./26.1#p2.t1.r2">m</mi><mo>≥</mo><mn>0</mn></mrow></math> of the generating
functions for partitions into distinct parts with largest part equal to <math class="ltx_Math" altimg="m61.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./26.1#p2.t1.r2">m</mi></math>.</p>
</div>
<div id="SS2.p2" class="ltx_para">
<table id="E3" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">26.10.3</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E3.png" altimg-height="70px" altimg-valign="-30px" altimg-width="667px" alttext="(1-x)\sum_{m,n=0}^{\infty}p_{m}\left(\leq k,\mathcal{D},n\right)x^{m}q^{n}=%
\sum_{m=0}^{k}\genfrac{[}{]}{0.0pt}{}{k}{m}_{q}q^{m(m+1)/2}x^{m}=\prod_{j=1}^{%
k}(1+x\,q^{j})," display="block"><mrow><mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><mi href="./26.1#p2.t1.r1">x</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mrow><mi href="./26.1#p2.t1.r2">m</mi><mo>,</mo><mi href="./26.1#p2.t1.r2">n</mi></mrow><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><mrow><msub><mi href="./26.9#i">p</mi><mi href="./26.1#p2.t1.r2">m</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi></mi><mo>≤</mo><mrow><mi href="./26.1#p2.t1.r2">k</mi><mo>,</mo><mi class="ltx_font_mathcaligraphic">𝒟</mi></mrow></mrow><mo>,</mo><mi href="./26.1#p2.t1.r2">n</mi><mo>)</mo></mrow></mrow><mo></mo><msup><mi href="./26.1#p2.t1.r1">x</mi><mi href="./26.1#p2.t1.r2">m</mi></msup><mo></mo><msup><mi href="./26.9#SS2.p1">q</mi><mi href="./26.1#p2.t1.r2">n</mi></msup></mrow></mrow></mrow><mo>=</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./26.1#p2.t1.r2">m</mi><mo>=</mo><mn>0</mn></mrow><mi href="./26.1#p2.t1.r2">k</mi></munderover><mrow><msub><mrow><mo href="./17.2#E27">[</mo><mfrac linethickness="0.0pt"><mi href="./26.1#p2.t1.r2">k</mi><mi href="./26.1#p2.t1.r2">m</mi></mfrac><mo href="./17.2#E27">]</mo></mrow><mi href="./26.9#SS2.p1">q</mi></msub><mo></mo><msup><mi href="./26.9#SS2.p1">q</mi><mrow><mrow><mi href="./26.1#p2.t1.r2">m</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./26.1#p2.t1.r2">m</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>/</mo><mn>2</mn></mrow></msup><mo></mo><msup><mi href="./26.1#p2.t1.r1">x</mi><mi href="./26.1#p2.t1.r2">m</mi></msup></mrow></mrow><mo>=</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∏</mo><mrow><mi href="./26.1#p2.t1.r2">j</mi><mo>=</mo><mn>1</mn></mrow><mi href="./26.1#p2.t1.r2">k</mi></munderover><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>+</mo><mrow><mpadded width="+1.7pt"><mi href="./26.1#p2.t1.r1">x</mi></mpadded><mo></mo><msup><mi href="./26.9#SS2.p1">q</mi><mi href="./26.1#p2.t1.r2">j</mi></msup></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m90.png" altimg-height="23px" altimg-valign="-7px" altimg-width="63px" alttext="|x|<1" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mi href="./26.1#p2.t1.r1">x</mi><mo stretchy="false">|</mo></mrow><mo><</mo><mn>1</mn></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./17.2#E27" title="(17.2.27) ‣ §17.2(ii) Binomial Coefficients ‣ §17.2 Calculus ‣ Properties ‣ Chapter 17 q -Hypergeometric and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="31px" altimg-valign="-13px" altimg-width="44px" alttext="\genfrac{[}{]}{0.0pt}{}{\NVar{n}}{\NVar{m}}_{\NVar{q}}" display="inline"><msub><mrow><mo href="./17.2#E27">[</mo><mfrac linethickness="0.0pt"><mi class="ltx_nvar" href="./17.1#p1.t1.r1">n</mi><mi class="ltx_nvar" href="./17.1#p1.t1.r1">m</mi></mfrac><mo href="./17.2#E27">]</mo></mrow><mi class="ltx_nvar" href="./17.1#p1.t1.r4">q</mi></msub></math>: <math class="ltx_Math" altimg="m86.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="q" display="inline"><mi href="./17.1#p1.t1.r4">q</mi></math>-binomial coefficient (or Gaussian polynomial)</a>,
<a href="./26.9#i" title="§26.9(i) Definitions ‣ §26.9 Integer Partitions: Restricted Number and Part Size ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m84.png" altimg-height="23px" altimg-valign="-7px" altimg-width="102px" alttext="p_{\NVar{k}}\left(\leq\NVar{m},\NVar{n}\right)" display="inline"><mrow><msub><mi href="./26.9#i">p</mi><mi class="ltx_nvar" href="./26.1#p2.t1.r2">k</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi></mi><mo>≤</mo><mi class="ltx_nvar" href="./26.1#p2.t1.r2">m</mi></mrow><mo>,</mo><mi class="ltx_nvar" href="./26.1#p2.t1.r2">n</mi><mo>)</mo></mrow></mrow></math>: number of partitions of <math class="ltx_Math" altimg="m66.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math> into at most <math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./26.1#p2.t1.r2">k</mi></math> parts, each less than or equal to <math class="ltx_Math" altimg="m61.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./26.1#p2.t1.r2">m</mi></math></a>,
<a href="./26.1#p2.t1.r1" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m88.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./26.1#p2.t1.r1">x</mi></math>: real variable</a>,
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="20px" altimg-valign="-6px" altimg-width="14px" alttext="j" display="inline"><mi href="./26.1#p2.t1.r2">j</mi></math>: nonnegative integer</a>,
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./26.1#p2.t1.r2">k</mi></math>: nonnegative integer</a>,
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m61.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./26.1#p2.t1.r2">m</mi></math>: nonnegative integer</a>,
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math>: nonnegative integer</a> and
<a href="./26.9#SS2.p1" title="§26.9(ii) Generating Functions ‣ §26.9 Integer Partitions: Restricted Number and Part Size ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m86.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="q" display="inline"><mi href="./26.9#SS2.p1">q</mi></math>: parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS2.p3" class="ltx_para">
<table id="E4" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">26.10.4</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E4.png" altimg-height="64px" altimg-valign="-27px" altimg-width="485px" alttext="\sum_{n=0}^{\infty}p\left(\mathcal{D}k,n\right)q^{n}={1+\sum_{m=1}^{\infty}%
\frac{q^{(km^{2}+(2-k)m)/2}}{(1-q)(1-q^{2})\cdots(1-q^{m})}}," display="block"><mrow><mrow><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./26.1#p2.t1.r2">n</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><mrow><mi href="./26.10#i">p</mi><mo></mo><mrow><mo>(</mo><mrow><mi class="ltx_font_mathcaligraphic">𝒟</mi><mo></mo><mi href="./26.1#p2.t1.r2">k</mi></mrow><mo>,</mo><mi href="./26.1#p2.t1.r2">n</mi><mo>)</mo></mrow></mrow><mo></mo><msup><mi href="./26.9#SS2.p1">q</mi><mi href="./26.1#p2.t1.r2">n</mi></msup></mrow></mrow><mo>=</mo><mrow><mn>1</mn><mo>+</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./26.1#p2.t1.r2">m</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mfrac><msup><mi href="./26.9#SS2.p1">q</mi><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mi href="./26.1#p2.t1.r2">k</mi><mo></mo><msup><mi href="./26.1#p2.t1.r2">m</mi><mn>2</mn></msup></mrow><mo>+</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mn>2</mn><mo>-</mo><mi href="./26.1#p2.t1.r2">k</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi href="./26.1#p2.t1.r2">m</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mo>/</mo><mn>2</mn></mrow></msup><mrow><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><mi href="./26.9#SS2.p1">q</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><msup><mi href="./26.9#SS2.p1">q</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi mathvariant="normal">⋯</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><msup><mi href="./26.9#SS2.p1">q</mi><mi href="./26.1#p2.t1.r2">m</mi></msup></mrow><mo stretchy="false">)</mo></mrow></mrow></mfrac></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E4.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./26.10#i" title="§26.10(i) Definitions ‣ §26.10 Integer Partitions: Other Restrictions ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="23px" altimg-valign="-7px" altimg-width="135px" alttext="p\left(\NVar{\mathrm{condition}},\NVar{n}\right)" display="inline"><mrow><mi href="./26.10#i">p</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">condition</mi><mo>,</mo><mi class="ltx_nvar" href="./26.1#p2.t1.r2">n</mi><mo>)</mo></mrow></mrow></math>: restricted number of partions of <math class="ltx_Math" altimg="m66.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math></a>,
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./26.1#p2.t1.r2">k</mi></math>: nonnegative integer</a>,
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m61.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./26.1#p2.t1.r2">m</mi></math>: nonnegative integer</a>,
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math>: nonnegative integer</a> and
<a href="./26.9#SS2.p1" title="§26.9(ii) Generating Functions ‣ §26.9 Integer Partitions: Restricted Number and Part Size ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m86.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="q" display="inline"><mi href="./26.9#SS2.p1">q</mi></math>: parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS2.p4" class="ltx_para">
<table id="E5" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">26.10.5</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E5.png" altimg-height="67px" altimg-valign="-31px" altimg-width="266px" alttext="\sum_{n=0}^{\infty}p\left(\in\!S,n\right)q^{n}=\prod_{j\in S}\frac{1}{1-q^{j}}." display="block"><mrow><mrow><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./26.1#p2.t1.r2">n</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><mrow><mi href="./26.10#i">p</mi><mo></mo><mrow><mo>(</mo><mrow><mi></mi><mo href="./front/introduction#Sx4.p1.t1.r9" rspace="0.8pt">∈</mo><mi href="./26.10#SS1.p1">S</mi></mrow><mo>,</mo><mi href="./26.1#p2.t1.r2">n</mi><mo>)</mo></mrow></mrow><mo></mo><msup><mi href="./26.9#SS2.p1">q</mi><mi href="./26.1#p2.t1.r2">n</mi></msup></mrow></mrow><mo>=</mo><mrow><munder><mo largeop="true" movablelimits="false" symmetric="true">∏</mo><mrow><mi href="./26.1#p2.t1.r2">j</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mi href="./26.10#SS1.p1">S</mi></mrow></munder><mfrac><mn>1</mn><mrow><mn>1</mn><mo>-</mo><msup><mi href="./26.9#SS2.p1">q</mi><mi href="./26.1#p2.t1.r2">j</mi></msup></mrow></mfrac></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E5.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./front/introduction#Sx4.p1.t1.r9" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="15px" altimg-valign="-3px" altimg-width="18px" alttext="\in" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo></math>: element of</a>,
<a href="./26.10#i" title="§26.10(i) Definitions ‣ §26.10 Integer Partitions: Other Restrictions ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="23px" altimg-valign="-7px" altimg-width="135px" alttext="p\left(\NVar{\mathrm{condition}},\NVar{n}\right)" display="inline"><mrow><mi href="./26.10#i">p</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">condition</mi><mo>,</mo><mi class="ltx_nvar" href="./26.1#p2.t1.r2">n</mi><mo>)</mo></mrow></mrow></math>: restricted number of partions of <math class="ltx_Math" altimg="m66.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math></a>,
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="20px" altimg-valign="-6px" altimg-width="14px" alttext="j" display="inline"><mi href="./26.1#p2.t1.r2">j</mi></math>: nonnegative integer</a>,
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math>: nonnegative integer</a>,
<a href="./26.10#SS1.p1" title="§26.10(i) Definitions ‣ §26.10 Integer Partitions: Other Restrictions ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m40.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi href="./26.10#SS1.p1">S</mi></math>: set</a> and
<a href="./26.9#SS2.p1" title="§26.9(ii) Generating Functions ‣ §26.9 Integer Partitions: Restricted Number and Part Size ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m86.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="q" display="inline"><mi href="./26.9#SS2.p1">q</mi></math>: parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="iii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§26.10(iii) </span>Recurrence Relations</h2>
<div id="SS3.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="SS3.p1" class="ltx_para">
<table id="E6" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">26.10.6</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E6.png" altimg-height="83px" altimg-valign="-47px" altimg-width="307px" alttext="p\left(\mathcal{D},n\right)=\frac{1}{n}\sum_{t=1}^{n}p\left(\mathcal{D},n-t%
\right)\sum_{\begin{subarray}{c}j\mathbin{|}t\\
\mbox{\scriptsize$j$ odd}\end{subarray}}j," display="block"><mrow><mrow><mrow><mi href="./26.10#i">p</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_font_mathcaligraphic">𝒟</mi><mo>,</mo><mi href="./26.1#p2.t1.r2">n</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mfrac><mn>1</mn><mi href="./26.1#p2.t1.r2">n</mi></mfrac><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi>t</mi><mo>=</mo><mn>1</mn></mrow><mi href="./26.1#p2.t1.r2">n</mi></munderover><mrow><mrow><mi href="./26.10#i">p</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_font_mathcaligraphic">𝒟</mi><mo>,</mo><mrow><mi href="./26.1#p2.t1.r2">n</mi><mo>-</mo><mi>t</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><munder><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mtable class="ltx_align_c" rowspacing="0.0pt"><mtr><mtd><mrow><mi href="./26.1#p2.t1.r2">j</mi><mo stretchy="false">|</mo><mi>t</mi></mrow></mtd></mtr><mtr><mtd><mrow><mi href="./26.1#p2.t1.r2">j</mi><mtext> odd</mtext></mrow></mtd></mtr></mtable></munder><mi href="./26.1#p2.t1.r2">j</mi></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E6.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./26.10#i" title="§26.10(i) Definitions ‣ §26.10 Integer Partitions: Other Restrictions ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="23px" altimg-valign="-7px" altimg-width="135px" alttext="p\left(\NVar{\mathrm{condition}},\NVar{n}\right)" display="inline"><mrow><mi href="./26.10#i">p</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">condition</mi><mo>,</mo><mi class="ltx_nvar" href="./26.1#p2.t1.r2">n</mi><mo>)</mo></mrow></mrow></math>: restricted number of partions of <math class="ltx_Math" altimg="m66.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math></a>,
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="20px" altimg-valign="-6px" altimg-width="14px" alttext="j" display="inline"><mi href="./26.1#p2.t1.r2">j</mi></math>: nonnegative integer</a> and
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math>: nonnegative integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where the inner sum is the sum of all positive odd divisors of <math class="ltx_Math" altimg="m87.png" altimg-height="17px" altimg-valign="-2px" altimg-width="11px" alttext="t" display="inline"><mi>t</mi></math>.</p>
</div>
<div id="SS3.p2" class="ltx_para">
<table id="E7" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">26.10.7</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E7.png" altimg-height="65px" altimg-valign="-27px" altimg-width="497px" alttext="\sum(-1)^{k}p\left(\mathcal{D},n-\tfrac{1}{2}(3k^{2}\pm k)\right)=\begin{cases%
}(-1)^{r},&n=3r^{2}\pm r,\\
0,&\mbox{otherwise},\end{cases}" display="block"><mrow><mrow><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./26.1#p2.t1.r2">k</mi></msup><mo></mo><mrow><mi href="./26.10#i">p</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_font_mathcaligraphic">𝒟</mi><mo>,</mo><mrow><mi href="./26.1#p2.t1.r2">n</mi><mo>-</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>3</mn><mo></mo><msup><mi href="./26.1#p2.t1.r2">k</mi><mn>2</mn></msup></mrow><mo>±</mo><mi href="./26.1#p2.t1.r2">k</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>=</mo><mrow><mo>{</mo><mtable columnspacing="5pt" displaystyle="true" rowspacing="0pt"><mtr><mtd columnalign="left"><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi>r</mi></msup><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mrow><mi href="./26.1#p2.t1.r2">n</mi><mo>=</mo><mrow><mrow><mn>3</mn><mo></mo><msup><mi>r</mi><mn>2</mn></msup></mrow><mo>±</mo><mi>r</mi></mrow></mrow><mo>,</mo></mrow></mtd></mtr><mtr><mtd columnalign="left"><mrow><mn>0</mn><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mtext>otherwise</mtext><mo>,</mo></mrow></mtd></mtr></mtable></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E7.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./26.10#i" title="§26.10(i) Definitions ‣ §26.10 Integer Partitions: Other Restrictions ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="23px" altimg-valign="-7px" altimg-width="135px" alttext="p\left(\NVar{\mathrm{condition}},\NVar{n}\right)" display="inline"><mrow><mi href="./26.10#i">p</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">condition</mi><mo>,</mo><mi class="ltx_nvar" href="./26.1#p2.t1.r2">n</mi><mo>)</mo></mrow></mrow></math>: restricted number of partions of <math class="ltx_Math" altimg="m66.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math></a>,
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./26.1#p2.t1.r2">k</mi></math>: nonnegative integer</a> and
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math>: nonnegative integer</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">24.2.2</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where the sum is over nonnegative integer values of <math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./26.1#p2.t1.r2">k</mi></math> for which
<math class="ltx_Math" altimg="m64.png" altimg-height="27px" altimg-valign="-9px" altimg-width="171px" alttext="n-\frac{1}{2}(3k^{2}\pm k)\geq 0" display="inline"><mrow><mrow><mi href="./26.1#p2.t1.r2">n</mi><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>3</mn><mo></mo><msup><mi href="./26.1#p2.t1.r2">k</mi><mn>2</mn></msup></mrow><mo>±</mo><mi href="./26.1#p2.t1.r2">k</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>≥</mo><mn>0</mn></mrow></math>.</p>
</div>
<div id="SS3.p3" class="ltx_para">
<table id="E8" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">26.10.8</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E8.png" altimg-height="65px" altimg-valign="-27px" altimg-width="464px" alttext="\sum(-1)^{k}p\left(\mathcal{D},n-(3k^{2}\pm k)\right)=\begin{cases}1,&n=\tfrac%
{1}{2}(r^{2}\pm r),\\
0,&\mbox{otherwise},\end{cases}" display="block"><mrow><mrow><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./26.1#p2.t1.r2">k</mi></msup><mo></mo><mrow><mi href="./26.10#i">p</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_font_mathcaligraphic">𝒟</mi><mo>,</mo><mrow><mi href="./26.1#p2.t1.r2">n</mi><mo>-</mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>3</mn><mo></mo><msup><mi href="./26.1#p2.t1.r2">k</mi><mn>2</mn></msup></mrow><mo>±</mo><mi href="./26.1#p2.t1.r2">k</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>=</mo><mrow><mo>{</mo><mtable columnspacing="5pt" displaystyle="true" rowspacing="0pt"><mtr><mtd columnalign="left"><mrow><mn>1</mn><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mrow><mi href="./26.1#p2.t1.r2">n</mi><mo>=</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mrow><mo stretchy="false">(</mo><mrow><msup><mi>r</mi><mn>2</mn></msup><mo>±</mo><mi>r</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>,</mo></mrow></mtd></mtr><mtr><mtd columnalign="left"><mrow><mn>0</mn><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mtext>otherwise</mtext><mo>,</mo></mrow></mtd></mtr></mtable></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E8.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./26.10#i" title="§26.10(i) Definitions ‣ §26.10 Integer Partitions: Other Restrictions ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="23px" altimg-valign="-7px" altimg-width="135px" alttext="p\left(\NVar{\mathrm{condition}},\NVar{n}\right)" display="inline"><mrow><mi href="./26.10#i">p</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">condition</mi><mo>,</mo><mi class="ltx_nvar" href="./26.1#p2.t1.r2">n</mi><mo>)</mo></mrow></mrow></math>: restricted number of partions of <math class="ltx_Math" altimg="m66.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math></a>,
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./26.1#p2.t1.r2">k</mi></math>: nonnegative integer</a> and
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math>: nonnegative integer</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">24.2.2</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where the sum is over nonnegative integer values of <math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./26.1#p2.t1.r2">k</mi></math> for which
<math class="ltx_Math" altimg="m63.png" altimg-height="25px" altimg-valign="-7px" altimg-width="158px" alttext="n-(3k^{2}\pm k)\geq 0" display="inline"><mrow><mrow><mi href="./26.1#p2.t1.r2">n</mi><mo>-</mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>3</mn><mo></mo><msup><mi href="./26.1#p2.t1.r2">k</mi><mn>2</mn></msup></mrow><mo>±</mo><mi href="./26.1#p2.t1.r2">k</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>≥</mo><mn>0</mn></mrow></math>.</p>
</div>
<div id="SS3.p4" class="ltx_para">
<p class="ltx_p">In exact analogy with (), we have</p>
</div>
<div id="SS3.p5" class="ltx_para">
<table id="EGx1" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E9">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">26.10.9</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m4.png" altimg-height="25px" altimg-valign="-7px" altimg-width="87px" alttext="\displaystyle p_{m}\left(\mathcal{D},n\right)" display="inline"><mrow><msub><mi href="./26.9#i">p</mi><mi href="./26.1#p2.t1.r2">m</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_font_mathcaligraphic">𝒟</mi><mo>,</mo><mi href="./26.1#p2.t1.r2">n</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m2.png" altimg-height="25px" altimg-valign="-7px" altimg-width="284px" alttext="\displaystyle=p_{m}\left(\mathcal{D},n-m\right)+p_{m-1}\left(\mathcal{D},n%
\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><msub><mi href="./26.9#i">p</mi><mi href="./26.1#p2.t1.r2">m</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_font_mathcaligraphic">𝒟</mi><mo>,</mo><mrow><mi href="./26.1#p2.t1.r2">n</mi><mo>-</mo><mi href="./26.1#p2.t1.r2">m</mi></mrow><mo>)</mo></mrow></mrow><mo>+</mo><mrow><msub><mi href="./26.9#i">p</mi><mrow><mi href="./26.1#p2.t1.r2">m</mi><mo>-</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_font_mathcaligraphic">𝒟</mi><mo>,</mo><mi href="./26.1#p2.t1.r2">n</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E9.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./26.9#i" title="§26.9(i) Definitions ‣ §26.9 Integer Partitions: Restricted Number and Part Size ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m84.png" altimg-height="23px" altimg-valign="-7px" altimg-width="102px" alttext="p_{\NVar{k}}\left(\leq\NVar{m},\NVar{n}\right)" display="inline"><mrow><msub><mi href="./26.9#i">p</mi><mi class="ltx_nvar" href="./26.1#p2.t1.r2">k</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi></mi><mo>≤</mo><mi class="ltx_nvar" href="./26.1#p2.t1.r2">m</mi></mrow><mo>,</mo><mi class="ltx_nvar" href="./26.1#p2.t1.r2">n</mi><mo>)</mo></mrow></mrow></math>: number of partitions of <math class="ltx_Math" altimg="m66.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math> into at most <math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./26.1#p2.t1.r2">k</mi></math> parts, each less than or equal to <math class="ltx_Math" altimg="m61.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./26.1#p2.t1.r2">m</mi></math></a>,
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m61.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./26.1#p2.t1.r2">m</mi></math>: nonnegative integer</a> and
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math>: nonnegative integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E10">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">26.10.10</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m3.png" altimg-height="25px" altimg-valign="-7px" altimg-width="83px" alttext="\displaystyle p\left(\mathcal{D}k,n\right)" display="inline"><mrow><mi href="./26.10#i">p</mi><mo></mo><mrow><mo>(</mo><mrow><mi class="ltx_font_mathcaligraphic">𝒟</mi><mo></mo><mi href="./26.1#p2.t1.r2">k</mi></mrow><mo>,</mo><mi href="./26.1#p2.t1.r2">n</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m1.png" altimg-height="37px" altimg-valign="-13px" altimg-width="309px" alttext="\displaystyle=\sum p_{m}\left(n-\tfrac{1}{2}km^{2}-m+\tfrac{1}{2}km\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mstyle displaystyle="true"><mrow><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><msub><mi href="./26.9#SS1.p1">p</mi><mi href="./26.1#p2.t1.r2">m</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mrow><mi href="./26.1#p2.t1.r2">n</mi><mo>-</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./26.1#p2.t1.r2">k</mi><mo></mo><msup><mi href="./26.1#p2.t1.r2">m</mi><mn>2</mn></msup></mrow><mo>-</mo><mi href="./26.1#p2.t1.r2">m</mi></mrow><mo>+</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./26.1#p2.t1.r2">k</mi><mo></mo><mi href="./26.1#p2.t1.r2">m</mi></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mstyle></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E10.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./26.9#SS1.p1" title="§26.9(i) Definitions ‣ §26.9 Integer Partitions: Restricted Number and Part Size ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m83.png" altimg-height="23px" altimg-valign="-7px" altimg-width="55px" alttext="p_{\NVar{k}}\left(\NVar{n}\right)" display="inline"><mrow><msub><mi href="./26.9#SS1.p1">p</mi><mi class="ltx_nvar" href="./26.1#p2.t1.r2">k</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./26.1#p2.t1.r2">n</mi><mo>)</mo></mrow></mrow></math>: total number of partitions of <math class="ltx_Math" altimg="m66.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math> into at most <math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./26.1#p2.t1.r2">k</mi></math> parts</a>,
<a href="./26.10#i" title="§26.10(i) Definitions ‣ §26.10 Integer Partitions: Other Restrictions ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="23px" altimg-valign="-7px" altimg-width="135px" alttext="p\left(\NVar{\mathrm{condition}},\NVar{n}\right)" display="inline"><mrow><mi href="./26.10#i">p</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">condition</mi><mo>,</mo><mi class="ltx_nvar" href="./26.1#p2.t1.r2">n</mi><mo>)</mo></mrow></mrow></math>: restricted number of partions of <math class="ltx_Math" altimg="m66.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math></a>,
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./26.1#p2.t1.r2">k</mi></math>: nonnegative integer</a>,
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m61.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./26.1#p2.t1.r2">m</mi></math>: nonnegative integer</a> and
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math>: nonnegative integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
<p class="ltx_p">where the sum is over nonnegative integer values of <math class="ltx_Math" altimg="m61.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./26.1#p2.t1.r2">m</mi></math> for which
<math class="ltx_Math" altimg="m65.png" altimg-height="27px" altimg-valign="-9px" altimg-width="235px" alttext="n-\tfrac{1}{2}km^{2}-m+\tfrac{1}{2}km\geq 0" display="inline"><mrow><mrow><mrow><mi href="./26.1#p2.t1.r2">n</mi><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./26.1#p2.t1.r2">k</mi><mo></mo><msup><mi href="./26.1#p2.t1.r2">m</mi><mn>2</mn></msup></mrow><mo>-</mo><mi href="./26.1#p2.t1.r2">m</mi></mrow><mo>+</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./26.1#p2.t1.r2">k</mi><mo></mo><mi href="./26.1#p2.t1.r2">m</mi></mrow></mrow><mo>≥</mo><mn>0</mn></mrow></math>.</p>
</div>
<div id="SS3.p6" class="ltx_para">
<table id="E11" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">26.10.11</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E11.png" altimg-height="83px" altimg-valign="-46px" altimg-width="320px" alttext="p\left(\in\!S,n\right)=\frac{1}{n}\sum_{t=1}^{n}p\left(\in\!S,n-t\right)\sum_{%
\begin{subarray}{c}j\mathbin{|}t\\
j\in S\end{subarray}}j," display="block"><mrow><mrow><mrow><mi href="./26.10#i">p</mi><mo></mo><mrow><mo>(</mo><mrow><mi></mi><mo href="./front/introduction#Sx4.p1.t1.r9" rspace="0.8pt">∈</mo><mi href="./26.10#SS1.p1">S</mi></mrow><mo>,</mo><mi href="./26.1#p2.t1.r2">n</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mfrac><mn>1</mn><mi href="./26.1#p2.t1.r2">n</mi></mfrac><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi>t</mi><mo>=</mo><mn>1</mn></mrow><mi href="./26.1#p2.t1.r2">n</mi></munderover><mrow><mrow><mi href="./26.10#i">p</mi><mo></mo><mrow><mo>(</mo><mrow><mi></mi><mo href="./front/introduction#Sx4.p1.t1.r9" rspace="0.8pt">∈</mo><mi href="./26.10#SS1.p1">S</mi></mrow><mo>,</mo><mrow><mi href="./26.1#p2.t1.r2">n</mi><mo>-</mo><mi>t</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><munder><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mtable class="ltx_align_c" rowspacing="0.0pt"><mtr><mtd><mrow><mi href="./26.1#p2.t1.r2">j</mi><mo stretchy="false">|</mo><mi>t</mi></mrow></mtd></mtr><mtr><mtd><mrow><mi href="./26.1#p2.t1.r2">j</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mi href="./26.10#SS1.p1">S</mi></mrow></mtd></mtr></mtable></munder><mi href="./26.1#p2.t1.r2">j</mi></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E11.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./front/introduction#Sx4.p1.t1.r9" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="15px" altimg-valign="-3px" altimg-width="18px" alttext="\in" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo></math>: element of</a>,
<a href="./26.10#i" title="§26.10(i) Definitions ‣ §26.10 Integer Partitions: Other Restrictions ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="23px" altimg-valign="-7px" altimg-width="135px" alttext="p\left(\NVar{\mathrm{condition}},\NVar{n}\right)" display="inline"><mrow><mi href="./26.10#i">p</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">condition</mi><mo>,</mo><mi class="ltx_nvar" href="./26.1#p2.t1.r2">n</mi><mo>)</mo></mrow></mrow></math>: restricted number of partions of <math class="ltx_Math" altimg="m66.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math></a>,
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="20px" altimg-valign="-6px" altimg-width="14px" alttext="j" display="inline"><mi href="./26.1#p2.t1.r2">j</mi></math>: nonnegative integer</a>,
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math>: nonnegative integer</a> and
<a href="./26.10#SS1.p1" title="§26.10(i) Definitions ‣ §26.10 Integer Partitions: Other Restrictions ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m40.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi href="./26.10#SS1.p1">S</mi></math>: set</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where the inner sum is the sum of all positive divisors of <math class="ltx_Math" altimg="m87.png" altimg-height="17px" altimg-valign="-2px" altimg-width="11px" alttext="t" display="inline"><mi>t</mi></math> that are in <math class="ltx_Math" altimg="m40.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi href="./26.10#SS1.p1">S</mi></math>.</p>
</div>
</section>
<section id="iv" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§26.10(iv) </span>Identities</h2>
<div id="SS4.info" class="ltx_metadata ltx_info">
.
</p>
<table id="E12" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">26.10.12</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E12.png" altimg-height="25px" altimg-valign="-7px" altimg-width="173px" alttext="p\left(\mathcal{D},n\right)=p\left(\mathcal{O},n\right)," display="block"><mrow><mrow><mrow><mi href="./26.10#i">p</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_font_mathcaligraphic">𝒟</mi><mo>,</mo><mi href="./26.1#p2.t1.r2">n</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mi href="./26.10#i">p</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_font_mathcaligraphic">𝒪</mi><mo>,</mo><mi href="./26.1#p2.t1.r2">n</mi><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E12.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./26.10#i" title="§26.10(i) Definitions ‣ §26.10 Integer Partitions: Other Restrictions ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="23px" altimg-valign="-7px" altimg-width="135px" alttext="p\left(\NVar{\mathrm{condition}},\NVar{n}\right)" display="inline"><mrow><mi href="./26.10#i">p</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">condition</mi><mo>,</mo><mi class="ltx_nvar" href="./26.1#p2.t1.r2">n</mi><mo>)</mo></mrow></mrow></math>: restricted number of partions of <math class="ltx_Math" altimg="m66.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math></a> and
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math>: nonnegative integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS4.p2" class="ltx_para">
<table id="E13" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">26.10.13</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E13.png" altimg-height="26px" altimg-valign="-8px" altimg-width="222px" alttext="p\left(\mathcal{D}2,n\right)=p\left(\in A_{1,5},n\right)," display="block"><mrow><mrow><mrow><mi href="./26.10#i">p</mi><mo></mo><mrow><mo>(</mo><mrow><mi class="ltx_font_mathcaligraphic">𝒟</mi><mo></mo><mn>2</mn></mrow><mo>,</mo><mi href="./26.1#p2.t1.r2">n</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mi href="./26.10#i">p</mi><mo></mo><mrow><mo>(</mo><mrow><mi></mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><msub><mi href="./26.10#SS1.p1">A</mi><mrow><mn>1</mn><mo>,</mo><mn>5</mn></mrow></msub></mrow><mo>,</mo><mi href="./26.1#p2.t1.r2">n</mi><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E13.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./front/introduction#Sx4.p1.t1.r9" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="15px" altimg-valign="-3px" altimg-width="18px" alttext="\in" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo></math>: element of</a>,
<a href="./26.10#i" title="§26.10(i) Definitions ‣ §26.10 Integer Partitions: Other Restrictions ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="23px" altimg-valign="-7px" altimg-width="135px" alttext="p\left(\NVar{\mathrm{condition}},\NVar{n}\right)" display="inline"><mrow><mi href="./26.10#i">p</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">condition</mi><mo>,</mo><mi class="ltx_nvar" href="./26.1#p2.t1.r2">n</mi><mo>)</mo></mrow></mrow></math>: restricted number of partions of <math class="ltx_Math" altimg="m66.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math></a>,
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math>: nonnegative integer</a> and
<a href="./26.10#SS1.p1" title="§26.10(i) Definitions ‣ §26.10 Integer Partitions: Other Restrictions ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m36.png" altimg-height="23px" altimg-valign="-8px" altimg-width="40px" alttext="A_{j,k}" display="inline"><msub><mi href="./26.10#SS1.p1">A</mi><mrow><mi href="./26.1#p2.t1.r2">j</mi><mo>,</mo><mi href="./26.1#p2.t1.r2">k</mi></mrow></msub></math>: set</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS4.p3" class="ltx_para">
<table id="E14" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">26.10.14</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E14.png" altimg-height="26px" altimg-valign="-8px" altimg-width="259px" alttext="p\left(\mathcal{D}2,\hbox{}\!\!\in T,n\right)=p\left(\in\!A_{2,5},n\right)," display="block"><mrow><mrow><mrow><mi href="./26.10#i">p</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mrow><mi class="ltx_font_mathcaligraphic">𝒟</mi><mo></mo><mn>2</mn></mrow><mo>,</mo><mpadded width="-3.3pt"><mrow></mrow></mpadded></mrow><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mi href="./26.10#SS1.p1">T</mi></mrow><mo>,</mo><mi href="./26.1#p2.t1.r2">n</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mi href="./26.10#i">p</mi><mo></mo><mrow><mo>(</mo><mrow><mi></mi><mo href="./front/introduction#Sx4.p1.t1.r9" rspace="0.8pt">∈</mo><msub><mi href="./26.10#SS1.p1">A</mi><mrow><mn>2</mn><mo>,</mo><mn>5</mn></mrow></msub></mrow><mo>,</mo><mi href="./26.1#p2.t1.r2">n</mi><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m41.png" altimg-height="23px" altimg-valign="-7px" altimg-width="145px" alttext="T=\{2,3,4,\ldots\}" display="inline"><mrow><mi href="./26.10#SS1.p1">T</mi><mo>=</mo><mrow><mo stretchy="false">{</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>,</mo><mi mathvariant="normal">…</mi><mo stretchy="false">}</mo></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E14.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./front/introduction#Sx4.p1.t1.r9" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="15px" altimg-valign="-3px" altimg-width="18px" alttext="\in" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo></math>: element of</a>,
<a href="./26.10#i" title="§26.10(i) Definitions ‣ §26.10 Integer Partitions: Other Restrictions ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="23px" altimg-valign="-7px" altimg-width="135px" alttext="p\left(\NVar{\mathrm{condition}},\NVar{n}\right)" display="inline"><mrow><mi href="./26.10#i">p</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">condition</mi><mo>,</mo><mi class="ltx_nvar" href="./26.1#p2.t1.r2">n</mi><mo>)</mo></mrow></mrow></math>: restricted number of partions of <math class="ltx_Math" altimg="m66.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math></a>,
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math>: nonnegative integer</a>,
<a href="./26.10#SS1.p1" title="§26.10(i) Definitions ‣ §26.10 Integer Partitions: Other Restrictions ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m36.png" altimg-height="23px" altimg-valign="-8px" altimg-width="40px" alttext="A_{j,k}" display="inline"><msub><mi href="./26.10#SS1.p1">A</mi><mrow><mi href="./26.1#p2.t1.r2">j</mi><mo>,</mo><mi href="./26.1#p2.t1.r2">k</mi></mrow></msub></math>: set</a> and
<a href="./26.10#SS1.p1" title="§26.10(i) Definitions ‣ §26.10 Integer Partitions: Other Restrictions ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m42.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="T" display="inline"><mi href="./26.10#SS1.p1">T</mi></math>: set</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS4.p4" class="ltx_para">
<table id="E15" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">26.10.15</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E15.png" altimg-height="27px" altimg-valign="-8px" altimg-width="228px" alttext="p\left(\mathcal{D}^{\prime}3,n\right)=p\left(\in A_{1,6},n\right)." display="block"><mrow><mrow><mrow><mi href="./26.10#i">p</mi><mo></mo><mrow><mo>(</mo><mrow><msup><mi class="ltx_font_mathcaligraphic">𝒟</mi><mo>′</mo></msup><mo></mo><mn>3</mn></mrow><mo>,</mo><mi href="./26.1#p2.t1.r2">n</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mi href="./26.10#i">p</mi><mo></mo><mrow><mo>(</mo><mrow><mi></mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><msub><mi href="./26.10#SS1.p1">A</mi><mrow><mn>1</mn><mo>,</mo><mn>6</mn></mrow></msub></mrow><mo>,</mo><mi href="./26.1#p2.t1.r2">n</mi><mo>)</mo></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E15.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./front/introduction#Sx4.p1.t1.r9" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="15px" altimg-valign="-3px" altimg-width="18px" alttext="\in" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo></math>: element of</a>,
<a href="./26.10#i" title="§26.10(i) Definitions ‣ §26.10 Integer Partitions: Other Restrictions ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="23px" altimg-valign="-7px" altimg-width="135px" alttext="p\left(\NVar{\mathrm{condition}},\NVar{n}\right)" display="inline"><mrow><mi href="./26.10#i">p</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">condition</mi><mo>,</mo><mi class="ltx_nvar" href="./26.1#p2.t1.r2">n</mi><mo>)</mo></mrow></mrow></math>: restricted number of partions of <math class="ltx_Math" altimg="m66.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math></a>,
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math>: nonnegative integer</a> and
<a href="./26.10#SS1.p1" title="§26.10(i) Definitions ‣ §26.10 Integer Partitions: Other Restrictions ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m36.png" altimg-height="23px" altimg-valign="-8px" altimg-width="40px" alttext="A_{j,k}" display="inline"><msub><mi href="./26.10#SS1.p1">A</mi><mrow><mi href="./26.1#p2.t1.r2">j</mi><mo>,</mo><mi href="./26.1#p2.t1.r2">k</mi></mrow></msub></math>: set</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS4.p5" class="ltx_para">
<p class="ltx_p">Note that <math class="ltx_Math" altimg="m78.png" altimg-height="24px" altimg-valign="-7px" altimg-width="188px" alttext="p\left(\mathcal{D}^{\prime}3,n\right)\leq p\left(\mathcal{D}3,n\right)" display="inline"><mrow><mrow><mi href="./26.10#i">p</mi><mo></mo><mrow><mo>(</mo><mrow><msup><mi class="ltx_font_mathcaligraphic">𝒟</mi><mo>′</mo></msup><mo></mo><mn>3</mn></mrow><mo>,</mo><mi href="./26.1#p2.t1.r2">n</mi><mo>)</mo></mrow></mrow><mo>≤</mo><mrow><mi href="./26.10#i">p</mi><mo></mo><mrow><mo>(</mo><mrow><mi class="ltx_font_mathcaligraphic">𝒟</mi><mo></mo><mn>3</mn></mrow><mo>,</mo><mi href="./26.1#p2.t1.r2">n</mi><mo>)</mo></mrow></mrow></mrow></math>, with strict inequality for
<math class="ltx_Math" altimg="m67.png" altimg-height="19px" altimg-valign="-5px" altimg-width="53px" alttext="n\geq 9" display="inline"><mrow><mi href="./26.1#p2.t1.r2">n</mi><mo>≥</mo><mn>9</mn></mrow></math>. It is known that for <math class="ltx_Math" altimg="m58.png" altimg-height="18px" altimg-valign="-3px" altimg-width="52px" alttext="k>3" display="inline"><mrow><mi href="./26.1#p2.t1.r2">k</mi><mo>></mo><mn>3</mn></mrow></math>,
<math class="ltx_Math" altimg="m80.png" altimg-height="24px" altimg-valign="-8px" altimg-width="230px" alttext="p\left(\mathcal{D}k,n\right)\geq p\left(\in\!A_{1,k+3},n\right)" display="inline"><mrow><mrow><mi href="./26.10#i">p</mi><mo></mo><mrow><mo>(</mo><mrow><mi class="ltx_font_mathcaligraphic">𝒟</mi><mo></mo><mi href="./26.1#p2.t1.r2">k</mi></mrow><mo>,</mo><mi href="./26.1#p2.t1.r2">n</mi><mo>)</mo></mrow></mrow><mo>≥</mo><mrow><mi href="./26.10#i">p</mi><mo></mo><mrow><mo>(</mo><mrow><mi></mi><mo href="./front/introduction#Sx4.p1.t1.r9" rspace="0.8pt">∈</mo><msub><mi href="./26.10#SS1.p1">A</mi><mrow><mn>1</mn><mo>,</mo><mrow><mi href="./26.1#p2.t1.r2">k</mi><mo>+</mo><mn>3</mn></mrow></mrow></msub></mrow><mo>,</mo><mi href="./26.1#p2.t1.r2">n</mi><mo>)</mo></mrow></mrow></mrow></math>, with strict inequality for <math class="ltx_Math" altimg="m66.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math>
sufficiently large, provided that <math class="ltx_Math" altimg="m57.png" altimg-height="21px" altimg-valign="-6px" altimg-width="202px" alttext="k=2^{m}-1,m=3,4,5" display="inline"><mrow><mrow><mi href="./26.1#p2.t1.r2">k</mi><mo>=</mo><mrow><msup><mn>2</mn><mi href="./26.1#p2.t1.r2">m</mi></msup><mo>-</mo><mn>1</mn></mrow></mrow><mo>,</mo><mrow><mi href="./26.1#p2.t1.r2">m</mi><mo>=</mo><mrow><mn>3</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>5</mn></mrow></mrow></mrow></math>, or <math class="ltx_Math" altimg="m60.png" altimg-height="20px" altimg-valign="-5px" altimg-width="62px" alttext="k\geq 32" display="inline"><mrow><mi href="./26.1#p2.t1.r2">k</mi><mo>≥</mo><mn>32</mn></mrow></math>; see <cite class="ltx_cite ltx_citemacro_citet">Yee (</dd>
</dl>
</div>
</div>
<div id="SS5.p1" class="ltx_para">
<table id="E16" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">26.10.16</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E16.png" altimg-height="62px" altimg-valign="-21px" altimg-width="200px" alttext="p\left(\mathcal{D},n\right)\sim\frac{{\mathrm{e}^{\pi\sqrt{n/3}}}}{(768n^{3})^%
{1/4}}," display="block"><mrow><mrow><mrow><mi href="./26.10#i">p</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_font_mathcaligraphic">𝒟</mi><mo>,</mo><mi href="./26.1#p2.t1.r2">n</mi><mo>)</mo></mrow></mrow><mo href="./2.1#E1">∼</mo><mfrac><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mi href="./3.12#E1">π</mi><mo></mo><msqrt><mrow><mi href="./26.1#p2.t1.r2">n</mi><mo>/</mo><mn>3</mn></mrow></msqrt></mrow></msup><msup><mrow><mo stretchy="false">(</mo><mrow><mn>768</mn><mo></mo><msup><mi href="./26.1#p2.t1.r2">n</mi><mn>3</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>4</mn></mrow></msup></mfrac></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m68.png" altimg-height="13px" altimg-valign="-2px" altimg-width="67px" alttext="n\to\infty" display="inline"><mrow><mi href="./26.1#p2.t1.r2">n</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E16.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./2.1#E1" title="(2.1.1) ‣ §2.1(i) Asymptotic and Order Symbols ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="12px" altimg-valign="-2px" altimg-width="20px" alttext="\sim" display="inline"><mo href="./2.1#E1">∼</mo></math>: asymptotic equality</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./26.10#i" title="§26.10(i) Definitions ‣ §26.10 Integer Partitions: Other Restrictions ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="23px" altimg-valign="-7px" altimg-width="135px" alttext="p\left(\NVar{\mathrm{condition}},\NVar{n}\right)" display="inline"><mrow><mi href="./26.10#i">p</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">condition</mi><mo>,</mo><mi class="ltx_nvar" href="./26.1#p2.t1.r2">n</mi><mo>)</mo></mrow></mrow></math>: restricted number of partions of <math class="ltx_Math" altimg="m66.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math></a> and
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math>: nonnegative integer</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">24.2.2</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="vi" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§26.10(vi) </span>Bessel-Function Expansion</h2>
<div id="SS6.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="SS6.p1" class="ltx_para">
<table id="E17" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">26.10.17</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E17.png" altimg-height="66px" altimg-valign="-28px" altimg-width="517px" alttext="p\left(\mathcal{D},n\right)=\pi\sum_{k=1}^{\infty}\frac{A_{2k-1}(n)}{(2k-1)%
\sqrt{24n+1}}I_{1}\left(\frac{\pi}{2k-1}\sqrt{\frac{24n+1}{72}}\right)," display="block"><mrow><mrow><mrow><mi href="./26.10#i">p</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_font_mathcaligraphic">𝒟</mi><mo>,</mo><mi href="./26.1#p2.t1.r2">n</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mi href="./3.12#E1">π</mi><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./26.1#p2.t1.r2">k</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><mfrac><mrow><msub><mi href="./26.10#SS6.p1">A</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./26.1#p2.t1.r2">k</mi></mrow><mo>-</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./26.1#p2.t1.r2">n</mi><mo stretchy="false">)</mo></mrow></mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>2</mn><mo></mo><mi href="./26.1#p2.t1.r2">k</mi></mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msqrt><mrow><mrow><mn>24</mn><mo></mo><mi href="./26.1#p2.t1.r2">n</mi></mrow><mo>+</mo><mn>1</mn></mrow></msqrt></mrow></mfrac><mo></mo><mrow><msub><mi href="./10.25#E2">I</mi><mn>1</mn></msub><mo></mo><mrow><mo>(</mo><mrow><mfrac><mi href="./3.12#E1">π</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./26.1#p2.t1.r2">k</mi></mrow><mo>-</mo><mn>1</mn></mrow></mfrac><mo></mo><msqrt><mfrac><mrow><mrow><mn>24</mn><mo></mo><mi href="./26.1#p2.t1.r2">n</mi></mrow><mo>+</mo><mn>1</mn></mrow><mn>72</mn></mfrac></msqrt></mrow><mo>)</mo></mrow></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E17.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./10.25#E2" title="(10.25.2) ‣ §10.25(ii) Standard Solutions ‣ §10.25 Definitions ‣ Modified Bessel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="23px" altimg-valign="-7px" altimg-width="52px" alttext="I_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.25#E2">I</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: modified Bessel function of the first kind</a>,
<a href="./26.10#i" title="§26.10(i) Definitions ‣ §26.10 Integer Partitions: Other Restrictions ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="23px" altimg-valign="-7px" altimg-width="135px" alttext="p\left(\NVar{\mathrm{condition}},\NVar{n}\right)" display="inline"><mrow><mi href="./26.10#i">p</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">condition</mi><mo>,</mo><mi class="ltx_nvar" href="./26.1#p2.t1.r2">n</mi><mo>)</mo></mrow></mrow></math>: restricted number of partions of <math class="ltx_Math" altimg="m66.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math></a>,
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./26.1#p2.t1.r2">k</mi></math>: nonnegative integer</a>,
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math>: nonnegative integer</a> and
<a href="./26.10#SS6.p1" title="§26.10(vi) Bessel-Function Expansion ‣ §26.10 Integer Partitions: Other Restrictions ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="23px" altimg-valign="-7px" altimg-width="57px" alttext="A_{k}(n)" display="inline"><mrow><msub><mi href="./26.10#SS6.p1">A</mi><mi href="./26.1#p2.t1.r2">k</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./26.1#p2.t1.r2">n</mi><mo stretchy="false">)</mo></mrow></mrow></math>: coefficient</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">24.2.2</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m38.png" altimg-height="23px" altimg-valign="-7px" altimg-width="52px" alttext="I_{1}\left(x\right)" display="inline"><mrow><msub><mi href="./10.25#E2">I</mi><mn>1</mn></msub><mo></mo><mrow><mo>(</mo><mi href="./26.1#p2.t1.r1">x</mi><mo>)</mo></mrow></mrow></math> is the modified Bessel function (§), and
</p>
<table id="E18" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">26.10.18</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E18.png" altimg-height="71px" altimg-valign="-47px" altimg-width="301px" alttext="A_{k}(n)=\sum_{\begin{subarray}{c}1<h\leq k\\
\left(h,k\right)=1\end{subarray}}{\mathrm{e}^{\pi\mathrm{i}f(h,k)-(2\pi\mathrm%
{i}nh/k)}}," display="block"><mrow><mrow><mrow><msub><mi href="./26.10#SS6.p1">A</mi><mi href="./26.1#p2.t1.r2">k</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./26.1#p2.t1.r2">n</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><munder><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mtable class="ltx_align_c" rowspacing="0.0pt"><mtr><mtd><mrow><mn>1</mn><mo><</mo><mi href="./26.1#p2.t1.r2">h</mi><mo>≤</mo><mi href="./26.1#p2.t1.r2">k</mi></mrow></mtd></mtr><mtr><mtd><mrow><mrow><mo href="./27.1#p2.t1.r3">(</mo><mrow><mi href="./26.1#p2.t1.r2">h</mi><mo>,</mo><mi href="./26.1#p2.t1.r2">k</mi></mrow><mo href="./27.1#p2.t1.r3">)</mo></mrow><mo>=</mo><mn>1</mn></mrow></mtd></mtr></mtable></munder><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi><mo></mo><mrow><mi href="./26.10#E19">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./26.1#p2.t1.r2">h</mi><mo>,</mo><mi href="./26.1#p2.t1.r2">k</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>-</mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi><mo></mo><mi href="./26.1#p2.t1.r2">n</mi><mo></mo><mi href="./26.1#p2.t1.r2">h</mi></mrow><mo>/</mo><mi href="./26.1#p2.t1.r2">k</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></msup></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E18.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./27.1#p2.t1.r3" title="§27.1 Special Notation ‣ Notation ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m45.png" altimg-height="23px" altimg-valign="-7px" altimg-width="58px" alttext="\left(\NVar{m},\NVar{n}\right)" display="inline"><mrow><mo href="./27.1#p2.t1.r3">(</mo><mrow><mi class="ltx_nvar" href="./27.1#p2.t1.r1">m</mi><mo>,</mo><mi class="ltx_nvar" href="./27.1#p2.t1.r1">n</mi></mrow><mo href="./27.1#p2.t1.r3">)</mo></mrow></math>: greatest common divisor (gcd)</a>,
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m55.png" altimg-height="18px" altimg-valign="-2px" altimg-width="16px" alttext="h" display="inline"><mi href="./26.1#p2.t1.r2">h</mi></math>: nonnegative integer</a>,
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./26.1#p2.t1.r2">k</mi></math>: nonnegative integer</a>,
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math>: nonnegative integer</a>,
<a href="./26.10#SS6.p1" title="§26.10(vi) Bessel-Function Expansion ‣ §26.10 Integer Partitions: Other Restrictions ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="23px" altimg-valign="-7px" altimg-width="57px" alttext="A_{k}(n)" display="inline"><mrow><msub><mi href="./26.10#SS6.p1">A</mi><mi href="./26.1#p2.t1.r2">k</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./26.1#p2.t1.r2">n</mi><mo stretchy="false">)</mo></mrow></mrow></math>: coefficient</a> and
<a href="./26.10#E19" title="(26.10.19) ‣ §26.10(vi) Bessel-Function Expansion ‣ §26.10 Integer Partitions: Other Restrictions ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="23px" altimg-valign="-7px" altimg-width="63px" alttext="f(h,k)" display="inline"><mrow><mi href="./26.10#E19">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./26.1#p2.t1.r2">h</mi><mo>,</mo><mi href="./26.1#p2.t1.r2">k</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">with</p>
<table id="E19" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">26.10.19</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E19.png" altimg-height="70px" altimg-valign="-30px" altimg-width="337px" alttext="f(h,k)=\sum_{j=1}^{k}\left[\!\!\left[\frac{2j-1}{2k}\right]\!\!\right]\left[\!%
\!\left[\frac{h(2j-1)}{k}\right]\!\!\right]," display="block"><mrow><mrow><mrow><mi href="./26.10#E19">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./26.1#p2.t1.r2">h</mi><mo>,</mo><mi href="./26.1#p2.t1.r2">k</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./26.1#p2.t1.r2">j</mi><mo>=</mo><mn>1</mn></mrow><mi href="./26.1#p2.t1.r2">k</mi></munderover><mrow><mrow><mo rspace="0pt">[</mo><mrow><mo>[</mo><mfrac><mrow><mrow><mn>2</mn><mo></mo><mi href="./26.1#p2.t1.r2">j</mi></mrow><mo>-</mo><mn>1</mn></mrow><mrow><mn>2</mn><mo></mo><mi href="./26.1#p2.t1.r2">k</mi></mrow></mfrac><mo rspace="0pt">]</mo></mrow><mo>]</mo></mrow><mo></mo><mrow><mo rspace="0pt">[</mo><mrow><mo>[</mo><mfrac><mrow><mi href="./26.1#p2.t1.r2">h</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>2</mn><mo></mo><mi href="./26.1#p2.t1.r2">j</mi></mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mi href="./26.1#p2.t1.r2">k</mi></mfrac><mo rspace="0pt">]</mo></mrow><mo>]</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E19.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text"><math class="ltx_Math" altimg="m54.png" altimg-height="23px" altimg-valign="-7px" altimg-width="63px" alttext="f(h,k)" display="inline"><mrow><mi href="./26.10#E19">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./26.1#p2.t1.r2">h</mi><mo>,</mo><mi href="./26.1#p2.t1.r2">k</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function (locally)</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m55.png" altimg-height="18px" altimg-valign="-2px" altimg-width="16px" alttext="h" display="inline"><mi href="./26.1#p2.t1.r2">h</mi></math>: nonnegative integer</a>,
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="20px" altimg-valign="-6px" altimg-width="14px" alttext="j" display="inline"><mi href="./26.1#p2.t1.r2">j</mi></math>: nonnegative integer</a> and
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./26.1#p2.t1.r2">k</mi></math>: nonnegative integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and</p>
<table id="E20" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">26.10.20</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E20.png" altimg-height="65px" altimg-valign="-27px" altimg-width="260px" alttext="[\![x]\!]=\begin{cases}x-\left\lfloor x\right\rfloor-\tfrac{1}{2},&x\notin%
\mathbb{Z},\\
0,&x\in\mathbb{Z}.\end{cases}" display="block"><mrow><mrow><mo rspace="0.8pt" stretchy="false">[</mo><mrow><mo stretchy="false">[</mo><mi href="./26.1#p2.t1.r1">x</mi><mo rspace="0.8pt" stretchy="false">]</mo></mrow><mo stretchy="false">]</mo></mrow><mo>=</mo><mrow><mo>{</mo><mtable columnspacing="5pt" displaystyle="true" rowspacing="0pt"><mtr><mtd columnalign="left"><mrow><mrow><mi href="./26.1#p2.t1.r1">x</mi><mo>-</mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r16">⌊</mo><mi href="./26.1#p2.t1.r1">x</mi><mo href="./front/introduction#Sx4.p1.t1.r16">⌋</mo></mrow><mo>-</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle></mrow><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mrow><mi href="./26.1#p2.t1.r1">x</mi><mo href="./front/introduction#Sx4.p1.t1.r10">∉</mo><mi href="./front/introduction#Sx4.p2.t1.r20">ℤ</mi></mrow><mo>,</mo></mrow></mtd></mtr><mtr><mtd columnalign="left"><mrow><mn>0</mn><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mrow><mi href="./26.1#p2.t1.r1">x</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mi href="./front/introduction#Sx4.p2.t1.r20">ℤ</mi></mrow><mo>.</mo></mrow></mtd></mtr></mtable></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E20.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./front/introduction#Sx4.p1.t1.r9" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="15px" altimg-valign="-3px" altimg-width="18px" alttext="\in" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo></math>: element of</a>,
<a href="./front/introduction#Sx4.p1.t1.r16" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m46.png" altimg-height="23px" altimg-valign="-7px" altimg-width="33px" alttext="\left\lfloor\NVar{x}\right\rfloor" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r16">⌊</mo><mi class="ltx_nvar">x</mi><mo href="./front/introduction#Sx4.p1.t1.r16">⌋</mo></mrow></math>: floor of <math class="ltx_Math" altimg="m88.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./front/introduction#Sx4.p2.t1.r20" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="\mathbb{Z}" display="inline"><mi href="./front/introduction#Sx4.p2.t1.r20">ℤ</mi></math>: set of all integers</a>,
<a href="./front/introduction#Sx4.p1.t1.r10" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="19px" altimg-valign="-3px" altimg-width="18px" alttext="\notin" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r10">∉</mo></math>: not an element of</a> and
<a href="./26.1#p2.t1.r1" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m88.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./26.1#p2.t1.r1">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">The quantity <math class="ltx_Math" altimg="m37.png" altimg-height="23px" altimg-valign="-7px" altimg-width="57px" alttext="A_{k}(n)" display="inline"><mrow><msub><mi href="./26.10#SS6.p1">A</mi><mi href="./26.1#p2.t1.r2">k</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./26.1#p2.t1.r2">n</mi><mo stretchy="false">)</mo></mrow></mrow></math> is real-valued.
</p>
</div>
</section>
</section>
</div>
<div class="ltx_page_footer">
<span></div>
</div>
</body></text>
</html>
</page>
<page>
<!DOCTYPE html><html>
<head>
<title>DLMF: 26.14 Permutations: Order Notation</title>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<!--GOOGLE BOOTSTRAP--></head>
<text><body class="color_default textfont_default titlefont_default fontsize_default navbar_default">
<div class="ltx_page_navbar">
<div class="ltx_page_navlogo"><div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m19.png" altimg-height="27px" altimg-valign="-9px" altimg-width="33px" alttext="\genfrac{<}{>}{0.0pt}{}{\NVar{n}}{\NVar{k}}" display="inline"><mrow><mo href="./26.14#SS1.p3">⟨</mo><mfrac linethickness="0.0pt"><mi class="ltx_nvar" href="./26.1#p2.t1.r2">n</mi><mi class="ltx_nvar" href="./26.1#p2.t1.r2">k</mi></mfrac><mo href="./26.14#SS1.p3">⟩</mo></mrow></math>: Eulerian number</span></dd>
<dt>Keywords:</dt>
<dd>
</dd>
</dl>
</div>
</div>
<div id="SS1.p1" class="ltx_para">
<p class="ltx_p">The set <math class="ltx_Math" altimg="m24.png" altimg-height="21px" altimg-valign="-5px" altimg-width="32px" alttext="\mathfrak{S}_{n}" display="inline"><msub><mi href="./26.13#p1">𝔖</mi><mi href="./26.1#p2.t1.r2">n</mi></msub></math> (§) can be viewed as the collection of all
ordered lists of elements of <math class="ltx_Math" altimg="m35.png" altimg-height="23px" altimg-valign="-7px" altimg-width="109px" alttext="\{1,2,\ldots,n\}" display="inline"><mrow><mo stretchy="false">{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><mi href="./26.1#p2.t1.r2">n</mi><mo stretchy="false">}</mo></mrow></math>:
<math class="ltx_Math" altimg="m36.png" altimg-height="23px" altimg-valign="-7px" altimg-width="169px" alttext="\{\sigma(1)\sigma(2)\cdots\sigma(n)\}" display="inline"><mrow><mo stretchy="false">{</mo><mrow><mrow><mi href="./26.14#SS1.p1">σ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mi href="./26.14#SS1.p1">σ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mi mathvariant="normal">⋯</mi><mo></mo><mrow><mi href="./26.14#SS1.p1">σ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./26.1#p2.t1.r2">n</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo stretchy="false">}</mo></mrow></math>. As an example, <math class="ltx_Math" altimg="m9.png" altimg-height="17px" altimg-valign="-2px" altimg-width="84px" alttext="35247816" display="inline"><mn>35247816</mn></math> is an
element of <math class="ltx_Math" altimg="m22.png" altimg-height="21px" altimg-valign="-5px" altimg-width="35px" alttext="\mathfrak{S}_{8}." display="inline"><mrow><msub><mi href="./26.13#p1">𝔖</mi><mn>8</mn></msub><mo>.</mo></mrow></math> The <em class="ltx_emph ltx_font_italic">inversion number</em> is the number of
pairs of elements for which the larger element precedes the smaller:
</p>
<table id="E1" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">26.14.1</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E1.png" altimg-height="71px" altimg-valign="-47px" altimg-width="179px" alttext="\mathop{\mathrm{inv}}(\sigma)=\sum_{\begin{subarray}{c}1\leq j<k\leq n\\
\sigma(j)>\sigma(k)\end{subarray}}1." display="block"><mrow><mrow><mrow><mo movablelimits="false">inv</mo><mrow><mo stretchy="false">(</mo><mi href="./26.14#SS1.p1">σ</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><munder><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mtable class="ltx_align_c" rowspacing="0.0pt"><mtr><mtd><mrow><mn>1</mn><mo>≤</mo><mi href="./26.1#p2.t1.r2">j</mi><mo><</mo><mi href="./26.1#p2.t1.r2">k</mi><mo>≤</mo><mi href="./26.1#p2.t1.r2">n</mi></mrow></mtd></mtr><mtr><mtd><mrow><mrow><mi href="./26.14#SS1.p1">σ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./26.1#p2.t1.r2">j</mi><mo stretchy="false">)</mo></mrow></mrow><mo>></mo><mrow><mi href="./26.14#SS1.p1">σ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./26.1#p2.t1.r2">k</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></mtd></mtr></mtable></munder><mn>1</mn></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="20px" altimg-valign="-6px" altimg-width="14px" alttext="j" display="inline"><mi href="./26.1#p2.t1.r2">j</mi></math>: nonnegative integer</a>,
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./26.1#p2.t1.r2">k</mi></math>: nonnegative integer</a>,
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math>: nonnegative integer</a> and
<a href="./26.14#SS1.p1" title="§26.14(i) Definitions ‣ §26.14 Permutations: Order Notation ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m33.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\sigma" display="inline"><mi href="./26.14#SS1.p1">σ</mi></math>: permutation</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Equivalently, this is the sum over <math class="ltx_Math" altimg="m8.png" altimg-height="20px" altimg-valign="-6px" altimg-width="89px" alttext="1\leq j<n" display="inline"><mrow><mn>1</mn><mo>≤</mo><mi href="./26.1#p2.t1.r2">j</mi><mo><</mo><mi href="./26.1#p2.t1.r2">n</mi></mrow></math> of the number of integers
less than <math class="ltx_Math" altimg="m31.png" altimg-height="23px" altimg-valign="-7px" altimg-width="41px" alttext="\sigma(j)" display="inline"><mrow><mi href="./26.14#SS1.p1">σ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./26.1#p2.t1.r2">j</mi><mo stretchy="false">)</mo></mrow></mrow></math> that lie in positions to the right of the <math class="ltx_Math" altimg="m37.png" altimg-height="20px" altimg-valign="-6px" altimg-width="14px" alttext="j" display="inline"><mi href="./26.1#p2.t1.r2">j</mi></math>th position:
<math class="ltx_Math" altimg="m25.png" altimg-height="23px" altimg-valign="-7px" altimg-width="421px" alttext="\mathop{\mathrm{inv}}(35247816)=2+3+1+1+2+2+0=11." display="inline"><mrow><mrow><mrow><mo>inv</mo><mrow><mo stretchy="false">(</mo><mn>35247816</mn><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mn>2</mn><mo>+</mo><mn>3</mn><mo>+</mo><mn>1</mn><mo>+</mo><mn>1</mn><mo>+</mo><mn>2</mn><mo>+</mo><mn>2</mn><mo>+</mo><mn>0</mn></mrow><mo>=</mo><mn>11</mn></mrow><mo>.</mo></mrow></math></p>
</div>
<div id="SS1.p2" class="ltx_para">
<p class="ltx_p">A <em class="ltx_emph ltx_font_italic">descent</em> of a permutation is a pair of adjacent elements for which the
first is larger than the second. The permutation <math class="ltx_Math" altimg="m9.png" altimg-height="17px" altimg-valign="-2px" altimg-width="84px" alttext="35247816" display="inline"><mn>35247816</mn></math> has two descents:
<math class="ltx_Math" altimg="m10.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="52" display="inline"><mn>52</mn></math> and <math class="ltx_Math" altimg="m11.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="81" display="inline"><mn>81</mn></math>. The <em class="ltx_emph ltx_font_italic">major index</em> is the sum of all positions that mark
the first element of a descent:
</p>
<table id="E2" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">26.14.2</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E2.png" altimg-height="71px" altimg-valign="-47px" altimg-width="203px" alttext="\mathop{\mathrm{maj}}(\sigma)=\sum_{\begin{subarray}{c}1\leq j<n\\
\sigma(j)>\sigma(j+1)\end{subarray}}j." display="block"><mrow><mrow><mrow><mo href="./26.14#SS1.p2" movablelimits="false">maj</mo><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./26.14#SS1.p1">σ</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><munder><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mtable class="ltx_align_c" rowspacing="0.0pt"><mtr><mtd><mrow><mn>1</mn><mo>≤</mo><mi href="./26.1#p2.t1.r2">j</mi><mo><</mo><mi href="./26.1#p2.t1.r2">n</mi></mrow></mtd></mtr><mtr><mtd><mrow><mrow><mi href="./26.14#SS1.p1">σ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./26.1#p2.t1.r2">j</mi><mo stretchy="false">)</mo></mrow></mrow><mo>></mo><mrow><mi href="./26.14#SS1.p1">σ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./26.1#p2.t1.r2">j</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></mtd></mtr></mtable></munder><mi href="./26.1#p2.t1.r2">j</mi></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="20px" altimg-valign="-6px" altimg-width="14px" alttext="j" display="inline"><mi href="./26.1#p2.t1.r2">j</mi></math>: nonnegative integer</a>,
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math>: nonnegative integer</a>,
<a href="./26.14#SS1.p1" title="§26.14(i) Definitions ‣ §26.14 Permutations: Order Notation ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m33.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\sigma" display="inline"><mi href="./26.14#SS1.p1">σ</mi></math>: permutation</a> and
<a href="./26.14#SS1.p2" title="§26.14(i) Definitions ‣ §26.14 Permutations: Order Notation ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m27.png" altimg-height="23px" altimg-valign="-7px" altimg-width="66px" alttext="\mathop{\mathrm{maj}}(\sigma)" display="inline"><mrow><mo href="./26.14#SS1.p2">maj</mo><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./26.14#SS1.p1">σ</mi><mo stretchy="false">)</mo></mrow></mrow></math>: major index</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">For example, <math class="ltx_Math" altimg="m26.png" altimg-height="23px" altimg-valign="-7px" altimg-width="241px" alttext="\mathop{\mathrm{maj}}(35247816)=2+6=8" display="inline"><mrow><mrow><mo>maj</mo><mrow><mo stretchy="false">(</mo><mn>35247816</mn><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mn>2</mn><mo>+</mo><mn>6</mn></mrow><mo>=</mo><mn>8</mn></mrow></math>. The major index is also called
the <em class="ltx_emph ltx_font_italic">greater index</em> of the permutation.
</p>
</div>
<div id="SS1.p3" class="ltx_para">
<p class="ltx_p">The <em class="ltx_emph ltx_font_italic">Eulerian number</em>, denoted <math class="ltx_Math" altimg="m20.png" altimg-height="27px" altimg-valign="-9px" altimg-width="33px" alttext="\genfrac{<}{>}{0.0pt}{}{n}{k}" display="inline"><mrow><mo href="./26.14#SS1.p3">⟨</mo><mfrac linethickness="0.0pt"><mi href="./26.1#p2.t1.r2">n</mi><mi href="./26.1#p2.t1.r2">k</mi></mfrac><mo href="./26.14#SS1.p3">⟩</mo></mrow></math>,
is the number of permutations in <math class="ltx_Math" altimg="m24.png" altimg-height="21px" altimg-valign="-5px" altimg-width="32px" alttext="\mathfrak{S}_{n}" display="inline"><msub><mi href="./26.13#p1">𝔖</mi><mi href="./26.1#p2.t1.r2">n</mi></msub></math> with exactly <math class="ltx_Math" altimg="m39.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./26.1#p2.t1.r2">k</mi></math> descents.
An <em class="ltx_emph ltx_font_italic">excedance</em> in
<math class="ltx_Math" altimg="m34.png" altimg-height="21px" altimg-valign="-5px" altimg-width="68px" alttext="\sigma\in\mathfrak{S}_{n}" display="inline"><mrow><mi href="./26.14#SS1.p1">σ</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><msub><mi href="./26.13#p1">𝔖</mi><mi href="./26.1#p2.t1.r2">n</mi></msub></mrow></math> is a position <math class="ltx_Math" altimg="m37.png" altimg-height="20px" altimg-valign="-6px" altimg-width="14px" alttext="j" display="inline"><mi href="./26.1#p2.t1.r2">j</mi></math> for which <math class="ltx_Math" altimg="m30.png" altimg-height="23px" altimg-valign="-7px" altimg-width="77px" alttext="\sigma(j)>j" display="inline"><mrow><mrow><mi href="./26.14#SS1.p1">σ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./26.1#p2.t1.r2">j</mi><mo stretchy="false">)</mo></mrow></mrow><mo>></mo><mi href="./26.1#p2.t1.r2">j</mi></mrow></math>. A
<em class="ltx_emph ltx_font_italic">weak excedance</em> is a position <math class="ltx_Math" altimg="m37.png" altimg-height="20px" altimg-valign="-6px" altimg-width="14px" alttext="j" display="inline"><mi href="./26.1#p2.t1.r2">j</mi></math> for which <math class="ltx_Math" altimg="m32.png" altimg-height="23px" altimg-valign="-7px" altimg-width="77px" alttext="\sigma(j)\geq j" display="inline"><mrow><mrow><mi href="./26.14#SS1.p1">σ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./26.1#p2.t1.r2">j</mi><mo stretchy="false">)</mo></mrow></mrow><mo>≥</mo><mi href="./26.1#p2.t1.r2">j</mi></mrow></math>. The
Eulerian number <math class="ltx_Math" altimg="m20.png" altimg-height="27px" altimg-valign="-9px" altimg-width="33px" alttext="\genfrac{<}{>}{0.0pt}{}{n}{k}" display="inline"><mrow><mo href="./26.14#SS1.p3">⟨</mo><mfrac linethickness="0.0pt"><mi href="./26.1#p2.t1.r2">n</mi><mi href="./26.1#p2.t1.r2">k</mi></mfrac><mo href="./26.14#SS1.p3">⟩</mo></mrow></math> is equal to the number of permutations in
<math class="ltx_Math" altimg="m24.png" altimg-height="21px" altimg-valign="-5px" altimg-width="32px" alttext="\mathfrak{S}_{n}" display="inline"><msub><mi href="./26.13#p1">𝔖</mi><mi href="./26.1#p2.t1.r2">n</mi></msub></math> with exactly <math class="ltx_Math" altimg="m39.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./26.1#p2.t1.r2">k</mi></math> excedances. It is also equal to the number of
permutations in <math class="ltx_Math" altimg="m24.png" altimg-height="21px" altimg-valign="-5px" altimg-width="32px" alttext="\mathfrak{S}_{n}" display="inline"><msub><mi href="./26.13#p1">𝔖</mi><mi href="./26.1#p2.t1.r2">n</mi></msub></math> with exactly <math class="ltx_Math" altimg="m38.png" altimg-height="19px" altimg-valign="-4px" altimg-width="50px" alttext="k+1" display="inline"><mrow><mi href="./26.1#p2.t1.r2">k</mi><mo>+</mo><mn>1</mn></mrow></math> weak excedances. See
Table . </p>
</div>
<figure id="T1" class="ltx_table">
<figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_table">Table 26.14.1: </span>Eulerian numbers <math class="ltx_Math" altimg="m20.png" altimg-height="27px" altimg-valign="-9px" altimg-width="33px" alttext="\genfrac{<}{>}{0.0pt}{}{n}{k}" display="inline"><mrow><mo href="./26.14#SS1.p3">⟨</mo><mfrac linethickness="0.0pt"><mi href="./26.1#p2.t1.r2">n</mi><mi href="./26.1#p2.t1.r2">k</mi></mfrac><mo href="./26.14#SS1.p3">⟩</mo></mrow></math>.
</figcaption>
<table id="T1.t1" class="ltx_tabular ltx_centering ltx_guessed_headers ltx_align_middle">
<thead class="ltx_thead">
<tr id="T1.t1.r1" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_th_row ltx_border_r ltx_border_tt" rowspan="2"><math class="ltx_Math" altimg="m43.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_l ltx_border_tt" colspan="10"><math class="ltx_Math" altimg="m39.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./26.1#p2.t1.r2">k</mi></math></th>
</tr>
<tr id="T1.t1.r2" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_column">0</th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column">1</th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column">2</th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column">3</th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column">4</th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column">5</th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column">6</th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column">7</th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column">8</th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column">9</th>
</tr>
</thead>
<tbody class="ltx_tbody">
<tr id="T1.t1.r3" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_r">0</th>
<td class="ltx_td ltx_align_right ltx_border_t">1</td>
<td class="ltx_td ltx_border_t"></td>
<td class="ltx_td ltx_border_t"></td>
<td class="ltx_td ltx_border_t"></td>
<td class="ltx_td ltx_border_t"></td>
<td class="ltx_td ltx_border_t"></td>
<td class="ltx_td ltx_border_t"></td>
<td class="ltx_td ltx_border_t"></td>
<td class="ltx_td ltx_border_t"></td>
<td class="ltx_td ltx_border_t"></td>
</tr>
<tr id="T1.t1.r4" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_r">1</th>
<td class="ltx_td ltx_align_right">1</td>
<td class="ltx_td"></td>
<td class="ltx_td"></td>
<td class="ltx_td"></td>
<td class="ltx_td"></td>
<td class="ltx_td"></td>
<td class="ltx_td"></td>
<td class="ltx_td"></td>
<td class="ltx_td"></td>
<td class="ltx_td"></td>
</tr>
<tr id="T1.t1.r5" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_r">2</th>
<td class="ltx_td ltx_align_right">1</td>
<td class="ltx_td ltx_align_right">1</td>
<td class="ltx_td"></td>
<td class="ltx_td"></td>
<td class="ltx_td"></td>
<td class="ltx_td"></td>
<td class="ltx_td"></td>
<td class="ltx_td"></td>
<td class="ltx_td"></td>
<td class="ltx_td"></td>
</tr>
<tr id="T1.t1.r6" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_r">3</th>
<td class="ltx_td ltx_align_right">1</td>
<td class="ltx_td ltx_align_right">4</td>
<td class="ltx_td ltx_align_right">1</td>
<td class="ltx_td"></td>
<td class="ltx_td"></td>
<td class="ltx_td"></td>
<td class="ltx_td"></td>
<td class="ltx_td"></td>
<td class="ltx_td"></td>
<td class="ltx_td"></td>
</tr>
<tr id="T1.t1.r7" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_T ltx_border_r">4</th>
<td class="ltx_td ltx_align_right ltx_border_T">1</td>
<td class="ltx_td ltx_align_right ltx_border_T">11</td>
<td class="ltx_td ltx_align_right ltx_border_T">11</td>
<td class="ltx_td ltx_align_right ltx_border_T">1</td>
<td class="ltx_td ltx_border_T"></td>
<td class="ltx_td ltx_border_T"></td>
<td class="ltx_td ltx_border_T"></td>
<td class="ltx_td ltx_border_T"></td>
<td class="ltx_td ltx_border_T"></td>
<td class="ltx_td ltx_border_T"></td>
</tr>
<tr id="T1.t1.r8" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_r">5</th>
<td class="ltx_td ltx_align_right">1</td>
<td class="ltx_td ltx_align_right">26</td>
<td class="ltx_td ltx_align_right">66</td>
<td class="ltx_td ltx_align_right">26</td>
<td class="ltx_td ltx_align_right">1</td>
<td class="ltx_td"></td>
<td class="ltx_td"></td>
<td class="ltx_td"></td>
<td class="ltx_td"></td>
<td class="ltx_td"></td>
</tr>
<tr id="T1.t1.r9" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_r">6</th>
<td class="ltx_td ltx_align_right">1</td>
<td class="ltx_td ltx_align_right">57</td>
<td class="ltx_td ltx_align_right">302</td>
<td class="ltx_td ltx_align_right">302</td>
<td class="ltx_td ltx_align_right">57</td>
<td class="ltx_td ltx_align_right">1</td>
<td class="ltx_td"></td>
<td class="ltx_td"></td>
<td class="ltx_td"></td>
<td class="ltx_td"></td>
</tr>
<tr id="T1.t1.r10" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_T ltx_border_r">7</th>
<td class="ltx_td ltx_align_right ltx_border_T">1</td>
<td class="ltx_td ltx_align_right ltx_border_T">120</td>
<td class="ltx_td ltx_align_right ltx_border_T">1191</td>
<td class="ltx_td ltx_align_right ltx_border_T">2416</td>
<td class="ltx_td ltx_align_right ltx_border_T">1191</td>
<td class="ltx_td ltx_align_right ltx_border_T">120</td>
<td class="ltx_td ltx_align_right ltx_border_T">1</td>
<td class="ltx_td ltx_border_T"></td>
<td class="ltx_td ltx_border_T"></td>
<td class="ltx_td ltx_border_T"></td>
</tr>
<tr id="T1.t1.r11" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_r">8</th>
<td class="ltx_td ltx_align_right">1</td>
<td class="ltx_td ltx_align_right">247</td>
<td class="ltx_td ltx_align_right">4293</td>
<td class="ltx_td ltx_align_right">15619</td>
<td class="ltx_td ltx_align_right">15619</td>
<td class="ltx_td ltx_align_right">4293</td>
<td class="ltx_td ltx_align_right">247</td>
<td class="ltx_td ltx_align_right">1</td>
<td class="ltx_td"></td>
<td class="ltx_td"></td>
</tr>
<tr id="T1.t1.r12" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_r">9</th>
<td class="ltx_td ltx_align_right">1</td>
<td class="ltx_td ltx_align_right">502</td>
<td class="ltx_td ltx_align_right">14608</td>
<td class="ltx_td ltx_align_right">88234</td>
<td class="ltx_td ltx_align_right">1 56190</td>
<td class="ltx_td ltx_align_right">88234</td>
<td class="ltx_td ltx_align_right">14608</td>
<td class="ltx_td ltx_align_right">502</td>
<td class="ltx_td ltx_align_right">1</td>
<td class="ltx_td"></td>
</tr>
<tr id="T1.t1.r13" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_b ltx_border_r">10</th>
<td class="ltx_td ltx_align_right ltx_border_b">1</td>
<td class="ltx_td ltx_align_right ltx_border_b">1013</td>
<td class="ltx_td ltx_align_right ltx_border_b">47840</td>
<td class="ltx_td ltx_align_right ltx_border_b">4 55192</td>
<td class="ltx_td ltx_align_right ltx_border_b">13 10354</td>
<td class="ltx_td ltx_align_right ltx_border_b">13 10354</td>
<td class="ltx_td ltx_align_right ltx_border_b">4 55192</td>
<td class="ltx_td ltx_align_right ltx_border_b">47840</td>
<td class="ltx_td ltx_align_right ltx_border_b">1013</td>
<td class="ltx_td ltx_align_right ltx_border_b">1</td>
</tr>
</tbody>
</table>
<div id="T1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./26.14#SS1.p3" title="§26.14(i) Definitions ‣ §26.14 Permutations: Order Notation ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="27px" altimg-valign="-9px" altimg-width="33px" alttext="\genfrac{<}{>}{0.0pt}{}{\NVar{n}}{\NVar{k}}" display="inline"><mrow><mo href="./26.14#SS1.p3">⟨</mo><mfrac linethickness="0.0pt"><mi class="ltx_nvar" href="./26.1#p2.t1.r2">n</mi><mi class="ltx_nvar" href="./26.1#p2.t1.r2">k</mi></mfrac><mo href="./26.14#SS1.p3">⟩</mo></mrow></math>: Eulerian number</a>,
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./26.1#p2.t1.r2">k</mi></math>: nonnegative integer</a> and
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math>: nonnegative integer</a>
</dd>
<dt>Keywords:</dt>
<dd></dd>
</dl>
</div>
</div>
<div id="SS2.p1" class="ltx_para">
<table id="E3" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">26.14.3</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E3.png" altimg-height="67px" altimg-valign="-30px" altimg-width="365px" alttext="\sum_{\sigma\in\mathfrak{S}_{n}}q^{\mathop{\mathrm{inv}}(\sigma)}=\sum_{\sigma%
\in\mathfrak{S}_{n}}q^{\mathop{\mathrm{maj}}(\sigma)}=\prod_{j=1}^{n}\frac{1-q%
^{j}}{1-q}." display="block"><mrow><mrow><mrow><munder><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi>σ</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><msub><mi href="./26.13#p1">𝔖</mi><mi href="./26.1#p2.t1.r2">n</mi></msub></mrow></munder><msup><mi href="./26.9#SS2.p1">q</mi><mrow><mo>inv</mo><mrow><mo stretchy="false">(</mo><mi>σ</mi><mo stretchy="false">)</mo></mrow></mrow></msup></mrow><mo>=</mo><mrow><munder><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi>σ</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><msub><mi href="./26.13#p1">𝔖</mi><mi href="./26.1#p2.t1.r2">n</mi></msub></mrow></munder><msup><mi href="./26.9#SS2.p1">q</mi><mrow><mo>maj</mo><mrow><mo stretchy="false">(</mo><mi>σ</mi><mo stretchy="false">)</mo></mrow></mrow></msup></mrow><mo>=</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∏</mo><mrow><mi href="./26.1#p2.t1.r2">j</mi><mo>=</mo><mn>1</mn></mrow><mi href="./26.1#p2.t1.r2">n</mi></munderover><mfrac><mrow><mn>1</mn><mo>-</mo><msup><mi href="./26.9#SS2.p1">q</mi><mi href="./26.1#p2.t1.r2">j</mi></msup></mrow><mrow><mn>1</mn><mo>-</mo><mi href="./26.9#SS2.p1">q</mi></mrow></mfrac></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./front/introduction#Sx4.p1.t1.r9" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m21.png" altimg-height="15px" altimg-valign="-3px" altimg-width="18px" alttext="\in" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo></math>: element of</a>,
<a href="./26.13#p1" title="§26.13 Permutations: Cycle Notation ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m23.png" altimg-height="21px" altimg-valign="-5px" altimg-width="32px" alttext="\mathfrak{S}_{\NVar{n}}" display="inline"><msub><mi href="./26.13#p1">𝔖</mi><mi class="ltx_nvar" href="./26.1#p2.t1.r2">n</mi></msub></math>: set of permutations of <math class="ltx_Math" altimg="m35.png" altimg-height="23px" altimg-valign="-7px" altimg-width="109px" alttext="\{1,2,\ldots,n\}" display="inline"><mrow><mo stretchy="false">{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><mi href="./26.1#p2.t1.r2">n</mi><mo stretchy="false">}</mo></mrow></math></a>,
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="20px" altimg-valign="-6px" altimg-width="14px" alttext="j" display="inline"><mi href="./26.1#p2.t1.r2">j</mi></math>: nonnegative integer</a>,
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math>: nonnegative integer</a> and
<a href="./26.9#SS2.p1" title="§26.9(ii) Generating Functions ‣ §26.9 Integer Partitions: Restricted Number and Part Size ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="q" display="inline"><mi href="./26.9#SS2.p1">q</mi></math>: parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS2.p2" class="ltx_para">
<table id="E4" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">26.14.4</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E4.png" altimg-height="67px" altimg-valign="-31px" altimg-width="334px" alttext="\sum_{n,k=0}^{\infty}\genfrac{<}{>}{0.0pt}{}{n}{k}x^{k}\,\frac{t^{n}}{n!}=%
\frac{1-x}{\exp((x-1)t)-x}," display="block"><mrow><mrow><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mrow><mi href="./26.1#p2.t1.r2">n</mi><mo>,</mo><mi href="./26.1#p2.t1.r2">k</mi></mrow><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><mrow><mo href="./26.14#SS1.p3">⟨</mo><mfrac linethickness="0.0pt"><mi href="./26.1#p2.t1.r2">n</mi><mi href="./26.1#p2.t1.r2">k</mi></mfrac><mo href="./26.14#SS1.p3">⟩</mo></mrow><mo></mo><mpadded width="+1.7pt"><msup><mi href="./26.1#p2.t1.r1">x</mi><mi href="./26.1#p2.t1.r2">k</mi></msup></mpadded><mo></mo><mfrac><msup><mi>t</mi><mi href="./26.1#p2.t1.r2">n</mi></msup><mrow><mi href="./26.1#p2.t1.r2">n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mfrac></mrow></mrow><mo>=</mo><mfrac><mrow><mn>1</mn><mo>-</mo><mi href="./26.1#p2.t1.r1">x</mi></mrow><mrow><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./26.1#p2.t1.r1">x</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi>t</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>-</mo><mi href="./26.1#p2.t1.r1">x</mi></mrow></mfrac></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m49.png" altimg-height="23px" altimg-valign="-7px" altimg-width="127px" alttext="|x|<1,|t|<1" display="inline"><mrow><mrow><mrow><mo stretchy="false">|</mo><mi href="./26.1#p2.t1.r1">x</mi><mo stretchy="false">|</mo></mrow><mo><</mo><mn>1</mn></mrow><mo>,</mo><mrow><mrow><mo stretchy="false">|</mo><mi>t</mi><mo stretchy="false">|</mo></mrow><mo><</mo><mn>1</mn></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E4.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./26.14#SS1.p3" title="§26.14(i) Definitions ‣ §26.14 Permutations: Order Notation ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="27px" altimg-valign="-9px" altimg-width="33px" alttext="\genfrac{<}{>}{0.0pt}{}{\NVar{n}}{\NVar{k}}" display="inline"><mrow><mo href="./26.14#SS1.p3">⟨</mo><mfrac linethickness="0.0pt"><mi class="ltx_nvar" href="./26.1#p2.t1.r2">n</mi><mi class="ltx_nvar" href="./26.1#p2.t1.r2">k</mi></mfrac><mo href="./26.14#SS1.p3">⟩</mo></mrow></math>: Eulerian number</a>,
<a href="./4.2#E19" title="(4.2.19) ‣ §4.2(iii) The Exponential Function ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m17.png" altimg-height="16px" altimg-valign="-6px" altimg-width="48px" alttext="\exp\NVar{z}" display="inline"><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: exponential function</a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m7.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m42.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./26.1#p2.t1.r1" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./26.1#p2.t1.r1">x</mi></math>: real variable</a>,
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./26.1#p2.t1.r2">k</mi></math>: nonnegative integer</a> and
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math>: nonnegative integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS2.p3" class="ltx_para">
<table id="E5" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">26.14.5</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E5.png" altimg-height="67px" altimg-valign="-28px" altimg-width="215px" alttext="\sum_{k=0}^{n-1}\genfrac{<}{>}{0.0pt}{}{n}{k}\genfrac{(}{)}{0.0pt}{}{x+k}{n}=x%
^{n}." display="block"><mrow><mrow><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./26.1#p2.t1.r2">k</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi href="./26.1#p2.t1.r2">n</mi><mo>-</mo><mn>1</mn></mrow></munderover><mrow><mrow><mo href="./26.14#SS1.p3">⟨</mo><mfrac linethickness="0.0pt"><mi href="./26.1#p2.t1.r2">n</mi><mi href="./26.1#p2.t1.r2">k</mi></mfrac><mo href="./26.14#SS1.p3">⟩</mo></mrow><mo></mo><mrow><mo href="./1.2#i">(</mo><mfrac linethickness="0.0pt"><mrow><mi href="./26.1#p2.t1.r1">x</mi><mo>+</mo><mi href="./26.1#p2.t1.r2">k</mi></mrow><mi href="./26.1#p2.t1.r2">n</mi></mfrac><mo href="./1.2#i">)</mo></mrow></mrow></mrow><mo>=</mo><msup><mi href="./26.1#p2.t1.r1">x</mi><mi href="./26.1#p2.t1.r2">n</mi></msup></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E5.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./26.14#SS1.p3" title="§26.14(i) Definitions ‣ §26.14 Permutations: Order Notation ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="27px" altimg-valign="-9px" altimg-width="33px" alttext="\genfrac{<}{>}{0.0pt}{}{\NVar{n}}{\NVar{k}}" display="inline"><mrow><mo href="./26.14#SS1.p3">⟨</mo><mfrac linethickness="0.0pt"><mi class="ltx_nvar" href="./26.1#p2.t1.r2">n</mi><mi class="ltx_nvar" href="./26.1#p2.t1.r2">k</mi></mfrac><mo href="./26.14#SS1.p3">⟩</mo></mrow></math>: Eulerian number</a>,
<a href="./1.2#i" title="§1.2(i) Binomial Coefficients ‣ §1.2 Elementary Algebra ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m18.png" altimg-height="27px" altimg-valign="-9px" altimg-width="37px" alttext="\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}" display="inline"><mrow><mo href="./1.2#i">(</mo><mfrac linethickness="0.0pt"><mi class="ltx_nvar" href="./1.1#p2.t1.r5">m</mi><mi class="ltx_nvar" href="./1.1#p2.t1.r5">n</mi></mfrac><mo href="./1.2#i">)</mo></mrow></math>: binomial coefficient</a>,
<a href="./26.1#p2.t1.r1" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./26.1#p2.t1.r1">x</mi></math>: real variable</a>,
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./26.1#p2.t1.r2">k</mi></math>: nonnegative integer</a> and
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math>: nonnegative integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="iii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§26.14(iii) </span>Identities</h2>
<div id="SS3.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="SS3.p1" class="ltx_para">
<p class="ltx_p">In this subsection <math class="ltx_Math" altimg="m15.png" altimg-height="23px" altimg-valign="-7px" altimg-width="68px" alttext="S\left(n,k\right)" display="inline"><mrow><mi href="./26.8#SS1.p3">S</mi><mo></mo><mrow><mo>(</mo><mi href="./26.1#p2.t1.r2">n</mi><mo>,</mo><mi href="./26.1#p2.t1.r2">k</mi><mo>)</mo></mrow></mrow></math> is again the Stirling number of the
second kind (§), and <math class="ltx_Math" altimg="m13.png" altimg-height="21px" altimg-valign="-5px" altimg-width="35px" alttext="B_{m}" display="inline"><msub><mi href="./24.2#i">B</mi><mi href="./26.1#p2.t1.r2">m</mi></msub></math> is the <math class="ltx_Math" altimg="m40.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./26.1#p2.t1.r2">m</mi></math>th Bernoulli
number (§).</p>
</div>
<div id="SS3.p2" class="ltx_para">
<table id="E6" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">26.14.6</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E6.png" altimg-height="70px" altimg-valign="-30px" altimg-width="339px" alttext="\genfrac{<}{>}{0.0pt}{}{n}{k}=\sum_{j=0}^{k}(-1)^{j}\genfrac{(}{)}{0.0pt}{}{n+%
1}{j}(k+1-j)^{n}," display="block"><mrow><mrow><mrow><mo href="./26.14#SS1.p3">⟨</mo><mfrac linethickness="0.0pt"><mi href="./26.1#p2.t1.r2">n</mi><mi href="./26.1#p2.t1.r2">k</mi></mfrac><mo href="./26.14#SS1.p3">⟩</mo></mrow><mo>=</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./26.1#p2.t1.r2">j</mi><mo>=</mo><mn>0</mn></mrow><mi href="./26.1#p2.t1.r2">k</mi></munderover><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./26.1#p2.t1.r2">j</mi></msup><mo></mo><mrow><mo href="./1.2#i">(</mo><mfrac linethickness="0.0pt"><mrow><mi href="./26.1#p2.t1.r2">n</mi><mo>+</mo><mn>1</mn></mrow><mi href="./26.1#p2.t1.r2">j</mi></mfrac><mo href="./1.2#i">)</mo></mrow><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mrow><mi href="./26.1#p2.t1.r2">k</mi><mo>+</mo><mn>1</mn></mrow><mo>-</mo><mi href="./26.1#p2.t1.r2">j</mi></mrow><mo stretchy="false">)</mo></mrow><mi href="./26.1#p2.t1.r2">n</mi></msup></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m44.png" altimg-height="19px" altimg-valign="-5px" altimg-width="53px" alttext="n\geq 1" display="inline"><mrow><mi href="./26.1#p2.t1.r2">n</mi><mo>≥</mo><mn>1</mn></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E6.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./26.14#SS1.p3" title="§26.14(i) Definitions ‣ §26.14 Permutations: Order Notation ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="27px" altimg-valign="-9px" altimg-width="33px" alttext="\genfrac{<}{>}{0.0pt}{}{\NVar{n}}{\NVar{k}}" display="inline"><mrow><mo href="./26.14#SS1.p3">⟨</mo><mfrac linethickness="0.0pt"><mi class="ltx_nvar" href="./26.1#p2.t1.r2">n</mi><mi class="ltx_nvar" href="./26.1#p2.t1.r2">k</mi></mfrac><mo href="./26.14#SS1.p3">⟩</mo></mrow></math>: Eulerian number</a>,
<a href="./1.2#i" title="§1.2(i) Binomial Coefficients ‣ §1.2 Elementary Algebra ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m18.png" altimg-height="27px" altimg-valign="-9px" altimg-width="37px" alttext="\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}" display="inline"><mrow><mo href="./1.2#i">(</mo><mfrac linethickness="0.0pt"><mi class="ltx_nvar" href="./1.1#p2.t1.r5">m</mi><mi class="ltx_nvar" href="./1.1#p2.t1.r5">n</mi></mfrac><mo href="./1.2#i">)</mo></mrow></math>: binomial coefficient</a>,
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="20px" altimg-valign="-6px" altimg-width="14px" alttext="j" display="inline"><mi href="./26.1#p2.t1.r2">j</mi></math>: nonnegative integer</a>,
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./26.1#p2.t1.r2">k</mi></math>: nonnegative integer</a> and
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math>: nonnegative integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS3.p3" class="ltx_para">
<table id="E7" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">26.14.7</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E7.png" altimg-height="70px" altimg-valign="-30px" altimg-width="360px" alttext="\genfrac{<}{>}{0.0pt}{}{n}{k}=\sum_{j=0}^{n-k}(-1)^{n-k-j}j!\genfrac{(}{)}{0.0%
pt}{}{n-j}{k}S\left(n,j\right)," display="block"><mrow><mrow><mrow><mo href="./26.14#SS1.p3">⟨</mo><mfrac linethickness="0.0pt"><mi href="./26.1#p2.t1.r2">n</mi><mi href="./26.1#p2.t1.r2">k</mi></mfrac><mo href="./26.14#SS1.p3">⟩</mo></mrow><mo>=</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./26.1#p2.t1.r2">j</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi href="./26.1#p2.t1.r2">n</mi><mo>-</mo><mi href="./26.1#p2.t1.r2">k</mi></mrow></munderover><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mrow><mi href="./26.1#p2.t1.r2">n</mi><mo>-</mo><mi href="./26.1#p2.t1.r2">k</mi><mo>-</mo><mi href="./26.1#p2.t1.r2">j</mi></mrow></msup><mo></mo><mrow><mi href="./26.1#p2.t1.r2">j</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow><mo></mo><mrow><mo href="./1.2#i">(</mo><mfrac linethickness="0.0pt"><mrow><mi href="./26.1#p2.t1.r2">n</mi><mo>-</mo><mi href="./26.1#p2.t1.r2">j</mi></mrow><mi href="./26.1#p2.t1.r2">k</mi></mfrac><mo href="./1.2#i">)</mo></mrow><mo></mo><mrow><mi href="./26.8#SS1.p3">S</mi><mo></mo><mrow><mo>(</mo><mi href="./26.1#p2.t1.r2">n</mi><mo>,</mo><mi href="./26.1#p2.t1.r2">j</mi><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E7.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./26.14#SS1.p3" title="§26.14(i) Definitions ‣ §26.14 Permutations: Order Notation ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="27px" altimg-valign="-9px" altimg-width="33px" alttext="\genfrac{<}{>}{0.0pt}{}{\NVar{n}}{\NVar{k}}" display="inline"><mrow><mo href="./26.14#SS1.p3">⟨</mo><mfrac linethickness="0.0pt"><mi class="ltx_nvar" href="./26.1#p2.t1.r2">n</mi><mi class="ltx_nvar" href="./26.1#p2.t1.r2">k</mi></mfrac><mo href="./26.14#SS1.p3">⟩</mo></mrow></math>: Eulerian number</a>,
<a href="./26.8#SS1.p3" title="§26.8(i) Definitions ‣ §26.8 Set Partitions: Stirling Numbers ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m14.png" altimg-height="23px" altimg-valign="-7px" altimg-width="68px" alttext="S\left(\NVar{n},\NVar{k}\right)" display="inline"><mrow><mi href="./26.8#SS1.p3">S</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./26.1#p2.t1.r2">n</mi><mo>,</mo><mi class="ltx_nvar" href="./26.1#p2.t1.r2">k</mi><mo>)</mo></mrow></mrow></math>: Stirling number of the second kind</a>,
<a href="./1.2#i" title="§1.2(i) Binomial Coefficients ‣ §1.2 Elementary Algebra ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m18.png" altimg-height="27px" altimg-valign="-9px" altimg-width="37px" alttext="\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}" display="inline"><mrow><mo href="./1.2#i">(</mo><mfrac linethickness="0.0pt"><mi class="ltx_nvar" href="./1.1#p2.t1.r5">m</mi><mi class="ltx_nvar" href="./1.1#p2.t1.r5">n</mi></mfrac><mo href="./1.2#i">)</mo></mrow></math>: binomial coefficient</a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m7.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m42.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="20px" altimg-valign="-6px" altimg-width="14px" alttext="j" display="inline"><mi href="./26.1#p2.t1.r2">j</mi></math>: nonnegative integer</a>,
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./26.1#p2.t1.r2">k</mi></math>: nonnegative integer</a> and
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math>: nonnegative integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS3.p4" class="ltx_para">
<table id="E8" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">26.14.8</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E8.png" altimg-height="53px" altimg-valign="-21px" altimg-width="380px" alttext="\genfrac{<}{>}{0.0pt}{}{n}{k}=(k+1)\genfrac{<}{>}{0.0pt}{}{n-1}{k}+(n-k)%
\genfrac{<}{>}{0.0pt}{}{n-1}{k-1}," display="block"><mrow><mrow><mrow><mo href="./26.14#SS1.p3">⟨</mo><mfrac linethickness="0.0pt"><mi href="./26.1#p2.t1.r2">n</mi><mi href="./26.1#p2.t1.r2">k</mi></mfrac><mo href="./26.14#SS1.p3">⟩</mo></mrow><mo>=</mo><mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./26.1#p2.t1.r2">k</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo href="./26.14#SS1.p3">⟨</mo><mfrac linethickness="0.0pt"><mrow><mi href="./26.1#p2.t1.r2">n</mi><mo>-</mo><mn>1</mn></mrow><mi href="./26.1#p2.t1.r2">k</mi></mfrac><mo href="./26.14#SS1.p3">⟩</mo></mrow></mrow><mo>+</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./26.1#p2.t1.r2">n</mi><mo>-</mo><mi href="./26.1#p2.t1.r2">k</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo href="./26.14#SS1.p3">⟨</mo><mfrac linethickness="0.0pt"><mrow><mi href="./26.1#p2.t1.r2">n</mi><mo>-</mo><mn>1</mn></mrow><mrow><mi href="./26.1#p2.t1.r2">k</mi><mo>-</mo><mn>1</mn></mrow></mfrac><mo href="./26.14#SS1.p3">⟩</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m45.png" altimg-height="19px" altimg-valign="-5px" altimg-width="53px" alttext="n\geq 2" display="inline"><mrow><mi href="./26.1#p2.t1.r2">n</mi><mo>≥</mo><mn>2</mn></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E8.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./26.14#SS1.p3" title="§26.14(i) Definitions ‣ §26.14 Permutations: Order Notation ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="27px" altimg-valign="-9px" altimg-width="33px" alttext="\genfrac{<}{>}{0.0pt}{}{\NVar{n}}{\NVar{k}}" display="inline"><mrow><mo href="./26.14#SS1.p3">⟨</mo><mfrac linethickness="0.0pt"><mi class="ltx_nvar" href="./26.1#p2.t1.r2">n</mi><mi class="ltx_nvar" href="./26.1#p2.t1.r2">k</mi></mfrac><mo href="./26.14#SS1.p3">⟩</mo></mrow></math>: Eulerian number</a>,
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./26.1#p2.t1.r2">k</mi></math>: nonnegative integer</a> and
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math>: nonnegative integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS3.p5" class="ltx_para">
<table id="E9" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">26.14.9</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E9.png" altimg-height="53px" altimg-valign="-21px" altimg-width="192px" alttext="\genfrac{<}{>}{0.0pt}{}{n}{k}=\genfrac{<}{>}{0.0pt}{}{n}{n-1-k}," display="block"><mrow><mrow><mrow><mo href="./26.14#SS1.p3">⟨</mo><mfrac linethickness="0.0pt"><mi href="./26.1#p2.t1.r2">n</mi><mi href="./26.1#p2.t1.r2">k</mi></mfrac><mo href="./26.14#SS1.p3">⟩</mo></mrow><mo>=</mo><mrow><mo href="./26.14#SS1.p3">⟨</mo><mfrac linethickness="0.0pt"><mi href="./26.1#p2.t1.r2">n</mi><mrow><mi href="./26.1#p2.t1.r2">n</mi><mo>-</mo><mn>1</mn><mo>-</mo><mi href="./26.1#p2.t1.r2">k</mi></mrow></mfrac><mo href="./26.14#SS1.p3">⟩</mo></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m44.png" altimg-height="19px" altimg-valign="-5px" altimg-width="53px" alttext="n\geq 1" display="inline"><mrow><mi href="./26.1#p2.t1.r2">n</mi><mo>≥</mo><mn>1</mn></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E9.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./26.14#SS1.p3" title="§26.14(i) Definitions ‣ §26.14 Permutations: Order Notation ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="27px" altimg-valign="-9px" altimg-width="33px" alttext="\genfrac{<}{>}{0.0pt}{}{\NVar{n}}{\NVar{k}}" display="inline"><mrow><mo href="./26.14#SS1.p3">⟨</mo><mfrac linethickness="0.0pt"><mi class="ltx_nvar" href="./26.1#p2.t1.r2">n</mi><mi class="ltx_nvar" href="./26.1#p2.t1.r2">k</mi></mfrac><mo href="./26.14#SS1.p3">⟩</mo></mrow></math>: Eulerian number</a>,
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./26.1#p2.t1.r2">k</mi></math>: nonnegative integer</a> and
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math>: nonnegative integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS3.p6" class="ltx_para">
<table id="E10" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">26.14.10</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E10.png" altimg-height="67px" altimg-valign="-28px" altimg-width="134px" alttext="\sum_{k=0}^{n-1}\genfrac{<}{>}{0.0pt}{}{n}{k}=n!," display="block"><mrow><mrow><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./26.1#p2.t1.r2">k</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi href="./26.1#p2.t1.r2">n</mi><mo>-</mo><mn>1</mn></mrow></munderover><mrow><mo href="./26.14#SS1.p3">⟨</mo><mfrac linethickness="0.0pt"><mi href="./26.1#p2.t1.r2">n</mi><mi href="./26.1#p2.t1.r2">k</mi></mfrac><mo href="./26.14#SS1.p3">⟩</mo></mrow></mrow><mo>=</mo><mrow><mi href="./26.1#p2.t1.r2">n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m44.png" altimg-height="19px" altimg-valign="-5px" altimg-width="53px" alttext="n\geq 1" display="inline"><mrow><mi href="./26.1#p2.t1.r2">n</mi><mo>≥</mo><mn>1</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E10.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./26.14#SS1.p3" title="§26.14(i) Definitions ‣ §26.14 Permutations: Order Notation ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="27px" altimg-valign="-9px" altimg-width="33px" alttext="\genfrac{<}{>}{0.0pt}{}{\NVar{n}}{\NVar{k}}" display="inline"><mrow><mo href="./26.14#SS1.p3">⟨</mo><mfrac linethickness="0.0pt"><mi class="ltx_nvar" href="./26.1#p2.t1.r2">n</mi><mi class="ltx_nvar" href="./26.1#p2.t1.r2">k</mi></mfrac><mo href="./26.14#SS1.p3">⟩</mo></mrow></math>: Eulerian number</a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m7.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m42.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./26.1#p2.t1.r2">k</mi></math>: nonnegative integer</a> and
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math>: nonnegative integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS3.p7" class="ltx_para">
<table id="E11" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">26.14.11</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E11.png" altimg-height="67px" altimg-valign="-28px" altimg-width="343px" alttext="B_{m}=\frac{m}{2^{m}(2^{m}-1)}\sum_{k=0}^{m-2}(-1)^{k}\genfrac{<}{>}{0.0pt}{}{%
m-1}{k}," display="block"><mrow><mrow><msub><mi href="./24.2#i">B</mi><mi href="./26.1#p2.t1.r2">m</mi></msub><mo>=</mo><mrow><mfrac><mi href="./26.1#p2.t1.r2">m</mi><mrow><msup><mn>2</mn><mi href="./26.1#p2.t1.r2">m</mi></msup><mo></mo><mrow><mo stretchy="false">(</mo><mrow><msup><mn>2</mn><mi href="./26.1#p2.t1.r2">m</mi></msup><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow></mfrac><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./26.1#p2.t1.r2">k</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi href="./26.1#p2.t1.r2">m</mi><mo>-</mo><mn>2</mn></mrow></munderover><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./26.1#p2.t1.r2">k</mi></msup><mo></mo><mrow><mo href="./26.14#SS1.p3">⟨</mo><mfrac linethickness="0.0pt"><mrow><mi href="./26.1#p2.t1.r2">m</mi><mo>-</mo><mn>1</mn></mrow><mi href="./26.1#p2.t1.r2">k</mi></mfrac><mo href="./26.14#SS1.p3">⟩</mo></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m41.png" altimg-height="19px" altimg-valign="-5px" altimg-width="58px" alttext="m\geq 2" display="inline"><mrow><mi href="./26.1#p2.t1.r2">m</mi><mo>≥</mo><mn>2</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E11.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./24.2#i" title="§24.2(i) Bernoulli Numbers and Polynomials ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m12.png" altimg-height="21px" altimg-valign="-5px" altimg-width="30px" alttext="B_{\NVar{n}}" display="inline"><msub><mi href="./24.2#i">B</mi><mi class="ltx_nvar" href="./24.1#p2.t1.r1">n</mi></msub></math>: Bernoulli numbers</a>,
<a href="./26.14#SS1.p3" title="§26.14(i) Definitions ‣ §26.14 Permutations: Order Notation ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="27px" altimg-valign="-9px" altimg-width="33px" alttext="\genfrac{<}{>}{0.0pt}{}{\NVar{n}}{\NVar{k}}" display="inline"><mrow><mo href="./26.14#SS1.p3">⟨</mo><mfrac linethickness="0.0pt"><mi class="ltx_nvar" href="./26.1#p2.t1.r2">n</mi><mi class="ltx_nvar" href="./26.1#p2.t1.r2">k</mi></mfrac><mo href="./26.14#SS1.p3">⟩</mo></mrow></math>: Eulerian number</a>,
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./26.1#p2.t1.r2">k</mi></math>: nonnegative integer</a> and
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m40.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./26.1#p2.t1.r2">m</mi></math>: nonnegative integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS3.p8" class="ltx_para">
<table id="E12" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">26.14.12</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E12.png" altimg-height="67px" altimg-valign="-28px" altimg-width="298px" alttext="S\left(n,m\right)=\frac{1}{m!}\sum_{k=0}^{n-1}\genfrac{<}{>}{0.0pt}{}{n}{k}%
\genfrac{(}{)}{0.0pt}{}{k}{n-m}," display="block"><mrow><mrow><mrow><mi href="./26.8#SS1.p3">S</mi><mo></mo><mrow><mo>(</mo><mi href="./26.1#p2.t1.r2">n</mi><mo>,</mo><mi href="./26.1#p2.t1.r2">m</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mfrac><mn>1</mn><mrow><mi href="./26.1#p2.t1.r2">m</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mfrac><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./26.1#p2.t1.r2">k</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi href="./26.1#p2.t1.r2">n</mi><mo>-</mo><mn>1</mn></mrow></munderover><mrow><mrow><mo href="./26.14#SS1.p3">⟨</mo><mfrac linethickness="0.0pt"><mi href="./26.1#p2.t1.r2">n</mi><mi href="./26.1#p2.t1.r2">k</mi></mfrac><mo href="./26.14#SS1.p3">⟩</mo></mrow><mo></mo><mrow><mo href="./1.2#i">(</mo><mfrac linethickness="0.0pt"><mi href="./26.1#p2.t1.r2">k</mi><mrow><mi href="./26.1#p2.t1.r2">n</mi><mo>-</mo><mi href="./26.1#p2.t1.r2">m</mi></mrow></mfrac><mo href="./1.2#i">)</mo></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m46.png" altimg-height="19px" altimg-valign="-5px" altimg-width="60px" alttext="n\geq m" display="inline"><mrow><mi href="./26.1#p2.t1.r2">n</mi><mo>≥</mo><mi href="./26.1#p2.t1.r2">m</mi></mrow></math>, <math class="ltx_Math" altimg="m44.png" altimg-height="19px" altimg-valign="-5px" altimg-width="53px" alttext="n\geq 1" display="inline"><mrow><mi href="./26.1#p2.t1.r2">n</mi><mo>≥</mo><mn>1</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E12.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./26.14#SS1.p3" title="§26.14(i) Definitions ‣ §26.14 Permutations: Order Notation ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="27px" altimg-valign="-9px" altimg-width="33px" alttext="\genfrac{<}{>}{0.0pt}{}{\NVar{n}}{\NVar{k}}" display="inline"><mrow><mo href="./26.14#SS1.p3">⟨</mo><mfrac linethickness="0.0pt"><mi class="ltx_nvar" href="./26.1#p2.t1.r2">n</mi><mi class="ltx_nvar" href="./26.1#p2.t1.r2">k</mi></mfrac><mo href="./26.14#SS1.p3">⟩</mo></mrow></math>: Eulerian number</a>,
<a href="./26.8#SS1.p3" title="§26.8(i) Definitions ‣ §26.8 Set Partitions: Stirling Numbers ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m14.png" altimg-height="23px" altimg-valign="-7px" altimg-width="68px" alttext="S\left(\NVar{n},\NVar{k}\right)" display="inline"><mrow><mi href="./26.8#SS1.p3">S</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./26.1#p2.t1.r2">n</mi><mo>,</mo><mi class="ltx_nvar" href="./26.1#p2.t1.r2">k</mi><mo>)</mo></mrow></mrow></math>: Stirling number of the second kind</a>,
<a href="./1.2#i" title="§1.2(i) Binomial Coefficients ‣ §1.2 Elementary Algebra ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m18.png" altimg-height="27px" altimg-valign="-9px" altimg-width="37px" alttext="\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}" display="inline"><mrow><mo href="./1.2#i">(</mo><mfrac linethickness="0.0pt"><mi class="ltx_nvar" href="./1.1#p2.t1.r5">m</mi><mi class="ltx_nvar" href="./1.1#p2.t1.r5">n</mi></mfrac><mo href="./1.2#i">)</mo></mrow></math>: binomial coefficient</a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m7.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m42.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./26.1#p2.t1.r2">k</mi></math>: nonnegative integer</a>,
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m40.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./26.1#p2.t1.r2">m</mi></math>: nonnegative integer</a> and
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math>: nonnegative integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="iv" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§26.14(iv) </span>Special Values</h2>
<div id="SS4.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="SS4.p1" class="ltx_para">
<table id="EGx1" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E13">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">26.14.13</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m4.png" altimg-height="53px" altimg-valign="-21px" altimg-width="47px" alttext="\displaystyle\genfrac{<}{>}{0.0pt}{}{0}{k}" display="inline"><mrow><mo href="./26.14#SS1.p3">⟨</mo><mstyle displaystyle="true"><mfrac linethickness="0.0pt"><mn>0</mn><mi href="./26.1#p2.t1.r2">k</mi></mfrac></mstyle><mo href="./26.14#SS1.p3">⟩</mo></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m3.png" altimg-height="25px" altimg-valign="-8px" altimg-width="64px" alttext="\displaystyle=\delta_{0,k}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><msub><mi href="./front/introduction#Sx4.p1.t1.r4">δ</mi><mrow><mn>0</mn><mo href="./front/introduction#Sx4.p1.t1.r4">,</mo><mi href="./26.1#p2.t1.r2">k</mi></mrow></msub></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E13.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./26.14#SS1.p3" title="§26.14(i) Definitions ‣ §26.14 Permutations: Order Notation ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="27px" altimg-valign="-9px" altimg-width="33px" alttext="\genfrac{<}{>}{0.0pt}{}{\NVar{n}}{\NVar{k}}" display="inline"><mrow><mo href="./26.14#SS1.p3">⟨</mo><mfrac linethickness="0.0pt"><mi class="ltx_nvar" href="./26.1#p2.t1.r2">n</mi><mi class="ltx_nvar" href="./26.1#p2.t1.r2">k</mi></mfrac><mo href="./26.14#SS1.p3">⟩</mo></mrow></math>: Eulerian number</a>,
<a href="./front/introduction#Sx4.p1.t1.r4" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m16.png" altimg-height="23px" altimg-valign="-8px" altimg-width="35px" alttext="\delta_{\NVar{j},\NVar{k}}" display="inline"><msub><mi href="./front/introduction#Sx4.p1.t1.r4">δ</mi><mrow><mi class="ltx_nvar">j</mi><mo href="./front/introduction#Sx4.p1.t1.r4">,</mo><mi class="ltx_nvar">k</mi></mrow></msub></math>: Kronecker delta</a> and
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./26.1#p2.t1.r2">k</mi></math>: nonnegative integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E14">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">26.14.14</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m5.png" altimg-height="53px" altimg-valign="-21px" altimg-width="48px" alttext="\displaystyle\genfrac{<}{>}{0.0pt}{}{n}{0}" display="inline"><mrow><mo href="./26.14#SS1.p3">⟨</mo><mstyle displaystyle="true"><mfrac linethickness="0.0pt"><mi href="./26.1#p2.t1.r2">n</mi><mn>0</mn></mfrac></mstyle><mo href="./26.14#SS1.p3">⟩</mo></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m1.png" altimg-height="22px" altimg-valign="-6px" altimg-width="43px" alttext="\displaystyle=1," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mn>1</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E14.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./26.14#SS1.p3" title="§26.14(i) Definitions ‣ §26.14 Permutations: Order Notation ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="27px" altimg-valign="-9px" altimg-width="33px" alttext="\genfrac{<}{>}{0.0pt}{}{\NVar{n}}{\NVar{k}}" display="inline"><mrow><mo href="./26.14#SS1.p3">⟨</mo><mfrac linethickness="0.0pt"><mi class="ltx_nvar" href="./26.1#p2.t1.r2">n</mi><mi class="ltx_nvar" href="./26.1#p2.t1.r2">k</mi></mfrac><mo href="./26.14#SS1.p3">⟩</mo></mrow></math>: Eulerian number</a> and
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math>: nonnegative integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E15">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">26.14.15</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m6.png" altimg-height="53px" altimg-valign="-21px" altimg-width="48px" alttext="\displaystyle\genfrac{<}{>}{0.0pt}{}{n}{1}" display="inline"><mrow><mo href="./26.14#SS1.p3">⟨</mo><mstyle displaystyle="true"><mfrac linethickness="0.0pt"><mi href="./26.1#p2.t1.r2">n</mi><mn>1</mn></mfrac></mstyle><mo href="./26.14#SS1.p3">⟩</mo></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m2.png" altimg-height="24px" altimg-valign="-6px" altimg-width="124px" alttext="\displaystyle=2^{n}-n-1," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msup><mn>2</mn><mi href="./26.1#p2.t1.r2">n</mi></msup><mo>-</mo><mi href="./26.1#p2.t1.r2">n</mi><mo>-</mo><mn>1</mn></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m44.png" altimg-height="19px" altimg-valign="-5px" altimg-width="53px" alttext="n\geq 1" display="inline"><mrow><mi href="./26.1#p2.t1.r2">n</mi><mo>≥</mo><mn>1</mn></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E15.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./26.14#SS1.p3" title="§26.14(i) Definitions ‣ §26.14 Permutations: Order Notation ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="27px" altimg-valign="-9px" altimg-width="33px" alttext="\genfrac{<}{>}{0.0pt}{}{\NVar{n}}{\NVar{k}}" display="inline"><mrow><mo href="./26.14#SS1.p3">⟨</mo><mfrac linethickness="0.0pt"><mi class="ltx_nvar" href="./26.1#p2.t1.r2">n</mi><mi class="ltx_nvar" href="./26.1#p2.t1.r2">k</mi></mfrac><mo href="./26.14#SS1.p3">⟩</mo></mrow></math>: Eulerian number</a> and
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math>: nonnegative integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
</div>
<div id="SS4.p2" class="ltx_para">
<table id="E16" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">26.14.16</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E16.png" altimg-height="53px" altimg-valign="-21px" altimg-width="308px" alttext="\genfrac{<}{>}{0.0pt}{}{n}{2}=3^{n}-(n+1)2^{n}+\genfrac{(}{)}{0.0pt}{}{n+1}{2}," display="block"><mrow><mrow><mrow><mo href="./26.14#SS1.p3">⟨</mo><mfrac linethickness="0.0pt"><mi href="./26.1#p2.t1.r2">n</mi><mn>2</mn></mfrac><mo href="./26.14#SS1.p3">⟩</mo></mrow><mo>=</mo><mrow><mrow><msup><mn>3</mn><mi href="./26.1#p2.t1.r2">n</mi></msup><mo>-</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./26.1#p2.t1.r2">n</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msup><mn>2</mn><mi href="./26.1#p2.t1.r2">n</mi></msup></mrow></mrow><mo>+</mo><mrow><mo href="./1.2#i">(</mo><mfrac linethickness="0.0pt"><mrow><mi href="./26.1#p2.t1.r2">n</mi><mo>+</mo><mn>1</mn></mrow><mn>2</mn></mfrac><mo href="./1.2#i">)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m44.png" altimg-height="19px" altimg-valign="-5px" altimg-width="53px" alttext="n\geq 1" display="inline"><mrow><mi href="./26.1#p2.t1.r2">n</mi><mo>≥</mo><mn>1</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E16.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./26.14#SS1.p3" title="§26.14(i) Definitions ‣ §26.14 Permutations: Order Notation ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="27px" altimg-valign="-9px" altimg-width="33px" alttext="\genfrac{<}{>}{0.0pt}{}{\NVar{n}}{\NVar{k}}" display="inline"><mrow><mo href="./26.14#SS1.p3">⟨</mo><mfrac linethickness="0.0pt"><mi class="ltx_nvar" href="./26.1#p2.t1.r2">n</mi><mi class="ltx_nvar" href="./26.1#p2.t1.r2">k</mi></mfrac><mo href="./26.14#SS1.p3">⟩</mo></mrow></math>: Eulerian number</a>,
<a href="./1.2#i" title="§1.2(i) Binomial Coefficients ‣ §1.2 Elementary Algebra ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m18.png" altimg-height="27px" altimg-valign="-9px" altimg-width="37px" alttext="\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}" display="inline"><mrow><mo href="./1.2#i">(</mo><mfrac linethickness="0.0pt"><mi class="ltx_nvar" href="./1.1#p2.t1.r5">m</mi><mi class="ltx_nvar" href="./1.1#p2.t1.r5">n</mi></mfrac><mo href="./1.2#i">)</mo></mrow></math>: binomial coefficient</a> and
<a href="./26.1#p2.t1.r2" title="§26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./26.1#p2.t1.r2">n</mi></math>: nonnegative integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
</section>
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<div id="SS1.p1" class="ltx_para">
<table id="E1" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.7.1</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E1.png" altimg-height="47px" altimg-valign="-16px" altimg-width="112px" alttext="\frac{\mathrm{d}}{\mathrm{d}z}\ln z=\frac{1}{z}," display="block"><mrow><mrow><mrow><mfrac><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mfrac><mo></mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mrow><mo>=</mo><mfrac><mn>1</mn><mi href="./4.1#p2.t1.r4">z</mi></mfrac></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#E4" title="(1.4.4) ‣ §1.4(iii) Derivatives ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m6.png" altimg-height="29px" altimg-valign="-9px" altimg-width="27px" alttext="\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: derivative of <math class="ltx_Math" altimg="m18.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m22.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m9.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m23.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.1.46</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E2" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.7.2</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E2.png" altimg-height="47px" altimg-valign="-16px" altimg-width="119px" alttext="\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{Ln}z=\frac{1}{z}," display="block"><mrow><mrow><mrow><mfrac><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mfrac><mo></mo><mrow><mi href="./4.2#E1">Ln</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mrow><mo>=</mo><mfrac><mn>1</mn><mi href="./4.1#p2.t1.r4">z</mi></mfrac></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#E4" title="(1.4.4) ‣ §1.4(iii) Derivatives ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m6.png" altimg-height="29px" altimg-valign="-9px" altimg-width="27px" alttext="\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: derivative of <math class="ltx_Math" altimg="m18.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m22.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E1" title="(4.2.1) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m12.png" altimg-height="18px" altimg-valign="-2px" altimg-width="41px" alttext="\operatorname{Ln}\NVar{z}" display="inline"><mrow><mi href="./4.2#E1">Ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: general logarithm function</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m23.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E3" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.7.3</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E3.png" altimg-height="47px" altimg-valign="-16px" altimg-width="281px" alttext="\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\ln z=(-1)^{n-1}(n-1)!z^{-n}," display="block"><mrow><mrow><mrow><mfrac><mpadded lspace="-1.7pt" width="-4.5pt"><msup><mo href="./1.4#E4">d</mo><mi href="./4.1#p2.t1.r1">n</mi></msup></mpadded><msup><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mi href="./1.4#E4">n</mi></msup></mfrac><mo></mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mrow><mo>=</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mrow><mi href="./4.1#p2.t1.r1">n</mi><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./4.1#p2.t1.r1">n</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow><mo></mo><msup><mi href="./4.1#p2.t1.r4">z</mi><mrow><mo>-</mo><mi href="./4.1#p2.t1.r1">n</mi></mrow></msup></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#E4" title="(1.4.4) ‣ §1.4(iii) Derivatives ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m6.png" altimg-height="29px" altimg-valign="-9px" altimg-width="27px" alttext="\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: derivative of <math class="ltx_Math" altimg="m18.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m22.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m1.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m19.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m9.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a>,
<a href="./4.1#p2.t1.r1" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./4.1#p2.t1.r1">n</mi></math>: integer</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m23.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.1.47</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E4" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.7.4</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E4.png" altimg-height="47px" altimg-valign="-16px" altimg-width="288px" alttext="\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\operatorname{Ln}z=(-1)^{n-1}(n-1)!z%
^{-n}." display="block"><mrow><mrow><mrow><mfrac><mpadded lspace="-1.7pt" width="-4.5pt"><msup><mo href="./1.4#E4">d</mo><mi href="./4.1#p2.t1.r1">n</mi></msup></mpadded><msup><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mi href="./1.4#E4">n</mi></msup></mfrac><mo></mo><mrow><mi href="./4.2#E1">Ln</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mrow><mo>=</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mrow><mi href="./4.1#p2.t1.r1">n</mi><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./4.1#p2.t1.r1">n</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow><mo></mo><msup><mi href="./4.1#p2.t1.r4">z</mi><mrow><mo>-</mo><mi href="./4.1#p2.t1.r1">n</mi></mrow></msup></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E4.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#E4" title="(1.4.4) ‣ §1.4(iii) Derivatives ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m6.png" altimg-height="29px" altimg-valign="-9px" altimg-width="27px" alttext="\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: derivative of <math class="ltx_Math" altimg="m18.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m22.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m1.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m19.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./4.2#E1" title="(4.2.1) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m12.png" altimg-height="18px" altimg-valign="-2px" altimg-width="41px" alttext="\operatorname{Ln}\NVar{z}" display="inline"><mrow><mi href="./4.2#E1">Ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: general logarithm function</a>,
<a href="./4.1#p2.t1.r1" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./4.1#p2.t1.r1">n</mi></math>: integer</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m23.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS1.p2" class="ltx_para">
<p class="ltx_p">For a nonvanishing analytic function <math class="ltx_Math" altimg="m17.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(z)" display="inline"><mrow><mi href="./4.7#SS1.p2">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./4.1#p2.t1.r4">z</mi><mo stretchy="false">)</mo></mrow></mrow></math>, the general solution of the
differential equation
</p>
<table id="E5" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.7.5</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E5.png" altimg-height="53px" altimg-valign="-21px" altimg-width="111px" alttext="\frac{\mathrm{d}w}{\mathrm{d}z}=\frac{f^{\prime}(z)}{f(z)}" display="block"><mrow><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi>w</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><msup><mi href="./4.7#SS1.p2">f</mi><mo>′</mo></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./4.1#p2.t1.r4">z</mi><mo stretchy="false">)</mo></mrow></mrow><mrow><mi href="./4.7#SS1.p2">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./4.1#p2.t1.r4">z</mi><mo stretchy="false">)</mo></mrow></mrow></mfrac></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E5.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#E4" title="(1.4.4) ‣ §1.4(iii) Derivatives ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m6.png" altimg-height="29px" altimg-valign="-9px" altimg-width="27px" alttext="\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: derivative of <math class="ltx_Math" altimg="m18.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m22.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m23.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a> and
<a href="./4.7#SS1.p2" title="§4.7(i) Logarithms ‣ §4.7 Derivatives and Differential Equations ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m17.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(z)" display="inline"><mrow><mi href="./4.7#SS1.p2">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./4.1#p2.t1.r4">z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: non-vanishing analytic function</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">is</p>
<table id="E6" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.7.6</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E6.png" altimg-height="25px" altimg-valign="-7px" altimg-width="263px" alttext="w(z)=\operatorname{Ln}\left(f(z)\right)+\hbox{ constant}." display="block"><mrow><mrow><mrow><mi href="./4.7#E6">w</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./4.1#p2.t1.r4">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mi href="./4.2#E1">Ln</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./4.7#SS1.p2">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./4.1#p2.t1.r4">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>)</mo></mrow></mrow><mo>+</mo><mtext> constant</mtext></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E6.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text"><math class="ltx_Math" altimg="m21.png" altimg-height="13px" altimg-valign="-2px" altimg-width="19px" alttext="w" display="inline"><mi href="./4.7#E6">w</mi></math>: solution (locally)</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./4.2#E1" title="(4.2.1) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m12.png" altimg-height="18px" altimg-valign="-2px" altimg-width="41px" alttext="\operatorname{Ln}\NVar{z}" display="inline"><mrow><mi href="./4.2#E1">Ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: general logarithm function</a>,
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m23.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a> and
<a href="./4.7#SS1.p2" title="§4.7(i) Logarithms ‣ §4.7 Derivatives and Differential Equations ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m17.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(z)" display="inline"><mrow><mi href="./4.7#SS1.p2">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./4.1#p2.t1.r4">z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: non-vanishing analytic function</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="ii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§4.7(ii) </span>Exponentials and Powers</h2>
<div id="SS2.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="SS2.p1" class="ltx_para">
<table id="E7" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.7.7</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E7.png" altimg-height="47px" altimg-valign="-16px" altimg-width="101px" alttext="\frac{\mathrm{d}}{\mathrm{d}z}e^{z}=e^{z}," display="block"><mrow><mrow><mrow><mfrac><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mfrac><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mi href="./4.1#p2.t1.r4">z</mi></msup></mrow><mo>=</mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mi href="./4.1#p2.t1.r4">z</mi></msup></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E7.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#E4" title="(1.4.4) ‣ §1.4(iii) Derivatives ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m6.png" altimg-height="29px" altimg-valign="-9px" altimg-width="27px" alttext="\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: derivative of <math class="ltx_Math" altimg="m18.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m22.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m11.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m23.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.2.49</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E8" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.7.8</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E8.png" altimg-height="47px" altimg-valign="-16px" altimg-width="129px" alttext="\frac{\mathrm{d}}{\mathrm{d}z}e^{az}=ae^{az}," display="block"><mrow><mrow><mrow><mfrac><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mfrac><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mi href="./4.1#p2.t1.r2">a</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></msup></mrow><mo>=</mo><mrow><mi href="./4.1#p2.t1.r2">a</mi><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mi href="./4.1#p2.t1.r2">a</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></msup></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E8.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#E4" title="(1.4.4) ‣ §1.4(iii) Derivatives ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m6.png" altimg-height="29px" altimg-valign="-9px" altimg-width="27px" alttext="\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: derivative of <math class="ltx_Math" altimg="m18.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m22.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m11.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./4.1#p2.t1.r2" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m14.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./4.1#p2.t1.r2">a</mi></math>: real or complex constant</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m23.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.2.50</span> (gives n-th derivative.)</span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E9" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.7.9</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E9.png" altimg-height="47px" altimg-valign="-16px" altimg-width="137px" alttext="\frac{\mathrm{d}}{\mathrm{d}z}a^{z}=a^{z}\ln a," display="block"><mrow><mrow><mrow><mfrac><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mfrac><mo></mo><msup><mi href="./4.1#p2.t1.r2">a</mi><mi href="./4.1#p2.t1.r4">z</mi></msup></mrow><mo>=</mo><mrow><msup><mi href="./4.1#p2.t1.r2">a</mi><mi href="./4.1#p2.t1.r4">z</mi></msup><mo></mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi href="./4.1#p2.t1.r2">a</mi></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m15.png" altimg-height="21px" altimg-valign="-6px" altimg-width="51px" alttext="a\neq 0" display="inline"><mrow><mi href="./4.1#p2.t1.r2">a</mi><mo>≠</mo><mn>0</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E9.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#E4" title="(1.4.4) ‣ §1.4(iii) Derivatives ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m6.png" altimg-height="29px" altimg-valign="-9px" altimg-width="27px" alttext="\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: derivative of <math class="ltx_Math" altimg="m18.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m22.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m9.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a>,
<a href="./4.1#p2.t1.r2" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m14.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./4.1#p2.t1.r2">a</mi></math>: real or complex constant</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m23.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.2.51</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">When <math class="ltx_Math" altimg="m16.png" altimg-height="18px" altimg-valign="-2px" altimg-width="24px" alttext="a^{z}" display="inline"><msup><mi href="./4.1#p2.t1.r2">a</mi><mi href="./4.1#p2.t1.r4">z</mi></msup></math> is a general power, <math class="ltx_Math" altimg="m8.png" altimg-height="18px" altimg-valign="-2px" altimg-width="35px" alttext="\ln a" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi href="./4.1#p2.t1.r2">a</mi></mrow></math> is replaced by the branch of
<math class="ltx_Math" altimg="m13.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\operatorname{Ln}a" display="inline"><mrow><mi href="./4.2#E1">Ln</mi><mo></mo><mi href="./4.1#p2.t1.r2">a</mi></mrow></math> used in constructing <math class="ltx_Math" altimg="m16.png" altimg-height="18px" altimg-valign="-2px" altimg-width="24px" alttext="a^{z}" display="inline"><msup><mi href="./4.1#p2.t1.r2">a</mi><mi href="./4.1#p2.t1.r4">z</mi></msup></math>.</p>
<table id="E10" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.7.10</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E10.png" altimg-height="47px" altimg-valign="-16px" altimg-width="135px" alttext="\frac{\mathrm{d}}{\mathrm{d}z}z^{a}=az^{a-1}," display="block"><mrow><mrow><mrow><mfrac><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mfrac><mo></mo><msup><mi href="./4.1#p2.t1.r4">z</mi><mi href="./4.1#p2.t1.r2">a</mi></msup></mrow><mo>=</mo><mrow><mi href="./4.1#p2.t1.r2">a</mi><mo></mo><msup><mi href="./4.1#p2.t1.r4">z</mi><mrow><mi href="./4.1#p2.t1.r2">a</mi><mo>-</mo><mn>1</mn></mrow></msup></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E10.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#E4" title="(1.4.4) ‣ §1.4(iii) Derivatives ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m6.png" altimg-height="29px" altimg-valign="-9px" altimg-width="27px" alttext="\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: derivative of <math class="ltx_Math" altimg="m18.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m22.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.1#p2.t1.r2" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m14.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./4.1#p2.t1.r2">a</mi></math>: real or complex constant</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m23.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.2.52</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E11" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.7.11</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E11.png" altimg-height="47px" altimg-valign="-16px" altimg-width="395px" alttext="\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}z^{a}=a(a-1)(a-2)\cdots(a-n+1)z^{a-n}." display="block"><mrow><mrow><mrow><mfrac><mpadded lspace="-1.7pt" width="-4.5pt"><msup><mo href="./1.4#E4">d</mo><mi href="./4.1#p2.t1.r1">n</mi></msup></mpadded><msup><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mi href="./1.4#E4">n</mi></msup></mfrac><mo></mo><msup><mi href="./4.1#p2.t1.r4">z</mi><mi href="./4.1#p2.t1.r2">a</mi></msup></mrow><mo>=</mo><mrow><mi href="./4.1#p2.t1.r2">a</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./4.1#p2.t1.r2">a</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./4.1#p2.t1.r2">a</mi><mo>-</mo><mn>2</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi mathvariant="normal">⋯</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mi href="./4.1#p2.t1.r2">a</mi><mo>-</mo><mi href="./4.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msup><mi href="./4.1#p2.t1.r4">z</mi><mrow><mi href="./4.1#p2.t1.r2">a</mi><mo>-</mo><mi href="./4.1#p2.t1.r1">n</mi></mrow></msup></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E11.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#E4" title="(1.4.4) ‣ §1.4(iii) Derivatives ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m6.png" altimg-height="29px" altimg-valign="-9px" altimg-width="27px" alttext="\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: derivative of <math class="ltx_Math" altimg="m18.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m22.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.1#p2.t1.r1" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./4.1#p2.t1.r1">n</mi></math>: integer</a>,
<a href="./4.1#p2.t1.r2" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m14.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./4.1#p2.t1.r2">a</mi></math>: real or complex constant</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m23.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS2.p2" class="ltx_para">
<p class="ltx_p">The general solution of the differential equation
</p>
<table id="E12" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.7.12</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E12.png" altimg-height="47px" altimg-valign="-16px" altimg-width="116px" alttext="\frac{\mathrm{d}w}{\mathrm{d}z}=f(z)w" display="block"><mrow><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi>w</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mfrac><mo>=</mo><mrow><mrow><mi href="./4.7#SS1.p2">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./4.1#p2.t1.r4">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mi>w</mi></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E12.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#E4" title="(1.4.4) ‣ §1.4(iii) Derivatives ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m6.png" altimg-height="29px" altimg-valign="-9px" altimg-width="27px" alttext="\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: derivative of <math class="ltx_Math" altimg="m18.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m22.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m23.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a> and
<a href="./4.7#SS1.p2" title="§4.7(i) Logarithms ‣ §4.7 Derivatives and Differential Equations ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m17.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(z)" display="inline"><mrow><mi href="./4.7#SS1.p2">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./4.1#p2.t1.r4">z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: non-vanishing analytic function</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">is
</p>
<table id="E13" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.7.13</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E13.png" altimg-height="53px" altimg-valign="-21px" altimg-width="296px" alttext="w=\exp\left(\int f(z)\mathrm{d}z\right)+{\rm constant}." display="block"><mrow><mrow><mi>w</mi><mo>=</mo><mrow><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mrow><mo>(</mo><mrow><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><mrow><mi href="./4.7#SS1.p2">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./4.1#p2.t1.r4">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mrow></mrow><mo>)</mo></mrow></mrow><mo>+</mo><mi>constant</mi></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E13.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m10.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m22.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E19" title="(4.2.19) ‣ §4.2(iii) The Exponential Function ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m5.png" altimg-height="16px" altimg-valign="-6px" altimg-width="48px" alttext="\exp\NVar{z}" display="inline"><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: exponential function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m7.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m23.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a> and
<a href="./4.7#SS1.p2" title="§4.7(i) Logarithms ‣ §4.7 Derivatives and Differential Equations ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m17.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(z)" display="inline"><mrow><mi href="./4.7#SS1.p2">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./4.1#p2.t1.r4">z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: non-vanishing analytic function</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS2.p3" class="ltx_para">
<p class="ltx_p">The general solution of the differential equation</p>
<table id="E14" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.7.14</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E14.png" altimg-height="51px" altimg-valign="-18px" altimg-width="103px" alttext="\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}=aw," display="block"><mrow><mrow><mfrac><mrow><mpadded lspace="-1.7pt" width="-4.5pt"><msup><mo href="./1.4#E4">d</mo><mn>2</mn></msup></mpadded><mi>w</mi></mrow><msup><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mn>2</mn></msup></mfrac><mo>=</mo><mrow><mi href="./4.1#p2.t1.r2">a</mi><mo></mo><mi>w</mi></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m15.png" altimg-height="21px" altimg-valign="-6px" altimg-width="51px" alttext="a\neq 0" display="inline"><mrow><mi href="./4.1#p2.t1.r2">a</mi><mo>≠</mo><mn>0</mn></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E14.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#E4" title="(1.4.4) ‣ §1.4(iii) Derivatives ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m6.png" altimg-height="29px" altimg-valign="-9px" altimg-width="27px" alttext="\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: derivative of <math class="ltx_Math" altimg="m18.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m22.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.1#p2.t1.r2" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m14.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./4.1#p2.t1.r2">a</mi></math>: real or complex constant</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m23.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">is</p>
<table id="E15" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.7.15</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E15.png" altimg-height="29px" altimg-valign="-6px" altimg-width="201px" alttext="w=Ae^{\sqrt{a}z}+Be^{-\sqrt{a}z}," display="block"><mrow><mrow><mi>w</mi><mo>=</mo><mrow><mrow><mi href="./4.7#SS2.p3">A</mi><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><msqrt><mi href="./4.1#p2.t1.r2">a</mi></msqrt><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></msup></mrow><mo>+</mo><mrow><mi href="./4.7#SS2.p3">B</mi><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><msqrt><mi href="./4.1#p2.t1.r2">a</mi></msqrt><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mrow></msup></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E15.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m11.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./4.1#p2.t1.r2" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m14.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./4.1#p2.t1.r2">a</mi></math>: real or complex constant</a>,
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m23.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>,
<a href="./4.7#SS2.p3" title="§4.7(ii) Exponentials and Powers ‣ §4.7 Derivatives and Differential Equations ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m2.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="A" display="inline"><mi href="./4.7#SS2.p3">A</mi></math>: arbitrary constant</a> and
<a href="./4.7#SS2.p3" title="§4.7(ii) Exponentials and Powers ‣ §4.7 Derivatives and Differential Equations ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m3.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="B" display="inline"><mi href="./4.7#SS2.p3">B</mi></math>: arbitrary constant</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m2.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="A" display="inline"><mi href="./4.7#SS2.p3">A</mi></math> and <math class="ltx_Math" altimg="m3.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="B" display="inline"><mi href="./4.7#SS2.p3">B</mi></math> are arbitrary constants.</p>
</div>
<div id="SS2.p4" class="ltx_para">
<p class="ltx_p">For other differential equations see <cite class="ltx_cite ltx_citemacro_citet">Kamke (</div>
</div>
</body></text>
</html>
</page>
<page>
<!DOCTYPE html><html>
<head>
<title>DLMF: 4.21 Identities</title>
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<div class="ltx_page_navlogo">)
with <math class="ltx_Math" altimg="m54.png" altimg-height="27px" altimg-valign="-9px" altimg-width="67px" alttext="\nu=\tfrac{1}{4}\pi" display="inline"><mrow><mi>ν</mi><mo>=</mo><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow></math>.</dd>
</dl>
</div>
</div>
<div id="SS1.p1" class="ltx_para">
<table id="E1" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.21.1</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E1.png" altimg-height="31px" altimg-valign="-9px" altimg-width="464px" alttext="\sin u\pm\cos u=\sqrt{2}\sin\left(u\pm\tfrac{1}{4}\pi\right)=\pm\sqrt{2}\cos%
\left(u\mp\tfrac{1}{4}\pi\right)." display="block"><mrow><mrow><mrow><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi>u</mi></mrow><mo>±</mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi>u</mi></mrow></mrow><mo>=</mo><mrow><msqrt><mn>2</mn></msqrt><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><mi>u</mi><mo>±</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>4</mn></mfrac></mstyle><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mrow><mo>±</mo><mrow><msqrt><mn>2</mn></msqrt><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mi>u</mi><mo>∓</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>4</mn></mfrac></mstyle><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m55.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a> and
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>
</dd>
<dt>Errata (effective with 1.0.7):</dt>
<dd>
Originally the symbol <math class="ltx_Math" altimg="m56.png" altimg-height="17px" altimg-valign="-4px" altimg-width="20px" alttext="\pm" display="inline"><mo>±</mo></math> was missing after the second equal sign.
<p><span class="ltx_font_italic">Reported 2012-09-27 by Dennis M. Heim</span></p>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="EGx1" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E2">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.21.2</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m40.png" altimg-height="25px" altimg-valign="-7px" altimg-width="95px" alttext="\displaystyle\sin\left(u\pm v\right)" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><mi>u</mi><mo>±</mo><mi>v</mi></mrow><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m22.png" altimg-height="23px" altimg-valign="-6px" altimg-width="223px" alttext="\displaystyle=\sin u\cos v\pm\cos u\sin v," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi>u</mi></mrow><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi>v</mi></mrow></mrow><mo>±</mo><mrow><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi>u</mi></mrow><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi>v</mi></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a> and
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.3.16</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E3">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.21.3</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m30.png" altimg-height="25px" altimg-valign="-7px" altimg-width="98px" alttext="\displaystyle\cos\left(u\pm v\right)" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mi>u</mi><mo>±</mo><mi>v</mi></mrow><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m12.png" altimg-height="23px" altimg-valign="-6px" altimg-width="223px" alttext="\displaystyle=\cos u\cos v\mp\sin u\sin v," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi>u</mi></mrow><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi>v</mi></mrow></mrow><mo>∓</mo><mrow><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi>u</mi></mrow><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi>v</mi></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a> and
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.3.17</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E4">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.21.4</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m43.png" altimg-height="25px" altimg-valign="-7px" altimg-width="100px" alttext="\displaystyle\tan\left(u\pm v\right)" display="inline"><mrow><mi href="./4.14#E4">tan</mi><mo></mo><mrow><mo>(</mo><mrow><mi>u</mi><mo>±</mo><mi>v</mi></mrow><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m21.png" altimg-height="47px" altimg-valign="-17px" altimg-width="161px" alttext="\displaystyle=\frac{\tan u\pm\tan v}{1\mp\tan u\tan v}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mstyle displaystyle="true"><mfrac><mrow><mrow><mi href="./4.14#E4">tan</mi><mo></mo><mi>u</mi></mrow><mo>±</mo><mrow><mi href="./4.14#E4">tan</mi><mo></mo><mi>v</mi></mrow></mrow><mrow><mn>1</mn><mo>∓</mo><mrow><mrow><mi href="./4.14#E4">tan</mi><mo></mo><mi>u</mi></mrow><mo></mo><mrow><mi href="./4.14#E4">tan</mi><mo></mo><mi>v</mi></mrow></mrow></mrow></mfrac></mstyle></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E4.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./4.14#E4" title="(4.14.4) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="17px" altimg-valign="-2px" altimg-width="47px" alttext="\tan\NVar{z}" display="inline"><mrow><mi href="./4.14#E4">tan</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: tangent function</a></dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.3.18</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E5">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.21.5</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m32.png" altimg-height="25px" altimg-valign="-7px" altimg-width="98px" alttext="\displaystyle\cot\left(u\pm v\right)" display="inline"><mrow><mi href="./4.14#E7">cot</mi><mo></mo><mrow><mo>(</mo><mrow><mi>u</mi><mo>±</mo><mi>v</mi></mrow><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m18.png" altimg-height="47px" altimg-valign="-17px" altimg-width="175px" alttext="\displaystyle=\frac{\pm\cot u\cot v-1}{\cot u\pm\cot v}." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mstyle displaystyle="true"><mfrac><mrow><mrow><mo>±</mo><mrow><mrow><mi href="./4.14#E7">cot</mi><mo></mo><mi>u</mi></mrow><mo></mo><mrow><mi href="./4.14#E7">cot</mi><mo></mo><mi>v</mi></mrow></mrow></mrow><mo>-</mo><mn>1</mn></mrow><mrow><mrow><mi href="./4.14#E7">cot</mi><mo></mo><mi>u</mi></mrow><mo>±</mo><mrow><mi href="./4.14#E7">cot</mi><mo></mo><mi>v</mi></mrow></mrow></mfrac></mstyle></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E5.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./4.14#E7" title="(4.14.7) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="17px" altimg-valign="-2px" altimg-width="44px" alttext="\cot\NVar{z}" display="inline"><mrow><mi href="./4.14#E7">cot</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cotangent function</a></dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.3.19</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
<table id="EGx2" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E6">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.21.6</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m34.png" altimg-height="21px" altimg-valign="-4px" altimg-width="108px" alttext="\displaystyle\sin u+\sin v" display="inline"><mrow><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi>u</mi></mrow><mo>+</mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi>v</mi></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m8.png" altimg-height="53px" altimg-valign="-21px" altimg-width="271px" alttext="\displaystyle=2\sin\left(\frac{u+v}{2}\right)\cos\left(\frac{u-v}{2}\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mn>2</mn><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac><mrow><mi>u</mi><mo>+</mo><mi>v</mi></mrow><mn>2</mn></mfrac></mstyle><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac><mrow><mi>u</mi><mo>-</mo><mi>v</mi></mrow><mn>2</mn></mfrac></mstyle><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E6.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a> and
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.3.34</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E7">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.21.7</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m35.png" altimg-height="21px" altimg-valign="-4px" altimg-width="108px" alttext="\displaystyle\sin u-\sin v" display="inline"><mrow><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi>u</mi></mrow><mo>-</mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi>v</mi></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m7.png" altimg-height="53px" altimg-valign="-21px" altimg-width="271px" alttext="\displaystyle=2\cos\left(\frac{u+v}{2}\right)\sin\left(\frac{u-v}{2}\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mn>2</mn><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac><mrow><mi>u</mi><mo>+</mo><mi>v</mi></mrow><mn>2</mn></mfrac></mstyle><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac><mrow><mi>u</mi><mo>-</mo><mi>v</mi></mrow><mn>2</mn></mfrac></mstyle><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E7.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a> and
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.3.35</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E8">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.21.8</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m24.png" altimg-height="19px" altimg-valign="-4px" altimg-width="112px" alttext="\displaystyle\cos u+\cos v" display="inline"><mrow><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi>u</mi></mrow><mo>+</mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi>v</mi></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m6.png" altimg-height="53px" altimg-valign="-21px" altimg-width="273px" alttext="\displaystyle=2\cos\left(\frac{u+v}{2}\right)\cos\left(\frac{u-v}{2}\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mn>2</mn><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac><mrow><mi>u</mi><mo>+</mo><mi>v</mi></mrow><mn>2</mn></mfrac></mstyle><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac><mrow><mi>u</mi><mo>-</mo><mi>v</mi></mrow><mn>2</mn></mfrac></mstyle><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E8.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a></dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.3.36</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E9">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.21.9</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m25.png" altimg-height="19px" altimg-valign="-4px" altimg-width="112px" alttext="\displaystyle\cos u-\cos v" display="inline"><mrow><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi>u</mi></mrow><mo>-</mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi>v</mi></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m1.png" altimg-height="53px" altimg-valign="-21px" altimg-width="284px" alttext="\displaystyle=-2\sin\left(\frac{u+v}{2}\right)\sin\left(\frac{u-v}{2}\right)." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo>-</mo><mrow><mn>2</mn><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac><mrow><mi>u</mi><mo>+</mo><mi>v</mi></mrow><mn>2</mn></mfrac></mstyle><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac><mrow><mi>u</mi><mo>-</mo><mi>v</mi></mrow><mn>2</mn></mfrac></mstyle><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E9.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a> and
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.3.37</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
<table id="EGx3" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E10">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.21.10</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m41.png" altimg-height="20px" altimg-valign="-4px" altimg-width="116px" alttext="\displaystyle\tan u\pm\tan v" display="inline"><mrow><mrow><mi href="./4.14#E4">tan</mi><mo></mo><mi>u</mi></mrow><mo>±</mo><mrow><mi href="./4.14#E4">tan</mi><mo></mo><mi>v</mi></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m19.png" altimg-height="48px" altimg-valign="-16px" altimg-width="127px" alttext="\displaystyle=\frac{\sin\left(u\pm v\right)}{\cos u\cos v}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mstyle displaystyle="true"><mfrac><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><mi>u</mi><mo>±</mo><mi>v</mi></mrow><mo>)</mo></mrow></mrow><mrow><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi>u</mi></mrow><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi>v</mi></mrow></mrow></mfrac></mstyle></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E10.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a> and
<a href="./4.14#E4" title="(4.14.4) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="17px" altimg-valign="-2px" altimg-width="47px" alttext="\tan\NVar{z}" display="inline"><mrow><mi href="./4.14#E4">tan</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: tangent function</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.3.38</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E11">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.21.11</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m31.png" altimg-height="20px" altimg-valign="-4px" altimg-width="112px" alttext="\displaystyle\cot u\pm\cot v" display="inline"><mrow><mrow><mi href="./4.14#E7">cot</mi><mo></mo><mi>u</mi></mrow><mo>±</mo><mrow><mi href="./4.14#E7">cot</mi><mo></mo><mi>v</mi></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m20.png" altimg-height="48px" altimg-valign="-16px" altimg-width="127px" alttext="\displaystyle=\frac{\sin\left(v\pm u\right)}{\sin u\sin v}." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mstyle displaystyle="true"><mfrac><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><mi>v</mi><mo>±</mo><mi>u</mi></mrow><mo>)</mo></mrow></mrow><mrow><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi>u</mi></mrow><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi>v</mi></mrow></mrow></mfrac></mstyle></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E11.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.14#E7" title="(4.14.7) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="17px" altimg-valign="-2px" altimg-width="44px" alttext="\cot\NVar{z}" display="inline"><mrow><mi href="./4.14#E7">cot</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cotangent function</a> and
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.3.39</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
</div>
</section>
<section id="ii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§4.21(ii) </span>Squares and Products</h2>
<div id="SS2.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="SS2.p1" class="ltx_para">
<table id="E12" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.21.12</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E12.png" altimg-height="27px" altimg-valign="-6px" altimg-width="162px" alttext="{\sin^{2}}z+{\cos^{2}}z=1," display="block"><mrow><mrow><mrow><mrow><msup><mi href="./4.14#E1">sin</mi><mn>2</mn></msup><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>+</mo><mrow><msup><mi href="./4.14#E2">cos</mi><mn>2</mn></msup><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mrow><mo>=</mo><mn>1</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E12.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.3.10</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E13" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.21.13</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E13.png" altimg-height="27px" altimg-valign="-6px" altimg-width="165px" alttext="{\sec^{2}}z=1+{\tan^{2}}z," display="block"><mrow><mrow><mrow><msup><mi href="./4.14#E6">sec</mi><mn>2</mn></msup><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>=</mo><mrow><mn>1</mn><mo>+</mo><mrow><msup><mi href="./4.14#E4">tan</mi><mn>2</mn></msup><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E13.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.14#E6" title="(4.14.6) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m57.png" altimg-height="13px" altimg-valign="-2px" altimg-width="43px" alttext="\sec\NVar{z}" display="inline"><mrow><mi href="./4.14#E6">sec</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: secant function</a>,
<a href="./4.14#E4" title="(4.14.4) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="17px" altimg-valign="-2px" altimg-width="47px" alttext="\tan\NVar{z}" display="inline"><mrow><mi href="./4.14#E4">tan</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: tangent function</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.3.11</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E14" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.21.14</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E14.png" altimg-height="25px" altimg-valign="-4px" altimg-width="163px" alttext="{\csc^{2}}z=1+{\cot^{2}}z." display="block"><mrow><mrow><mrow><msup><mi href="./4.14#E5">csc</mi><mn>2</mn></msup><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>=</mo><mrow><mn>1</mn><mo>+</mo><mrow><msup><mi href="./4.14#E7">cot</mi><mn>2</mn></msup><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E14.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.14#E5" title="(4.14.5) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="13px" altimg-valign="-2px" altimg-width="43px" alttext="\csc\NVar{z}" display="inline"><mrow><mi href="./4.14#E5">csc</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosecant function</a>,
<a href="./4.14#E7" title="(4.14.7) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="17px" altimg-valign="-2px" altimg-width="44px" alttext="\cot\NVar{z}" display="inline"><mrow><mi href="./4.14#E7">cot</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cotangent function</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.3.12</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E15" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.21.15</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E15.png" altimg-height="25px" altimg-valign="-7px" altimg-width="343px" alttext="2\sin u\sin v=\cos\left(u-v\right)-\cos\left(u+v\right)," display="block"><mrow><mrow><mrow><mn>2</mn><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi>u</mi></mrow><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi>v</mi></mrow></mrow><mo>=</mo><mrow><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mi>u</mi><mo>-</mo><mi>v</mi></mrow><mo>)</mo></mrow></mrow><mo>-</mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mi>u</mi><mo>+</mo><mi>v</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E15.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a> and
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.3.31</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E16" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.21.16</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E16.png" altimg-height="25px" altimg-valign="-7px" altimg-width="347px" alttext="2\cos u\cos v=\cos\left(u-v\right)+\cos\left(u+v\right)," display="block"><mrow><mrow><mrow><mn>2</mn><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi>u</mi></mrow><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi>v</mi></mrow></mrow><mo>=</mo><mrow><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mi>u</mi><mo>-</mo><mi>v</mi></mrow><mo>)</mo></mrow></mrow><mo>+</mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mi>u</mi><mo>+</mo><mi>v</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E16.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a></dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.3.32</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E17" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.21.17</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E17.png" altimg-height="25px" altimg-valign="-7px" altimg-width="341px" alttext="2\sin u\cos v=\sin\left(u-v\right)+\sin\left(u+v\right)." display="block"><mrow><mrow><mrow><mn>2</mn><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi>u</mi></mrow><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi>v</mi></mrow></mrow><mo>=</mo><mrow><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><mi>u</mi><mo>-</mo><mi>v</mi></mrow><mo>)</mo></mrow></mrow><mo>+</mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><mi>u</mi><mo>+</mo><mi>v</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E17.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a> and
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.3.33</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="EGx4" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E18">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.21.18</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m46.png" altimg-height="25px" altimg-valign="-4px" altimg-width="119px" alttext="\displaystyle{\sin^{2}}u-{\sin^{2}}v" display="inline"><mrow><mrow><msup><mi href="./4.14#E1">sin</mi><mn>2</mn></msup><mo></mo><mi>u</mi></mrow><mo>-</mo><mrow><msup><mi href="./4.14#E1">sin</mi><mn>2</mn></msup><mo></mo><mi>v</mi></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m23.png" altimg-height="25px" altimg-valign="-7px" altimg-width="218px" alttext="\displaystyle=\sin\left(u+v\right)\sin\left(u-v\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><mi>u</mi><mo>+</mo><mi>v</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><mi>u</mi><mo>-</mo><mi>v</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E18.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a></dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.3.40</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E19">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.21.19</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m44.png" altimg-height="25px" altimg-valign="-4px" altimg-width="123px" alttext="\displaystyle{\cos^{2}}u-{\cos^{2}}v" display="inline"><mrow><mrow><msup><mi href="./4.14#E2">cos</mi><mn>2</mn></msup><mo></mo><mi>u</mi></mrow><mo>-</mo><mrow><msup><mi href="./4.14#E2">cos</mi><mn>2</mn></msup><mo></mo><mi>v</mi></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m4.png" altimg-height="25px" altimg-valign="-7px" altimg-width="237px" alttext="\displaystyle=-\sin\left(u+v\right)\sin\left(u-v\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo>-</mo><mrow><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><mi>u</mi><mo>+</mo><mi>v</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><mi>u</mi><mo>-</mo><mi>v</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E19.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a> and
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.3.41</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E20">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.21.20</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m45.png" altimg-height="25px" altimg-valign="-4px" altimg-width="121px" alttext="\displaystyle{\cos^{2}}u-{\sin^{2}}v" display="inline"><mrow><mrow><msup><mi href="./4.14#E2">cos</mi><mn>2</mn></msup><mo></mo><mi>u</mi></mrow><mo>-</mo><mrow><msup><mi href="./4.14#E1">sin</mi><mn>2</mn></msup><mo></mo><mi>v</mi></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m14.png" altimg-height="25px" altimg-valign="-7px" altimg-width="223px" alttext="\displaystyle=\cos\left(u+v\right)\cos\left(u-v\right)." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mi>u</mi><mo>+</mo><mi>v</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mi>u</mi><mo>-</mo><mi>v</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E20.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a> and
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.3.42</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
</div>
</section>
<section id="iii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§4.21(iii) </span>Multiples of the Argument</h2>
<div id="SS3.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="SS3.p1" class="ltx_para">
<table id="E21" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.21.21</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E21.png" altimg-height="60px" altimg-valign="-21px" altimg-width="230px" alttext="\sin\frac{z}{2}=\pm\left(\frac{1-\cos z}{2}\right)^{1/2}," display="block"><mrow><mrow><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mfrac><mi href="./4.1#p2.t1.r4">z</mi><mn>2</mn></mfrac></mrow><mo>=</mo><mrow><mo>±</mo><msup><mrow><mo>(</mo><mfrac><mrow><mn>1</mn><mo>-</mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mrow><mn>2</mn></mfrac><mo>)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E21.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.3.20</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E22" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.21.22</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E22.png" altimg-height="60px" altimg-valign="-21px" altimg-width="233px" alttext="\cos\frac{z}{2}=\pm\left(\frac{1+\cos z}{2}\right)^{1/2}," display="block"><mrow><mrow><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mfrac><mi href="./4.1#p2.t1.r4">z</mi><mn>2</mn></mfrac></mrow><mo>=</mo><mrow><mo>±</mo><msup><mrow><mo>(</mo><mfrac><mrow><mn>1</mn><mo>+</mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mrow><mn>2</mn></mfrac><mo>)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E22.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.3.21</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E23" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.21.23</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E23.png" altimg-height="60px" altimg-valign="-21px" altimg-width="443px" alttext="\tan\frac{z}{2}=\pm\left(\frac{1-\cos z}{1+\cos z}\right)^{1/2}=\frac{1-\cos z%
}{\sin z}=\frac{\sin z}{1+\cos z}." display="block"><mrow><mrow><mrow><mi href="./4.14#E4">tan</mi><mo></mo><mfrac><mi href="./4.1#p2.t1.r4">z</mi><mn>2</mn></mfrac></mrow><mo>=</mo><mrow><mo>±</mo><msup><mrow><mo>(</mo><mfrac><mrow><mn>1</mn><mo>-</mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mrow><mrow><mn>1</mn><mo>+</mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mrow></mfrac><mo>)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow><mo>=</mo><mfrac><mrow><mn>1</mn><mo>-</mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mrow><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mrow><mn>1</mn><mo>+</mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mrow></mfrac></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E23.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./4.14#E4" title="(4.14.4) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="17px" altimg-valign="-2px" altimg-width="47px" alttext="\tan\NVar{z}" display="inline"><mrow><mi href="./4.14#E4">tan</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: tangent function</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.3.22</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">In ( and
analytic continuation will assist in resolving sign ambiguities.</p>
<table id="EGx5" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E24">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.21.24</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m37.png" altimg-height="25px" altimg-valign="-7px" altimg-width="75px" alttext="\displaystyle\sin\left(-z\right)" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m3.png" altimg-height="23px" altimg-valign="-6px" altimg-width="89px" alttext="\displaystyle=-\sin z," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo>-</mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E24.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.3.13</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E25">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.21.25</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m27.png" altimg-height="25px" altimg-valign="-7px" altimg-width="77px" alttext="\displaystyle\cos\left(-z\right)" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m13.png" altimg-height="18px" altimg-valign="-6px" altimg-width="73px" alttext="\displaystyle=\cos z," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E25.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.3.14</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E26">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.21.26</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m42.png" altimg-height="25px" altimg-valign="-7px" altimg-width="79px" alttext="\displaystyle\tan\left(-z\right)" display="inline"><mrow><mi href="./4.14#E4">tan</mi><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m5.png" altimg-height="20px" altimg-valign="-4px" altimg-width="94px" alttext="\displaystyle=-\tan z." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo>-</mo><mrow><mi href="./4.14#E4">tan</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E26.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.14#E4" title="(4.14.4) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="17px" altimg-valign="-2px" altimg-width="47px" alttext="\tan\NVar{z}" display="inline"><mrow><mi href="./4.14#E4">tan</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: tangent function</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.3.15</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
<table id="E27" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.21.27</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E27.png" altimg-height="48px" altimg-valign="-18px" altimg-width="310px" alttext="\sin\left(2z\right)=2\sin z\cos z=\frac{2\tan z}{1+{\tan^{2}}z}," display="block"><mrow><mrow><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mn>2</mn><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mrow><mo>=</mo><mfrac><mrow><mn>2</mn><mo></mo><mrow><mi href="./4.14#E4">tan</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mrow><mrow><mn>1</mn><mo>+</mo><mrow><msup><mi href="./4.14#E4">tan</mi><mn>2</mn></msup><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mrow></mfrac></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E27.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./4.14#E4" title="(4.14.4) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="17px" altimg-valign="-2px" altimg-width="47px" alttext="\tan\NVar{z}" display="inline"><mrow><mi href="./4.14#E4">tan</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: tangent function</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.3.24</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E28" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.21.28</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E28.png" altimg-height="51px" altimg-valign="-18px" altimg-width="562px" alttext="\cos\left(2z\right)=2{\cos^{2}}z-1=1-2{\sin^{2}}z={\cos^{2}}z-{\sin^{2}}z=%
\frac{1-{\tan^{2}}z}{1+{\tan^{2}}z}," display="block"><mrow><mrow><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mn>2</mn><mo></mo><mrow><msup><mi href="./4.14#E2">cos</mi><mn>2</mn></msup><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mrow><mo>-</mo><mn>1</mn></mrow><mo>=</mo><mrow><mn>1</mn><mo>-</mo><mrow><mn>2</mn><mo></mo><mrow><msup><mi href="./4.14#E1">sin</mi><mn>2</mn></msup><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mrow></mrow><mo>=</mo><mrow><mrow><msup><mi href="./4.14#E2">cos</mi><mn>2</mn></msup><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>-</mo><mrow><msup><mi href="./4.14#E1">sin</mi><mn>2</mn></msup><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mrow><mo>=</mo><mfrac><mrow><mn>1</mn><mo>-</mo><mrow><msup><mi href="./4.14#E4">tan</mi><mn>2</mn></msup><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mrow><mrow><mn>1</mn><mo>+</mo><mrow><msup><mi href="./4.14#E4">tan</mi><mn>2</mn></msup><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mrow></mfrac></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E28.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./4.14#E4" title="(4.14.4) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="17px" altimg-valign="-2px" altimg-width="47px" alttext="\tan\NVar{z}" display="inline"><mrow><mi href="./4.14#E4">tan</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: tangent function</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.3.25</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E29" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.21.29</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E29.png" altimg-height="48px" altimg-valign="-18px" altimg-width="442px" alttext="\tan\left(2z\right)=\frac{2\tan z}{1-{\tan^{2}}z}=\frac{2\cot z}{{\cot^{2}}z-1%
}=\frac{2}{\cot z-\tan z}." display="block"><mrow><mrow><mrow><mi href="./4.14#E4">tan</mi><mo></mo><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mfrac><mrow><mn>2</mn><mo></mo><mrow><mi href="./4.14#E4">tan</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mrow><mrow><mn>1</mn><mo>-</mo><mrow><msup><mi href="./4.14#E4">tan</mi><mn>2</mn></msup><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>2</mn><mo></mo><mrow><mi href="./4.14#E7">cot</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mrow><mrow><mrow><msup><mi href="./4.14#E7">cot</mi><mn>2</mn></msup><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>-</mo><mn>1</mn></mrow></mfrac><mo>=</mo><mfrac><mn>2</mn><mrow><mrow><mi href="./4.14#E7">cot</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>-</mo><mrow><mi href="./4.14#E4">tan</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mrow></mfrac></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E29.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.14#E7" title="(4.14.7) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="17px" altimg-valign="-2px" altimg-width="44px" alttext="\cot\NVar{z}" display="inline"><mrow><mi href="./4.14#E7">cot</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cotangent function</a>,
<a href="./4.14#E4" title="(4.14.4) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="17px" altimg-valign="-2px" altimg-width="47px" alttext="\tan\NVar{z}" display="inline"><mrow><mi href="./4.14#E4">tan</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: tangent function</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.3.26</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="EGx6" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E30">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.21.30</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m38.png" altimg-height="25px" altimg-valign="-7px" altimg-width="69px" alttext="\displaystyle\sin\left(3z\right)" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><mn>3</mn><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m9.png" altimg-height="27px" altimg-valign="-6px" altimg-width="162px" alttext="\displaystyle=3\sin z-4{\sin^{3}}z," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mn>3</mn><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mrow><mo>-</mo><mrow><mn>4</mn><mo></mo><mrow><msup><mi href="./4.14#E1">sin</mi><mn>3</mn></msup><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E30.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.3.27</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E31">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.21.31</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m28.png" altimg-height="25px" altimg-valign="-7px" altimg-width="72px" alttext="\displaystyle\cos\left(3z\right)" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mn>3</mn><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m2.png" altimg-height="27px" altimg-valign="-6px" altimg-width="182px" alttext="\displaystyle=-3\cos z+4{\cos^{3}}z," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mo>-</mo><mrow><mn>3</mn><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mrow></mrow><mo>+</mo><mrow><mn>4</mn><mo></mo><mrow><msup><mi href="./4.14#E2">cos</mi><mn>3</mn></msup><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E31.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.3.28</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E32">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.21.32</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m39.png" altimg-height="25px" altimg-valign="-7px" altimg-width="69px" alttext="\displaystyle\sin\left(4z\right)" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><mn>4</mn><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m10.png" altimg-height="27px" altimg-valign="-6px" altimg-width="249px" alttext="\displaystyle=8{\cos^{3}}z\sin z-4\cos z\sin z," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mn>8</mn><mo></mo><mrow><msup><mi href="./4.14#E2">cos</mi><mn>3</mn></msup><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mrow><mo>-</mo><mrow><mn>4</mn><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E32.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.3.29</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E33">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.21.33</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m29.png" altimg-height="25px" altimg-valign="-7px" altimg-width="72px" alttext="\displaystyle\cos\left(4z\right)" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mn>4</mn><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m11.png" altimg-height="25px" altimg-valign="-4px" altimg-width="203px" alttext="\displaystyle=8{\cos^{4}}z-8{\cos^{2}}z+1." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mrow><mn>8</mn><mo></mo><mrow><msup><mi href="./4.14#E2">cos</mi><mn>4</mn></msup><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mrow><mo>-</mo><mrow><mn>8</mn><mo></mo><mrow><msup><mi href="./4.14#E2">cos</mi><mn>2</mn></msup><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mrow></mrow><mo>+</mo><mn>1</mn></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E33.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.3.30</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
</div>
<section id="Px1" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">De Moivre’s Theorem</h3>
<div id="Px1.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px1.p1" class="ltx_para">
<p class="ltx_p">When <math class="ltx_Math" altimg="m64.png" altimg-height="18px" altimg-valign="-3px" altimg-width="54px" alttext="n\in\mathbb{Z}" display="inline"><mrow><mi href="./4.1#p2.t1.r1">n</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mi href="./front/introduction#Sx4.p2.t1.r20">ℤ</mi></mrow></math></p>
<table id="E34" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.21.34</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E34.png" altimg-height="25px" altimg-valign="-7px" altimg-width="345px" alttext="\cos\left(nz\right)+i\sin\left(nz\right)=(\cos z+i\sin z)^{n}." display="block"><mrow><mrow><mrow><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./4.1#p2.t1.r1">n</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./4.1#p2.t1.r1">n</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mrow><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mrow></mrow><mo stretchy="false">)</mo></mrow><mi href="./4.1#p2.t1.r1">n</mi></msup></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E34.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./4.1#p2.t1.r1" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./4.1#p2.t1.r1">n</mi></math>: integer</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.3.48</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">This result is also valid when <math class="ltx_Math" altimg="m63.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./4.1#p2.t1.r1">n</mi></math> is fractional or complex, provided that
<math class="ltx_Math" altimg="m47.png" altimg-height="20px" altimg-valign="-5px" altimg-width="122px" alttext="-\pi\leq\Re z\leq\pi" display="inline"><mrow><mrow><mo>-</mo><mi href="./3.12#E1">π</mi></mrow><mo>≤</mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>≤</mo><mi href="./3.12#E1">π</mi></mrow></math>.</p>
<table id="E35" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.21.35</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E35.png" altimg-height="67px" altimg-valign="-28px" altimg-width="304px" alttext="\sin\left(nz\right)=2^{n-1}\prod_{k=0}^{n-1}\sin\left(z+\frac{k\pi}{n}\right)," display="block"><mrow><mrow><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./4.1#p2.t1.r1">n</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msup><mn>2</mn><mrow><mi href="./4.1#p2.t1.r1">n</mi><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∏</mo><mrow><mi href="./4.1#p2.t1.r1">k</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi href="./4.1#p2.t1.r1">n</mi><mo>-</mo><mn>1</mn></mrow></munderover><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./4.1#p2.t1.r4">z</mi><mo>+</mo><mfrac><mrow><mi href="./4.1#p2.t1.r1">k</mi><mo></mo><mi href="./3.12#E1">π</mi></mrow><mi href="./4.1#p2.t1.r1">n</mi></mfrac></mrow><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m62.png" altimg-height="20px" altimg-valign="-6px" altimg-width="123px" alttext="n=1,2,3,\dots" display="inline"><mrow><mi href="./4.1#p2.t1.r1">n</mi><mo>=</mo><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E35.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m55.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./4.1#p2.t1.r1" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m61.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./4.1#p2.t1.r1">k</mi></math>: integer</a>,
<a href="./4.1#p2.t1.r1" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./4.1#p2.t1.r1">n</mi></math>: integer</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="Px1.p2" class="ltx_para">
<p class="ltx_p">If <math class="ltx_Math" altimg="m65.png" altimg-height="27px" altimg-valign="-9px" altimg-width="111px" alttext="t=\tan\left(\frac{1}{2}z\right)" display="inline"><mrow><mi>t</mi><mo>=</mo><mrow><mi href="./4.14#E4">tan</mi><mo></mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></mrow></math>, then</p>
<table id="E36" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex1" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="3" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">4.21.36</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m36.png" altimg-height="19px" altimg-valign="-2px" altimg-width="44px" alttext="\displaystyle\sin z" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m16.png" altimg-height="47px" altimg-valign="-17px" altimg-width="88px" alttext="\displaystyle=\frac{2t}{1+t^{2}}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mstyle displaystyle="true"><mfrac><mrow><mn>2</mn><mo></mo><mi>t</mi></mrow><mrow><mn>1</mn><mo>+</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac></mstyle></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex2" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m26.png" altimg-height="14px" altimg-valign="-2px" altimg-width="46px" alttext="\displaystyle\cos z" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m15.png" altimg-height="51px" altimg-valign="-17px" altimg-width="88px" alttext="\displaystyle=\frac{1-t^{2}}{1+t^{2}}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mstyle displaystyle="true"><mfrac><mrow><mn>1</mn><mo>-</mo><msup><mi>t</mi><mn>2</mn></msup></mrow><mrow><mn>1</mn><mo>+</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac></mstyle></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex3" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m33.png" altimg-height="19px" altimg-valign="-2px" altimg-width="27px" alttext="\displaystyle\mathrm{d}z" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m17.png" altimg-height="47px" altimg-valign="-17px" altimg-width="106px" alttext="\displaystyle=\frac{2}{1+t^{2}}\mathrm{d}t." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><mfrac><mn>2</mn><mrow><mn>1</mn><mo>+</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac></mstyle><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E36.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m66.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.3.23</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
</section>
<section id="iv" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§4.21(iv) </span>Real and Imaginary Parts; Moduli</h2>
<div id="SS4.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="SS4.p1" class="ltx_para">
<p class="ltx_p">With <math class="ltx_Math" altimg="m68.png" altimg-height="20px" altimg-valign="-6px" altimg-width="94px" alttext="z=x+iy" display="inline"><mrow><mi href="./4.1#p2.t1.r4">z</mi><mo>=</mo><mrow><mi href="./4.1#p2.t1.r3">x</mi><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./4.1#p2.t1.r3">y</mi></mrow></mrow></mrow></math></p>
<table id="E37" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.21.37</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E37.png" altimg-height="23px" altimg-valign="-6px" altimg-width="297px" alttext="\sin z=\sin x\cosh y+\mathrm{i}\cos x\sinh y," display="block"><mrow><mrow><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>=</mo><mrow><mrow><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi href="./4.1#p2.t1.r3">x</mi></mrow><mo></mo><mrow><mi href="./4.28#E2">cosh</mi><mo></mo><mi href="./4.1#p2.t1.r3">y</mi></mrow></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi href="./4.1#p2.t1.r3">x</mi></mrow><mo></mo><mrow><mi href="./4.28#E1">sinh</mi><mo></mo><mi href="./4.1#p2.t1.r3">y</mi></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E37.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./4.28#E2" title="(4.28.2) ‣ §4.28 Definitions and Periodicity ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="18px" altimg-valign="-2px" altimg-width="56px" alttext="\cosh\NVar{z}" display="inline"><mrow><mi href="./4.28#E2">cosh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: hyperbolic cosine function</a>,
<a href="./4.28#E1" title="(4.28.1) ‣ §4.28 Definitions and Periodicity ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="53px" alttext="\sinh\NVar{z}" display="inline"><mrow><mi href="./4.28#E1">sinh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: hyperbolic sine function</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./4.1#p2.t1.r3" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./4.1#p2.t1.r3">x</mi></math>: real variable</a>,
<a href="./4.1#p2.t1.r3" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m67.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./4.1#p2.t1.r3">y</mi></math>: real variable</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.3.55</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E38" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.21.38</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E38.png" altimg-height="23px" altimg-valign="-6px" altimg-width="300px" alttext="\cos z=\cos x\cosh y-\mathrm{i}\sin x\sinh y," display="block"><mrow><mrow><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>=</mo><mrow><mrow><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi href="./4.1#p2.t1.r3">x</mi></mrow><mo></mo><mrow><mi href="./4.28#E2">cosh</mi><mo></mo><mi href="./4.1#p2.t1.r3">y</mi></mrow></mrow><mo>-</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi href="./4.1#p2.t1.r3">x</mi></mrow><mo></mo><mrow><mi href="./4.28#E1">sinh</mi><mo></mo><mi href="./4.1#p2.t1.r3">y</mi></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E38.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./4.28#E2" title="(4.28.2) ‣ §4.28 Definitions and Periodicity ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="18px" altimg-valign="-2px" altimg-width="56px" alttext="\cosh\NVar{z}" display="inline"><mrow><mi href="./4.28#E2">cosh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: hyperbolic cosine function</a>,
<a href="./4.28#E1" title="(4.28.1) ‣ §4.28 Definitions and Periodicity ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="53px" alttext="\sinh\NVar{z}" display="inline"><mrow><mi href="./4.28#E1">sinh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: hyperbolic sine function</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./4.1#p2.t1.r3" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./4.1#p2.t1.r3">x</mi></math>: real variable</a>,
<a href="./4.1#p2.t1.r3" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m67.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./4.1#p2.t1.r3">y</mi></math>: real variable</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.3.56</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E39" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.21.39</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E39.png" altimg-height="53px" altimg-valign="-21px" altimg-width="258px" alttext="\tan z=\frac{\sin\left(2x\right)+\mathrm{i}\sinh\left(2y\right)}{\cos\left(2x%
\right)+\cosh\left(2y\right)}," display="block"><mrow><mrow><mrow><mi href="./4.14#E4">tan</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>=</mo><mfrac><mrow><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><mi href="./4.1#p2.t1.r3">x</mi></mrow><mo>)</mo></mrow></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mrow><mi href="./4.28#E1">sinh</mi><mo></mo><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><mi href="./4.1#p2.t1.r3">y</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mrow><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><mi href="./4.1#p2.t1.r3">x</mi></mrow><mo>)</mo></mrow></mrow><mo>+</mo><mrow><mi href="./4.28#E2">cosh</mi><mo></mo><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><mi href="./4.1#p2.t1.r3">y</mi></mrow><mo>)</mo></mrow></mrow></mrow></mfrac></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E39.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./4.28#E2" title="(4.28.2) ‣ §4.28 Definitions and Periodicity ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="18px" altimg-valign="-2px" altimg-width="56px" alttext="\cosh\NVar{z}" display="inline"><mrow><mi href="./4.28#E2">cosh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: hyperbolic cosine function</a>,
<a href="./4.28#E1" title="(4.28.1) ‣ §4.28 Definitions and Periodicity ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="53px" alttext="\sinh\NVar{z}" display="inline"><mrow><mi href="./4.28#E1">sinh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: hyperbolic sine function</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./4.14#E4" title="(4.14.4) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="17px" altimg-valign="-2px" altimg-width="47px" alttext="\tan\NVar{z}" display="inline"><mrow><mi href="./4.14#E4">tan</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: tangent function</a>,
<a href="./4.1#p2.t1.r3" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./4.1#p2.t1.r3">x</mi></math>: real variable</a>,
<a href="./4.1#p2.t1.r3" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m67.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./4.1#p2.t1.r3">y</mi></math>: real variable</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.3.57</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E40" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.21.40</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E40.png" altimg-height="53px" altimg-valign="-21px" altimg-width="256px" alttext="\cot z=\frac{\sin\left(2x\right)-\mathrm{i}\sinh\left(2y\right)}{\cosh\left(2y%
\right)-\cos\left(2x\right)}." display="block"><mrow><mrow><mrow><mi href="./4.14#E7">cot</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>=</mo><mfrac><mrow><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><mi href="./4.1#p2.t1.r3">x</mi></mrow><mo>)</mo></mrow></mrow><mo>-</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mrow><mi href="./4.28#E1">sinh</mi><mo></mo><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><mi href="./4.1#p2.t1.r3">y</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mrow><mrow><mi href="./4.28#E2">cosh</mi><mo></mo><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><mi href="./4.1#p2.t1.r3">y</mi></mrow><mo>)</mo></mrow></mrow><mo>-</mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><mi href="./4.1#p2.t1.r3">x</mi></mrow><mo>)</mo></mrow></mrow></mrow></mfrac></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E40.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./4.14#E7" title="(4.14.7) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="17px" altimg-valign="-2px" altimg-width="44px" alttext="\cot\NVar{z}" display="inline"><mrow><mi href="./4.14#E7">cot</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cotangent function</a>,
<a href="./4.28#E2" title="(4.28.2) ‣ §4.28 Definitions and Periodicity ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="18px" altimg-valign="-2px" altimg-width="56px" alttext="\cosh\NVar{z}" display="inline"><mrow><mi href="./4.28#E2">cosh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: hyperbolic cosine function</a>,
<a href="./4.28#E1" title="(4.28.1) ‣ §4.28 Definitions and Periodicity ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="53px" alttext="\sinh\NVar{z}" display="inline"><mrow><mi href="./4.28#E1">sinh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: hyperbolic sine function</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./4.1#p2.t1.r3" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./4.1#p2.t1.r3">x</mi></math>: real variable</a>,
<a href="./4.1#p2.t1.r3" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m67.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./4.1#p2.t1.r3">y</mi></math>: real variable</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.3.58</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E41" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.21.41</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E41.png" altimg-height="35px" altimg-valign="-9px" altimg-width="526px" alttext="|\sin z|=({\sin^{2}}x+{\sinh^{2}}y)^{1/2}=\left(\tfrac{1}{2}\left(\cosh\left(2%
y\right)-\cos\left(2x\right)\right)\right)^{1/2}," display="block"><mrow><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo stretchy="false">|</mo></mrow><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mrow><mrow><msup><mi href="./4.14#E1">sin</mi><mn>2</mn></msup><mo></mo><mi href="./4.1#p2.t1.r3">x</mi></mrow><mo>+</mo><mrow><msup><mi href="./4.28#E1">sinh</mi><mn>2</mn></msup><mo></mo><mi href="./4.1#p2.t1.r3">y</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>=</mo><msup><mrow><mo>(</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mrow><mo>(</mo><mrow><mrow><mi href="./4.28#E2">cosh</mi><mo></mo><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><mi href="./4.1#p2.t1.r3">y</mi></mrow><mo>)</mo></mrow></mrow><mo>-</mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><mi href="./4.1#p2.t1.r3">x</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>)</mo></mrow></mrow><mo>)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E41.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./4.28#E2" title="(4.28.2) ‣ §4.28 Definitions and Periodicity ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="18px" altimg-valign="-2px" altimg-width="56px" alttext="\cosh\NVar{z}" display="inline"><mrow><mi href="./4.28#E2">cosh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: hyperbolic cosine function</a>,
<a href="./4.28#E1" title="(4.28.1) ‣ §4.28 Definitions and Periodicity ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="53px" alttext="\sinh\NVar{z}" display="inline"><mrow><mi href="./4.28#E1">sinh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: hyperbolic sine function</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./4.1#p2.t1.r3" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./4.1#p2.t1.r3">x</mi></math>: real variable</a>,
<a href="./4.1#p2.t1.r3" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m67.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./4.1#p2.t1.r3">y</mi></math>: real variable</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.3.59</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E42" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.21.42</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E42.png" altimg-height="35px" altimg-valign="-9px" altimg-width="527px" alttext="|\cos z|=({\cos^{2}}x+{\sinh^{2}}y)^{1/2}=\left(\tfrac{1}{2}(\cosh\left(2y%
\right)+\cos\left(2x\right))\right)^{1/2}," display="block"><mrow><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo stretchy="false">|</mo></mrow><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mrow><mrow><msup><mi href="./4.14#E2">cos</mi><mn>2</mn></msup><mo></mo><mi href="./4.1#p2.t1.r3">x</mi></mrow><mo>+</mo><mrow><msup><mi href="./4.28#E1">sinh</mi><mn>2</mn></msup><mo></mo><mi href="./4.1#p2.t1.r3">y</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>=</mo><msup><mrow><mo>(</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mi href="./4.28#E2">cosh</mi><mo></mo><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><mi href="./4.1#p2.t1.r3">y</mi></mrow><mo>)</mo></mrow></mrow><mo>+</mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><mi href="./4.1#p2.t1.r3">x</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E42.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./4.28#E2" title="(4.28.2) ‣ §4.28 Definitions and Periodicity ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="18px" altimg-valign="-2px" altimg-width="56px" alttext="\cosh\NVar{z}" display="inline"><mrow><mi href="./4.28#E2">cosh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: hyperbolic cosine function</a>,
<a href="./4.28#E1" title="(4.28.1) ‣ §4.28 Definitions and Periodicity ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="53px" alttext="\sinh\NVar{z}" display="inline"><mrow><mi href="./4.28#E1">sinh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: hyperbolic sine function</a>,
<a href="./4.1#p2.t1.r3" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./4.1#p2.t1.r3">x</mi></math>: real variable</a>,
<a href="./4.1#p2.t1.r3" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m67.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./4.1#p2.t1.r3">y</mi></math>: real variable</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.3.61</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E43" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.21.43</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E43.png" altimg-height="60px" altimg-valign="-21px" altimg-width="319px" alttext="|\tan z|=\left(\frac{\cosh\left(2y\right)-\cos\left(2x\right)}{\cosh\left(2y%
\right)+\cos\left(2x\right)}\right)^{1/2}." display="block"><mrow><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./4.14#E4">tan</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo stretchy="false">|</mo></mrow><mo>=</mo><msup><mrow><mo>(</mo><mfrac><mrow><mrow><mi href="./4.28#E2">cosh</mi><mo></mo><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><mi href="./4.1#p2.t1.r3">y</mi></mrow><mo>)</mo></mrow></mrow><mo>-</mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><mi href="./4.1#p2.t1.r3">x</mi></mrow><mo>)</mo></mrow></mrow></mrow><mrow><mrow><mi href="./4.28#E2">cosh</mi><mo></mo><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><mi href="./4.1#p2.t1.r3">y</mi></mrow><mo>)</mo></mrow></mrow><mo>+</mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><mi href="./4.1#p2.t1.r3">x</mi></mrow><mo>)</mo></mrow></mrow></mrow></mfrac><mo>)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow><mo>.</mo></mrow></math></td>
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<div id="E43.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./4.28#E2" title="(4.28.2) ‣ §4.28 Definitions and Periodicity ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="18px" altimg-valign="-2px" altimg-width="56px" alttext="\cosh\NVar{z}" display="inline"><mrow><mi href="./4.28#E2">cosh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: hyperbolic cosine function</a>,
<a href="./4.14#E4" title="(4.14.4) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="17px" altimg-valign="-2px" altimg-width="47px" alttext="\tan\NVar{z}" display="inline"><mrow><mi href="./4.14#E4">tan</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: tangent function</a>,
<a href="./4.1#p2.t1.r3" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./4.1#p2.t1.r3">x</mi></math>: real variable</a>,
<a href="./4.1#p2.t1.r3" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m67.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./4.1#p2.t1.r3">y</mi></math>: real variable</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.3.63</span></span>
</dd>
</dl>
</div>
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</td></tr>
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<title>DLMF: 4.23 Inverse Trigonometric Functions</title>
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<p class="ltx_p">The general values of the inverse trigonometric functions are defined by</p>
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<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.23.1</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m40.png" altimg-height="19px" altimg-valign="-2px" altimg-width="76px" alttext="\displaystyle\operatorname{Arcsin}z" display="inline"><mrow><mi href="./4.23#E1">Arcsin</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m7.png" altimg-height="53px" altimg-valign="-21px" altimg-width="161px" alttext="\displaystyle=\int_{0}^{z}\frac{\mathrm{d}t}{(1-t^{2})^{1/2}}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi href="./4.1#p2.t1.r4">z</mi></msubsup></mstyle><mstyle displaystyle="true"><mfrac><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><msup><mi>t</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mfrac></mstyle></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
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<div id="E1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m104.png" altimg-height="18px" altimg-valign="-2px" altimg-width="74px" alttext="\operatorname{Arcsin}\NVar{z}" display="inline"><mrow><mi href="./4.23#E1">Arcsin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: general arcsine function</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m95.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m161.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m90.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m173.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.4.1</span> (modified)</span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E2">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.23.2</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m36.png" altimg-height="19px" altimg-valign="-2px" altimg-width="78px" alttext="\displaystyle\operatorname{Arccos}z" display="inline"><mrow><mi href="./4.23#E2">Arccos</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m9.png" altimg-height="56px" altimg-valign="-21px" altimg-width="161px" alttext="\displaystyle=\int_{z}^{1}\frac{\mathrm{d}t}{(1-t^{2})^{1/2}}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi href="./4.1#p2.t1.r4">z</mi><mn>1</mn></msubsup></mstyle><mstyle displaystyle="true"><mfrac><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><msup><mi>t</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mfrac></mstyle></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
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<div id="E2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m99.png" altimg-height="18px" altimg-valign="-2px" altimg-width="76px" alttext="\operatorname{Arccos}\NVar{z}" display="inline"><mrow><mi href="./4.23#E2">Arccos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: general arccosine function</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m95.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m161.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m90.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m173.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.4.2</span> (modified)</span>
</dd>
</dl>
</div>
</div>
</td></tr>
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<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.23.3</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m41.png" altimg-height="19px" altimg-valign="-2px" altimg-width="80px" alttext="\displaystyle\operatorname{Arctan}z" display="inline"><mrow><mi href="./4.23#E3">Arctan</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m8.png" altimg-height="53px" altimg-valign="-20px" altimg-width="120px" alttext="\displaystyle=\int_{0}^{z}\frac{\mathrm{d}t}{1+t^{2}}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi href="./4.1#p2.t1.r4">z</mi></msubsup></mstyle><mstyle displaystyle="true"><mfrac><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow><mrow><mn>1</mn><mo>+</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac></mstyle></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m175.png" altimg-height="21px" altimg-valign="-6px" altimg-width="62px" alttext="z\neq\pm\mathrm{i}" display="inline"><mrow><mi href="./4.1#p2.t1.r4">z</mi><mo>≠</mo><mrow><mo>±</mo><mi mathvariant="normal">i</mi></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m106.png" altimg-height="18px" altimg-valign="-2px" altimg-width="78px" alttext="\operatorname{Arctan}\NVar{z}" display="inline"><mrow><mi href="./4.23#E3">Arctan</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: general arctangent function</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m95.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m161.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m90.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m173.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.4.3</span> (modified)</span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E4">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.23.4</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m38.png" altimg-height="19px" altimg-valign="-2px" altimg-width="77px" alttext="\displaystyle\operatorname{Arccsc}z" display="inline"><mrow><mi href="./4.23#E4">Arccsc</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m13.png" altimg-height="25px" altimg-valign="-7px" altimg-width="141px" alttext="\displaystyle=\operatorname{Arcsin}\left(1/z\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./4.23#E1">Arcsin</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>/</mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
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<div id="E4.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m102.png" altimg-height="18px" altimg-valign="-2px" altimg-width="75px" alttext="\operatorname{Arccsc}\NVar{z}" display="inline"><mrow><mi href="./4.23#E4">Arccsc</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: general arccosecant function</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./4.23#E1" title="(4.23.1) ‣ §4.23(i) General Definitions ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m104.png" altimg-height="18px" altimg-valign="-2px" altimg-width="74px" alttext="\operatorname{Arcsin}\NVar{z}" display="inline"><mrow><mi href="./4.23#E1">Arcsin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: general arcsine function</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m173.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.4.6</span> (modified)</span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E5">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.23.5</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m39.png" altimg-height="19px" altimg-valign="-2px" altimg-width="77px" alttext="\displaystyle\operatorname{Arcsec}z" display="inline"><mrow><mi href="./4.23#E5">Arcsec</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m11.png" altimg-height="25px" altimg-valign="-7px" altimg-width="143px" alttext="\displaystyle=\operatorname{Arccos}\left(1/z\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./4.23#E2">Arccos</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>/</mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E5.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m103.png" altimg-height="18px" altimg-valign="-2px" altimg-width="75px" alttext="\operatorname{Arcsec}\NVar{z}" display="inline"><mrow><mi href="./4.23#E5">Arcsec</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: general arcsecant function</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./4.23#E2" title="(4.23.2) ‣ §4.23(i) General Definitions ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m99.png" altimg-height="18px" altimg-valign="-2px" altimg-width="76px" alttext="\operatorname{Arccos}\NVar{z}" display="inline"><mrow><mi href="./4.23#E2">Arccos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: general arccosine function</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m173.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.4.7</span> (modified)</span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E6">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.23.6</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m37.png" altimg-height="19px" altimg-valign="-2px" altimg-width="78px" alttext="\displaystyle\operatorname{Arccot}z" display="inline"><mrow><mi href="./4.23#E6">Arccot</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m15.png" altimg-height="25px" altimg-valign="-7px" altimg-width="145px" alttext="\displaystyle=\operatorname{Arctan}\left(1/z\right)." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./4.23#E3">Arctan</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>/</mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E6.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m100.png" altimg-height="18px" altimg-valign="-2px" altimg-width="76px" alttext="\operatorname{Arccot}\NVar{z}" display="inline"><mrow><mi href="./4.23#E6">Arccot</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: general arccotangent function</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./4.23#E3" title="(4.23.3) ‣ §4.23(i) General Definitions ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m106.png" altimg-height="18px" altimg-valign="-2px" altimg-width="78px" alttext="\operatorname{Arctan}\NVar{z}" display="inline"><mrow><mi href="./4.23#E3">Arctan</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: general arctangent function</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m173.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.4.8</span> (modified)</span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
<p class="ltx_p">In () the integration paths may
not pass through either of the points <math class="ltx_Math" altimg="m158.png" altimg-height="18px" altimg-valign="-4px" altimg-width="64px" alttext="t=\pm 1" display="inline"><mrow><mi>t</mi><mo>=</mo><mrow><mo>±</mo><mn>1</mn></mrow></mrow></math>. The function
<math class="ltx_Math" altimg="m58.png" altimg-height="26px" altimg-valign="-7px" altimg-width="95px" alttext="(1-t^{2})^{1/2}" display="inline"><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><msup><mi>t</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></math> assumes its principal value when <math class="ltx_Math" altimg="m159.png" altimg-height="23px" altimg-valign="-7px" altimg-width="96px" alttext="t\in(-1,1)" display="inline"><mrow><mi>t</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mn>1</mn><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></mrow></math>; elsewhere on
the integration paths the branch is determined by continuity. In
() the integration path may not intersect <math class="ltx_Math" altimg="m145.png" altimg-height="18px" altimg-valign="-4px" altimg-width="27px" alttext="\pm i" display="inline"><mrow><mo>±</mo><mi mathvariant="normal">i</mi></mrow></math>. Each of
the six functions is a multivalued function of <math class="ltx_Math" altimg="m173.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>. <math class="ltx_Math" altimg="m107.png" altimg-height="18px" altimg-valign="-2px" altimg-width="78px" alttext="\operatorname{Arctan}z" display="inline"><mrow><mi href="./4.23#E3">Arctan</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></math> and
<math class="ltx_Math" altimg="m101.png" altimg-height="18px" altimg-valign="-2px" altimg-width="76px" alttext="\operatorname{Arccot}z" display="inline"><mrow><mi href="./4.23#E6">Arccot</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></math> have branch points at <math class="ltx_Math" altimg="m170.png" altimg-height="19px" altimg-valign="-4px" altimg-width="62px" alttext="z=\pm\mathrm{i}" display="inline"><mrow><mi href="./4.1#p2.t1.r4">z</mi><mo>=</mo><mrow><mo>±</mo><mi mathvariant="normal">i</mi></mrow></mrow></math>; the other four functions
have branch points at <math class="ltx_Math" altimg="m169.png" altimg-height="18px" altimg-valign="-4px" altimg-width="66px" alttext="z=\pm 1" display="inline"><mrow><mi href="./4.1#p2.t1.r4">z</mi><mo>=</mo><mrow><mo>±</mo><mn>1</mn></mrow></mrow></math>.</p>
</div>
</section>
<section id="ii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§4.23(ii) </span>Principal Values</h2>
<div id="SS2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd>
<span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m109.png" altimg-height="13px" altimg-valign="-2px" altimg-width="71px" alttext="\operatorname{arccos}\NVar{z}" display="inline"><mrow><mi href="./4.23#SS2.p1">arccos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: arccosine function</span>,
<span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m126.png" altimg-height="18px" altimg-valign="-2px" altimg-width="69px" alttext="\operatorname{arcsin}\NVar{z}" display="inline"><mrow><mi href="./4.23#SS2.p1">arcsin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: arcsine function</span> and
<span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m131.png" altimg-height="17px" altimg-valign="-2px" altimg-width="73px" alttext="\operatorname{arctan}\NVar{z}" display="inline"><mrow><mi href="./4.23#SS2.p1">arctan</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: arctangent function</span>
</dd>
<dt>Keywords:</dt>
<dd>
</dd>
</dl>
</div>
</div>
<div id="SS2.p1" class="ltx_para">
<p class="ltx_p">The <em class="ltx_emph ltx_font_italic">principal values</em> (or <em class="ltx_emph ltx_font_italic">principal branches</em>) of the inverse sine,
cosine, and tangent are obtained by introducing cuts in the <math class="ltx_Math" altimg="m173.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>-plane as
indicated in Figures ). The principal branches are denoted by <math class="ltx_Math" altimg="m129.png" altimg-height="18px" altimg-valign="-2px" altimg-width="69px" alttext="\operatorname{arcsin}z" display="inline"><mrow><mi href="./4.23#SS2.p1">arcsin</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></math>,
<math class="ltx_Math" altimg="m111.png" altimg-height="13px" altimg-valign="-2px" altimg-width="71px" alttext="\operatorname{arccos}z" display="inline"><mrow><mi href="./4.23#SS2.p1">arccos</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></math>, <math class="ltx_Math" altimg="m133.png" altimg-height="17px" altimg-valign="-2px" altimg-width="73px" alttext="\operatorname{arctan}z" display="inline"><mrow><mi href="./4.23#SS2.p1">arctan</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></math>, respectively.
Each is two-valued on the corresponding cuts,
and each is real on the part of the real
axis that remains after deleting the intersections with the corresponding cuts.</p>
</div>
<div id="SS2.p2" class="ltx_para">
<p class="ltx_p">The principal values of the inverse cosecant, secant, and cotangent are given
by</p>
<table id="EGx2" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E7">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.23.7</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m48.png" altimg-height="14px" altimg-valign="-2px" altimg-width="72px" alttext="\displaystyle\operatorname{arccsc}z" display="inline"><mrow><mi href="./4.23#E7">arccsc</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m20.png" altimg-height="25px" altimg-valign="-7px" altimg-width="136px" alttext="\displaystyle=\operatorname{arcsin}\left(1/z\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./4.23#SS2.p1">arcsin</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>/</mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E7.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m117.png" altimg-height="13px" altimg-valign="-2px" altimg-width="70px" alttext="\operatorname{arccsc}\NVar{z}" display="inline"><mrow><mi href="./4.23#E7">arccsc</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: arccosecant function</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./4.23#SS2.p1" title="§4.23(ii) Principal Values ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m126.png" altimg-height="18px" altimg-valign="-2px" altimg-width="69px" alttext="\operatorname{arcsin}\NVar{z}" display="inline"><mrow><mi href="./4.23#SS2.p1">arcsin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: arcsine function</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m173.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E8">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.23.8</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m50.png" altimg-height="14px" altimg-valign="-2px" altimg-width="72px" alttext="\displaystyle\operatorname{arcsec}z" display="inline"><mrow><mi href="./4.23#E8">arcsec</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m18.png" altimg-height="25px" altimg-valign="-7px" altimg-width="138px" alttext="\displaystyle=\operatorname{arccos}\left(1/z\right)." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./4.23#SS2.p1">arccos</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>/</mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E8.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m121.png" altimg-height="13px" altimg-valign="-2px" altimg-width="70px" alttext="\operatorname{arcsec}\NVar{z}" display="inline"><mrow><mi href="./4.23#E8">arcsec</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: arcsecant function</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./4.23#SS2.p1" title="§4.23(ii) Principal Values ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m109.png" altimg-height="13px" altimg-valign="-2px" altimg-width="71px" alttext="\operatorname{arccos}\NVar{z}" display="inline"><mrow><mi href="./4.23#SS2.p1">arccos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: arccosine function</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m173.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E9">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.23.9</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m46.png" altimg-height="18px" altimg-valign="-2px" altimg-width="73px" alttext="\displaystyle\operatorname{arccot}z" display="inline"><mrow><mi href="./4.23#E9">arccot</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m21.png" altimg-height="25px" altimg-valign="-7px" altimg-width="140px" alttext="\displaystyle=\operatorname{arctan}\left(1/z\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./4.23#SS2.p1">arctan</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>/</mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m175.png" altimg-height="21px" altimg-valign="-6px" altimg-width="62px" alttext="z\neq\pm\mathrm{i}" display="inline"><mrow><mi href="./4.1#p2.t1.r4">z</mi><mo>≠</mo><mrow><mo>±</mo><mi mathvariant="normal">i</mi></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E9.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m113.png" altimg-height="17px" altimg-valign="-2px" altimg-width="71px" alttext="\operatorname{arccot}\NVar{z}" display="inline"><mrow><mi href="./4.23#E9">arccot</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: arccotangent function</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./4.23#SS2.p1" title="§4.23(ii) Principal Values ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m131.png" altimg-height="17px" altimg-valign="-2px" altimg-width="73px" alttext="\operatorname{arctan}\NVar{z}" display="inline"><mrow><mi href="./4.23#SS2.p1">arctan</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: arctangent function</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m173.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
<p class="ltx_p">These functions are analytic in the cut plane depicted in Figures
</td>
</tr>
<tr id="F1.t1.r2" class="ltx_tr">
<td class="ltx_td ltx_align_center">(i) <math class="ltx_Math" altimg="m129.png" altimg-height="18px" altimg-valign="-2px" altimg-width="69px" alttext="\operatorname{arcsin}z" display="inline"><mrow><mi href="./4.23#SS2.p1">arcsin</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></math> and <math class="ltx_Math" altimg="m111.png" altimg-height="13px" altimg-valign="-2px" altimg-width="71px" alttext="\operatorname{arccos}z" display="inline"><mrow><mi href="./4.23#SS2.p1">arccos</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></math>
</td>
<td class="ltx_td ltx_align_center">(ii) <math class="ltx_Math" altimg="m133.png" altimg-height="17px" altimg-valign="-2px" altimg-width="73px" alttext="\operatorname{arctan}z" display="inline"><mrow><mi href="./4.23#SS2.p1">arctan</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></math>
</td>
<td class="ltx_td ltx_align_center">(iii) <math class="ltx_Math" altimg="m119.png" altimg-height="13px" altimg-valign="-2px" altimg-width="70px" alttext="\operatorname{arccsc}z" display="inline"><mrow><mi href="./4.23#E7">arccsc</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></math> and <math class="ltx_Math" altimg="m124.png" altimg-height="13px" altimg-valign="-2px" altimg-width="70px" alttext="\operatorname{arcsec}z" display="inline"><mrow><mi href="./4.23#E8">arcsec</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></math>
</td>
<td class="ltx_td ltx_align_center">(iv) <math class="ltx_Math" altimg="m115.png" altimg-height="17px" altimg-valign="-2px" altimg-width="71px" alttext="\operatorname{arccot}z" display="inline"><mrow><mi href="./4.23#E9">arccot</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></math>
</td>
</tr>
</tbody>
</table>
<figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 4.23.1: </span><math class="ltx_Math" altimg="m173.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>-plane. Branch cuts for the inverse trigonometric functions.
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.23#E7" title="(4.23.7) ‣ §4.23(ii) Principal Values ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m117.png" altimg-height="13px" altimg-valign="-2px" altimg-width="70px" alttext="\operatorname{arccsc}\NVar{z}" display="inline"><mrow><mi href="./4.23#E7">arccsc</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: arccosecant function</a>,
<a href="./4.23#SS2.p1" title="§4.23(ii) Principal Values ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m109.png" altimg-height="13px" altimg-valign="-2px" altimg-width="71px" alttext="\operatorname{arccos}\NVar{z}" display="inline"><mrow><mi href="./4.23#SS2.p1">arccos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: arccosine function</a>,
<a href="./4.23#E9" title="(4.23.9) ‣ §4.23(ii) Principal Values ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m113.png" altimg-height="17px" altimg-valign="-2px" altimg-width="71px" alttext="\operatorname{arccot}\NVar{z}" display="inline"><mrow><mi href="./4.23#E9">arccot</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: arccotangent function</a>,
<a href="./4.23#E8" title="(4.23.8) ‣ §4.23(ii) Principal Values ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m121.png" altimg-height="13px" altimg-valign="-2px" altimg-width="70px" alttext="\operatorname{arcsec}\NVar{z}" display="inline"><mrow><mi href="./4.23#E8">arcsec</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: arcsecant function</a>,
<a href="./4.23#SS2.p1" title="§4.23(ii) Principal Values ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m126.png" altimg-height="18px" altimg-valign="-2px" altimg-width="69px" alttext="\operatorname{arcsin}\NVar{z}" display="inline"><mrow><mi href="./4.23#SS2.p1">arcsin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: arcsine function</a>,
<a href="./4.23#SS2.p1" title="§4.23(ii) Principal Values ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m131.png" altimg-height="17px" altimg-valign="-2px" altimg-width="73px" alttext="\operatorname{arctan}\NVar{z}" display="inline"><mrow><mi href="./4.23#SS2.p1">arctan</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: arctangent function</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m173.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>Keywords:</dt>
<dd></dd>
</dl>
</div>
</div>
<div id="SS3.p1" class="ltx_para">
<table id="EGx3" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E10">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.23.10</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m51.png" altimg-height="25px" altimg-valign="-7px" altimg-width="102px" alttext="\displaystyle\operatorname{arcsin}\left(-z\right)" display="inline"><mrow><mi href="./4.23#SS2.p1">arcsin</mi><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m3.png" altimg-height="23px" altimg-valign="-6px" altimg-width="116px" alttext="\displaystyle=-\operatorname{arcsin}z," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo>-</mo><mrow><mi href="./4.23#SS2.p1">arcsin</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E10.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.23#SS2.p1" title="§4.23(ii) Principal Values ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m126.png" altimg-height="18px" altimg-valign="-2px" altimg-width="69px" alttext="\operatorname{arcsin}\NVar{z}" display="inline"><mrow><mi href="./4.23#SS2.p1">arcsin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: arcsine function</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m173.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.4.14</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E11">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.23.11</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m42.png" altimg-height="25px" altimg-valign="-7px" altimg-width="104px" alttext="\displaystyle\operatorname{arccos}\left(-z\right)" display="inline"><mrow><mi href="./4.23#SS2.p1">arccos</mi><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m22.png" altimg-height="19px" altimg-valign="-4px" altimg-width="136px" alttext="\displaystyle=\pi-\operatorname{arccos}z." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./3.12#E1">π</mi><mo>-</mo><mrow><mi href="./4.23#SS2.p1">arccos</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E11.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m143.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.23#SS2.p1" title="§4.23(ii) Principal Values ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m109.png" altimg-height="13px" altimg-valign="-2px" altimg-width="71px" alttext="\operatorname{arccos}\NVar{z}" display="inline"><mrow><mi href="./4.23#SS2.p1">arccos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: arccosine function</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m173.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.4.15</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E12">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.23.12</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m54.png" altimg-height="25px" altimg-valign="-7px" altimg-width="106px" alttext="\displaystyle\operatorname{arctan}\left(-z\right)" display="inline"><mrow><mi href="./4.23#SS2.p1">arctan</mi><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m4.png" altimg-height="22px" altimg-valign="-6px" altimg-width="120px" alttext="\displaystyle=-\operatorname{arctan}z," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo>-</mo><mrow><mi href="./4.23#SS2.p1">arctan</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m175.png" altimg-height="21px" altimg-valign="-6px" altimg-width="62px" alttext="z\neq\pm\mathrm{i}" display="inline"><mrow><mi href="./4.1#p2.t1.r4">z</mi><mo>≠</mo><mrow><mo>±</mo><mi mathvariant="normal">i</mi></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E12.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.23#SS2.p1" title="§4.23(ii) Principal Values ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m131.png" altimg-height="17px" altimg-valign="-2px" altimg-width="73px" alttext="\operatorname{arctan}\NVar{z}" display="inline"><mrow><mi href="./4.23#SS2.p1">arctan</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: arctangent function</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m173.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.4.16</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E13">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.23.13</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m47.png" altimg-height="25px" altimg-valign="-7px" altimg-width="103px" alttext="\displaystyle\operatorname{arccsc}\left(-z\right)" display="inline"><mrow><mi href="./4.23#E7">arccsc</mi><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m2.png" altimg-height="21px" altimg-valign="-6px" altimg-width="117px" alttext="\displaystyle=-\operatorname{arccsc}z," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo>-</mo><mrow><mi href="./4.23#E7">arccsc</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E13.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.23#E7" title="(4.23.7) ‣ §4.23(ii) Principal Values ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m117.png" altimg-height="13px" altimg-valign="-2px" altimg-width="70px" alttext="\operatorname{arccsc}\NVar{z}" display="inline"><mrow><mi href="./4.23#E7">arccsc</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: arccosecant function</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m173.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.4.17</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E14">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.23.14</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m49.png" altimg-height="25px" altimg-valign="-7px" altimg-width="103px" alttext="\displaystyle\operatorname{arcsec}\left(-z\right)" display="inline"><mrow><mi href="./4.23#E8">arcsec</mi><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m23.png" altimg-height="19px" altimg-valign="-4px" altimg-width="135px" alttext="\displaystyle=\pi-\operatorname{arcsec}z." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./3.12#E1">π</mi><mo>-</mo><mrow><mi href="./4.23#E8">arcsec</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E14.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m143.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.23#E8" title="(4.23.8) ‣ §4.23(ii) Principal Values ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m121.png" altimg-height="13px" altimg-valign="-2px" altimg-width="70px" alttext="\operatorname{arcsec}\NVar{z}" display="inline"><mrow><mi href="./4.23#E8">arcsec</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: arcsecant function</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m173.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.4.18</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E15">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.23.15</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m45.png" altimg-height="25px" altimg-valign="-7px" altimg-width="104px" alttext="\displaystyle\operatorname{arccot}\left(-z\right)" display="inline"><mrow><mi href="./4.23#E9">arccot</mi><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m1.png" altimg-height="22px" altimg-valign="-6px" altimg-width="118px" alttext="\displaystyle=-\operatorname{arccot}z," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo>-</mo><mrow><mi href="./4.23#E9">arccot</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m175.png" altimg-height="21px" altimg-valign="-6px" altimg-width="62px" alttext="z\neq\pm\mathrm{i}" display="inline"><mrow><mi href="./4.1#p2.t1.r4">z</mi><mo>≠</mo><mrow><mo>±</mo><mi mathvariant="normal">i</mi></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E15.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.23#E9" title="(4.23.9) ‣ §4.23(ii) Principal Values ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m113.png" altimg-height="17px" altimg-valign="-2px" altimg-width="71px" alttext="\operatorname{arccot}\NVar{z}" display="inline"><mrow><mi href="./4.23#E9">arccot</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: arccotangent function</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m173.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.4.19</span> (Early printings had an error.)</span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E16">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.23.16</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m44.png" altimg-height="14px" altimg-valign="-2px" altimg-width="73px" alttext="\displaystyle\operatorname{arccos}z" display="inline"><mrow><mi href="./4.23#SS2.p1">arccos</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m32.png" altimg-height="29px" altimg-valign="-9px" altimg-width="146px" alttext="\displaystyle=\tfrac{1}{2}\pi-\operatorname{arcsin}z," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo>-</mo><mrow><mi href="./4.23#SS2.p1">arcsin</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E16.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m143.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.23#SS2.p1" title="§4.23(ii) Principal Values ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m109.png" altimg-height="13px" altimg-valign="-2px" altimg-width="71px" alttext="\operatorname{arccos}\NVar{z}" display="inline"><mrow><mi href="./4.23#SS2.p1">arccos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: arccosine function</a>,
<a href="./4.23#SS2.p1" title="§4.23(ii) Principal Values ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m126.png" altimg-height="18px" altimg-valign="-2px" altimg-width="69px" alttext="\operatorname{arcsin}\NVar{z}" display="inline"><mrow><mi href="./4.23#SS2.p1">arcsin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: arcsine function</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m173.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.4.2</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E17">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.23.17</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m50.png" altimg-height="14px" altimg-valign="-2px" altimg-width="72px" alttext="\displaystyle\operatorname{arcsec}z" display="inline"><mrow><mi href="./4.23#E8">arcsec</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m31.png" altimg-height="29px" altimg-valign="-9px" altimg-width="147px" alttext="\displaystyle=\tfrac{1}{2}\pi-\operatorname{arccsc}z." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo>-</mo><mrow><mi href="./4.23#E7">arccsc</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E17.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m143.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.23#E7" title="(4.23.7) ‣ §4.23(ii) Principal Values ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m117.png" altimg-height="13px" altimg-valign="-2px" altimg-width="70px" alttext="\operatorname{arccsc}\NVar{z}" display="inline"><mrow><mi href="./4.23#E7">arccsc</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: arccosecant function</a>,
<a href="./4.23#E8" title="(4.23.8) ‣ §4.23(ii) Principal Values ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m121.png" altimg-height="13px" altimg-valign="-2px" altimg-width="70px" alttext="\operatorname{arcsec}\NVar{z}" display="inline"><mrow><mi href="./4.23#E8">arcsec</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: arcsecant function</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m173.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.4.9</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E18">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.23.18</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m46.png" altimg-height="18px" altimg-valign="-2px" altimg-width="73px" alttext="\displaystyle\operatorname{arccot}z" display="inline"><mrow><mi href="./4.23#E9">arccot</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m25.png" altimg-height="29px" altimg-valign="-9px" altimg-width="166px" alttext="\displaystyle=\pm\tfrac{1}{2}\pi-\operatorname{arctan}z," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mo>±</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow><mo>-</mo><mrow><mi href="./4.23#SS2.p1">arctan</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m78.png" altimg-height="21px" altimg-valign="-6px" altimg-width="65px" alttext="\Re z\gtrless 0" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>≷</mo><mn>0</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E18.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m143.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.23#E9" title="(4.23.9) ‣ §4.23(ii) Principal Values ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m113.png" altimg-height="17px" altimg-valign="-2px" altimg-width="71px" alttext="\operatorname{arccot}\NVar{z}" display="inline"><mrow><mi href="./4.23#E9">arccot</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: arccotangent function</a>,
<a href="./4.23#SS2.p1" title="§4.23(ii) Principal Values ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m131.png" altimg-height="17px" altimg-valign="-2px" altimg-width="73px" alttext="\operatorname{arctan}\NVar{z}" display="inline"><mrow><mi href="./4.23#SS2.p1">arctan</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: arctangent function</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m79.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m173.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.4.5</span> (The first printing had an error.)</span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
</div>
</section>
<section id="iv" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§4.23(iv) </span>Logarithmic Forms</h2>
<div id="SS4.info" class="ltx_metadata ltx_info">
), denote the right-hand side by <math class="ltx_Math" altimg="m141.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="\phi(z)" display="inline"><mrow><mi href="./4.12#p1">ϕ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./4.1#p2.t1.r4">z</mi><mo stretchy="false">)</mo></mrow></mrow></math>,
and the domain <math class="ltx_Math" altimg="m94.png" altimg-height="23px" altimg-valign="-7px" altimg-width="195px" alttext="\mathbb{C}\setminus(-\infty,-1]\cup[1,\infty)" display="inline"><mrow><mrow><mi href="./front/introduction#Sx4.p1.t1.r1">ℂ</mi><mo href="./front/introduction#Sx4.p2.t1.r19">∖</mo><mrow><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">(</mo><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mo href="./front/introduction#Sx4.p2.t1.r1">,</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">]</mo></mrow></mrow><mo href="./front/introduction#Sx4.p1.t1.r27">∪</mo><mrow><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">[</mo><mn>1</mn><mo href="./front/introduction#Sx4.p2.t1.r1">,</mo><mi mathvariant="normal">∞</mi><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">)</mo></mrow></mrow></math> by <math class="ltx_Math" altimg="m74.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./4.23#SS4.info">D</mi></math>.
If <math class="ltx_Math" altimg="m172.png" altimg-height="23px" altimg-valign="-7px" altimg-width="137px" alttext="z=x\in(-1,1)" display="inline"><mrow><mi href="./4.1#p2.t1.r4">z</mi><mo>=</mo><mi href="./4.1#p2.t1.r3">x</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mn>1</mn><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></mrow></math>, then <math class="ltx_Math" altimg="m142.png" altimg-height="26px" altimg-valign="-7px" altimg-width="183px" alttext="\phi^{\prime}(x)=(1-x^{2})^{-1/2}" display="inline"><mrow><mrow><msup><mi href="./4.12#p1">ϕ</mi><mo>′</mo></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./4.1#p2.t1.r3">x</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><msup><mi href="./4.1#p2.t1.r3">x</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></mrow></msup></mrow></math> and
<math class="ltx_Math" altimg="m140.png" altimg-height="23px" altimg-valign="-7px" altimg-width="78px" alttext="\phi(0)=0" display="inline"><mrow><mrow><mi href="./4.12#p1">ϕ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mn>0</mn></mrow></math>. Hence () with <math class="ltx_Math" altimg="m105.png" altimg-height="18px" altimg-valign="-2px" altimg-width="60px" alttext="\operatorname{Arcsin}" display="inline"><mi href="./4.23#E1">Arcsin</mi></math> replaced by <math class="ltx_Math" altimg="m127.png" altimg-height="18px" altimg-valign="-2px" altimg-width="55px" alttext="\operatorname{arcsin}" display="inline"><mi href="./4.23#SS2.p1">arcsin</mi></math>. We may now
extend () to the rest of <math class="ltx_Math" altimg="m74.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./4.23#SS4.info">D</mi></math> simply by showing that
<math class="ltx_Math" altimg="m141.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="\phi(z)" display="inline"><mrow><mi href="./4.12#p1">ϕ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./4.1#p2.t1.r4">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> is analytic on <math class="ltx_Math" altimg="m74.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./4.23#SS4.info">D</mi></math>; compare §. Since the
principal value of <math class="ltx_Math" altimg="m61.png" altimg-height="26px" altimg-valign="-7px" altimg-width="98px" alttext="(1-z^{2})^{1/2}" display="inline"><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><msup><mi href="./4.1#p2.t1.r4">z</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></math> is analytic on <math class="ltx_Math" altimg="m74.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./4.23#SS4.info">D</mi></math>, the only
possible singularities of <math class="ltx_Math" altimg="m141.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="\phi(z)" display="inline"><mrow><mi href="./4.12#p1">ϕ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./4.1#p2.t1.r4">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> occur on the branch cut of the
logarithm, that is, when <math class="ltx_Math" altimg="m60.png" altimg-height="26px" altimg-valign="-7px" altimg-width="189px" alttext="(1-z^{2})^{1/2}=-iz-t" display="inline"><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><msup><mi href="./4.1#p2.t1.r4">z</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>=</mo><mrow><mrow><mo>-</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mrow><mo>-</mo><mi>t</mi></mrow></mrow></math> with
<math class="ltx_Math" altimg="m160.png" altimg-height="23px" altimg-valign="-7px" altimg-width="88px" alttext="t\in[0,\infty)" display="inline"><mrow><mi>t</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mrow><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">[</mo><mn>0</mn><mo href="./front/introduction#Sx4.p2.t1.r1">,</mo><mi mathvariant="normal">∞</mi><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">)</mo></mrow></mrow></math>. By squaring the last equation we see that
<math class="ltx_Math" altimg="m59.png" altimg-height="26px" altimg-valign="-7px" altimg-width="140px" alttext="(1-z^{2})^{1/2}+iz" display="inline"><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><msup><mi href="./4.1#p2.t1.r4">z</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mrow></math> is real only when <math class="ltx_Math" altimg="m173.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math> lies on the imaginary axis,
and it is then positive.
The proofs of (</dd>
</dl>
</div>
</div>
<div id="Px1.p1" class="ltx_para">
<table id="E19" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.23.19</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E19.png" altimg-height="41px" altimg-valign="-15px" altimg-width="311px" alttext="\operatorname{arcsin}z=-i\ln\left((1-z^{2})^{1/2}+iz\right)," display="block"><mrow><mrow><mrow><mi href="./4.23#SS2.p1">arcsin</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>=</mo><mrow><mo>-</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><mo>(</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><msup><mi href="./4.1#p2.t1.r4">z</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m174.png" altimg-height="23px" altimg-valign="-7px" altimg-width="234px" alttext="z\in\mathbb{C}\setminus(-\infty,-1)\cup(1,\infty)" display="inline"><mrow><mi href="./4.1#p2.t1.r4">z</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mrow><mrow><mi href="./front/introduction#Sx4.p1.t1.r1">ℂ</mi><mo href="./front/introduction#Sx4.p2.t1.r19">∖</mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></mrow><mo href="./front/introduction#Sx4.p1.t1.r27">∪</mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mn>1</mn><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi mathvariant="normal">∞</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></mrow></mrow></math>;</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E19.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./front/introduction#Sx4.p1.t1.r1" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m93.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\mathbb{C}" display="inline"><mi href="./front/introduction#Sx4.p1.t1.r1">ℂ</mi></math>: complex plane</a>,
<a href="./front/introduction#Sx4.p1.t1.r9" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m88.png" altimg-height="15px" altimg-valign="-3px" altimg-width="18px" alttext="\in" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo></math>: element of</a>,
<a href="./4.23#SS2.p1" title="§4.23(ii) Principal Values ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m126.png" altimg-height="18px" altimg-valign="-2px" altimg-width="69px" alttext="\operatorname{arcsin}\NVar{z}" display="inline"><mrow><mi href="./4.23#SS2.p1">arcsin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: arcsine function</a>,
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m92.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a>,
<a href="./front/introduction#Sx4.p1.t1.r28" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m62.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="(\NVar{a},\NVar{b})" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi class="ltx_nvar">a</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi class="ltx_nvar">b</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math>: open interval</a>,
<a href="./front/introduction#Sx4.p2.t1.r19" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m149.png" altimg-height="23px" altimg-valign="-7px" altimg-width="14px" alttext="\setminus" display="inline"><mo href="./front/introduction#Sx4.p2.t1.r19">∖</mo></math>: set subtraction</a>,
<a href="./front/introduction#Sx4.p1.t1.r27" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="15px" altimg-valign="-2px" altimg-width="18px" alttext="\cup" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r27">∪</mo></math>: union</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m173.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">compare Figure (i). On the cuts</p>
<table id="EGx4" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E20">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.23.20</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m52.png" altimg-height="19px" altimg-valign="-2px" altimg-width="72px" alttext="\displaystyle\operatorname{arcsin}x" display="inline"><mrow><mi href="./4.23#SS2.p1">arcsin</mi><mo></mo><mi href="./4.1#p2.t1.r3">x</mi></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m33.png" altimg-height="41px" altimg-valign="-15px" altimg-width="270px" alttext="\displaystyle=\tfrac{1}{2}\pi\pm i\ln\left((x^{2}-1)^{1/2}+x\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo>±</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><mo>(</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><msup><mi href="./4.1#p2.t1.r3">x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>+</mo><mi href="./4.1#p2.t1.r3">x</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m164.png" altimg-height="23px" altimg-valign="-7px" altimg-width="92px" alttext="x\in[1,\infty)" display="inline"><mrow><mi href="./4.1#p2.t1.r3">x</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mrow><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">[</mo><mn>1</mn><mo href="./front/introduction#Sx4.p2.t1.r1">,</mo><mi mathvariant="normal">∞</mi><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">)</mo></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E20.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m143.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./front/introduction#Sx4.p2.t1.r1" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m76.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="[\NVar{a},\NVar{b})" display="inline"><mrow><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">[</mo><mi class="ltx_nvar">a</mi><mo href="./front/introduction#Sx4.p2.t1.r1">,</mo><mi class="ltx_nvar">b</mi><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">)</mo></mrow></math>: half-closed interval</a>,
<a href="./front/introduction#Sx4.p1.t1.r9" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m88.png" altimg-height="15px" altimg-valign="-3px" altimg-width="18px" alttext="\in" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo></math>: element of</a>,
<a href="./4.23#SS2.p1" title="§4.23(ii) Principal Values ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m126.png" altimg-height="18px" altimg-valign="-2px" altimg-width="69px" alttext="\operatorname{arcsin}\NVar{z}" display="inline"><mrow><mi href="./4.23#SS2.p1">arcsin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: arcsine function</a>,
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m92.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a> and
<a href="./4.1#p2.t1.r3" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m161.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./4.1#p2.t1.r3">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E21">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.23.21</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m52.png" altimg-height="19px" altimg-valign="-2px" altimg-width="72px" alttext="\displaystyle\operatorname{arcsin}x" display="inline"><mrow><mi href="./4.23#SS2.p1">arcsin</mi><mo></mo><mi href="./4.1#p2.t1.r3">x</mi></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m5.png" altimg-height="41px" altimg-valign="-15px" altimg-width="285px" alttext="\displaystyle=-\tfrac{1}{2}\pi\pm i\ln\left((x^{2}-1)^{1/2}-x\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow><mo>±</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><mo>(</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><msup><mi href="./4.1#p2.t1.r3">x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>-</mo><mi href="./4.1#p2.t1.r3">x</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m162.png" altimg-height="23px" altimg-valign="-7px" altimg-width="123px" alttext="x\in(-\infty,-1]" display="inline"><mrow><mi href="./4.1#p2.t1.r3">x</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mrow><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">(</mo><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mo href="./front/introduction#Sx4.p2.t1.r1">,</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">]</mo></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E21.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m143.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./front/introduction#Sx4.p1.t1.r9" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m88.png" altimg-height="15px" altimg-valign="-3px" altimg-width="18px" alttext="\in" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo></math>: element of</a>,
<a href="./4.23#SS2.p1" title="§4.23(ii) Principal Values ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m126.png" altimg-height="18px" altimg-valign="-2px" altimg-width="69px" alttext="\operatorname{arcsin}\NVar{z}" display="inline"><mrow><mi href="./4.23#SS2.p1">arcsin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: arcsine function</a>,
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m92.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a>,
<a href="./front/introduction#Sx4.p2.t1.r1" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="(\NVar{a},\NVar{b}]" display="inline"><mrow><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">(</mo><mi class="ltx_nvar">a</mi><mo href="./front/introduction#Sx4.p2.t1.r1">,</mo><mi class="ltx_nvar">b</mi><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">]</mo></mrow></math>: half-closed interval</a> and
<a href="./4.1#p2.t1.r3" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m161.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./4.1#p2.t1.r3">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
<p class="ltx_p">upper signs being taken on upper sides, and lower signs on lower sides.</p>
</div>
</section>
<section id="Px2" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Inverse Cosine</h3>
<div id="Px2.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px2.p1" class="ltx_para">
<table id="E22" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.23.22</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E22.png" altimg-height="41px" altimg-valign="-15px" altimg-width="347px" alttext="\operatorname{arccos}z=\tfrac{1}{2}\pi+i\ln\left((1-z^{2})^{1/2}+iz\right)," display="block"><mrow><mrow><mrow><mi href="./4.23#SS2.p1">arccos</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>=</mo><mrow><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><mo>(</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><msup><mi href="./4.1#p2.t1.r4">z</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m174.png" altimg-height="23px" altimg-valign="-7px" altimg-width="234px" alttext="z\in\mathbb{C}\setminus(-\infty,-1)\cup(1,\infty)" display="inline"><mrow><mi href="./4.1#p2.t1.r4">z</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mrow><mrow><mi href="./front/introduction#Sx4.p1.t1.r1">ℂ</mi><mo href="./front/introduction#Sx4.p2.t1.r19">∖</mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></mrow><mo href="./front/introduction#Sx4.p1.t1.r27">∪</mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mn>1</mn><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi mathvariant="normal">∞</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></mrow></mrow></math>;</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E22.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m143.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./front/introduction#Sx4.p1.t1.r1" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m93.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\mathbb{C}" display="inline"><mi href="./front/introduction#Sx4.p1.t1.r1">ℂ</mi></math>: complex plane</a>,
<a href="./front/introduction#Sx4.p1.t1.r9" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m88.png" altimg-height="15px" altimg-valign="-3px" altimg-width="18px" alttext="\in" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo></math>: element of</a>,
<a href="./4.23#SS2.p1" title="§4.23(ii) Principal Values ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m109.png" altimg-height="13px" altimg-valign="-2px" altimg-width="71px" alttext="\operatorname{arccos}\NVar{z}" display="inline"><mrow><mi href="./4.23#SS2.p1">arccos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: arccosine function</a>,
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m92.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a>,
<a href="./front/introduction#Sx4.p1.t1.r28" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m62.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="(\NVar{a},\NVar{b})" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi class="ltx_nvar">a</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi class="ltx_nvar">b</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math>: open interval</a>,
<a href="./front/introduction#Sx4.p2.t1.r19" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m149.png" altimg-height="23px" altimg-valign="-7px" altimg-width="14px" alttext="\setminus" display="inline"><mo href="./front/introduction#Sx4.p2.t1.r19">∖</mo></math>: set subtraction</a>,
<a href="./front/introduction#Sx4.p1.t1.r27" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="15px" altimg-valign="-2px" altimg-width="18px" alttext="\cup" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r27">∪</mo></math>: union</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m173.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">compare Figure (i). An equivalent definition is</p>
<table id="E23" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.23.23</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E23.png" altimg-height="65px" altimg-valign="-27px" altimg-width="425px" alttext="\operatorname{arccos}z=-2i\ln\left(\left(\frac{1+z}{2}\right)^{1/2}+i\left(%
\frac{1-z}{2}\right)^{1/2}\right)," display="block"><mrow><mrow><mrow><mi href="./4.23#SS2.p1">arccos</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>=</mo><mrow><mo>-</mo><mrow><mn>2</mn><mo></mo><mi mathvariant="normal">i</mi><mo></mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><mo>(</mo><mrow><msup><mrow><mo>(</mo><mfrac><mrow><mn>1</mn><mo>+</mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mn>2</mn></mfrac><mo>)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><msup><mrow><mo>(</mo><mfrac><mrow><mn>1</mn><mo>-</mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mn>2</mn></mfrac><mo>)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m174.png" altimg-height="23px" altimg-valign="-7px" altimg-width="234px" alttext="z\in\mathbb{C}\setminus(-\infty,-1)\cup(1,\infty)" display="inline"><mrow><mi href="./4.1#p2.t1.r4">z</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mrow><mrow><mi href="./front/introduction#Sx4.p1.t1.r1">ℂ</mi><mo href="./front/introduction#Sx4.p2.t1.r19">∖</mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></mrow><mo href="./front/introduction#Sx4.p1.t1.r27">∪</mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mn>1</mn><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi mathvariant="normal">∞</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></mrow></mrow></math>;</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E23.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./front/introduction#Sx4.p1.t1.r1" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m93.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\mathbb{C}" display="inline"><mi href="./front/introduction#Sx4.p1.t1.r1">ℂ</mi></math>: complex plane</a>,
<a href="./front/introduction#Sx4.p1.t1.r9" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m88.png" altimg-height="15px" altimg-valign="-3px" altimg-width="18px" alttext="\in" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo></math>: element of</a>,
<a href="./4.23#SS2.p1" title="§4.23(ii) Principal Values ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m109.png" altimg-height="13px" altimg-valign="-2px" altimg-width="71px" alttext="\operatorname{arccos}\NVar{z}" display="inline"><mrow><mi href="./4.23#SS2.p1">arccos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: arccosine function</a>,
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m92.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a>,
<a href="./front/introduction#Sx4.p1.t1.r28" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m62.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="(\NVar{a},\NVar{b})" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi class="ltx_nvar">a</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi class="ltx_nvar">b</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math>: open interval</a>,
<a href="./front/introduction#Sx4.p2.t1.r19" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m149.png" altimg-height="23px" altimg-valign="-7px" altimg-width="14px" alttext="\setminus" display="inline"><mo href="./front/introduction#Sx4.p2.t1.r19">∖</mo></math>: set subtraction</a>,
<a href="./front/introduction#Sx4.p1.t1.r27" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="15px" altimg-valign="-2px" altimg-width="18px" alttext="\cup" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r27">∪</mo></math>: union</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m173.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">see <cite class="ltx_cite ltx_citemacro_citet">Kahan ()</cite>.</p>
</div>
<div id="Px2.p2" class="ltx_para">
<p class="ltx_p">On the cuts</p>
<table id="EGx5" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E24">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.23.24</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m43.png" altimg-height="14px" altimg-valign="-2px" altimg-width="74px" alttext="\displaystyle\operatorname{arccos}x" display="inline"><mrow><mi href="./4.23#SS2.p1">arccos</mi><mo></mo><mi href="./4.1#p2.t1.r3">x</mi></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m10.png" altimg-height="41px" altimg-valign="-15px" altimg-width="236px" alttext="\displaystyle=\mp i\ln\left((x^{2}-1)^{1/2}+x\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo>∓</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><mo>(</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><msup><mi href="./4.1#p2.t1.r3">x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>+</mo><mi href="./4.1#p2.t1.r3">x</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m164.png" altimg-height="23px" altimg-valign="-7px" altimg-width="92px" alttext="x\in[1,\infty)" display="inline"><mrow><mi href="./4.1#p2.t1.r3">x</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mrow><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">[</mo><mn>1</mn><mo href="./front/introduction#Sx4.p2.t1.r1">,</mo><mi mathvariant="normal">∞</mi><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">)</mo></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E24.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./front/introduction#Sx4.p2.t1.r1" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m76.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="[\NVar{a},\NVar{b})" display="inline"><mrow><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">[</mo><mi class="ltx_nvar">a</mi><mo href="./front/introduction#Sx4.p2.t1.r1">,</mo><mi class="ltx_nvar">b</mi><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">)</mo></mrow></math>: half-closed interval</a>,
<a href="./front/introduction#Sx4.p1.t1.r9" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m88.png" altimg-height="15px" altimg-valign="-3px" altimg-width="18px" alttext="\in" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo></math>: element of</a>,
<a href="./4.23#SS2.p1" title="§4.23(ii) Principal Values ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m109.png" altimg-height="13px" altimg-valign="-2px" altimg-width="71px" alttext="\operatorname{arccos}\NVar{z}" display="inline"><mrow><mi href="./4.23#SS2.p1">arccos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: arccosine function</a>,
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m92.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a> and
<a href="./4.1#p2.t1.r3" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m161.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./4.1#p2.t1.r3">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E25">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.23.25</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m43.png" altimg-height="14px" altimg-valign="-2px" altimg-width="74px" alttext="\displaystyle\operatorname{arccos}x" display="inline"><mrow><mi href="./4.23#SS2.p1">arccos</mi><mo></mo><mi href="./4.1#p2.t1.r3">x</mi></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m24.png" altimg-height="41px" altimg-valign="-15px" altimg-width="257px" alttext="\displaystyle=\pi\mp i\ln\left((x^{2}-1)^{1/2}-x\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./3.12#E1">π</mi><mo>∓</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><mo>(</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><msup><mi href="./4.1#p2.t1.r3">x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>-</mo><mi href="./4.1#p2.t1.r3">x</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m162.png" altimg-height="23px" altimg-valign="-7px" altimg-width="123px" alttext="x\in(-\infty,-1]" display="inline"><mrow><mi href="./4.1#p2.t1.r3">x</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mrow><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">(</mo><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mo href="./front/introduction#Sx4.p2.t1.r1">,</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">]</mo></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E25.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m143.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./front/introduction#Sx4.p1.t1.r9" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m88.png" altimg-height="15px" altimg-valign="-3px" altimg-width="18px" alttext="\in" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo></math>: element of</a>,
<a href="./4.23#SS2.p1" title="§4.23(ii) Principal Values ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m109.png" altimg-height="13px" altimg-valign="-2px" altimg-width="71px" alttext="\operatorname{arccos}\NVar{z}" display="inline"><mrow><mi href="./4.23#SS2.p1">arccos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: arccosine function</a>,
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m92.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a>,
<a href="./front/introduction#Sx4.p2.t1.r1" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="(\NVar{a},\NVar{b}]" display="inline"><mrow><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">(</mo><mi class="ltx_nvar">a</mi><mo href="./front/introduction#Sx4.p2.t1.r1">,</mo><mi class="ltx_nvar">b</mi><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">]</mo></mrow></math>: half-closed interval</a> and
<a href="./4.1#p2.t1.r3" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m161.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./4.1#p2.t1.r3">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
<p class="ltx_p">the upper/lower signs corresponding to the upper/lower sides.</p>
</div>
</section>
<section id="Px3" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Inverse Tangent</h3>
<div id="Px3.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px3.p1" class="ltx_para">
<table id="E26" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.23.26</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E26.png" altimg-height="53px" altimg-valign="-21px" altimg-width="224px" alttext="\operatorname{arctan}z=\frac{i}{2}\ln\left(\frac{i+z}{i-z}\right)," display="block"><mrow><mrow><mrow><mi href="./4.23#SS2.p1">arctan</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>=</mo><mrow><mfrac><mi mathvariant="normal">i</mi><mn>2</mn></mfrac><mo></mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><mo>(</mo><mfrac><mrow><mi mathvariant="normal">i</mi><mo>+</mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mrow><mi mathvariant="normal">i</mi><mo>-</mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mfrac><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m168.png" altimg-height="23px" altimg-valign="-7px" altimg-width="246px" alttext="z/i\in\mathbb{C}\setminus(-\infty,-1]\cup[1,\infty)" display="inline"><mrow><mrow><mi href="./4.1#p2.t1.r4">z</mi><mo>/</mo><mi mathvariant="normal">i</mi></mrow><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mrow><mrow><mi href="./front/introduction#Sx4.p1.t1.r1">ℂ</mi><mo href="./front/introduction#Sx4.p2.t1.r19">∖</mo><mrow><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">(</mo><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mo href="./front/introduction#Sx4.p2.t1.r1">,</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">]</mo></mrow></mrow><mo href="./front/introduction#Sx4.p1.t1.r27">∪</mo><mrow><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">[</mo><mn>1</mn><mo href="./front/introduction#Sx4.p2.t1.r1">,</mo><mi mathvariant="normal">∞</mi><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">)</mo></mrow></mrow></mrow></math>;</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E26.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./front/introduction#Sx4.p2.t1.r1" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m76.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="[\NVar{a},\NVar{b})" display="inline"><mrow><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">[</mo><mi class="ltx_nvar">a</mi><mo href="./front/introduction#Sx4.p2.t1.r1">,</mo><mi class="ltx_nvar">b</mi><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">)</mo></mrow></math>: half-closed interval</a>,
<a href="./front/introduction#Sx4.p1.t1.r1" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m93.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\mathbb{C}" display="inline"><mi href="./front/introduction#Sx4.p1.t1.r1">ℂ</mi></math>: complex plane</a>,
<a href="./front/introduction#Sx4.p1.t1.r9" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m88.png" altimg-height="15px" altimg-valign="-3px" altimg-width="18px" alttext="\in" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo></math>: element of</a>,
<a href="./4.23#SS2.p1" title="§4.23(ii) Principal Values ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m131.png" altimg-height="17px" altimg-valign="-2px" altimg-width="73px" alttext="\operatorname{arctan}\NVar{z}" display="inline"><mrow><mi href="./4.23#SS2.p1">arctan</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: arctangent function</a>,
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m92.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a>,
<a href="./front/introduction#Sx4.p2.t1.r1" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="(\NVar{a},\NVar{b}]" display="inline"><mrow><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">(</mo><mi class="ltx_nvar">a</mi><mo href="./front/introduction#Sx4.p2.t1.r1">,</mo><mi class="ltx_nvar">b</mi><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">]</mo></mrow></math>: half-closed interval</a>,
<a href="./front/introduction#Sx4.p2.t1.r19" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m149.png" altimg-height="23px" altimg-valign="-7px" altimg-width="14px" alttext="\setminus" display="inline"><mo href="./front/introduction#Sx4.p2.t1.r19">∖</mo></math>: set subtraction</a>,
<a href="./front/introduction#Sx4.p1.t1.r27" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="15px" altimg-valign="-2px" altimg-width="18px" alttext="\cup" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r27">∪</mo></math>: union</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m173.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">compare Figure (ii). On the cuts</p>
<table id="E27" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.23.27</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E27.png" altimg-height="53px" altimg-valign="-21px" altimg-width="317px" alttext="\operatorname{arctan}\left(iy\right)=\pm\frac{1}{2}\pi+\frac{i}{2}\ln\left(%
\frac{y+1}{y-1}\right)," display="block"><mrow><mrow><mrow><mi href="./4.23#SS2.p1">arctan</mi><mo></mo><mrow><mo>(</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./4.1#p2.t1.r3">y</mi></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mo>±</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow><mo>+</mo><mrow><mfrac><mi mathvariant="normal">i</mi><mn>2</mn></mfrac><mo></mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><mo>(</mo><mfrac><mrow><mi href="./4.1#p2.t1.r3">y</mi><mo>+</mo><mn>1</mn></mrow><mrow><mi href="./4.1#p2.t1.r3">y</mi><mo>-</mo><mn>1</mn></mrow></mfrac><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m167.png" altimg-height="23px" altimg-valign="-7px" altimg-width="201px" alttext="y\in(-\infty,-1)\cup(1,\infty)" display="inline"><mrow><mi href="./4.1#p2.t1.r3">y</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mrow><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow><mo href="./front/introduction#Sx4.p1.t1.r27">∪</mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mn>1</mn><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi mathvariant="normal">∞</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E27.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m143.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./front/introduction#Sx4.p1.t1.r9" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m88.png" altimg-height="15px" altimg-valign="-3px" altimg-width="18px" alttext="\in" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo></math>: element of</a>,
<a href="./4.23#SS2.p1" title="§4.23(ii) Principal Values ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m131.png" altimg-height="17px" altimg-valign="-2px" altimg-width="73px" alttext="\operatorname{arctan}\NVar{z}" display="inline"><mrow><mi href="./4.23#SS2.p1">arctan</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: arctangent function</a>,
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m92.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a>,
<a href="./front/introduction#Sx4.p1.t1.r28" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m62.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="(\NVar{a},\NVar{b})" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi class="ltx_nvar">a</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi class="ltx_nvar">b</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math>: open interval</a>,
<a href="./front/introduction#Sx4.p1.t1.r27" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="15px" altimg-valign="-2px" altimg-width="18px" alttext="\cup" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r27">∪</mo></math>: union</a> and
<a href="./4.1#p2.t1.r3" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m166.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./4.1#p2.t1.r3">y</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">the upper/lower sign corresponding to the right/left side.</p>
</div>
</section>
<section id="Px4" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Other Inverse Functions</h3>
<div id="Px4.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px4.p1" class="ltx_para">
<p class="ltx_p">For the corresponding results for <math class="ltx_Math" altimg="m119.png" altimg-height="13px" altimg-valign="-2px" altimg-width="70px" alttext="\operatorname{arccsc}z" display="inline"><mrow><mi href="./4.23#E7">arccsc</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></math>, <math class="ltx_Math" altimg="m124.png" altimg-height="13px" altimg-valign="-2px" altimg-width="70px" alttext="\operatorname{arcsec}z" display="inline"><mrow><mi href="./4.23#E8">arcsec</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></math>, and <math class="ltx_Math" altimg="m115.png" altimg-height="17px" altimg-valign="-2px" altimg-width="71px" alttext="\operatorname{arccot}z" display="inline"><mrow><mi href="./4.23#E9">arccot</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></math>,
use (). Care needs to be taken on
the cuts, for example, if <math class="ltx_Math" altimg="m70.png" altimg-height="17px" altimg-valign="-3px" altimg-width="99px" alttext="0<x<\infty" display="inline"><mrow><mn>0</mn><mo><</mo><mi href="./4.1#p2.t1.r3">x</mi><mo><</mo><mi mathvariant="normal">∞</mi></mrow></math> then <math class="ltx_Math" altimg="m72.png" altimg-height="23px" altimg-valign="-7px" altimg-width="207px" alttext="1/(x+i0)=(1/x)-i0" display="inline"><mrow><mrow><mn>1</mn><mo>/</mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./4.1#p2.t1.r3">x</mi><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mn>0</mn></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>/</mo><mi href="./4.1#p2.t1.r3">x</mi></mrow><mo stretchy="false">)</mo></mrow><mo>-</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mn>0</mn></mrow></mrow></mrow></math>.</p>
</div>
</section>
</section>
<section id="v" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§4.23(v) </span>Fundamental Property</h2>
<div id="SS5.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="SS5.p1" class="ltx_para">
<p class="ltx_p">With <math class="ltx_Math" altimg="m157.png" altimg-height="18px" altimg-valign="-3px" altimg-width="53px" alttext="k\in\mathbb{Z}" display="inline"><mrow><mi href="./4.1#p2.t1.r1">k</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mi href="./front/introduction#Sx4.p2.t1.r20">ℤ</mi></mrow></math>, the general solutions of the equations</p>
<table id="EGx6" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E28">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.23.28</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m57.png" altimg-height="14px" altimg-valign="-2px" altimg-width="16px" alttext="\displaystyle z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m26.png" altimg-height="23px" altimg-valign="-6px" altimg-width="75px" alttext="\displaystyle=\sin w," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi>w</mi></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E28.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m150.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m173.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E29">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.23.29</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m57.png" altimg-height="14px" altimg-valign="-2px" altimg-width="16px" alttext="\displaystyle z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m6.png" altimg-height="18px" altimg-valign="-6px" altimg-width="77px" alttext="\displaystyle=\cos w," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi>w</mi></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E29.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m173.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E30">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.23.30</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m57.png" altimg-height="14px" altimg-valign="-2px" altimg-width="16px" alttext="\displaystyle z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m27.png" altimg-height="22px" altimg-valign="-6px" altimg-width="80px" alttext="\displaystyle=\tan w," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./4.14#E4">tan</mi><mo></mo><mi>w</mi></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E30.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.14#E4" title="(4.14.4) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m152.png" altimg-height="17px" altimg-valign="-2px" altimg-width="47px" alttext="\tan\NVar{z}" display="inline"><mrow><mi href="./4.14#E4">tan</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: tangent function</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m173.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
<p class="ltx_p">are respectively</p>
<table id="EGx7" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E31">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.23.31</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m56.png" altimg-height="14px" altimg-valign="-2px" altimg-width="21px" alttext="\displaystyle w" display="inline"><mi>w</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m14.png" altimg-height="28px" altimg-valign="-7px" altimg-width="295px" alttext="\displaystyle=\operatorname{Arcsin}z=(-1)^{k}\operatorname{arcsin}z+k\pi," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./4.23#E1">Arcsin</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>=</mo><mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./4.1#p2.t1.r1">k</mi></msup><mo></mo><mrow><mi href="./4.23#SS2.p1">arcsin</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mrow><mo>+</mo><mrow><mi href="./4.1#p2.t1.r1">k</mi><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E31.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m143.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.23#SS2.p1" title="§4.23(ii) Principal Values ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m126.png" altimg-height="18px" altimg-valign="-2px" altimg-width="69px" alttext="\operatorname{arcsin}\NVar{z}" display="inline"><mrow><mi href="./4.23#SS2.p1">arcsin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: arcsine function</a>,
<a href="./4.23#E1" title="(4.23.1) ‣ §4.23(i) General Definitions ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m104.png" altimg-height="18px" altimg-valign="-2px" altimg-width="74px" alttext="\operatorname{Arcsin}\NVar{z}" display="inline"><mrow><mi href="./4.23#E1">Arcsin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: general arcsine function</a>,
<a href="./4.1#p2.t1.r1" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m156.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./4.1#p2.t1.r1">k</mi></math>: integer</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m173.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.4.10</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E32">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.23.32</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m56.png" altimg-height="14px" altimg-valign="-2px" altimg-width="21px" alttext="\displaystyle w" display="inline"><mi>w</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m12.png" altimg-height="23px" altimg-valign="-6px" altimg-width="274px" alttext="\displaystyle=\operatorname{Arccos}z=\pm\operatorname{arccos}z+2k\pi," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./4.23#E2">Arccos</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>=</mo><mrow><mrow><mo>±</mo><mrow><mi href="./4.23#SS2.p1">arccos</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mrow><mo>+</mo><mrow><mn>2</mn><mo></mo><mi href="./4.1#p2.t1.r1">k</mi><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E32.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m143.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.23#SS2.p1" title="§4.23(ii) Principal Values ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m109.png" altimg-height="13px" altimg-valign="-2px" altimg-width="71px" alttext="\operatorname{arccos}\NVar{z}" display="inline"><mrow><mi href="./4.23#SS2.p1">arccos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: arccosine function</a>,
<a href="./4.23#E2" title="(4.23.2) ‣ §4.23(i) General Definitions ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m99.png" altimg-height="18px" altimg-valign="-2px" altimg-width="76px" alttext="\operatorname{Arccos}\NVar{z}" display="inline"><mrow><mi href="./4.23#E2">Arccos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: general arccosine function</a>,
<a href="./4.1#p2.t1.r1" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m156.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./4.1#p2.t1.r1">k</mi></math>: integer</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m173.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.4.11</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E33">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.23.33</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m56.png" altimg-height="14px" altimg-valign="-2px" altimg-width="21px" alttext="\displaystyle w" display="inline"><mi>w</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m16.png" altimg-height="23px" altimg-valign="-6px" altimg-width="249px" alttext="\displaystyle=\operatorname{Arctan}z=\operatorname{arctan}z+k\pi," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./4.23#E3">Arctan</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>=</mo><mrow><mrow><mi href="./4.23#SS2.p1">arctan</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>+</mo><mrow><mi href="./4.1#p2.t1.r1">k</mi><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m175.png" altimg-height="21px" altimg-valign="-6px" altimg-width="62px" alttext="z\neq\pm\mathrm{i}" display="inline"><mrow><mi href="./4.1#p2.t1.r4">z</mi><mo>≠</mo><mrow><mo>±</mo><mi mathvariant="normal">i</mi></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E33.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m143.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.23#SS2.p1" title="§4.23(ii) Principal Values ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m131.png" altimg-height="17px" altimg-valign="-2px" altimg-width="73px" alttext="\operatorname{arctan}\NVar{z}" display="inline"><mrow><mi href="./4.23#SS2.p1">arctan</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: arctangent function</a>,
<a href="./4.23#E3" title="(4.23.3) ‣ §4.23(i) General Definitions ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m106.png" altimg-height="18px" altimg-valign="-2px" altimg-width="78px" alttext="\operatorname{Arctan}\NVar{z}" display="inline"><mrow><mi href="./4.23#E3">Arctan</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: general arctangent function</a>,
<a href="./4.1#p2.t1.r1" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m156.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./4.1#p2.t1.r1">k</mi></math>: integer</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m173.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.4.12</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
</div>
</section>
<section id="vi" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§4.23(vi) </span>Real and Imaginary Parts</h2>
<div id="SS6.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="SS6.p1" class="ltx_para">
<table id="EGx8" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E34">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.23.34</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m53.png" altimg-height="19px" altimg-valign="-2px" altimg-width="71px" alttext="\displaystyle\operatorname{arcsin}z" display="inline"><mrow><mi href="./4.23#SS2.p1">arcsin</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m19.png" altimg-height="41px" altimg-valign="-15px" altimg-width="380px" alttext="\displaystyle=\operatorname{arcsin}\beta+\mathrm{i}\operatorname{sign}\left(y%
\right)\ln\left(\alpha+(\alpha^{2}-1)^{1/2}\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mi href="./4.23#SS2.p1">arcsin</mi><mo></mo><mi href="./4.23#E38">β</mi></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mrow><mi href="./front/introduction#Sx4.p2.t1.r18">sign</mi><mo></mo><mrow><mo>(</mo><mi href="./4.1#p2.t1.r3">y</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./4.23#E37">α</mi><mo>+</mo><msup><mrow><mo stretchy="false">(</mo><mrow><msup><mi href="./4.23#E37">α</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E34.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./front/introduction#Sx4.p1.t1.r29" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="[\NVar{a},\NVar{b}]" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">[</mo><mi class="ltx_nvar">a</mi><mo href="./front/introduction#Sx4.p1.t1.r29">,</mo><mi class="ltx_nvar">b</mi><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">]</mo></mrow></math>: closed interval</a>,
<a href="./front/introduction#Sx4.p1.t1.r9" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m88.png" altimg-height="15px" altimg-valign="-3px" altimg-width="18px" alttext="\in" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo></math>: element of</a>,
<a href="./4.23#SS2.p1" title="§4.23(ii) Principal Values ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m126.png" altimg-height="18px" altimg-valign="-2px" altimg-width="69px" alttext="\operatorname{arcsin}\NVar{z}" display="inline"><mrow><mi href="./4.23#SS2.p1">arcsin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: arcsine function</a>,
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m92.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a>,
<a href="./front/introduction#Sx4.p1.t1.r10" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m98.png" altimg-height="19px" altimg-valign="-3px" altimg-width="18px" alttext="\notin" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r10">∉</mo></math>: not an element of</a>,
<a href="./front/introduction#Sx4.p1.t1.r28" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m62.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="(\NVar{a},\NVar{b})" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi class="ltx_nvar">a</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi class="ltx_nvar">b</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math>: open interval</a>,
<a href="./front/introduction#Sx4.p2.t1.r18" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m138.png" altimg-height="21px" altimg-valign="-6px" altimg-width="53px" alttext="\operatorname{sign}\NVar{x}" display="inline"><mrow><mi href="./front/introduction#Sx4.p2.t1.r18">sign</mi><mo></mo><mi class="ltx_nvar">x</mi></mrow></math>: sign of <math class="ltx_Math" altimg="m161.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.1#p2.t1.r3" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m161.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./4.1#p2.t1.r3">x</mi></math>: real variable</a>,
<a href="./4.1#p2.t1.r3" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m166.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./4.1#p2.t1.r3">y</mi></math>: real variable</a>,
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m173.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>,
<a href="./4.23#E37" title="(4.23.37) ‣ §4.23(vi) Real and Imaginary Parts ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m80.png" altimg-height="13px" altimg-valign="-2px" altimg-width="17px" alttext="\alpha" display="inline"><mi href="./4.23#E37">α</mi></math></a> and
<a href="./4.23#E38" title="(4.23.38) ‣ §4.23(vi) Real and Imaginary Parts ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m81.png" altimg-height="21px" altimg-valign="-6px" altimg-width="17px" alttext="\beta" display="inline"><mi href="./4.23#E38">β</mi></math></a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.4.37</span> (with the general value.)</span>
</dd>
<dt>Errata (effective with 1.0.7):</dt>
<dd>
Originally the factor <math class="ltx_Math" altimg="m139.png" altimg-height="23px" altimg-valign="-7px" altimg-width="68px" alttext="\operatorname{sign}\left(y\right)" display="inline"><mrow><mi href="./front/introduction#Sx4.p2.t1.r18">sign</mi><mo></mo><mrow><mo>(</mo><mi href="./4.1#p2.t1.r3">y</mi><mo>)</mo></mrow></mrow></math> was missing from the second term on the right side
of this equation. Also, the originally stated condition <math class="ltx_Math" altimg="m163.png" altimg-height="23px" altimg-valign="-7px" altimg-width="95px" alttext="x\in[-1,1]" display="inline"><mrow><mi href="./4.1#p2.t1.r3">x</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">[</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo href="./front/introduction#Sx4.p1.t1.r29">,</mo><mn>1</mn><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">]</mo></mrow></mrow></math> for this equation,
stated on the line following (), was replaced with the more general
condition <math class="ltx_Math" altimg="m146.png" altimg-height="23px" altimg-valign="-7px" altimg-width="109px" alttext="\pm z\notin(1,\infty)" display="inline"><mrow><mrow><mo>±</mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo href="./front/introduction#Sx4.p1.t1.r10">∉</mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mn>1</mn><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi mathvariant="normal">∞</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></mrow></math>.
<p><span class="ltx_font_italic">Reported 2013-07-01 by Volker Thürey</span></p>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E35">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.23.35</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m44.png" altimg-height="14px" altimg-valign="-2px" altimg-width="73px" alttext="\displaystyle\operatorname{arccos}z" display="inline"><mrow><mi href="./4.23#SS2.p1">arccos</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m17.png" altimg-height="41px" altimg-valign="-15px" altimg-width="383px" alttext="\displaystyle=\operatorname{arccos}\beta-\mathrm{i}\operatorname{sign}\left(y%
\right)\ln\left(\alpha+(\alpha^{2}-1)^{1/2}\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mi href="./4.23#SS2.p1">arccos</mi><mo></mo><mi href="./4.23#E38">β</mi></mrow><mo>-</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mrow><mi href="./front/introduction#Sx4.p2.t1.r18">sign</mi><mo></mo><mrow><mo>(</mo><mi href="./4.1#p2.t1.r3">y</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./4.23#E37">α</mi><mo>+</mo><msup><mrow><mo stretchy="false">(</mo><mrow><msup><mi href="./4.23#E37">α</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E35.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./front/introduction#Sx4.p1.t1.r29" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="[\NVar{a},\NVar{b}]" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">[</mo><mi class="ltx_nvar">a</mi><mo href="./front/introduction#Sx4.p1.t1.r29">,</mo><mi class="ltx_nvar">b</mi><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">]</mo></mrow></math>: closed interval</a>,
<a href="./front/introduction#Sx4.p1.t1.r9" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m88.png" altimg-height="15px" altimg-valign="-3px" altimg-width="18px" alttext="\in" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo></math>: element of</a>,
<a href="./4.23#SS2.p1" title="§4.23(ii) Principal Values ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m109.png" altimg-height="13px" altimg-valign="-2px" altimg-width="71px" alttext="\operatorname{arccos}\NVar{z}" display="inline"><mrow><mi href="./4.23#SS2.p1">arccos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: arccosine function</a>,
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m92.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a>,
<a href="./front/introduction#Sx4.p1.t1.r10" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m98.png" altimg-height="19px" altimg-valign="-3px" altimg-width="18px" alttext="\notin" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r10">∉</mo></math>: not an element of</a>,
<a href="./front/introduction#Sx4.p1.t1.r28" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m62.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="(\NVar{a},\NVar{b})" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi class="ltx_nvar">a</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi class="ltx_nvar">b</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math>: open interval</a>,
<a href="./front/introduction#Sx4.p2.t1.r18" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m138.png" altimg-height="21px" altimg-valign="-6px" altimg-width="53px" alttext="\operatorname{sign}\NVar{x}" display="inline"><mrow><mi href="./front/introduction#Sx4.p2.t1.r18">sign</mi><mo></mo><mi class="ltx_nvar">x</mi></mrow></math>: sign of <math class="ltx_Math" altimg="m161.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.1#p2.t1.r3" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m161.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./4.1#p2.t1.r3">x</mi></math>: real variable</a>,
<a href="./4.1#p2.t1.r3" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m166.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./4.1#p2.t1.r3">y</mi></math>: real variable</a>,
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m173.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>,
<a href="./4.23#E37" title="(4.23.37) ‣ §4.23(vi) Real and Imaginary Parts ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m80.png" altimg-height="13px" altimg-valign="-2px" altimg-width="17px" alttext="\alpha" display="inline"><mi href="./4.23#E37">α</mi></math></a> and
<a href="./4.23#E38" title="(4.23.38) ‣ §4.23(vi) Real and Imaginary Parts ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m81.png" altimg-height="21px" altimg-valign="-6px" altimg-width="17px" alttext="\beta" display="inline"><mi href="./4.23#E38">β</mi></math></a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.4.38</span> (with the general value.)</span>
</dd>
<dt>Errata (effective with 1.0.7):</dt>
<dd>
Originally the factor <math class="ltx_Math" altimg="m139.png" altimg-height="23px" altimg-valign="-7px" altimg-width="68px" alttext="\operatorname{sign}\left(y\right)" display="inline"><mrow><mi href="./front/introduction#Sx4.p2.t1.r18">sign</mi><mo></mo><mrow><mo>(</mo><mi href="./4.1#p2.t1.r3">y</mi><mo>)</mo></mrow></mrow></math> was missing from the second term on the right side
of this equation. Also, the originally stated condition <math class="ltx_Math" altimg="m163.png" altimg-height="23px" altimg-valign="-7px" altimg-width="95px" alttext="x\in[-1,1]" display="inline"><mrow><mi href="./4.1#p2.t1.r3">x</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">[</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo href="./front/introduction#Sx4.p1.t1.r29">,</mo><mn>1</mn><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">]</mo></mrow></mrow></math> for this equation,
stated on the line following (), was replaced with the more general
condition <math class="ltx_Math" altimg="m146.png" altimg-height="23px" altimg-valign="-7px" altimg-width="109px" alttext="\pm z\notin(1,\infty)" display="inline"><mrow><mrow><mo>±</mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo href="./front/introduction#Sx4.p1.t1.r10">∉</mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mn>1</mn><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi mathvariant="normal">∞</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></mrow></math>.
<p><span class="ltx_font_italic">Reported 2013-07-01 by Volker Thürey</span></p>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E36">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.23.36</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m55.png" altimg-height="18px" altimg-valign="-2px" altimg-width="75px" alttext="\displaystyle\operatorname{arctan}z" display="inline"><mrow><mi href="./4.23#SS2.p1">arctan</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m30.png" altimg-height="54px" altimg-valign="-21px" altimg-width="458px" alttext="\displaystyle=\tfrac{1}{2}\operatorname{arctan}\left(\frac{2x}{1-x^{2}-y^{2}}%
\right)+\tfrac{1}{4}i\ln\left(\frac{x^{2}+(y+1)^{2}}{x^{2}+(y-1)^{2}}\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mrow><mi href="./4.23#SS2.p1">arctan</mi><mo></mo><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac><mrow><mn>2</mn><mo></mo><mi href="./4.1#p2.t1.r3">x</mi></mrow><mrow><mn>1</mn><mo>-</mo><msup><mi href="./4.1#p2.t1.r3">x</mi><mn>2</mn></msup><mo>-</mo><msup><mi href="./4.1#p2.t1.r3">y</mi><mn>2</mn></msup></mrow></mfrac></mstyle><mo>)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac><mo></mo><mi mathvariant="normal">i</mi><mo></mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac><mrow><msup><mi href="./4.1#p2.t1.r3">x</mi><mn>2</mn></msup><mo>+</mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./4.1#p2.t1.r3">y</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow><mrow><msup><mi href="./4.1#p2.t1.r3">x</mi><mn>2</mn></msup><mo>+</mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./4.1#p2.t1.r3">y</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow></mfrac></mstyle><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E36.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.23#SS2.p1" title="§4.23(ii) Principal Values ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m131.png" altimg-height="17px" altimg-valign="-2px" altimg-width="73px" alttext="\operatorname{arctan}\NVar{z}" display="inline"><mrow><mi href="./4.23#SS2.p1">arctan</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: arctangent function</a>,
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m92.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a>,
<a href="./4.1#p2.t1.r3" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m161.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./4.1#p2.t1.r3">x</mi></math>: real variable</a>,
<a href="./4.1#p2.t1.r3" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m166.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./4.1#p2.t1.r3">y</mi></math>: real variable</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m173.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.4.39</span> (with the general value.)</span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m171.png" altimg-height="21px" altimg-valign="-6px" altimg-width="93px" alttext="z=x+\mathrm{i}y" display="inline"><mrow><mi href="./4.1#p2.t1.r4">z</mi><mo>=</mo><mrow><mi href="./4.1#p2.t1.r3">x</mi><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./4.1#p2.t1.r3">y</mi></mrow></mrow></mrow></math> and <math class="ltx_Math" altimg="m146.png" altimg-height="23px" altimg-valign="-7px" altimg-width="109px" alttext="\pm z\notin(1,\infty)" display="inline"><mrow><mrow><mo>±</mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo href="./front/introduction#Sx4.p1.t1.r10">∉</mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mn>1</mn><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi mathvariant="normal">∞</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></mrow></math> in (), and <math class="ltx_Math" altimg="m91.png" altimg-height="23px" altimg-valign="-7px" altimg-width="62px" alttext="\left|z\right|<1" display="inline"><mrow><mrow><mo>|</mo><mi href="./4.1#p2.t1.r4">z</mi><mo>|</mo></mrow><mo><</mo><mn>1</mn></mrow></math> in (). Also,</p>
<table id="EGx9" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E37">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.23.37</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m34.png" altimg-height="14px" altimg-valign="-2px" altimg-width="19px" alttext="\displaystyle\alpha" display="inline"><mi href="./4.23#E37">α</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m28.png" altimg-height="35px" altimg-valign="-9px" altimg-width="402px" alttext="\displaystyle=\tfrac{1}{2}\left((x+1)^{2}+y^{2}\right)^{1/2}+\tfrac{1}{2}\left%
((x-1)^{2}+y^{2}\right)^{1/2}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><msup><mrow><mo>(</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./4.1#p2.t1.r3">x</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>+</mo><msup><mi href="./4.1#p2.t1.r3">y</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow><mo>+</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><msup><mrow><mo>(</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./4.1#p2.t1.r3">x</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>+</mo><msup><mi href="./4.1#p2.t1.r3">y</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E37.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text"><math class="ltx_Math" altimg="m80.png" altimg-height="13px" altimg-valign="-2px" altimg-width="17px" alttext="\alpha" display="inline"><mi href="./4.23#E37">α</mi></math> (locally)</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./4.1#p2.t1.r3" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m161.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./4.1#p2.t1.r3">x</mi></math>: real variable</a> and
<a href="./4.1#p2.t1.r3" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m166.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./4.1#p2.t1.r3">y</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E38">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.23.38</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m35.png" altimg-height="23px" altimg-valign="-6px" altimg-width="18px" alttext="\displaystyle\beta" display="inline"><mi href="./4.23#E38">β</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m29.png" altimg-height="35px" altimg-valign="-9px" altimg-width="402px" alttext="\displaystyle=\tfrac{1}{2}\left((x+1)^{2}+y^{2}\right)^{1/2}-\tfrac{1}{2}\left%
((x-1)^{2}+y^{2}\right)^{1/2}." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><msup><mrow><mo>(</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./4.1#p2.t1.r3">x</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>+</mo><msup><mi href="./4.1#p2.t1.r3">y</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><msup><mrow><mo>(</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./4.1#p2.t1.r3">x</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>+</mo><msup><mi href="./4.1#p2.t1.r3">y</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E38.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text"><math class="ltx_Math" altimg="m81.png" altimg-height="21px" altimg-valign="-6px" altimg-width="17px" alttext="\beta" display="inline"><mi href="./4.23#E38">β</mi></math> (locally)</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./4.1#p2.t1.r3" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m161.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./4.1#p2.t1.r3">x</mi></math>: real variable</a> and
<a href="./4.1#p2.t1.r3" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m166.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./4.1#p2.t1.r3">y</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
</div>
</section>
<section id="vii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§4.23(vii) </span>Special Values and Interrelations</h2>
<div id="SS7.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<figure id="T1" class="ltx_table">
<figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_table">Table 4.23.1: </span>Inverse trigonometric functions: principal values at 0, <math class="ltx_Math" altimg="m144.png" altimg-height="18px" altimg-valign="-4px" altimg-width="30px" alttext="\pm 1" display="inline"><mrow><mo>±</mo><mn>1</mn></mrow></math>, <math class="ltx_Math" altimg="m147.png" altimg-height="17px" altimg-valign="-4px" altimg-width="40px" alttext="\pm\infty" display="inline"><mrow><mo>±</mo><mi mathvariant="normal">∞</mi></mrow></math>.
</figcaption>
<table id="T1.t1" class="ltx_tabular ltx_centering ltx_guessed_headers ltx_align_middle">
<thead class="ltx_thead">
<tr id="T1.t1.r1" class="ltx_tr">
<th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m161.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./4.1#p2.t1.r3">x</mi></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m128.png" altimg-height="18px" altimg-valign="-2px" altimg-width="70px" alttext="\operatorname{arcsin}x" display="inline"><mrow><mi href="./4.23#SS2.p1">arcsin</mi><mo></mo><mi href="./4.1#p2.t1.r3">x</mi></mrow></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m110.png" altimg-height="13px" altimg-valign="-2px" altimg-width="72px" alttext="\operatorname{arccos}x" display="inline"><mrow><mi href="./4.23#SS2.p1">arccos</mi><mo></mo><mi href="./4.1#p2.t1.r3">x</mi></mrow></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m132.png" altimg-height="17px" altimg-valign="-2px" altimg-width="74px" alttext="\operatorname{arctan}x" display="inline"><mrow><mi href="./4.23#SS2.p1">arctan</mi><mo></mo><mi href="./4.1#p2.t1.r3">x</mi></mrow></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m118.png" altimg-height="13px" altimg-valign="-2px" altimg-width="71px" alttext="\operatorname{arccsc}x" display="inline"><mrow><mi href="./4.23#E7">arccsc</mi><mo></mo><mi href="./4.1#p2.t1.r3">x</mi></mrow></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m123.png" altimg-height="13px" altimg-valign="-2px" altimg-width="71px" alttext="\operatorname{arcsec}x" display="inline"><mrow><mi href="./4.23#E8">arcsec</mi><mo></mo><mi href="./4.1#p2.t1.r3">x</mi></mrow></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m114.png" altimg-height="17px" altimg-valign="-2px" altimg-width="72px" alttext="\operatorname{arccot}x" display="inline"><mrow><mi href="./4.23#E9">arccot</mi><mo></mo><mi href="./4.1#p2.t1.r3">x</mi></mrow></math></th>
</tr>
</thead>
<tbody class="ltx_tbody">
<tr id="T1.t1.r2" class="ltx_tr">
<td class="ltx_td ltx_align_center ltx_border_t" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m67.png" altimg-height="17px" altimg-valign="-4px" altimg-width="40px" alttext="-\infty" display="inline"><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow></math></td>
<td class="ltx_td ltx_align_center ltx_border_t" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;">–</td>
<td class="ltx_td ltx_align_center ltx_border_t" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;">–</td>
<td class="ltx_td ltx_align_center ltx_border_t" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m68.png" altimg-height="27px" altimg-valign="-9px" altimg-width="45px" alttext="-\tfrac{1}{2}\pi" display="inline"><mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow></math></td>
<td class="ltx_td ltx_align_center ltx_border_t" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m71.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_center ltx_border_t" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m154.png" altimg-height="27px" altimg-valign="-9px" altimg-width="29px" alttext="\tfrac{1}{2}\pi" display="inline"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow></math></td>
<td class="ltx_td ltx_align_center ltx_border_t" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m71.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
</tr>
<tr id="T1.t1.r3" class="ltx_tr">
<td class="ltx_td ltx_align_center" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m64.png" altimg-height="18px" altimg-valign="-4px" altimg-width="30px" alttext="-1" display="inline"><mrow><mo>-</mo><mn>1</mn></mrow></math></td>
<td class="ltx_td ltx_align_center" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m68.png" altimg-height="27px" altimg-valign="-9px" altimg-width="45px" alttext="-\tfrac{1}{2}\pi" display="inline"><mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow></math></td>
<td class="ltx_td ltx_align_center" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m143.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math></td>
<td class="ltx_td ltx_align_center" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m69.png" altimg-height="27px" altimg-valign="-9px" altimg-width="45px" alttext="-\tfrac{1}{4}\pi" display="inline"><mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow></math></td>
<td class="ltx_td ltx_align_center" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m68.png" altimg-height="27px" altimg-valign="-9px" altimg-width="45px" alttext="-\tfrac{1}{2}\pi" display="inline"><mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow></math></td>
<td class="ltx_td ltx_align_center" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m143.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math></td>
<td class="ltx_td ltx_align_center" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m69.png" altimg-height="27px" altimg-valign="-9px" altimg-width="45px" alttext="-\tfrac{1}{4}\pi" display="inline"><mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow></math></td>
</tr>
<tr id="T1.t1.r4" class="ltx_tr">
<td class="ltx_td ltx_align_center" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m71.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_center" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m71.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_center" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m154.png" altimg-height="27px" altimg-valign="-9px" altimg-width="29px" alttext="\tfrac{1}{2}\pi" display="inline"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow></math></td>
<td class="ltx_td ltx_align_center" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m71.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_center" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;">–</td>
<td class="ltx_td ltx_align_center" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;">–</td>
<td class="ltx_td ltx_align_center" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m97.png" altimg-height="27px" altimg-valign="-9px" altimg-width="45px" alttext="\mp\tfrac{1}{2}\pi" display="inline"><mrow><mo>∓</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow></math></td>
</tr>
<tr id="T1.t1.r5" class="ltx_tr">
<td class="ltx_td ltx_align_center" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m73.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="1" display="inline"><mn>1</mn></math></td>
<td class="ltx_td ltx_align_center" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m154.png" altimg-height="27px" altimg-valign="-9px" altimg-width="29px" alttext="\tfrac{1}{2}\pi" display="inline"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow></math></td>
<td class="ltx_td ltx_align_center" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m71.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_center" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m155.png" altimg-height="27px" altimg-valign="-9px" altimg-width="29px" alttext="\tfrac{1}{4}\pi" display="inline"><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow></math></td>
<td class="ltx_td ltx_align_center" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m154.png" altimg-height="27px" altimg-valign="-9px" altimg-width="29px" alttext="\tfrac{1}{2}\pi" display="inline"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow></math></td>
<td class="ltx_td ltx_align_center" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m71.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_center" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m155.png" altimg-height="27px" altimg-valign="-9px" altimg-width="29px" alttext="\tfrac{1}{4}\pi" display="inline"><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow></math></td>
</tr>
<tr id="T1.t1.r6" class="ltx_tr">
<td class="ltx_td ltx_align_center ltx_border_b" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m89.png" altimg-height="13px" altimg-valign="-2px" altimg-width="24px" alttext="\infty" display="inline"><mi mathvariant="normal">∞</mi></math></td>
<td class="ltx_td ltx_align_center ltx_border_b" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;">–</td>
<td class="ltx_td ltx_align_center ltx_border_b" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;">–</td>
<td class="ltx_td ltx_align_center ltx_border_b" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m154.png" altimg-height="27px" altimg-valign="-9px" altimg-width="29px" alttext="\tfrac{1}{2}\pi" display="inline"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow></math></td>
<td class="ltx_td ltx_align_center ltx_border_b" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m71.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_center ltx_border_b" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m154.png" altimg-height="27px" altimg-valign="-9px" altimg-width="29px" alttext="\tfrac{1}{2}\pi" display="inline"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow></math></td>
<td class="ltx_td ltx_align_center ltx_border_b" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m71.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math></td>
</tr>
</tbody>
</table>
<div id="T1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m143.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.23#E7" title="(4.23.7) ‣ §4.23(ii) Principal Values ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m117.png" altimg-height="13px" altimg-valign="-2px" altimg-width="70px" alttext="\operatorname{arccsc}\NVar{z}" display="inline"><mrow><mi href="./4.23#E7">arccsc</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: arccosecant function</a>,
<a href="./4.23#SS2.p1" title="§4.23(ii) Principal Values ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m109.png" altimg-height="13px" altimg-valign="-2px" altimg-width="71px" alttext="\operatorname{arccos}\NVar{z}" display="inline"><mrow><mi href="./4.23#SS2.p1">arccos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: arccosine function</a>,
<a href="./4.23#E9" title="(4.23.9) ‣ §4.23(ii) Principal Values ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m113.png" altimg-height="17px" altimg-valign="-2px" altimg-width="71px" alttext="\operatorname{arccot}\NVar{z}" display="inline"><mrow><mi href="./4.23#E9">arccot</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: arccotangent function</a>,
<a href="./4.23#E8" title="(4.23.8) ‣ §4.23(ii) Principal Values ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m121.png" altimg-height="13px" altimg-valign="-2px" altimg-width="70px" alttext="\operatorname{arcsec}\NVar{z}" display="inline"><mrow><mi href="./4.23#E8">arcsec</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: arcsecant function</a>,
<a href="./4.23#SS2.p1" title="§4.23(ii) Principal Values ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m126.png" altimg-height="18px" altimg-valign="-2px" altimg-width="69px" alttext="\operatorname{arcsin}\NVar{z}" display="inline"><mrow><mi href="./4.23#SS2.p1">arcsin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: arcsine function</a>,
<a href="./4.23#SS2.p1" title="§4.23(ii) Principal Values ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m131.png" altimg-height="17px" altimg-valign="-2px" altimg-width="73px" alttext="\operatorname{arctan}\NVar{z}" display="inline"><mrow><mi href="./4.23#SS2.p1">arctan</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: arctangent function</a> and
<a href="./4.1#p2.t1.r3" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m161.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./4.1#p2.t1.r3">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</figure>
<div id="SS7.p1" class="ltx_para">
<p class="ltx_p">For interrelations see Table . For example, from the heading
and last entry in the penultimate column we have <math class="ltx_Math" altimg="m122.png" altimg-height="28px" altimg-valign="-9px" altimg-width="278px" alttext="\operatorname{arcsec}a=\operatorname{arccot}\left((a^{2}-1)^{-1/2}\right)" display="inline"><mrow><mrow><mi href="./4.23#E8">arcsec</mi><mo></mo><mi href="./4.1#p2.t1.r2">a</mi></mrow><mo>=</mo><mrow><mi href="./4.23#E9">arccot</mi><mo></mo><mrow><mo>(</mo><msup><mrow><mo stretchy="false">(</mo><mrow><msup><mi href="./4.1#p2.t1.r2">a</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></mrow></msup><mo>)</mo></mrow></mrow></mrow></math>.</p>
</div>
</section>
<section id="viii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§4.23(viii) </span>Gudermannian Function</h2>
<div id="SS8.info" class="ltx_metadata ltx_info">
) may be verified
by differentiation plus comparison of values as <math class="ltx_Math" altimg="m165.png" altimg-height="17px" altimg-valign="-2px" altimg-width="57px" alttext="x\to 0" display="inline"><mrow><mi href="./4.1#p2.t1.r3">x</mi><mo>→</mo><mn>0</mn></mrow></math>.</dd>
</dl>
</div>
</div>
<div id="SS8.p1" class="ltx_para">
<p class="ltx_p">The <em class="ltx_emph ltx_font_italic">Gudermannian</em> <math class="ltx_Math" altimg="m136.png" altimg-height="23px" altimg-valign="-7px" altimg-width="56px" alttext="\operatorname{gd}\left(x\right)" display="inline"><mrow><mi href="./4.23#E39">gd</mi><mo></mo><mrow><mo>(</mo><mi href="./4.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math> is defined by</p>
<table id="E39" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.23.39</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E39.png" altimg-height="53px" altimg-valign="-20px" altimg-width="188px" alttext="\operatorname{gd}\left(x\right)=\int_{0}^{x}\operatorname{sech}t\mathrm{d}t," display="block"><mrow><mrow><mrow><mi href="./4.23#E39">gd</mi><mo></mo><mrow><mo>(</mo><mi href="./4.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi href="./4.1#p2.t1.r3">x</mi></msubsup><mrow><mrow><mi href="./4.28#E6">sech</mi><mo></mo><mi>t</mi></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m66.png" altimg-height="17px" altimg-valign="-4px" altimg-width="124px" alttext="-\infty<x<\infty" display="inline"><mrow><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mo><</mo><mi href="./4.1#p2.t1.r3">x</mi><mo><</mo><mi mathvariant="normal">∞</mi></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E39.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m135.png" altimg-height="21px" altimg-valign="-6px" altimg-width="40px" alttext="\operatorname{gd}\NVar{x}" display="inline"><mrow><mi href="./4.23#E39">gd</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r3">x</mi></mrow></math>: Gudermannian function</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m95.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m161.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.28#E6" title="(4.28.6) ‣ §4.28 Definitions and Periodicity ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m137.png" altimg-height="18px" altimg-valign="-2px" altimg-width="54px" alttext="\operatorname{sech}\NVar{z}" display="inline"><mrow><mi href="./4.28#E6">sech</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: hyperbolic secant function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m90.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a> and
<a href="./4.1#p2.t1.r3" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m161.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./4.1#p2.t1.r3">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Equivalently,</p>
<table id="E40" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.23.40</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_Math ltx_eqn_cell ltx_align_center"><math class="ltx_Math" alttext="\operatorname{gd}\left(x\right)=2\operatorname{arctan}\left(e^{x}\right)-%
\tfrac{1}{2}\pi\\
=\operatorname{arcsin}\left(\tanh x\right)=\operatorname{arccsc}\left(\coth x%
\right)\\
=\operatorname{arccos}\left(\operatorname{sech}x\right)=\operatorname{arcsec}%
\left(\cosh x\right)\\
=\operatorname{arctan}\left(\sinh x\right)=\operatorname{arccot}\left(%
\operatorname{csch}x\right)." display="block"><mrow><mtable align="baseline 1" columnalign="left" columnspacing="0.1em" displaystyle="true"><mtr><mtd><mrow><mi href="./4.23#E39">gd</mi><mo></mo><mrow><mo>(</mo><mi href="./4.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mtd><mtd><mo>=</mo><mrow><mrow><mn>2</mn><mo></mo><mrow><mi href="./4.23#SS2.p1">arctan</mi><mo></mo><mrow><mo>(</mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mi href="./4.1#p2.t1.r3">x</mi></msup><mo>)</mo></mrow></mrow></mrow><mo>-</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow><mo>=</mo><mrow><mi href="./4.23#SS2.p1">arcsin</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./4.28#E4">tanh</mi><mo></mo><mi href="./4.1#p2.t1.r3">x</mi></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mi href="./4.23#E7">arccsc</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./4.28#E7">coth</mi><mo></mo><mi href="./4.1#p2.t1.r3">x</mi></mrow><mo>)</mo></mrow></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo><mrow><mi href="./4.23#SS2.p1">arccos</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./4.28#E6">sech</mi><mo></mo><mi href="./4.1#p2.t1.r3">x</mi></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mi href="./4.23#E8">arcsec</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./4.28#E2">cosh</mi><mo></mo><mi href="./4.1#p2.t1.r3">x</mi></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mi href="./4.23#SS2.p1">arctan</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./4.28#E1">sinh</mi><mo></mo><mi href="./4.1#p2.t1.r3">x</mi></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mi href="./4.23#E9">arccot</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./4.28#E5">csch</mi><mo></mo><mi href="./4.1#p2.t1.r3">x</mi></mrow><mo>)</mo></mrow></mrow><mo>.</mo></mtd></mtr></mtable></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E40.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.23#E39" title="(4.23.39) ‣ §4.23(viii) Gudermannian Function ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m135.png" altimg-height="21px" altimg-valign="-6px" altimg-width="40px" alttext="\operatorname{gd}\NVar{x}" display="inline"><mrow><mi href="./4.23#E39">gd</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r3">x</mi></mrow></math>: Gudermannian function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m143.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m96.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./4.28#E5" title="(4.28.5) ‣ §4.28 Definitions and Periodicity ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m134.png" altimg-height="18px" altimg-valign="-2px" altimg-width="54px" alttext="\operatorname{csch}\NVar{z}" display="inline"><mrow><mi href="./4.28#E5">csch</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: hyperbolic cosecant function</a>,
<a href="./4.28#E2" title="(4.28.2) ‣ §4.28 Definitions and Periodicity ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m83.png" altimg-height="18px" altimg-valign="-2px" altimg-width="56px" alttext="\cosh\NVar{z}" display="inline"><mrow><mi href="./4.28#E2">cosh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: hyperbolic cosine function</a>,
<a href="./4.28#E7" title="(4.28.7) ‣ §4.28 Definitions and Periodicity ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="18px" altimg-valign="-2px" altimg-width="55px" alttext="\coth\NVar{z}" display="inline"><mrow><mi href="./4.28#E7">coth</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: hyperbolic cotangent function</a>,
<a href="./4.28#E6" title="(4.28.6) ‣ §4.28 Definitions and Periodicity ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m137.png" altimg-height="18px" altimg-valign="-2px" altimg-width="54px" alttext="\operatorname{sech}\NVar{z}" display="inline"><mrow><mi href="./4.28#E6">sech</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: hyperbolic secant function</a>,
<a href="./4.28#E1" title="(4.28.1) ‣ §4.28 Definitions and Periodicity ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m151.png" altimg-height="18px" altimg-valign="-2px" altimg-width="53px" alttext="\sinh\NVar{z}" display="inline"><mrow><mi href="./4.28#E1">sinh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: hyperbolic sine function</a>,
<a href="./4.28#E4" title="(4.28.4) ‣ §4.28 Definitions and Periodicity ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m153.png" altimg-height="18px" altimg-valign="-2px" altimg-width="58px" alttext="\tanh\NVar{z}" display="inline"><mrow><mi href="./4.28#E4">tanh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: hyperbolic tangent function</a>,
<a href="./4.23#E7" title="(4.23.7) ‣ §4.23(ii) Principal Values ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m117.png" altimg-height="13px" altimg-valign="-2px" altimg-width="70px" alttext="\operatorname{arccsc}\NVar{z}" display="inline"><mrow><mi href="./4.23#E7">arccsc</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: arccosecant function</a>,
<a href="./4.23#SS2.p1" title="§4.23(ii) Principal Values ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m109.png" altimg-height="13px" altimg-valign="-2px" altimg-width="71px" alttext="\operatorname{arccos}\NVar{z}" display="inline"><mrow><mi href="./4.23#SS2.p1">arccos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: arccosine function</a>,
<a href="./4.23#E9" title="(4.23.9) ‣ §4.23(ii) Principal Values ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m113.png" altimg-height="17px" altimg-valign="-2px" altimg-width="71px" alttext="\operatorname{arccot}\NVar{z}" display="inline"><mrow><mi href="./4.23#E9">arccot</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: arccotangent function</a>,
<a href="./4.23#E8" title="(4.23.8) ‣ §4.23(ii) Principal Values ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m121.png" altimg-height="13px" altimg-valign="-2px" altimg-width="70px" alttext="\operatorname{arcsec}\NVar{z}" display="inline"><mrow><mi href="./4.23#E8">arcsec</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: arcsecant function</a>,
<a href="./4.23#SS2.p1" title="§4.23(ii) Principal Values ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m126.png" altimg-height="18px" altimg-valign="-2px" altimg-width="69px" alttext="\operatorname{arcsin}\NVar{z}" display="inline"><mrow><mi href="./4.23#SS2.p1">arcsin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: arcsine function</a>,
<a href="./4.23#SS2.p1" title="§4.23(ii) Principal Values ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m131.png" altimg-height="17px" altimg-valign="-2px" altimg-width="73px" alttext="\operatorname{arctan}\NVar{z}" display="inline"><mrow><mi href="./4.23#SS2.p1">arctan</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: arctangent function</a> and
<a href="./4.1#p2.t1.r3" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m161.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./4.1#p2.t1.r3">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS8.p2" class="ltx_para">
<p class="ltx_p">The inverse Gudermannian function is given by
</p>
<table id="E41" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.23.41</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E41.png" altimg-height="53px" altimg-valign="-20px" altimg-width="198px" alttext="{\operatorname{gd}^{-1}}\left(x\right)=\int_{0}^{x}\sec t\mathrm{d}t," display="block"><mrow><mrow><mrow><msup><mi href="./4.23#E41">gd</mi><mrow><mo href="./4.23#E41">-</mo><mn>1</mn></mrow></msup><mo></mo><mrow><mo>(</mo><mi href="./4.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi href="./4.1#p2.t1.r3">x</mi></msubsup><mrow><mrow><mi href="./4.14#E6">sec</mi><mo></mo><mi>t</mi></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m65.png" altimg-height="27px" altimg-valign="-9px" altimg-width="134px" alttext="-\frac{1}{2}\pi<x<\frac{1}{2}\pi" display="inline"><mrow><mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow><mo><</mo><mi href="./4.1#p2.t1.r3">x</mi><mo><</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E41.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m176.png" altimg-height="26px" altimg-valign="-7px" altimg-width="77px" alttext="{\operatorname{gd}^{-1}}\left(\NVar{x}\right)" display="inline"><mrow><msup><mi href="./4.23#E41">gd</mi><mrow><mo href="./4.23#E41">-</mo><mn>1</mn></mrow></msup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./4.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: inverse Gudermannian function</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m143.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m95.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m161.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m90.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./4.14#E6" title="(4.14.6) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m148.png" altimg-height="13px" altimg-valign="-2px" altimg-width="43px" alttext="\sec\NVar{z}" display="inline"><mrow><mi href="./4.14#E6">sec</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: secant function</a> and
<a href="./4.1#p2.t1.r3" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m161.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./4.1#p2.t1.r3">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Equivalently, and again when <math class="ltx_Math" altimg="m65.png" altimg-height="27px" altimg-valign="-9px" altimg-width="134px" alttext="-\frac{1}{2}\pi<x<\frac{1}{2}\pi" display="inline"><mrow><mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow><mo><</mo><mi href="./4.1#p2.t1.r3">x</mi><mo><</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow></math>,</p>
<table id="E42" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.23.42</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_Math ltx_eqn_cell ltx_align_center"><math class="ltx_Math" alttext="{\operatorname{gd}^{-1}}\left(x\right)=\ln\tan\left(\tfrac{1}{2}x+\tfrac{1}{4}%
\pi\right)=\ln\left(\sec x+\tan x\right)=\operatorname{arcsinh}\left(\tan x%
\right)=\operatorname{arccsch}\left(\cot x\right)=\operatorname{arccosh}\left(%
\sec x\right)=\operatorname{arcsech}\left(\cos x\right)=\operatorname{arctanh}%
\left(\sin x\right)=\operatorname{arccoth}\left(\csc x\right)." display="block"><mrow><mtable align="baseline 1" columnalign="left" columnspacing="0.1em" displaystyle="true"><mtr><mtd><mrow><msup><mi href="./4.23#E41">gd</mi><mrow><mo href="./4.23#E41">-</mo><mn>1</mn></mrow></msup><mo></mo><mrow><mo>(</mo><mi href="./4.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mtd><mtd><mo>=</mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><mi href="./4.14#E4">tan</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./4.1#p2.t1.r3">x</mi></mrow><mo>+</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>4</mn></mfrac></mstyle><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mi href="./4.14#E6">sec</mi><mo></mo><mi href="./4.1#p2.t1.r3">x</mi></mrow><mo>+</mo><mrow><mi href="./4.14#E4">tan</mi><mo></mo><mi href="./4.1#p2.t1.r3">x</mi></mrow></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mi href="./4.37#SS2.p1">arcsinh</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./4.14#E4">tan</mi><mo></mo><mi href="./4.1#p2.t1.r3">x</mi></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mi href="./4.37#E7">arccsch</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./4.14#E7">cot</mi><mo></mo><mi href="./4.1#p2.t1.r3">x</mi></mrow><mo>)</mo></mrow></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo><mrow><mi href="./4.37#SS2.p1">arccosh</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./4.14#E6">sec</mi><mo></mo><mi href="./4.1#p2.t1.r3">x</mi></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mi href="./4.37#E8">arcsech</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi href="./4.1#p2.t1.r3">x</mi></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mi href="./4.37#SS2.p1">arctanh</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi href="./4.1#p2.t1.r3">x</mi></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mi href="./4.37#E9">arccoth</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./4.14#E5">csc</mi><mo></mo><mi href="./4.1#p2.t1.r3">x</mi></mrow><mo>)</mo></mrow></mrow><mo>.</mo></mtd></mtr></mtable></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E42.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m143.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.14#E5" title="(4.14.5) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m86.png" altimg-height="13px" altimg-valign="-2px" altimg-width="43px" alttext="\csc\NVar{z}" display="inline"><mrow><mi href="./4.14#E5">csc</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosecant function</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./4.14#E7" title="(4.14.7) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m84.png" altimg-height="17px" altimg-valign="-2px" altimg-width="44px" alttext="\cot\NVar{z}" display="inline"><mrow><mi href="./4.14#E7">cot</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cotangent function</a>,
<a href="./4.37#E7" title="(4.37.7) ‣ §4.37(ii) Principal Values ‣ §4.37 Inverse Hyperbolic Functions ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m116.png" altimg-height="18px" altimg-valign="-2px" altimg-width="80px" alttext="\operatorname{arccsch}\NVar{z}" display="inline"><mrow><mi href="./4.37#E7">arccsch</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: inverse hyperbolic cosecant function</a>,
<a href="./4.37#SS2.p1" title="§4.37(ii) Principal Values ‣ §4.37 Inverse Hyperbolic Functions ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m108.png" altimg-height="18px" altimg-valign="-2px" altimg-width="82px" alttext="\operatorname{arccosh}\NVar{z}" display="inline"><mrow><mi href="./4.37#SS2.p1">arccosh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: inverse hyperbolic cosine function</a>,
<a href="./4.37#E9" title="(4.37.9) ‣ §4.37(ii) Principal Values ‣ §4.37 Inverse Hyperbolic Functions ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m112.png" altimg-height="18px" altimg-valign="-2px" altimg-width="82px" alttext="\operatorname{arccoth}\NVar{z}" display="inline"><mrow><mi href="./4.37#E9">arccoth</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: inverse hyperbolic cotangent function</a>,
<a href="./4.37#E8" title="(4.37.8) ‣ §4.37(ii) Principal Values ‣ §4.37 Inverse Hyperbolic Functions ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m120.png" altimg-height="18px" altimg-valign="-2px" altimg-width="80px" alttext="\operatorname{arcsech}\NVar{z}" display="inline"><mrow><mi href="./4.37#E8">arcsech</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: inverse hyperbolic secant function</a>,
<a href="./4.37#SS2.p1" title="§4.37(ii) Principal Values ‣ §4.37 Inverse Hyperbolic Functions ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m125.png" altimg-height="18px" altimg-valign="-2px" altimg-width="80px" alttext="\operatorname{arcsinh}\NVar{z}" display="inline"><mrow><mi href="./4.37#SS2.p1">arcsinh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: inverse hyperbolic sine function</a>,
<a href="./4.37#SS2.p1" title="§4.37(ii) Principal Values ‣ §4.37 Inverse Hyperbolic Functions ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m130.png" altimg-height="18px" altimg-valign="-2px" altimg-width="84px" alttext="\operatorname{arctanh}\NVar{z}" display="inline"><mrow><mi href="./4.37#SS2.p1">arctanh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: inverse hyperbolic tangent function</a>,
<a href="./4.23#E41" title="(4.23.41) ‣ §4.23(viii) Gudermannian Function ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="26px" altimg-valign="-7px" altimg-width="77px" alttext="{\operatorname{gd}^{-1}}\left(\NVar{x}\right)" display="inline"><mrow><msup><mi href="./4.23#E41">gd</mi><mrow><mo href="./4.23#E41">-</mo><mn>1</mn></mrow></msup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./4.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math>: inverse Gudermannian function</a>,
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m92.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a>,
<a href="./4.14#E6" title="(4.14.6) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m148.png" altimg-height="13px" altimg-valign="-2px" altimg-width="43px" alttext="\sec\NVar{z}" display="inline"><mrow><mi href="./4.14#E6">sec</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: secant function</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m150.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./4.14#E4" title="(4.14.4) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m152.png" altimg-height="17px" altimg-valign="-2px" altimg-width="47px" alttext="\tan\NVar{z}" display="inline"><mrow><mi href="./4.14#E4">tan</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: tangent function</a> and
<a href="./4.1#p2.t1.r3" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m161.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./4.1#p2.t1.r3">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
</section>
</div>
<div class="ltx_page_footer">
<span></div>
</div>
</body></text>
</html>
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<page>
<!DOCTYPE html><html>
<head>
<title>DLMF: 4.37 Inverse Hyperbolic Functions</title>
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<div id="SS1.p1" class="ltx_para">
<p class="ltx_p">The general values of the inverse hyperbolic functions are defined by</p>
<table id="EGx1" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E1">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.37.1</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m23.png" altimg-height="19px" altimg-valign="-2px" altimg-width="87px" alttext="\displaystyle\operatorname{Arcsinh}z" display="inline"><mrow><mi href="./4.37#E1">Arcsinh</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m5.png" altimg-height="53px" altimg-valign="-21px" altimg-width="161px" alttext="\displaystyle=\int_{0}^{z}\frac{\mathrm{d}t}{(1+t^{2})^{1/2}}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi href="./4.1#p2.t1.r4">z</mi></msubsup></mstyle><mstyle displaystyle="true"><mfrac><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>+</mo><msup><mi>t</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mfrac></mstyle></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
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<tr><td class="ltx_align_right" colspan="5">
<div id="E1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m62.png" altimg-height="18px" altimg-valign="-2px" altimg-width="85px" alttext="\operatorname{Arcsinh}\NVar{z}" display="inline"><mrow><mi href="./4.37#E1">Arcsinh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: general inverse hyperbolic sine function</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m94.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m105.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E2">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.37.2</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m19.png" altimg-height="19px" altimg-valign="-2px" altimg-width="89px" alttext="\displaystyle\operatorname{Arccosh}z" display="inline"><mrow><mi href="./4.37#E2">Arccosh</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m7.png" altimg-height="53px" altimg-valign="-21px" altimg-width="161px" alttext="\displaystyle=\int_{1}^{z}\frac{\mathrm{d}t}{(t^{2}-1)^{1/2}}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>1</mn><mi href="./4.1#p2.t1.r4">z</mi></msubsup></mstyle><mstyle displaystyle="true"><mfrac><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow><msup><mrow><mo stretchy="false">(</mo><mrow><msup><mi>t</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mfrac></mstyle></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m57.png" altimg-height="18px" altimg-valign="-2px" altimg-width="87px" alttext="\operatorname{Arccosh}\NVar{z}" display="inline"><mrow><mi href="./4.37#E2">Arccosh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: general inverse hyperbolic cosine function</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m94.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m105.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E3">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.37.3</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m24.png" altimg-height="19px" altimg-valign="-2px" altimg-width="91px" alttext="\displaystyle\operatorname{Arctanh}z" display="inline"><mrow><mi href="./4.37#E3">Arctanh</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m6.png" altimg-height="53px" altimg-valign="-20px" altimg-width="120px" alttext="\displaystyle=\int_{0}^{z}\frac{\mathrm{d}t}{1-t^{2}}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi href="./4.1#p2.t1.r4">z</mi></msubsup></mstyle><mstyle displaystyle="true"><mfrac><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow><mrow><mn>1</mn><mo>-</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac></mstyle></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m108.png" altimg-height="21px" altimg-valign="-6px" altimg-width="66px" alttext="z\neq\pm 1" display="inline"><mrow><mi href="./4.1#p2.t1.r4">z</mi><mo>≠</mo><mrow><mo>±</mo><mn>1</mn></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m64.png" altimg-height="18px" altimg-valign="-2px" altimg-width="89px" alttext="\operatorname{Arctanh}\NVar{z}" display="inline"><mrow><mi href="./4.37#E3">Arctanh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: general inverse hyperbolic tangent function</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m94.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m105.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E4">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.37.4</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m21.png" altimg-height="19px" altimg-valign="-2px" altimg-width="87px" alttext="\displaystyle\operatorname{Arccsch}z" display="inline"><mrow><mi href="./4.37#E4">Arccsch</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m11.png" altimg-height="25px" altimg-valign="-7px" altimg-width="152px" alttext="\displaystyle=\operatorname{Arcsinh}\left(1/z\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./4.37#E1">Arcsinh</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>/</mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E4.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="85px" alttext="\operatorname{Arccsch}\NVar{z}" display="inline"><mrow><mi href="./4.37#E4">Arccsch</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: general inverse hyperbolic cosecant function</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./4.37#E1" title="(4.37.1) ‣ §4.37(i) General Definitions ‣ §4.37 Inverse Hyperbolic Functions ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m62.png" altimg-height="18px" altimg-valign="-2px" altimg-width="85px" alttext="\operatorname{Arcsinh}\NVar{z}" display="inline"><mrow><mi href="./4.37#E1">Arcsinh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: general inverse hyperbolic sine function</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m105.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E5">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.37.5</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m22.png" altimg-height="19px" altimg-valign="-2px" altimg-width="87px" alttext="\displaystyle\operatorname{Arcsech}z" display="inline"><mrow><mi href="./4.37#E5">Arcsech</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m9.png" altimg-height="25px" altimg-valign="-7px" altimg-width="154px" alttext="\displaystyle=\operatorname{Arccosh}\left(1/z\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./4.37#E2">Arccosh</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>/</mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E5.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m61.png" altimg-height="18px" altimg-valign="-2px" altimg-width="85px" alttext="\operatorname{Arcsech}\NVar{z}" display="inline"><mrow><mi href="./4.37#E5">Arcsech</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: general inverse hyperbolic secant function</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./4.37#E2" title="(4.37.2) ‣ §4.37(i) General Definitions ‣ §4.37 Inverse Hyperbolic Functions ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m57.png" altimg-height="18px" altimg-valign="-2px" altimg-width="87px" alttext="\operatorname{Arccosh}\NVar{z}" display="inline"><mrow><mi href="./4.37#E2">Arccosh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: general inverse hyperbolic cosine function</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m105.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E6">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.37.6</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m20.png" altimg-height="19px" altimg-valign="-2px" altimg-width="89px" alttext="\displaystyle\operatorname{Arccoth}z" display="inline"><mrow><mi href="./4.37#E6">Arccoth</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m13.png" altimg-height="25px" altimg-valign="-7px" altimg-width="156px" alttext="\displaystyle=\operatorname{Arctanh}\left(1/z\right)." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./4.37#E3">Arctanh</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>/</mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E6.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m58.png" altimg-height="18px" altimg-valign="-2px" altimg-width="87px" alttext="\operatorname{Arccoth}\NVar{z}" display="inline"><mrow><mi href="./4.37#E6">Arccoth</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: general inverse hyperbolic cotangent function</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./4.37#E3" title="(4.37.3) ‣ §4.37(i) General Definitions ‣ §4.37 Inverse Hyperbolic Functions ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m64.png" altimg-height="18px" altimg-valign="-2px" altimg-width="89px" alttext="\operatorname{Arctanh}\NVar{z}" display="inline"><mrow><mi href="./4.37#E3">Arctanh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: general inverse hyperbolic tangent function</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m105.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
<p class="ltx_p">In () the integration path may not pass through either of
the points <math class="ltx_Math" altimg="m91.png" altimg-height="18px" altimg-valign="-4px" altimg-width="60px" alttext="t=\pm i" display="inline"><mrow><mi>t</mi><mo>=</mo><mrow><mo>±</mo><mi mathvariant="normal">i</mi></mrow></mrow></math>, and the function <math class="ltx_Math" altimg="m36.png" altimg-height="26px" altimg-valign="-7px" altimg-width="95px" alttext="(1+t^{2})^{1/2}" display="inline"><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>+</mo><msup><mi>t</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></math> assumes its
principal value when <math class="ltx_Math" altimg="m92.png" altimg-height="17px" altimg-valign="-2px" altimg-width="11px" alttext="t" display="inline"><mi>t</mi></math> is real. In () the integration
path may not pass through either of the points <math class="ltx_Math" altimg="m82.png" altimg-height="18px" altimg-valign="-4px" altimg-width="30px" alttext="\pm 1" display="inline"><mrow><mo>±</mo><mn>1</mn></mrow></math>, and the function
<math class="ltx_Math" altimg="m39.png" altimg-height="26px" altimg-valign="-7px" altimg-width="95px" alttext="(t^{2}-1)^{1/2}" display="inline"><msup><mrow><mo stretchy="false">(</mo><mrow><msup><mi>t</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></math> assumes its principal value when <math class="ltx_Math" altimg="m93.png" altimg-height="23px" altimg-valign="-7px" altimg-width="90px" alttext="t\in(1,\infty)" display="inline"><mrow><mi>t</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mn>1</mn><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi mathvariant="normal">∞</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></mrow></math>.
Elsewhere on the integration paths in () the integration path may not intersect <math class="ltx_Math" altimg="m82.png" altimg-height="18px" altimg-valign="-4px" altimg-width="30px" alttext="\pm 1" display="inline"><mrow><mo>±</mo><mn>1</mn></mrow></math>. Each of
the six functions is a multivalued function of <math class="ltx_Math" altimg="m105.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>. <math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="85px" alttext="\operatorname{Arcsinh}z" display="inline"><mrow><mi href="./4.37#E1">Arcsinh</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></math> and
<math class="ltx_Math" altimg="m60.png" altimg-height="18px" altimg-valign="-2px" altimg-width="85px" alttext="\operatorname{Arccsch}z" display="inline"><mrow><mi href="./4.37#E4">Arccsch</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></math> have branch points at <math class="ltx_Math" altimg="m104.png" altimg-height="18px" altimg-valign="-4px" altimg-width="63px" alttext="z=\pm i" display="inline"><mrow><mi href="./4.1#p2.t1.r4">z</mi><mo>=</mo><mrow><mo>±</mo><mi mathvariant="normal">i</mi></mrow></mrow></math>; the other four functions have
branch points at <math class="ltx_Math" altimg="m103.png" altimg-height="18px" altimg-valign="-4px" altimg-width="66px" alttext="z=\pm 1" display="inline"><mrow><mi href="./4.1#p2.t1.r4">z</mi><mo>=</mo><mrow><mo>±</mo><mn>1</mn></mrow></mrow></math>.</p>
</div>
</section>
<section id="ii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§4.37(ii) </span>Principal Values</h2>
<div id="SS2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd>
<span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m65.png" altimg-height="18px" altimg-valign="-2px" altimg-width="82px" alttext="\operatorname{arccosh}\NVar{z}" display="inline"><mrow><mi href="./4.37#SS2.p1">arccosh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: inverse hyperbolic cosine function</span>,
<span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m75.png" altimg-height="18px" altimg-valign="-2px" altimg-width="80px" alttext="\operatorname{arcsinh}\NVar{z}" display="inline"><mrow><mi href="./4.37#SS2.p1">arcsinh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: inverse hyperbolic sine function</span> and
<span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m78.png" altimg-height="18px" altimg-valign="-2px" altimg-width="84px" alttext="\operatorname{arctanh}\NVar{z}" display="inline"><mrow><mi href="./4.37#SS2.p1">arctanh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: inverse hyperbolic tangent function</span>
</dd>
<dt>Keywords:</dt>
<dd>
</dd>
</dl>
</div>
</div>
<div id="SS2.p1" class="ltx_para">
<p class="ltx_p">The <em class="ltx_emph ltx_font_italic">principal values</em> (or <em class="ltx_emph ltx_font_italic">principal branches</em>) of the inverse
<math class="ltx_Math" altimg="m86.png" altimg-height="18px" altimg-valign="-2px" altimg-width="40px" alttext="\sinh" display="inline"><mi href="./4.28#E1">sinh</mi></math>, <math class="ltx_Math" altimg="m50.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\cosh" display="inline"><mi href="./4.28#E2">cosh</mi></math>, and <math class="ltx_Math" altimg="m88.png" altimg-height="18px" altimg-valign="-2px" altimg-width="44px" alttext="\tanh" display="inline"><mi href="./4.28#E4">tanh</mi></math> are obtained by introducing cuts in the <math class="ltx_Math" altimg="m105.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>-plane
as indicated in Figure ). The principal branches are denoted by <math class="ltx_Math" altimg="m76.png" altimg-height="18px" altimg-valign="-2px" altimg-width="66px" alttext="\operatorname{arcsinh}" display="inline"><mi href="./4.37#SS2.p1">arcsinh</mi></math>,
<math class="ltx_Math" altimg="m66.png" altimg-height="18px" altimg-valign="-2px" altimg-width="69px" alttext="\operatorname{arccosh}" display="inline"><mi href="./4.37#SS2.p1">arccosh</mi></math>, <math class="ltx_Math" altimg="m79.png" altimg-height="18px" altimg-valign="-2px" altimg-width="71px" alttext="\operatorname{arctanh}" display="inline"><mi href="./4.37#SS2.p1">arctanh</mi></math> respectively.
Each is two-valued on the corresponding cut(s),
and each is real on the part of the real axis that
remains after deleting the intersections with the corresponding cuts.
</p>
</div>
<div id="SS2.p2" class="ltx_para">
<p class="ltx_p">The principal values of the inverse hyperbolic cosecant, hyperbolic secant, and
hyperbolic tangent are given by</p>
<table id="EGx2" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E7">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.37.7</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m29.png" altimg-height="19px" altimg-valign="-2px" altimg-width="82px" alttext="\displaystyle\operatorname{arccsch}z" display="inline"><mrow><mi href="./4.37#E7">arccsch</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m16.png" altimg-height="25px" altimg-valign="-7px" altimg-width="147px" alttext="\displaystyle=\operatorname{arcsinh}\left(1/z\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./4.37#SS2.p1">arcsinh</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>/</mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E7.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m70.png" altimg-height="18px" altimg-valign="-2px" altimg-width="80px" alttext="\operatorname{arccsch}\NVar{z}" display="inline"><mrow><mi href="./4.37#E7">arccsch</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: inverse hyperbolic cosecant function</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./4.37#SS2.p1" title="§4.37(ii) Principal Values ‣ §4.37 Inverse Hyperbolic Functions ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m75.png" altimg-height="18px" altimg-valign="-2px" altimg-width="80px" alttext="\operatorname{arcsinh}\NVar{z}" display="inline"><mrow><mi href="./4.37#SS2.p1">arcsinh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: inverse hyperbolic sine function</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m105.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.6.4</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E8">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.37.8</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m31.png" altimg-height="19px" altimg-valign="-2px" altimg-width="82px" alttext="\displaystyle\operatorname{arcsech}z" display="inline"><mrow><mi href="./4.37#E8">arcsech</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m15.png" altimg-height="25px" altimg-valign="-7px" altimg-width="149px" alttext="\displaystyle=\operatorname{arccosh}\left(1/z\right)." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./4.37#SS2.p1">arccosh</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>/</mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E8.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m72.png" altimg-height="18px" altimg-valign="-2px" altimg-width="80px" alttext="\operatorname{arcsech}\NVar{z}" display="inline"><mrow><mi href="./4.37#E8">arcsech</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: inverse hyperbolic secant function</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./4.37#SS2.p1" title="§4.37(ii) Principal Values ‣ §4.37 Inverse Hyperbolic Functions ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m65.png" altimg-height="18px" altimg-valign="-2px" altimg-width="82px" alttext="\operatorname{arccosh}\NVar{z}" display="inline"><mrow><mi href="./4.37#SS2.p1">arccosh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: inverse hyperbolic cosine function</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m105.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.6.5</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E9">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.37.9</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m27.png" altimg-height="19px" altimg-valign="-2px" altimg-width="84px" alttext="\displaystyle\operatorname{arccoth}z" display="inline"><mrow><mi href="./4.37#E9">arccoth</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m17.png" altimg-height="25px" altimg-valign="-7px" altimg-width="151px" alttext="\displaystyle=\operatorname{arctanh}\left(1/z\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./4.37#SS2.p1">arctanh</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>/</mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m108.png" altimg-height="21px" altimg-valign="-6px" altimg-width="66px" alttext="z\neq\pm 1" display="inline"><mrow><mi href="./4.1#p2.t1.r4">z</mi><mo>≠</mo><mrow><mo>±</mo><mn>1</mn></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E9.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m68.png" altimg-height="18px" altimg-valign="-2px" altimg-width="82px" alttext="\operatorname{arccoth}\NVar{z}" display="inline"><mrow><mi href="./4.37#E9">arccoth</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: inverse hyperbolic cotangent function</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./4.37#SS2.p1" title="§4.37(ii) Principal Values ‣ §4.37 Inverse Hyperbolic Functions ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="18px" altimg-valign="-2px" altimg-width="84px" alttext="\operatorname{arctanh}\NVar{z}" display="inline"><mrow><mi href="./4.37#SS2.p1">arctanh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: inverse hyperbolic tangent function</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m105.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.6.6</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
<p class="ltx_p">These functions are analytic in the cut plane depicted in Figure
</td>
</tr>
<tr id="F1.t1.r2" class="ltx_tr">
<td class="ltx_td ltx_align_center">(i) <math class="ltx_Math" altimg="m77.png" altimg-height="18px" altimg-valign="-2px" altimg-width="80px" alttext="\operatorname{arcsinh}z" display="inline"><mrow><mi href="./4.37#SS2.p1">arcsinh</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></math>
</td>
<td class="ltx_td ltx_align_center">(ii) <math class="ltx_Math" altimg="m67.png" altimg-height="18px" altimg-valign="-2px" altimg-width="82px" alttext="\operatorname{arccosh}z" display="inline"><mrow><mi href="./4.37#SS2.p1">arccosh</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></math>
</td>
<td class="ltx_td ltx_align_center">(iii) <math class="ltx_Math" altimg="m80.png" altimg-height="18px" altimg-valign="-2px" altimg-width="84px" alttext="\operatorname{arctanh}z" display="inline"><mrow><mi href="./4.37#SS2.p1">arctanh</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></math>
</td>
</tr>
<tr id="F1.t1.r3" class="ltx_tr">
<td class="ltx_td ltx_align_center"></td>
</tr>
<tr id="F1.t1.r4" class="ltx_tr">
<td class="ltx_td ltx_align_center">(iv) <math class="ltx_Math" altimg="m71.png" altimg-height="18px" altimg-valign="-2px" altimg-width="80px" alttext="\operatorname{arccsch}z" display="inline"><mrow><mi href="./4.37#E7">arccsch</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></math>
</td>
<td class="ltx_td ltx_align_center">(v) <math class="ltx_Math" altimg="m74.png" altimg-height="18px" altimg-valign="-2px" altimg-width="80px" alttext="\operatorname{arcsech}z" display="inline"><mrow><mi href="./4.37#E8">arcsech</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></math>
</td>
<td class="ltx_td ltx_align_center">(vi) <math class="ltx_Math" altimg="m69.png" altimg-height="18px" altimg-valign="-2px" altimg-width="82px" alttext="\operatorname{arccoth}z" display="inline"><mrow><mi href="./4.37#E9">arccoth</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></math>
</td>
</tr>
</tbody>
</table>
<figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 4.37.1: </span><math class="ltx_Math" altimg="m105.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>-plane. Branch cuts for the inverse hyperbolic functions.
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.37#E7" title="(4.37.7) ‣ §4.37(ii) Principal Values ‣ §4.37 Inverse Hyperbolic Functions ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="18px" altimg-valign="-2px" altimg-width="80px" alttext="\operatorname{arccsch}\NVar{z}" display="inline"><mrow><mi href="./4.37#E7">arccsch</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: inverse hyperbolic cosecant function</a>,
<a href="./4.37#SS2.p1" title="§4.37(ii) Principal Values ‣ §4.37 Inverse Hyperbolic Functions ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m65.png" altimg-height="18px" altimg-valign="-2px" altimg-width="82px" alttext="\operatorname{arccosh}\NVar{z}" display="inline"><mrow><mi href="./4.37#SS2.p1">arccosh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: inverse hyperbolic cosine function</a>,
<a href="./4.37#E9" title="(4.37.9) ‣ §4.37(ii) Principal Values ‣ §4.37 Inverse Hyperbolic Functions ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m68.png" altimg-height="18px" altimg-valign="-2px" altimg-width="82px" alttext="\operatorname{arccoth}\NVar{z}" display="inline"><mrow><mi href="./4.37#E9">arccoth</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: inverse hyperbolic cotangent function</a>,
<a href="./4.37#E8" title="(4.37.8) ‣ §4.37(ii) Principal Values ‣ §4.37 Inverse Hyperbolic Functions ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="18px" altimg-valign="-2px" altimg-width="80px" alttext="\operatorname{arcsech}\NVar{z}" display="inline"><mrow><mi href="./4.37#E8">arcsech</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: inverse hyperbolic secant function</a>,
<a href="./4.37#SS2.p1" title="§4.37(ii) Principal Values ‣ §4.37 Inverse Hyperbolic Functions ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m75.png" altimg-height="18px" altimg-valign="-2px" altimg-width="80px" alttext="\operatorname{arcsinh}\NVar{z}" display="inline"><mrow><mi href="./4.37#SS2.p1">arcsinh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: inverse hyperbolic sine function</a>,
<a href="./4.37#SS2.p1" title="§4.37(ii) Principal Values ‣ §4.37 Inverse Hyperbolic Functions ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="18px" altimg-valign="-2px" altimg-width="84px" alttext="\operatorname{arctanh}\NVar{z}" display="inline"><mrow><mi href="./4.37#SS2.p1">arctanh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: inverse hyperbolic tangent function</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m105.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>Keywords:</dt>
<dd></dd>
</dl>
</div>
</div>
<div id="SS3.p1" class="ltx_para">
<table id="EGx3" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E10">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.37.10</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m32.png" altimg-height="25px" altimg-valign="-7px" altimg-width="113px" alttext="\displaystyle\operatorname{arcsinh}\left(-z\right)" display="inline"><mrow><mi href="./4.37#SS2.p1">arcsinh</mi><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m3.png" altimg-height="21px" altimg-valign="-4px" altimg-width="127px" alttext="\displaystyle=-\operatorname{arcsinh}z." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo>-</mo><mrow><mi href="./4.37#SS2.p1">arcsinh</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E10.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.37#SS2.p1" title="§4.37(ii) Principal Values ‣ §4.37 Inverse Hyperbolic Functions ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m75.png" altimg-height="18px" altimg-valign="-2px" altimg-width="80px" alttext="\operatorname{arcsinh}\NVar{z}" display="inline"><mrow><mi href="./4.37#SS2.p1">arcsinh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: inverse hyperbolic sine function</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m105.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.6.11</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E11">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.37.11</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m25.png" altimg-height="25px" altimg-valign="-7px" altimg-width="115px" alttext="\displaystyle\operatorname{arccosh}\left(-z\right)" display="inline"><mrow><mi href="./4.37#SS2.p1">arccosh</mi><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m18.png" altimg-height="23px" altimg-valign="-6px" altimg-width="169px" alttext="\displaystyle=\pm\pi i+\operatorname{arccosh}z," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mo>±</mo><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi></mrow></mrow><mo>+</mo><mrow><mi href="./4.37#SS2.p1">arccosh</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m46.png" altimg-height="21px" altimg-valign="-6px" altimg-width="65px" alttext="\Im z\gtrless 0" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℑ</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>≷</mo><mn>0</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E11.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.37#SS2.p1" title="§4.37(ii) Principal Values ‣ §4.37 Inverse Hyperbolic Functions ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m65.png" altimg-height="18px" altimg-valign="-2px" altimg-width="82px" alttext="\operatorname{arccosh}\NVar{z}" display="inline"><mrow><mi href="./4.37#SS2.p1">arccosh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: inverse hyperbolic cosine function</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Im" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℑ</mi><mo></mo></mrow></math>: imaginary part</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m105.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.6.12</span> (has an error even in the tenth printing.)</span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E12">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.37.12</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m33.png" altimg-height="25px" altimg-valign="-7px" altimg-width="117px" alttext="\displaystyle\operatorname{arctanh}\left(-z\right)" display="inline"><mrow><mi href="./4.37#SS2.p1">arctanh</mi><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m4.png" altimg-height="23px" altimg-valign="-6px" altimg-width="131px" alttext="\displaystyle=-\operatorname{arctanh}z," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo>-</mo><mrow><mi href="./4.37#SS2.p1">arctanh</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m108.png" altimg-height="21px" altimg-valign="-6px" altimg-width="66px" alttext="z\neq\pm 1" display="inline"><mrow><mi href="./4.1#p2.t1.r4">z</mi><mo>≠</mo><mrow><mo>±</mo><mn>1</mn></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E12.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.37#SS2.p1" title="§4.37(ii) Principal Values ‣ §4.37 Inverse Hyperbolic Functions ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="18px" altimg-valign="-2px" altimg-width="84px" alttext="\operatorname{arctanh}\NVar{z}" display="inline"><mrow><mi href="./4.37#SS2.p1">arctanh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: inverse hyperbolic tangent function</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m105.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.6.13</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E13">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.37.13</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m28.png" altimg-height="25px" altimg-valign="-7px" altimg-width="113px" alttext="\displaystyle\operatorname{arccsch}\left(-z\right)" display="inline"><mrow><mi href="./4.37#E7">arccsch</mi><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m2.png" altimg-height="21px" altimg-valign="-4px" altimg-width="128px" alttext="\displaystyle=-\operatorname{arccsch}z." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo>-</mo><mrow><mi href="./4.37#E7">arccsch</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E13.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.37#E7" title="(4.37.7) ‣ §4.37(ii) Principal Values ‣ §4.37 Inverse Hyperbolic Functions ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="18px" altimg-valign="-2px" altimg-width="80px" alttext="\operatorname{arccsch}\NVar{z}" display="inline"><mrow><mi href="./4.37#E7">arccsch</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: inverse hyperbolic cosecant function</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m105.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E14">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.37.14</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m30.png" altimg-height="25px" altimg-valign="-7px" altimg-width="113px" alttext="\displaystyle\operatorname{arcsech}\left(-z\right)" display="inline"><mrow><mi href="./4.37#E8">arcsech</mi><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m8.png" altimg-height="23px" altimg-valign="-6px" altimg-width="168px" alttext="\displaystyle=\mp\pi i+\operatorname{arcsech}z," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mo>∓</mo><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi></mrow></mrow><mo>+</mo><mrow><mi href="./4.37#E8">arcsech</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m46.png" altimg-height="21px" altimg-valign="-6px" altimg-width="65px" alttext="\Im z\gtrless 0" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℑ</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>≷</mo><mn>0</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E14.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.37#E8" title="(4.37.8) ‣ §4.37(ii) Principal Values ‣ §4.37 Inverse Hyperbolic Functions ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="18px" altimg-valign="-2px" altimg-width="80px" alttext="\operatorname{arcsech}\NVar{z}" display="inline"><mrow><mi href="./4.37#E8">arcsech</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: inverse hyperbolic secant function</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Im" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℑ</mi><mo></mo></mrow></math>: imaginary part</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m105.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E15">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.37.15</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m26.png" altimg-height="25px" altimg-valign="-7px" altimg-width="115px" alttext="\displaystyle\operatorname{arccoth}\left(-z\right)" display="inline"><mrow><mi href="./4.37#E9">arccoth</mi><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m1.png" altimg-height="23px" altimg-valign="-6px" altimg-width="129px" alttext="\displaystyle=-\operatorname{arccoth}z," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo>-</mo><mrow><mi href="./4.37#E9">arccoth</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m108.png" altimg-height="21px" altimg-valign="-6px" altimg-width="66px" alttext="z\neq\pm 1" display="inline"><mrow><mi href="./4.1#p2.t1.r4">z</mi><mo>≠</mo><mrow><mo>±</mo><mn>1</mn></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E15.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.37#E9" title="(4.37.9) ‣ §4.37(ii) Principal Values ‣ §4.37 Inverse Hyperbolic Functions ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m68.png" altimg-height="18px" altimg-valign="-2px" altimg-width="82px" alttext="\operatorname{arccoth}\NVar{z}" display="inline"><mrow><mi href="./4.37#E9">arccoth</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: inverse hyperbolic cotangent function</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m105.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
</div>
</section>
<section id="iv" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§4.37(iv) </span>Logarithmic Forms</h2>
<div id="SS4.info" class="ltx_metadata ltx_info">
) the principal value of <math class="ltx_Math" altimg="m40.png" altimg-height="26px" altimg-valign="-7px" altimg-width="98px" alttext="(z^{2}-1)^{1/2}" display="inline"><msup><mrow><mo stretchy="false">(</mo><mrow><msup><mi href="./4.1#p2.t1.r4">z</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></math> is
discontinuous on the imaginary axis, hence to continue <math class="ltx_Math" altimg="m40.png" altimg-height="26px" altimg-valign="-7px" altimg-width="98px" alttext="(z^{2}-1)^{1/2}" display="inline"><msup><mrow><mo stretchy="false">(</mo><mrow><msup><mi href="./4.1#p2.t1.r4">z</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></math>
analytically we switch to the other branch. This accounts for the <math class="ltx_Math" altimg="m83.png" altimg-height="17px" altimg-valign="-4px" altimg-width="20px" alttext="\pm" display="inline"><mo>±</mo></math>
sign in (</dd>
</dl>
</div>
</div>
<div id="Px1.p1" class="ltx_para">
<table id="E16" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.37.16</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E16.png" altimg-height="41px" altimg-valign="-15px" altimg-width="289px" alttext="\operatorname{arcsinh}z=\ln\left((z^{2}+1)^{1/2}+z\right)," display="block"><mrow><mrow><mrow><mi href="./4.37#SS2.p1">arcsinh</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>=</mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><mo>(</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><msup><mi href="./4.1#p2.t1.r4">z</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>+</mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m102.png" altimg-height="23px" altimg-valign="-7px" altimg-width="250px" alttext="z/i\in\mathbb{C}\setminus(-\infty,-1)\cup(1,\infty)" display="inline"><mrow><mrow><mi href="./4.1#p2.t1.r4">z</mi><mo>/</mo><mi mathvariant="normal">i</mi></mrow><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mrow><mrow><mi href="./front/introduction#Sx4.p1.t1.r1">ℂ</mi><mo href="./front/introduction#Sx4.p2.t1.r19">∖</mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></mrow><mo href="./front/introduction#Sx4.p1.t1.r27">∪</mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mn>1</mn><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi mathvariant="normal">∞</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></mrow></mrow></math>;</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E16.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./front/introduction#Sx4.p1.t1.r1" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m55.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\mathbb{C}" display="inline"><mi href="./front/introduction#Sx4.p1.t1.r1">ℂ</mi></math>: complex plane</a>,
<a href="./front/introduction#Sx4.p1.t1.r9" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="15px" altimg-valign="-3px" altimg-width="18px" alttext="\in" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo></math>: element of</a>,
<a href="./4.37#SS2.p1" title="§4.37(ii) Principal Values ‣ §4.37 Inverse Hyperbolic Functions ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m75.png" altimg-height="18px" altimg-valign="-2px" altimg-width="80px" alttext="\operatorname{arcsinh}\NVar{z}" display="inline"><mrow><mi href="./4.37#SS2.p1">arcsinh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: inverse hyperbolic sine function</a>,
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a>,
<a href="./front/introduction#Sx4.p1.t1.r28" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="(\NVar{a},\NVar{b})" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi class="ltx_nvar">a</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi class="ltx_nvar">b</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math>: open interval</a>,
<a href="./front/introduction#Sx4.p2.t1.r19" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m84.png" altimg-height="23px" altimg-valign="-7px" altimg-width="14px" alttext="\setminus" display="inline"><mo href="./front/introduction#Sx4.p2.t1.r19">∖</mo></math>: set subtraction</a>,
<a href="./front/introduction#Sx4.p1.t1.r27" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="15px" altimg-valign="-2px" altimg-width="18px" alttext="\cup" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r27">∪</mo></math>: union</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m105.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">compare Figure (i). On the cuts</p>
<table id="E17" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.37.17</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E17.png" altimg-height="41px" altimg-valign="-15px" altimg-width="369px" alttext="\operatorname{arcsinh}\left(iy\right)=\tfrac{1}{2}\pi i\pm\ln\left((y^{2}-1)^{%
1/2}+y\right)," display="block"><mrow><mrow><mrow><mi href="./4.37#SS2.p1">arcsinh</mi><mo></mo><mrow><mo>(</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./4.1#p2.t1.r3">y</mi></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi></mrow><mo>±</mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><mo>(</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><msup><mi href="./4.1#p2.t1.r3">y</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>+</mo><mi href="./4.1#p2.t1.r3">y</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m101.png" altimg-height="23px" altimg-valign="-7px" altimg-width="91px" alttext="y\in[1,\infty)" display="inline"><mrow><mi href="./4.1#p2.t1.r3">y</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mrow><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">[</mo><mn>1</mn><mo href="./front/introduction#Sx4.p2.t1.r1">,</mo><mi mathvariant="normal">∞</mi><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">)</mo></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E17.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./front/introduction#Sx4.p2.t1.r1" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m45.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="[\NVar{a},\NVar{b})" display="inline"><mrow><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">[</mo><mi class="ltx_nvar">a</mi><mo href="./front/introduction#Sx4.p2.t1.r1">,</mo><mi class="ltx_nvar">b</mi><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">)</mo></mrow></math>: half-closed interval</a>,
<a href="./front/introduction#Sx4.p1.t1.r9" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="15px" altimg-valign="-3px" altimg-width="18px" alttext="\in" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo></math>: element of</a>,
<a href="./4.37#SS2.p1" title="§4.37(ii) Principal Values ‣ §4.37 Inverse Hyperbolic Functions ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m75.png" altimg-height="18px" altimg-valign="-2px" altimg-width="80px" alttext="\operatorname{arcsinh}\NVar{z}" display="inline"><mrow><mi href="./4.37#SS2.p1">arcsinh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: inverse hyperbolic sine function</a>,
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a> and
<a href="./4.1#p2.t1.r3" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m98.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./4.1#p2.t1.r3">y</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E18" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.37.18</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E18.png" altimg-height="41px" altimg-valign="-15px" altimg-width="384px" alttext="\operatorname{arcsinh}\left(iy\right)=-\tfrac{1}{2}\pi i\pm\ln\left((y^{2}-1)^%
{1/2}-y\right)," display="block"><mrow><mrow><mrow><mi href="./4.37#SS2.p1">arcsinh</mi><mo></mo><mrow><mo>(</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./4.1#p2.t1.r3">y</mi></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mo>-</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi></mrow></mrow><mo>±</mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><mo>(</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><msup><mi href="./4.1#p2.t1.r3">y</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>-</mo><mi href="./4.1#p2.t1.r3">y</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m100.png" altimg-height="23px" altimg-valign="-7px" altimg-width="122px" alttext="y\in(-\infty,-1]" display="inline"><mrow><mi href="./4.1#p2.t1.r3">y</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mrow><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">(</mo><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mo href="./front/introduction#Sx4.p2.t1.r1">,</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">]</mo></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E18.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./front/introduction#Sx4.p1.t1.r9" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="15px" altimg-valign="-3px" altimg-width="18px" alttext="\in" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo></math>: element of</a>,
<a href="./4.37#SS2.p1" title="§4.37(ii) Principal Values ‣ §4.37 Inverse Hyperbolic Functions ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m75.png" altimg-height="18px" altimg-valign="-2px" altimg-width="80px" alttext="\operatorname{arcsinh}\NVar{z}" display="inline"><mrow><mi href="./4.37#SS2.p1">arcsinh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: inverse hyperbolic sine function</a>,
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a>,
<a href="./front/introduction#Sx4.p2.t1.r1" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="(\NVar{a},\NVar{b}]" display="inline"><mrow><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">(</mo><mi class="ltx_nvar">a</mi><mo href="./front/introduction#Sx4.p2.t1.r1">,</mo><mi class="ltx_nvar">b</mi><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">]</mo></mrow></math>: half-closed interval</a> and
<a href="./4.1#p2.t1.r3" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m98.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./4.1#p2.t1.r3">y</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">the upper/lower signs corresponding to the right/left sides.</p>
</div>
</section>
<section id="Px2" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Inverse Hyperbolic Cosine</h3>
<div id="Px2.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px2.p1" class="ltx_para">
<table id="E19" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.37.19</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E19.png" altimg-height="41px" altimg-valign="-15px" altimg-width="307px" alttext="\operatorname{arccosh}z=\ln\left(\pm(z^{2}-1)^{1/2}+z\right)," display="block"><mrow><mrow><mrow><mi href="./4.37#SS2.p1">arccosh</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>=</mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mo>±</mo><msup><mrow><mo stretchy="false">(</mo><mrow><msup><mi href="./4.1#p2.t1.r4">z</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow><mo>+</mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m107.png" altimg-height="23px" altimg-valign="-7px" altimg-width="142px" alttext="z\in\mathbb{C}\setminus(-\infty,1)" display="inline"><mrow><mi href="./4.1#p2.t1.r4">z</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mrow><mi href="./front/introduction#Sx4.p1.t1.r1">ℂ</mi><mo href="./front/introduction#Sx4.p2.t1.r19">∖</mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mn>1</mn><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E19.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./front/introduction#Sx4.p1.t1.r1" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m55.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\mathbb{C}" display="inline"><mi href="./front/introduction#Sx4.p1.t1.r1">ℂ</mi></math>: complex plane</a>,
<a href="./front/introduction#Sx4.p1.t1.r9" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="15px" altimg-valign="-3px" altimg-width="18px" alttext="\in" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo></math>: element of</a>,
<a href="./4.37#SS2.p1" title="§4.37(ii) Principal Values ‣ §4.37 Inverse Hyperbolic Functions ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m65.png" altimg-height="18px" altimg-valign="-2px" altimg-width="82px" alttext="\operatorname{arccosh}\NVar{z}" display="inline"><mrow><mi href="./4.37#SS2.p1">arccosh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: inverse hyperbolic cosine function</a>,
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a>,
<a href="./front/introduction#Sx4.p1.t1.r28" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="(\NVar{a},\NVar{b})" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi class="ltx_nvar">a</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi class="ltx_nvar">b</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math>: open interval</a>,
<a href="./front/introduction#Sx4.p2.t1.r19" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m84.png" altimg-height="23px" altimg-valign="-7px" altimg-width="14px" alttext="\setminus" display="inline"><mo href="./front/introduction#Sx4.p2.t1.r19">∖</mo></math>: set subtraction</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m105.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">the upper or lower sign being taken according as <math class="ltx_Math" altimg="m48.png" altimg-height="21px" altimg-valign="-6px" altimg-width="65px" alttext="\Re z\gtrless 0" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>≷</mo><mn>0</mn></mrow></math>;
compare Figure (ii). Also,</p>
<table id="E20" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.37.20</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E20.png" altimg-height="41px" altimg-valign="-15px" altimg-width="384px" alttext="\operatorname{arccosh}\left(\mathrm{i}y\right)=\pm\tfrac{1}{2}\pi\mathrm{i}+%
\ln\left((y^{2}+1)^{1/2}\pm y\right)," display="block"><mrow><mrow><mrow><mi href="./4.37#SS2.p1">arccosh</mi><mo></mo><mrow><mo>(</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./4.1#p2.t1.r3">y</mi></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mo>±</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi></mrow></mrow><mo>+</mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><mo>(</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><msup><mi href="./4.1#p2.t1.r3">y</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>±</mo><mi href="./4.1#p2.t1.r3">y</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m99.png" altimg-height="21px" altimg-valign="-6px" altimg-width="51px" alttext="y\gtrless 0" display="inline"><mrow><mi href="./4.1#p2.t1.r3">y</mi><mo>≷</mo><mn>0</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E20.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.37#SS2.p1" title="§4.37(ii) Principal Values ‣ §4.37 Inverse Hyperbolic Functions ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m65.png" altimg-height="18px" altimg-valign="-2px" altimg-width="82px" alttext="\operatorname{arccosh}\NVar{z}" display="inline"><mrow><mi href="./4.37#SS2.p1">arccosh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: inverse hyperbolic cosine function</a>,
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a> and
<a href="./4.1#p2.t1.r3" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m98.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./4.1#p2.t1.r3">y</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">It should be noted that the imaginary axis is not a cut; the function defined
by () is analytic everywhere
except on <math class="ltx_Math" altimg="m35.png" altimg-height="23px" altimg-valign="-7px" altimg-width="72px" alttext="(-\infty,1]" display="inline"><mrow><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">(</mo><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mo href="./front/introduction#Sx4.p2.t1.r1">,</mo><mn>1</mn><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">]</mo></mrow></math>. Compare Figure (ii).</p>
</div>
<div id="Px2.p2" class="ltx_para">
<p class="ltx_p">An equivalent definition is</p>
<table id="E21" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.37.21</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E21.png" altimg-height="65px" altimg-valign="-27px" altimg-width="403px" alttext="\operatorname{arccosh}z=2\ln\left(\left(\frac{z+1}{2}\right)^{1/2}+\left(\frac%
{z-1}{2}\right)^{1/2}\right)," display="block"><mrow><mrow><mrow><mi href="./4.37#SS2.p1">arccosh</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>=</mo><mrow><mn>2</mn><mo></mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><mo>(</mo><mrow><msup><mrow><mo>(</mo><mfrac><mrow><mi href="./4.1#p2.t1.r4">z</mi><mo>+</mo><mn>1</mn></mrow><mn>2</mn></mfrac><mo>)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>+</mo><msup><mrow><mo>(</mo><mfrac><mrow><mi href="./4.1#p2.t1.r4">z</mi><mo>-</mo><mn>1</mn></mrow><mn>2</mn></mfrac><mo>)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m107.png" altimg-height="23px" altimg-valign="-7px" altimg-width="142px" alttext="z\in\mathbb{C}\setminus(-\infty,1)" display="inline"><mrow><mi href="./4.1#p2.t1.r4">z</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mrow><mi href="./front/introduction#Sx4.p1.t1.r1">ℂ</mi><mo href="./front/introduction#Sx4.p2.t1.r19">∖</mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mn>1</mn><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></mrow></mrow></math>;</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E21.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./front/introduction#Sx4.p1.t1.r1" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m55.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\mathbb{C}" display="inline"><mi href="./front/introduction#Sx4.p1.t1.r1">ℂ</mi></math>: complex plane</a>,
<a href="./front/introduction#Sx4.p1.t1.r9" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="15px" altimg-valign="-3px" altimg-width="18px" alttext="\in" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo></math>: element of</a>,
<a href="./4.37#SS2.p1" title="§4.37(ii) Principal Values ‣ §4.37 Inverse Hyperbolic Functions ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m65.png" altimg-height="18px" altimg-valign="-2px" altimg-width="82px" alttext="\operatorname{arccosh}\NVar{z}" display="inline"><mrow><mi href="./4.37#SS2.p1">arccosh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: inverse hyperbolic cosine function</a>,
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a>,
<a href="./front/introduction#Sx4.p1.t1.r28" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="(\NVar{a},\NVar{b})" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi class="ltx_nvar">a</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi class="ltx_nvar">b</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math>: open interval</a>,
<a href="./front/introduction#Sx4.p2.t1.r19" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m84.png" altimg-height="23px" altimg-valign="-7px" altimg-width="14px" alttext="\setminus" display="inline"><mo href="./front/introduction#Sx4.p2.t1.r19">∖</mo></math>: set subtraction</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m105.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">see <cite class="ltx_cite ltx_citemacro_citet">Kahan ()</cite>.</p>
</div>
<div id="Px2.p3" class="ltx_para">
<p class="ltx_p">On the part of the cuts from <math class="ltx_Math" altimg="m41.png" altimg-height="18px" altimg-valign="-4px" altimg-width="30px" alttext="-1" display="inline"><mrow><mo>-</mo><mn>1</mn></mrow></math> to <math class="ltx_Math" altimg="m43.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="1" display="inline"><mn>1</mn></math></p>
<table id="E22" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.37.22</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E22.png" altimg-height="41px" altimg-valign="-15px" altimg-width="321px" alttext="\operatorname{arccosh}x=\pm\ln\left(i(1-x^{2})^{1/2}+x\right)," display="block"><mrow><mrow><mrow><mi href="./4.37#SS2.p1">arccosh</mi><mo></mo><mi href="./4.1#p2.t1.r3">x</mi></mrow><mo>=</mo><mrow><mo>±</mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mi mathvariant="normal">i</mi><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><msup><mi href="./4.1#p2.t1.r3">x</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow><mo>+</mo><mi href="./4.1#p2.t1.r3">x</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m95.png" altimg-height="23px" altimg-valign="-7px" altimg-width="98px" alttext="x\in(-1,1]" display="inline"><mrow><mi href="./4.1#p2.t1.r3">x</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mrow><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo href="./front/introduction#Sx4.p2.t1.r1">,</mo><mn>1</mn><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">]</mo></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E22.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./front/introduction#Sx4.p1.t1.r9" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="15px" altimg-valign="-3px" altimg-width="18px" alttext="\in" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo></math>: element of</a>,
<a href="./4.37#SS2.p1" title="§4.37(ii) Principal Values ‣ §4.37 Inverse Hyperbolic Functions ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m65.png" altimg-height="18px" altimg-valign="-2px" altimg-width="82px" alttext="\operatorname{arccosh}\NVar{z}" display="inline"><mrow><mi href="./4.37#SS2.p1">arccosh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: inverse hyperbolic cosine function</a>,
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a>,
<a href="./front/introduction#Sx4.p2.t1.r1" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="(\NVar{a},\NVar{b}]" display="inline"><mrow><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">(</mo><mi class="ltx_nvar">a</mi><mo href="./front/introduction#Sx4.p2.t1.r1">,</mo><mi class="ltx_nvar">b</mi><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">]</mo></mrow></math>: half-closed interval</a> and
<a href="./4.1#p2.t1.r3" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m94.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./4.1#p2.t1.r3">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">the upper/lower sign corresponding to the upper/lower side.</p>
</div>
<div id="Px2.p4" class="ltx_para">
<p class="ltx_p">On the part of the cut from <math class="ltx_Math" altimg="m42.png" altimg-height="17px" altimg-valign="-4px" altimg-width="40px" alttext="-\infty" display="inline"><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow></math> to <math class="ltx_Math" altimg="m41.png" altimg-height="18px" altimg-valign="-4px" altimg-width="30px" alttext="-1" display="inline"><mrow><mo>-</mo><mn>1</mn></mrow></math></p>
<table id="E23" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.37.23</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E23.png" altimg-height="41px" altimg-valign="-15px" altimg-width="354px" alttext="\operatorname{arccosh}x=\pm\pi i+\ln\left((x^{2}-1)^{1/2}-x\right)," display="block"><mrow><mrow><mrow><mi href="./4.37#SS2.p1">arccosh</mi><mo></mo><mi href="./4.1#p2.t1.r3">x</mi></mrow><mo>=</mo><mrow><mrow><mo>±</mo><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi></mrow></mrow><mo>+</mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><mo>(</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><msup><mi href="./4.1#p2.t1.r3">x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>-</mo><mi href="./4.1#p2.t1.r3">x</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m97.png" altimg-height="23px" altimg-valign="-7px" altimg-width="123px" alttext="x\in(-\infty,-1]" display="inline"><mrow><mi href="./4.1#p2.t1.r3">x</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mrow><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">(</mo><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mo href="./front/introduction#Sx4.p2.t1.r1">,</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">]</mo></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E23.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./front/introduction#Sx4.p1.t1.r9" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="15px" altimg-valign="-3px" altimg-width="18px" alttext="\in" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo></math>: element of</a>,
<a href="./4.37#SS2.p1" title="§4.37(ii) Principal Values ‣ §4.37 Inverse Hyperbolic Functions ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m65.png" altimg-height="18px" altimg-valign="-2px" altimg-width="82px" alttext="\operatorname{arccosh}\NVar{z}" display="inline"><mrow><mi href="./4.37#SS2.p1">arccosh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: inverse hyperbolic cosine function</a>,
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a>,
<a href="./front/introduction#Sx4.p2.t1.r1" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="(\NVar{a},\NVar{b}]" display="inline"><mrow><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">(</mo><mi class="ltx_nvar">a</mi><mo href="./front/introduction#Sx4.p2.t1.r1">,</mo><mi class="ltx_nvar">b</mi><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">]</mo></mrow></math>: half-closed interval</a> and
<a href="./4.1#p2.t1.r3" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m94.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./4.1#p2.t1.r3">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">the upper/lower sign corresponding to the upper/lower side.</p>
</div>
</section>
<section id="Px3" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Inverse Hyperbolic Tangent</h3>
<div id="Px3.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px3.p1" class="ltx_para">
<table id="E24" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.37.24</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E24.png" altimg-height="53px" altimg-valign="-21px" altimg-width="236px" alttext="\operatorname{arctanh}z=\tfrac{1}{2}\ln\left(\frac{1+z}{1-z}\right)," display="block"><mrow><mrow><mrow><mi href="./4.37#SS2.p1">arctanh</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>=</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><mo>(</mo><mfrac><mrow><mn>1</mn><mo>+</mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mrow><mn>1</mn><mo>-</mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mfrac><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m106.png" altimg-height="23px" altimg-valign="-7px" altimg-width="229px" alttext="z\in\mathbb{C}\setminus(-\infty,-1]\cup[1,\infty)" display="inline"><mrow><mi href="./4.1#p2.t1.r4">z</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mrow><mrow><mi href="./front/introduction#Sx4.p1.t1.r1">ℂ</mi><mo href="./front/introduction#Sx4.p2.t1.r19">∖</mo><mrow><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">(</mo><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mo href="./front/introduction#Sx4.p2.t1.r1">,</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">]</mo></mrow></mrow><mo href="./front/introduction#Sx4.p1.t1.r27">∪</mo><mrow><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">[</mo><mn>1</mn><mo href="./front/introduction#Sx4.p2.t1.r1">,</mo><mi mathvariant="normal">∞</mi><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">)</mo></mrow></mrow></mrow></math>;</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E24.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./front/introduction#Sx4.p2.t1.r1" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m45.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="[\NVar{a},\NVar{b})" display="inline"><mrow><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">[</mo><mi class="ltx_nvar">a</mi><mo href="./front/introduction#Sx4.p2.t1.r1">,</mo><mi class="ltx_nvar">b</mi><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">)</mo></mrow></math>: half-closed interval</a>,
<a href="./front/introduction#Sx4.p1.t1.r1" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m55.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\mathbb{C}" display="inline"><mi href="./front/introduction#Sx4.p1.t1.r1">ℂ</mi></math>: complex plane</a>,
<a href="./front/introduction#Sx4.p1.t1.r9" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="15px" altimg-valign="-3px" altimg-width="18px" alttext="\in" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo></math>: element of</a>,
<a href="./4.37#SS2.p1" title="§4.37(ii) Principal Values ‣ §4.37 Inverse Hyperbolic Functions ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="18px" altimg-valign="-2px" altimg-width="84px" alttext="\operatorname{arctanh}\NVar{z}" display="inline"><mrow><mi href="./4.37#SS2.p1">arctanh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: inverse hyperbolic tangent function</a>,
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a>,
<a href="./front/introduction#Sx4.p2.t1.r1" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="(\NVar{a},\NVar{b}]" display="inline"><mrow><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">(</mo><mi class="ltx_nvar">a</mi><mo href="./front/introduction#Sx4.p2.t1.r1">,</mo><mi class="ltx_nvar">b</mi><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">]</mo></mrow></math>: half-closed interval</a>,
<a href="./front/introduction#Sx4.p2.t1.r19" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m84.png" altimg-height="23px" altimg-valign="-7px" altimg-width="14px" alttext="\setminus" display="inline"><mo href="./front/introduction#Sx4.p2.t1.r19">∖</mo></math>: set subtraction</a>,
<a href="./front/introduction#Sx4.p1.t1.r27" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="15px" altimg-valign="-2px" altimg-width="18px" alttext="\cup" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r27">∪</mo></math>: union</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m105.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">compare Figure (iii). On the cuts</p>
<table id="E25" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.37.25</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E25.png" altimg-height="53px" altimg-valign="-21px" altimg-width="310px" alttext="\operatorname{arctanh}x=\pm\tfrac{1}{2}\pi i+\tfrac{1}{2}\ln\left(\frac{x+1}{x%
-1}\right)," display="block"><mrow><mrow><mrow><mi href="./4.37#SS2.p1">arctanh</mi><mo></mo><mi href="./4.1#p2.t1.r3">x</mi></mrow><mo>=</mo><mrow><mrow><mo>±</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi></mrow></mrow><mo>+</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><mo>(</mo><mfrac><mrow><mi href="./4.1#p2.t1.r3">x</mi><mo>+</mo><mn>1</mn></mrow><mrow><mi href="./4.1#p2.t1.r3">x</mi><mo>-</mo><mn>1</mn></mrow></mfrac><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m96.png" altimg-height="23px" altimg-valign="-7px" altimg-width="202px" alttext="x\in(-\infty,-1)\cup(1,\infty)" display="inline"><mrow><mi href="./4.1#p2.t1.r3">x</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mrow><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow><mo href="./front/introduction#Sx4.p1.t1.r27">∪</mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mn>1</mn><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi mathvariant="normal">∞</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E25.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./front/introduction#Sx4.p1.t1.r9" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="15px" altimg-valign="-3px" altimg-width="18px" alttext="\in" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo></math>: element of</a>,
<a href="./4.37#SS2.p1" title="§4.37(ii) Principal Values ‣ §4.37 Inverse Hyperbolic Functions ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="18px" altimg-valign="-2px" altimg-width="84px" alttext="\operatorname{arctanh}\NVar{z}" display="inline"><mrow><mi href="./4.37#SS2.p1">arctanh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: inverse hyperbolic tangent function</a>,
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a>,
<a href="./front/introduction#Sx4.p1.t1.r28" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="(\NVar{a},\NVar{b})" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi class="ltx_nvar">a</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi class="ltx_nvar">b</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math>: open interval</a>,
<a href="./front/introduction#Sx4.p1.t1.r27" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="15px" altimg-valign="-2px" altimg-width="18px" alttext="\cup" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r27">∪</mo></math>: union</a> and
<a href="./4.1#p2.t1.r3" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m94.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./4.1#p2.t1.r3">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">the upper/lower sign corresponding to the upper/lower sides.</p>
</div>
</section>
<section id="Px4" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Other Inverse Functions</h3>
<div id="Px4.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px4.p1" class="ltx_para">
<p class="ltx_p">For the corresponding results for <math class="ltx_Math" altimg="m71.png" altimg-height="18px" altimg-valign="-2px" altimg-width="80px" alttext="\operatorname{arccsch}z" display="inline"><mrow><mi href="./4.37#E7">arccsch</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></math>, <math class="ltx_Math" altimg="m74.png" altimg-height="18px" altimg-valign="-2px" altimg-width="80px" alttext="\operatorname{arcsech}z" display="inline"><mrow><mi href="./4.37#E8">arcsech</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></math>,
and <math class="ltx_Math" altimg="m69.png" altimg-height="18px" altimg-valign="-2px" altimg-width="82px" alttext="\operatorname{arccoth}z" display="inline"><mrow><mi href="./4.37#E9">arccoth</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></math>, use (</dd>
</dl>
</div>
</div>
<div id="SS5.p1" class="ltx_para">
<p class="ltx_p">With <math class="ltx_Math" altimg="m90.png" altimg-height="18px" altimg-valign="-3px" altimg-width="53px" alttext="k\in\mathbb{Z}" display="inline"><mrow><mi href="./4.1#p2.t1.r1">k</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mi href="./front/introduction#Sx4.p2.t1.r20">ℤ</mi></mrow></math>, the general solutions of the equations</p>
<table id="E26" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.37.26</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E26.png" altimg-height="23px" altimg-valign="-6px" altimg-width="102px" alttext="z=\sinh w," display="block"><mrow><mrow><mi href="./4.1#p2.t1.r4">z</mi><mo>=</mo><mrow><mi href="./4.28#E1">sinh</mi><mo></mo><mi>w</mi></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E26.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.28#E1" title="(4.28.1) ‣ §4.28 Definitions and Periodicity ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="18px" altimg-valign="-2px" altimg-width="53px" alttext="\sinh\NVar{z}" display="inline"><mrow><mi href="./4.28#E1">sinh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: hyperbolic sine function</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m105.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E27" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.37.27</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E27.png" altimg-height="23px" altimg-valign="-6px" altimg-width="104px" alttext="z=\cosh w," display="block"><mrow><mrow><mi href="./4.1#p2.t1.r4">z</mi><mo>=</mo><mrow><mi href="./4.28#E2">cosh</mi><mo></mo><mi>w</mi></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E27.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.28#E2" title="(4.28.2) ‣ §4.28 Definitions and Periodicity ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="18px" altimg-valign="-2px" altimg-width="56px" alttext="\cosh\NVar{z}" display="inline"><mrow><mi href="./4.28#E2">cosh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: hyperbolic cosine function</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m105.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E28" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.37.28</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E28.png" altimg-height="23px" altimg-valign="-6px" altimg-width="106px" alttext="z=\tanh w," display="block"><mrow><mrow><mi href="./4.1#p2.t1.r4">z</mi><mo>=</mo><mrow><mi href="./4.28#E4">tanh</mi><mo></mo><mi>w</mi></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E28.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.28#E4" title="(4.28.4) ‣ §4.28 Definitions and Periodicity ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="18px" altimg-valign="-2px" altimg-width="58px" alttext="\tanh\NVar{z}" display="inline"><mrow><mi href="./4.28#E4">tanh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: hyperbolic tangent function</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m105.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">are respectively given by
</p>
<table id="EGx4" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E29">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.37.29</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m34.png" altimg-height="14px" altimg-valign="-2px" altimg-width="21px" alttext="\displaystyle w" display="inline"><mi>w</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m12.png" altimg-height="28px" altimg-valign="-7px" altimg-width="324px" alttext="\displaystyle=\operatorname{Arcsinh}z=(-1)^{k}\operatorname{arcsinh}z+k\pi i," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./4.37#E1">Arcsinh</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>=</mo><mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./4.1#p2.t1.r1">k</mi></msup><mo></mo><mrow><mi href="./4.37#SS2.p1">arcsinh</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mrow><mo>+</mo><mrow><mi href="./4.1#p2.t1.r1">k</mi><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E29.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.37#SS2.p1" title="§4.37(ii) Principal Values ‣ §4.37 Inverse Hyperbolic Functions ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m75.png" altimg-height="18px" altimg-valign="-2px" altimg-width="80px" alttext="\operatorname{arcsinh}\NVar{z}" display="inline"><mrow><mi href="./4.37#SS2.p1">arcsinh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: inverse hyperbolic sine function</a>,
<a href="./4.37#E1" title="(4.37.1) ‣ §4.37(i) General Definitions ‣ §4.37 Inverse Hyperbolic Functions ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m62.png" altimg-height="18px" altimg-valign="-2px" altimg-width="85px" alttext="\operatorname{Arcsinh}\NVar{z}" display="inline"><mrow><mi href="./4.37#E1">Arcsinh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: general inverse hyperbolic sine function</a>,
<a href="./4.1#p2.t1.r1" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m89.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./4.1#p2.t1.r1">k</mi></math>: integer</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m105.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.6.8</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E30">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.37.30</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m34.png" altimg-height="14px" altimg-valign="-2px" altimg-width="21px" alttext="\displaystyle w" display="inline"><mi>w</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m10.png" altimg-height="23px" altimg-valign="-6px" altimg-width="303px" alttext="\displaystyle=\operatorname{Arccosh}z=\pm\operatorname{arccosh}z+2k\pi i," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./4.37#E2">Arccosh</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>=</mo><mrow><mrow><mo>±</mo><mrow><mi href="./4.37#SS2.p1">arccosh</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mrow><mo>+</mo><mrow><mn>2</mn><mo></mo><mi href="./4.1#p2.t1.r1">k</mi><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E30.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.37#SS2.p1" title="§4.37(ii) Principal Values ‣ §4.37 Inverse Hyperbolic Functions ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m65.png" altimg-height="18px" altimg-valign="-2px" altimg-width="82px" alttext="\operatorname{arccosh}\NVar{z}" display="inline"><mrow><mi href="./4.37#SS2.p1">arccosh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: inverse hyperbolic cosine function</a>,
<a href="./4.37#E2" title="(4.37.2) ‣ §4.37(i) General Definitions ‣ §4.37 Inverse Hyperbolic Functions ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m57.png" altimg-height="18px" altimg-valign="-2px" altimg-width="87px" alttext="\operatorname{Arccosh}\NVar{z}" display="inline"><mrow><mi href="./4.37#E2">Arccosh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: general inverse hyperbolic cosine function</a>,
<a href="./4.1#p2.t1.r1" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m89.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./4.1#p2.t1.r1">k</mi></math>: integer</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m105.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.6.9</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E31">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.37.31</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m34.png" altimg-height="14px" altimg-valign="-2px" altimg-width="21px" alttext="\displaystyle w" display="inline"><mi>w</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m14.png" altimg-height="23px" altimg-valign="-6px" altimg-width="278px" alttext="\displaystyle=\operatorname{Arctanh}z=\operatorname{arctanh}z+k\pi i," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./4.37#E3">Arctanh</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>=</mo><mrow><mrow><mi href="./4.37#SS2.p1">arctanh</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>+</mo><mrow><mi href="./4.1#p2.t1.r1">k</mi><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m108.png" altimg-height="21px" altimg-valign="-6px" altimg-width="66px" alttext="z\neq\pm 1" display="inline"><mrow><mi href="./4.1#p2.t1.r4">z</mi><mo>≠</mo><mrow><mo>±</mo><mn>1</mn></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E31.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.37#SS2.p1" title="§4.37(ii) Principal Values ‣ §4.37 Inverse Hyperbolic Functions ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="18px" altimg-valign="-2px" altimg-width="84px" alttext="\operatorname{arctanh}\NVar{z}" display="inline"><mrow><mi href="./4.37#SS2.p1">arctanh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: inverse hyperbolic tangent function</a>,
<a href="./4.37#E3" title="(4.37.3) ‣ §4.37(i) General Definitions ‣ §4.37 Inverse Hyperbolic Functions ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m64.png" altimg-height="18px" altimg-valign="-2px" altimg-width="89px" alttext="\operatorname{Arctanh}\NVar{z}" display="inline"><mrow><mi href="./4.37#E3">Arctanh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: general inverse hyperbolic tangent function</a>,
<a href="./4.1#p2.t1.r1" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m89.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./4.1#p2.t1.r1">k</mi></math>: integer</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m105.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.6.10</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
</div>
</section>
<section id="vi" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§4.37(vi) </span>Interrelations</h2>
<div id="SS6.info" class="ltx_metadata ltx_info">
can also be used to find interrelations between
inverse hyperbolic functions. For example, <math class="ltx_Math" altimg="m73.png" altimg-height="28px" altimg-valign="-9px" altimg-width="300px" alttext="\operatorname{arcsech}a=\operatorname{arccoth}\left((1-a^{2})^{-1/2}\right)" display="inline"><mrow><mrow><mi href="./4.37#E8">arcsech</mi><mo></mo><mi href="./4.1#p2.t1.r2">a</mi></mrow><mo>=</mo><mrow><mi href="./4.37#E9">arccoth</mi><mo></mo><mrow><mo>(</mo><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><msup><mi href="./4.1#p2.t1.r2">a</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></mrow></msup><mo>)</mo></mrow></mrow></mrow></math>.</p>
</div>
</section>
</section>
</div>
<div class="ltx_page_footer">
<span></div>
</div>
</body></text>
</html>
</page>
<page>
<!DOCTYPE html><html>
<head>
<title>DLMF: 4.38 Inverse Hyperbolic Functions: Further Properties</title>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<!--GOOGLE BOOTSTRAP--></head>
<text><body class="color_default textfont_default titlefont_default fontsize_default navbar_default">
<div class="ltx_page_navbar">
<div class="ltx_page_navlogo">) expand <math class="ltx_Math" altimg="m14.png" altimg-height="26px" altimg-valign="-7px" altimg-width="111px" alttext="(1+z^{2})^{-1/2}" display="inline"><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>+</mo><msup><mi href="./4.1#p2.t1.r4">z</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></mrow></msup></math> by the binomial
theorem and integrate term by term. For (), write
<math class="ltx_Math" altimg="m13.png" altimg-height="26px" altimg-valign="-7px" altimg-width="311px" alttext="(1+z^{2})^{-1/2}=z^{-1}(1+(1/z^{2}))^{-1/2}" display="inline"><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>+</mo><msup><mi href="./4.1#p2.t1.r4">z</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></mrow></msup><mo>=</mo><mrow><msup><mi href="./4.1#p2.t1.r4">z</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>+</mo><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>/</mo><msup><mi href="./4.1#p2.t1.r4">z</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></mrow></msup></mrow></mrow></math>, <math class="ltx_Math" altimg="m19.png" altimg-height="18px" altimg-valign="-3px" altimg-width="65px" alttext="\Re z>0" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>></mo><mn>0</mn></mrow></math>, and
then expand and integrate. For the constant of integration note that for
large <math class="ltx_Math" altimg="m46.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>, <math class="ltx_Math" altimg="m38.png" altimg-height="18px" altimg-valign="-2px" altimg-width="80px" alttext="\operatorname{arcsinh}z" display="inline"><mrow><mi href="./4.37#SS2.p1">arcsinh</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></math> behaves like <math class="ltx_Math" altimg="m28.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="\ln\left(2z\right)" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></math> and the constant is
<math class="ltx_Math" altimg="m26.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln 2" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mn>2</mn></mrow></math>. (</dd>
</dl>
</div>
</div>
<div id="SS1.p1" class="ltx_para">
<table id="E1" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.38.1</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E1.png" altimg-height="50px" altimg-valign="-16px" altimg-width="437px" alttext="\operatorname{arcsinh}z=z-\frac{1}{2}\frac{z^{3}}{3}+\frac{1\cdot 3}{2\cdot 4}%
\frac{z^{5}}{5}-\frac{1\cdot 3\cdot 5}{2\cdot 4\cdot 6}\frac{z^{7}}{7}+\cdots," display="block"><mrow><mrow><mrow><mi href="./4.37#SS2.p1">arcsinh</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>=</mo><mrow><mrow><mrow><mrow><mi href="./4.1#p2.t1.r4">z</mi><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mfrac><msup><mi href="./4.1#p2.t1.r4">z</mi><mn>3</mn></msup><mn>3</mn></mfrac></mrow></mrow><mo>+</mo><mrow><mfrac><mrow><mn>1</mn><mo>⋅</mo><mn>3</mn></mrow><mrow><mn>2</mn><mo>⋅</mo><mn>4</mn></mrow></mfrac><mo></mo><mfrac><msup><mi href="./4.1#p2.t1.r4">z</mi><mn>5</mn></msup><mn>5</mn></mfrac></mrow></mrow><mo>-</mo><mrow><mfrac><mrow><mn>1</mn><mo>⋅</mo><mn>3</mn><mo>⋅</mo><mn>5</mn></mrow><mrow><mn>2</mn><mo>⋅</mo><mn>4</mn><mo>⋅</mo><mn>6</mn></mrow></mfrac><mo></mo><mfrac><msup><mi href="./4.1#p2.t1.r4">z</mi><mn>7</mn></msup><mn>7</mn></mfrac></mrow></mrow><mo>+</mo><mi mathvariant="normal">⋯</mi></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m49.png" altimg-height="23px" altimg-valign="-7px" altimg-width="62px" alttext="|z|<1" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mi href="./4.1#p2.t1.r4">z</mi><mo stretchy="false">|</mo></mrow><mo><</mo><mn>1</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.37#SS2.p1" title="§4.37(ii) Principal Values ‣ §4.37 Inverse Hyperbolic Functions ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="18px" altimg-valign="-2px" altimg-width="80px" alttext="\operatorname{arcsinh}\NVar{z}" display="inline"><mrow><mi href="./4.37#SS2.p1">arcsinh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: inverse hyperbolic sine function</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m46.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.6.31</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E2" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.38.2</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E2.png" altimg-height="46px" altimg-valign="-16px" altimg-width="512px" alttext="\operatorname{arcsinh}z=\ln\left(2z\right)+\frac{1}{2}\frac{1}{2z^{2}}-\frac{1%
\cdot 3}{2\cdot 4}\frac{1}{4z^{4}}+\frac{1\cdot 3\cdot 5}{2\cdot 4\cdot 6}%
\frac{1}{6z^{6}}-\cdots," display="block"><mrow><mrow><mrow><mi href="./4.37#SS2.p1">arcsinh</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>=</mo><mrow><mrow><mrow><mrow><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow><mo>+</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mfrac><mn>1</mn><mrow><mn>2</mn><mo></mo><msup><mi href="./4.1#p2.t1.r4">z</mi><mn>2</mn></msup></mrow></mfrac></mrow></mrow><mo>-</mo><mrow><mfrac><mrow><mn>1</mn><mo>⋅</mo><mn>3</mn></mrow><mrow><mn>2</mn><mo>⋅</mo><mn>4</mn></mrow></mfrac><mo></mo><mfrac><mn>1</mn><mrow><mn>4</mn><mo></mo><msup><mi href="./4.1#p2.t1.r4">z</mi><mn>4</mn></msup></mrow></mfrac></mrow></mrow><mo>+</mo><mrow><mfrac><mrow><mn>1</mn><mo>⋅</mo><mn>3</mn><mo>⋅</mo><mn>5</mn></mrow><mrow><mn>2</mn><mo>⋅</mo><mn>4</mn><mo>⋅</mo><mn>6</mn></mrow></mfrac><mo></mo><mfrac><mn>1</mn><mrow><mn>6</mn><mo></mo><msup><mi href="./4.1#p2.t1.r4">z</mi><mn>6</mn></msup></mrow></mfrac></mrow></mrow><mo>-</mo><mi mathvariant="normal">⋯</mi></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m19.png" altimg-height="18px" altimg-valign="-3px" altimg-width="65px" alttext="\Re z>0" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>></mo><mn>0</mn></mrow></math>, <math class="ltx_Math" altimg="m50.png" altimg-height="23px" altimg-valign="-7px" altimg-width="62px" alttext="|z|>1" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mi href="./4.1#p2.t1.r4">z</mi><mo stretchy="false">|</mo></mrow><mo>></mo><mn>1</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.37#SS2.p1" title="§4.37(ii) Principal Values ‣ §4.37 Inverse Hyperbolic Functions ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="18px" altimg-valign="-2px" altimg-width="80px" alttext="\operatorname{arcsinh}\NVar{z}" display="inline"><mrow><mi href="./4.37#SS2.p1">arcsinh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: inverse hyperbolic sine function</a>,
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m27.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m46.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.6.31</span> (misses a condition on <math class="ltx_Math" altimg="m46.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>.)</span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E3" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.38.3</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E3.png" altimg-height="46px" altimg-valign="-16px" altimg-width="514px" alttext="\operatorname{arccosh}z=\ln\left(2z\right)-\frac{1}{2}\frac{1}{2z^{2}}-\frac{1%
\cdot 3}{2\cdot 4}\frac{1}{4z^{4}}-\frac{1\cdot 3\cdot 5}{2\cdot 4\cdot 6}%
\frac{1}{6z^{6}}-\cdots," display="block"><mrow><mrow><mrow><mi href="./4.37#SS2.p1">arccosh</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>=</mo><mrow><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mfrac><mn>1</mn><mrow><mn>2</mn><mo></mo><msup><mi href="./4.1#p2.t1.r4">z</mi><mn>2</mn></msup></mrow></mfrac></mrow><mo>-</mo><mrow><mfrac><mrow><mn>1</mn><mo>⋅</mo><mn>3</mn></mrow><mrow><mn>2</mn><mo>⋅</mo><mn>4</mn></mrow></mfrac><mo></mo><mfrac><mn>1</mn><mrow><mn>4</mn><mo></mo><msup><mi href="./4.1#p2.t1.r4">z</mi><mn>4</mn></msup></mrow></mfrac></mrow><mo>-</mo><mrow><mfrac><mrow><mn>1</mn><mo>⋅</mo><mn>3</mn><mo>⋅</mo><mn>5</mn></mrow><mrow><mn>2</mn><mo>⋅</mo><mn>4</mn><mo>⋅</mo><mn>6</mn></mrow></mfrac><mo></mo><mfrac><mn>1</mn><mrow><mn>6</mn><mo></mo><msup><mi href="./4.1#p2.t1.r4">z</mi><mn>6</mn></msup></mrow></mfrac></mrow><mo>-</mo><mi mathvariant="normal">⋯</mi></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m50.png" altimg-height="23px" altimg-valign="-7px" altimg-width="62px" alttext="|z|>1" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mi href="./4.1#p2.t1.r4">z</mi><mo stretchy="false">|</mo></mrow><mo>></mo><mn>1</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.37#SS2.p1" title="§4.37(ii) Principal Values ‣ §4.37 Inverse Hyperbolic Functions ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m33.png" altimg-height="18px" altimg-valign="-2px" altimg-width="82px" alttext="\operatorname{arccosh}\NVar{z}" display="inline"><mrow><mi href="./4.37#SS2.p1">arccosh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: inverse hyperbolic cosine function</a>,
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m27.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m46.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.6.32</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E4" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.38.4</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E4.png" altimg-height="66px" altimg-valign="-27px" altimg-width="613px" alttext="\operatorname{arccosh}z=(2(z-1))^{1/2}\*{\left(1+\sum_{n=1}^{\infty}(-1)^{n}%
\frac{1\cdot 3\cdot 5\cdots(2n-1)}{2^{2n}n!(2n+1)}(z-1)^{n}\right)}," display="block"><mrow><mrow><mrow><mi href="./4.37#SS2.p1">arccosh</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>=</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mn>2</mn><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./4.1#p2.t1.r4">z</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./4.1#p2.t1.r1">n</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./4.1#p2.t1.r1">n</mi></msup><mo></mo><mfrac><mrow><mrow><mn>1</mn><mo>⋅</mo><mn>3</mn><mo>⋅</mo><mn>5</mn></mrow><mo></mo><mi mathvariant="normal">⋯</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>2</mn><mo></mo><mi href="./4.1#p2.t1.r1">n</mi></mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mrow><msup><mn>2</mn><mrow><mn>2</mn><mo></mo><mi href="./4.1#p2.t1.r1">n</mi></mrow></msup><mo></mo><mrow><mi href="./4.1#p2.t1.r1">n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>2</mn><mo></mo><mi href="./4.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow></mfrac><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./4.1#p2.t1.r4">z</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./4.1#p2.t1.r1">n</mi></msup></mrow></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m19.png" altimg-height="18px" altimg-valign="-3px" altimg-width="65px" alttext="\Re z>0" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>></mo><mn>0</mn></mrow></math>, <math class="ltx_Math" altimg="m48.png" altimg-height="23px" altimg-valign="-7px" altimg-width="96px" alttext="|z-1|\leq 2" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./4.1#p2.t1.r4">z</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">|</mo></mrow><mo>≤</mo><mn>2</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E4.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m12.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m42.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./4.37#SS2.p1" title="§4.37(ii) Principal Values ‣ §4.37 Inverse Hyperbolic Functions ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m33.png" altimg-height="18px" altimg-valign="-2px" altimg-width="82px" alttext="\operatorname{arccosh}\NVar{z}" display="inline"><mrow><mi href="./4.37#SS2.p1">arccosh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: inverse hyperbolic cosine function</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./4.1#p2.t1.r1" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./4.1#p2.t1.r1">n</mi></math>: integer</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m46.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E5" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.38.5</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E5.png" altimg-height="50px" altimg-valign="-16px" altimg-width="324px" alttext="\operatorname{arctanh}z=z+\frac{z^{3}}{3}+\frac{z^{5}}{5}+\frac{z^{7}}{7}+\cdots," display="block"><mrow><mrow><mrow><mi href="./4.37#SS2.p1">arctanh</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>=</mo><mrow><mi href="./4.1#p2.t1.r4">z</mi><mo>+</mo><mfrac><msup><mi href="./4.1#p2.t1.r4">z</mi><mn>3</mn></msup><mn>3</mn></mfrac><mo>+</mo><mfrac><msup><mi href="./4.1#p2.t1.r4">z</mi><mn>5</mn></msup><mn>5</mn></mfrac><mo>+</mo><mfrac><msup><mi href="./4.1#p2.t1.r4">z</mi><mn>7</mn></msup><mn>7</mn></mfrac><mo>+</mo><mi mathvariant="normal">⋯</mi></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m25.png" altimg-height="23px" altimg-valign="-7px" altimg-width="62px" alttext="\left|z\right|\leq 1" display="inline"><mrow><mrow><mo>|</mo><mi href="./4.1#p2.t1.r4">z</mi><mo>|</mo></mrow><mo>≤</mo><mn>1</mn></mrow></math>, <math class="ltx_Math" altimg="m47.png" altimg-height="21px" altimg-valign="-6px" altimg-width="66px" alttext="z\neq\pm 1" display="inline"><mrow><mi href="./4.1#p2.t1.r4">z</mi><mo>≠</mo><mrow><mo>±</mo><mn>1</mn></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E5.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.37#SS2.p1" title="§4.37(ii) Principal Values ‣ §4.37 Inverse Hyperbolic Functions ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="18px" altimg-valign="-2px" altimg-width="84px" alttext="\operatorname{arctanh}\NVar{z}" display="inline"><mrow><mi href="./4.37#SS2.p1">arctanh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: inverse hyperbolic tangent function</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m46.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.6.33</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E6" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.38.6</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E6.png" altimg-height="46px" altimg-valign="-16px" altimg-width="363px" alttext="\operatorname{arctanh}z=\pm\mathrm{i}\frac{\pi}{2}+\frac{1}{z}+\frac{1}{3z^{3}%
}+\frac{1}{5z^{5}}+\cdots," display="block"><mrow><mrow><mrow><mi href="./4.37#SS2.p1">arctanh</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>=</mo><mrow><mrow><mo>±</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mfrac><mi href="./3.12#E1">π</mi><mn>2</mn></mfrac></mrow></mrow><mo>+</mo><mfrac><mn>1</mn><mi href="./4.1#p2.t1.r4">z</mi></mfrac><mo>+</mo><mfrac><mn>1</mn><mrow><mn>3</mn><mo></mo><msup><mi href="./4.1#p2.t1.r4">z</mi><mn>3</mn></msup></mrow></mfrac><mo>+</mo><mfrac><mn>1</mn><mrow><mn>5</mn><mo></mo><msup><mi href="./4.1#p2.t1.r4">z</mi><mn>5</mn></msup></mrow></mfrac><mo>+</mo><mi mathvariant="normal">⋯</mi></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m17.png" altimg-height="21px" altimg-valign="-6px" altimg-width="65px" alttext="\Im z\gtrless 0" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℑ</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>≷</mo><mn>0</mn></mrow></math>, <math class="ltx_Math" altimg="m24.png" altimg-height="23px" altimg-valign="-7px" altimg-width="62px" alttext="\left|z\right|\geq 1" display="inline"><mrow><mrow><mo>|</mo><mi href="./4.1#p2.t1.r4">z</mi><mo>|</mo></mrow><mo>≥</mo><mn>1</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E6.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m40.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.37#SS2.p1" title="§4.37(ii) Principal Values ‣ §4.37 Inverse Hyperbolic Functions ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="18px" altimg-valign="-2px" altimg-width="84px" alttext="\operatorname{arctanh}\NVar{z}" display="inline"><mrow><mi href="./4.37#SS2.p1">arctanh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: inverse hyperbolic tangent function</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m18.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Im" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℑ</mi><mo></mo></mrow></math>: imaginary part</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m46.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.6.33</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E7" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.38.7</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E7.png" altimg-height="65px" altimg-valign="-27px" altimg-width="529px" alttext="\operatorname{arctanh}z=\frac{z}{1-z^{2}}\*{\left(1+\frac{2}{3}\frac{z^{2}}{z^%
{2}-1}+\frac{2\cdot 4}{3\cdot 5}\left(\frac{z^{2}}{z^{2}-1}\right)^{2}+\cdots%
\right)}," display="block"><mrow><mrow><mrow><mi href="./4.37#SS2.p1">arctanh</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>=</mo><mrow><mfrac><mi href="./4.1#p2.t1.r4">z</mi><mrow><mn>1</mn><mo>-</mo><msup><mi href="./4.1#p2.t1.r4">z</mi><mn>2</mn></msup></mrow></mfrac><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mrow><mfrac><mn>2</mn><mn>3</mn></mfrac><mo></mo><mfrac><msup><mi href="./4.1#p2.t1.r4">z</mi><mn>2</mn></msup><mrow><msup><mi href="./4.1#p2.t1.r4">z</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow></mfrac></mrow><mo>+</mo><mrow><mfrac><mrow><mn>2</mn><mo>⋅</mo><mn>4</mn></mrow><mrow><mn>3</mn><mo>⋅</mo><mn>5</mn></mrow></mfrac><mo></mo><msup><mrow><mo>(</mo><mfrac><msup><mi href="./4.1#p2.t1.r4">z</mi><mn>2</mn></msup><mrow><msup><mi href="./4.1#p2.t1.r4">z</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow></mfrac><mo>)</mo></mrow><mn>2</mn></msup></mrow><mo>+</mo><mi mathvariant="normal">⋯</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m21.png" altimg-height="27px" altimg-valign="-9px" altimg-width="93px" alttext="\Re(z^{2})<\tfrac{1}{2}" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mrow><mo stretchy="false">(</mo><msup><mi href="./4.1#p2.t1.r4">z</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow></mrow><mo><</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E7.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.37#SS2.p1" title="§4.37(ii) Principal Values ‣ §4.37 Inverse Hyperbolic Functions ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="18px" altimg-valign="-2px" altimg-width="84px" alttext="\operatorname{arctanh}\NVar{z}" display="inline"><mrow><mi href="./4.37#SS2.p1">arctanh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: inverse hyperbolic tangent function</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m46.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">which requires <math class="ltx_Math" altimg="m46.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math> <math class="ltx_Math" altimg="m15.png" altimg-height="23px" altimg-valign="-7px" altimg-width="94px" alttext="(=x+iy)" display="inline"><mrow><mo stretchy="false">(</mo><mrow><mi></mi><mo>=</mo><mrow><mi href="./4.1#p2.t1.r3">x</mi><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./4.1#p2.t1.r3">y</mi></mrow></mrow></mrow><mo stretchy="false">)</mo></mrow></math> to lie between the two rectangular hyperbolas given by</p>
<table id="E8" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.38.8</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E8.png" altimg-height="30px" altimg-valign="-9px" altimg-width="115px" alttext="x^{2}-y^{2}=\tfrac{1}{2}." display="block"><mrow><mrow><mrow><msup><mi href="./4.1#p2.t1.r3">x</mi><mn>2</mn></msup><mo>-</mo><msup><mi href="./4.1#p2.t1.r3">y</mi><mn>2</mn></msup></mrow><mo>=</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E8.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.1#p2.t1.r3" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./4.1#p2.t1.r3">x</mi></math>: real variable</a> and
<a href="./4.1#p2.t1.r3" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m45.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./4.1#p2.t1.r3">y</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="ii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§4.38(ii) </span>Derivatives</h2>
<div id="SS2.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="SS2.p1" class="ltx_para">
<p class="ltx_p">In the following equations square roots have their principal values.
</p>
<table id="EGx1" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E9">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.38.9</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m10.png" altimg-height="47px" altimg-valign="-16px" altimg-width="111px" alttext="\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{arcsinh}z" display="inline"><mrow><mstyle displaystyle="true"><mfrac><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mfrac></mstyle><mo></mo><mrow><mi href="./4.37#SS2.p1">arcsinh</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m1.png" altimg-height="29px" altimg-valign="-7px" altimg-width="139px" alttext="\displaystyle=(1+z^{2})^{-1/2}." display="inline"><mrow><mrow><mi></mi><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>+</mo><msup><mi href="./4.1#p2.t1.r4">z</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></mrow></msup></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E9.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#E4" title="(1.4.4) ‣ §1.4(iii) Derivatives ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m23.png" altimg-height="29px" altimg-valign="-9px" altimg-width="27px" alttext="\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: derivative of <math class="ltx_Math" altimg="m41.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m44.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.37#SS2.p1" title="§4.37(ii) Principal Values ‣ §4.37 Inverse Hyperbolic Functions ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="18px" altimg-valign="-2px" altimg-width="80px" alttext="\operatorname{arcsinh}\NVar{z}" display="inline"><mrow><mi href="./4.37#SS2.p1">arcsinh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: inverse hyperbolic sine function</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m46.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.6.37</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E10">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.38.10</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m6.png" altimg-height="47px" altimg-valign="-16px" altimg-width="113px" alttext="\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{arccosh}z" display="inline"><mrow><mstyle displaystyle="true"><mfrac><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mfrac></mstyle><mo></mo><mrow><mi href="./4.37#SS2.p1">arccosh</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m5.png" altimg-height="29px" altimg-valign="-7px" altimg-width="155px" alttext="\displaystyle=\pm(z^{2}-1)^{-1/2}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo>±</mo><msup><mrow><mo stretchy="false">(</mo><mrow><msup><mi href="./4.1#p2.t1.r4">z</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></mrow></msup></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m20.png" altimg-height="21px" altimg-valign="-6px" altimg-width="65px" alttext="\Re z\gtrless 0" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>≷</mo><mn>0</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E10.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#E4" title="(1.4.4) ‣ §1.4(iii) Derivatives ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m23.png" altimg-height="29px" altimg-valign="-9px" altimg-width="27px" alttext="\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: derivative of <math class="ltx_Math" altimg="m41.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m44.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.37#SS2.p1" title="§4.37(ii) Principal Values ‣ §4.37 Inverse Hyperbolic Functions ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m33.png" altimg-height="18px" altimg-valign="-2px" altimg-width="82px" alttext="\operatorname{arccosh}\NVar{z}" display="inline"><mrow><mi href="./4.37#SS2.p1">arccosh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: inverse hyperbolic cosine function</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m46.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.6.38</span> (has an error.)</span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E11">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.38.11</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m11.png" altimg-height="47px" altimg-valign="-16px" altimg-width="115px" alttext="\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{arctanh}z" display="inline"><mrow><mstyle displaystyle="true"><mfrac><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mfrac></mstyle><mo></mo><mrow><mi href="./4.37#SS2.p1">arctanh</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m3.png" altimg-height="47px" altimg-valign="-17px" altimg-width="91px" alttext="\displaystyle=\frac{1}{1-z^{2}}." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mstyle displaystyle="true"><mfrac><mn>1</mn><mrow><mn>1</mn><mo>-</mo><msup><mi href="./4.1#p2.t1.r4">z</mi><mn>2</mn></msup></mrow></mfrac></mstyle></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E11.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#E4" title="(1.4.4) ‣ §1.4(iii) Derivatives ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m23.png" altimg-height="29px" altimg-valign="-9px" altimg-width="27px" alttext="\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: derivative of <math class="ltx_Math" altimg="m41.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m44.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.37#SS2.p1" title="§4.37(ii) Principal Values ‣ §4.37 Inverse Hyperbolic Functions ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="18px" altimg-valign="-2px" altimg-width="84px" alttext="\operatorname{arctanh}\NVar{z}" display="inline"><mrow><mi href="./4.37#SS2.p1">arctanh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: inverse hyperbolic tangent function</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m46.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.6.39</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E12">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.38.12</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m8.png" altimg-height="47px" altimg-valign="-16px" altimg-width="112px" alttext="\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{arccsch}z" display="inline"><mrow><mstyle displaystyle="true"><mfrac><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mfrac></mstyle><mo></mo><mrow><mi href="./4.37#E7">arccsch</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m4.png" altimg-height="52px" altimg-valign="-21px" altimg-width="157px" alttext="\displaystyle=\mp\frac{1}{z(1+z^{2})^{1/2}}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo>∓</mo><mstyle displaystyle="true"><mfrac><mn>1</mn><mrow><mi href="./4.1#p2.t1.r4">z</mi><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>+</mo><msup><mi href="./4.1#p2.t1.r4">z</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></mfrac></mstyle></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m20.png" altimg-height="21px" altimg-valign="-6px" altimg-width="65px" alttext="\Re z\gtrless 0" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>≷</mo><mn>0</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E12.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#E4" title="(1.4.4) ‣ §1.4(iii) Derivatives ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m23.png" altimg-height="29px" altimg-valign="-9px" altimg-width="27px" alttext="\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: derivative of <math class="ltx_Math" altimg="m41.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m44.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.37#E7" title="(4.37.7) ‣ §4.37(ii) Principal Values ‣ §4.37 Inverse Hyperbolic Functions ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m35.png" altimg-height="18px" altimg-valign="-2px" altimg-width="80px" alttext="\operatorname{arccsch}\NVar{z}" display="inline"><mrow><mi href="./4.37#E7">arccsch</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: inverse hyperbolic cosecant function</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m46.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.6.40</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E13">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.38.13</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m9.png" altimg-height="47px" altimg-valign="-16px" altimg-width="112px" alttext="\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{arcsech}z" display="inline"><mrow><mstyle displaystyle="true"><mfrac><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mfrac></mstyle><mo></mo><mrow><mi href="./4.37#E8">arcsech</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m2.png" altimg-height="52px" altimg-valign="-21px" altimg-width="157px" alttext="\displaystyle=-\frac{1}{z(1-z^{2})^{1/2}}." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo>-</mo><mstyle displaystyle="true"><mfrac><mn>1</mn><mrow><mi href="./4.1#p2.t1.r4">z</mi><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><msup><mi href="./4.1#p2.t1.r4">z</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></mfrac></mstyle></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E13.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#E4" title="(1.4.4) ‣ §1.4(iii) Derivatives ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m23.png" altimg-height="29px" altimg-valign="-9px" altimg-width="27px" alttext="\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: derivative of <math class="ltx_Math" altimg="m41.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m44.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.37#E8" title="(4.37.8) ‣ §4.37(ii) Principal Values ‣ §4.37 Inverse Hyperbolic Functions ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m36.png" altimg-height="18px" altimg-valign="-2px" altimg-width="80px" alttext="\operatorname{arcsech}\NVar{z}" display="inline"><mrow><mi href="./4.37#E8">arcsech</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: inverse hyperbolic secant function</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m46.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.6.41</span> (has an error.)</span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E14">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.38.14</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m7.png" altimg-height="47px" altimg-valign="-16px" altimg-width="113px" alttext="\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{arccoth}z" display="inline"><mrow><mstyle displaystyle="true"><mfrac><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mfrac></mstyle><mo></mo><mrow><mi href="./4.37#E9">arccoth</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m3.png" altimg-height="47px" altimg-valign="-17px" altimg-width="91px" alttext="\displaystyle=\frac{1}{1-z^{2}}." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mstyle displaystyle="true"><mfrac><mn>1</mn><mrow><mn>1</mn><mo>-</mo><msup><mi href="./4.1#p2.t1.r4">z</mi><mn>2</mn></msup></mrow></mfrac></mstyle></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E14.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#E4" title="(1.4.4) ‣ §1.4(iii) Derivatives ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m23.png" altimg-height="29px" altimg-valign="-9px" altimg-width="27px" alttext="\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: derivative of <math class="ltx_Math" altimg="m41.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m44.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.37#E9" title="(4.37.9) ‣ §4.37(ii) Principal Values ‣ §4.37 Inverse Hyperbolic Functions ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m34.png" altimg-height="18px" altimg-valign="-2px" altimg-width="82px" alttext="\operatorname{arccoth}\NVar{z}" display="inline"><mrow><mi href="./4.37#E9">arccoth</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: inverse hyperbolic cotangent function</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m46.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.6.42</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
</div>
</section>
<section id="iii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§4.38(iii) </span>Addition Formulas</h2>
<div id="SS3.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="SS3.p1" class="ltx_para">
<table id="E15" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.38.15</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E15.png" altimg-height="41px" altimg-valign="-15px" altimg-width="558px" alttext="\operatorname{Arcsinh}u\pm\operatorname{Arcsinh}v=\operatorname{Arcsinh}\left(%
u(1+v^{2})^{1/2}\pm v(1+u^{2})^{1/2}\right)," display="block"><mrow><mrow><mrow><mrow><mi href="./4.37#E1">Arcsinh</mi><mo></mo><mi>u</mi></mrow><mo>±</mo><mrow><mi href="./4.37#E1">Arcsinh</mi><mo></mo><mi>v</mi></mrow></mrow><mo>=</mo><mrow><mi href="./4.37#E1">Arcsinh</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mi>u</mi><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>+</mo><msup><mi>v</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow><mo>±</mo><mrow><mi>v</mi><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>+</mo><msup><mi>u</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E15.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./4.37#E1" title="(4.37.1) ‣ §4.37(i) General Definitions ‣ §4.37 Inverse Hyperbolic Functions ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m31.png" altimg-height="18px" altimg-valign="-2px" altimg-width="85px" alttext="\operatorname{Arcsinh}\NVar{z}" display="inline"><mrow><mi href="./4.37#E1">Arcsinh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: general inverse hyperbolic sine function</a></dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.6.26</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E16" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.38.16</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E16.png" altimg-height="41px" altimg-valign="-15px" altimg-width="555px" alttext="\operatorname{Arccosh}u\pm\operatorname{Arccosh}v=\operatorname{Arccosh}\left(%
uv\pm((u^{2}-1)(v^{2}-1))^{1/2}\right)," display="block"><mrow><mrow><mrow><mrow><mi href="./4.37#E2">Arccosh</mi><mo></mo><mi>u</mi></mrow><mo>±</mo><mrow><mi href="./4.37#E2">Arccosh</mi><mo></mo><mi>v</mi></mrow></mrow><mo>=</mo><mrow><mi href="./4.37#E2">Arccosh</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mi>u</mi><mo></mo><mi>v</mi></mrow><mo>±</mo><msup><mrow><mo stretchy="false">(</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><msup><mi>u</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><msup><mi>v</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E16.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./4.37#E2" title="(4.37.2) ‣ §4.37(i) General Definitions ‣ §4.37 Inverse Hyperbolic Functions ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="18px" altimg-valign="-2px" altimg-width="87px" alttext="\operatorname{Arccosh}\NVar{z}" display="inline"><mrow><mi href="./4.37#E2">Arccosh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: general inverse hyperbolic cosine function</a></dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.6.27</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E17" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.38.17</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E17.png" altimg-height="53px" altimg-valign="-21px" altimg-width="402px" alttext="\operatorname{Arctanh}u\pm\operatorname{Arctanh}v=\operatorname{Arctanh}\left(%
\frac{u\pm v}{1\pm uv}\right)," display="block"><mrow><mrow><mrow><mrow><mi href="./4.37#E3">Arctanh</mi><mo></mo><mi>u</mi></mrow><mo>±</mo><mrow><mi href="./4.37#E3">Arctanh</mi><mo></mo><mi>v</mi></mrow></mrow><mo>=</mo><mrow><mi href="./4.37#E3">Arctanh</mi><mo></mo><mrow><mo>(</mo><mfrac><mrow><mi>u</mi><mo>±</mo><mi>v</mi></mrow><mrow><mn>1</mn><mo>±</mo><mrow><mi>u</mi><mo></mo><mi>v</mi></mrow></mrow></mfrac><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E17.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./4.37#E3" title="(4.37.3) ‣ §4.37(i) General Definitions ‣ §4.37 Inverse Hyperbolic Functions ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m32.png" altimg-height="18px" altimg-valign="-2px" altimg-width="89px" alttext="\operatorname{Arctanh}\NVar{z}" display="inline"><mrow><mi href="./4.37#E3">Arctanh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: general inverse hyperbolic tangent function</a></dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.6.28</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E18" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.38.18</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_Math ltx_eqn_cell ltx_align_center"><math class="ltx_Math" alttext="\operatorname{Arcsinh}u\pm\operatorname{Arccosh}v=\operatorname{Arcsinh}\left(%
uv\pm((1+u^{2})(v^{2}-1))^{1/2}\right)=\operatorname{Arccosh}\left(v(1+u^{2})^%
{1/2}\pm u(v^{2}-1)^{1/2}\right)," display="block"><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mrow><mrow><mi href="./4.37#E1">Arcsinh</mi><mo></mo><mi>u</mi></mrow><mo>±</mo><mrow><mi href="./4.37#E2">Arccosh</mi><mo></mo><mi>v</mi></mrow></mrow><mo>=</mo><mrow><mi href="./4.37#E1">Arcsinh</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mi>u</mi><mo></mo><mi>v</mi></mrow><mo>±</mo><msup><mrow><mo stretchy="false">(</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>+</mo><msup><mi>u</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><msup><mi>v</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow><mo>)</mo></mrow></mrow></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>=</mo><mrow><mi href="./4.37#E2">Arccosh</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mi>v</mi><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>+</mo><msup><mi>u</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow><mo>±</mo><mrow><mi>u</mi><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><msup><mi>v</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></mrow><mo>)</mo></mrow></mrow><mo>,</mo></mtd></mtr></mtable></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E18.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.37#E2" title="(4.37.2) ‣ §4.37(i) General Definitions ‣ §4.37 Inverse Hyperbolic Functions ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="18px" altimg-valign="-2px" altimg-width="87px" alttext="\operatorname{Arccosh}\NVar{z}" display="inline"><mrow><mi href="./4.37#E2">Arccosh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: general inverse hyperbolic cosine function</a> and
<a href="./4.37#E1" title="(4.37.1) ‣ §4.37(i) General Definitions ‣ §4.37 Inverse Hyperbolic Functions ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m31.png" altimg-height="18px" altimg-valign="-2px" altimg-width="85px" alttext="\operatorname{Arcsinh}\NVar{z}" display="inline"><mrow><mi href="./4.37#E1">Arcsinh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: general inverse hyperbolic sine function</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.6.29</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E19" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.38.19</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E19.png" altimg-height="53px" altimg-valign="-21px" altimg-width="589px" alttext="\operatorname{Arctanh}u\pm\operatorname{Arccoth}v=\operatorname{Arctanh}\left(%
\frac{uv\pm 1}{v\pm u}\right)=\operatorname{Arccoth}\left(\frac{v\pm u}{uv\pm 1%
}\right)." display="block"><mrow><mrow><mrow><mrow><mi href="./4.37#E3">Arctanh</mi><mo></mo><mi>u</mi></mrow><mo>±</mo><mrow><mi href="./4.37#E6">Arccoth</mi><mo></mo><mi>v</mi></mrow></mrow><mo>=</mo><mrow><mi href="./4.37#E3">Arctanh</mi><mo></mo><mrow><mo>(</mo><mfrac><mrow><mrow><mi>u</mi><mo></mo><mi>v</mi></mrow><mo>±</mo><mn>1</mn></mrow><mrow><mi>v</mi><mo>±</mo><mi>u</mi></mrow></mfrac><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mi href="./4.37#E6">Arccoth</mi><mo></mo><mrow><mo>(</mo><mfrac><mrow><mi>v</mi><mo>±</mo><mi>u</mi></mrow><mrow><mrow><mi>u</mi><mo></mo><mi>v</mi></mrow><mo>±</mo><mn>1</mn></mrow></mfrac><mo>)</mo></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E19.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.37#E6" title="(4.37.6) ‣ §4.37(i) General Definitions ‣ §4.37 Inverse Hyperbolic Functions ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m30.png" altimg-height="18px" altimg-valign="-2px" altimg-width="87px" alttext="\operatorname{Arccoth}\NVar{z}" display="inline"><mrow><mi href="./4.37#E6">Arccoth</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: general inverse hyperbolic cotangent function</a> and
<a href="./4.37#E3" title="(4.37.3) ‣ §4.37(i) General Definitions ‣ §4.37 Inverse Hyperbolic Functions ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m32.png" altimg-height="18px" altimg-valign="-2px" altimg-width="89px" alttext="\operatorname{Arctanh}\NVar{z}" display="inline"><mrow><mi href="./4.37#E3">Arctanh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: general inverse hyperbolic tangent function</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">4.6.30</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">The above equations are interpreted in the sense that every value of the
left-hand side is a value of the right-hand side and vice-versa. All square
roots have either possible value.</p>
</div>
</section>
</section>
</div>
<div class="ltx_page_footer">
<span></div>
</div>
</body></text>
</html>
</page>
<page>
<!DOCTYPE html><html>
<head>
<title>DLMF: 4.45 Methods of Computation</title>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<!--GOOGLE BOOTSTRAP--></head>
<text><body class="color_default textfont_default titlefont_default fontsize_default navbar_default">
<div class="ltx_page_navbar">
<div class="ltx_page_navlogo"></li>
<li class="ltx_tocentry"><a href="#iii"><span class="ltx_tag ltx_tag_subsection">§4.45(iii) </span>Lambert <math class="ltx_Math" altimg="m30.png" altimg-height="18px" altimg-valign="-2px" altimg-width="26px" alttext="W" display="inline"><mi href="./4.13#E1">W</mi></math>-Function</a></li>
</ul>
<section id="i" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§4.45(i) </span>Real Variables</h2>
<div id="SS1.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px1.p1" class="ltx_para">
<p class="ltx_p">The function <math class="ltx_Math" altimg="m38.png" altimg-height="18px" altimg-valign="-2px" altimg-width="36px" alttext="\ln x" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi href="./4.1#p2.t1.r3">x</mi></mrow></math> can always be computed from its ascending power series
after preliminary scaling. Suppose first <math class="ltx_Math" altimg="m27.png" altimg-height="23px" altimg-valign="-7px" altimg-width="129px" alttext="1/10\leq x\leq 10" display="inline"><mrow><mrow><mn>1</mn><mo>/</mo><mn>10</mn></mrow><mo>≤</mo><mi href="./4.1#p2.t1.r3">x</mi><mo>≤</mo><mn>10</mn></mrow></math>. Then we take
square roots repeatedly until <math class="ltx_Math" altimg="m80.png" altimg-height="23px" altimg-valign="-7px" altimg-width="26px" alttext="|y|" display="inline"><mrow><mo stretchy="false">|</mo><mi href="./4.1#p2.t1.r3">y</mi><mo stretchy="false">|</mo></mrow></math> is sufficiently small, where</p>
<table id="E1" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.45.1</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E1.png" altimg-height="30px" altimg-valign="-6px" altimg-width="127px" alttext="y=x^{2^{-m}}-1." display="block"><mrow><mrow><mi href="./4.1#p2.t1.r3">y</mi><mo>=</mo><mrow><msup><mi href="./4.1#p2.t1.r3">x</mi><msup><mn>2</mn><mrow><mo>-</mo><mi href="./4.1#p2.t1.r1">m</mi></mrow></msup></msup><mo>-</mo><mn>1</mn></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.1#p2.t1.r1" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m64.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./4.1#p2.t1.r1">m</mi></math>: integer</a>,
<a href="./4.1#p2.t1.r3" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./4.1#p2.t1.r3">x</mi></math>: real variable</a> and
<a href="./4.1#p2.t1.r3" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./4.1#p2.t1.r3">y</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">After computing <math class="ltx_Math" altimg="m41.png" altimg-height="23px" altimg-valign="-7px" altimg-width="85px" alttext="\ln\left(1+y\right)" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mi href="./4.1#p2.t1.r3">y</mi></mrow><mo>)</mo></mrow></mrow></math> from ()</p>
<table id="E2" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.45.2</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E2.png" altimg-height="25px" altimg-valign="-7px" altimg-width="181px" alttext="\ln x=2^{m}\ln\left(1+y\right)." display="block"><mrow><mrow><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi href="./4.1#p2.t1.r3">x</mi></mrow><mo>=</mo><mrow><msup><mn>2</mn><mi href="./4.1#p2.t1.r1">m</mi></msup><mo></mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mi href="./4.1#p2.t1.r3">y</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m40.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a>,
<a href="./4.1#p2.t1.r1" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m64.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./4.1#p2.t1.r1">m</mi></math>: integer</a>,
<a href="./4.1#p2.t1.r3" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./4.1#p2.t1.r3">x</mi></math>: real variable</a> and
<a href="./4.1#p2.t1.r3" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./4.1#p2.t1.r3">y</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="Px1.p2" class="ltx_para">
<p class="ltx_p">For other values of <math class="ltx_Math" altimg="m72.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./4.1#p2.t1.r3">x</mi></math> set <math class="ltx_Math" altimg="m70.png" altimg-height="21px" altimg-valign="-6px" altimg-width="87px" alttext="x=10^{m}\xi" display="inline"><mrow><mi href="./4.1#p2.t1.r3">x</mi><mo>=</mo><mrow><msup><mn>10</mn><mi href="./4.1#p2.t1.r1">m</mi></msup><mo></mo><mi>ξ</mi></mrow></mrow></math>, where <math class="ltx_Math" altimg="m28.png" altimg-height="23px" altimg-valign="-7px" altimg-width="127px" alttext="1/10\leq\xi\leq 10" display="inline"><mrow><mrow><mn>1</mn><mo>/</mo><mn>10</mn></mrow><mo>≤</mo><mi>ξ</mi><mo>≤</mo><mn>10</mn></mrow></math> and
<math class="ltx_Math" altimg="m65.png" altimg-height="18px" altimg-valign="-3px" altimg-width="59px" alttext="m\in\mathbb{Z}" display="inline"><mrow><mi href="./4.1#p2.t1.r1">m</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mi href="./front/introduction#Sx4.p2.t1.r20">ℤ</mi></mrow></math>. Then</p>
<table id="E3" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.45.3</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E3.png" altimg-height="23px" altimg-valign="-6px" altimg-width="184px" alttext="\ln x=\ln\xi+m\ln 10." display="block"><mrow><mrow><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi href="./4.1#p2.t1.r3">x</mi></mrow><mo>=</mo><mrow><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi>ξ</mi></mrow><mo>+</mo><mrow><mi href="./4.1#p2.t1.r1">m</mi><mo></mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mn>10</mn></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m40.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a>,
<a href="./4.1#p2.t1.r1" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m64.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./4.1#p2.t1.r1">m</mi></math>: integer</a> and
<a href="./4.1#p2.t1.r3" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./4.1#p2.t1.r3">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px2" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Exponentials</h3>
<div id="Px2.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px2.p1" class="ltx_para">
<p class="ltx_p">Let <math class="ltx_Math" altimg="m72.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./4.1#p2.t1.r3">x</mi></math> have any real value. First, rescale via</p>
<table id="E4" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex1" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">4.45.4</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m18.png" altimg-height="14px" altimg-valign="-2px" altimg-width="24px" alttext="\displaystyle m" display="inline"><mi href="./4.1#p2.t1.r1">m</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m7.png" altimg-height="53px" altimg-valign="-21px" altimg-width="143px" alttext="\displaystyle=\left\lfloor\frac{x}{\ln 10}+\frac{1}{2}\right\rfloor," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r16">⌊</mo><mrow><mstyle displaystyle="true"><mfrac><mi href="./4.1#p2.t1.r3">x</mi><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mn>10</mn></mrow></mfrac></mstyle><mo>+</mo><mstyle displaystyle="true"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle></mrow><mo href="./front/introduction#Sx4.p1.t1.r16">⌋</mo></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex2" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m23.png" altimg-height="18px" altimg-valign="-6px" altimg-width="17px" alttext="\displaystyle y" display="inline"><mi href="./4.1#p2.t1.r3">y</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m12.png" altimg-height="21px" altimg-valign="-4px" altimg-width="129px" alttext="\displaystyle=x-m\ln 10." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./4.1#p2.t1.r3">x</mi><mo>-</mo><mrow><mi href="./4.1#p2.t1.r1">m</mi><mo></mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mn>10</mn></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E4.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./front/introduction#Sx4.p1.t1.r16" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="23px" altimg-valign="-7px" altimg-width="33px" alttext="\left\lfloor\NVar{x}\right\rfloor" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r16">⌊</mo><mi class="ltx_nvar">x</mi><mo href="./front/introduction#Sx4.p1.t1.r16">⌋</mo></mrow></math>: floor of <math class="ltx_Math" altimg="m72.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m40.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a>,
<a href="./4.1#p2.t1.r1" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m64.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./4.1#p2.t1.r1">m</mi></math>: integer</a>,
<a href="./4.1#p2.t1.r3" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./4.1#p2.t1.r3">x</mi></math>: real variable</a> and
<a href="./4.1#p2.t1.r3" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./4.1#p2.t1.r3">y</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Then</p>
<table id="E5" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.45.5</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E5.png" altimg-height="24px" altimg-valign="-6px" altimg-width="111px" alttext="e^{x}=10^{m}e^{y}," display="block"><mrow><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mi href="./4.1#p2.t1.r3">x</mi></msup><mo>=</mo><mrow><msup><mn>10</mn><mi href="./4.1#p2.t1.r1">m</mi></msup><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mi href="./4.1#p2.t1.r3">y</mi></msup></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E5.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./4.1#p2.t1.r1" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m64.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./4.1#p2.t1.r1">m</mi></math>: integer</a>,
<a href="./4.1#p2.t1.r3" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./4.1#p2.t1.r3">x</mi></math>: real variable</a> and
<a href="./4.1#p2.t1.r3" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./4.1#p2.t1.r3">y</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and since <math class="ltx_Math" altimg="m81.png" altimg-height="27px" altimg-valign="-9px" altimg-width="197px" alttext="|y|\leq\frac{1}{2}\ln 10=1.15\dots" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mi href="./4.1#p2.t1.r3">y</mi><mo stretchy="false">|</mo></mrow><mo>≤</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mn>10</mn></mrow></mrow><mo>=</mo><mrow><mn>1.15</mn><mo></mo><mi mathvariant="normal">…</mi></mrow></mrow></math>, <math class="ltx_Math" altimg="m62.png" altimg-height="18px" altimg-valign="-2px" altimg-width="23px" alttext="e^{y}" display="inline"><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mi href="./4.1#p2.t1.r3">y</mi></msup></math> can be computed
straightforwardly from (</dd>
</dl>
</div>
</div>
<div id="Px3.p1" class="ltx_para">
<p class="ltx_p">Let <math class="ltx_Math" altimg="m72.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./4.1#p2.t1.r3">x</mi></math> have any real value. We first compute <math class="ltx_Math" altimg="m61.png" altimg-height="23px" altimg-valign="-7px" altimg-width="74px" alttext="\xi=x/\pi" display="inline"><mrow><mi>ξ</mi><mo>=</mo><mrow><mi href="./4.1#p2.t1.r3">x</mi><mo>/</mo><mi href="./3.12#E1">π</mi></mrow></mrow></math>, followed by</p>
<table id="E6" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex3" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">4.45.6</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m18.png" altimg-height="14px" altimg-valign="-2px" altimg-width="24px" alttext="\displaystyle m" display="inline"><mi href="./4.1#p2.t1.r1">m</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m8.png" altimg-height="29px" altimg-valign="-9px" altimg-width="101px" alttext="\displaystyle=\left\lfloor\xi+\tfrac{1}{2}\right\rfloor," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r16">⌊</mo><mrow><mi>ξ</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo href="./front/introduction#Sx4.p1.t1.r16">⌋</mo></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex4" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m16.png" altimg-height="19px" altimg-valign="-2px" altimg-width="16px" alttext="\displaystyle\theta" display="inline"><mi>θ</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m10.png" altimg-height="25px" altimg-valign="-7px" altimg-width="112px" alttext="\displaystyle=\pi(\xi-m)." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./3.12#E1">π</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>ξ</mi><mo>-</mo><mi href="./4.1#p2.t1.r1">m</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E6.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./front/introduction#Sx4.p1.t1.r16" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="23px" altimg-valign="-7px" altimg-width="33px" alttext="\left\lfloor\NVar{x}\right\rfloor" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r16">⌊</mo><mi class="ltx_nvar">x</mi><mo href="./front/introduction#Sx4.p1.t1.r16">⌋</mo></mrow></math>: floor of <math class="ltx_Math" altimg="m72.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./4.1#p2.t1.r1" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m64.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./4.1#p2.t1.r1">m</mi></math>: integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Then</p>
<table id="E7" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex5" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">4.45.7</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m15.png" altimg-height="19px" altimg-valign="-2px" altimg-width="45px" alttext="\displaystyle\sin x" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi href="./4.1#p2.t1.r3">x</mi></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m2.png" altimg-height="25px" altimg-valign="-7px" altimg-width="130px" alttext="\displaystyle=(-1)^{m}\sin\theta," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./4.1#p2.t1.r1">m</mi></msup><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi>θ</mi></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex6" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m13.png" altimg-height="14px" altimg-valign="-2px" altimg-width="47px" alttext="\displaystyle\cos x" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi href="./4.1#p2.t1.r3">x</mi></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m1.png" altimg-height="25px" altimg-valign="-7px" altimg-width="132px" alttext="\displaystyle=(-1)^{m}\cos\theta," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./4.1#p2.t1.r1">m</mi></msup><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi>θ</mi></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E7.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m35.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./4.1#p2.t1.r1" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m64.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./4.1#p2.t1.r1">m</mi></math>: integer</a> and
<a href="./4.1#p2.t1.r3" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./4.1#p2.t1.r3">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and since <math class="ltx_Math" altimg="m79.png" altimg-height="27px" altimg-valign="-9px" altimg-width="165px" alttext="|\theta|\leq\frac{1}{2}\pi=1.57\dots" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mi>θ</mi><mo stretchy="false">|</mo></mrow><mo>≤</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo>=</mo><mrow><mn>1.57</mn><mo></mo><mi mathvariant="normal">…</mi></mrow></mrow></math>, <math class="ltx_Math" altimg="m60.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\theta" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi>θ</mi></mrow></math> and
<math class="ltx_Math" altimg="m36.png" altimg-height="18px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\theta" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi>θ</mi></mrow></math> can be computed straightforwardly from (</dd>
</dl>
</div>
</div>
<div id="Px4.p1" class="ltx_para">
<p class="ltx_p">The function <math class="ltx_Math" altimg="m53.png" altimg-height="17px" altimg-valign="-2px" altimg-width="74px" alttext="\operatorname{arctan}x" display="inline"><mrow><mi href="./4.23#SS2.p1">arctan</mi><mo></mo><mi href="./4.1#p2.t1.r3">x</mi></mrow></math> can always be computed from its ascending power
series after preliminary transformations to reduce the size of <math class="ltx_Math" altimg="m72.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./4.1#p2.t1.r3">x</mi></math>. From
() with <math class="ltx_Math" altimg="m69.png" altimg-height="26px" altimg-valign="-7px" altimg-width="246px" alttext="u=v=((1+x^{2})^{1/2}-1)/x" display="inline"><mrow><mi>u</mi><mo>=</mo><mi>v</mi><mo>=</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>+</mo><msup><mi href="./4.1#p2.t1.r3">x</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo>/</mo><mi href="./4.1#p2.t1.r3">x</mi></mrow></mrow></math>, we have</p>
<table id="E8" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.45.8</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E8.png" altimg-height="47px" altimg-valign="-21px" altimg-width="315px" alttext="2\operatorname{arctan}\frac{x}{1+(1+x^{2})^{1/2}}=\operatorname{arctan}x," display="block"><mrow><mrow><mrow><mn>2</mn><mo></mo><mrow><mi href="./4.23#SS2.p1">arctan</mi><mo></mo><mfrac><mi href="./4.1#p2.t1.r3">x</mi><mrow><mn>1</mn><mo>+</mo><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>+</mo><msup><mi href="./4.1#p2.t1.r3">x</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></mfrac></mrow></mrow><mo>=</mo><mrow><mi href="./4.23#SS2.p1">arctan</mi><mo></mo><mi href="./4.1#p2.t1.r3">x</mi></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m26.png" altimg-height="17px" altimg-valign="-3px" altimg-width="99px" alttext="0<x<\infty" display="inline"><mrow><mn>0</mn><mo><</mo><mi href="./4.1#p2.t1.r3">x</mi><mo><</mo><mi mathvariant="normal">∞</mi></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E8.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.23#SS2.p1" title="§4.23(ii) Principal Values ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="17px" altimg-valign="-2px" altimg-width="73px" alttext="\operatorname{arctan}\NVar{z}" display="inline"><mrow><mi href="./4.23#SS2.p1">arctan</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: arctangent function</a> and
<a href="./4.1#p2.t1.r3" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./4.1#p2.t1.r3">x</mi></math>: real variable</a>
</dd>
<dt>Substitution (effective with 1.0.7):</dt>
<dd>
The equation that was given originally,
<math class="ltx_Math" altimg="m29.png" altimg-height="34px" altimg-valign="-9px" altimg-width="269px" alttext="2\operatorname{arctan}\frac{(1+x^{2})^{1/2}-1}{x}=\operatorname{arctan}x" display="inline"><mrow><mrow><mn>2</mn><mo></mo><mrow><mi href="./4.23#SS2.p1">arctan</mi><mo></mo><mfrac><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>+</mo><msup><mi href="./4.1#p2.t1.r3">x</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>-</mo><mn>1</mn></mrow><mi href="./4.1#p2.t1.r3">x</mi></mfrac></mrow></mrow><mo>=</mo><mrow><mi href="./4.23#SS2.p1">arctan</mi><mo></mo><mi href="./4.1#p2.t1.r3">x</mi></mrow></mrow></math>,
has been replaced with an equation that leads to a better recurrence
for numerically computing <math class="ltx_Math" altimg="m53.png" altimg-height="17px" altimg-valign="-2px" altimg-width="74px" alttext="\operatorname{arctan}x" display="inline"><mrow><mi href="./4.23#SS2.p1">arctan</mi><mo></mo><mi href="./4.1#p2.t1.r3">x</mi></mrow></math>; see (</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Beginning with <math class="ltx_Math" altimg="m74.png" altimg-height="16px" altimg-valign="-5px" altimg-width="63px" alttext="x_{0}=x" display="inline"><mrow><msub><mi href="./4.1#p2.t1.r3">x</mi><mn>0</mn></msub><mo>=</mo><mi href="./4.1#p2.t1.r3">x</mi></mrow></math>, generate the sequence</p>
<table id="E9" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.45.9</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E9.png" altimg-height="49px" altimg-valign="-23px" altimg-width="217px" alttext="x_{n}=\frac{x_{n-1}}{1+(1+x^{2}_{n-1})^{1/2}}," display="block"><mrow><mrow><msub><mi href="./4.1#p2.t1.r3">x</mi><mi href="./4.1#p2.t1.r1">n</mi></msub><mo>=</mo><mfrac><msub><mi href="./4.1#p2.t1.r3">x</mi><mrow><mi href="./4.1#p2.t1.r1">n</mi><mo>-</mo><mn>1</mn></mrow></msub><mrow><mn>1</mn><mo>+</mo><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>+</mo><msubsup><mi href="./4.1#p2.t1.r3">x</mi><mrow><mi href="./4.1#p2.t1.r1">n</mi><mo>-</mo><mn>1</mn></mrow><mn>2</mn></msubsup></mrow><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></mfrac></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m66.png" altimg-height="20px" altimg-valign="-6px" altimg-width="123px" alttext="n=1,2,3,\dots" display="inline"><mrow><mi href="./4.1#p2.t1.r1">n</mi><mo>=</mo><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E9.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.23#SS2.p1" title="§4.23(ii) Principal Values ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="17px" altimg-valign="-2px" altimg-width="73px" alttext="\operatorname{arctan}\NVar{z}" display="inline"><mrow><mi href="./4.23#SS2.p1">arctan</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: arctangent function</a>,
<a href="./4.1#p2.t1.r1" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m67.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./4.1#p2.t1.r1">n</mi></math>: integer</a> and
<a href="./4.1#p2.t1.r3" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./4.1#p2.t1.r3">x</mi></math>: real variable</a>
</dd>
<dt>Substitution (effective with 1.0.7):</dt>
<dd>
The original recurrence given in this equation,
<math class="ltx_Math" altimg="m75.png" altimg-height="39px" altimg-valign="-13px" altimg-width="169px" alttext="x_{n}=\frac{(1+x^{2}_{n-1})^{1/2}-1}{x_{n-1}}" display="inline"><mrow><msub><mi href="./4.1#p2.t1.r3">x</mi><mi href="./4.1#p2.t1.r1">n</mi></msub><mo>=</mo><mfrac><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>+</mo><msubsup><mi href="./4.1#p2.t1.r3">x</mi><mrow><mi href="./4.1#p2.t1.r1">n</mi><mo>-</mo><mn>1</mn></mrow><mn>2</mn></msubsup></mrow><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>-</mo><mn>1</mn></mrow><msub><mi href="./4.1#p2.t1.r3">x</mi><mrow><mi href="./4.1#p2.t1.r1">n</mi><mo>-</mo><mn>1</mn></mrow></msub></mfrac></mrow></math>
is susceptible to cancellation error when <math class="ltx_Math" altimg="m76.png" altimg-height="16px" altimg-valign="-5px" altimg-width="27px" alttext="x_{n}" display="inline"><msub><mi href="./4.1#p2.t1.r3">x</mi><mi href="./4.1#p2.t1.r1">n</mi></msub></math> is small. It has been replaced with
another recurrence that is better for numerically computing <math class="ltx_Math" altimg="m53.png" altimg-height="17px" altimg-valign="-2px" altimg-width="74px" alttext="\operatorname{arctan}x" display="inline"><mrow><mi href="./4.23#SS2.p1">arctan</mi><mo></mo><mi href="./4.1#p2.t1.r3">x</mi></mrow></math>.
<p><span class="ltx_font_italic">Suggested 2014-02-05 by Masataka Urago</span></p>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">until <math class="ltx_Math" altimg="m76.png" altimg-height="16px" altimg-valign="-5px" altimg-width="27px" alttext="x_{n}" display="inline"><msub><mi href="./4.1#p2.t1.r3">x</mi><mi href="./4.1#p2.t1.r1">n</mi></msub></math> is sufficiently small. We then compute <math class="ltx_Math" altimg="m55.png" altimg-height="20px" altimg-valign="-5px" altimg-width="85px" alttext="\operatorname{arctan}x_{n}" display="inline"><mrow><mi href="./4.23#SS2.p1">arctan</mi><mo></mo><msub><mi href="./4.1#p2.t1.r3">x</mi><mi href="./4.1#p2.t1.r1">n</mi></msub></mrow></math> from
(), followed by</p>
<table id="E10" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.45.10</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E10.png" altimg-height="23px" altimg-valign="-5px" altimg-width="213px" alttext="\operatorname{arctan}x=2^{n}\operatorname{arctan}x_{n}." display="block"><mrow><mrow><mrow><mi href="./4.23#SS2.p1">arctan</mi><mo></mo><mi href="./4.1#p2.t1.r3">x</mi></mrow><mo>=</mo><mrow><msup><mn>2</mn><mi href="./4.1#p2.t1.r1">n</mi></msup><mo></mo><mrow><mi href="./4.23#SS2.p1">arctan</mi><mo></mo><msub><mi href="./4.1#p2.t1.r3">x</mi><mi href="./4.1#p2.t1.r1">n</mi></msub></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E10.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.23#SS2.p1" title="§4.23(ii) Principal Values ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="17px" altimg-valign="-2px" altimg-width="73px" alttext="\operatorname{arctan}\NVar{z}" display="inline"><mrow><mi href="./4.23#SS2.p1">arctan</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: arctangent function</a>,
<a href="./4.1#p2.t1.r1" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m67.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./4.1#p2.t1.r1">n</mi></math>: integer</a> and
<a href="./4.1#p2.t1.r3" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./4.1#p2.t1.r3">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="Px4.p2" class="ltx_para">
<p class="ltx_p">Another method, when <math class="ltx_Math" altimg="m72.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./4.1#p2.t1.r3">x</mi></math> is large, is to sum</p>
<table id="E11" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.45.11</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E11.png" altimg-height="46px" altimg-valign="-16px" altimg-width="335px" alttext="\operatorname{arctan}x=\frac{\pi}{2}-\frac{1}{x}+\frac{1}{3x^{3}}-\frac{1}{5x^%
{5}}+\dots;" display="block"><mrow><mrow><mrow><mi href="./4.23#SS2.p1">arctan</mi><mo></mo><mi href="./4.1#p2.t1.r3">x</mi></mrow><mo>=</mo><mrow><mrow><mrow><mrow><mfrac><mi href="./3.12#E1">π</mi><mn>2</mn></mfrac><mo>-</mo><mfrac><mn>1</mn><mi href="./4.1#p2.t1.r3">x</mi></mfrac></mrow><mo>+</mo><mfrac><mn>1</mn><mrow><mn>3</mn><mo></mo><msup><mi href="./4.1#p2.t1.r3">x</mi><mn>3</mn></msup></mrow></mfrac></mrow><mo>-</mo><mfrac><mn>1</mn><mrow><mn>5</mn><mo></mo><msup><mi href="./4.1#p2.t1.r3">x</mi><mn>5</mn></msup></mrow></mfrac></mrow><mo>+</mo><mi mathvariant="normal">…</mi></mrow></mrow><mo>;</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E11.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.23#SS2.p1" title="§4.23(ii) Principal Values ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="17px" altimg-valign="-2px" altimg-width="73px" alttext="\operatorname{arctan}\NVar{z}" display="inline"><mrow><mi href="./4.23#SS2.p1">arctan</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: arctangent function</a> and
<a href="./4.1#p2.t1.r3" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./4.1#p2.t1.r3">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">compare ().</p>
</div>
<div id="Px4.p3" class="ltx_para">
<p class="ltx_p">As an example, take <math class="ltx_Math" altimg="m71.png" altimg-height="17px" altimg-valign="-2px" altimg-width="108px" alttext="x=9.47376" display="inline"><mrow><mi href="./4.1#p2.t1.r3">x</mi><mo>=</mo><mn>9.47376</mn></mrow></math>. Then</p>
<table id="E12" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex7" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="4" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">4.45.12</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m19.png" altimg-height="17px" altimg-valign="-5px" altimg-width="26px" alttext="\displaystyle x_{1}" display="inline"><msub><mi href="./4.1#p2.t1.r3">x</mi><mn>1</mn></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m6.png" altimg-height="22px" altimg-valign="-6px" altimg-width="128px" alttext="\displaystyle=0.90000\dots," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mn>0.90000</mn><mo></mo><mi mathvariant="normal">…</mi></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex8" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m20.png" altimg-height="17px" altimg-valign="-5px" altimg-width="26px" alttext="\displaystyle x_{2}" display="inline"><msub><mi href="./4.1#p2.t1.r3">x</mi><mn>2</mn></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m5.png" altimg-height="22px" altimg-valign="-6px" altimg-width="128px" alttext="\displaystyle=0.38373\dots," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mn>0.38373</mn><mo></mo><mi mathvariant="normal">…</mi></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex9" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m21.png" altimg-height="17px" altimg-valign="-5px" altimg-width="26px" alttext="\displaystyle x_{3}" display="inline"><msub><mi href="./4.1#p2.t1.r3">x</mi><mn>3</mn></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m4.png" altimg-height="22px" altimg-valign="-6px" altimg-width="128px" alttext="\displaystyle=0.18528\dots," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mn>0.18528</mn><mo></mo><mi mathvariant="normal">…</mi></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex10" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m22.png" altimg-height="17px" altimg-valign="-5px" altimg-width="26px" alttext="\displaystyle x_{4}" display="inline"><msub><mi href="./4.1#p2.t1.r3">x</mi><mn>4</mn></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m3.png" altimg-height="18px" altimg-valign="-2px" altimg-width="128px" alttext="\displaystyle=0.09185\dots." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mn>0.09185</mn><mo></mo><mi mathvariant="normal">…</mi></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E12.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./4.1#p2.t1.r3" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./4.1#p2.t1.r3">x</mi></math>: real variable</a></dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="Px4.p4" class="ltx_para">
<p class="ltx_p">From () <math class="ltx_Math" altimg="m54.png" altimg-height="20px" altimg-valign="-5px" altimg-width="202px" alttext="\operatorname{arctan}x_{4}=0.09160\dots" display="inline"><mrow><mrow><mi href="./4.23#SS2.p1">arctan</mi><mo></mo><msub><mi href="./4.1#p2.t1.r3">x</mi><mn>4</mn></msub></mrow><mo>=</mo><mrow><mn>0.09160</mn><mo></mo><mi mathvariant="normal">…</mi></mrow></mrow></math>. From
()</p>
<table id="E13" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.45.13</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E13.png" altimg-height="21px" altimg-valign="-5px" altimg-width="332px" alttext="\operatorname{arctan}x=16\operatorname{arctan}x_{4}=1.46563\dots." display="block"><mrow><mrow><mrow><mi href="./4.23#SS2.p1">arctan</mi><mo></mo><mi href="./4.1#p2.t1.r3">x</mi></mrow><mo>=</mo><mrow><mn>16</mn><mo></mo><mrow><mi href="./4.23#SS2.p1">arctan</mi><mo></mo><msub><mi href="./4.1#p2.t1.r3">x</mi><mn>4</mn></msub></mrow></mrow><mo>=</mo><mrow><mn>1.46563</mn><mo></mo><mi mathvariant="normal">…</mi></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E13.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.23#SS2.p1" title="§4.23(ii) Principal Values ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="17px" altimg-valign="-2px" altimg-width="73px" alttext="\operatorname{arctan}\NVar{z}" display="inline"><mrow><mi href="./4.23#SS2.p1">arctan</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: arctangent function</a> and
<a href="./4.1#p2.t1.r3" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./4.1#p2.t1.r3">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="Px4.p5" class="ltx_para">
<p class="ltx_p">As a check, from ()</p>
<table id="E14" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.45.14</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E14.png" altimg-height="20px" altimg-valign="-4px" altimg-width="602px" alttext="\operatorname{arctan}x=1.57079\ldots-0.10555\ldots+0.00039\ldots-\cdots=1.4656%
3\dots." display="block"><mrow><mrow><mrow><mi href="./4.23#SS2.p1">arctan</mi><mo></mo><mi href="./4.1#p2.t1.r3">x</mi></mrow><mo>=</mo><mrow><mrow><mrow><mrow><mn>1.57079</mn><mo></mo><mi mathvariant="normal">…</mi></mrow><mo>-</mo><mrow><mn>0.10555</mn><mo></mo><mi mathvariant="normal">…</mi></mrow></mrow><mo>+</mo><mrow><mn>0.00039</mn><mo></mo><mi mathvariant="normal">…</mi></mrow></mrow><mo>-</mo><mi mathvariant="normal">⋯</mi></mrow><mo>=</mo><mrow><mn>1.46563</mn><mo></mo><mi mathvariant="normal">…</mi></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E14.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.23#SS2.p1" title="§4.23(ii) Principal Values ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="17px" altimg-valign="-2px" altimg-width="73px" alttext="\operatorname{arctan}\NVar{z}" display="inline"><mrow><mi href="./4.23#SS2.p1">arctan</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: arctangent function</a> and
<a href="./4.1#p2.t1.r3" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./4.1#p2.t1.r3">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="Px4.p6" class="ltx_para">
<p class="ltx_p">For the remaining inverse trigonometric functions, we may use the identities
provided by the fourth row of Table . For example,
<math class="ltx_Math" altimg="m50.png" altimg-height="28px" altimg-valign="-9px" altimg-width="293px" alttext="\operatorname{arcsin}x=\operatorname{arctan}\left(x(1-x^{2})^{-1/2}\right)" display="inline"><mrow><mrow><mi href="./4.23#SS2.p1">arcsin</mi><mo></mo><mi href="./4.1#p2.t1.r3">x</mi></mrow><mo>=</mo><mrow><mi href="./4.23#SS2.p1">arctan</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./4.1#p2.t1.r3">x</mi><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><msup><mi href="./4.1#p2.t1.r3">x</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></mrow></msup></mrow><mo>)</mo></mrow></mrow></mrow></math>.</p>
</div>
</section>
<section id="Px5" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Hyperbolic and Inverse Hyperbolic Functions</h3>
<div id="Px5.info" class="ltx_metadata ltx_info">
). The inverses <math class="ltx_Math" altimg="m49.png" altimg-height="18px" altimg-valign="-2px" altimg-width="66px" alttext="\operatorname{arcsinh}" display="inline"><mi href="./4.37#SS2.p1">arcsinh</mi></math>, <math class="ltx_Math" altimg="m45.png" altimg-height="18px" altimg-valign="-2px" altimg-width="69px" alttext="\operatorname{arccosh}" display="inline"><mi href="./4.37#SS2.p1">arccosh</mi></math>, and
<math class="ltx_Math" altimg="m51.png" altimg-height="18px" altimg-valign="-2px" altimg-width="71px" alttext="\operatorname{arctanh}" display="inline"><mi href="./4.37#SS2.p1">arctanh</mi></math> can be computed from the logarithmic forms given in
§, with real arguments. For <math class="ltx_Math" altimg="m47.png" altimg-height="18px" altimg-valign="-2px" altimg-width="67px" alttext="\operatorname{arccsch}" display="inline"><mi href="./4.37#E7">arccsch</mi></math>, <math class="ltx_Math" altimg="m48.png" altimg-height="18px" altimg-valign="-2px" altimg-width="67px" alttext="\operatorname{arcsech}" display="inline"><mi href="./4.37#E8">arcsech</mi></math>, and <math class="ltx_Math" altimg="m46.png" altimg-height="18px" altimg-valign="-2px" altimg-width="69px" alttext="\operatorname{arccoth}" display="inline"><mi href="./4.37#E9">arccoth</mi></math>
we have (</dd>
</dl>
</div>
</div>
<div id="SS2.p1" class="ltx_para">
<p class="ltx_p">For <math class="ltx_Math" altimg="m39.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln z" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></math> and <math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="23px" alttext="e^{z}" display="inline"><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mi href="./4.1#p2.t1.r4">z</mi></msup></math></p>
<table id="EGx1" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E15">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.45.15</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m14.png" altimg-height="19px" altimg-valign="-2px" altimg-width="36px" alttext="\displaystyle\ln z" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m9.png" altimg-height="25px" altimg-valign="-7px" altimg-width="144px" alttext="\displaystyle=\ln|z|+i\operatorname{ph}z," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><mo stretchy="false">|</mo><mi href="./4.1#p2.t1.r4">z</mi><mo stretchy="false">|</mo></mrow></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m25.png" altimg-height="21px" altimg-valign="-6px" altimg-width="133px" alttext="-\pi\leq\operatorname{ph}z\leq\pi" display="inline"><mrow><mrow><mo>-</mo><mi href="./3.12#E1">π</mi></mrow><mo>≤</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>≤</mo><mi href="./3.12#E1">π</mi></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E15.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m40.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a>,
<a href="./1.9#E7" title="(1.9.7) ‣ Modulus and Phase ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="21px" altimg-valign="-6px" altimg-width="26px" alttext="\operatorname{ph}" display="inline"><mi href="./1.9#E7">ph</mi></math>: phase</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E16">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">4.45.16</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m17.png" altimg-height="20px" altimg-valign="-2px" altimg-width="25px" alttext="\displaystyle e^{z}" display="inline"><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mi href="./4.1#p2.t1.r4">z</mi></msup></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m11.png" altimg-height="28px" altimg-valign="-7px" altimg-width="251px" alttext="\displaystyle=e^{\Re z}(\cos\left(\Im z\right)+i\sin\left(\Im z\right))." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></msup><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℑ</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℑ</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E16.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m35.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m33.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Im" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℑ</mi><mo></mo></mrow></math>: imaginary part</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m34.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a> and
<a href="./4.1#p2.t1.r4" title="§4.1 Special Notation ‣ Notation ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./4.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
<p class="ltx_p">See § for the precise relationship of <math class="ltx_Math" altimg="m57.png" altimg-height="21px" altimg-valign="-6px" altimg-width="40px" alttext="\operatorname{ph}z" display="inline"><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mi href="./4.1#p2.t1.r4">z</mi></mrow></math> to the
arctangent function.</p>
</div>
<div id="SS2.p2" class="ltx_para">
<p class="ltx_p">The trigonometric functions may be computed from the definitions
()</cite>.</p>
</div>
</section>
<section id="iii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§4.45(iii) </span>Lambert <math class="ltx_Math" altimg="m30.png" altimg-height="18px" altimg-valign="-2px" altimg-width="26px" alttext="W" display="inline"><mi href="./4.13#E1">W</mi></math>-Function</h2>
<div id="SS3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Keywords:</dt>
<dd><a class="ltx_keyword" href="./idx/L#LambertWfunction">Lambert <math class="ltx_Math" altimg="m30.png" altimg-height="18px" altimg-valign="-2px" altimg-width="26px" alttext="W" display="inline"><mi href="./4.13#E1">W</mi></math>-function</a></dd>
</dl>
</div>
</div>
<div id="SS3.p1" class="ltx_para">
<p class="ltx_p">For <math class="ltx_Math" altimg="m73.png" altimg-height="23px" altimg-valign="-7px" altimg-width="127px" alttext="x\in[-1/e,\infty)" display="inline"><mrow><mi href="./4.1#p2.t1.r3">x</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mrow><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">[</mo><mrow><mo>-</mo><mrow><mn>1</mn><mo>/</mo><mi href="./4.2#E11" mathvariant="normal">e</mi></mrow></mrow><mo href="./front/introduction#Sx4.p2.t1.r1">,</mo><mi mathvariant="normal">∞</mi><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">)</mo></mrow></mrow></math> the principal branch <math class="ltx_Math" altimg="m43.png" altimg-height="23px" altimg-valign="-7px" altimg-width="66px" alttext="\mathrm{Wp}\left(x\right)" display="inline"><mrow><mi href="./4.13#p2">Wp</mi><mo></mo><mrow><mo>(</mo><mi href="./4.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math> can be computed
by solving the defining equation
<math class="ltx_Math" altimg="m31.png" altimg-height="20px" altimg-valign="-2px" altimg-width="91px" alttext="We^{W}=x" display="inline"><mrow><mrow><mi>W</mi><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mi>W</mi></msup></mrow><mo>=</mo><mi href="./4.1#p2.t1.r3">x</mi></mrow></math> numerically, for example, by Newton’s rule
(§) (with <math class="ltx_Math" altimg="m68.png" altimg-height="19px" altimg-valign="-5px" altimg-width="48px" alttext="t\geq 0" display="inline"><mrow><mi>t</mi><mo>≥</mo><mn>0</mn></mrow></math>) when
<math class="ltx_Math" altimg="m72.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./4.1#p2.t1.r3">x</mi></math> is close to <math class="ltx_Math" altimg="m24.png" altimg-height="23px" altimg-valign="-7px" altimg-width="49px" alttext="-1/e" display="inline"><mrow><mo>-</mo><mrow><mn>1</mn><mo>/</mo><mi href="./4.2#E11" mathvariant="normal">e</mi></mrow></mrow></math>, from the asymptotic expansion ()
when <math class="ltx_Math" altimg="m72.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./4.1#p2.t1.r3">x</mi></math> is large, and by numerical integration of the differential equation
() for other values of <math class="ltx_Math" altimg="m72.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./4.1#p2.t1.r3">x</mi></math>.</p>
</div>
<div id="SS3.p2" class="ltx_para">
<p class="ltx_p">Similarly for <math class="ltx_Math" altimg="m42.png" altimg-height="23px" altimg-valign="-7px" altimg-width="72px" alttext="\mathrm{Wm}\left(x\right)" display="inline"><mrow><mi href="./4.13#p2">Wm</mi><mo></mo><mrow><mo>(</mo><mi href="./4.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></math> in the interval <math class="ltx_Math" altimg="m32.png" altimg-height="23px" altimg-valign="-7px" altimg-width="81px" alttext="[-1/e,0)" display="inline"><mrow><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">[</mo><mrow><mo>-</mo><mrow><mn>1</mn><mo>/</mo><mi href="./4.2#E11" mathvariant="normal">e</mi></mrow></mrow><mo href="./front/introduction#Sx4.p2.t1.r1">,</mo><mn>0</mn><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">)</mo></mrow></math>.</p>
</div>
<div id="SS3.p3" class="ltx_para">
<p class="ltx_p">See also <cite class="ltx_cite ltx_citemacro_citet">Barry<span class="ltx_text ltx_bib_etal"> et al.</span> (</div>
</div>
</body></text>
</html>
</page>
<page>
<!DOCTYPE html><html>
<head>
<title>DLMF: 19.7 Connection Formulas</title>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<!--GOOGLE BOOTSTRAP--></head>
<text><body class="color_default textfont_default titlefont_default fontsize_default navbar_default">
<div class="ltx_page_navbar">
<div class="ltx_page_navlogo"></li>
<li class="ltx_tocentry"><a href="#iii"><span class="ltx_tag ltx_tag_subsection">§19.7(iii) </span>Change of Parameter of <math class="ltx_Math" altimg="m54.png" altimg-height="27px" altimg-valign="-9px" altimg-width="103px" alttext="\Pi\left(\phi,\alpha^{2},k\right)" display="inline"><mrow><mi href="./19.2#E7" mathvariant="normal">Π</mi><mo></mo><mrow><mo>(</mo><mi href="./19.1#p1.t1.r2">ϕ</mi><mo>,</mo><msup><mi href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup><mo>,</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math></a></li>
</ul>
<section id="i" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§19.7(i) </span>Complete Integrals of the First and Second Kinds</h2>
<div id="SS1.info" class="ltx_metadata ltx_info">
) put <math class="ltx_Math" altimg="m80.png" altimg-height="25px" altimg-valign="-7px" altimg-width="115px" alttext="z+1=1/k^{2}" display="inline"><mrow><mrow><mi>z</mi><mo>+</mo><mn>1</mn></mrow><mo>=</mo><mrow><mn>1</mn><mo>/</mo><msup><mi href="./19.1#p1.t1.r3">k</mi><mn>2</mn></msup></mrow></mrow></math>, use homogeneity,
and apply the penultimate equation in (</dd>
</dl>
</div>
</div>
<div id="Px1.p1" class="ltx_para">
<table id="E1" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.7.1</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E1.png" altimg-height="29px" altimg-valign="-9px" altimg-width="422px" alttext="E\left(k\right){K^{\prime}}\left(k\right)+{E^{\prime}}\left(k\right)K\left(k%
\right)-K\left(k\right){K^{\prime}}\left(k\right)=\tfrac{1}{2}\pi." display="block"><mrow><mrow><mrow><mrow><mrow><mrow><mi href="./19.2#E8">E</mi><mo></mo><mrow><mo>(</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><msup><mi href="./19.2#E9">K</mi><mo href="./19.2#E9">′</mo></msup><mo></mo><mrow><mo>(</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mrow><msup><mi href="./19.2#E9">E</mi><mo href="./19.2#E9">′</mo></msup><mo></mo><mrow><mo>(</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./19.2#E8">K</mi><mo></mo><mrow><mo>(</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>-</mo><mrow><mrow><mi href="./19.2#E8">K</mi><mo></mo><mrow><mo>(</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><msup><mi href="./19.2#E9">K</mi><mo href="./19.2#E9">′</mo></msup><mo></mo><mrow><mo>(</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>=</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m68.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./19.2#E9" title="(19.2.9) ‣ §19.2(ii) Legendre’s Integrals ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="24px" altimg-valign="-7px" altimg-width="58px" alttext="{K^{\prime}}\left(\NVar{k}\right)" display="inline"><mrow><msup><mi href="./19.2#E9">K</mi><mo href="./19.2#E9">′</mo></msup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Legendre’s complementary complete elliptic integral of the first kind</a>,
<a href="./19.2#E9" title="(19.2.9) ‣ §19.2(ii) Legendre’s Integrals ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m81.png" altimg-height="24px" altimg-valign="-7px" altimg-width="56px" alttext="{E^{\prime}}\left(\NVar{k}\right)" display="inline"><mrow><msup><mi href="./19.2#E9">E</mi><mo href="./19.2#E9">′</mo></msup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Legendre’s complementary complete elliptic integral of the second kind</a>,
<a href="./19.2#E8" title="(19.2.8) ‣ §19.2(ii) Legendre’s Integrals ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="23px" altimg-valign="-7px" altimg-width="52px" alttext="K\left(\NVar{k}\right)" display="inline"><mrow><mi href="./19.2#E8">K</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Legendre’s complete elliptic integral of the first kind</a>,
<a href="./19.2#E8" title="(19.2.8) ‣ §19.2(ii) Legendre’s Integrals ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="23px" altimg-valign="-7px" altimg-width="50px" alttext="E\left(\NVar{k}\right)" display="inline"><mrow><mi href="./19.2#E8">E</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Legendre’s complete elliptic integral of the second kind</a> and
<a href="./19.1#p1.t1.r3" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./19.1#p1.t1.r3">k</mi></math>: real or complex modulus</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="Px1.p2" class="ltx_para">
<p class="ltx_p">Also,</p>
<table id="E2" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex1" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="4" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">19.7.2</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m35.png" altimg-height="26px" altimg-valign="-7px" altimg-width="88px" alttext="\displaystyle K\left(ik/k^{\prime}\right)" display="inline"><mrow><mi href="./19.2#E8">K</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./19.1#p1.t1.r3">k</mi></mrow><mo>/</mo><msup><mi href="./19.1#p1.t1.r4">k</mi><mo>′</mo></msup></mrow><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m21.png" altimg-height="26px" altimg-valign="-7px" altimg-width="101px" alttext="\displaystyle=k^{\prime}K\left(k\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msup><mi href="./19.1#p1.t1.r4">k</mi><mo>′</mo></msup><mo></mo><mrow><mi href="./19.2#E8">K</mi><mo></mo><mrow><mo>(</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex2" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m32.png" altimg-height="26px" altimg-valign="-7px" altimg-width="103px" alttext="\displaystyle K\left(-ik^{\prime}/k\right)" display="inline"><mrow><mi href="./19.2#E8">K</mi><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mrow><mrow><mi mathvariant="normal">i</mi><mo></mo><msup><mi href="./19.1#p1.t1.r4">k</mi><mo>′</mo></msup></mrow><mo>/</mo><mi href="./19.1#p1.t1.r3">k</mi></mrow></mrow><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m18.png" altimg-height="26px" altimg-valign="-7px" altimg-width="101px" alttext="\displaystyle=kK\left(k^{\prime}\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./19.1#p1.t1.r3">k</mi><mo></mo><mrow><mi href="./19.2#E8">K</mi><mo></mo><mrow><mo>(</mo><msup><mi href="./19.1#p1.t1.r4">k</mi><mo>′</mo></msup><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex3" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m28.png" altimg-height="26px" altimg-valign="-7px" altimg-width="85px" alttext="\displaystyle E\left(ik/k^{\prime}\right)" display="inline"><mrow><mi href="./19.2#E8">E</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./19.1#p1.t1.r3">k</mi></mrow><mo>/</mo><msup><mi href="./19.1#p1.t1.r4">k</mi><mo>′</mo></msup></mrow><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m4.png" altimg-height="26px" altimg-valign="-7px" altimg-width="134px" alttext="\displaystyle=(1/k^{\prime})E\left(k\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>/</mo><msup><mi href="./19.1#p1.t1.r4">k</mi><mo>′</mo></msup></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mi href="./19.2#E8">E</mi><mo></mo><mrow><mo>(</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex4" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m22.png" altimg-height="26px" altimg-valign="-7px" altimg-width="101px" alttext="\displaystyle E\left(-ik^{\prime}/k\right)" display="inline"><mrow><mi href="./19.2#E8">E</mi><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mrow><mrow><mi mathvariant="normal">i</mi><mo></mo><msup><mi href="./19.1#p1.t1.r4">k</mi><mo>′</mo></msup></mrow><mo>/</mo><mi href="./19.1#p1.t1.r3">k</mi></mrow></mrow><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m2.png" altimg-height="26px" altimg-valign="-7px" altimg-width="134px" alttext="\displaystyle=(1/k)E\left(k^{\prime}\right)." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>/</mo><mi href="./19.1#p1.t1.r3">k</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mi href="./19.2#E8">E</mi><mo></mo><mrow><mo>(</mo><msup><mi href="./19.1#p1.t1.r4">k</mi><mo>′</mo></msup><mo>)</mo></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.2#E8" title="(19.2.8) ‣ §19.2(ii) Legendre’s Integrals ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="23px" altimg-valign="-7px" altimg-width="52px" alttext="K\left(\NVar{k}\right)" display="inline"><mrow><mi href="./19.2#E8">K</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Legendre’s complete elliptic integral of the first kind</a>,
<a href="./19.2#E8" title="(19.2.8) ‣ §19.2(ii) Legendre’s Integrals ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="23px" altimg-valign="-7px" altimg-width="50px" alttext="E\left(\NVar{k}\right)" display="inline"><mrow><mi href="./19.2#E8">E</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Legendre’s complete elliptic integral of the second kind</a>,
<a href="./19.1#p1.t1.r3" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./19.1#p1.t1.r3">k</mi></math>: real or complex modulus</a> and
<a href="./19.1#p1.t1.r4" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="19px" altimg-valign="-2px" altimg-width="21px" alttext="k^{\prime}" display="inline"><msup><mi href="./19.1#p1.t1.r4">k</mi><mo>′</mo></msup></math>: complementary modulus</a>
</dd>
<dt>Clarification (effective with 1.0.17):</dt>
<dd>
The argument <math class="ltx_Math" altimg="m77.png" altimg-height="24px" altimg-valign="-7px" altimg-width="49px" alttext="k^{\prime}/ik" display="inline"><mrow><mrow><msup><mi href="./19.1#p1.t1.r4">k</mi><mo>′</mo></msup><mo>/</mo><mi mathvariant="normal">i</mi></mrow><mo></mo><mi href="./19.1#p1.t1.r3">k</mi></mrow></math> has been replaced with <math class="ltx_Math" altimg="m45.png" altimg-height="24px" altimg-valign="-7px" altimg-width="64px" alttext="-ik^{\prime}/k" display="inline"><mrow><mo>-</mo><mrow><mrow><mi mathvariant="normal">i</mi><mo></mo><msup><mi href="./19.1#p1.t1.r4">k</mi><mo>′</mo></msup></mrow><mo>/</mo><mi href="./19.1#p1.t1.r3">k</mi></mrow></mrow></math> twice to clarify the meaning.
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E3" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex5" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="4" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">19.7.3</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m33.png" altimg-height="25px" altimg-valign="-7px" altimg-width="74px" alttext="\displaystyle K\left(1/k\right)" display="inline"><mrow><mi href="./19.2#E8">K</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>/</mo><mi href="./19.1#p1.t1.r3">k</mi></mrow><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m16.png" altimg-height="26px" altimg-valign="-7px" altimg-width="191px" alttext="\displaystyle=k(K\left(k\right)\mp\mathrm{i}K\left(k^{\prime}\right))," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./19.1#p1.t1.r3">k</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mi href="./19.2#E8">K</mi><mo></mo><mrow><mo>(</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo>∓</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mrow><mi href="./19.2#E8">K</mi><mo></mo><mrow><mo>(</mo><msup><mi href="./19.1#p1.t1.r4">k</mi><mo>′</mo></msup><mo>)</mo></mrow></mrow></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex6" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m34.png" altimg-height="26px" altimg-valign="-7px" altimg-width="80px" alttext="\displaystyle K\left(1/k^{\prime}\right)" display="inline"><mrow><mi href="./19.2#E8">K</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>/</mo><msup><mi href="./19.1#p1.t1.r4">k</mi><mo>′</mo></msup></mrow><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m20.png" altimg-height="26px" altimg-valign="-7px" altimg-width="197px" alttext="\displaystyle=k^{\prime}(K\left(k^{\prime}\right)\pm\mathrm{i}K\left(k\right))," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msup><mi href="./19.1#p1.t1.r4">k</mi><mo>′</mo></msup><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mi href="./19.2#E8">K</mi><mo></mo><mrow><mo>(</mo><msup><mi href="./19.1#p1.t1.r4">k</mi><mo>′</mo></msup><mo>)</mo></mrow></mrow><mo>±</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mrow><mi href="./19.2#E8">K</mi><mo></mo><mrow><mo>(</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex7" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m23.png" altimg-height="25px" altimg-valign="-7px" altimg-width="72px" alttext="\displaystyle E\left(1/k\right)" display="inline"><mrow><mi href="./19.2#E8">E</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>/</mo><mi href="./19.1#p1.t1.r3">k</mi></mrow><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m3.png" altimg-height="41px" altimg-valign="-15px" altimg-width="438px" alttext="\displaystyle=(1/k)\left(E\left(k\right)\pm\mathrm{i}E\left(k^{\prime}\right)-%
{k^{\prime}}^{2}K\left(k\right)\mp\mathrm{i}k^{2}K\left(k^{\prime}\right)%
\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>/</mo><mi href="./19.1#p1.t1.r3">k</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo>(</mo><mrow><mrow><mrow><mrow><mi href="./19.2#E8">E</mi><mo></mo><mrow><mo>(</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo>±</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mrow><mi href="./19.2#E8">E</mi><mo></mo><mrow><mo>(</mo><msup><mi href="./19.1#p1.t1.r4">k</mi><mo>′</mo></msup><mo>)</mo></mrow></mrow></mrow></mrow><mo>-</mo><mrow><mmultiscripts><mi href="./19.1#p1.t1.r4">k</mi><none></none><mo>′</mo><none></none><mn>2</mn></mmultiscripts><mo></mo><mrow><mi href="./19.2#E8">K</mi><mo></mo><mrow><mo>(</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>∓</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><msup><mi href="./19.1#p1.t1.r3">k</mi><mn>2</mn></msup><mo></mo><mrow><mi href="./19.2#E8">K</mi><mo></mo><mrow><mo>(</mo><msup><mi href="./19.1#p1.t1.r4">k</mi><mo>′</mo></msup><mo>)</mo></mrow></mrow></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex8" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m24.png" altimg-height="26px" altimg-valign="-7px" altimg-width="77px" alttext="\displaystyle E\left(1/k^{\prime}\right)" display="inline"><mrow><mi href="./19.2#E8">E</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>/</mo><msup><mi href="./19.1#p1.t1.r4">k</mi><mo>′</mo></msup></mrow><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m5.png" altimg-height="41px" altimg-valign="-15px" altimg-width="444px" alttext="\displaystyle=(1/k^{\prime})\left(E\left(k^{\prime}\right)\mp\mathrm{i}E\left(%
k\right)-k^{2}K\left(k^{\prime}\right)\pm\mathrm{i}{k^{\prime}}^{2}K\left(k%
\right)\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>/</mo><msup><mi href="./19.1#p1.t1.r4">k</mi><mo>′</mo></msup></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo>(</mo><mrow><mrow><mrow><mrow><mi href="./19.2#E8">E</mi><mo></mo><mrow><mo>(</mo><msup><mi href="./19.1#p1.t1.r4">k</mi><mo>′</mo></msup><mo>)</mo></mrow></mrow><mo>∓</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mrow><mi href="./19.2#E8">E</mi><mo></mo><mrow><mo>(</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>-</mo><mrow><msup><mi href="./19.1#p1.t1.r3">k</mi><mn>2</mn></msup><mo></mo><mrow><mi href="./19.2#E8">K</mi><mo></mo><mrow><mo>(</mo><msup><mi href="./19.1#p1.t1.r4">k</mi><mo>′</mo></msup><mo>)</mo></mrow></mrow></mrow></mrow><mo>±</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mmultiscripts><mi href="./19.1#p1.t1.r4">k</mi><none></none><mo>′</mo><none></none><mn>2</mn></mmultiscripts><mo></mo><mrow><mi href="./19.2#E8">K</mi><mo></mo><mrow><mo>(</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.2#E8" title="(19.2.8) ‣ §19.2(ii) Legendre’s Integrals ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="23px" altimg-valign="-7px" altimg-width="52px" alttext="K\left(\NVar{k}\right)" display="inline"><mrow><mi href="./19.2#E8">K</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Legendre’s complete elliptic integral of the first kind</a>,
<a href="./19.2#E8" title="(19.2.8) ‣ §19.2(ii) Legendre’s Integrals ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="23px" altimg-valign="-7px" altimg-width="50px" alttext="E\left(\NVar{k}\right)" display="inline"><mrow><mi href="./19.2#E8">E</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Legendre’s complete elliptic integral of the second kind</a>,
<a href="./19.1#p1.t1.r3" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./19.1#p1.t1.r3">k</mi></math>: real or complex modulus</a> and
<a href="./19.1#p1.t1.r4" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="19px" altimg-valign="-2px" altimg-width="21px" alttext="k^{\prime}" display="inline"><msup><mi href="./19.1#p1.t1.r4">k</mi><mo>′</mo></msup></math>: complementary modulus</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where upper signs apply if <math class="ltx_Math" altimg="m52.png" altimg-height="21px" altimg-valign="-3px" altimg-width="75px" alttext="\Im k^{2}>0" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℑ</mi><mo></mo><msup><mi href="./19.1#p1.t1.r3">k</mi><mn>2</mn></msup></mrow><mo>></mo><mn>0</mn></mrow></math> and lower signs if
<math class="ltx_Math" altimg="m51.png" altimg-height="21px" altimg-valign="-3px" altimg-width="75px" alttext="\Im k^{2}<0" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℑ</mi><mo></mo><msup><mi href="./19.1#p1.t1.r3">k</mi><mn>2</mn></msup></mrow><mo><</mo><mn>0</mn></mrow></math>. This dichotomy of signs (missing in several references)
is due to <cite class="ltx_cite ltx_citemacro_citet">Fettis (</dd>
</dl>
</div>
</div>
<div id="Px2.p1" class="ltx_para">
<table id="E4" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex9" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="4" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">19.7.4</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m30.png" altimg-height="25px" altimg-valign="-7px" altimg-width="80px" alttext="\displaystyle F\left(\phi,k_{1}\right)" display="inline"><mrow><mi href="./19.2#E4">F</mi><mo></mo><mrow><mo>(</mo><mi href="./19.1#p1.t1.r2">ϕ</mi><mo>,</mo><msub><mi href="./19.1#p1.t1.r3">k</mi><mn>1</mn></msub><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m17.png" altimg-height="25px" altimg-valign="-7px" altimg-width="114px" alttext="\displaystyle=kF\left(\beta,k\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./19.1#p1.t1.r3">k</mi><mo></mo><mrow><mi href="./19.2#E4">F</mi><mo></mo><mrow><mo>(</mo><mi>β</mi><mo>,</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex10" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m26.png" altimg-height="25px" altimg-valign="-7px" altimg-width="81px" alttext="\displaystyle E\left(\phi,k_{1}\right)" display="inline"><mrow><mi href="./19.2#E5">E</mi><mo></mo><mrow><mo>(</mo><mi href="./19.1#p1.t1.r2">ϕ</mi><mo>,</mo><msub><mi href="./19.1#p1.t1.r3">k</mi><mn>1</mn></msub><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m6.png" altimg-height="31px" altimg-valign="-7px" altimg-width="252px" alttext="\displaystyle=(E\left(\beta,k\right)-{k^{\prime}}^{2}F\left(\beta,k\right))/k," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mi href="./19.2#E5">E</mi><mo></mo><mrow><mo>(</mo><mi>β</mi><mo>,</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo>-</mo><mrow><mmultiscripts><mi href="./19.1#p1.t1.r4">k</mi><none></none><mo>′</mo><none></none><mn>2</mn></mmultiscripts><mo></mo><mrow><mi href="./19.2#E4">F</mi><mo></mo><mrow><mo>(</mo><mi>β</mi><mo>,</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo stretchy="false">)</mo></mrow><mo>/</mo><mi href="./19.1#p1.t1.r3">k</mi></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex11" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m37.png" altimg-height="30px" altimg-valign="-9px" altimg-width="113px" alttext="\displaystyle\Pi\left(\phi,\alpha^{2},k_{1}\right)" display="inline"><mrow><mi href="./19.2#E7" mathvariant="normal">Π</mi><mo></mo><mrow><mo>(</mo><mi href="./19.1#p1.t1.r2">ϕ</mi><mo>,</mo><msup><mi href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup><mo>,</mo><msub><mi href="./19.1#p1.t1.r3">k</mi><mn>1</mn></msub><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m19.png" altimg-height="30px" altimg-valign="-9px" altimg-width="166px" alttext="\displaystyle=k\Pi\left(\beta,k^{2}\alpha^{2},k\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./19.1#p1.t1.r3">k</mi><mo></mo><mrow><mi href="./19.2#E7" mathvariant="normal">Π</mi><mo></mo><mrow><mo>(</mo><mi>β</mi><mo>,</mo><mrow><msup><mi href="./19.1#p1.t1.r3">k</mi><mn>2</mn></msup><mo></mo><msup><mi href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup></mrow><mo>,</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m79.png" altimg-height="23px" altimg-valign="-7px" altimg-width="81px" alttext="k_{1}=1/k" display="inline"><mrow><msub><mi href="./19.1#p1.t1.r3">k</mi><mn>1</mn></msub><mo>=</mo><mrow><mn>1</mn><mo>/</mo><mi href="./19.1#p1.t1.r3">k</mi></mrow></mrow></math>, <math class="ltx_Math" altimg="m70.png" altimg-height="21px" altimg-valign="-6px" altimg-width="170px" alttext="\sin\beta=k_{1}\sin\phi\leq 1" display="inline"><mrow><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi>β</mi></mrow><mo>=</mo><mrow><msub><mi href="./19.1#p1.t1.r3">k</mi><mn>1</mn></msub><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi href="./19.1#p1.t1.r2">ϕ</mi></mrow></mrow><mo>≤</mo><mn>1</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E4.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.2#E4" title="(19.2.4) ‣ §19.2(ii) Legendre’s Integrals ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="23px" altimg-valign="-7px" altimg-width="70px" alttext="F\left(\NVar{\phi},\NVar{k}\right)" display="inline"><mrow><mi href="./19.2#E4">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r2">ϕ</mi><mo>,</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Legendre’s incomplete elliptic integral of the first kind</a>,
<a href="./19.2#E5" title="(19.2.5) ‣ §19.2(ii) Legendre’s Integrals ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m46.png" altimg-height="23px" altimg-valign="-7px" altimg-width="71px" alttext="E\left(\NVar{\phi},\NVar{k}\right)" display="inline"><mrow><mi href="./19.2#E5">E</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r2">ϕ</mi><mo>,</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Legendre’s incomplete elliptic integral of the second kind</a>,
<a href="./19.2#E7" title="(19.2.7) ‣ §19.2(ii) Legendre’s Integrals ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="27px" altimg-valign="-9px" altimg-width="103px" alttext="\Pi\left(\NVar{\phi},\NVar{\alpha}^{2},\NVar{k}\right)" display="inline"><mrow><mi href="./19.2#E7" mathvariant="normal">Π</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r2">ϕ</mi><mo>,</mo><msup><mi class="ltx_nvar" href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup><mo>,</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Legendre’s incomplete elliptic integral of the third kind</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./19.1#p1.t1.r2" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m67.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="\phi" display="inline"><mi href="./19.1#p1.t1.r2">ϕ</mi></math>: real or complex argument</a>,
<a href="./19.1#p1.t1.r3" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./19.1#p1.t1.r3">k</mi></math>: real or complex modulus</a>,
<a href="./19.1#p1.t1.r4" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="19px" altimg-valign="-2px" altimg-width="21px" alttext="k^{\prime}" display="inline"><msup><mi href="./19.1#p1.t1.r4">k</mi><mo>′</mo></msup></math>: complementary modulus</a> and
<a href="./19.1#p1.t1.r5" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="20px" altimg-valign="-2px" altimg-width="26px" alttext="\alpha^{2}" display="inline"><msup><mi href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup></math>: real or complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="Px2.p2" class="ltx_para">
<p class="ltx_p">Provided the functions in these identities are correctly analytically continued
in the complex <math class="ltx_Math" altimg="m61.png" altimg-height="21px" altimg-valign="-6px" altimg-width="17px" alttext="\beta" display="inline"><mi>β</mi></math>-plane, then the identities will also hold in the
complex <math class="ltx_Math" altimg="m61.png" altimg-height="21px" altimg-valign="-6px" altimg-width="17px" alttext="\beta" display="inline"><mi>β</mi></math>-plane.</p>
</div>
</section>
<section id="Px3" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Imaginary-Modulus Transformation</h3>
<div id="Px3.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px3.p1" class="ltx_para">
<table id="E5" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex12" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="3" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">19.7.5</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m29.png" altimg-height="25px" altimg-valign="-7px" altimg-width="79px" alttext="\displaystyle F\left(\phi,ik\right)" display="inline"><mrow><mi href="./19.2#E4">F</mi><mo></mo><mrow><mo>(</mo><mi href="./19.1#p1.t1.r2">ϕ</mi><mo>,</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./19.1#p1.t1.r3">k</mi></mrow><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m12.png" altimg-height="26px" altimg-valign="-7px" altimg-width="118px" alttext="\displaystyle=\kappa^{\prime}F\left(\theta,\kappa\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msup><mi href="./19.7#Px3.p1">κ</mi><mo>′</mo></msup><mo></mo><mrow><mi href="./19.2#E4">F</mi><mo></mo><mrow><mo>(</mo><mi>θ</mi><mo>,</mo><mi href="./19.7#Px3.p1">κ</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex13" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m25.png" altimg-height="25px" altimg-valign="-7px" altimg-width="79px" alttext="\displaystyle E\left(\phi,ik\right)" display="inline"><mrow><mi href="./19.2#E5">E</mi><mo></mo><mrow><mo>(</mo><mi href="./19.1#p1.t1.r2">ϕ</mi><mo>,</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./19.1#p1.t1.r3">k</mi></mrow><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m1.png" altimg-height="41px" altimg-valign="-15px" altimg-width="484px" alttext="\displaystyle=(1/\kappa^{\prime})\left(E\left(\theta,\kappa\right)-\kappa^{2}%
\*(\sin\theta\cos\theta)\*(1-\kappa^{2}{\sin^{2}}\theta)^{-\ifrac{1}{2}}\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>/</mo><msup><mi href="./19.7#Px3.p1">κ</mi><mo>′</mo></msup></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo>(</mo><mrow><mrow><mi href="./19.2#E5">E</mi><mo></mo><mrow><mo>(</mo><mi>θ</mi><mo>,</mo><mi href="./19.7#Px3.p1">κ</mi><mo>)</mo></mrow></mrow><mo>-</mo><mrow><msup><mi href="./19.7#Px3.p1">κ</mi><mn>2</mn></msup><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi>θ</mi></mrow><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi>θ</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><mrow><msup><mi href="./19.7#Px3.p1">κ</mi><mn>2</mn></msup><mo></mo><mrow><msup><mi href="./4.14#E1">sin</mi><mn>2</mn></msup><mo></mo><mi>θ</mi></mrow></mrow></mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></mrow></msup></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex14" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m36.png" altimg-height="30px" altimg-valign="-9px" altimg-width="112px" alttext="\displaystyle\Pi\left(\phi,\alpha^{2},ik\right)" display="inline"><mrow><mi href="./19.2#E7" mathvariant="normal">Π</mi><mo></mo><mrow><mo>(</mo><mi href="./19.1#p1.t1.r2">ϕ</mi><mo>,</mo><msup><mi href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup><mo>,</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./19.1#p1.t1.r3">k</mi></mrow><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m7.png" altimg-height="41px" altimg-valign="-15px" altimg-width="382px" alttext="\displaystyle=(\kappa^{\prime}/\alpha_{1}^{2})\left(\kappa^{2}F\left(\theta,%
\kappa\right)+{\kappa^{\prime}}^{2}\alpha^{2}\Pi\left(\theta,\alpha_{1}^{2},%
\kappa\right)\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><msup><mi href="./19.7#Px3.p1">κ</mi><mo>′</mo></msup><mo>/</mo><msubsup><mi href="./19.1#p1.t1.r5">α</mi><mn>1</mn><mn>2</mn></msubsup></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo>(</mo><mrow><mrow><msup><mi href="./19.7#Px3.p1">κ</mi><mn>2</mn></msup><mo></mo><mrow><mi href="./19.2#E4">F</mi><mo></mo><mrow><mo>(</mo><mi>θ</mi><mo>,</mo><mi href="./19.7#Px3.p1">κ</mi><mo>)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mmultiscripts><mi href="./19.7#Px3.p1">κ</mi><none></none><mo>′</mo><none></none><mn>2</mn></mmultiscripts><mo></mo><msup><mi href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup><mo></mo><mrow><mi href="./19.2#E7" mathvariant="normal">Π</mi><mo></mo><mrow><mo>(</mo><mi>θ</mi><mo>,</mo><msubsup><mi href="./19.1#p1.t1.r5">α</mi><mn>1</mn><mn>2</mn></msubsup><mo>,</mo><mi href="./19.7#Px3.p1">κ</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E5.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m62.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./19.2#E4" title="(19.2.4) ‣ §19.2(ii) Legendre’s Integrals ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="23px" altimg-valign="-7px" altimg-width="70px" alttext="F\left(\NVar{\phi},\NVar{k}\right)" display="inline"><mrow><mi href="./19.2#E4">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r2">ϕ</mi><mo>,</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Legendre’s incomplete elliptic integral of the first kind</a>,
<a href="./19.2#E5" title="(19.2.5) ‣ §19.2(ii) Legendre’s Integrals ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m46.png" altimg-height="23px" altimg-valign="-7px" altimg-width="71px" alttext="E\left(\NVar{\phi},\NVar{k}\right)" display="inline"><mrow><mi href="./19.2#E5">E</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r2">ϕ</mi><mo>,</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Legendre’s incomplete elliptic integral of the second kind</a>,
<a href="./19.2#E7" title="(19.2.7) ‣ §19.2(ii) Legendre’s Integrals ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="27px" altimg-valign="-9px" altimg-width="103px" alttext="\Pi\left(\NVar{\phi},\NVar{\alpha}^{2},\NVar{k}\right)" display="inline"><mrow><mi href="./19.2#E7" mathvariant="normal">Π</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r2">ϕ</mi><mo>,</mo><msup><mi class="ltx_nvar" href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup><mo>,</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Legendre’s incomplete elliptic integral of the third kind</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./19.1#p1.t1.r2" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m67.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="\phi" display="inline"><mi href="./19.1#p1.t1.r2">ϕ</mi></math>: real or complex argument</a>,
<a href="./19.1#p1.t1.r3" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./19.1#p1.t1.r3">k</mi></math>: real or complex modulus</a>,
<a href="./19.1#p1.t1.r5" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="20px" altimg-valign="-2px" altimg-width="26px" alttext="\alpha^{2}" display="inline"><msup><mi href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup></math>: real or complex parameter</a>,
<a href="./19.7#Px3.p1" title="Imaginary-Modulus Transformation ‣ §19.7(ii) Change of Modulus and Amplitude ‣ §19.7 Connection Formulas ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\kappa" display="inline"><mi href="./19.7#Px3.p1">κ</mi></math>: modulus</a> and
<a href="./19.7#Px3.p1" title="Imaginary-Modulus Transformation ‣ §19.7(ii) Change of Modulus and Amplitude ‣ §19.7 Connection Formulas ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m64.png" altimg-height="19px" altimg-valign="-2px" altimg-width="21px" alttext="\kappa^{\prime}" display="inline"><msup><mi href="./19.7#Px3.p1">κ</mi><mo>′</mo></msup></math>: complementary modulus</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where
</p>
<table id="E6" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex15" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="4" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">19.7.6</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m40.png" altimg-height="14px" altimg-valign="-2px" altimg-width="18px" alttext="\displaystyle\kappa" display="inline"><mi href="./19.7#Px3.p1">κ</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m11.png" altimg-height="51px" altimg-valign="-21px" altimg-width="108px" alttext="\displaystyle=\frac{k}{\sqrt{1+k^{2}}}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mstyle displaystyle="true"><mfrac><mi href="./19.1#p1.t1.r3">k</mi><msqrt><mrow><mn>1</mn><mo>+</mo><msup><mi href="./19.1#p1.t1.r3">k</mi><mn>2</mn></msup></mrow></msqrt></mfrac></mstyle></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex16" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m41.png" altimg-height="21px" altimg-valign="-2px" altimg-width="23px" alttext="\displaystyle\kappa^{\prime}" display="inline"><msup><mi href="./19.7#Px3.p1">κ</mi><mo>′</mo></msup></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m8.png" altimg-height="50px" altimg-valign="-21px" altimg-width="108px" alttext="\displaystyle=\frac{1}{\sqrt{1+k^{2}}}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mstyle displaystyle="true"><mfrac><mn>1</mn><msqrt><mrow><mn>1</mn><mo>+</mo><msup><mi href="./19.1#p1.t1.r3">k</mi><mn>2</mn></msup></mrow></msqrt></mfrac></mstyle></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex17" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m42.png" altimg-height="19px" altimg-valign="-2px" altimg-width="44px" alttext="\displaystyle\sin\theta" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi>θ</mi></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m10.png" altimg-height="60px" altimg-valign="-25px" altimg-width="157px" alttext="\displaystyle=\frac{\sqrt{1+k^{2}}\sin\phi}{\sqrt{1+k^{2}{\sin^{2}}\phi}}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mstyle displaystyle="true"><mfrac><mrow><msqrt><mrow><mn>1</mn><mo>+</mo><msup><mi href="./19.1#p1.t1.r3">k</mi><mn>2</mn></msup></mrow></msqrt><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi href="./19.1#p1.t1.r2">ϕ</mi></mrow></mrow><msqrt><mrow><mn>1</mn><mo>+</mo><mrow><msup><mi href="./19.1#p1.t1.r3">k</mi><mn>2</mn></msup><mo></mo><mrow><msup><mi href="./4.14#E1">sin</mi><mn>2</mn></msup><mo></mo><mi href="./19.1#p1.t1.r2">ϕ</mi></mrow></mrow></mrow></msqrt></mfrac></mstyle></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex18" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m39.png" altimg-height="28px" altimg-valign="-7px" altimg-width="28px" alttext="\displaystyle\alpha_{1}^{2}" display="inline"><msubsup><mi href="./19.1#p1.t1.r5">α</mi><mn>1</mn><mn>2</mn></msubsup></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m9.png" altimg-height="51px" altimg-valign="-17px" altimg-width="103px" alttext="\displaystyle=\frac{\alpha^{2}+k^{2}}{1+k^{2}}." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mstyle displaystyle="true"><mfrac><mrow><msup><mi href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup><mo>+</mo><msup><mi href="./19.1#p1.t1.r3">k</mi><mn>2</mn></msup></mrow><mrow><mn>1</mn><mo>+</mo><msup><mi href="./19.1#p1.t1.r3">k</mi><mn>2</mn></msup></mrow></mfrac></mstyle></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E6.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./19.1#p1.t1.r2" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m67.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="\phi" display="inline"><mi href="./19.1#p1.t1.r2">ϕ</mi></math>: real or complex argument</a>,
<a href="./19.1#p1.t1.r3" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./19.1#p1.t1.r3">k</mi></math>: real or complex modulus</a>,
<a href="./19.1#p1.t1.r5" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="20px" altimg-valign="-2px" altimg-width="26px" alttext="\alpha^{2}" display="inline"><msup><mi href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup></math>: real or complex parameter</a>,
<a href="./19.7#Px3.p1" title="Imaginary-Modulus Transformation ‣ §19.7(ii) Change of Modulus and Amplitude ‣ §19.7 Connection Formulas ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\kappa" display="inline"><mi href="./19.7#Px3.p1">κ</mi></math>: modulus</a> and
<a href="./19.7#Px3.p1" title="Imaginary-Modulus Transformation ‣ §19.7(ii) Change of Modulus and Amplitude ‣ §19.7 Connection Formulas ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m64.png" altimg-height="19px" altimg-valign="-2px" altimg-width="21px" alttext="\kappa^{\prime}" display="inline"><msup><mi href="./19.7#Px3.p1">κ</mi><mo>′</mo></msup></math>: complementary modulus</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px4" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Imaginary-Argument Transformation</h3>
<div id="Px4.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px4.p1" class="ltx_para">
<p class="ltx_p">With <math class="ltx_Math" altimg="m71.png" altimg-height="21px" altimg-valign="-6px" altimg-width="127px" alttext="\sinh\phi=\tan\psi" display="inline"><mrow><mrow><mi href="./4.28#E1">sinh</mi><mo></mo><mi href="./19.1#p1.t1.r2">ϕ</mi></mrow><mo>=</mo><mrow><mi href="./4.14#E4">tan</mi><mo></mo><mi>ψ</mi></mrow></mrow></math>,</p>
<table id="E7" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex19" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="3" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">19.7.7</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m31.png" altimg-height="25px" altimg-valign="-7px" altimg-width="79px" alttext="\displaystyle F\left(i\phi,k\right)" display="inline"><mrow><mi href="./19.2#E4">F</mi><mo></mo><mrow><mo>(</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./19.1#p1.t1.r2">ϕ</mi></mrow><mo>,</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m13.png" altimg-height="26px" altimg-valign="-7px" altimg-width="116px" alttext="\displaystyle=iF\left(\psi,k^{\prime}\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mrow><mi href="./19.2#E4">F</mi><mo></mo><mrow><mo>(</mo><mi>ψ</mi><mo>,</mo><msup><mi href="./19.1#p1.t1.r4">k</mi><mo>′</mo></msup><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex20" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m27.png" altimg-height="25px" altimg-valign="-7px" altimg-width="79px" alttext="\displaystyle E\left(i\phi,k\right)" display="inline"><mrow><mi href="./19.2#E5">E</mi><mo></mo><mrow><mo>(</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./19.1#p1.t1.r2">ϕ</mi></mrow><mo>,</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m14.png" altimg-height="53px" altimg-valign="-21px" altimg-width="460px" alttext="\displaystyle=i\left(F\left(\psi,k^{\prime}\right)-E\left(\psi,k^{\prime}%
\right)+(\tan\psi)\sqrt{1-{k^{\prime}}^{2}{\sin^{2}}\psi}\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mrow><mi href="./19.2#E4">F</mi><mo></mo><mrow><mo>(</mo><mi>ψ</mi><mo>,</mo><msup><mi href="./19.1#p1.t1.r4">k</mi><mo>′</mo></msup><mo>)</mo></mrow></mrow><mo>-</mo><mrow><mi href="./19.2#E5">E</mi><mo></mo><mrow><mo>(</mo><mi>ψ</mi><mo>,</mo><msup><mi href="./19.1#p1.t1.r4">k</mi><mo>′</mo></msup><mo>)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./4.14#E4">tan</mi><mo></mo><mi>ψ</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msqrt><mrow><mn>1</mn><mo>-</mo><mrow><mmultiscripts><mi href="./19.1#p1.t1.r4">k</mi><none></none><mo>′</mo><none></none><mn>2</mn></mmultiscripts><mo></mo><mrow><msup><mi href="./4.14#E1">sin</mi><mn>2</mn></msup><mo></mo><mi>ψ</mi></mrow></mrow></mrow></msqrt></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex21" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m38.png" altimg-height="30px" altimg-valign="-9px" altimg-width="112px" alttext="\displaystyle\Pi\left(i\phi,\alpha^{2},k\right)" display="inline"><mrow><mi href="./19.2#E7" mathvariant="normal">Π</mi><mo></mo><mrow><mo>(</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./19.1#p1.t1.r2">ϕ</mi></mrow><mo>,</mo><msup><mi href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup><mo>,</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m15.png" altimg-height="30px" altimg-valign="-9px" altimg-width="406px" alttext="\displaystyle=i\left(F\left(\psi,k^{\prime}\right)-\alpha^{2}\Pi\left(\psi,1-%
\alpha^{2},k^{\prime}\right)\right)/{(1-\alpha^{2})}." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mi mathvariant="normal">i</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mi href="./19.2#E4">F</mi><mo></mo><mrow><mo>(</mo><mi>ψ</mi><mo>,</mo><msup><mi href="./19.1#p1.t1.r4">k</mi><mo>′</mo></msup><mo>)</mo></mrow></mrow><mo>-</mo><mrow><msup><mi href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup><mo></mo><mrow><mi href="./19.2#E7" mathvariant="normal">Π</mi><mo></mo><mrow><mo>(</mo><mi>ψ</mi><mo>,</mo><mrow><mn>1</mn><mo>-</mo><msup><mi href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup></mrow><mo>,</mo><msup><mi href="./19.1#p1.t1.r4">k</mi><mo>′</mo></msup><mo>)</mo></mrow></mrow></mrow></mrow><mo>)</mo></mrow></mrow><mo>/</mo><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><msup><mi href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E7.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.2#E4" title="(19.2.4) ‣ §19.2(ii) Legendre’s Integrals ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="23px" altimg-valign="-7px" altimg-width="70px" alttext="F\left(\NVar{\phi},\NVar{k}\right)" display="inline"><mrow><mi href="./19.2#E4">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r2">ϕ</mi><mo>,</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Legendre’s incomplete elliptic integral of the first kind</a>,
<a href="./19.2#E5" title="(19.2.5) ‣ §19.2(ii) Legendre’s Integrals ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m46.png" altimg-height="23px" altimg-valign="-7px" altimg-width="71px" alttext="E\left(\NVar{\phi},\NVar{k}\right)" display="inline"><mrow><mi href="./19.2#E5">E</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r2">ϕ</mi><mo>,</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Legendre’s incomplete elliptic integral of the second kind</a>,
<a href="./19.2#E7" title="(19.2.7) ‣ §19.2(ii) Legendre’s Integrals ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="27px" altimg-valign="-9px" altimg-width="103px" alttext="\Pi\left(\NVar{\phi},\NVar{\alpha}^{2},\NVar{k}\right)" display="inline"><mrow><mi href="./19.2#E7" mathvariant="normal">Π</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r2">ϕ</mi><mo>,</mo><msup><mi class="ltx_nvar" href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup><mo>,</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Legendre’s incomplete elliptic integral of the third kind</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./4.14#E4" title="(4.14.4) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="17px" altimg-valign="-2px" altimg-width="47px" alttext="\tan\NVar{z}" display="inline"><mrow><mi href="./4.14#E4">tan</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: tangent function</a>,
<a href="./19.1#p1.t1.r2" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m67.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="\phi" display="inline"><mi href="./19.1#p1.t1.r2">ϕ</mi></math>: real or complex argument</a>,
<a href="./19.1#p1.t1.r3" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./19.1#p1.t1.r3">k</mi></math>: real or complex modulus</a>,
<a href="./19.1#p1.t1.r4" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="19px" altimg-valign="-2px" altimg-width="21px" alttext="k^{\prime}" display="inline"><msup><mi href="./19.1#p1.t1.r4">k</mi><mo>′</mo></msup></math>: complementary modulus</a> and
<a href="./19.1#p1.t1.r5" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="20px" altimg-valign="-2px" altimg-width="26px" alttext="\alpha^{2}" display="inline"><msup><mi href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup></math>: real or complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="Px4.p2" class="ltx_para">
<p class="ltx_p">For two further transformations of this type see
<cite class="ltx_cite ltx_citemacro_citet">Erdélyi<span class="ltx_text ltx_bib_etal"> et al.</span> (, p. 316)</cite>.</p>
</div>
</section>
</section>
<section id="iii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§19.7(iii) </span>Change of Parameter of <math class="ltx_Math" altimg="m54.png" altimg-height="27px" altimg-valign="-9px" altimg-width="103px" alttext="\Pi\left(\phi,\alpha^{2},k\right)" display="inline"><mrow><mi href="./19.2#E7" mathvariant="normal">Π</mi><mo></mo><mrow><mo>(</mo><mi href="./19.1#p1.t1.r2">ϕ</mi><mo>,</mo><msup><mi href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup><mo>,</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>
</h2>
<div id="SS3.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="SS3.p1" class="ltx_para">
<p class="ltx_p">There are three relations connecting <math class="ltx_Math" altimg="m54.png" altimg-height="27px" altimg-valign="-9px" altimg-width="103px" alttext="\Pi\left(\phi,\alpha^{2},k\right)" display="inline"><mrow><mi href="./19.2#E7" mathvariant="normal">Π</mi><mo></mo><mrow><mo>(</mo><mi href="./19.1#p1.t1.r2">ϕ</mi><mo>,</mo><msup><mi href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup><mo>,</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math> and
<math class="ltx_Math" altimg="m55.png" altimg-height="27px" altimg-valign="-9px" altimg-width="103px" alttext="\Pi\left(\phi,\omega^{2},k\right)" display="inline"><mrow><mi href="./19.2#E7" mathvariant="normal">Π</mi><mo></mo><mrow><mo>(</mo><mi href="./19.1#p1.t1.r2">ϕ</mi><mo>,</mo><msup><mi href="./19.7#SS3.p1">ω</mi><mn>2</mn></msup><mo>,</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>, where <math class="ltx_Math" altimg="m66.png" altimg-height="20px" altimg-valign="-2px" altimg-width="26px" alttext="\omega^{2}" display="inline"><msup><mi href="./19.7#SS3.p1">ω</mi><mn>2</mn></msup></math> is a rational function of
<math class="ltx_Math" altimg="m58.png" altimg-height="20px" altimg-valign="-2px" altimg-width="26px" alttext="\alpha^{2}" display="inline"><msup><mi href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup></math>. If <math class="ltx_Math" altimg="m75.png" altimg-height="20px" altimg-valign="-2px" altimg-width="24px" alttext="k^{2}" display="inline"><msup><mi href="./19.1#p1.t1.r3">k</mi><mn>2</mn></msup></math> and <math class="ltx_Math" altimg="m58.png" altimg-height="20px" altimg-valign="-2px" altimg-width="26px" alttext="\alpha^{2}" display="inline"><msup><mi href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup></math> are real, then both integrals are circular
cases or both are hyperbolic cases (see §).</p>
</div>
<div id="SS3.p2" class="ltx_para">
<p class="ltx_p">The first of the three relations maps each circular region onto itself and each
hyperbolic region onto the other; in particular, it gives the Cauchy principal
value of <math class="ltx_Math" altimg="m54.png" altimg-height="27px" altimg-valign="-9px" altimg-width="103px" alttext="\Pi\left(\phi,\alpha^{2},k\right)" display="inline"><mrow><mi href="./19.2#E7" mathvariant="normal">Π</mi><mo></mo><mrow><mo>(</mo><mi href="./19.1#p1.t1.r2">ϕ</mi><mo>,</mo><msup><mi href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup><mo>,</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math> when <math class="ltx_Math" altimg="m57.png" altimg-height="24px" altimg-valign="-6px" altimg-width="99px" alttext="\alpha^{2}>{\csc^{2}}\phi" display="inline"><mrow><msup><mi href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup><mo>></mo><mrow><msup><mi href="./4.14#E5">csc</mi><mn>2</mn></msup><mo></mo><mi href="./19.1#p1.t1.r2">ϕ</mi></mrow></mrow></math> (see
() for the complete case). Let <math class="ltx_Math" altimg="m73.png" altimg-height="24px" altimg-valign="-6px" altimg-width="134px" alttext="c={\csc^{2}}\phi\neq\alpha^{2}" display="inline"><mrow><mi>c</mi><mo>=</mo><mrow><msup><mi href="./4.14#E5">csc</mi><mn>2</mn></msup><mo></mo><mi href="./19.1#p1.t1.r2">ϕ</mi></mrow><mo>≠</mo><msup><mi href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup></mrow></math>. Then</p>
<table id="E8" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.7.8</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E8.png" altimg-height="30px" altimg-valign="-9px" altimg-width="706px" alttext="\Pi\left(\phi,\alpha^{2},k\right)+\Pi\left(\phi,\omega^{2},k\right)=F\left(%
\phi,k\right)+\sqrt{c}R_{C}\left((c-1)(c-k^{2}),(c-\alpha^{2})(c-\omega^{2})%
\right)," display="block"><mrow><mrow><mrow><mrow><mi href="./19.2#E7" mathvariant="normal">Π</mi><mo></mo><mrow><mo>(</mo><mi href="./19.1#p1.t1.r2">ϕ</mi><mo>,</mo><msup><mi href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup><mo>,</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo>+</mo><mrow><mi href="./19.2#E7" mathvariant="normal">Π</mi><mo></mo><mrow><mo>(</mo><mi href="./19.1#p1.t1.r2">ϕ</mi><mo>,</mo><msup><mi href="./19.7#SS3.p1">ω</mi><mn>2</mn></msup><mo>,</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mrow><mrow><mi href="./19.2#E4">F</mi><mo></mo><mrow><mo>(</mo><mi href="./19.1#p1.t1.r2">ϕ</mi><mo>,</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo>+</mo><mrow><msqrt><mi>c</mi></msqrt><mo></mo><mrow><msub><mi href="./19.2#E17">R</mi><mi href="./19.2#E17">C</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mi>c</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>c</mi><mo>-</mo><msup><mi href="./19.1#p1.t1.r3">k</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mi>c</mi><mo>-</mo><msup><mi href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>c</mi><mo>-</mo><msup><mi href="./19.7#SS3.p1">ω</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m59.png" altimg-height="20px" altimg-valign="-2px" altimg-width="95px" alttext="\alpha^{2}\omega^{2}=k^{2}" display="inline"><mrow><mrow><msup><mi href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup><mo></mo><msup><mi href="./19.7#SS3.p1">ω</mi><mn>2</mn></msup></mrow><mo>=</mo><msup><mi href="./19.1#p1.t1.r3">k</mi><mn>2</mn></msup></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E8.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.2#E17" title="(19.2.17) ‣ §19.2(iv) A Related Function: R C ( x , y ) ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="23px" altimg-valign="-7px" altimg-width="82px" alttext="R_{C}\left(\NVar{x},\NVar{y}\right)" display="inline"><mrow><msub><mi href="./19.2#E17">R</mi><mi href="./19.2#E17">C</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>)</mo></mrow></mrow></math>: Carlson’s combination of inverse circular and inverse hyperbolic functions</a>,
<a href="./19.2#E4" title="(19.2.4) ‣ §19.2(ii) Legendre’s Integrals ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="23px" altimg-valign="-7px" altimg-width="70px" alttext="F\left(\NVar{\phi},\NVar{k}\right)" display="inline"><mrow><mi href="./19.2#E4">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r2">ϕ</mi><mo>,</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Legendre’s incomplete elliptic integral of the first kind</a>,
<a href="./19.2#E7" title="(19.2.7) ‣ §19.2(ii) Legendre’s Integrals ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="27px" altimg-valign="-9px" altimg-width="103px" alttext="\Pi\left(\NVar{\phi},\NVar{\alpha}^{2},\NVar{k}\right)" display="inline"><mrow><mi href="./19.2#E7" mathvariant="normal">Π</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r2">ϕ</mi><mo>,</mo><msup><mi class="ltx_nvar" href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup><mo>,</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Legendre’s incomplete elliptic integral of the third kind</a>,
<a href="./19.1#p1.t1.r2" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m67.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="\phi" display="inline"><mi href="./19.1#p1.t1.r2">ϕ</mi></math>: real or complex argument</a>,
<a href="./19.1#p1.t1.r3" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./19.1#p1.t1.r3">k</mi></math>: real or complex modulus</a>,
<a href="./19.1#p1.t1.r5" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="20px" altimg-valign="-2px" altimg-width="26px" alttext="\alpha^{2}" display="inline"><msup><mi href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup></math>: real or complex parameter</a> and
<a href="./19.7#SS3.p1" title="§19.7(iii) Change of Parameter of Π ( ϕ , α 2 , k ) ‣ §19.7 Connection Formulas ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="20px" altimg-valign="-2px" altimg-width="26px" alttext="\omega^{2}" display="inline"><msup><mi href="./19.7#SS3.p1">ω</mi><mn>2</mn></msup></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Since <math class="ltx_Math" altimg="m76.png" altimg-height="22px" altimg-valign="-5px" altimg-width="59px" alttext="k^{2}\leq c" display="inline"><mrow><msup><mi href="./19.1#p1.t1.r3">k</mi><mn>2</mn></msup><mo>≤</mo><mi>c</mi></mrow></math> we have <math class="ltx_Math" altimg="m60.png" altimg-height="22px" altimg-valign="-5px" altimg-width="83px" alttext="\alpha^{2}\omega^{2}\leq c" display="inline"><mrow><mrow><msup><mi href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup><mo></mo><msup><mi href="./19.7#SS3.p1">ω</mi><mn>2</mn></msup></mrow><mo>≤</mo><mi>c</mi></mrow></math>; hence <math class="ltx_Math" altimg="m56.png" altimg-height="21px" altimg-valign="-3px" altimg-width="61px" alttext="\alpha^{2}>c" display="inline"><mrow><msup><mi href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup><mo>></mo><mi>c</mi></mrow></math>
implies <math class="ltx_Math" altimg="m65.png" altimg-height="22px" altimg-valign="-5px" altimg-width="98px" alttext="\omega^{2}<1\leq c" display="inline"><mrow><msup><mi href="./19.7#SS3.p1">ω</mi><mn>2</mn></msup><mo><</mo><mn>1</mn><mo>≤</mo><mi>c</mi></mrow></math>.</p>
</div>
<div id="SS3.p3" class="ltx_para">
<p class="ltx_p">The second relation maps each hyperbolic region onto itself and each circular
region onto the other:</p>
<table id="E9" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.7.9</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_Math ltx_eqn_cell ltx_align_center"><math class="ltx_Math" alttext="(k^{2}-\alpha^{2})\Pi\left(\phi,\alpha^{2},k\right)+(k^{2}-\omega^{2})\Pi\left%
(\phi,\omega^{2},k\right)=k^{2}F\left(\phi,k\right)-\alpha^{2}\omega^{2}\sqrt{%
c-1}R_{C}\left(c(c-k^{2}),(c-\alpha^{2})(c-\omega^{2})\right)," display="block"><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><msup><mi href="./19.1#p1.t1.r3">k</mi><mn>2</mn></msup><mo>-</mo><msup><mi href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mi href="./19.2#E7" mathvariant="normal">Π</mi><mo></mo><mrow><mo>(</mo><mi href="./19.1#p1.t1.r2">ϕ</mi><mo>,</mo><msup><mi href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup><mo>,</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><msup><mi href="./19.1#p1.t1.r3">k</mi><mn>2</mn></msup><mo>-</mo><msup><mi href="./19.7#SS3.p1">ω</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mi href="./19.2#E7" mathvariant="normal">Π</mi><mo></mo><mrow><mo>(</mo><mi href="./19.1#p1.t1.r2">ϕ</mi><mo>,</mo><msup><mi href="./19.7#SS3.p1">ω</mi><mn>2</mn></msup><mo>,</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>=</mo><mrow><mrow><msup><mi href="./19.1#p1.t1.r3">k</mi><mn>2</mn></msup><mo></mo><mrow><mi href="./19.2#E4">F</mi><mo></mo><mrow><mo>(</mo><mi href="./19.1#p1.t1.r2">ϕ</mi><mo>,</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></mrow><mo>-</mo><mrow><msup><mi href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup><mo></mo><msup><mi href="./19.7#SS3.p1">ω</mi><mn>2</mn></msup><mo></mo><msqrt><mrow><mi>c</mi><mo>-</mo><mn>1</mn></mrow></msqrt><mo></mo><mrow><msub><mi href="./19.2#E17">R</mi><mi href="./19.2#E17">C</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi>c</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>c</mi><mo>-</mo><msup><mi href="./19.1#p1.t1.r3">k</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mi>c</mi><mo>-</mo><msup><mi href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>c</mi><mo>-</mo><msup><mi href="./19.7#SS3.p1">ω</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mtd></mtr></mtable></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m43.png" altimg-height="25px" altimg-valign="-7px" altimg-width="229px" alttext="(1-\alpha^{2})(1-\omega^{2})=1-k^{2}" display="inline"><mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><msup><mi href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><msup><mi href="./19.7#SS3.p1">ω</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mn>1</mn><mo>-</mo><msup><mi href="./19.1#p1.t1.r3">k</mi><mn>2</mn></msup></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E9.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.2#E17" title="(19.2.17) ‣ §19.2(iv) A Related Function: R C ( x , y ) ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="23px" altimg-valign="-7px" altimg-width="82px" alttext="R_{C}\left(\NVar{x},\NVar{y}\right)" display="inline"><mrow><msub><mi href="./19.2#E17">R</mi><mi href="./19.2#E17">C</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>)</mo></mrow></mrow></math>: Carlson’s combination of inverse circular and inverse hyperbolic functions</a>,
<a href="./19.2#E4" title="(19.2.4) ‣ §19.2(ii) Legendre’s Integrals ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="23px" altimg-valign="-7px" altimg-width="70px" alttext="F\left(\NVar{\phi},\NVar{k}\right)" display="inline"><mrow><mi href="./19.2#E4">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r2">ϕ</mi><mo>,</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Legendre’s incomplete elliptic integral of the first kind</a>,
<a href="./19.2#E7" title="(19.2.7) ‣ §19.2(ii) Legendre’s Integrals ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="27px" altimg-valign="-9px" altimg-width="103px" alttext="\Pi\left(\NVar{\phi},\NVar{\alpha}^{2},\NVar{k}\right)" display="inline"><mrow><mi href="./19.2#E7" mathvariant="normal">Π</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r2">ϕ</mi><mo>,</mo><msup><mi class="ltx_nvar" href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup><mo>,</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Legendre’s incomplete elliptic integral of the third kind</a>,
<a href="./19.1#p1.t1.r2" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m67.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="\phi" display="inline"><mi href="./19.1#p1.t1.r2">ϕ</mi></math>: real or complex argument</a>,
<a href="./19.1#p1.t1.r3" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./19.1#p1.t1.r3">k</mi></math>: real or complex modulus</a>,
<a href="./19.1#p1.t1.r5" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="20px" altimg-valign="-2px" altimg-width="26px" alttext="\alpha^{2}" display="inline"><msup><mi href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup></math>: real or complex parameter</a> and
<a href="./19.7#SS3.p1" title="§19.7(iii) Change of Parameter of Π ( ϕ , α 2 , k ) ‣ §19.7 Connection Formulas ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="20px" altimg-valign="-2px" altimg-width="26px" alttext="\omega^{2}" display="inline"><msup><mi href="./19.7#SS3.p1">ω</mi><mn>2</mn></msup></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS3.p4" class="ltx_para">
<p class="ltx_p">The third relation (missing from the literature of Legendre’s integrals) maps
each circular region onto the other and each hyperbolic region onto the other:</p>
<table id="E10" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.7.10</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_Math ltx_eqn_cell ltx_align_center"><math class="ltx_Math" alttext="(1-\alpha^{2})\Pi\left(\phi,\alpha^{2},k\right)+(1-\omega^{2})\Pi\left(\phi,%
\omega^{2},k\right)=F\left(\phi,k\right)+(1-\alpha^{2}-\omega^{2})\sqrt{c-k^{2%
}}\*R_{C}\left(c(c-1),(c-\alpha^{2})(c-\omega^{2})\right)," display="block"><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><msup><mi href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mi href="./19.2#E7" mathvariant="normal">Π</mi><mo></mo><mrow><mo>(</mo><mi href="./19.1#p1.t1.r2">ϕ</mi><mo>,</mo><msup><mi href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup><mo>,</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><msup><mi href="./19.7#SS3.p1">ω</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mi href="./19.2#E7" mathvariant="normal">Π</mi><mo></mo><mrow><mo>(</mo><mi href="./19.1#p1.t1.r2">ϕ</mi><mo>,</mo><msup><mi href="./19.7#SS3.p1">ω</mi><mn>2</mn></msup><mo>,</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>=</mo><mrow><mrow><mi href="./19.2#E4">F</mi><mo></mo><mrow><mo>(</mo><mi href="./19.1#p1.t1.r2">ϕ</mi><mo>,</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo>+</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><msup><mi href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup><mo>-</mo><msup><mi href="./19.7#SS3.p1">ω</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msqrt><mrow><mi>c</mi><mo>-</mo><msup><mi href="./19.1#p1.t1.r3">k</mi><mn>2</mn></msup></mrow></msqrt><mo></mo><mrow><msub><mi href="./19.2#E17">R</mi><mi href="./19.2#E17">C</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi>c</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>c</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mi>c</mi><mo>-</mo><msup><mi href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>c</mi><mo>-</mo><msup><mi href="./19.7#SS3.p1">ω</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mtd></mtr></mtable></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m44.png" altimg-height="25px" altimg-valign="-7px" altimg-width="284px" alttext="(k^{2}-\alpha^{2})(k^{2}-\omega^{2})=k^{2}(k^{2}-1)" display="inline"><mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><msup><mi href="./19.1#p1.t1.r3">k</mi><mn>2</mn></msup><mo>-</mo><msup><mi href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><msup><mi href="./19.1#p1.t1.r3">k</mi><mn>2</mn></msup><mo>-</mo><msup><mi href="./19.7#SS3.p1">ω</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><msup><mi href="./19.1#p1.t1.r3">k</mi><mn>2</mn></msup><mo></mo><mrow><mo stretchy="false">(</mo><mrow><msup><mi href="./19.1#p1.t1.r3">k</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E10.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.2#E17" title="(19.2.17) ‣ §19.2(iv) A Related Function: R C ( x , y ) ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="23px" altimg-valign="-7px" altimg-width="82px" alttext="R_{C}\left(\NVar{x},\NVar{y}\right)" display="inline"><mrow><msub><mi href="./19.2#E17">R</mi><mi href="./19.2#E17">C</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>)</mo></mrow></mrow></math>: Carlson’s combination of inverse circular and inverse hyperbolic functions</a>,
<a href="./19.2#E4" title="(19.2.4) ‣ §19.2(ii) Legendre’s Integrals ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="23px" altimg-valign="-7px" altimg-width="70px" alttext="F\left(\NVar{\phi},\NVar{k}\right)" display="inline"><mrow><mi href="./19.2#E4">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r2">ϕ</mi><mo>,</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Legendre’s incomplete elliptic integral of the first kind</a>,
<a href="./19.2#E7" title="(19.2.7) ‣ §19.2(ii) Legendre’s Integrals ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="27px" altimg-valign="-9px" altimg-width="103px" alttext="\Pi\left(\NVar{\phi},\NVar{\alpha}^{2},\NVar{k}\right)" display="inline"><mrow><mi href="./19.2#E7" mathvariant="normal">Π</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r2">ϕ</mi><mo>,</mo><msup><mi class="ltx_nvar" href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup><mo>,</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Legendre’s incomplete elliptic integral of the third kind</a>,
<a href="./19.1#p1.t1.r2" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m67.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="\phi" display="inline"><mi href="./19.1#p1.t1.r2">ϕ</mi></math>: real or complex argument</a>,
<a href="./19.1#p1.t1.r3" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./19.1#p1.t1.r3">k</mi></math>: real or complex modulus</a>,
<a href="./19.1#p1.t1.r5" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="20px" altimg-valign="-2px" altimg-width="26px" alttext="\alpha^{2}" display="inline"><msup><mi href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup></math>: real or complex parameter</a> and
<a href="./19.7#SS3.p1" title="§19.7(iii) Change of Parameter of Π ( ϕ , α 2 , k ) ‣ §19.7 Connection Formulas ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="20px" altimg-valign="-2px" altimg-width="26px" alttext="\omega^{2}" display="inline"><msup><mi href="./19.7#SS3.p1">ω</mi><mn>2</mn></msup></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
</section>
</div>
<div class="ltx_page_footer">
<span></div>
</div>
</body></text>
</html>
</page>
<page>
<!DOCTYPE html><html>
<head>
<title>DLMF: 19.18 Derivatives and Differential Equations</title>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
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<text><body class="color_default textfont_default titlefont_default fontsize_default navbar_default">
<div class="ltx_page_navbar">
<div class="ltx_page_navlogo">, <a class="ltx_keyword" href="./idx/H#hypergeometricRfunction">hypergeometric <math class="ltx_Math" altimg="m6.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="R" display="inline"><mi>R</mi></math>-function</a>, ) put <math class="ltx_Math" altimg="m37.png" altimg-height="18px" altimg-valign="-4px" altimg-width="64px" alttext="t=-a" display="inline"><mrow><mi>t</mi><mo>=</mo><mrow><mo>-</mo><mi>a</mi></mrow></mrow></math> and
<math class="ltx_Math" altimg="m26.png" altimg-height="20px" altimg-valign="-4px" altimg-width="90px" alttext="c=a+a^{\prime}" display="inline"><mrow><mi>c</mi><mo>=</mo><mrow><mi>a</mi><mo>+</mo><msup><mi>a</mi><mo>′</mo></msup></mrow></mrow></math> in <cite class="ltx_cite ltx_citemacro_citet">Carlson (</dd>
</dl>
</div>
</div>
<div id="SS1.p1" class="ltx_para">
<table id="E1" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.18.1</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E1.png" altimg-height="48px" altimg-valign="-16px" altimg-width="281px" alttext="\frac{\partial R_{F}\left(x,y,z\right)}{\partial z}=-\tfrac{1}{6}R_{D}\left(x,%
y,z\right)," display="block"><mrow><mrow><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>z</mi></mrow></mfrac><mo>=</mo><mrow><mo>-</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>6</mn></mfrac></mstyle><mo></mo><mrow><msub><mi href="./19.16#E5">R</mi><mi href="./19.16#E5">D</mi></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.16#E5" title="(19.16.5) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m8.png" altimg-height="23px" altimg-valign="-7px" altimg-width="102px" alttext="R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E5">R</mi><mi href="./19.16#E5">D</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: elliptic integral symmetric in only two variables</a>,
<a href="./19.16#E1" title="(19.16.1) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m10.png" altimg-height="23px" altimg-valign="-7px" altimg-width="101px" alttext="R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: symmetric elliptic integral of first kind</a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m27.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m44.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m44.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E2" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.18.2</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E2.png" altimg-height="47px" altimg-valign="-16px" altimg-width="481px" alttext="\frac{\mathrm{d}}{\mathrm{d}x}R_{G}\left(x+a,x+b,x+c\right)=\tfrac{1}{2}R_{F}%
\left(x+a,x+b,x+c\right)." display="block"><mrow><mrow><mrow><mfrac><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi>x</mi></mrow></mfrac><mo></mo><mrow><msub><mi href="./19.16#E3">R</mi><mi href="./19.16#E3">G</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mi>a</mi></mrow><mo>,</mo><mrow><mi>x</mi><mo>+</mo><mi>b</mi></mrow><mo>,</mo><mrow><mi>x</mi><mo>+</mo><mi>c</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mi>a</mi></mrow><mo>,</mo><mrow><mi>x</mi><mo>+</mo><mi>b</mi></mrow><mo>,</mo><mrow><mi>x</mi><mo>+</mo><mi>c</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.16#E1" title="(19.16.1) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m10.png" altimg-height="23px" altimg-valign="-7px" altimg-width="101px" alttext="R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: symmetric elliptic integral of first kind</a>,
<a href="./19.16#E3" title="(19.16.3) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m12.png" altimg-height="23px" altimg-valign="-7px" altimg-width="101px" alttext="R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E3">R</mi><mi href="./19.16#E3">G</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: symmetric elliptic integral of second kind</a> and
<a href="./1.4#E4" title="(1.4.4) ‣ §1.4(iii) Derivatives ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="29px" altimg-valign="-9px" altimg-width="27px" alttext="\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: derivative of <math class="ltx_Math" altimg="m27.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m44.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Let <math class="ltx_Math" altimg="m24.png" altimg-height="24px" altimg-valign="-8px" altimg-width="106px" alttext="\partial_{j}=\ifrac{\partial}{\partial z_{j}}" display="inline"><mrow><msub><mo href="./1.5#E3">∂</mo><mi>j</mi></msub><mo>=</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3">/</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><msub><mi>z</mi><mi>j</mi></msub></mrow></mrow></mrow></math>, and <math class="ltx_Math" altimg="m21.png" altimg-height="18px" altimg-valign="-8px" altimg-width="23px" alttext="\mathbf{e}_{j}" display="inline"><msub><mi mathvariant="bold">e</mi><mi>j</mi></msub></math> be an <math class="ltx_Math" altimg="m35.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./19.1#p1.t1.r1">n</mi></math>-tuple with 1
in the <math class="ltx_Math" altimg="m29.png" altimg-height="20px" altimg-valign="-6px" altimg-width="14px" alttext="j" display="inline"><mi>j</mi></math>th place and 0’s elsewhere. Also define
</p>
<table id="E3" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex1" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">19.18.3</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m4.png" altimg-height="20px" altimg-valign="-8px" altimg-width="29px" alttext="\displaystyle w_{j}" display="inline"><msub><mi href="./19.18#SS1.p1">w</mi><mi>j</mi></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m2.png" altimg-height="67px" altimg-valign="-30px" altimg-width="130px" alttext="\displaystyle=b_{j}\biggm{/}\sum_{j=1}^{n}b_{j}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msub><mi>b</mi><mi>j</mi></msub><mo mathsize="210%" stretchy="false">/</mo><mrow><mstyle displaystyle="true"><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi href="./19.1#p1.t1.r1">n</mi></munderover></mstyle><msub><mi>b</mi><mi>j</mi></msub></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex2" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m3.png" altimg-height="21px" altimg-valign="-2px" altimg-width="22px" alttext="\displaystyle a^{\prime}" display="inline"><msup><mi>a</mi><mo>′</mo></msup></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m1.png" altimg-height="67px" altimg-valign="-30px" altimg-width="132px" alttext="\displaystyle=-a+\sum_{j=1}^{n}b_{j}." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mo>-</mo><mi>a</mi></mrow><mo>+</mo><mrow><mstyle displaystyle="true"><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi href="./19.1#p1.t1.r1">n</mi></munderover></mstyle><msub><mi>b</mi><mi>j</mi></msub></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.1#p1.t1.r1" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m35.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./19.1#p1.t1.r1">n</mi></math>: nonnegative integer</a> and
<a href="./19.18#SS1.p1" title="§19.18(i) Derivatives ‣ §19.18 Derivatives and Differential Equations ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m42.png" altimg-height="18px" altimg-valign="-8px" altimg-width="27px" alttext="w_{j}" display="inline"><msub><mi href="./19.18#SS1.p1">w</mi><mi>j</mi></msub></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">The next two equations apply to ().</p>
<table id="E4" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.18.4</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E4.png" altimg-height="26px" altimg-valign="-8px" altimg-width="348px" alttext="\partial_{j}R_{-a}\left(\mathbf{b};\mathbf{z}\right)=-aw_{j}R_{-a-1}\left(%
\mathbf{b}+\mathbf{e}_{j};\mathbf{z}\right)," display="block"><mrow><mrow><mrow><msub><mo href="./1.5#E3">∂</mo><mi>j</mi></msub><mo></mo><mrow><msub><mi href="./19.16#E9">R</mi><mrow><mo>-</mo><mi>a</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi mathvariant="bold">b</mi><mo>;</mo><mi mathvariant="bold">z</mi><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mrow><mo>-</mo><mrow><mi>a</mi><mo></mo><msub><mi href="./19.18#SS1.p1">w</mi><mi>j</mi></msub><mo></mo><mrow><msub><mi href="./19.16#E9">R</mi><mrow><mrow><mo>-</mo><mi>a</mi></mrow><mo>-</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mrow><mi mathvariant="bold">b</mi><mo>+</mo><msub><mi mathvariant="bold">e</mi><mi>j</mi></msub></mrow><mo>;</mo><mi mathvariant="bold">z</mi><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E4.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.16#E9" title="(19.16.9) ‣ §19.16(ii) R - a ( b ; z ) ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m14.png" altimg-height="23px" altimg-valign="-7px" altimg-width="233px" alttext="R_{\NVar{-a}}\left(\NVar{b_{1}},\dots,\NVar{b_{n}};\NVar{z_{1}},\dots,\NVar{z_%
{n}}\right)" display="inline"><mrow><msub><mi href="./19.16#E9">R</mi><mrow><mo class="ltx_nvar">-</mo><mi class="ltx_nvar">a</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mrow><msub><mi class="ltx_nvar">b</mi><mn class="ltx_nvar">1</mn></msub><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><msub><mi class="ltx_nvar">b</mi><mi class="ltx_nvar" href="./19.1#p1.t1.r1">n</mi></msub></mrow><mo>;</mo><mrow><msub><mi class="ltx_nvar">z</mi><mn class="ltx_nvar">1</mn></msub><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><msub><mi class="ltx_nvar">z</mi><mi class="ltx_nvar" href="./19.1#p1.t1.r1">n</mi></msub></mrow><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m13.png" altimg-height="23px" altimg-valign="-7px" altimg-width="92px" alttext="R_{\NVar{-a}}\left(\NVar{\mathbf{b}};\NVar{\mathbf{z}}\right)" display="inline"><mrow><msub><mi href="./19.16#E9">R</mi><mrow><mo class="ltx_nvar">-</mo><mi class="ltx_nvar">a</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" mathvariant="bold">b</mi><mo>;</mo><mi class="ltx_nvar" mathvariant="bold">z</mi><mo>)</mo></mrow></mrow></math>: multivariate hypergeometric function</a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m44.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./19.18#SS1.p1" title="§19.18(i) Derivatives ‣ §19.18 Derivatives and Differential Equations ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m42.png" altimg-height="18px" altimg-valign="-8px" altimg-width="27px" alttext="w_{j}" display="inline"><msub><mi href="./19.18#SS1.p1">w</mi><mi>j</mi></msub></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E5" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.18.5</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E5.png" altimg-height="27px" altimg-valign="-8px" altimg-width="393px" alttext="(z_{j}\partial_{j}+b_{j})R_{-a}\left(\mathbf{b};\mathbf{z}\right)=w_{j}a^{%
\prime}R_{-a}\left(\mathbf{b}+\mathbf{e}_{j};\mathbf{z}\right)." display="block"><mrow><mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><msub><mi>z</mi><mi>j</mi></msub><mo></mo><msub><mo href="./1.5#E3">∂</mo><mi>j</mi></msub></mrow><mo>+</mo><msub><mi>b</mi><mi>j</mi></msub></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><msub><mi href="./19.16#E9">R</mi><mrow><mo>-</mo><mi>a</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi mathvariant="bold">b</mi><mo>;</mo><mi mathvariant="bold">z</mi><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mrow><msub><mi href="./19.18#SS1.p1">w</mi><mi>j</mi></msub><mo></mo><msup><mi>a</mi><mo>′</mo></msup><mo></mo><mrow><msub><mi href="./19.16#E9">R</mi><mrow><mo>-</mo><mi>a</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mrow><mi mathvariant="bold">b</mi><mo>+</mo><msub><mi mathvariant="bold">e</mi><mi>j</mi></msub></mrow><mo>;</mo><mi mathvariant="bold">z</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E5.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.16#E9" title="(19.16.9) ‣ §19.16(ii) R - a ( b ; z ) ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m14.png" altimg-height="23px" altimg-valign="-7px" altimg-width="233px" alttext="R_{\NVar{-a}}\left(\NVar{b_{1}},\dots,\NVar{b_{n}};\NVar{z_{1}},\dots,\NVar{z_%
{n}}\right)" display="inline"><mrow><msub><mi href="./19.16#E9">R</mi><mrow><mo class="ltx_nvar">-</mo><mi class="ltx_nvar">a</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mrow><msub><mi class="ltx_nvar">b</mi><mn class="ltx_nvar">1</mn></msub><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><msub><mi class="ltx_nvar">b</mi><mi class="ltx_nvar" href="./19.1#p1.t1.r1">n</mi></msub></mrow><mo>;</mo><mrow><msub><mi class="ltx_nvar">z</mi><mn class="ltx_nvar">1</mn></msub><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><msub><mi class="ltx_nvar">z</mi><mi class="ltx_nvar" href="./19.1#p1.t1.r1">n</mi></msub></mrow><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m13.png" altimg-height="23px" altimg-valign="-7px" altimg-width="92px" alttext="R_{\NVar{-a}}\left(\NVar{\mathbf{b}};\NVar{\mathbf{z}}\right)" display="inline"><mrow><msub><mi href="./19.16#E9">R</mi><mrow><mo class="ltx_nvar">-</mo><mi class="ltx_nvar">a</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" mathvariant="bold">b</mi><mo>;</mo><mi class="ltx_nvar" mathvariant="bold">z</mi><mo>)</mo></mrow></mrow></math>: multivariate hypergeometric function</a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m44.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./19.18#SS1.p1" title="§19.18(i) Derivatives ‣ §19.18 Derivatives and Differential Equations ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m42.png" altimg-height="18px" altimg-valign="-8px" altimg-width="27px" alttext="w_{j}" display="inline"><msub><mi href="./19.18#SS1.p1">w</mi><mi>j</mi></msub></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="ii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§19.18(ii) </span>Differential Equations</h2>
<div id="SS2.info" class="ltx_metadata ltx_info">
, <a class="ltx_keyword" href="./idx/H#hypergeometricRfunction">hypergeometric <math class="ltx_Math" altimg="m6.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="R" display="inline"><mi>R</mi></math>-function</a>, </dd>
</dl>
</div>
</div>
<div id="SS2.p1" class="ltx_para">
<table id="E6" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.18.6</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E6.png" altimg-height="56px" altimg-valign="-22px" altimg-width="361px" alttext="\left(\frac{\partial}{\partial x}+\frac{\partial}{\partial y}+\frac{\partial}{%
\partial z}\right)R_{F}\left(x,y,z\right)=\frac{-1}{2\sqrt{xyz}}," display="block"><mrow><mrow><mrow><mrow><mo>(</mo><mrow><mfrac><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow></mfrac><mo>+</mo><mfrac><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow></mfrac><mo>+</mo><mfrac><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>z</mi></mrow></mfrac></mrow><mo>)</mo></mrow><mo></mo><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mfrac><mrow><mo>-</mo><mn>1</mn></mrow><mrow><mn>2</mn><mo></mo><msqrt><mrow><mi>x</mi><mo></mo><mi>y</mi><mo></mo><mi>z</mi></mrow></msqrt></mrow></mfrac></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E6.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.16#E1" title="(19.16.1) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m10.png" altimg-height="23px" altimg-valign="-7px" altimg-width="101px" alttext="R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: symmetric elliptic integral of first kind</a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m27.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m44.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m44.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E7" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.18.7</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E7.png" altimg-height="53px" altimg-valign="-21px" altimg-width="411px" alttext="\left(\frac{\partial}{\partial x}+\frac{\partial}{\partial y}+\frac{\partial}{%
\partial z}\right)R_{G}\left(x,y,z\right)=\tfrac{1}{2}R_{F}\left(x,y,z\right)." display="block"><mrow><mrow><mrow><mrow><mo>(</mo><mrow><mfrac><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow></mfrac><mo>+</mo><mfrac><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow></mfrac><mo>+</mo><mfrac><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>z</mi></mrow></mfrac></mrow><mo>)</mo></mrow><mo></mo><mrow><msub><mi href="./19.16#E3">R</mi><mi href="./19.16#E3">G</mi></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E7.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.16#E1" title="(19.16.1) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m10.png" altimg-height="23px" altimg-valign="-7px" altimg-width="101px" alttext="R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: symmetric elliptic integral of first kind</a>,
<a href="./19.16#E3" title="(19.16.3) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m12.png" altimg-height="23px" altimg-valign="-7px" altimg-width="101px" alttext="R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E3">R</mi><mi href="./19.16#E3">G</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: symmetric elliptic integral of second kind</a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m27.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m44.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m44.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E8" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.18.8</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E8.png" altimg-height="67px" altimg-valign="-30px" altimg-width="318px" alttext="\sum_{j=1}^{n}\partial_{j}R_{-a}\left(\mathbf{b};\mathbf{z}\right)=-aR_{-a-1}%
\left(\mathbf{b};\mathbf{z}\right)." display="block"><mrow><mrow><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi href="./19.1#p1.t1.r1">n</mi></munderover><mrow><msub><mo href="./1.5#E3">∂</mo><mi>j</mi></msub><mo></mo><mrow><msub><mi href="./19.16#E9">R</mi><mrow><mo>-</mo><mi>a</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi mathvariant="bold">b</mi><mo>;</mo><mi mathvariant="bold">z</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>=</mo><mrow><mo>-</mo><mrow><mi>a</mi><mo></mo><mrow><msub><mi href="./19.16#E9">R</mi><mrow><mrow><mo>-</mo><mi>a</mi></mrow><mo>-</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mi mathvariant="bold">b</mi><mo>;</mo><mi mathvariant="bold">z</mi><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E8.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.16#E9" title="(19.16.9) ‣ §19.16(ii) R - a ( b ; z ) ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m14.png" altimg-height="23px" altimg-valign="-7px" altimg-width="233px" alttext="R_{\NVar{-a}}\left(\NVar{b_{1}},\dots,\NVar{b_{n}};\NVar{z_{1}},\dots,\NVar{z_%
{n}}\right)" display="inline"><mrow><msub><mi href="./19.16#E9">R</mi><mrow><mo class="ltx_nvar">-</mo><mi class="ltx_nvar">a</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mrow><msub><mi class="ltx_nvar">b</mi><mn class="ltx_nvar">1</mn></msub><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><msub><mi class="ltx_nvar">b</mi><mi class="ltx_nvar" href="./19.1#p1.t1.r1">n</mi></msub></mrow><mo>;</mo><mrow><msub><mi class="ltx_nvar">z</mi><mn class="ltx_nvar">1</mn></msub><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><msub><mi class="ltx_nvar">z</mi><mi class="ltx_nvar" href="./19.1#p1.t1.r1">n</mi></msub></mrow><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m13.png" altimg-height="23px" altimg-valign="-7px" altimg-width="92px" alttext="R_{\NVar{-a}}\left(\NVar{\mathbf{b}};\NVar{\mathbf{z}}\right)" display="inline"><mrow><msub><mi href="./19.16#E9">R</mi><mrow><mo class="ltx_nvar">-</mo><mi class="ltx_nvar">a</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" mathvariant="bold">b</mi><mo>;</mo><mi class="ltx_nvar" mathvariant="bold">z</mi><mo>)</mo></mrow></mrow></math>: multivariate hypergeometric function</a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m44.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./19.1#p1.t1.r1" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m35.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./19.1#p1.t1.r1">n</mi></math>: nonnegative integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E9" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.18.9</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E9.png" altimg-height="53px" altimg-valign="-21px" altimg-width="458px" alttext="\left(x\frac{\partial}{\partial x}+y\frac{\partial}{\partial y}+z\frac{%
\partial}{\partial z}\right)R_{F}\left(x,y,z\right)=-\tfrac{1}{2}R_{F}\left(x,%
y,z\right)," display="block"><mrow><mrow><mrow><mrow><mo>(</mo><mrow><mrow><mi>x</mi><mo></mo><mfrac><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow></mfrac></mrow><mo>+</mo><mrow><mi>y</mi><mo></mo><mfrac><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow></mfrac></mrow><mo>+</mo><mrow><mi>z</mi><mo></mo><mfrac><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>z</mi></mrow></mfrac></mrow></mrow><mo>)</mo></mrow><mo></mo><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mrow><mo>-</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E9.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.16#E1" title="(19.16.1) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m10.png" altimg-height="23px" altimg-valign="-7px" altimg-width="101px" alttext="R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: symmetric elliptic integral of first kind</a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m27.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m44.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m44.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E10" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.18.10</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E10.png" altimg-height="54px" altimg-valign="-21px" altimg-width="440px" alttext="\left((x-y)\frac{{\partial}^{2}}{\partial x\partial y}+\frac{1}{2}\left(\frac{%
\partial}{\partial y}-\frac{\partial}{\partial x}\right)\right)R_{F}\left(x,y,%
z\right)=0," display="block"><mrow><mrow><mrow><mrow><mo>(</mo><mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mi>x</mi><mo>-</mo><mi>y</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mfrac><mrow><mpadded width="-1.1pt"><msup><mo href="./1.5#E3">∂</mo><mn>2</mn></msup></mpadded><mo href="./1.5#E3"></mo></mrow><mrow><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow><mo></mo><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow></mrow></mfrac></mrow><mo>+</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mrow><mo>(</mo><mrow><mfrac><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow></mfrac><mo>-</mo><mfrac><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow></mfrac></mrow><mo>)</mo></mrow></mrow></mrow><mo>)</mo></mrow><mo></mo><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mn>0</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E10.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.16#E1" title="(19.16.1) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m10.png" altimg-height="23px" altimg-valign="-7px" altimg-width="101px" alttext="R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: symmetric elliptic integral of first kind</a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m27.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m44.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m44.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and two similar equations obtained by permuting <math class="ltx_Math" altimg="m43.png" altimg-height="16px" altimg-valign="-6px" altimg-width="54px" alttext="x,y,z" display="inline"><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi></mrow></math> in ().</p>
</div>
<div id="SS2.p2" class="ltx_para">
<p class="ltx_p">More concisely, if <math class="ltx_Math" altimg="m39.png" altimg-height="23px" altimg-valign="-7px" altimg-width="129px" alttext="v=R_{-a}\left(\mathbf{b};\mathbf{z}\right)" display="inline"><mrow><mi href="./19.18#SS2.p2">v</mi><mo>=</mo><mrow><msub><mi href="./19.16#E9">R</mi><mrow><mo>-</mo><mi>a</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi mathvariant="bold">b</mi><mo>;</mo><mi mathvariant="bold">z</mi><mo>)</mo></mrow></mrow></mrow></math>, then each of
()
satisfies <em class="ltx_emph ltx_font_italic">Euler’s homogeneity relation</em>:
</p>
<table id="E11" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.18.11</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E11.png" altimg-height="67px" altimg-valign="-30px" altimg-width="157px" alttext="\sum_{j=1}^{n}z_{j}\partial_{j}v=-av," display="block"><mrow><mrow><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi href="./19.1#p1.t1.r1">n</mi></munderover><mrow><msub><mi>z</mi><mi>j</mi></msub><mo></mo><mrow><msub><mo href="./1.5#E3">∂</mo><mi>j</mi></msub><mo></mo><mi href="./19.18#SS2.p2">v</mi></mrow></mrow></mrow><mo>=</mo><mrow><mo>-</mo><mrow><mi>a</mi><mo></mo><mi href="./19.18#SS2.p2">v</mi></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E11.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m44.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./19.1#p1.t1.r1" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m35.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./19.1#p1.t1.r1">n</mi></math>: nonnegative integer</a> and
<a href="./19.18#SS2.p2" title="§19.18(ii) Differential Equations ‣ §19.18 Derivatives and Differential Equations ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m40.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="v" display="inline"><mi href="./19.18#SS2.p2">v</mi></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and also a system of <math class="ltx_Math" altimg="m32.png" altimg-height="23px" altimg-valign="-7px" altimg-width="98px" alttext="n(n-1)/2" display="inline"><mrow><mrow><mi href="./19.1#p1.t1.r1">n</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./19.1#p1.t1.r1">n</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>/</mo><mn>2</mn></mrow></math> <em class="ltx_emph ltx_font_italic">Euler–Poisson differential equations</em> (of
which only <math class="ltx_Math" altimg="m33.png" altimg-height="18px" altimg-valign="-4px" altimg-width="51px" alttext="n-1" display="inline"><mrow><mi href="./19.1#p1.t1.r1">n</mi><mo>-</mo><mn>1</mn></mrow></math> are independent):</p>
<table id="E12" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.18.12</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E12.png" altimg-height="26px" altimg-valign="-8px" altimg-width="275px" alttext="(z_{j}\partial_{j}+b_{j})\partial_{l}v=(z_{l}\partial_{l}+b_{l})\partial_{j}v," display="block"><mrow><mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><msub><mi>z</mi><mi>j</mi></msub><mo></mo><msub><mo href="./1.5#E3">∂</mo><mi>j</mi></msub></mrow><mo>+</mo><msub><mi>b</mi><mi>j</mi></msub></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><msub><mo href="./1.5#E3">∂</mo><mi href="./19.1#p1.t1.r1">l</mi></msub><mo></mo><mi href="./19.18#SS2.p2">v</mi></mrow></mrow><mo>=</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><msub><mi>z</mi><mi href="./19.1#p1.t1.r1">l</mi></msub><mo></mo><msub><mo href="./1.5#E3">∂</mo><mi href="./19.1#p1.t1.r1">l</mi></msub></mrow><mo>+</mo><msub><mi>b</mi><mi href="./19.1#p1.t1.r1">l</mi></msub></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><msub><mo href="./1.5#E3">∂</mo><mi>j</mi></msub><mo></mo><mi href="./19.18#SS2.p2">v</mi></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E12.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m44.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./19.1#p1.t1.r1" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m31.png" altimg-height="18px" altimg-valign="-2px" altimg-width="11px" alttext="l" display="inline"><mi href="./19.1#p1.t1.r1">l</mi></math>: nonnegative integer</a> and
<a href="./19.18#SS2.p2" title="§19.18(ii) Differential Equations ‣ §19.18 Derivatives and Differential Equations ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m40.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="v" display="inline"><mi href="./19.18#SS2.p2">v</mi></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">or equivalently,</p>
<table id="E13" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.18.13</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E13.png" altimg-height="26px" altimg-valign="-8px" altimg-width="299px" alttext="((z_{j}-z_{l})\partial_{j}\partial_{l}+b_{j}\partial_{l}-b_{l}\partial_{j})v=0." display="block"><mrow><mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><msub><mi>z</mi><mi>j</mi></msub><mo>-</mo><msub><mi>z</mi><mi href="./19.1#p1.t1.r1">l</mi></msub></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><msub><mo href="./1.5#E3">∂</mo><mi>j</mi></msub><mo></mo><msub><mo href="./1.5#E3">∂</mo><mi href="./19.1#p1.t1.r1">l</mi></msub></mrow></mrow><mo>+</mo><mrow><msub><mi>b</mi><mi>j</mi></msub><mo></mo><msub><mo href="./1.5#E3">∂</mo><mi href="./19.1#p1.t1.r1">l</mi></msub></mrow></mrow><mo>-</mo><mrow><msub><mi>b</mi><mi href="./19.1#p1.t1.r1">l</mi></msub><mo></mo><msub><mo href="./1.5#E3">∂</mo><mi>j</mi></msub></mrow></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi href="./19.18#SS2.p2">v</mi></mrow><mo>=</mo><mn>0</mn></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E13.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m44.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./19.1#p1.t1.r1" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m31.png" altimg-height="18px" altimg-valign="-2px" altimg-width="11px" alttext="l" display="inline"><mi href="./19.1#p1.t1.r1">l</mi></math>: nonnegative integer</a> and
<a href="./19.18#SS2.p2" title="§19.18(ii) Differential Equations ‣ §19.18 Derivatives and Differential Equations ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m40.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="v" display="inline"><mi href="./19.18#SS2.p2">v</mi></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Here <math class="ltx_Math" altimg="m28.png" altimg-height="21px" altimg-valign="-6px" altimg-width="139px" alttext="j,l=1,2,\dots,n" display="inline"><mrow><mrow><mrow><mi>j</mi><mo>,</mo><mi href="./19.1#p1.t1.r1">l</mi></mrow><mo>=</mo><mn>1</mn></mrow><mo>,</mo><mrow><mn>2</mn><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><mi href="./19.1#p1.t1.r1">n</mi></mrow></mrow></math> and <math class="ltx_Math" altimg="m30.png" altimg-height="21px" altimg-valign="-6px" altimg-width="47px" alttext="j\neq l" display="inline"><mrow><mi>j</mi><mo>≠</mo><mi href="./19.1#p1.t1.r1">l</mi></mrow></math>. For group-theoretical aspects of this
system see <cite class="ltx_cite ltx_citemacro_citet">Carlson (, §VI)</cite>. If <math class="ltx_Math" altimg="m34.png" altimg-height="17px" altimg-valign="-2px" altimg-width="53px" alttext="n=2" display="inline"><mrow><mi href="./19.1#p1.t1.r1">n</mi><mo>=</mo><mn>2</mn></mrow></math>, then elimination of
<math class="ltx_Math" altimg="m23.png" altimg-height="21px" altimg-valign="-5px" altimg-width="34px" alttext="\partial_{2}v" display="inline"><mrow><msub><mo href="./1.5#E3">∂</mo><mn>2</mn></msub><mo></mo><mi href="./19.18#SS2.p2">v</mi></mrow></math> between (), followed by
the substitution <math class="ltx_Math" altimg="m5.png" altimg-height="23px" altimg-valign="-7px" altimg-width="291px" alttext="(b_{1},b_{2},z_{1},z_{2})=(b,c-b,1-z,1)" display="inline"><mrow><mrow><mo stretchy="false">(</mo><msub><mi>b</mi><mn>1</mn></msub><mo>,</mo><msub><mi>b</mi><mn>2</mn></msub><mo>,</mo><msub><mi>z</mi><mn>1</mn></msub><mo>,</mo><msub><mi>z</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mrow><mo>=</mo><mrow><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mrow><mi>c</mi><mo>-</mo><mi>b</mi></mrow><mo>,</mo><mrow><mn>1</mn><mo>-</mo><mi>z</mi></mrow><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mrow></math>, produces the Gauss
hypergeometric equation ().</p>
</div>
<div id="SS2.p3" class="ltx_para">
<p class="ltx_p">The next four differential equations apply to the complete case of <math class="ltx_Math" altimg="m9.png" altimg-height="21px" altimg-valign="-5px" altimg-width="33px" alttext="R_{F}" display="inline"><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub></math>
and <math class="ltx_Math" altimg="m11.png" altimg-height="21px" altimg-valign="-5px" altimg-width="33px" alttext="R_{G}" display="inline"><msub><mi href="./19.16#E3">R</mi><mi href="./19.16#E3">G</mi></msub></math> in the form
<math class="ltx_Math" altimg="m7.png" altimg-height="27px" altimg-valign="-9px" altimg-width="152px" alttext="R_{-a}\left(\tfrac{1}{2},\tfrac{1}{2};z_{1},z_{2}\right)" display="inline"><mrow><msub><mi href="./19.16#E9">R</mi><mrow><mo>-</mo><mi>a</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>,</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>;</mo><mrow><msub><mi>z</mi><mn>1</mn></msub><mo>,</mo><msub><mi>z</mi><mn>2</mn></msub></mrow><mo>)</mo></mrow></mrow></math> (see ()).</p>
</div>
<div id="SS2.p4" class="ltx_para">
<p class="ltx_p">The function <math class="ltx_Math" altimg="m41.png" altimg-height="27px" altimg-valign="-9px" altimg-width="249px" alttext="w=R_{-a}\left(\tfrac{1}{2},\tfrac{1}{2};x+y,x-y\right)" display="inline"><mrow><mi>w</mi><mo>=</mo><mrow><msub><mi href="./19.16#E9">R</mi><mrow><mo>-</mo><mi>a</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>,</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>;</mo><mrow><mrow><mi>x</mi><mo>+</mo><mi>y</mi></mrow><mo>,</mo><mrow><mi>x</mi><mo>-</mo><mi>y</mi></mrow></mrow><mo>)</mo></mrow></mrow></mrow></math> satisfies
an <em class="ltx_emph ltx_font_italic">Euler–Poisson–Darboux equation</em>:</p>
<table id="E14" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.18.14</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E14.png" altimg-height="56px" altimg-valign="-22px" altimg-width="189px" alttext="\frac{{\partial}^{2}w}{{\partial x}^{2}}=\frac{{\partial}^{2}w}{{\partial y}^{%
2}}+\frac{1}{y}\frac{\partial w}{\partial y}." display="block"><mrow><mrow><mfrac><mrow><mpadded width="-1.7pt"><msup><mo href="./1.5#E3">∂</mo><mn>2</mn></msup></mpadded><mo href="./1.5#E3"></mo><mi>w</mi></mrow><msup><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow><mn>2</mn></msup></mfrac><mo>=</mo><mrow><mfrac><mrow><mpadded width="-1.7pt"><msup><mo href="./1.5#E3">∂</mo><mn>2</mn></msup></mpadded><mo href="./1.5#E3"></mo><mi>w</mi></mrow><msup><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow><mn>2</mn></msup></mfrac><mo>+</mo><mrow><mfrac><mn>1</mn><mi>y</mi></mfrac><mo></mo><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>w</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow></mfrac></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E14.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m27.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m44.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m44.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Also <math class="ltx_Math" altimg="m17.png" altimg-height="27px" altimg-valign="-9px" altimg-width="244px" alttext="W=R_{-a}\left(\tfrac{1}{2},\tfrac{1}{2};t+r,t-r\right)" display="inline"><mrow><mi href="./19.18#SS2.p4">W</mi><mo>=</mo><mrow><msub><mi href="./19.16#E9">R</mi><mrow><mo>-</mo><mi>a</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>,</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>;</mo><mrow><mrow><mi>t</mi><mo>+</mo><mi>r</mi></mrow><mo>,</mo><mrow><mi>t</mi><mo>-</mo><mi>r</mi></mrow></mrow><mo>)</mo></mrow></mrow></mrow></math>, with
<math class="ltx_Math" altimg="m36.png" altimg-height="28px" altimg-valign="-8px" altimg-width="124px" alttext="r=\sqrt{x^{2}+y^{2}}" display="inline"><mrow><mi>r</mi><mo>=</mo><msqrt><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup></mrow></msqrt></mrow></math>, satisfies a <em class="ltx_emph ltx_font_italic">wave equation</em>:
</p>
<table id="E15" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.18.15</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E15.png" altimg-height="56px" altimg-valign="-22px" altimg-width="203px" alttext="\frac{{\partial}^{2}W}{{\partial t}^{2}}=\frac{{\partial}^{2}W}{{\partial x}^{%
2}}+\frac{{\partial}^{2}W}{{\partial y}^{2}}." display="block"><mrow><mrow><mfrac><mrow><mpadded width="-1.7pt"><msup><mo href="./1.5#E3">∂</mo><mn>2</mn></msup></mpadded><mo href="./1.5#E3"></mo><mi href="./19.18#SS2.p4">W</mi></mrow><msup><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>t</mi></mrow><mn>2</mn></msup></mfrac><mo>=</mo><mrow><mfrac><mrow><mpadded width="-1.7pt"><msup><mo href="./1.5#E3">∂</mo><mn>2</mn></msup></mpadded><mo href="./1.5#E3"></mo><mi href="./19.18#SS2.p4">W</mi></mrow><msup><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow><mn>2</mn></msup></mfrac><mo>+</mo><mfrac><mrow><mpadded width="-1.7pt"><msup><mo href="./1.5#E3">∂</mo><mn>2</mn></msup></mpadded><mo href="./1.5#E3"></mo><mi href="./19.18#SS2.p4">W</mi></mrow><msup><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow><mn>2</mn></msup></mfrac></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E15.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m27.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m44.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m44.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./19.18#SS2.p4" title="§19.18(ii) Differential Equations ‣ §19.18 Derivatives and Differential Equations ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m18.png" altimg-height="18px" altimg-valign="-2px" altimg-width="26px" alttext="W" display="inline"><mi href="./19.18#SS2.p4">W</mi></math>: wave function</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Similarly, the function
<math class="ltx_Math" altimg="m38.png" altimg-height="27px" altimg-valign="-9px" altimg-width="259px" alttext="u=R_{-a}\left(\tfrac{1}{2},\tfrac{1}{2};x+iy,x-iy\right)" display="inline"><mrow><mi>u</mi><mo>=</mo><mrow><msub><mi href="./19.16#E9">R</mi><mrow><mo>-</mo><mi>a</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>,</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>;</mo><mrow><mrow><mi>x</mi><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi>y</mi></mrow></mrow><mo>,</mo><mrow><mi>x</mi><mo>-</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi>y</mi></mrow></mrow></mrow><mo>)</mo></mrow></mrow></mrow></math> satisfies an equation
of <em class="ltx_emph ltx_font_italic">axially symmetric potential theory</em>:
</p>
<table id="E16" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.18.16</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E16.png" altimg-height="56px" altimg-valign="-22px" altimg-width="214px" alttext="\frac{{\partial}^{2}u}{{\partial x}^{2}}+\frac{{\partial}^{2}u}{{\partial y}^{%
2}}+\frac{1}{y}\frac{\partial u}{\partial y}=0," display="block"><mrow><mrow><mrow><mfrac><mrow><mpadded width="-1.7pt"><msup><mo href="./1.5#E3">∂</mo><mn>2</mn></msup></mpadded><mo href="./1.5#E3"></mo><mi>u</mi></mrow><msup><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow><mn>2</mn></msup></mfrac><mo>+</mo><mfrac><mrow><mpadded width="-1.7pt"><msup><mo href="./1.5#E3">∂</mo><mn>2</mn></msup></mpadded><mo href="./1.5#E3"></mo><mi>u</mi></mrow><msup><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow><mn>2</mn></msup></mfrac><mo>+</mo><mrow><mfrac><mn>1</mn><mi>y</mi></mfrac><mo></mo><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>u</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow></mfrac></mrow></mrow><mo>=</mo><mn>0</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E16.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m27.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m44.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m44.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and <math class="ltx_Math" altimg="m15.png" altimg-height="27px" altimg-valign="-9px" altimg-width="261px" alttext="U=R_{-a}\left(\tfrac{1}{2},\tfrac{1}{2};z+i\rho,z-i\rho\right)" display="inline"><mrow><mi href="./19.18#SS2.p4">U</mi><mo>=</mo><mrow><msub><mi href="./19.16#E9">R</mi><mrow><mo>-</mo><mi>a</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>,</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>;</mo><mrow><mrow><mi>z</mi><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi>ρ</mi></mrow></mrow><mo>,</mo><mrow><mi>z</mi><mo>-</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi>ρ</mi></mrow></mrow></mrow><mo>)</mo></mrow></mrow></mrow></math>, with
<math class="ltx_Math" altimg="m25.png" altimg-height="28px" altimg-valign="-8px" altimg-width="125px" alttext="\rho=\sqrt{x^{2}+y^{2}}" display="inline"><mrow><mi>ρ</mi><mo>=</mo><msqrt><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup></mrow></msqrt></mrow></math>, satisfies <em class="ltx_emph ltx_font_italic">Laplace’s equation</em>:</p>
<table id="E17" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.18.17</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E17.png" altimg-height="56px" altimg-valign="-22px" altimg-width="220px" alttext="\frac{{\partial}^{2}U}{{\partial x}^{2}}+\frac{{\partial}^{2}U}{{\partial y}^{%
2}}+\frac{{\partial}^{2}U}{{\partial z}^{2}}=0." display="block"><mrow><mrow><mrow><mfrac><mrow><mpadded width="-1.7pt"><msup><mo href="./1.5#E3">∂</mo><mn>2</mn></msup></mpadded><mo href="./1.5#E3"></mo><mi href="./19.18#SS2.p4">U</mi></mrow><msup><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow><mn>2</mn></msup></mfrac><mo>+</mo><mfrac><mrow><mpadded width="-1.7pt"><msup><mo href="./1.5#E3">∂</mo><mn>2</mn></msup></mpadded><mo href="./1.5#E3"></mo><mi href="./19.18#SS2.p4">U</mi></mrow><msup><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow><mn>2</mn></msup></mfrac><mo>+</mo><mfrac><mrow><mpadded width="-1.7pt"><msup><mo href="./1.5#E3">∂</mo><mn>2</mn></msup></mpadded><mo href="./1.5#E3"></mo><mi href="./19.18#SS2.p4">U</mi></mrow><msup><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>z</mi></mrow><mn>2</mn></msup></mfrac></mrow><mo>=</mo><mn>0</mn></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E17.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m27.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m44.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m44.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./19.18#SS2.p4" title="§19.18(ii) Differential Equations ‣ §19.18 Derivatives and Differential Equations ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m16.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="U" display="inline"><mi href="./19.18#SS2.p4">U</mi></math></a>
</dd>
</dl>
</div>
</div>
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<title>DLMF: 19.20 Special Cases</title>
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<h6>Contents</h6>
<ul class="ltx_toclist ltx_toclist_section">
<li class="ltx_tocentry"><a href="#i"><span class="ltx_tag ltx_tag_subsection">§19.20(i) </span><math class="ltx_Math" altimg="m71.png" altimg-height="23px" altimg-valign="-7px" altimg-width="101px" alttext="R_{F}\left(x,y,z\right)" display="inline"><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow></mrow></math></a></li>
<li class="ltx_tocentry"><a href="#ii"><span class="ltx_tag ltx_tag_subsection">§19.20(ii) </span><math class="ltx_Math" altimg="m73.png" altimg-height="23px" altimg-valign="-7px" altimg-width="101px" alttext="R_{G}\left(x,y,z\right)" display="inline"><mrow><msub><mi href="./19.16#E3">R</mi><mi href="./19.16#E3">G</mi></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow></mrow></math></a></li>
<li class="ltx_tocentry"><a href="#iii"><span class="ltx_tag ltx_tag_subsection">§19.20(iii) </span><math class="ltx_Math" altimg="m76.png" altimg-height="23px" altimg-valign="-7px" altimg-width="118px" alttext="R_{J}\left(x,y,z,p\right)" display="inline"><mrow><msub><mi href="./19.16#E2">R</mi><mi href="./19.16#E2">J</mi></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>,</mo><mi>p</mi><mo>)</mo></mrow></mrow></math></a></li>
<li class="ltx_tocentry"><a href="#iv"><span class="ltx_tag ltx_tag_subsection">§19.20(iv) </span><math class="ltx_Math" altimg="m69.png" altimg-height="23px" altimg-valign="-7px" altimg-width="102px" alttext="R_{D}\left(x,y,z\right)" display="inline"><mrow><msub><mi href="./19.16#E5">R</mi><mi href="./19.16#E5">D</mi></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow></mrow></math></a></li>
<li class="ltx_tocentry"><a href="#v"><span class="ltx_tag ltx_tag_subsection">§19.20(v) </span><math class="ltx_Math" altimg="m66.png" altimg-height="23px" altimg-valign="-7px" altimg-width="92px" alttext="R_{-a}\left(\mathbf{b};\mathbf{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E9">R</mi><mrow><mo>-</mo><mi>a</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi mathvariant="bold">b</mi><mo>;</mo><mi mathvariant="bold">z</mi><mo>)</mo></mrow></mrow></math></a></li>
</ul>
<section id="i" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§19.20(i) </span><math class="ltx_Math" altimg="m71.png" altimg-height="23px" altimg-valign="-7px" altimg-width="101px" alttext="R_{F}\left(x,y,z\right)" display="inline"><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow></mrow></math>
</h2>
<div id="SS1.info" class="ltx_metadata ltx_info">
) put <math class="ltx_Math" altimg="m105.png" altimg-height="26px" altimg-valign="-7px" altimg-width="118px" alttext="t=1/\sqrt{s+1}" display="inline"><mrow><mi>t</mi><mo>=</mo><mrow><mn>1</mn><mo>/</mo><msqrt><mrow><mi>s</mi><mo>+</mo><mn>1</mn></mrow></msqrt></mrow></mrow></math>; alternatively use
().
For the second equality replace <math class="ltx_Math" altimg="m108.png" altimg-height="20px" altimg-valign="-2px" altimg-width="20px" alttext="t^{4}" display="inline"><msup><mi>t</mi><mn>4</mn></msup></math> by <math class="ltx_Math" altimg="m107.png" altimg-height="17px" altimg-valign="-2px" altimg-width="11px" alttext="t" display="inline"><mi>t</mi></math> and apply
(, the
variables of all <math class="ltx_Math" altimg="m65.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="R" display="inline"><mi>R</mi></math>-functions satisfy the constraints specified in
§ unless other conditions are stated.</p>
<table id="E1" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex1" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="5" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">19.20.1</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m39.png" altimg-height="25px" altimg-valign="-7px" altimg-width="105px" alttext="\displaystyle R_{F}\left(x,x,x\right)" display="inline"><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m27.png" altimg-height="28px" altimg-valign="-6px" altimg-width="81px" alttext="\displaystyle=x^{-1/2}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><msup><mi>x</mi><mrow><mo>-</mo><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></mrow></msup></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex2" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m38.png" altimg-height="25px" altimg-valign="-7px" altimg-width="138px" alttext="\displaystyle R_{F}\left(\lambda x,\lambda y,\lambda z\right)" display="inline"><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi>λ</mi><mo></mo><mi>x</mi></mrow><mo>,</mo><mrow><mi>λ</mi><mo></mo><mi>y</mi></mrow><mo>,</mo><mrow><mi>λ</mi><mo></mo><mi>z</mi></mrow><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m18.png" altimg-height="29px" altimg-valign="-7px" altimg-width="182px" alttext="\displaystyle=\lambda^{-1/2}R_{F}\left(x,y,z\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msup><mi>λ</mi><mrow><mo>-</mo><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></mrow></msup><mo></mo><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex3" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m40.png" altimg-height="25px" altimg-valign="-7px" altimg-width="103px" alttext="\displaystyle R_{F}\left(x,y,y\right)" display="inline"><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m5.png" altimg-height="25px" altimg-valign="-7px" altimg-width="114px" alttext="\displaystyle=R_{C}\left(x,y\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msub><mi href="./19.2#E17">R</mi><mi href="./19.2#E17">C</mi></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex4" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m37.png" altimg-height="25px" altimg-valign="-7px" altimg-width="102px" alttext="\displaystyle R_{F}\left(0,y,y\right)" display="inline"><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>y</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m22.png" altimg-height="31px" altimg-valign="-9px" altimg-width="105px" alttext="\displaystyle=\tfrac{1}{2}\pi y^{-1/2}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><msup><mi>y</mi><mrow><mo>-</mo><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></mrow></msup></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex5" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m36.png" altimg-height="25px" altimg-valign="-7px" altimg-width="101px" alttext="\displaystyle R_{F}\left(0,0,z\right)" display="inline"><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>,</mo><mi>z</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m17.png" altimg-height="14px" altimg-valign="-2px" altimg-width="53px" alttext="\displaystyle=\infty." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mi mathvariant="normal">∞</mi></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.2#E17" title="(19.2.17) ‣ §19.2(iv) A Related Function: R C ( x , y ) ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m67.png" altimg-height="23px" altimg-valign="-7px" altimg-width="82px" alttext="R_{C}\left(\NVar{x},\NVar{y}\right)" display="inline"><mrow><msub><mi href="./19.2#E17">R</mi><mi href="./19.2#E17">C</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>)</mo></mrow></mrow></math>: Carlson’s combination of inverse circular and inverse hyperbolic functions</a>,
<a href="./19.16#E1" title="(19.16.1) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="23px" altimg-valign="-7px" altimg-width="101px" alttext="R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: symmetric elliptic integral of first kind</a> and
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS1.p2" class="ltx_para">
<p class="ltx_p">The <em class="ltx_emph ltx_font_italic">first lemniscate constant</em> is given by</p>
<table id="E2" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.20.2</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E2.png" altimg-height="60px" altimg-valign="-21px" altimg-width="636px" alttext="\int_{0}^{1}\frac{\mathrm{d}t}{\sqrt{1-t^{4}}}=R_{F}\left(0,1,2\right)=\frac{%
\left(\Gamma\left(\frac{1}{4}\right)\right)^{2}}{4(2\pi)^{1/2}}=1.31102\;87771%
\;46059\;90523\;\dots." display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mn>1</mn></msubsup><mfrac><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow><msqrt><mrow><mn>1</mn><mo>-</mo><msup><mi>t</mi><mn>4</mn></msup></mrow></msqrt></mfrac></mrow><mo>=</mo><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></mrow><mo>=</mo><mfrac><msup><mrow><mo>(</mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mo>)</mo></mrow></mrow><mo>)</mo></mrow><mn>2</mn></msup><mrow><mn>4</mn><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></mfrac><mo>=</mo><mrow><mpadded width="+2.8pt"><mn>1.31102 87771 46059 90523</mn></mpadded><mo></mo><mi mathvariant="normal">…</mi></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.16#E1" title="(19.16.1) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="23px" altimg-valign="-7px" altimg-width="101px" alttext="R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: symmetric elliptic integral of first kind</a>,
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m81.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m114.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m83.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p"><cite class="ltx_cite ltx_citemacro_citet">Todd ()</cite> refers to a proof by T. Schneider that this is a
transcendental number. The <em class="ltx_emph ltx_font_italic">general lemniscatic case</em> is
</p>
<table id="E3" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.20.3</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E3.png" altimg-height="33px" altimg-valign="-12px" altimg-width="294px" alttext="R_{F}\left(x,a,y\right)=R_{-\frac{1}{4}}\left(\tfrac{3}{4},\tfrac{1}{2};a^{2},%
xy\right)," display="block"><mrow><mrow><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>a</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msub><mi href="./19.16#E9">R</mi><mrow><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></msub><mo></mo><mrow><mo>(</mo><mrow><mstyle displaystyle="false"><mfrac><mn>3</mn><mn>4</mn></mfrac></mstyle><mo>,</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle></mrow><mo>;</mo><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>,</mo><mrow><mi>x</mi><mo></mo><mi>y</mi></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m90.png" altimg-height="27px" altimg-valign="-9px" altimg-width="116px" alttext="a=\frac{1}{2}(x+y)" display="inline"><mrow><mi>a</mi><mo>=</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>x</mi><mo>+</mo><mi>y</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.16#E9" title="(19.16.9) ‣ §19.16(ii) R - a ( b ; z ) ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="23px" altimg-valign="-7px" altimg-width="233px" alttext="R_{\NVar{-a}}\left(\NVar{b_{1}},\dots,\NVar{b_{n}};\NVar{z_{1}},\dots,\NVar{z_%
{n}}\right)" display="inline"><mrow><msub><mi href="./19.16#E9">R</mi><mrow><mo class="ltx_nvar">-</mo><mi class="ltx_nvar">a</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mrow><msub><mi class="ltx_nvar">b</mi><mn class="ltx_nvar">1</mn></msub><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><msub><mi class="ltx_nvar">b</mi><mi class="ltx_nvar" href="./19.1#p1.t1.r1">n</mi></msub></mrow><mo>;</mo><mrow><msub><mi class="ltx_nvar">z</mi><mn class="ltx_nvar">1</mn></msub><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><msub><mi class="ltx_nvar">z</mi><mi class="ltx_nvar" href="./19.1#p1.t1.r1">n</mi></msub></mrow><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m77.png" altimg-height="23px" altimg-valign="-7px" altimg-width="92px" alttext="R_{\NVar{-a}}\left(\NVar{\mathbf{b}};\NVar{\mathbf{z}}\right)" display="inline"><mrow><msub><mi href="./19.16#E9">R</mi><mrow><mo class="ltx_nvar">-</mo><mi class="ltx_nvar">a</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" mathvariant="bold">b</mi><mo>;</mo><mi class="ltx_nvar" mathvariant="bold">z</mi><mo>)</mo></mrow></mrow></math>: multivariate hypergeometric function</a> and
<a href="./19.16#E1" title="(19.16.1) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="23px" altimg-valign="-7px" altimg-width="101px" alttext="R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: symmetric elliptic integral of first kind</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="ii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§19.20(ii) </span><math class="ltx_Math" altimg="m73.png" altimg-height="23px" altimg-valign="-7px" altimg-width="101px" alttext="R_{G}\left(x,y,z\right)" display="inline"><mrow><msub><mi href="./19.16#E3">R</mi><mi href="./19.16#E3">G</mi></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow></mrow></math>
</h2>
<div id="SS2.info" class="ltx_metadata ltx_info">
) put <math class="ltx_Math" altimg="m123.png" altimg-height="16px" altimg-valign="-6px" altimg-width="51px" alttext="z=y" display="inline"><mrow><mi>z</mi><mo>=</mo><mi>y</mi></mrow></math> in (</dd>
</dl>
</div>
</div>
<div id="SS2.p1" class="ltx_para">
<table id="E4" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex6" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="4" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">19.20.4</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m44.png" altimg-height="25px" altimg-valign="-7px" altimg-width="105px" alttext="\displaystyle R_{G}\left(x,x,x\right)" display="inline"><mrow><msub><mi href="./19.16#E3">R</mi><mi href="./19.16#E3">G</mi></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m29.png" altimg-height="28px" altimg-valign="-6px" altimg-width="69px" alttext="\displaystyle=x^{1/2}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><msup><mi>x</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex7" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m43.png" altimg-height="25px" altimg-valign="-7px" altimg-width="138px" alttext="\displaystyle R_{G}\left(\lambda x,\lambda y,\lambda z\right)" display="inline"><mrow><msub><mi href="./19.16#E3">R</mi><mi href="./19.16#E3">G</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi>λ</mi><mo></mo><mi>x</mi></mrow><mo>,</mo><mrow><mi>λ</mi><mo></mo><mi>y</mi></mrow><mo>,</mo><mrow><mi>λ</mi><mo></mo><mi>z</mi></mrow><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m21.png" altimg-height="29px" altimg-valign="-7px" altimg-width="170px" alttext="\displaystyle=\lambda^{1/2}R_{G}\left(x,y,z\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msup><mi>λ</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo></mo><mrow><msub><mi href="./19.16#E3">R</mi><mi href="./19.16#E3">G</mi></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex8" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m42.png" altimg-height="25px" altimg-valign="-7px" altimg-width="102px" alttext="\displaystyle R_{G}\left(0,y,y\right)" display="inline"><mrow><msub><mi href="./19.16#E3">R</mi><mi href="./19.16#E3">G</mi></msub><mo></mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>y</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m24.png" altimg-height="31px" altimg-valign="-9px" altimg-width="93px" alttext="\displaystyle=\tfrac{1}{4}\pi y^{1/2}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><msup><mi>y</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex9" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m41.png" altimg-height="25px" altimg-valign="-7px" altimg-width="101px" alttext="\displaystyle R_{G}\left(0,0,z\right)" display="inline"><mrow><msub><mi href="./19.16#E3">R</mi><mi href="./19.16#E3">G</mi></msub><mo></mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>,</mo><mi>z</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m23.png" altimg-height="31px" altimg-valign="-9px" altimg-width="81px" alttext="\displaystyle=\tfrac{1}{2}z^{1/2}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><msup><mi>z</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E4.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.16#E3" title="(19.16.3) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="23px" altimg-valign="-7px" altimg-width="101px" alttext="R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E3">R</mi><mi href="./19.16#E3">G</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: symmetric elliptic integral of second kind</a> and
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E5" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.20.5</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E5.png" altimg-height="27px" altimg-valign="-7px" altimg-width="283px" alttext="2\!R_{G}\left(x,y,y\right)=yR_{C}\left(x,y\right)+\sqrt{x}." display="block"><mrow><mrow><mrow><mpadded width="-1.7pt"><mn>2</mn></mpadded><mo></mo><mrow><msub><mi href="./19.16#E3">R</mi><mi href="./19.16#E3">G</mi></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mrow><mrow><mi>y</mi><mo></mo><mrow><msub><mi href="./19.2#E17">R</mi><mi href="./19.2#E17">C</mi></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></mrow></mrow><mo>+</mo><msqrt><mi>x</mi></msqrt></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E5.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.2#E17" title="(19.2.17) ‣ §19.2(iv) A Related Function: R C ( x , y ) ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m67.png" altimg-height="23px" altimg-valign="-7px" altimg-width="82px" alttext="R_{C}\left(\NVar{x},\NVar{y}\right)" display="inline"><mrow><msub><mi href="./19.2#E17">R</mi><mi href="./19.2#E17">C</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>)</mo></mrow></mrow></math>: Carlson’s combination of inverse circular and inverse hyperbolic functions</a> and
<a href="./19.16#E3" title="(19.16.3) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="23px" altimg-valign="-7px" altimg-width="101px" alttext="R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E3">R</mi><mi href="./19.16#E3">G</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: symmetric elliptic integral of second kind</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="iii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§19.20(iii) </span><math class="ltx_Math" altimg="m76.png" altimg-height="23px" altimg-valign="-7px" altimg-width="118px" alttext="R_{J}\left(x,y,z,p\right)" display="inline"><mrow><msub><mi href="./19.16#E2">R</mi><mi href="./19.16#E2">J</mi></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>,</mo><mi>p</mi><mo>)</mo></mrow></mrow></math>
</h2>
<div id="SS3.info" class="ltx_metadata ltx_info">
) for <math class="ltx_Math" altimg="m100.png" altimg-height="20px" altimg-valign="-6px" altimg-width="71px" alttext="p\rightarrow 0+" display="inline"><mrow><mi>p</mi><mo>→</mo><mrow><mn>0</mn><mo>+</mo></mrow></mrow></math>;
for <math class="ltx_Math" altimg="m101.png" altimg-height="20px" altimg-valign="-6px" altimg-width="71px" alttext="p\rightarrow 0-" display="inline"><mrow><mi>p</mi><mo>→</mo><mrow><mn>0</mn><mo>-</mo></mrow></mrow></math> use () put <math class="ltx_Math" altimg="m113.png" altimg-height="17px" altimg-valign="-2px" altimg-width="52px" alttext="x=0" display="inline"><mrow><mi>x</mi><mo>=</mo><mn>0</mn></mrow></math> in () interchange <math class="ltx_Math" altimg="m114.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math> and <math class="ltx_Math" altimg="m124.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi>z</mi></math> in () let <math class="ltx_Math" altimg="m103.png" altimg-height="16px" altimg-valign="-6px" altimg-width="50px" alttext="q=p" display="inline"><mrow><mi>q</mi><mo>=</mo><mi>p</mi></mrow></math> in () exchange <math class="ltx_Math" altimg="m114.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math> and <math class="ltx_Math" altimg="m124.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi>z</mi></math> in (</dd>
</dl>
</div>
</div>
<div id="SS3.p1" class="ltx_para">
<table id="E6" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex10" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="6" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">19.20.6</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m50.png" altimg-height="25px" altimg-valign="-7px" altimg-width="123px" alttext="\displaystyle R_{J}\left(x,x,x,x\right)" display="inline"><mrow><msub><mi href="./19.16#E2">R</mi><mi href="./19.16#E2">J</mi></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m28.png" altimg-height="28px" altimg-valign="-6px" altimg-width="81px" alttext="\displaystyle=x^{-3/2}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><msup><mi>x</mi><mrow><mo>-</mo><mrow><mn>3</mn><mo>/</mo><mn>2</mn></mrow></mrow></msup></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex11" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m48.png" altimg-height="25px" altimg-valign="-7px" altimg-width="166px" alttext="\displaystyle R_{J}\left(\lambda x,\lambda y,\lambda z,\lambda p\right)" display="inline"><mrow><msub><mi href="./19.16#E2">R</mi><mi href="./19.16#E2">J</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi>λ</mi><mo></mo><mi>x</mi></mrow><mo>,</mo><mrow><mi>λ</mi><mo></mo><mi>y</mi></mrow><mo>,</mo><mrow><mi>λ</mi><mo></mo><mi>z</mi></mrow><mo>,</mo><mrow><mi>λ</mi><mo></mo><mi>p</mi></mrow><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m20.png" altimg-height="29px" altimg-valign="-7px" altimg-width="199px" alttext="\displaystyle=\lambda^{-3/2}R_{J}\left(x,y,z,p\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msup><mi>λ</mi><mrow><mo>-</mo><mrow><mn>3</mn><mo>/</mo><mn>2</mn></mrow></mrow></msup><mo></mo><mrow><msub><mi href="./19.16#E2">R</mi><mi href="./19.16#E2">J</mi></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>,</mo><mi>p</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex12" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m53.png" altimg-height="25px" altimg-valign="-7px" altimg-width="120px" alttext="\displaystyle R_{J}\left(x,y,z,z\right)" display="inline"><mrow><msub><mi href="./19.16#E2">R</mi><mi href="./19.16#E2">J</mi></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m8.png" altimg-height="25px" altimg-valign="-7px" altimg-width="134px" alttext="\displaystyle=R_{D}\left(x,y,z\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msub><mi href="./19.16#E5">R</mi><mi href="./19.16#E5">D</mi></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex13" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m45.png" altimg-height="25px" altimg-valign="-7px" altimg-width="118px" alttext="\displaystyle R_{J}\left(0,0,z,p\right)" display="inline"><mrow><msub><mi href="./19.16#E2">R</mi><mi href="./19.16#E2">J</mi></msub><mo></mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>,</mo><mi>z</mi><mo>,</mo><mi>p</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m16.png" altimg-height="18px" altimg-valign="-6px" altimg-width="53px" alttext="\displaystyle=\infty," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mi mathvariant="normal">∞</mi></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex14" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m49.png" altimg-height="25px" altimg-valign="-7px" altimg-width="122px" alttext="\displaystyle R_{J}\left(x,x,x,p\right)" display="inline"><mrow><msub><mi href="./19.16#E2">R</mi><mi href="./19.16#E2">J</mi></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>p</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m6.png" altimg-height="53px" altimg-valign="-21px" altimg-width="378px" alttext="\displaystyle=R_{D}\left(p,p,x\right)=\frac{3}{x-p}\left(R_{C}\left(x,p\right)%
-\frac{1}{\sqrt{x}}\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msub><mi href="./19.16#E5">R</mi><mi href="./19.16#E5">D</mi></msub><mo></mo><mrow><mo>(</mo><mi>p</mi><mo>,</mo><mi>p</mi><mo>,</mo><mi>x</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mstyle displaystyle="true"><mfrac><mn>3</mn><mrow><mi>x</mi><mo>-</mo><mi>p</mi></mrow></mfrac></mstyle><mo></mo><mrow><mo>(</mo><mrow><mrow><msub><mi href="./19.2#E17">R</mi><mi href="./19.2#E17">C</mi></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>p</mi><mo>)</mo></mrow></mrow><mo>-</mo><mstyle displaystyle="true"><mfrac><mn>1</mn><msqrt><mi>x</mi></msqrt></mfrac></mstyle></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m115.png" altimg-height="21px" altimg-valign="-6px" altimg-width="52px" alttext="x\neq p" display="inline"><mrow><mi>x</mi><mo>≠</mo><mi>p</mi></mrow></math>, <math class="ltx_Math" altimg="m118.png" altimg-height="21px" altimg-valign="-6px" altimg-width="62px" alttext="xp\neq 0" display="inline"><mrow><mrow><mi>x</mi><mo></mo><mi>p</mi></mrow><mo>≠</mo><mn>0</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E6.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.2#E17" title="(19.2.17) ‣ §19.2(iv) A Related Function: R C ( x , y ) ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m67.png" altimg-height="23px" altimg-valign="-7px" altimg-width="82px" alttext="R_{C}\left(\NVar{x},\NVar{y}\right)" display="inline"><mrow><msub><mi href="./19.2#E17">R</mi><mi href="./19.2#E17">C</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>)</mo></mrow></mrow></math>: Carlson’s combination of inverse circular and inverse hyperbolic functions</a>,
<a href="./19.16#E5" title="(19.16.5) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m68.png" altimg-height="23px" altimg-valign="-7px" altimg-width="102px" alttext="R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E5">R</mi><mi href="./19.16#E5">D</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: elliptic integral symmetric in only two variables</a> and
<a href="./19.16#E2" title="(19.16.2) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m75.png" altimg-height="23px" altimg-valign="-7px" altimg-width="118px" alttext="R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)" display="inline"><mrow><msub><mi href="./19.16#E2">R</mi><mi href="./19.16#E2">J</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>,</mo><mi class="ltx_nvar">p</mi><mo>)</mo></mrow></mrow></math>: symmetric elliptic integral of third kind</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E7" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.20.7</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E7.png" altimg-height="25px" altimg-valign="-7px" altimg-width="192px" alttext="R_{J}\left(x,y,z,p\right)\to+\infty," display="block"><mrow><mrow><mrow><msub><mi href="./19.16#E2">R</mi><mi href="./19.16#E2">J</mi></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>,</mo><mi>p</mi><mo>)</mo></mrow></mrow><mo>→</mo><mrow><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m102.png" altimg-height="20px" altimg-valign="-6px" altimg-width="71px" alttext="p\to 0+" display="inline"><mrow><mi>p</mi><mo>→</mo><mrow><mn>0</mn><mo>+</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m62.png" altimg-height="18px" altimg-valign="-4px" altimg-width="30px" alttext="0-" display="inline"><mrow><mn>0</mn><mo>-</mo></mrow></math>; <math class="ltx_Math" altimg="m110.png" altimg-height="20px" altimg-valign="-6px" altimg-width="91px" alttext="x,y,z>0" display="inline"><mrow><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi></mrow><mo>></mo><mn>0</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E7.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./19.16#E2" title="(19.16.2) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m75.png" altimg-height="23px" altimg-valign="-7px" altimg-width="118px" alttext="R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)" display="inline"><mrow><msub><mi href="./19.16#E2">R</mi><mi href="./19.16#E2">J</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>,</mo><mi class="ltx_nvar">p</mi><mo>)</mo></mrow></mrow></math>: symmetric elliptic integral of third kind</a></dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E8" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex15" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="7" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">19.20.8</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m47.png" altimg-height="25px" altimg-valign="-7px" altimg-width="119px" alttext="\displaystyle R_{J}\left(0,y,y,p\right)" display="inline"><mrow><msub><mi href="./19.16#E2">R</mi><mi href="./19.16#E2">J</mi></msub><mo></mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>y</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>p</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m10.png" altimg-height="52px" altimg-valign="-22px" altimg-width="161px" alttext="\displaystyle=\frac{3\pi}{2(y\sqrt{p}+p\sqrt{y})}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mstyle displaystyle="true"><mfrac><mrow><mn>3</mn><mo></mo><mi href="./3.12#E1">π</mi></mrow><mrow><mn>2</mn><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mi>y</mi><mo></mo><msqrt><mi>p</mi></msqrt></mrow><mo>+</mo><mrow><mi>p</mi><mo></mo><msqrt><mi>y</mi></msqrt></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></mfrac></mstyle></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m97.png" altimg-height="20px" altimg-valign="-6px" altimg-width="51px" alttext="p>0" display="inline"><mrow><mi>p</mi><mo>></mo><mn>0</mn></mrow></math>,</span></td></tr>
<tr id="Ex16" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m46.png" altimg-height="25px" altimg-valign="-7px" altimg-width="134px" alttext="\displaystyle R_{J}\left(0,y,y,-q\right)" display="inline"><mrow><msub><mi href="./19.16#E2">R</mi><mi href="./19.16#E2">J</mi></msub><mo></mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>y</mi><mo>,</mo><mi>y</mi><mo>,</mo><mrow><mo>-</mo><mi>q</mi></mrow><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m9.png" altimg-height="52px" altimg-valign="-22px" altimg-width="134px" alttext="\displaystyle=\frac{-3\pi}{2\sqrt{y}(y+q)}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mstyle displaystyle="true"><mfrac><mrow><mo>-</mo><mrow><mn>3</mn><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow><mrow><mn>2</mn><mo></mo><msqrt><mi>y</mi></msqrt><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>y</mi><mo>+</mo><mi>q</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mfrac></mstyle></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m104.png" altimg-height="20px" altimg-valign="-6px" altimg-width="50px" alttext="q>0" display="inline"><mrow><mi>q</mi><mo>></mo><mn>0</mn></mrow></math>,</span></td></tr>
<tr id="Ex17" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m51.png" altimg-height="25px" altimg-valign="-7px" altimg-width="120px" alttext="\displaystyle R_{J}\left(x,y,y,p\right)" display="inline"><mrow><msub><mi href="./19.16#E2">R</mi><mi href="./19.16#E2">J</mi></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>p</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m12.png" altimg-height="49px" altimg-valign="-20px" altimg-width="278px" alttext="\displaystyle=\frac{3}{p-y}(R_{C}\left(x,y\right)-R_{C}\left(x,p\right))," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><mfrac><mn>3</mn><mrow><mi>p</mi><mo>-</mo><mi>y</mi></mrow></mfrac></mstyle><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><msub><mi href="./19.2#E17">R</mi><mi href="./19.2#E17">C</mi></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></mrow><mo>-</mo><mrow><msub><mi href="./19.2#E17">R</mi><mi href="./19.2#E17">C</mi></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>p</mi><mo>)</mo></mrow></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m99.png" altimg-height="21px" altimg-valign="-6px" altimg-width="51px" alttext="p\neq y" display="inline"><mrow><mi>p</mi><mo>≠</mo><mi>y</mi></mrow></math>,</span></td></tr>
<tr id="Ex18" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m52.png" altimg-height="25px" altimg-valign="-7px" altimg-width="121px" alttext="\displaystyle R_{J}\left(x,y,y,y\right)" display="inline"><mrow><msub><mi href="./19.16#E2">R</mi><mi href="./19.16#E2">J</mi></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m7.png" altimg-height="25px" altimg-valign="-7px" altimg-width="134px" alttext="\displaystyle=R_{D}\left(x,y,y\right)." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msub><mi href="./19.16#E5">R</mi><mi href="./19.16#E5">D</mi></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E8.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.2#E17" title="(19.2.17) ‣ §19.2(iv) A Related Function: R C ( x , y ) ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m67.png" altimg-height="23px" altimg-valign="-7px" altimg-width="82px" alttext="R_{C}\left(\NVar{x},\NVar{y}\right)" display="inline"><mrow><msub><mi href="./19.2#E17">R</mi><mi href="./19.2#E17">C</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>)</mo></mrow></mrow></math>: Carlson’s combination of inverse circular and inverse hyperbolic functions</a>,
<a href="./19.16#E5" title="(19.16.5) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m68.png" altimg-height="23px" altimg-valign="-7px" altimg-width="102px" alttext="R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E5">R</mi><mi href="./19.16#E5">D</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: elliptic integral symmetric in only two variables</a>,
<a href="./19.16#E2" title="(19.16.2) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m75.png" altimg-height="23px" altimg-valign="-7px" altimg-width="118px" alttext="R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)" display="inline"><mrow><msub><mi href="./19.16#E2">R</mi><mi href="./19.16#E2">J</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>,</mo><mi class="ltx_nvar">p</mi><mo>)</mo></mrow></mrow></math>: symmetric elliptic integral of third kind</a> and
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E9" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.20.9</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E9.png" altimg-height="52px" altimg-valign="-22px" altimg-width="359px" alttext="R_{J}\left(0,y,z,\pm\sqrt{yz}\right)=\pm\frac{3}{2\sqrt{yz}}R_{F}\left(0,y,z%
\right)." display="block"><mrow><mrow><mrow><msub><mi href="./19.16#E2">R</mi><mi href="./19.16#E2">J</mi></msub><mo></mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>,</mo><mrow><mo>±</mo><msqrt><mrow><mi>y</mi><mo></mo><mi>z</mi></mrow></msqrt></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mo>±</mo><mrow><mfrac><mn>3</mn><mrow><mn>2</mn><mo></mo><msqrt><mrow><mi>y</mi><mo></mo><mi>z</mi></mrow></msqrt></mrow></mfrac><mo></mo><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E9.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.16#E1" title="(19.16.1) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="23px" altimg-valign="-7px" altimg-width="101px" alttext="R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: symmetric elliptic integral of first kind</a> and
<a href="./19.16#E2" title="(19.16.2) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m75.png" altimg-height="23px" altimg-valign="-7px" altimg-width="118px" alttext="R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)" display="inline"><mrow><msub><mi href="./19.16#E2">R</mi><mi href="./19.16#E2">J</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>,</mo><mi class="ltx_nvar">p</mi><mo>)</mo></mrow></mrow></math>: symmetric elliptic integral of third kind</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E10" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex19" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">19.20.10</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m55.png" altimg-height="37px" altimg-valign="-19px" altimg-width="197px" alttext="\displaystyle\lim_{p\to 0+}\sqrt{p}R_{J}\left(0,y,z,p\right)" display="inline"><mrow><munder><mo movablelimits="false">lim</mo><mrow><mi>p</mi><mo>→</mo><mrow><mn>0</mn><mo>+</mo></mrow></mrow></munder><mo></mo><mrow><msqrt><mi>p</mi></msqrt><mo></mo><mrow><msub><mi href="./19.16#E2">R</mi><mi href="./19.16#E2">J</mi></msub><mo></mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>,</mo><mi>p</mi><mo>)</mo></mrow></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m11.png" altimg-height="52px" altimg-valign="-22px" altimg-width="85px" alttext="\displaystyle=\frac{3\pi}{2\sqrt{yz}}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mstyle displaystyle="true"><mfrac><mrow><mn>3</mn><mo></mo><mi href="./3.12#E1">π</mi></mrow><mrow><mn>2</mn><mo></mo><msqrt><mrow><mi>y</mi><mo></mo><mi>z</mi></mrow></msqrt></mrow></mfrac></mstyle></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex20" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m56.png" altimg-height="37px" altimg-valign="-19px" altimg-width="171px" alttext="\displaystyle\lim_{p\to 0-}R_{J}\left(0,y,z,p\right)" display="inline"><mrow><munder><mo movablelimits="false">lim</mo><mrow><mi>p</mi><mo>→</mo><mrow><mn>0</mn><mo>-</mo></mrow></mrow></munder><mo></mo><mrow><msub><mi href="./19.16#E2">R</mi><mi href="./19.16#E2">J</mi></msub><mo></mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>,</mo><mi>p</mi><mo>)</mo></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m3.png" altimg-height="49px" altimg-valign="-20px" altimg-width="422px" alttext="\displaystyle=-R_{D}\left(0,y,z\right)-R_{D}\left(0,z,y\right)=\frac{-6}{yz}R_%
{G}\left(0,y,z\right)." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mo>-</mo><mrow><msub><mi href="./19.16#E5">R</mi><mi href="./19.16#E5">D</mi></msub><mo></mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow><mo>-</mo><mrow><msub><mi href="./19.16#E5">R</mi><mi href="./19.16#E5">D</mi></msub><mo></mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>z</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mrow><mstyle displaystyle="true"><mfrac><mrow><mo>-</mo><mn>6</mn></mrow><mrow><mi>y</mi><mo></mo><mi>z</mi></mrow></mfrac></mstyle><mo></mo><mrow><msub><mi href="./19.16#E3">R</mi><mi href="./19.16#E3">G</mi></msub><mo></mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E10.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.16#E5" title="(19.16.5) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m68.png" altimg-height="23px" altimg-valign="-7px" altimg-width="102px" alttext="R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E5">R</mi><mi href="./19.16#E5">D</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: elliptic integral symmetric in only two variables</a>,
<a href="./19.16#E3" title="(19.16.3) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="23px" altimg-valign="-7px" altimg-width="101px" alttext="R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E3">R</mi><mi href="./19.16#E3">G</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: symmetric elliptic integral of second kind</a>,
<a href="./19.16#E2" title="(19.16.2) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m75.png" altimg-height="23px" altimg-valign="-7px" altimg-width="118px" alttext="R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)" display="inline"><mrow><msub><mi href="./19.16#E2">R</mi><mi href="./19.16#E2">J</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>,</mo><mi class="ltx_nvar">p</mi><mo>)</mo></mrow></mrow></math>: symmetric elliptic integral of third kind</a> and
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E11" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.20.11</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E11.png" altimg-height="53px" altimg-valign="-21px" altimg-width="293px" alttext="R_{J}\left(0,y,z,p\right)\sim\frac{3}{2p\sqrt{z}}\ln\left(\frac{16z}{y}\right)," display="block"><mrow><mrow><mrow><msub><mi href="./19.16#E2">R</mi><mi href="./19.16#E2">J</mi></msub><mo></mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>,</mo><mi>p</mi><mo>)</mo></mrow></mrow><mo href="./2.1#E1">∼</mo><mrow><mfrac><mn>3</mn><mrow><mn>2</mn><mo></mo><mi>p</mi><mo></mo><msqrt><mi>z</mi></msqrt></mrow></mfrac><mo></mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><mo>(</mo><mfrac><mrow><mn>16</mn><mo></mo><mi>z</mi></mrow><mi>y</mi></mfrac><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m121.png" altimg-height="20px" altimg-valign="-6px" altimg-width="71px" alttext="y\to 0+" display="inline"><mrow><mi>y</mi><mo>→</mo><mrow><mn>0</mn><mo>+</mo></mrow></mrow></math>; <math class="ltx_Math" altimg="m98.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="p" display="inline"><mi>p</mi></math> (<math class="ltx_Math" altimg="m86.png" altimg-height="21px" altimg-valign="-6px" altimg-width="35px" alttext="\neq 0" display="inline"><mrow><mi></mi><mo>≠</mo><mn>0</mn></mrow></math>) real.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E11.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.16#E2" title="(19.16.2) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m75.png" altimg-height="23px" altimg-valign="-7px" altimg-width="118px" alttext="R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)" display="inline"><mrow><msub><mi href="./19.16#E2">R</mi><mi href="./19.16#E2">J</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>,</mo><mi class="ltx_nvar">p</mi><mo>)</mo></mrow></mrow></math>: symmetric elliptic integral of third kind</a>,
<a href="./2.1#E1" title="(2.1.1) ‣ §2.1(i) Asymptotic and Order Symbols ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m88.png" altimg-height="12px" altimg-valign="-2px" altimg-width="20px" alttext="\sim" display="inline"><mo href="./2.1#E1">∼</mo></math>: asymptotic equality</a> and
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m84.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E12" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.20.12</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E12.png" altimg-height="37px" altimg-valign="-19px" altimg-width="329px" alttext="\lim_{p\to\pm\infty}pR_{J}\left(x,y,z,p\right)=3\!R_{F}\left(x,y,z\right)." display="block"><mrow><mrow><mrow><munder><mo movablelimits="false">lim</mo><mrow><mi>p</mi><mo>→</mo><mrow><mo>±</mo><mi mathvariant="normal">∞</mi></mrow></mrow></munder><mo></mo><mrow><mi>p</mi><mo></mo><mrow><msub><mi href="./19.16#E2">R</mi><mi href="./19.16#E2">J</mi></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>,</mo><mi>p</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>=</mo><mrow><mpadded width="-1.7pt"><mn>3</mn></mpadded><mo></mo><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E12.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.16#E1" title="(19.16.1) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="23px" altimg-valign="-7px" altimg-width="101px" alttext="R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: symmetric elliptic integral of first kind</a> and
<a href="./19.16#E2" title="(19.16.2) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m75.png" altimg-height="23px" altimg-valign="-7px" altimg-width="118px" alttext="R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)" display="inline"><mrow><msub><mi href="./19.16#E2">R</mi><mi href="./19.16#E2">J</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>,</mo><mi class="ltx_nvar">p</mi><mo>)</mo></mrow></mrow></math>: symmetric elliptic integral of third kind</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E13" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.20.13</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E13.png" altimg-height="30px" altimg-valign="-9px" altimg-width="491px" alttext="2(p-x)R_{J}\left(x,y,z,p\right)=3\!R_{F}\left(x,y,z\right)-3\sqrt{x}R_{C}\left%
(yz,p^{2}\right)," display="block"><mrow><mrow><mrow><mn>2</mn><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>p</mi><mo>-</mo><mi>x</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><msub><mi href="./19.16#E2">R</mi><mi href="./19.16#E2">J</mi></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>,</mo><mi>p</mi><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mrow><mrow><mpadded width="-1.7pt"><mn>3</mn></mpadded><mo></mo><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow><mo>-</mo><mrow><mn>3</mn><mo></mo><msqrt><mi>x</mi></msqrt><mo></mo><mrow><msub><mi href="./19.2#E17">R</mi><mi href="./19.2#E17">C</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi>y</mi><mo></mo><mi>z</mi></mrow><mo>,</mo><msup><mi>p</mi><mn>2</mn></msup><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m95.png" altimg-height="28px" altimg-valign="-8px" altimg-width="220px" alttext="p=x\pm\sqrt{(y-x)(z-x)}" display="inline"><mrow><mi>p</mi><mo>=</mo><mrow><mi>x</mi><mo>±</mo><msqrt><mrow><mrow><mo stretchy="false">(</mo><mrow><mi>y</mi><mo>-</mo><mi>x</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>z</mi><mo>-</mo><mi>x</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></msqrt></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E13.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.2#E17" title="(19.2.17) ‣ §19.2(iv) A Related Function: R C ( x , y ) ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m67.png" altimg-height="23px" altimg-valign="-7px" altimg-width="82px" alttext="R_{C}\left(\NVar{x},\NVar{y}\right)" display="inline"><mrow><msub><mi href="./19.2#E17">R</mi><mi href="./19.2#E17">C</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>)</mo></mrow></mrow></math>: Carlson’s combination of inverse circular and inverse hyperbolic functions</a>,
<a href="./19.16#E1" title="(19.16.1) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="23px" altimg-valign="-7px" altimg-width="101px" alttext="R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: symmetric elliptic integral of first kind</a> and
<a href="./19.16#E2" title="(19.16.2) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m75.png" altimg-height="23px" altimg-valign="-7px" altimg-width="118px" alttext="R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)" display="inline"><mrow><msub><mi href="./19.16#E2">R</mi><mi href="./19.16#E2">J</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>,</mo><mi class="ltx_nvar">p</mi><mo>)</mo></mrow></mrow></math>: symmetric elliptic integral of third kind</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m111.png" altimg-height="16px" altimg-valign="-6px" altimg-width="54px" alttext="x,y,z" display="inline"><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi></mrow></math> may be permuted.</p>
</div>
<div id="SS3.p2" class="ltx_para">
<p class="ltx_p">When the variables are real and distinct, the various cases of
<math class="ltx_Math" altimg="m76.png" altimg-height="23px" altimg-valign="-7px" altimg-width="118px" alttext="R_{J}\left(x,y,z,p\right)" display="inline"><mrow><msub><mi href="./19.16#E2">R</mi><mi href="./19.16#E2">J</mi></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>,</mo><mi>p</mi><mo>)</mo></mrow></mrow></math> are called <em class="ltx_emph ltx_font_italic">circular (hyperbolic) cases</em> if
<math class="ltx_Math" altimg="m61.png" altimg-height="23px" altimg-valign="-7px" altimg-width="186px" alttext="(p-x)(p-y)(p-z)" display="inline"><mrow><mrow><mo stretchy="false">(</mo><mrow><mi>p</mi><mo>-</mo><mi>x</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>p</mi><mo>-</mo><mi>y</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>p</mi><mo>-</mo><mi>z</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></math> is positive (negative), because they typically occur in
conjunction with inverse circular (hyperbolic) functions. Cases encountered in
dynamical problems are usually circular; hyperbolic cases include Cauchy
principal values. If <math class="ltx_Math" altimg="m111.png" altimg-height="16px" altimg-valign="-6px" altimg-width="54px" alttext="x,y,z" display="inline"><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi></mrow></math> are permuted so that <math class="ltx_Math" altimg="m63.png" altimg-height="20px" altimg-valign="-6px" altimg-width="126px" alttext="0\leq x<y<z" display="inline"><mrow><mn>0</mn><mo>≤</mo><mi>x</mi><mo><</mo><mi>y</mi><mo><</mo><mi>z</mi></mrow></math>, then the
Cauchy principal value of <math class="ltx_Math" altimg="m74.png" altimg-height="21px" altimg-valign="-5px" altimg-width="31px" alttext="R_{J}" display="inline"><msub><mi href="./19.16#E2">R</mi><mi href="./19.16#E2">J</mi></msub></math> is given by</p>
<table id="E14" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.20.14</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_Math ltx_eqn_cell ltx_align_center"><math class="ltx_Math" alttext="(q+z)R_{J}\left(x,y,z,-q\right)=(p-z)R_{J}\left(x,y,z,p\right)-3\!R_{F}\left(x%
,y,z\right)+3\left(\frac{xyz}{xy+pq}\right)^{1/2}R_{C}\left(xy+pq,pq\right)," display="block"><mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mi>q</mi><mo>+</mo><mi>z</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><msub><mi href="./19.16#E2">R</mi><mi href="./19.16#E2">J</mi></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>,</mo><mrow><mo>-</mo><mi>q</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mi>p</mi><mo>-</mo><mi>z</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><msub><mi href="./19.16#E2">R</mi><mi href="./19.16#E2">J</mi></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>,</mo><mi>p</mi><mo>)</mo></mrow></mrow></mrow><mo>-</mo><mrow><mpadded width="-1.7pt"><mn>3</mn></mpadded><mo></mo><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>+</mo><mrow><mn>3</mn><mo></mo><msup><mrow><mo>(</mo><mfrac><mrow><mi>x</mi><mo></mo><mi>y</mi><mo></mo><mi>z</mi></mrow><mrow><mrow><mi>x</mi><mo></mo><mi>y</mi></mrow><mo>+</mo><mrow><mi>p</mi><mo></mo><mi>q</mi></mrow></mrow></mfrac><mo>)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo></mo><mrow><msub><mi href="./19.2#E17">R</mi><mi href="./19.2#E17">C</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mrow><mi>x</mi><mo></mo><mi>y</mi></mrow><mo>+</mo><mrow><mi>p</mi><mo></mo><mi>q</mi></mrow></mrow><mo>,</mo><mrow><mi>p</mi><mo></mo><mi>q</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mtd></mtr></mtable></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E14.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.2#E17" title="(19.2.17) ‣ §19.2(iv) A Related Function: R C ( x , y ) ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m67.png" altimg-height="23px" altimg-valign="-7px" altimg-width="82px" alttext="R_{C}\left(\NVar{x},\NVar{y}\right)" display="inline"><mrow><msub><mi href="./19.2#E17">R</mi><mi href="./19.2#E17">C</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>)</mo></mrow></mrow></math>: Carlson’s combination of inverse circular and inverse hyperbolic functions</a>,
<a href="./19.16#E1" title="(19.16.1) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="23px" altimg-valign="-7px" altimg-width="101px" alttext="R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: symmetric elliptic integral of first kind</a> and
<a href="./19.16#E2" title="(19.16.2) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m75.png" altimg-height="23px" altimg-valign="-7px" altimg-width="118px" alttext="R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)" display="inline"><mrow><msub><mi href="./19.16#E2">R</mi><mi href="./19.16#E2">J</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>,</mo><mi class="ltx_nvar">p</mi><mo>)</mo></mrow></mrow></math>: symmetric elliptic integral of third kind</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">valid when</p>
<table id="E15" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex21" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">19.20.15</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m58.png" altimg-height="18px" altimg-valign="-6px" altimg-width="16px" alttext="\displaystyle q" display="inline"><mi>q</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m30.png" altimg-height="22px" altimg-valign="-6px" altimg-width="43px" alttext="\displaystyle>0," display="inline"><mrow><mrow><mi></mi><mo>></mo><mn>0</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex22" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m57.png" altimg-height="18px" altimg-valign="-6px" altimg-width="16px" alttext="\displaystyle p" display="inline"><mi>p</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m14.png" altimg-height="51px" altimg-valign="-20px" altimg-width="189px" alttext="\displaystyle=\frac{z(x+y+q)-xy}{z+q}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mstyle displaystyle="true"><mfrac><mrow><mrow><mi>z</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>x</mi><mo>+</mo><mi>y</mi><mo>+</mo><mi>q</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>-</mo><mrow><mi>x</mi><mo></mo><mi>y</mi></mrow></mrow><mrow><mi>z</mi><mo>+</mo><mi>q</mi></mrow></mfrac></mstyle></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E15.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">or
</p>
<table id="E16" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex23" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="3" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">19.20.16</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m57.png" altimg-height="18px" altimg-valign="-6px" altimg-width="16px" alttext="\displaystyle p" display="inline"><mi>p</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m26.png" altimg-height="25px" altimg-valign="-7px" altimg-width="157px" alttext="\displaystyle=wy+(1-w)z," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mi>w</mi><mo></mo><mi>y</mi></mrow><mo>+</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><mi>w</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi>z</mi></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex24" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m59.png" altimg-height="14px" altimg-valign="-2px" altimg-width="21px" alttext="\displaystyle w" display="inline"><mi>w</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m15.png" altimg-height="48px" altimg-valign="-20px" altimg-width="83px" alttext="\displaystyle=\frac{z-x}{z+q}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mstyle displaystyle="true"><mfrac><mrow><mi>z</mi><mo>-</mo><mi>x</mi></mrow><mrow><mi>z</mi><mo>+</mo><mi>q</mi></mrow></mfrac></mstyle></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex25" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m1.png" altimg-height="18px" altimg-valign="-2px" altimg-width="16px" alttext="\displaystyle 0" display="inline"><mn>0</mn></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m2.png" altimg-height="19px" altimg-valign="-3px" altimg-width="84px" alttext="\displaystyle<w<1." display="inline"><mrow><mrow><mi></mi><mo><</mo><mi>w</mi><mo><</mo><mn>1</mn></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E16.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Since <math class="ltx_Math" altimg="m112.png" altimg-height="18px" altimg-valign="-6px" altimg-width="126px" alttext="x<y<p<z" display="inline"><mrow><mi>x</mi><mo><</mo><mi>y</mi><mo><</mo><mi>p</mi><mo><</mo><mi>z</mi></mrow></math>, <math class="ltx_Math" altimg="m98.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="p" display="inline"><mi>p</mi></math> is in a hyperbolic region. In the complete case
(<math class="ltx_Math" altimg="m113.png" altimg-height="17px" altimg-valign="-2px" altimg-width="52px" alttext="x=0" display="inline"><mrow><mi>x</mi><mo>=</mo><mn>0</mn></mrow></math>) () reduces to</p>
<table id="E17" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.20.17</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E17.png" altimg-height="25px" altimg-valign="-7px" altimg-width="527px" alttext="(q+z)R_{J}\left(0,y,z,-q\right)=(p-z)R_{J}\left(0,y,z,p\right)-3\!R_{F}\left(0%
,y,z\right)," display="block"><mrow><mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mi>q</mi><mo>+</mo><mi>z</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><msub><mi href="./19.16#E2">R</mi><mi href="./19.16#E2">J</mi></msub><mo></mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>,</mo><mrow><mo>-</mo><mi>q</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mi>p</mi><mo>-</mo><mi>z</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><msub><mi href="./19.16#E2">R</mi><mi href="./19.16#E2">J</mi></msub><mo></mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>,</mo><mi>p</mi><mo>)</mo></mrow></mrow></mrow><mo>-</mo><mrow><mpadded width="-1.7pt"><mn>3</mn></mpadded><mo></mo><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m96.png" altimg-height="23px" altimg-valign="-7px" altimg-width="181px" alttext="p=z(y+q)/(z+q)" display="inline"><mrow><mi>p</mi><mo>=</mo><mrow><mrow><mi>z</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>y</mi><mo>+</mo><mi>q</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>/</mo><mrow><mo stretchy="false">(</mo><mrow><mi>z</mi><mo>+</mo><mi>q</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></math>, <math class="ltx_Math" altimg="m109.png" altimg-height="23px" altimg-valign="-7px" altimg-width="125px" alttext="w=z/(z+q)" display="inline"><mrow><mi>w</mi><mo>=</mo><mrow><mi>z</mi><mo>/</mo><mrow><mo stretchy="false">(</mo><mrow><mi>z</mi><mo>+</mo><mi>q</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E17.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.16#E1" title="(19.16.1) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="23px" altimg-valign="-7px" altimg-width="101px" alttext="R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: symmetric elliptic integral of first kind</a> and
<a href="./19.16#E2" title="(19.16.2) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m75.png" altimg-height="23px" altimg-valign="-7px" altimg-width="118px" alttext="R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)" display="inline"><mrow><msub><mi href="./19.16#E2">R</mi><mi href="./19.16#E2">J</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>,</mo><mi class="ltx_nvar">p</mi><mo>)</mo></mrow></mrow></math>: symmetric elliptic integral of third kind</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="iv" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§19.20(iv) </span><math class="ltx_Math" altimg="m69.png" altimg-height="23px" altimg-valign="-7px" altimg-width="102px" alttext="R_{D}\left(x,y,z\right)" display="inline"><mrow><msub><mi href="./19.16#E5">R</mi><mi href="./19.16#E5">D</mi></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow></mrow></math>
</h2>
<div id="SS4.info" class="ltx_metadata ltx_info">
) put <math class="ltx_Math" altimg="m106.png" altimg-height="24px" altimg-valign="-6px" altimg-width="96px" alttext="t=y{\tan^{2}}\theta" display="inline"><mrow><mi>t</mi><mo>=</mo><mrow><mi>y</mi><mo></mo><mrow><msup><mi href="./4.14#E4">tan</mi><mn>2</mn></msup><mo></mo><mi>θ</mi></mrow></mrow></mrow></math>
in () put <math class="ltx_Math" altimg="m105.png" altimg-height="26px" altimg-valign="-7px" altimg-width="118px" alttext="t=1/\sqrt{s+1}" display="inline"><mrow><mi>t</mi><mo>=</mo><mrow><mn>1</mn><mo>/</mo><msqrt><mrow><mi>s</mi><mo>+</mo><mn>1</mn></mrow></msqrt></mrow></mrow></math>; alternatively use
().
For the second equality replace <math class="ltx_Math" altimg="m108.png" altimg-height="20px" altimg-valign="-2px" altimg-width="20px" alttext="t^{4}" display="inline"><msup><mi>t</mi><mn>4</mn></msup></math> by <math class="ltx_Math" altimg="m107.png" altimg-height="17px" altimg-valign="-2px" altimg-width="11px" alttext="t" display="inline"><mi>t</mi></math> and apply
(</dd>
</dl>
</div>
</div>
<div id="SS4.p1" class="ltx_para">
<table id="E18" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex26" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="4" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">19.20.18</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m35.png" altimg-height="25px" altimg-valign="-7px" altimg-width="106px" alttext="\displaystyle R_{D}\left(x,x,x\right)" display="inline"><mrow><msub><mi href="./19.16#E5">R</mi><mi href="./19.16#E5">D</mi></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m28.png" altimg-height="28px" altimg-valign="-6px" altimg-width="81px" alttext="\displaystyle=x^{-3/2}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><msup><mi>x</mi><mrow><mo>-</mo><mrow><mn>3</mn><mo>/</mo><mn>2</mn></mrow></mrow></msup></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex27" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m34.png" altimg-height="25px" altimg-valign="-7px" altimg-width="139px" alttext="\displaystyle R_{D}\left(\lambda x,\lambda y,\lambda z\right)" display="inline"><mrow><msub><mi href="./19.16#E5">R</mi><mi href="./19.16#E5">D</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi>λ</mi><mo></mo><mi>x</mi></mrow><mo>,</mo><mrow><mi>λ</mi><mo></mo><mi>y</mi></mrow><mo>,</mo><mrow><mi>λ</mi><mo></mo><mi>z</mi></mrow><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m19.png" altimg-height="29px" altimg-valign="-7px" altimg-width="183px" alttext="\displaystyle=\lambda^{-3/2}R_{D}\left(x,y,z\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msup><mi>λ</mi><mrow><mo>-</mo><mrow><mn>3</mn><mo>/</mo><mn>2</mn></mrow></mrow></msup><mo></mo><mrow><msub><mi href="./19.16#E5">R</mi><mi href="./19.16#E5">D</mi></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex28" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m33.png" altimg-height="25px" altimg-valign="-7px" altimg-width="103px" alttext="\displaystyle R_{D}\left(0,y,y\right)" display="inline"><mrow><msub><mi href="./19.16#E5">R</mi><mi href="./19.16#E5">D</mi></msub><mo></mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>y</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m25.png" altimg-height="31px" altimg-valign="-9px" altimg-width="109px" alttext="\displaystyle=\tfrac{3}{4}\pi\,y^{-3/2}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mfrac><mn>3</mn><mn>4</mn></mfrac><mo></mo><mpadded width="+1.7pt"><mi href="./3.12#E1">π</mi></mpadded><mo></mo><msup><mi>y</mi><mrow><mo>-</mo><mrow><mn>3</mn><mo>/</mo><mn>2</mn></mrow></mrow></msup></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex29" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m32.png" altimg-height="25px" altimg-valign="-7px" altimg-width="102px" alttext="\displaystyle R_{D}\left(0,0,z\right)" display="inline"><mrow><msub><mi href="./19.16#E5">R</mi><mi href="./19.16#E5">D</mi></msub><mo></mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>,</mo><mi>z</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m17.png" altimg-height="14px" altimg-valign="-2px" altimg-width="53px" alttext="\displaystyle=\infty." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mi mathvariant="normal">∞</mi></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E18.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.16#E5" title="(19.16.5) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m68.png" altimg-height="23px" altimg-valign="-7px" altimg-width="102px" alttext="R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E5">R</mi><mi href="./19.16#E5">D</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: elliptic integral symmetric in only two variables</a> and
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E19" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.20.19</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E19.png" altimg-height="29px" altimg-valign="-7px" altimg-width="231px" alttext="R_{D}\left(x,y,z\right)\sim 3(xyz)^{-1/2}," display="block"><mrow><mrow><mrow><msub><mi href="./19.16#E5">R</mi><mi href="./19.16#E5">D</mi></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow></mrow><mo href="./2.1#E1">∼</mo><mrow><mn>3</mn><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi>x</mi><mo></mo><mi>y</mi><mo></mo><mi>z</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></mrow></msup></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m122.png" altimg-height="25px" altimg-valign="-9px" altimg-width="104px" alttext="z/\sqrt{xy}\to 0" display="inline"><mrow><mrow><mi>z</mi><mo>/</mo><msqrt><mrow><mi>x</mi><mo></mo><mi>y</mi></mrow></msqrt></mrow><mo>→</mo><mn>0</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E19.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.16#E5" title="(19.16.5) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m68.png" altimg-height="23px" altimg-valign="-7px" altimg-width="102px" alttext="R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E5">R</mi><mi href="./19.16#E5">D</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: elliptic integral symmetric in only two variables</a> and
<a href="./2.1#E1" title="(2.1.1) ‣ §2.1(i) Asymptotic and Order Symbols ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m88.png" altimg-height="12px" altimg-valign="-2px" altimg-width="20px" alttext="\sim" display="inline"><mo href="./2.1#E1">∼</mo></math>: asymptotic equality</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E20" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.20.20</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E20.png" altimg-height="55px" altimg-valign="-21px" altimg-width="384px" alttext="R_{D}\left(x,y,y\right)=\frac{3}{2(y-x)}\left(R_{C}\left(x,y\right)-\frac{%
\sqrt{x}}{y}\right)," display="block"><mrow><mrow><mrow><msub><mi href="./19.16#E5">R</mi><mi href="./19.16#E5">D</mi></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mfrac><mn>3</mn><mrow><mn>2</mn><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>y</mi><mo>-</mo><mi>x</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mfrac><mo></mo><mrow><mo>(</mo><mrow><mrow><msub><mi href="./19.2#E17">R</mi><mi href="./19.2#E17">C</mi></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></mrow><mo>-</mo><mfrac><msqrt><mi>x</mi></msqrt><mi>y</mi></mfrac></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m116.png" altimg-height="21px" altimg-valign="-6px" altimg-width="53px" alttext="x\neq y" display="inline"><mrow><mi>x</mi><mo>≠</mo><mi>y</mi></mrow></math>, <math class="ltx_Math" altimg="m120.png" altimg-height="21px" altimg-valign="-6px" altimg-width="51px" alttext="y\neq 0" display="inline"><mrow><mi>y</mi><mo>≠</mo><mn>0</mn></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E20.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.2#E17" title="(19.2.17) ‣ §19.2(iv) A Related Function: R C ( x , y ) ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m67.png" altimg-height="23px" altimg-valign="-7px" altimg-width="82px" alttext="R_{C}\left(\NVar{x},\NVar{y}\right)" display="inline"><mrow><msub><mi href="./19.2#E17">R</mi><mi href="./19.2#E17">C</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>)</mo></mrow></mrow></math>: Carlson’s combination of inverse circular and inverse hyperbolic functions</a> and
<a href="./19.16#E5" title="(19.16.5) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m68.png" altimg-height="23px" altimg-valign="-7px" altimg-width="102px" alttext="R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E5">R</mi><mi href="./19.16#E5">D</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: elliptic integral symmetric in only two variables</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E21" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.20.21</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E21.png" altimg-height="53px" altimg-valign="-21px" altimg-width="357px" alttext="R_{D}\left(x,x,z\right)=\frac{3}{z-x}\left(R_{C}\left(z,x\right)-\frac{1}{%
\sqrt{z}}\right)," display="block"><mrow><mrow><mrow><msub><mi href="./19.16#E5">R</mi><mi href="./19.16#E5">D</mi></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mfrac><mn>3</mn><mrow><mi>z</mi><mo>-</mo><mi>x</mi></mrow></mfrac><mo></mo><mrow><mo>(</mo><mrow><mrow><msub><mi href="./19.2#E17">R</mi><mi href="./19.2#E17">C</mi></msub><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>,</mo><mi>x</mi><mo>)</mo></mrow></mrow><mo>-</mo><mfrac><mn>1</mn><msqrt><mi>z</mi></msqrt></mfrac></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m117.png" altimg-height="21px" altimg-valign="-6px" altimg-width="52px" alttext="x\neq z" display="inline"><mrow><mi>x</mi><mo>≠</mo><mi>z</mi></mrow></math>, <math class="ltx_Math" altimg="m119.png" altimg-height="21px" altimg-valign="-6px" altimg-width="62px" alttext="xz\neq 0" display="inline"><mrow><mrow><mi>x</mi><mo></mo><mi>z</mi></mrow><mo>≠</mo><mn>0</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E21.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.2#E17" title="(19.2.17) ‣ §19.2(iv) A Related Function: R C ( x , y ) ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m67.png" altimg-height="23px" altimg-valign="-7px" altimg-width="82px" alttext="R_{C}\left(\NVar{x},\NVar{y}\right)" display="inline"><mrow><msub><mi href="./19.2#E17">R</mi><mi href="./19.2#E17">C</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>)</mo></mrow></mrow></math>: Carlson’s combination of inverse circular and inverse hyperbolic functions</a> and
<a href="./19.16#E5" title="(19.16.5) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m68.png" altimg-height="23px" altimg-valign="-7px" altimg-width="102px" alttext="R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E5">R</mi><mi href="./19.16#E5">D</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: elliptic integral symmetric in only two variables</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS4.p2" class="ltx_para">
<p class="ltx_p">The <em class="ltx_emph ltx_font_italic">second lemniscate constant</em> is given by</p>
<table id="E22" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.20.22</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E22.png" altimg-height="60px" altimg-valign="-21px" altimg-width="644px" alttext="\int_{0}^{1}\frac{t^{2}\mathrm{d}t}{\sqrt{1-t^{4}}}=\tfrac{1}{3}R_{D}\left(0,2%
,1\right)=\frac{\left(\Gamma\left(\frac{3}{4}\right)\right)^{2}}{(2\pi)^{1/2}}%
=0.59907\;01173\;67796\;10371\dots." display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mn>1</mn></msubsup><mfrac><mrow><msup><mi>t</mi><mn>2</mn></msup><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow><msqrt><mrow><mn>1</mn><mo>-</mo><msup><mi>t</mi><mn>4</mn></msup></mrow></msqrt></mfrac></mrow><mo>=</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>3</mn></mfrac></mstyle><mo></mo><mrow><msub><mi href="./19.16#E5">R</mi><mi href="./19.16#E5">D</mi></msub><mo></mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mfrac><msup><mrow><mo>(</mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mfrac><mn>3</mn><mn>4</mn></mfrac><mo>)</mo></mrow></mrow><mo>)</mo></mrow><mn>2</mn></msup><msup><mrow><mo stretchy="false">(</mo><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mfrac><mo>=</mo><mrow><mn>0.59907 01173 67796 10371</mn><mo></mo><mi mathvariant="normal">…</mi></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E22.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.16#E5" title="(19.16.5) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m68.png" altimg-height="23px" altimg-valign="-7px" altimg-width="102px" alttext="R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E5">R</mi><mi href="./19.16#E5">D</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: elliptic integral symmetric in only two variables</a>,
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m81.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m114.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m83.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p"><cite class="ltx_cite ltx_citemacro_citet">Todd (). The <em class="ltx_emph ltx_font_italic">general
lemniscatic case</em> is
</p>
<table id="E23" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.20.23</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E23.png" altimg-height="33px" altimg-valign="-12px" altimg-width="295px" alttext="R_{D}\left(x,y,a\right)=R_{-\frac{3}{4}}\left(\tfrac{5}{4},\tfrac{1}{2};a^{2},%
xy\right)," display="block"><mrow><mrow><mrow><msub><mi href="./19.16#E5">R</mi><mi href="./19.16#E5">D</mi></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>a</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msub><mi href="./19.16#E9">R</mi><mrow><mo>-</mo><mfrac><mn>3</mn><mn>4</mn></mfrac></mrow></msub><mo></mo><mrow><mo>(</mo><mrow><mstyle displaystyle="false"><mfrac><mn>5</mn><mn>4</mn></mfrac></mstyle><mo>,</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle></mrow><mo>;</mo><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>,</mo><mrow><mi>x</mi><mo></mo><mi>y</mi></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m91.png" altimg-height="27px" altimg-valign="-9px" altimg-width="113px" alttext="a=\tfrac{1}{2}x+\tfrac{1}{2}y" display="inline"><mrow><mi>a</mi><mo>=</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi>x</mi></mrow><mo>+</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi>y</mi></mrow></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E23.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.16#E9" title="(19.16.9) ‣ §19.16(ii) R - a ( b ; z ) ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="23px" altimg-valign="-7px" altimg-width="233px" alttext="R_{\NVar{-a}}\left(\NVar{b_{1}},\dots,\NVar{b_{n}};\NVar{z_{1}},\dots,\NVar{z_%
{n}}\right)" display="inline"><mrow><msub><mi href="./19.16#E9">R</mi><mrow><mo class="ltx_nvar">-</mo><mi class="ltx_nvar">a</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mrow><msub><mi class="ltx_nvar">b</mi><mn class="ltx_nvar">1</mn></msub><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><msub><mi class="ltx_nvar">b</mi><mi class="ltx_nvar" href="./19.1#p1.t1.r1">n</mi></msub></mrow><mo>;</mo><mrow><msub><mi class="ltx_nvar">z</mi><mn class="ltx_nvar">1</mn></msub><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><msub><mi class="ltx_nvar">z</mi><mi class="ltx_nvar" href="./19.1#p1.t1.r1">n</mi></msub></mrow><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m77.png" altimg-height="23px" altimg-valign="-7px" altimg-width="92px" alttext="R_{\NVar{-a}}\left(\NVar{\mathbf{b}};\NVar{\mathbf{z}}\right)" display="inline"><mrow><msub><mi href="./19.16#E9">R</mi><mrow><mo class="ltx_nvar">-</mo><mi class="ltx_nvar">a</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" mathvariant="bold">b</mi><mo>;</mo><mi class="ltx_nvar" mathvariant="bold">z</mi><mo>)</mo></mrow></mrow></math>: multivariate hypergeometric function</a> and
<a href="./19.16#E5" title="(19.16.5) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m68.png" altimg-height="23px" altimg-valign="-7px" altimg-width="102px" alttext="R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E5">R</mi><mi href="./19.16#E5">D</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: elliptic integral symmetric in only two variables</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="v" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§19.20(v) </span><math class="ltx_Math" altimg="m66.png" altimg-height="23px" altimg-valign="-7px" altimg-width="92px" alttext="R_{-a}\left(\mathbf{b};\mathbf{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E9">R</mi><mrow><mo>-</mo><mi>a</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi mathvariant="bold">b</mi><mo>;</mo><mi mathvariant="bold">z</mi><mo>)</mo></mrow></mrow></math>
</h2>
<div id="SS5.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="SS5.p1" class="ltx_para">
<p class="ltx_p">Define <math class="ltx_Math" altimg="m92.png" altimg-height="28px" altimg-valign="-11px" altimg-width="109px" alttext="c=\sum_{j=1}^{n}b_{j}" display="inline"><mrow><mi>c</mi><mo>=</mo><mrow><msubsup><mo largeop="true" symmetric="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi href="./19.1#p1.t1.r1">n</mi></msubsup><msub><mi>b</mi><mi>j</mi></msub></mrow></mrow></math>. Then</p>
<table id="E24" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex30" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="3" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">19.20.24</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m31.png" altimg-height="25px" altimg-valign="-7px" altimg-width="81px" alttext="\displaystyle R_{0}\left(\mathbf{b};\mathbf{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E9">R</mi><mn>0</mn></msub><mo></mo><mrow><mo>(</mo><mi mathvariant="bold">b</mi><mo>;</mo><mi mathvariant="bold">z</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m4.png" altimg-height="22px" altimg-valign="-6px" altimg-width="43px" alttext="\displaystyle=1," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mn>1</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex31" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m54.png" altimg-height="25px" altimg-valign="-7px" altimg-width="87px" alttext="\displaystyle R_{N}\left(\mathbf{b};\mathbf{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E9">R</mi><mi>N</mi></msub><mo></mo><mrow><mo>(</mo><mi mathvariant="bold">b</mi><mo>;</mo><mi mathvariant="bold">z</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m13.png" altimg-height="52px" altimg-valign="-22px" altimg-width="151px" alttext="\displaystyle=\frac{N!}{{\left(c\right)_{N}}}T_{N}(\mathbf{b},\mathbf{z})," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><mfrac><mrow><mi>N</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow><msub><mrow><mo href="./5.2#iii">(</mo><mi>c</mi><mo href="./5.2#iii">)</mo></mrow><mi>N</mi></msub></mfrac></mstyle><mo></mo><mrow><msub><mi href="./19.19#p1">T</mi><mi>N</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi mathvariant="bold">b</mi><mo>,</mo><mi mathvariant="bold">z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m64.png" altimg-height="21px" altimg-valign="-6px" altimg-width="129px" alttext="N=0,1,2,\dots" display="inline"><mrow><mi>N</mi><mo>=</mo><mrow><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E24.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.16#E9" title="(19.16.9) ‣ §19.16(ii) R - a ( b ; z ) ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="23px" altimg-valign="-7px" altimg-width="233px" alttext="R_{\NVar{-a}}\left(\NVar{b_{1}},\dots,\NVar{b_{n}};\NVar{z_{1}},\dots,\NVar{z_%
{n}}\right)" display="inline"><mrow><msub><mi href="./19.16#E9">R</mi><mrow><mo class="ltx_nvar">-</mo><mi class="ltx_nvar">a</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mrow><msub><mi class="ltx_nvar">b</mi><mn class="ltx_nvar">1</mn></msub><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><msub><mi class="ltx_nvar">b</mi><mi class="ltx_nvar" href="./19.1#p1.t1.r1">n</mi></msub></mrow><mo>;</mo><mrow><msub><mi class="ltx_nvar">z</mi><mn class="ltx_nvar">1</mn></msub><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><msub><mi class="ltx_nvar">z</mi><mi class="ltx_nvar" href="./19.1#p1.t1.r1">n</mi></msub></mrow><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m77.png" altimg-height="23px" altimg-valign="-7px" altimg-width="92px" alttext="R_{\NVar{-a}}\left(\NVar{\mathbf{b}};\NVar{\mathbf{z}}\right)" display="inline"><mrow><msub><mi href="./19.16#E9">R</mi><mrow><mo class="ltx_nvar">-</mo><mi class="ltx_nvar">a</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" mathvariant="bold">b</mi><mo>;</mo><mi class="ltx_nvar" mathvariant="bold">z</mi><mo>)</mo></mrow></mrow></math>: multivariate hypergeometric function</a>,
<a href="./5.2#iii" title="§5.2(iii) Pochhammer’s Symbol ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m125.png" altimg-height="24px" altimg-valign="-8px" altimg-width="41px" alttext="{\left(\NVar{a}\right)_{\NVar{n}}}" display="inline"><msub><mrow><mo href="./5.2#iii">(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r5">a</mi><mo href="./5.2#iii">)</mo></mrow><mi class="ltx_nvar" href="./5.1#p2.t1.r1">n</mi></msub></math>: Pochhammer’s symbol (or shifted factorial)</a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m93.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a> and
<a href="./19.19#p1" title="§19.19 Taylor and Related Series ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m79.png" altimg-height="23px" altimg-valign="-7px" altimg-width="78px" alttext="T_{N}(\mathbf{b},\mathbf{z})" display="inline"><mrow><msub><mi href="./19.19#p1">T</mi><mi href="./19.19#p1">N</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi mathvariant="bold">b</mi><mo>,</mo><mi mathvariant="bold">z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: polynomial</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m80.png" altimg-height="21px" altimg-valign="-5px" altimg-width="31px" alttext="T_{N}" display="inline"><msub><mi href="./19.19#p1">T</mi><mi>N</mi></msub></math> is defined by (). Also,</p>
<table id="E25" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.20.25</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E25.png" altimg-height="67px" altimg-valign="-30px" altimg-width="193px" alttext="R_{-c}\left(\mathbf{b};\mathbf{z}\right)=\prod_{j=1}^{n}z_{j}^{-b_{j}}," display="block"><mrow><mrow><mrow><msub><mi href="./19.16#E9">R</mi><mrow><mo>-</mo><mi>c</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi mathvariant="bold">b</mi><mo>;</mo><mi mathvariant="bold">z</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∏</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi href="./19.1#p1.t1.r1">n</mi></munderover><msubsup><mi>z</mi><mi>j</mi><mrow><mo>-</mo><msub><mi>b</mi><mi>j</mi></msub></mrow></msubsup></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E25.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.16#E9" title="(19.16.9) ‣ §19.16(ii) R - a ( b ; z ) ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="23px" altimg-valign="-7px" altimg-width="233px" alttext="R_{\NVar{-a}}\left(\NVar{b_{1}},\dots,\NVar{b_{n}};\NVar{z_{1}},\dots,\NVar{z_%
{n}}\right)" display="inline"><mrow><msub><mi href="./19.16#E9">R</mi><mrow><mo class="ltx_nvar">-</mo><mi class="ltx_nvar">a</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mrow><msub><mi class="ltx_nvar">b</mi><mn class="ltx_nvar">1</mn></msub><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><msub><mi class="ltx_nvar">b</mi><mi class="ltx_nvar" href="./19.1#p1.t1.r1">n</mi></msub></mrow><mo>;</mo><mrow><msub><mi class="ltx_nvar">z</mi><mn class="ltx_nvar">1</mn></msub><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><msub><mi class="ltx_nvar">z</mi><mi class="ltx_nvar" href="./19.1#p1.t1.r1">n</mi></msub></mrow><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m77.png" altimg-height="23px" altimg-valign="-7px" altimg-width="92px" alttext="R_{\NVar{-a}}\left(\NVar{\mathbf{b}};\NVar{\mathbf{z}}\right)" display="inline"><mrow><msub><mi href="./19.16#E9">R</mi><mrow><mo class="ltx_nvar">-</mo><mi class="ltx_nvar">a</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" mathvariant="bold">b</mi><mo>;</mo><mi class="ltx_nvar" mathvariant="bold">z</mi><mo>)</mo></mrow></mrow></math>: multivariate hypergeometric function</a> and
<a href="./19.1#p1.t1.r1" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m94.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./19.1#p1.t1.r1">n</mi></math>: nonnegative integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E26" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.20.26</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E26.png" altimg-height="67px" altimg-valign="-30px" altimg-width="320px" alttext="R_{-a}\left(\mathbf{b};\mathbf{z}\right)=\prod_{j=1}^{n}z_{j}^{-b_{j}}R_{-a^{%
\prime}}\left(\mathbf{b};\boldsymbol{{z^{-1}}}\right)," display="block"><mrow><mrow><mrow><msub><mi href="./19.16#E9">R</mi><mrow><mo>-</mo><mi>a</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi mathvariant="bold">b</mi><mo>;</mo><mi mathvariant="bold">z</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∏</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi href="./19.1#p1.t1.r1">n</mi></munderover><mrow><msubsup><mi>z</mi><mi>j</mi><mrow><mo>-</mo><msub><mi>b</mi><mi>j</mi></msub></mrow></msubsup><mo></mo><mrow><msub><mi href="./19.16#E9">R</mi><mrow><mo>-</mo><msup><mi>a</mi><mo>′</mo></msup></mrow></msub><mo></mo><mrow><mo>(</mo><mi mathvariant="bold">b</mi><mo>;</mo><msup><mi mathvariant="bold-italic">z</mi><mrow><mo mathvariant="bold">-</mo><mn mathvariant="bold">1</mn></mrow></msup><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m89.png" altimg-height="20px" altimg-valign="-4px" altimg-width="90px" alttext="a+a^{\prime}=c" display="inline"><mrow><mrow><mi>a</mi><mo>+</mo><msup><mi>a</mi><mo>′</mo></msup></mrow><mo>=</mo><mi>c</mi></mrow></math>, <math class="ltx_Math" altimg="m82.png" altimg-height="26px" altimg-valign="-7px" altimg-width="190px" alttext="\boldsymbol{{z^{-1}}}=(z_{1}^{-1},\dots,z_{n}^{-1})" display="inline"><mrow><msup><mi mathvariant="bold-italic">z</mi><mrow><mo mathvariant="bold">-</mo><mn mathvariant="bold">1</mn></mrow></msup><mo>=</mo><mrow><mo stretchy="false">(</mo><msubsup><mi>z</mi><mn>1</mn><mrow><mo>-</mo><mn>1</mn></mrow></msubsup><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><msubsup><mi>z</mi><mi href="./19.1#p1.t1.r1">n</mi><mrow><mo>-</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">)</mo></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E26.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.16#E9" title="(19.16.9) ‣ §19.16(ii) R - a ( b ; z ) ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="23px" altimg-valign="-7px" altimg-width="233px" alttext="R_{\NVar{-a}}\left(\NVar{b_{1}},\dots,\NVar{b_{n}};\NVar{z_{1}},\dots,\NVar{z_%
{n}}\right)" display="inline"><mrow><msub><mi href="./19.16#E9">R</mi><mrow><mo class="ltx_nvar">-</mo><mi class="ltx_nvar">a</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mrow><msub><mi class="ltx_nvar">b</mi><mn class="ltx_nvar">1</mn></msub><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><msub><mi class="ltx_nvar">b</mi><mi class="ltx_nvar" href="./19.1#p1.t1.r1">n</mi></msub></mrow><mo>;</mo><mrow><msub><mi class="ltx_nvar">z</mi><mn class="ltx_nvar">1</mn></msub><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><msub><mi class="ltx_nvar">z</mi><mi class="ltx_nvar" href="./19.1#p1.t1.r1">n</mi></msub></mrow><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m77.png" altimg-height="23px" altimg-valign="-7px" altimg-width="92px" alttext="R_{\NVar{-a}}\left(\NVar{\mathbf{b}};\NVar{\mathbf{z}}\right)" display="inline"><mrow><msub><mi href="./19.16#E9">R</mi><mrow><mo class="ltx_nvar">-</mo><mi class="ltx_nvar">a</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" mathvariant="bold">b</mi><mo>;</mo><mi class="ltx_nvar" mathvariant="bold">z</mi><mo>)</mo></mrow></mrow></math>: multivariate hypergeometric function</a> and
<a href="./19.1#p1.t1.r1" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m94.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./19.1#p1.t1.r1">n</mi></math>: nonnegative integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">See also (</div>
</div>
</body></text>
</html>
</page>
<page>
<!DOCTYPE html><html>
<head>
<title>DLMF: 19.24 Inequalities</title>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<!--GOOGLE BOOTSTRAP--></head>
<text><body class="color_default textfont_default titlefont_default fontsize_default navbar_default">
<div class="ltx_page_navbar">
<div class="ltx_page_navlogo"></dd>
</dl>
</div>
</div>
<div id="SS1.p1" class="ltx_para">
<p class="ltx_p">The condition <math class="ltx_Math" altimg="m55.png" altimg-height="20px" altimg-valign="-6px" altimg-width="51px" alttext="y\leq z" display="inline"><mrow><mi>y</mi><mo>≤</mo><mi>z</mi></mrow></math> for () serves
only to identify <math class="ltx_Math" altimg="m54.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi>y</mi></math> as the smaller of the two nonzero variables of a symmetric
function; it does not restrict validity.</p>
<table id="E1" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.24.1</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E1.png" altimg-height="32px" altimg-valign="-9px" altimg-width="336px" alttext="\ln 4\leq\sqrt{z}R_{F}\left(0,y,z\right)+\ln\sqrt{y/z}\leq\tfrac{1}{2}\pi," display="block"><mrow><mrow><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mn>4</mn></mrow><mo>≤</mo><mrow><mrow><msqrt><mi>z</mi></msqrt><mo></mo><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><msqrt><mrow><mi>y</mi><mo>/</mo><mi>z</mi></mrow></msqrt></mrow></mrow><mo>≤</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m16.png" altimg-height="20px" altimg-valign="-6px" altimg-width="88px" alttext="0<y\leq z" display="inline"><mrow><mn>0</mn><mo><</mo><mi>y</mi><mo>≤</mo><mi>z</mi></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.16#E1" title="(19.16.1) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m25.png" altimg-height="23px" altimg-valign="-7px" altimg-width="101px" alttext="R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: symmetric elliptic integral of first kind</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a> and
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m33.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E2" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.24.2</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E2.png" altimg-height="31px" altimg-valign="-9px" altimg-width="246px" alttext="\tfrac{1}{2}\leq z^{-1/2}R_{G}\left(0,y,z\right)\leq\tfrac{1}{4}\pi," display="block"><mrow><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo>≤</mo><mrow><msup><mi>z</mi><mrow><mo>-</mo><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></mrow></msup><mo></mo><mrow><msub><mi href="./19.16#E3">R</mi><mi href="./19.16#E3">G</mi></msub><mo></mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow><mo>≤</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>4</mn></mfrac></mstyle><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m17.png" altimg-height="20px" altimg-valign="-6px" altimg-width="88px" alttext="0\leq y\leq z" display="inline"><mrow><mn>0</mn><mo>≤</mo><mi>y</mi><mo>≤</mo><mi>z</mi></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.16#E3" title="(19.16.3) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="23px" altimg-valign="-7px" altimg-width="101px" alttext="R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E3">R</mi><mi href="./19.16#E3">G</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: symmetric elliptic integral of second kind</a> and
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E3" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.24.3</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E3.png" altimg-height="61px" altimg-valign="-21px" altimg-width="464px" alttext="\left(\frac{y^{3/2}+z^{3/2}}{2}\right)^{2/3}\leq\frac{4}{\pi}R_{G}\left(0,y^{2%
},z^{2}\right)\leq\left(\frac{y^{2}+z^{2}}{2}\right)^{1/2}," display="block"><mrow><mrow><msup><mrow><mo>(</mo><mfrac><mrow><msup><mi>y</mi><mrow><mn>3</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>+</mo><msup><mi>z</mi><mrow><mn>3</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow><mn>2</mn></mfrac><mo>)</mo></mrow><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>≤</mo><mrow><mfrac><mn>4</mn><mi href="./3.12#E1">π</mi></mfrac><mo></mo><mrow><msub><mi href="./19.16#E3">R</mi><mi href="./19.16#E3">G</mi></msub><mo></mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><msup><mi>y</mi><mn>2</mn></msup><mo>,</mo><msup><mi>z</mi><mn>2</mn></msup><mo>)</mo></mrow></mrow></mrow><mo>≤</mo><msup><mrow><mo>(</mo><mfrac><mrow><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><msup><mi>z</mi><mn>2</mn></msup></mrow><mn>2</mn></mfrac><mo>)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m53.png" altimg-height="20px" altimg-valign="-6px" altimg-width="51px" alttext="y>0" display="inline"><mrow><mi>y</mi><mo>></mo><mn>0</mn></mrow></math>, <math class="ltx_Math" altimg="m56.png" altimg-height="17px" altimg-valign="-3px" altimg-width="51px" alttext="z>0" display="inline"><mrow><mi>z</mi><mo>></mo><mn>0</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.16#E3" title="(19.16.3) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="23px" altimg-valign="-7px" altimg-width="101px" alttext="R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E3">R</mi><mi href="./19.16#E3">G</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: symmetric elliptic integral of second kind</a> and
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS1.p2" class="ltx_para">
<p class="ltx_p">If <math class="ltx_Math" altimg="m54.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi>y</mi></math>, <math class="ltx_Math" altimg="m57.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi>z</mi></math>, and <math class="ltx_Math" altimg="m48.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="p" display="inline"><mi>p</mi></math> are positive, then</p>
<table id="E4" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.24.4</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E4.png" altimg-height="52px" altimg-valign="-22px" altimg-width="501px" alttext="\frac{2}{\sqrt{p}}(2yz+yp+zp)^{-1/2}\leq\frac{4}{3\pi}R_{J}\left(0,y,z,p\right%
)\leq(yzp^{2})^{-3/8}." display="block"><mrow><mrow><mrow><mfrac><mn>2</mn><msqrt><mi>p</mi></msqrt></mfrac><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>2</mn><mo></mo><mi>y</mi><mo></mo><mi>z</mi></mrow><mo>+</mo><mrow><mi>y</mi><mo></mo><mi>p</mi></mrow><mo>+</mo><mrow><mi>z</mi><mo></mo><mi>p</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></mrow></msup></mrow><mo>≤</mo><mrow><mfrac><mn>4</mn><mrow><mn>3</mn><mo></mo><mi href="./3.12#E1">π</mi></mrow></mfrac><mo></mo><mrow><msub><mi href="./19.16#E2">R</mi><mi href="./19.16#E2">J</mi></msub><mo></mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>,</mo><mi>p</mi><mo>)</mo></mrow></mrow></mrow><mo>≤</mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi>y</mi><mo></mo><mi>z</mi><mo></mo><msup><mi>p</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mrow><mn>3</mn><mo>/</mo><mn>8</mn></mrow></mrow></msup></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E4.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.16#E2" title="(19.16.2) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m30.png" altimg-height="23px" altimg-valign="-7px" altimg-width="118px" alttext="R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)" display="inline"><mrow><msub><mi href="./19.16#E2">R</mi><mi href="./19.16#E2">J</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>,</mo><mi class="ltx_nvar">p</mi><mo>)</mo></mrow></mrow></math>: symmetric elliptic integral of third kind</a> and
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Inequalities for <math class="ltx_Math" altimg="m22.png" altimg-height="23px" altimg-valign="-7px" altimg-width="101px" alttext="R_{D}\left(0,y,z\right)" display="inline"><mrow><msub><mi href="./19.16#E5">R</mi><mi href="./19.16#E5">D</mi></msub><mo></mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow></mrow></math> are included as the case <math class="ltx_Math" altimg="m47.png" altimg-height="16px" altimg-valign="-6px" altimg-width="51px" alttext="p=z" display="inline"><mrow><mi>p</mi><mo>=</mo><mi>z</mi></mrow></math>.</p>
</div>
<div id="SS1.p3" class="ltx_para">
<p class="ltx_p">A series of successively sharper inequalities is obtained from the AGM process
(§) with <math class="ltx_Math" altimg="m41.png" altimg-height="20px" altimg-valign="-6px" altimg-width="105px" alttext="a_{0}\geq g_{0}>0" display="inline"><mrow><msub><mi href="./19.8#SS1.p1">a</mi><mn>0</mn></msub><mo>≥</mo><msub><mi href="./19.8#SS1.p1">g</mi><mn>0</mn></msub><mo>></mo><mn>0</mn></mrow></math>:</p>
<table id="E5" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.24.5</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E5.png" altimg-height="49px" altimg-valign="-20px" altimg-width="249px" alttext="\frac{1}{a_{n}}\leq\frac{2}{\pi}R_{F}\left(0,a_{0}^{2},g_{0}^{2}\right)\leq%
\frac{1}{g_{n}}," display="block"><mrow><mrow><mfrac><mn>1</mn><msub><mi href="./19.8#SS1.p1">a</mi><mi href="./19.1#p1.t1.r1">n</mi></msub></mfrac><mo>≤</mo><mrow><mfrac><mn>2</mn><mi href="./3.12#E1">π</mi></mfrac><mo></mo><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><msubsup><mi href="./19.8#SS1.p1">a</mi><mn>0</mn><mn>2</mn></msubsup><mo>,</mo><msubsup><mi href="./19.8#SS1.p1">g</mi><mn>0</mn><mn>2</mn></msubsup><mo>)</mo></mrow></mrow></mrow><mo>≤</mo><mfrac><mn>1</mn><msub><mi href="./19.8#SS1.p1">g</mi><mi href="./19.1#p1.t1.r1">n</mi></msub></mfrac></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m45.png" altimg-height="20px" altimg-valign="-6px" altimg-width="123px" alttext="n=0,1,2,\dots" display="inline"><mrow><mi href="./19.1#p1.t1.r1">n</mi><mo>=</mo><mrow><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E5.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.16#E1" title="(19.16.1) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m25.png" altimg-height="23px" altimg-valign="-7px" altimg-width="101px" alttext="R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: symmetric elliptic integral of first kind</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./19.1#p1.t1.r1" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m46.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./19.1#p1.t1.r1">n</mi></math>: nonnegative integer</a>,
<a href="./19.8#SS1.p1" title="§19.8(i) Gauss’s Arithmetic-Geometric Mean (AGM) ‣ §19.8 Quadratic Transformations ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m42.png" altimg-height="16px" altimg-valign="-5px" altimg-width="26px" alttext="a_{n}" display="inline"><msub><mi href="./19.8#SS1.p1">a</mi><mi href="./19.1#p1.t1.r1">n</mi></msub></math>: iterate</a> and
<a href="./19.8#SS1.p1" title="§19.8(i) Gauss’s Arithmetic-Geometric Mean (AGM) ‣ §19.8 Quadratic Transformations ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="16px" altimg-valign="-6px" altimg-width="25px" alttext="g_{n}" display="inline"><msub><mi href="./19.8#SS1.p1">g</mi><mi href="./19.1#p1.t1.r1">n</mi></msub></math>: iterate</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where</p>
<table id="E6" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex1" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">19.24.6</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m14.png" altimg-height="19px" altimg-valign="-7px" altimg-width="48px" alttext="\displaystyle a_{n+1}" display="inline"><msub><mi href="./19.8#SS1.p1">a</mi><mrow><mi href="./19.1#p1.t1.r1">n</mi><mo>+</mo><mn>1</mn></mrow></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m1.png" altimg-height="25px" altimg-valign="-7px" altimg-width="134px" alttext="\displaystyle=(a_{n}+g_{n})/2," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><msub><mi href="./19.8#SS1.p1">a</mi><mi href="./19.1#p1.t1.r1">n</mi></msub><mo>+</mo><msub><mi href="./19.8#SS1.p1">g</mi><mi href="./19.1#p1.t1.r1">n</mi></msub></mrow><mo stretchy="false">)</mo></mrow><mo>/</mo><mn>2</mn></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex2" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m15.png" altimg-height="19px" altimg-valign="-7px" altimg-width="47px" alttext="\displaystyle g_{n+1}" display="inline"><msub><mi href="./19.8#SS1.p1">g</mi><mrow><mi href="./19.1#p1.t1.r1">n</mi><mo>+</mo><mn>1</mn></mrow></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m2.png" altimg-height="26px" altimg-valign="-8px" altimg-width="91px" alttext="\displaystyle=\sqrt{a_{n}g_{n}}." display="inline"><mrow><mrow><mi></mi><mo>=</mo><msqrt><mrow><msub><mi href="./19.8#SS1.p1">a</mi><mi href="./19.1#p1.t1.r1">n</mi></msub><mo></mo><msub><mi href="./19.8#SS1.p1">g</mi><mi href="./19.1#p1.t1.r1">n</mi></msub></mrow></msqrt></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E6.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.1#p1.t1.r1" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m46.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./19.1#p1.t1.r1">n</mi></math>: nonnegative integer</a>,
<a href="./19.8#SS1.p1" title="§19.8(i) Gauss’s Arithmetic-Geometric Mean (AGM) ‣ §19.8 Quadratic Transformations ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m42.png" altimg-height="16px" altimg-valign="-5px" altimg-width="26px" alttext="a_{n}" display="inline"><msub><mi href="./19.8#SS1.p1">a</mi><mi href="./19.1#p1.t1.r1">n</mi></msub></math>: iterate</a> and
<a href="./19.8#SS1.p1" title="§19.8(i) Gauss’s Arithmetic-Geometric Mean (AGM) ‣ §19.8 Quadratic Transformations ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="16px" altimg-valign="-6px" altimg-width="25px" alttext="g_{n}" display="inline"><msub><mi href="./19.8#SS1.p1">g</mi><mi href="./19.1#p1.t1.r1">n</mi></msub></math>: iterate</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS1.p4" class="ltx_para">
<p class="ltx_p">Other inequalities can be obtained by applying
<cite class="ltx_cite ltx_citemacro_citet">Carlson ().
Approximations and one-sided inequalities for
<math class="ltx_Math" altimg="m28.png" altimg-height="23px" altimg-valign="-7px" altimg-width="100px" alttext="R_{G}\left(0,y,z\right)" display="inline"><mrow><msub><mi href="./19.16#E3">R</mi><mi href="./19.16#E3">G</mi></msub><mo></mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow></mrow></math> follow from those given in § for
the length <math class="ltx_Math" altimg="m18.png" altimg-height="23px" altimg-valign="-7px" altimg-width="61px" alttext="L(a,b)" display="inline"><mrow><mi href="./19.24#SS1.p4">L</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></mrow></math> of an ellipse with semiaxes <math class="ltx_Math" altimg="m39.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi>a</mi></math> and <math class="ltx_Math" altimg="m43.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="b" display="inline"><mi>b</mi></math>, since
</p>
<table id="E7" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.24.7</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E7.png" altimg-height="30px" altimg-valign="-9px" altimg-width="220px" alttext="L(a,b)=8\!R_{G}\left(0,a^{2},b^{2}\right)." display="block"><mrow><mrow><mrow><mi href="./19.24#SS1.p4">L</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mpadded width="-1.7pt"><mn>8</mn></mpadded><mo></mo><mrow><msub><mi href="./19.16#E3">R</mi><mi href="./19.16#E3">G</mi></msub><mo></mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><msup><mi>a</mi><mn>2</mn></msup><mo>,</mo><msup><mi>b</mi><mn>2</mn></msup><mo>)</mo></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E7.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.16#E3" title="(19.16.3) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="23px" altimg-valign="-7px" altimg-width="101px" alttext="R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E3">R</mi><mi href="./19.16#E3">G</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: symmetric elliptic integral of second kind</a> and
<a href="./19.24#SS1.p4" title="§19.24(i) Complete Integrals ‣ §19.24 Inequalities ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m18.png" altimg-height="23px" altimg-valign="-7px" altimg-width="61px" alttext="L(a,b)" display="inline"><mrow><mi href="./19.24#SS1.p4">L</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></mrow></math>: length</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS1.p5" class="ltx_para">
<p class="ltx_p">For <math class="ltx_Math" altimg="m49.png" altimg-height="17px" altimg-valign="-3px" altimg-width="52px" alttext="x>0" display="inline"><mrow><mi>x</mi><mo>></mo><mn>0</mn></mrow></math>, <math class="ltx_Math" altimg="m53.png" altimg-height="20px" altimg-valign="-6px" altimg-width="51px" alttext="y>0" display="inline"><mrow><mi>y</mi><mo>></mo><mn>0</mn></mrow></math>, and <math class="ltx_Math" altimg="m51.png" altimg-height="21px" altimg-valign="-6px" altimg-width="53px" alttext="x\neq y" display="inline"><mrow><mi>x</mi><mo>≠</mo><mi>y</mi></mrow></math>, the complete cases of <math class="ltx_Math" altimg="m24.png" altimg-height="21px" altimg-valign="-5px" altimg-width="33px" alttext="R_{F}" display="inline"><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub></math> and
<math class="ltx_Math" altimg="m27.png" altimg-height="21px" altimg-valign="-5px" altimg-width="33px" alttext="R_{G}" display="inline"><msub><mi href="./19.16#E3">R</mi><mi href="./19.16#E3">G</mi></msub></math> satisfy
</p>
<table id="E8" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex3" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">19.24.8</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m9.png" altimg-height="25px" altimg-valign="-7px" altimg-width="203px" alttext="\displaystyle R_{F}\left(x,y,0\right)R_{G}\left(x,y,0\right)" display="inline"><mrow><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mn>0</mn><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./19.16#E3">R</mi><mi href="./19.16#E3">G</mi></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mn>0</mn><mo>)</mo></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m7.png" altimg-height="30px" altimg-valign="-9px" altimg-width="66px" alttext="\displaystyle>\tfrac{1}{8}\pi^{2}," display="inline"><mrow><mrow><mi></mi><mo>></mo><mrow><mfrac><mn>1</mn><mn>8</mn></mfrac><mo></mo><msup><mi href="./3.12#E1">π</mi><mn>2</mn></msup></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex4" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m8.png" altimg-height="25px" altimg-valign="-7px" altimg-width="231px" alttext="\displaystyle R_{F}\left(x,y,0\right)+2\!R_{G}\left(x,y,0\right)" display="inline"><mrow><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mn>0</mn><mo>)</mo></mrow></mrow><mo>+</mo><mrow><mpadded width="-1.7pt"><mn>2</mn></mpadded><mo></mo><mrow><msub><mi href="./19.16#E3">R</mi><mi href="./19.16#E3">G</mi></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mn>0</mn><mo>)</mo></mrow></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m6.png" altimg-height="17px" altimg-valign="-3px" altimg-width="45px" alttext="\displaystyle>\pi." display="inline"><mrow><mrow><mi></mi><mo>></mo><mi href="./3.12#E1">π</mi></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E8.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.16#E1" title="(19.16.1) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m25.png" altimg-height="23px" altimg-valign="-7px" altimg-width="101px" alttext="R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: symmetric elliptic integral of first kind</a>,
<a href="./19.16#E3" title="(19.16.3) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="23px" altimg-valign="-7px" altimg-width="101px" alttext="R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E3">R</mi><mi href="./19.16#E3">G</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: symmetric elliptic integral of second kind</a> and
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Also, with the notation of (),</p>
<table id="E9" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.24.9</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E9.png" altimg-height="56px" altimg-valign="-21px" altimg-width="261px" alttext="\frac{1}{2}\,g_{1}^{2}\leq\frac{R_{G}\left(a_{0}^{2},g_{0}^{2},0\right)}{R_{F}%
\left(a_{0}^{2},g_{0}^{2},0\right)}\leq\frac{1}{2}\,a_{1}^{2}," display="block"><mrow><mrow><mrow><mpadded width="+1.7pt"><mfrac><mn>1</mn><mn>2</mn></mfrac></mpadded><mo></mo><msubsup><mi href="./19.8#SS1.p1">g</mi><mn>1</mn><mn>2</mn></msubsup></mrow><mo>≤</mo><mfrac><mrow><msub><mi href="./19.16#E3">R</mi><mi href="./19.16#E3">G</mi></msub><mo></mo><mrow><mo>(</mo><msubsup><mi href="./19.8#SS1.p1">a</mi><mn>0</mn><mn>2</mn></msubsup><mo>,</mo><msubsup><mi href="./19.8#SS1.p1">g</mi><mn>0</mn><mn>2</mn></msubsup><mo>,</mo><mn>0</mn><mo>)</mo></mrow></mrow><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><msubsup><mi href="./19.8#SS1.p1">a</mi><mn>0</mn><mn>2</mn></msubsup><mo>,</mo><msubsup><mi href="./19.8#SS1.p1">g</mi><mn>0</mn><mn>2</mn></msubsup><mo>,</mo><mn>0</mn><mo>)</mo></mrow></mrow></mfrac><mo>≤</mo><mrow><mpadded width="+1.7pt"><mfrac><mn>1</mn><mn>2</mn></mfrac></mpadded><mo></mo><msubsup><mi href="./19.8#SS1.p1">a</mi><mn>1</mn><mn>2</mn></msubsup></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E9.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.16#E1" title="(19.16.1) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m25.png" altimg-height="23px" altimg-valign="-7px" altimg-width="101px" alttext="R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: symmetric elliptic integral of first kind</a>,
<a href="./19.16#E3" title="(19.16.3) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="23px" altimg-valign="-7px" altimg-width="101px" alttext="R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E3">R</mi><mi href="./19.16#E3">G</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: symmetric elliptic integral of second kind</a>,
<a href="./19.8#SS1.p1" title="§19.8(i) Gauss’s Arithmetic-Geometric Mean (AGM) ‣ §19.8 Quadratic Transformations ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m42.png" altimg-height="16px" altimg-valign="-5px" altimg-width="26px" alttext="a_{n}" display="inline"><msub><mi href="./19.8#SS1.p1">a</mi><mi href="./19.1#p1.t1.r1">n</mi></msub></math>: iterate</a> and
<a href="./19.8#SS1.p1" title="§19.8(i) Gauss’s Arithmetic-Geometric Mean (AGM) ‣ §19.8 Quadratic Transformations ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="16px" altimg-valign="-6px" altimg-width="25px" alttext="g_{n}" display="inline"><msub><mi href="./19.8#SS1.p1">g</mi><mi href="./19.1#p1.t1.r1">n</mi></msub></math>: iterate</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">with equality iff <math class="ltx_Math" altimg="m40.png" altimg-height="16px" altimg-valign="-6px" altimg-width="69px" alttext="a_{0}=g_{0}" display="inline"><mrow><msub><mi href="./19.8#SS1.p1">a</mi><mn>0</mn></msub><mo>=</mo><msub><mi href="./19.8#SS1.p1">g</mi><mn>0</mn></msub></mrow></math>.
</p>
</div>
</section>
<section id="ii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§19.24(ii) </span>Incomplete Integrals</h2>
<div id="SS2.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="SS2.p1" class="ltx_para">
<p class="ltx_p">Inequalities for <math class="ltx_Math" altimg="m19.png" altimg-height="23px" altimg-valign="-7px" altimg-width="92px" alttext="R_{-a}\left(\mathbf{b};\mathbf{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E9">R</mi><mrow><mo>-</mo><mi>a</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi mathvariant="bold">b</mi><mo>;</mo><mi mathvariant="bold">z</mi><mo>)</mo></mrow></mrow></math> in
<cite class="ltx_cite ltx_citemacro_citet">Carlson (</dd>
</dl>
</div>
</div>
<div id="Px1.p1" class="ltx_para">
<table id="E10" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.24.10</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E10.png" altimg-height="52px" altimg-valign="-22px" altimg-width="374px" alttext="\frac{3}{\sqrt{x}+\sqrt{y}+\sqrt{z}}\leq R_{F}\left(x,y,z\right)\leq\frac{1}{(%
xyz)^{1/6}}," display="block"><mrow><mrow><mfrac><mn>3</mn><mrow><msqrt><mi>x</mi></msqrt><mo>+</mo><msqrt><mi>y</mi></msqrt><mo>+</mo><msqrt><mi>z</mi></msqrt></mrow></mfrac><mo>≤</mo><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow></mrow><mo>≤</mo><mfrac><mn>1</mn><msup><mrow><mo stretchy="false">(</mo><mrow><mi>x</mi><mo></mo><mi>y</mi><mo></mo><mi>z</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>6</mn></mrow></msup></mfrac></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E10.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./19.16#E1" title="(19.16.1) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m25.png" altimg-height="23px" altimg-valign="-7px" altimg-width="101px" alttext="R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: symmetric elliptic integral of first kind</a></dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E11" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.24.11</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E11.png" altimg-height="59px" altimg-valign="-22px" altimg-width="519px" alttext="\left(\frac{5}{\sqrt{x}+\sqrt{y}+\sqrt{z}+2\sqrt{p}}\right)^{3}\leq R_{J}\left%
(x,y,z,p\right)\leq(xyzp^{2})^{-3/10}," display="block"><mrow><mrow><msup><mrow><mo>(</mo><mfrac><mn>5</mn><mrow><msqrt><mi>x</mi></msqrt><mo>+</mo><msqrt><mi>y</mi></msqrt><mo>+</mo><msqrt><mi>z</mi></msqrt><mo>+</mo><mrow><mn>2</mn><mo></mo><msqrt><mi>p</mi></msqrt></mrow></mrow></mfrac><mo>)</mo></mrow><mn>3</mn></msup><mo>≤</mo><mrow><msub><mi href="./19.16#E2">R</mi><mi href="./19.16#E2">J</mi></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>,</mo><mi>p</mi><mo>)</mo></mrow></mrow><mo>≤</mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi>x</mi><mo></mo><mi>y</mi><mo></mo><mi>z</mi><mo></mo><msup><mi>p</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mrow><mn>3</mn><mo>/</mo><mn>10</mn></mrow></mrow></msup></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E11.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./19.16#E2" title="(19.16.2) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m30.png" altimg-height="23px" altimg-valign="-7px" altimg-width="118px" alttext="R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)" display="inline"><mrow><msub><mi href="./19.16#E2">R</mi><mi href="./19.16#E2">J</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>,</mo><mi class="ltx_nvar">p</mi><mo>)</mo></mrow></mrow></math>: symmetric elliptic integral of third kind</a></dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E12" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.24.12</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E12.png" altimg-height="65px" altimg-valign="-27px" altimg-width="619px" alttext="\tfrac{1}{3}(\sqrt{x}+\sqrt{y}+\sqrt{z})\leq R_{G}\left(x,y,z\right)\leq\min%
\left(\sqrt{\frac{x+y+z}{3}},\frac{x^{2}+y^{2}+z^{2}}{3\sqrt{xyz}}\right)." display="block"><mrow><mrow><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>3</mn></mfrac></mstyle><mo></mo><mrow><mo stretchy="false">(</mo><mrow><msqrt><mi>x</mi></msqrt><mo>+</mo><msqrt><mi>y</mi></msqrt><mo>+</mo><msqrt><mi>z</mi></msqrt></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>≤</mo><mrow><msub><mi href="./19.16#E3">R</mi><mi href="./19.16#E3">G</mi></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow></mrow><mo>≤</mo><mrow><mi>min</mi><mo></mo><mrow><mo>(</mo><msqrt><mfrac><mrow><mi>x</mi><mo>+</mo><mi>y</mi><mo>+</mo><mi>z</mi></mrow><mn>3</mn></mfrac></msqrt><mo>,</mo><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><msup><mi>z</mi><mn>2</mn></msup></mrow><mrow><mn>3</mn><mo></mo><msqrt><mrow><mi>x</mi><mo></mo><mi>y</mi><mo></mo><mi>z</mi></mrow></msqrt></mrow></mfrac><mo>)</mo></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E12.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./19.16#E3" title="(19.16.3) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="23px" altimg-valign="-7px" altimg-width="101px" alttext="R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E3">R</mi><mi href="./19.16#E3">G</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: symmetric elliptic integral of second kind</a></dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Inequalities for <math class="ltx_Math" altimg="m21.png" altimg-height="23px" altimg-valign="-7px" altimg-width="82px" alttext="R_{C}\left(x,y\right)" display="inline"><mrow><msub><mi href="./19.2#E17">R</mi><mi href="./19.2#E17">C</mi></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></mrow></math> and <math class="ltx_Math" altimg="m23.png" altimg-height="23px" altimg-valign="-7px" altimg-width="102px" alttext="R_{D}\left(x,y,z\right)" display="inline"><mrow><msub><mi href="./19.16#E5">R</mi><mi href="./19.16#E5">D</mi></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow></mrow></math> are included as
special cases (see ()).</p>
</div>
<div id="Px1.p2" class="ltx_para">
<p class="ltx_p">Other inequalities for <math class="ltx_Math" altimg="m26.png" altimg-height="23px" altimg-valign="-7px" altimg-width="101px" alttext="R_{F}\left(x,y,z\right)" display="inline"><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow></mrow></math> are given in
<cite class="ltx_cite ltx_citemacro_citet">Carlson ()</cite>.</p>
</div>
<div id="Px1.p3" class="ltx_para">
<p class="ltx_p">If <math class="ltx_Math" altimg="m39.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi>a</mi></math> (<math class="ltx_Math" altimg="m36.png" altimg-height="21px" altimg-valign="-6px" altimg-width="35px" alttext="\neq 0" display="inline"><mrow><mi></mi><mo>≠</mo><mn>0</mn></mrow></math>) is real, all components of <math class="ltx_Math" altimg="m34.png" altimg-height="18px" altimg-valign="-2px" altimg-width="17px" alttext="\mathbf{b}" display="inline"><mi mathvariant="bold">b</mi></math> and <math class="ltx_Math" altimg="m35.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="\mathbf{z}" display="inline"><mi mathvariant="bold">z</mi></math> are
positive, and the components of <math class="ltx_Math" altimg="m57.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi>z</mi></math> are not all equal, then</p>
<table id="E13" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex5" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">19.24.13</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m13.png" altimg-height="25px" altimg-valign="-7px" altimg-width="173px" alttext="\displaystyle R_{a}\left(\mathbf{b};\mathbf{z}\right)R_{-a}\left(\mathbf{b};%
\mathbf{z}\right)" display="inline"><mrow><mrow><msub><mi href="./19.16#E9">R</mi><mi>a</mi></msub><mo></mo><mrow><mo>(</mo><mi mathvariant="bold">b</mi><mo>;</mo><mi mathvariant="bold">z</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./19.16#E9">R</mi><mrow><mo>-</mo><mi>a</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi mathvariant="bold">b</mi><mo>;</mo><mi mathvariant="bold">z</mi><mo>)</mo></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m3.png" altimg-height="22px" altimg-valign="-6px" altimg-width="43px" alttext="\displaystyle>1," display="inline"><mrow><mrow><mi></mi><mo>></mo><mn>1</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex6" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m12.png" altimg-height="25px" altimg-valign="-7px" altimg-width="194px" alttext="\displaystyle R_{a}\left(\mathbf{b};\mathbf{z}\right)+R_{-a}\left(\mathbf{b};%
\mathbf{z}\right)" display="inline"><mrow><mrow><msub><mi href="./19.16#E9">R</mi><mi>a</mi></msub><mo></mo><mrow><mo>(</mo><mi mathvariant="bold">b</mi><mo>;</mo><mi mathvariant="bold">z</mi><mo>)</mo></mrow></mrow><mo>+</mo><mrow><msub><mi href="./19.16#E9">R</mi><mrow><mo>-</mo><mi>a</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi mathvariant="bold">b</mi><mo>;</mo><mi mathvariant="bold">z</mi><mo>)</mo></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m5.png" altimg-height="22px" altimg-valign="-6px" altimg-width="43px" alttext="\displaystyle>2;" display="inline"><mrow><mrow><mi></mi><mo>></mo><mn>2</mn></mrow><mo>;</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E13.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./19.16#E9" title="(19.16.9) ‣ §19.16(ii) R - a ( b ; z ) ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m32.png" altimg-height="23px" altimg-valign="-7px" altimg-width="233px" alttext="R_{\NVar{-a}}\left(\NVar{b_{1}},\dots,\NVar{b_{n}};\NVar{z_{1}},\dots,\NVar{z_%
{n}}\right)" display="inline"><mrow><msub><mi href="./19.16#E9">R</mi><mrow><mo class="ltx_nvar">-</mo><mi class="ltx_nvar">a</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mrow><msub><mi class="ltx_nvar">b</mi><mn class="ltx_nvar">1</mn></msub><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><msub><mi class="ltx_nvar">b</mi><mi class="ltx_nvar" href="./19.1#p1.t1.r1">n</mi></msub></mrow><mo>;</mo><mrow><msub><mi class="ltx_nvar">z</mi><mn class="ltx_nvar">1</mn></msub><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><msub><mi class="ltx_nvar">z</mi><mi class="ltx_nvar" href="./19.1#p1.t1.r1">n</mi></msub></mrow><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m31.png" altimg-height="23px" altimg-valign="-7px" altimg-width="92px" alttext="R_{\NVar{-a}}\left(\NVar{\mathbf{b}};\NVar{\mathbf{z}}\right)" display="inline"><mrow><msub><mi href="./19.16#E9">R</mi><mrow><mo class="ltx_nvar">-</mo><mi class="ltx_nvar">a</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" mathvariant="bold">b</mi><mo>;</mo><mi class="ltx_nvar" mathvariant="bold">z</mi><mo>)</mo></mrow></mrow></math>: multivariate hypergeometric function</a></dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">see <cite class="ltx_cite ltx_citemacro_citet">Neuman (, (2.13))</cite>. Special cases with <math class="ltx_Math" altimg="m38.png" altimg-height="27px" altimg-valign="-9px" altimg-width="70px" alttext="a=\pm\frac{1}{2}" display="inline"><mrow><mi>a</mi><mo>=</mo><mrow><mo>±</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow></math>
are ()),
and</p>
<table id="E14" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex7" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">19.24.14</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m11.png" altimg-height="25px" altimg-valign="-7px" altimg-width="203px" alttext="\displaystyle R_{F}\left(x,y,z\right)R_{G}\left(x,y,z\right)" display="inline"><mrow><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./19.16#E3">R</mi><mi href="./19.16#E3">G</mi></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m3.png" altimg-height="22px" altimg-valign="-6px" altimg-width="43px" alttext="\displaystyle>1," display="inline"><mrow><mrow><mi></mi><mo>></mo><mn>1</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex8" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m10.png" altimg-height="25px" altimg-valign="-7px" altimg-width="224px" alttext="\displaystyle R_{F}\left(x,y,z\right)+R_{G}\left(x,y,z\right)" display="inline"><mrow><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow></mrow><mo>+</mo><mrow><msub><mi href="./19.16#E3">R</mi><mi href="./19.16#E3">G</mi></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m4.png" altimg-height="19px" altimg-valign="-3px" altimg-width="43px" alttext="\displaystyle>2." display="inline"><mrow><mrow><mi></mi><mo>></mo><mn>2</mn></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E14.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.16#E1" title="(19.16.1) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m25.png" altimg-height="23px" altimg-valign="-7px" altimg-width="101px" alttext="R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: symmetric elliptic integral of first kind</a> and
<a href="./19.16#E3" title="(19.16.3) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="23px" altimg-valign="-7px" altimg-width="101px" alttext="R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E3">R</mi><mi href="./19.16#E3">G</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: symmetric elliptic integral of second kind</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">The same reference also gives upper and lower bounds for symmetric integrals
in terms of their elementary degenerate cases. These bounds include a sharper
but more complicated lower bound than that supplied in the next result:</p>
<table id="E15" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.24.15</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E15.png" altimg-height="29px" altimg-valign="-9px" altimg-width="413px" alttext="R_{C}\left(x,\tfrac{1}{2}(y+z)\right)\leq R_{F}\left(x,y,z\right)\leq R_{C}%
\left(x,\sqrt{yz}\right)," display="block"><mrow><mrow><mrow><msub><mi href="./19.2#E17">R</mi><mi href="./19.2#E17">C</mi></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>y</mi><mo>+</mo><mi>z</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>)</mo></mrow></mrow><mo>≤</mo><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow></mrow><mo>≤</mo><mrow><msub><mi href="./19.2#E17">R</mi><mi href="./19.2#E17">C</mi></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><msqrt><mrow><mi>y</mi><mo></mo><mi>z</mi></mrow></msqrt><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m50.png" altimg-height="19px" altimg-valign="-5px" altimg-width="52px" alttext="x\geq 0" display="inline"><mrow><mi>x</mi><mo>≥</mo><mn>0</mn></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E15.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.2#E17" title="(19.2.17) ‣ §19.2(iv) A Related Function: R C ( x , y ) ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="23px" altimg-valign="-7px" altimg-width="82px" alttext="R_{C}\left(\NVar{x},\NVar{y}\right)" display="inline"><mrow><msub><mi href="./19.2#E17">R</mi><mi href="./19.2#E17">C</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>)</mo></mrow></mrow></math>: Carlson’s combination of inverse circular and inverse hyperbolic functions</a> and
<a href="./19.16#E1" title="(19.16.1) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m25.png" altimg-height="23px" altimg-valign="-7px" altimg-width="101px" alttext="R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: symmetric elliptic integral of first kind</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">with equality iff <math class="ltx_Math" altimg="m52.png" altimg-height="16px" altimg-valign="-6px" altimg-width="51px" alttext="y=z" display="inline"><mrow><mi>y</mi><mo>=</mo><mi>z</mi></mrow></math>.</p>
</div>
</section>
</section>
</section>
</div>
<div class="ltx_page_footer">
<span></div>
</div>
</body></text>
</html>
</page>
<page>
<!DOCTYPE html><html>
<head>
<title>DLMF: 19.25 Relations to Other Functions</title>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<!--GOOGLE BOOTSTRAP--></head>
<text><body class="color_default textfont_default titlefont_default fontsize_default navbar_default">
<div class="ltx_page_navbar">
<div class="ltx_page_navlogo">) put <math class="ltx_Math" altimg="m125.png" altimg-height="17px" altimg-valign="-2px" altimg-width="49px" alttext="c=1" display="inline"><mrow><mi>c</mi><mo>=</mo><mn>1</mn></mrow></math> in (), let <math class="ltx_Math" altimg="m36.png" altimg-height="25px" altimg-valign="-7px" altimg-width="234px" alttext="(c-1,c-k^{2},c)=(x,y,z)" display="inline"><mrow><mrow><mo stretchy="false">(</mo><mrow><mi>c</mi><mo>-</mo><mn>1</mn></mrow><mo>,</mo><mrow><mi>c</mi><mo>-</mo><msup><mi href="./19.1#p1.t1.r3">k</mi><mn>2</mn></msup></mrow><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><mo>=</mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math> and eliminate
<math class="ltx_Math" altimg="m70.png" altimg-height="21px" altimg-valign="-5px" altimg-width="33px" alttext="R_{G}" display="inline"><msub><mi href="./19.16#E3">R</mi><mi href="./19.16#E3">G</mi></msub></math> between () obtained by permuting <math class="ltx_Math" altimg="m145.png" altimg-height="16px" altimg-valign="-6px" altimg-width="35px" alttext="x,y" display="inline"><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow></math> and <math class="ltx_Math" altimg="m160.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi>z</mi></math>.
For (</dd>
</dl>
</div>
</div>
<div id="SS1.p1" class="ltx_para">
<p class="ltx_p">Let <math class="ltx_Math" altimg="m163.png" altimg-height="24px" altimg-valign="-4px" altimg-width="111px" alttext="{k^{\prime}}^{2}=1-k^{2}" display="inline"><mrow><mmultiscripts><mi href="./19.1#p1.t1.r4">k</mi><none></none><mo>′</mo><none></none><mn>2</mn></mmultiscripts><mo>=</mo><mrow><mn>1</mn><mo>-</mo><msup><mi href="./19.1#p1.t1.r3">k</mi><mn>2</mn></msup></mrow></mrow></math> and <math class="ltx_Math" altimg="m126.png" altimg-height="24px" altimg-valign="-6px" altimg-width="86px" alttext="c={\csc^{2}}\phi" display="inline"><mrow><mi>c</mi><mo>=</mo><mrow><msup><mi href="./4.14#E5">csc</mi><mn>2</mn></msup><mo></mo><mi href="./19.1#p1.t1.r2">ϕ</mi></mrow></mrow></math>. Then
</p>
<table id="E1" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex1" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="5" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">19.25.1</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m22.png" altimg-height="25px" altimg-valign="-7px" altimg-width="54px" alttext="\displaystyle K\left(k\right)" display="inline"><mrow><mi href="./19.2#E8">K</mi><mo></mo><mrow><mo>(</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m3.png" altimg-height="41px" altimg-valign="-15px" altimg-width="155px" alttext="\displaystyle=R_{F}\left(0,{k^{\prime}}^{2},1\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mmultiscripts><mi href="./19.1#p1.t1.r4">k</mi><none></none><mo>′</mo><none></none><mn>2</mn></mmultiscripts><mo>,</mo><mn>1</mn><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex2" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m19.png" altimg-height="25px" altimg-valign="-7px" altimg-width="52px" alttext="\displaystyle E\left(k\right)" display="inline"><mrow><mi href="./19.2#E8">E</mi><mo></mo><mrow><mo>(</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m2.png" altimg-height="41px" altimg-valign="-15px" altimg-width="161px" alttext="\displaystyle=2\!R_{G}\left(0,{k^{\prime}}^{2},1\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mpadded width="-1.7pt"><mn>2</mn></mpadded><mo></mo><mrow><msub><mi href="./19.16#E3">R</mi><mi href="./19.16#E3">G</mi></msub><mo></mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mmultiscripts><mi href="./19.1#p1.t1.r4">k</mi><none></none><mo>′</mo><none></none><mn>2</mn></mmultiscripts><mo>,</mo><mn>1</mn><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex3" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m19.png" altimg-height="25px" altimg-valign="-7px" altimg-width="52px" alttext="\displaystyle E\left(k\right)" display="inline"><mrow><mi href="./19.2#E8">E</mi><mo></mo><mrow><mo>(</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m12.png" altimg-height="41px" altimg-valign="-15px" altimg-width="365px" alttext="\displaystyle=\tfrac{1}{3}{k^{\prime}}^{2}\left(R_{D}\left(0,{k^{\prime}}^{2},%
1\right)+R_{D}\left(0,1,{k^{\prime}}^{2}\right)\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac><mo></mo><mmultiscripts><mi href="./19.1#p1.t1.r4">k</mi><none></none><mo>′</mo><none></none><mn>2</mn></mmultiscripts><mo></mo><mrow><mo>(</mo><mrow><mrow><msub><mi href="./19.16#E5">R</mi><mi href="./19.16#E5">D</mi></msub><mo></mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mmultiscripts><mi href="./19.1#p1.t1.r4">k</mi><none></none><mo>′</mo><none></none><mn>2</mn></mmultiscripts><mo>,</mo><mn>1</mn><mo>)</mo></mrow></mrow><mo>+</mo><mrow><msub><mi href="./19.16#E5">R</mi><mi href="./19.16#E5">D</mi></msub><mo></mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mmultiscripts><mi href="./19.1#p1.t1.r4">k</mi><none></none><mo>′</mo><none></none><mn>2</mn></mmultiscripts><mo>)</mo></mrow></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex4" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m21.png" altimg-height="25px" altimg-valign="-7px" altimg-width="124px" alttext="\displaystyle K\left(k\right)-E\left(k\right)" display="inline"><mrow><mrow><mi href="./19.2#E8">K</mi><mo></mo><mrow><mo>(</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo>-</mo><mrow><mi href="./19.2#E8">E</mi><mo></mo><mrow><mo>(</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m16.png" altimg-height="41px" altimg-valign="-15px" altimg-width="282px" alttext="\displaystyle=k^{2}D\left(k\right)=\tfrac{1}{3}k^{2}R_{D}\left(0,{k^{\prime}}^%
{2},1\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msup><mi href="./19.1#p1.t1.r3">k</mi><mn>2</mn></msup><mo></mo><mrow><mi href="./19.2#E8">D</mi><mo></mo><mrow><mo>(</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac><mo></mo><msup><mi href="./19.1#p1.t1.r3">k</mi><mn>2</mn></msup><mo></mo><mrow><msub><mi href="./19.16#E5">R</mi><mi href="./19.16#E5">D</mi></msub><mo></mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mmultiscripts><mi href="./19.1#p1.t1.r4">k</mi><none></none><mo>′</mo><none></none><mn>2</mn></mmultiscripts><mo>,</mo><mn>1</mn><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex5" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m18.png" altimg-height="31px" altimg-valign="-7px" altimg-width="150px" alttext="\displaystyle E\left(k\right)-{k^{\prime}}^{2}K\left(k\right)" display="inline"><mrow><mrow><mi href="./19.2#E8">E</mi><mo></mo><mrow><mo>(</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo>-</mo><mrow><mmultiscripts><mi href="./19.1#p1.t1.r4">k</mi><none></none><mo>′</mo><none></none><mn>2</mn></mmultiscripts><mo></mo><mrow><mi href="./19.2#E8">K</mi><mo></mo><mrow><mo>(</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m11.png" altimg-height="41px" altimg-valign="-15px" altimg-width="214px" alttext="\displaystyle=\tfrac{1}{3}k^{2}{k^{\prime}}^{2}R_{D}\left(0,1,{k^{\prime}}^{2}%
\right)." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac><mo></mo><msup><mi href="./19.1#p1.t1.r3">k</mi><mn>2</mn></msup><mo></mo><mmultiscripts><mi href="./19.1#p1.t1.r4">k</mi><none></none><mo>′</mo><none></none><mn>2</mn></mmultiscripts><mo></mo><mrow><msub><mi href="./19.16#E5">R</mi><mi href="./19.16#E5">D</mi></msub><mo></mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mmultiscripts><mi href="./19.1#p1.t1.r4">k</mi><none></none><mo>′</mo><none></none><mn>2</mn></mmultiscripts><mo>)</mo></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.16#E5" title="(19.16.5) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="23px" altimg-valign="-7px" altimg-width="102px" alttext="R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E5">R</mi><mi href="./19.16#E5">D</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: elliptic integral symmetric in only two variables</a>,
<a href="./19.16#E1" title="(19.16.1) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m68.png" altimg-height="23px" altimg-valign="-7px" altimg-width="101px" alttext="R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: symmetric elliptic integral of first kind</a>,
<a href="./19.16#E3" title="(19.16.3) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m71.png" altimg-height="23px" altimg-valign="-7px" altimg-width="101px" alttext="R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E3">R</mi><mi href="./19.16#E3">G</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: symmetric elliptic integral of second kind</a>,
<a href="./19.2#E8" title="(19.2.8) ‣ §19.2(ii) Legendre’s Integrals ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m57.png" altimg-height="23px" altimg-valign="-7px" altimg-width="51px" alttext="D\left(\NVar{k}\right)" display="inline"><mrow><mi href="./19.2#E8">D</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: complete elliptic integral of Legendre’s type</a>,
<a href="./19.2#E8" title="(19.2.8) ‣ §19.2(ii) Legendre’s Integrals ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m62.png" altimg-height="23px" altimg-valign="-7px" altimg-width="52px" alttext="K\left(\NVar{k}\right)" display="inline"><mrow><mi href="./19.2#E8">K</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Legendre’s complete elliptic integral of the first kind</a>,
<a href="./19.2#E8" title="(19.2.8) ‣ §19.2(ii) Legendre’s Integrals ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="23px" altimg-valign="-7px" altimg-width="50px" alttext="E\left(\NVar{k}\right)" display="inline"><mrow><mi href="./19.2#E8">E</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Legendre’s complete elliptic integral of the second kind</a>,
<a href="./19.1#p1.t1.r3" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m132.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./19.1#p1.t1.r3">k</mi></math>: real or complex modulus</a> and
<a href="./19.1#p1.t1.r4" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m136.png" altimg-height="19px" altimg-valign="-2px" altimg-width="21px" alttext="k^{\prime}" display="inline"><msup><mi href="./19.1#p1.t1.r4">k</mi><mo>′</mo></msup></math>: complementary modulus</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E2" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.25.2</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E2.png" altimg-height="41px" altimg-valign="-15px" altimg-width="408px" alttext="\Pi\left(\alpha^{2},k\right)-K\left(k\right)=\tfrac{1}{3}\alpha^{2}R_{J}\left(%
0,{k^{\prime}}^{2},1,1-\alpha^{2}\right)." display="block"><mrow><mrow><mrow><mrow><mi href="./19.2#E8" mathvariant="normal">Π</mi><mo></mo><mrow><mo>(</mo><msup><mi href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup><mo>,</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo>-</mo><mrow><mi href="./19.2#E8">K</mi><mo></mo><mrow><mo>(</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>3</mn></mfrac></mstyle><mo></mo><msup><mi href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup><mo></mo><mrow><msub><mi href="./19.16#E2">R</mi><mi href="./19.16#E2">J</mi></msub><mo></mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mmultiscripts><mi href="./19.1#p1.t1.r4">k</mi><none></none><mo>′</mo><none></none><mn>2</mn></mmultiscripts><mo>,</mo><mn>1</mn><mo>,</mo><mrow><mn>1</mn><mo>-</mo><msup><mi href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.16#E2" title="(19.16.2) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="23px" altimg-valign="-7px" altimg-width="118px" alttext="R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)" display="inline"><mrow><msub><mi href="./19.16#E2">R</mi><mi href="./19.16#E2">J</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>,</mo><mi class="ltx_nvar">p</mi><mo>)</mo></mrow></mrow></math>: symmetric elliptic integral of third kind</a>,
<a href="./19.2#E8" title="(19.2.8) ‣ §19.2(ii) Legendre’s Integrals ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m62.png" altimg-height="23px" altimg-valign="-7px" altimg-width="52px" alttext="K\left(\NVar{k}\right)" display="inline"><mrow><mi href="./19.2#E8">K</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Legendre’s complete elliptic integral of the first kind</a>,
<a href="./19.2#E8" title="(19.2.8) ‣ §19.2(ii) Legendre’s Integrals ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m76.png" altimg-height="27px" altimg-valign="-9px" altimg-width="82px" alttext="\Pi\left(\NVar{\alpha}^{2},\NVar{k}\right)" display="inline"><mrow><mi href="./19.2#E8" mathvariant="normal">Π</mi><mo></mo><mrow><mo>(</mo><msup><mi class="ltx_nvar" href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup><mo>,</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Legendre’s complete elliptic integral of the third kind</a>,
<a href="./19.1#p1.t1.r3" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m132.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./19.1#p1.t1.r3">k</mi></math>: real or complex modulus</a>,
<a href="./19.1#p1.t1.r4" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m136.png" altimg-height="19px" altimg-valign="-2px" altimg-width="21px" alttext="k^{\prime}" display="inline"><msup><mi href="./19.1#p1.t1.r4">k</mi><mo>′</mo></msup></math>: complementary modulus</a> and
<a href="./19.1#p1.t1.r5" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m81.png" altimg-height="20px" altimg-valign="-2px" altimg-width="26px" alttext="\alpha^{2}" display="inline"><msup><mi href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup></math>: real or complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E3" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.25.3</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E3.png" altimg-height="41px" altimg-valign="-15px" altimg-width="398px" alttext="\Pi\left(\alpha^{2},k\right)=\tfrac{1}{2}\pi R_{-\frac{1}{2}}\left(\tfrac{1}{2%
},-\tfrac{1}{2},1;{k^{\prime}}^{2},1,1-\alpha^{2}\right)," display="block"><mrow><mrow><mrow><mi href="./19.2#E8" mathvariant="normal">Π</mi><mo></mo><mrow><mo>(</mo><msup><mi href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup><mo>,</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mrow><msub><mi href="./19.16#E9">R</mi><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msub><mo></mo><mrow><mo>(</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo>,</mo><mrow><mo>-</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle></mrow><mo>,</mo><mn>1</mn></mrow><mo>;</mo><mrow><mmultiscripts><mi href="./19.1#p1.t1.r4">k</mi><none></none><mo>′</mo><none></none><mn>2</mn></mmultiscripts><mo>,</mo><mn>1</mn><mo>,</mo><mrow><mn>1</mn><mo>-</mo><msup><mi href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.16#E9" title="(19.16.9) ‣ §19.16(ii) R - a ( b ; z ) ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="23px" altimg-valign="-7px" altimg-width="233px" alttext="R_{\NVar{-a}}\left(\NVar{b_{1}},\dots,\NVar{b_{n}};\NVar{z_{1}},\dots,\NVar{z_%
{n}}\right)" display="inline"><mrow><msub><mi href="./19.16#E9">R</mi><mrow><mo class="ltx_nvar">-</mo><mi class="ltx_nvar">a</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mrow><msub><mi class="ltx_nvar">b</mi><mn class="ltx_nvar">1</mn></msub><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><msub><mi class="ltx_nvar">b</mi><mi class="ltx_nvar" href="./19.1#p1.t1.r1">n</mi></msub></mrow><mo>;</mo><mrow><msub><mi class="ltx_nvar">z</mi><mn class="ltx_nvar">1</mn></msub><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><msub><mi class="ltx_nvar">z</mi><mi class="ltx_nvar" href="./19.1#p1.t1.r1">n</mi></msub></mrow><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m73.png" altimg-height="23px" altimg-valign="-7px" altimg-width="92px" alttext="R_{\NVar{-a}}\left(\NVar{\mathbf{b}};\NVar{\mathbf{z}}\right)" display="inline"><mrow><msub><mi href="./19.16#E9">R</mi><mrow><mo class="ltx_nvar">-</mo><mi class="ltx_nvar">a</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" mathvariant="bold">b</mi><mo>;</mo><mi class="ltx_nvar" mathvariant="bold">z</mi><mo>)</mo></mrow></mrow></math>: multivariate hypergeometric function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m112.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./19.2#E8" title="(19.2.8) ‣ §19.2(ii) Legendre’s Integrals ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m76.png" altimg-height="27px" altimg-valign="-9px" altimg-width="82px" alttext="\Pi\left(\NVar{\alpha}^{2},\NVar{k}\right)" display="inline"><mrow><mi href="./19.2#E8" mathvariant="normal">Π</mi><mo></mo><mrow><mo>(</mo><msup><mi class="ltx_nvar" href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup><mo>,</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Legendre’s complete elliptic integral of the third kind</a>,
<a href="./19.1#p1.t1.r3" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m132.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./19.1#p1.t1.r3">k</mi></math>: real or complex modulus</a>,
<a href="./19.1#p1.t1.r4" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m136.png" altimg-height="19px" altimg-valign="-2px" altimg-width="21px" alttext="k^{\prime}" display="inline"><msup><mi href="./19.1#p1.t1.r4">k</mi><mo>′</mo></msup></math>: complementary modulus</a> and
<a href="./19.1#p1.t1.r5" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m81.png" altimg-height="20px" altimg-valign="-2px" altimg-width="26px" alttext="\alpha^{2}" display="inline"><msup><mi href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup></math>: real or complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">with Cauchy principal value</p>
<table id="E4" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.25.4</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E4.png" altimg-height="30px" altimg-valign="-9px" altimg-width="465px" alttext="\Pi\left(\alpha^{2},k\right)=-\tfrac{1}{3}(k^{2}/\alpha^{2})R_{J}\left(0,1-k^{%
2},1,1-(k^{2}/\alpha^{2})\right)," display="block"><mrow><mrow><mrow><mi href="./19.2#E8" mathvariant="normal">Π</mi><mo></mo><mrow><mo>(</mo><msup><mi href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup><mo>,</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mo>-</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>3</mn></mfrac></mstyle><mo></mo><mrow><mo stretchy="false">(</mo><mrow><msup><mi href="./19.1#p1.t1.r3">k</mi><mn>2</mn></msup><mo>/</mo><msup><mi href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><msub><mi href="./19.16#E2">R</mi><mi href="./19.16#E2">J</mi></msub><mo></mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mrow><mn>1</mn><mo>-</mo><msup><mi href="./19.1#p1.t1.r3">k</mi><mn>2</mn></msup></mrow><mo>,</mo><mn>1</mn><mo>,</mo><mrow><mn>1</mn><mo>-</mo><mrow><mo stretchy="false">(</mo><mrow><msup><mi href="./19.1#p1.t1.r3">k</mi><mn>2</mn></msup><mo>/</mo><msup><mi href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m38.png" altimg-height="21px" altimg-valign="-4px" altimg-width="171px" alttext="-\infty<k^{2}<1<\alpha^{2}" display="inline"><mrow><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mo><</mo><msup><mi href="./19.1#p1.t1.r3">k</mi><mn>2</mn></msup><mo><</mo><mn>1</mn><mo><</mo><msup><mi href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E4.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.16#E2" title="(19.16.2) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="23px" altimg-valign="-7px" altimg-width="118px" alttext="R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)" display="inline"><mrow><msub><mi href="./19.16#E2">R</mi><mi href="./19.16#E2">J</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>,</mo><mi class="ltx_nvar">p</mi><mo>)</mo></mrow></mrow></math>: symmetric elliptic integral of third kind</a>,
<a href="./19.2#E8" title="(19.2.8) ‣ §19.2(ii) Legendre’s Integrals ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m76.png" altimg-height="27px" altimg-valign="-9px" altimg-width="82px" alttext="\Pi\left(\NVar{\alpha}^{2},\NVar{k}\right)" display="inline"><mrow><mi href="./19.2#E8" mathvariant="normal">Π</mi><mo></mo><mrow><mo>(</mo><msup><mi class="ltx_nvar" href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup><mo>,</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Legendre’s complete elliptic integral of the third kind</a>,
<a href="./19.1#p1.t1.r3" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m132.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./19.1#p1.t1.r3">k</mi></math>: real or complex modulus</a> and
<a href="./19.1#p1.t1.r5" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m81.png" altimg-height="20px" altimg-valign="-2px" altimg-width="26px" alttext="\alpha^{2}" display="inline"><msup><mi href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup></math>: real or complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="EGx1" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E5">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.25.5</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m20.png" altimg-height="25px" altimg-valign="-7px" altimg-width="72px" alttext="\displaystyle F\left(\phi,k\right)" display="inline"><mrow><mi href="./19.2#E4">F</mi><mo></mo><mrow><mo>(</mo><mi href="./19.1#p1.t1.r2">ϕ</mi><mo>,</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m4.png" altimg-height="30px" altimg-valign="-9px" altimg-width="208px" alttext="\displaystyle=R_{F}\left(c-1,c-k^{2},c\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi>c</mi><mo>-</mo><mn>1</mn></mrow><mo>,</mo><mrow><mi>c</mi><mo>-</mo><msup><mi href="./19.1#p1.t1.r3">k</mi><mn>2</mn></msup></mrow><mo>,</mo><mi>c</mi><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E5.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.16#E1" title="(19.16.1) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m68.png" altimg-height="23px" altimg-valign="-7px" altimg-width="101px" alttext="R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: symmetric elliptic integral of first kind</a>,
<a href="./19.2#E4" title="(19.2.4) ‣ §19.2(ii) Legendre’s Integrals ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="23px" altimg-valign="-7px" altimg-width="70px" alttext="F\left(\NVar{\phi},\NVar{k}\right)" display="inline"><mrow><mi href="./19.2#E4">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r2">ϕ</mi><mo>,</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Legendre’s incomplete elliptic integral of the first kind</a>,
<a href="./19.1#p1.t1.r2" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m110.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="\phi" display="inline"><mi href="./19.1#p1.t1.r2">ϕ</mi></math>: real or complex argument</a> and
<a href="./19.1#p1.t1.r3" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m132.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./19.1#p1.t1.r3">k</mi></math>: real or complex modulus</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E6">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.25.6</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m27.png" altimg-height="48px" altimg-valign="-16px" altimg-width="89px" alttext="\displaystyle\frac{\partial F\left(\phi,k\right)}{\partial k}" display="inline"><mstyle displaystyle="true"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mrow><mi href="./19.2#E4">F</mi><mo></mo><mrow><mo>(</mo><mi href="./19.1#p1.t1.r2">ϕ</mi><mo>,</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./19.1#p1.t1.r3">k</mi></mrow></mfrac></mstyle></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m10.png" altimg-height="30px" altimg-valign="-9px" altimg-width="233px" alttext="\displaystyle=\tfrac{1}{3}kR_{D}\left(c-1,c,c-k^{2}\right)." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac><mo></mo><mi href="./19.1#p1.t1.r3">k</mi><mo></mo><mrow><msub><mi href="./19.16#E5">R</mi><mi href="./19.16#E5">D</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi>c</mi><mo>-</mo><mn>1</mn></mrow><mo>,</mo><mi>c</mi><mo>,</mo><mrow><mi>c</mi><mo>-</mo><msup><mi href="./19.1#p1.t1.r3">k</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E6.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.16#E5" title="(19.16.5) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="23px" altimg-valign="-7px" altimg-width="102px" alttext="R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E5">R</mi><mi href="./19.16#E5">D</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: elliptic integral symmetric in only two variables</a>,
<a href="./19.2#E4" title="(19.2.4) ‣ §19.2(ii) Legendre’s Integrals ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="23px" altimg-valign="-7px" altimg-width="70px" alttext="F\left(\NVar{\phi},\NVar{k}\right)" display="inline"><mrow><mi href="./19.2#E4">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r2">ϕ</mi><mo>,</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Legendre’s incomplete elliptic integral of the first kind</a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m84.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m128.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m155.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m109.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m155.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./19.1#p1.t1.r2" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m110.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="\phi" display="inline"><mi href="./19.1#p1.t1.r2">ϕ</mi></math>: real or complex argument</a> and
<a href="./19.1#p1.t1.r3" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m132.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./19.1#p1.t1.r3">k</mi></math>: real or complex modulus</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
<table id="E7" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.25.7</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E7.png" altimg-height="32px" altimg-valign="-9px" altimg-width="728px" alttext="E\left(\phi,k\right)=2\!R_{G}\left(c-1,c-k^{2},c\right)-(c-1)R_{F}\left(c-1,c-%
k^{2},c\right)-\sqrt{(c-1)(c-k^{2})/c}," display="block"><mrow><mrow><mrow><mi href="./19.2#E5">E</mi><mo></mo><mrow><mo>(</mo><mi href="./19.1#p1.t1.r2">ϕ</mi><mo>,</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mpadded width="-1.7pt"><mn>2</mn></mpadded><mo></mo><mrow><msub><mi href="./19.16#E3">R</mi><mi href="./19.16#E3">G</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi>c</mi><mo>-</mo><mn>1</mn></mrow><mo>,</mo><mrow><mi>c</mi><mo>-</mo><msup><mi href="./19.1#p1.t1.r3">k</mi><mn>2</mn></msup></mrow><mo>,</mo><mi>c</mi><mo>)</mo></mrow></mrow></mrow><mo>-</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mi>c</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi>c</mi><mo>-</mo><mn>1</mn></mrow><mo>,</mo><mrow><mi>c</mi><mo>-</mo><msup><mi href="./19.1#p1.t1.r3">k</mi><mn>2</mn></msup></mrow><mo>,</mo><mi>c</mi><mo>)</mo></mrow></mrow></mrow><mo>-</mo><msqrt><mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mi>c</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>c</mi><mo>-</mo><msup><mi href="./19.1#p1.t1.r3">k</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>/</mo><mi>c</mi></mrow></msqrt></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E7.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.16#E1" title="(19.16.1) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m68.png" altimg-height="23px" altimg-valign="-7px" altimg-width="101px" alttext="R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: symmetric elliptic integral of first kind</a>,
<a href="./19.16#E3" title="(19.16.3) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m71.png" altimg-height="23px" altimg-valign="-7px" altimg-width="101px" alttext="R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E3">R</mi><mi href="./19.16#E3">G</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: symmetric elliptic integral of second kind</a>,
<a href="./19.2#E5" title="(19.2.5) ‣ §19.2(ii) Legendre’s Integrals ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="23px" altimg-valign="-7px" altimg-width="71px" alttext="E\left(\NVar{\phi},\NVar{k}\right)" display="inline"><mrow><mi href="./19.2#E5">E</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r2">ϕ</mi><mo>,</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Legendre’s incomplete elliptic integral of the second kind</a>,
<a href="./19.1#p1.t1.r2" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m110.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="\phi" display="inline"><mi href="./19.1#p1.t1.r2">ϕ</mi></math>: real or complex argument</a> and
<a href="./19.1#p1.t1.r3" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m132.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./19.1#p1.t1.r3">k</mi></math>: real or complex modulus</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E8" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.25.8</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E8.png" altimg-height="33px" altimg-valign="-12px" altimg-width="372px" alttext="E\left(\phi,k\right)=R_{-\frac{1}{2}}\left(\tfrac{1}{2},-\tfrac{1}{2},\tfrac{3%
}{2};c-1,c-k^{2},c\right)," display="block"><mrow><mrow><mrow><mi href="./19.2#E5">E</mi><mo></mo><mrow><mo>(</mo><mi href="./19.1#p1.t1.r2">ϕ</mi><mo>,</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msub><mi href="./19.16#E9">R</mi><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msub><mo></mo><mrow><mo>(</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo>,</mo><mrow><mo>-</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle></mrow><mo>,</mo><mstyle displaystyle="false"><mfrac><mn>3</mn><mn>2</mn></mfrac></mstyle></mrow><mo>;</mo><mrow><mrow><mi>c</mi><mo>-</mo><mn>1</mn></mrow><mo>,</mo><mrow><mi>c</mi><mo>-</mo><msup><mi href="./19.1#p1.t1.r3">k</mi><mn>2</mn></msup></mrow><mo>,</mo><mi>c</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E8.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.16#E9" title="(19.16.9) ‣ §19.16(ii) R - a ( b ; z ) ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="23px" altimg-valign="-7px" altimg-width="233px" alttext="R_{\NVar{-a}}\left(\NVar{b_{1}},\dots,\NVar{b_{n}};\NVar{z_{1}},\dots,\NVar{z_%
{n}}\right)" display="inline"><mrow><msub><mi href="./19.16#E9">R</mi><mrow><mo class="ltx_nvar">-</mo><mi class="ltx_nvar">a</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mrow><msub><mi class="ltx_nvar">b</mi><mn class="ltx_nvar">1</mn></msub><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><msub><mi class="ltx_nvar">b</mi><mi class="ltx_nvar" href="./19.1#p1.t1.r1">n</mi></msub></mrow><mo>;</mo><mrow><msub><mi class="ltx_nvar">z</mi><mn class="ltx_nvar">1</mn></msub><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><msub><mi class="ltx_nvar">z</mi><mi class="ltx_nvar" href="./19.1#p1.t1.r1">n</mi></msub></mrow><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m73.png" altimg-height="23px" altimg-valign="-7px" altimg-width="92px" alttext="R_{\NVar{-a}}\left(\NVar{\mathbf{b}};\NVar{\mathbf{z}}\right)" display="inline"><mrow><msub><mi href="./19.16#E9">R</mi><mrow><mo class="ltx_nvar">-</mo><mi class="ltx_nvar">a</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" mathvariant="bold">b</mi><mo>;</mo><mi class="ltx_nvar" mathvariant="bold">z</mi><mo>)</mo></mrow></mrow></math>: multivariate hypergeometric function</a>,
<a href="./19.2#E5" title="(19.2.5) ‣ §19.2(ii) Legendre’s Integrals ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="23px" altimg-valign="-7px" altimg-width="71px" alttext="E\left(\NVar{\phi},\NVar{k}\right)" display="inline"><mrow><mi href="./19.2#E5">E</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r2">ϕ</mi><mo>,</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Legendre’s incomplete elliptic integral of the second kind</a>,
<a href="./19.1#p1.t1.r2" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m110.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="\phi" display="inline"><mi href="./19.1#p1.t1.r2">ϕ</mi></math>: real or complex argument</a> and
<a href="./19.1#p1.t1.r3" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m132.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./19.1#p1.t1.r3">k</mi></math>: real or complex modulus</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E9" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.25.9</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E9.png" altimg-height="30px" altimg-valign="-9px" altimg-width="510px" alttext="E\left(\phi,k\right)=R_{F}\left(c-1,c-k^{2},c\right)-\tfrac{1}{3}k^{2}R_{D}%
\left(c-1,c-k^{2},c\right)," display="block"><mrow><mrow><mrow><mi href="./19.2#E5">E</mi><mo></mo><mrow><mo>(</mo><mi href="./19.1#p1.t1.r2">ϕ</mi><mo>,</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi>c</mi><mo>-</mo><mn>1</mn></mrow><mo>,</mo><mrow><mi>c</mi><mo>-</mo><msup><mi href="./19.1#p1.t1.r3">k</mi><mn>2</mn></msup></mrow><mo>,</mo><mi>c</mi><mo>)</mo></mrow></mrow><mo>-</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>3</mn></mfrac></mstyle><mo></mo><msup><mi href="./19.1#p1.t1.r3">k</mi><mn>2</mn></msup><mo></mo><mrow><msub><mi href="./19.16#E5">R</mi><mi href="./19.16#E5">D</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi>c</mi><mo>-</mo><mn>1</mn></mrow><mo>,</mo><mrow><mi>c</mi><mo>-</mo><msup><mi href="./19.1#p1.t1.r3">k</mi><mn>2</mn></msup></mrow><mo>,</mo><mi>c</mi><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E9.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.16#E5" title="(19.16.5) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="23px" altimg-valign="-7px" altimg-width="102px" alttext="R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E5">R</mi><mi href="./19.16#E5">D</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: elliptic integral symmetric in only two variables</a>,
<a href="./19.16#E1" title="(19.16.1) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m68.png" altimg-height="23px" altimg-valign="-7px" altimg-width="101px" alttext="R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: symmetric elliptic integral of first kind</a>,
<a href="./19.2#E5" title="(19.2.5) ‣ §19.2(ii) Legendre’s Integrals ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="23px" altimg-valign="-7px" altimg-width="71px" alttext="E\left(\NVar{\phi},\NVar{k}\right)" display="inline"><mrow><mi href="./19.2#E5">E</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r2">ϕ</mi><mo>,</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Legendre’s incomplete elliptic integral of the second kind</a>,
<a href="./19.1#p1.t1.r2" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m110.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="\phi" display="inline"><mi href="./19.1#p1.t1.r2">ϕ</mi></math>: real or complex argument</a> and
<a href="./19.1#p1.t1.r3" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m132.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./19.1#p1.t1.r3">k</mi></math>: real or complex modulus</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E10" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.25.10</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E10.png" altimg-height="33px" altimg-valign="-9px" altimg-width="783px" alttext="E\left(\phi,k\right)={k^{\prime}}^{2}R_{F}\left(c-1,c-k^{2},c\right)+\tfrac{1}%
{3}k^{2}{k^{\prime}}^{2}R_{D}\left(c-1,c,c-k^{2}\right)+k^{2}\sqrt{(c-1)/(c(c-%
k^{2}))}," display="block"><mrow><mrow><mrow><mi href="./19.2#E5">E</mi><mo></mo><mrow><mo>(</mo><mi href="./19.1#p1.t1.r2">ϕ</mi><mo>,</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mmultiscripts><mi href="./19.1#p1.t1.r4">k</mi><none></none><mo>′</mo><none></none><mn>2</mn></mmultiscripts><mo></mo><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi>c</mi><mo>-</mo><mn>1</mn></mrow><mo>,</mo><mrow><mi>c</mi><mo>-</mo><msup><mi href="./19.1#p1.t1.r3">k</mi><mn>2</mn></msup></mrow><mo>,</mo><mi>c</mi><mo>)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>3</mn></mfrac></mstyle><mo></mo><msup><mi href="./19.1#p1.t1.r3">k</mi><mn>2</mn></msup><mo></mo><mmultiscripts><mi href="./19.1#p1.t1.r4">k</mi><none></none><mo>′</mo><none></none><mn>2</mn></mmultiscripts><mo></mo><mrow><msub><mi href="./19.16#E5">R</mi><mi href="./19.16#E5">D</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi>c</mi><mo>-</mo><mn>1</mn></mrow><mo>,</mo><mi>c</mi><mo>,</mo><mrow><mi>c</mi><mo>-</mo><msup><mi href="./19.1#p1.t1.r3">k</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow></mrow></mrow><mo>+</mo><mrow><msup><mi href="./19.1#p1.t1.r3">k</mi><mn>2</mn></msup><mo></mo><msqrt><mrow><mrow><mo stretchy="false">(</mo><mrow><mi>c</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo>/</mo><mrow><mo stretchy="false">(</mo><mrow><mi>c</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>c</mi><mo>-</mo><msup><mi href="./19.1#p1.t1.r3">k</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></msqrt></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m127.png" altimg-height="21px" altimg-valign="-3px" altimg-width="59px" alttext="c>k^{2}" display="inline"><mrow><mi>c</mi><mo>></mo><msup><mi href="./19.1#p1.t1.r3">k</mi><mn>2</mn></msup></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E10.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.16#E5" title="(19.16.5) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="23px" altimg-valign="-7px" altimg-width="102px" alttext="R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E5">R</mi><mi href="./19.16#E5">D</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: elliptic integral symmetric in only two variables</a>,
<a href="./19.16#E1" title="(19.16.1) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m68.png" altimg-height="23px" altimg-valign="-7px" altimg-width="101px" alttext="R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: symmetric elliptic integral of first kind</a>,
<a href="./19.2#E5" title="(19.2.5) ‣ §19.2(ii) Legendre’s Integrals ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="23px" altimg-valign="-7px" altimg-width="71px" alttext="E\left(\NVar{\phi},\NVar{k}\right)" display="inline"><mrow><mi href="./19.2#E5">E</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r2">ϕ</mi><mo>,</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Legendre’s incomplete elliptic integral of the second kind</a>,
<a href="./19.1#p1.t1.r2" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m110.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="\phi" display="inline"><mi href="./19.1#p1.t1.r2">ϕ</mi></math>: real or complex argument</a>,
<a href="./19.1#p1.t1.r3" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m132.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./19.1#p1.t1.r3">k</mi></math>: real or complex modulus</a> and
<a href="./19.1#p1.t1.r4" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m136.png" altimg-height="19px" altimg-valign="-2px" altimg-width="21px" alttext="k^{\prime}" display="inline"><msup><mi href="./19.1#p1.t1.r4">k</mi><mo>′</mo></msup></math>: complementary modulus</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E11" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.25.11</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E11.png" altimg-height="33px" altimg-valign="-9px" altimg-width="537px" alttext="E\left(\phi,k\right)=-\tfrac{1}{3}{k^{\prime}}^{2}R_{D}\left(c-k^{2},c,c-1%
\right)+\sqrt{(c-k^{2})/(c(c-1))}," display="block"><mrow><mrow><mrow><mi href="./19.2#E5">E</mi><mo></mo><mrow><mo>(</mo><mi href="./19.1#p1.t1.r2">ϕ</mi><mo>,</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mo>-</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>3</mn></mfrac></mstyle><mo></mo><mmultiscripts><mi href="./19.1#p1.t1.r4">k</mi><none></none><mo>′</mo><none></none><mn>2</mn></mmultiscripts><mo></mo><mrow><msub><mi href="./19.16#E5">R</mi><mi href="./19.16#E5">D</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi>c</mi><mo>-</mo><msup><mi href="./19.1#p1.t1.r3">k</mi><mn>2</mn></msup></mrow><mo>,</mo><mi>c</mi><mo>,</mo><mrow><mi>c</mi><mo>-</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>+</mo><msqrt><mrow><mrow><mo stretchy="false">(</mo><mrow><mi>c</mi><mo>-</mo><msup><mi href="./19.1#p1.t1.r3">k</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mo>/</mo><mrow><mo stretchy="false">(</mo><mrow><mi>c</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>c</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></msqrt></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m111.png" altimg-height="27px" altimg-valign="-9px" altimg-width="68px" alttext="\phi\neq\tfrac{1}{2}\pi" display="inline"><mrow><mi href="./19.1#p1.t1.r2">ϕ</mi><mo>≠</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E11.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.16#E5" title="(19.16.5) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="23px" altimg-valign="-7px" altimg-width="102px" alttext="R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E5">R</mi><mi href="./19.16#E5">D</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: elliptic integral symmetric in only two variables</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m112.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./19.2#E5" title="(19.2.5) ‣ §19.2(ii) Legendre’s Integrals ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="23px" altimg-valign="-7px" altimg-width="71px" alttext="E\left(\NVar{\phi},\NVar{k}\right)" display="inline"><mrow><mi href="./19.2#E5">E</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r2">ϕ</mi><mo>,</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Legendre’s incomplete elliptic integral of the second kind</a>,
<a href="./19.1#p1.t1.r2" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m110.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="\phi" display="inline"><mi href="./19.1#p1.t1.r2">ϕ</mi></math>: real or complex argument</a>,
<a href="./19.1#p1.t1.r3" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m132.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./19.1#p1.t1.r3">k</mi></math>: real or complex modulus</a> and
<a href="./19.1#p1.t1.r4" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m136.png" altimg-height="19px" altimg-valign="-2px" altimg-width="21px" alttext="k^{\prime}" display="inline"><msup><mi href="./19.1#p1.t1.r4">k</mi><mo>′</mo></msup></math>: complementary modulus</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS1.p2" class="ltx_para">
<p class="ltx_p">Equations () correspond to three
(nonzero) choices for the last variable of <math class="ltx_Math" altimg="m65.png" altimg-height="21px" altimg-valign="-5px" altimg-width="34px" alttext="R_{D}" display="inline"><msub><mi href="./19.16#E5">R</mi><mi href="./19.16#E5">D</mi></msub></math>; see
(). All terms on the right-hand sides are nonnegative when
<math class="ltx_Math" altimg="m135.png" altimg-height="22px" altimg-valign="-5px" altimg-width="61px" alttext="k^{2}\leq 0" display="inline"><mrow><msup><mi href="./19.1#p1.t1.r3">k</mi><mn>2</mn></msup><mo>≤</mo><mn>0</mn></mrow></math>, <math class="ltx_Math" altimg="m44.png" altimg-height="22px" altimg-valign="-5px" altimg-width="97px" alttext="0\leq k^{2}\leq 1" display="inline"><mrow><mn>0</mn><mo>≤</mo><msup><mi href="./19.1#p1.t1.r3">k</mi><mn>2</mn></msup><mo>≤</mo><mn>1</mn></mrow></math>, or <math class="ltx_Math" altimg="m51.png" altimg-height="22px" altimg-valign="-5px" altimg-width="96px" alttext="1\leq k^{2}\leq c" display="inline"><mrow><mn>1</mn><mo>≤</mo><msup><mi href="./19.1#p1.t1.r3">k</mi><mn>2</mn></msup><mo>≤</mo><mi>c</mi></mrow></math>, respectively.</p>
<table id="E12" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.25.12</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E12.png" altimg-height="48px" altimg-valign="-16px" altimg-width="337px" alttext="\frac{\partial E\left(\phi,k\right)}{\partial k}=-\tfrac{1}{3}kR_{D}\left(c-1,%
c-k^{2},c\right)." display="block"><mrow><mrow><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mrow><mi href="./19.2#E5">E</mi><mo></mo><mrow><mo>(</mo><mi href="./19.1#p1.t1.r2">ϕ</mi><mo>,</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./19.1#p1.t1.r3">k</mi></mrow></mfrac><mo>=</mo><mrow><mo>-</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>3</mn></mfrac></mstyle><mo></mo><mi href="./19.1#p1.t1.r3">k</mi><mo></mo><mrow><msub><mi href="./19.16#E5">R</mi><mi href="./19.16#E5">D</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi>c</mi><mo>-</mo><mn>1</mn></mrow><mo>,</mo><mrow><mi>c</mi><mo>-</mo><msup><mi href="./19.1#p1.t1.r3">k</mi><mn>2</mn></msup></mrow><mo>,</mo><mi>c</mi><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E12.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.16#E5" title="(19.16.5) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="23px" altimg-valign="-7px" altimg-width="102px" alttext="R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E5">R</mi><mi href="./19.16#E5">D</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: elliptic integral symmetric in only two variables</a>,
<a href="./19.2#E5" title="(19.2.5) ‣ §19.2(ii) Legendre’s Integrals ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="23px" altimg-valign="-7px" altimg-width="71px" alttext="E\left(\NVar{\phi},\NVar{k}\right)" display="inline"><mrow><mi href="./19.2#E5">E</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r2">ϕ</mi><mo>,</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Legendre’s incomplete elliptic integral of the second kind</a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m84.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m128.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m155.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m109.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m155.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./19.1#p1.t1.r2" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m110.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="\phi" display="inline"><mi href="./19.1#p1.t1.r2">ϕ</mi></math>: real or complex argument</a> and
<a href="./19.1#p1.t1.r3" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m132.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./19.1#p1.t1.r3">k</mi></math>: real or complex modulus</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E13" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.25.13</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E13.png" altimg-height="30px" altimg-valign="-9px" altimg-width="295px" alttext="D\left(\phi,k\right)=\tfrac{1}{3}R_{D}\left(c-1,c-k^{2},c\right)." display="block"><mrow><mrow><mrow><mi href="./19.2#E6">D</mi><mo></mo><mrow><mo>(</mo><mi href="./19.1#p1.t1.r2">ϕ</mi><mo>,</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>3</mn></mfrac></mstyle><mo></mo><mrow><msub><mi href="./19.16#E5">R</mi><mi href="./19.16#E5">D</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi>c</mi><mo>-</mo><mn>1</mn></mrow><mo>,</mo><mrow><mi>c</mi><mo>-</mo><msup><mi href="./19.1#p1.t1.r3">k</mi><mn>2</mn></msup></mrow><mo>,</mo><mi>c</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E13.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.16#E5" title="(19.16.5) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="23px" altimg-valign="-7px" altimg-width="102px" alttext="R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E5">R</mi><mi href="./19.16#E5">D</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: elliptic integral symmetric in only two variables</a>,
<a href="./19.2#E6" title="(19.2.6) ‣ §19.2(ii) Legendre’s Integrals ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="23px" altimg-valign="-7px" altimg-width="72px" alttext="D\left(\NVar{\phi},\NVar{k}\right)" display="inline"><mrow><mi href="./19.2#E6">D</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r2">ϕ</mi><mo>,</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: incomplete elliptic integral of Legendre’s type</a>,
<a href="./19.1#p1.t1.r2" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m110.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="\phi" display="inline"><mi href="./19.1#p1.t1.r2">ϕ</mi></math>: real or complex argument</a> and
<a href="./19.1#p1.t1.r3" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m132.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./19.1#p1.t1.r3">k</mi></math>: real or complex modulus</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E14" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.25.14</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E14.png" altimg-height="30px" altimg-valign="-9px" altimg-width="499px" alttext="\Pi\left(\phi,\alpha^{2},k\right)-F\left(\phi,k\right)=\tfrac{1}{3}\alpha^{2}R%
_{J}\left(c-1,c-k^{2},c,c-\alpha^{2}\right)," display="block"><mrow><mrow><mrow><mrow><mi href="./19.2#E7" mathvariant="normal">Π</mi><mo></mo><mrow><mo>(</mo><mi href="./19.1#p1.t1.r2">ϕ</mi><mo>,</mo><msup><mi href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup><mo>,</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo>-</mo><mrow><mi href="./19.2#E4">F</mi><mo></mo><mrow><mo>(</mo><mi href="./19.1#p1.t1.r2">ϕ</mi><mo>,</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>3</mn></mfrac></mstyle><mo></mo><msup><mi href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup><mo></mo><mrow><msub><mi href="./19.16#E2">R</mi><mi href="./19.16#E2">J</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi>c</mi><mo>-</mo><mn>1</mn></mrow><mo>,</mo><mrow><mi>c</mi><mo>-</mo><msup><mi href="./19.1#p1.t1.r3">k</mi><mn>2</mn></msup></mrow><mo>,</mo><mi>c</mi><mo>,</mo><mrow><mi>c</mi><mo>-</mo><msup><mi href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E14.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.16#E2" title="(19.16.2) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="23px" altimg-valign="-7px" altimg-width="118px" alttext="R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)" display="inline"><mrow><msub><mi href="./19.16#E2">R</mi><mi href="./19.16#E2">J</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>,</mo><mi class="ltx_nvar">p</mi><mo>)</mo></mrow></mrow></math>: symmetric elliptic integral of third kind</a>,
<a href="./19.2#E4" title="(19.2.4) ‣ §19.2(ii) Legendre’s Integrals ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="23px" altimg-valign="-7px" altimg-width="70px" alttext="F\left(\NVar{\phi},\NVar{k}\right)" display="inline"><mrow><mi href="./19.2#E4">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r2">ϕ</mi><mo>,</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Legendre’s incomplete elliptic integral of the first kind</a>,
<a href="./19.2#E7" title="(19.2.7) ‣ §19.2(ii) Legendre’s Integrals ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="27px" altimg-valign="-9px" altimg-width="103px" alttext="\Pi\left(\NVar{\phi},\NVar{\alpha}^{2},\NVar{k}\right)" display="inline"><mrow><mi href="./19.2#E7" mathvariant="normal">Π</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r2">ϕ</mi><mo>,</mo><msup><mi class="ltx_nvar" href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup><mo>,</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Legendre’s incomplete elliptic integral of the third kind</a>,
<a href="./19.1#p1.t1.r2" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m110.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="\phi" display="inline"><mi href="./19.1#p1.t1.r2">ϕ</mi></math>: real or complex argument</a>,
<a href="./19.1#p1.t1.r3" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m132.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./19.1#p1.t1.r3">k</mi></math>: real or complex modulus</a> and
<a href="./19.1#p1.t1.r5" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m81.png" altimg-height="20px" altimg-valign="-2px" altimg-width="26px" alttext="\alpha^{2}" display="inline"><msup><mi href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup></math>: real or complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E15" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.25.15</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E15.png" altimg-height="33px" altimg-valign="-12px" altimg-width="487px" alttext="\Pi\left(\phi,\alpha^{2},k\right)=R_{-\frac{1}{2}}\left(\tfrac{1}{2},\tfrac{1}%
{2},-\tfrac{1}{2},1;c-1,c-k^{2},c,c-\alpha^{2}\right)." display="block"><mrow><mrow><mrow><mi href="./19.2#E7" mathvariant="normal">Π</mi><mo></mo><mrow><mo>(</mo><mi href="./19.1#p1.t1.r2">ϕ</mi><mo>,</mo><msup><mi href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup><mo>,</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msub><mi href="./19.16#E9">R</mi><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msub><mo></mo><mrow><mo>(</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo>,</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo>,</mo><mrow><mo>-</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle></mrow><mo>,</mo><mn>1</mn></mrow><mo>;</mo><mrow><mrow><mi>c</mi><mo>-</mo><mn>1</mn></mrow><mo>,</mo><mrow><mi>c</mi><mo>-</mo><msup><mi href="./19.1#p1.t1.r3">k</mi><mn>2</mn></msup></mrow><mo>,</mo><mi>c</mi><mo>,</mo><mrow><mi>c</mi><mo>-</mo><msup><mi href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E15.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.16#E9" title="(19.16.9) ‣ §19.16(ii) R - a ( b ; z ) ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="23px" altimg-valign="-7px" altimg-width="233px" alttext="R_{\NVar{-a}}\left(\NVar{b_{1}},\dots,\NVar{b_{n}};\NVar{z_{1}},\dots,\NVar{z_%
{n}}\right)" display="inline"><mrow><msub><mi href="./19.16#E9">R</mi><mrow><mo class="ltx_nvar">-</mo><mi class="ltx_nvar">a</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mrow><msub><mi class="ltx_nvar">b</mi><mn class="ltx_nvar">1</mn></msub><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><msub><mi class="ltx_nvar">b</mi><mi class="ltx_nvar" href="./19.1#p1.t1.r1">n</mi></msub></mrow><mo>;</mo><mrow><msub><mi class="ltx_nvar">z</mi><mn class="ltx_nvar">1</mn></msub><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><msub><mi class="ltx_nvar">z</mi><mi class="ltx_nvar" href="./19.1#p1.t1.r1">n</mi></msub></mrow><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m73.png" altimg-height="23px" altimg-valign="-7px" altimg-width="92px" alttext="R_{\NVar{-a}}\left(\NVar{\mathbf{b}};\NVar{\mathbf{z}}\right)" display="inline"><mrow><msub><mi href="./19.16#E9">R</mi><mrow><mo class="ltx_nvar">-</mo><mi class="ltx_nvar">a</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" mathvariant="bold">b</mi><mo>;</mo><mi class="ltx_nvar" mathvariant="bold">z</mi><mo>)</mo></mrow></mrow></math>: multivariate hypergeometric function</a>,
<a href="./19.2#E7" title="(19.2.7) ‣ §19.2(ii) Legendre’s Integrals ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="27px" altimg-valign="-9px" altimg-width="103px" alttext="\Pi\left(\NVar{\phi},\NVar{\alpha}^{2},\NVar{k}\right)" display="inline"><mrow><mi href="./19.2#E7" mathvariant="normal">Π</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r2">ϕ</mi><mo>,</mo><msup><mi class="ltx_nvar" href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup><mo>,</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Legendre’s incomplete elliptic integral of the third kind</a>,
<a href="./19.1#p1.t1.r2" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m110.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="\phi" display="inline"><mi href="./19.1#p1.t1.r2">ϕ</mi></math>: real or complex argument</a>,
<a href="./19.1#p1.t1.r3" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m132.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./19.1#p1.t1.r3">k</mi></math>: real or complex modulus</a> and
<a href="./19.1#p1.t1.r5" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m81.png" altimg-height="20px" altimg-valign="-2px" altimg-width="26px" alttext="\alpha^{2}" display="inline"><msup><mi href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup></math>: real or complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">If <math class="ltx_Math" altimg="m80.png" altimg-height="21px" altimg-valign="-3px" altimg-width="61px" alttext="\alpha^{2}>c" display="inline"><mrow><msup><mi href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup><mo>></mo><mi>c</mi></mrow></math>, then the Cauchy principal value is</p>
<table id="E16" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.25.16</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_Math ltx_eqn_cell ltx_align_center"><math class="ltx_Math" alttext="\Pi\left(\phi,\alpha^{2},k\right)=-\tfrac{1}{3}\omega^{2}R_{J}\left(c-1,c-k^{2%
},c,c-\omega^{2}\right)+\sqrt{\frac{(c-1)(c-k^{2})}{(\alpha^{2}-1)(1-\omega^{2%
})}}\*R_{C}\left(c(\alpha^{2}-1)(1-\omega^{2}),(\alpha^{2}-c)(c-\omega^{2})%
\right)," display="block"><mrow><mrow><mi href="./19.2#E7" mathvariant="normal">Π</mi><mo></mo><mrow><mo>(</mo><mi href="./19.1#p1.t1.r2">ϕ</mi><mo>,</mo><msup><mi href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup><mo>,</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mrow><mo>-</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>3</mn></mfrac></mstyle><mo></mo><msup><mi>ω</mi><mn>2</mn></msup><mo></mo><mrow><msub><mi href="./19.16#E2">R</mi><mi href="./19.16#E2">J</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi>c</mi><mo>-</mo><mn>1</mn></mrow><mo>,</mo><mrow><mi>c</mi><mo>-</mo><msup><mi href="./19.1#p1.t1.r3">k</mi><mn>2</mn></msup></mrow><mo>,</mo><mi>c</mi><mo>,</mo><mrow><mi>c</mi><mo>-</mo><msup><mi>ω</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>+</mo><mrow><msqrt><mfrac><mrow><mrow><mo stretchy="false">(</mo><mrow><mi>c</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>c</mi><mo>-</mo><msup><mi href="./19.1#p1.t1.r3">k</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow></mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><msup><mi href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><msup><mi>ω</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow></mrow></mfrac></msqrt><mo></mo><mrow><msub><mi href="./19.2#E17">R</mi><mi href="./19.2#E17">C</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi>c</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><msup><mi href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><msup><mi>ω</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><msup><mi href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup><mo>-</mo><mi>c</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>c</mi><mo>-</mo><msup><mi>ω</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mtd></mtr></mtable></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m95.png" altimg-height="25px" altimg-valign="-7px" altimg-width="105px" alttext="\omega^{2}=k^{2}/\alpha^{2}" display="inline"><mrow><msup><mi>ω</mi><mn>2</mn></msup><mo>=</mo><mrow><msup><mi href="./19.1#p1.t1.r3">k</mi><mn>2</mn></msup><mo>/</mo><msup><mi href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E16.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.2#E17" title="(19.2.17) ‣ §19.2(iv) A Related Function: R C ( x , y ) ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m64.png" altimg-height="23px" altimg-valign="-7px" altimg-width="82px" alttext="R_{C}\left(\NVar{x},\NVar{y}\right)" display="inline"><mrow><msub><mi href="./19.2#E17">R</mi><mi href="./19.2#E17">C</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>)</mo></mrow></mrow></math>: Carlson’s combination of inverse circular and inverse hyperbolic functions</a>,
<a href="./19.16#E2" title="(19.16.2) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="23px" altimg-valign="-7px" altimg-width="118px" alttext="R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)" display="inline"><mrow><msub><mi href="./19.16#E2">R</mi><mi href="./19.16#E2">J</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>,</mo><mi class="ltx_nvar">p</mi><mo>)</mo></mrow></mrow></math>: symmetric elliptic integral of third kind</a>,
<a href="./19.2#E7" title="(19.2.7) ‣ §19.2(ii) Legendre’s Integrals ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="27px" altimg-valign="-9px" altimg-width="103px" alttext="\Pi\left(\NVar{\phi},\NVar{\alpha}^{2},\NVar{k}\right)" display="inline"><mrow><mi href="./19.2#E7" mathvariant="normal">Π</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r2">ϕ</mi><mo>,</mo><msup><mi class="ltx_nvar" href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup><mo>,</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Legendre’s incomplete elliptic integral of the third kind</a>,
<a href="./19.1#p1.t1.r2" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m110.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="\phi" display="inline"><mi href="./19.1#p1.t1.r2">ϕ</mi></math>: real or complex argument</a>,
<a href="./19.1#p1.t1.r3" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m132.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./19.1#p1.t1.r3">k</mi></math>: real or complex modulus</a> and
<a href="./19.1#p1.t1.r5" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m81.png" altimg-height="20px" altimg-valign="-2px" altimg-width="26px" alttext="\alpha^{2}" display="inline"><msup><mi href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup></math>: real or complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS1.p3" class="ltx_para">
<p class="ltx_p">The transformations in §) as</p>
<table id="E17" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.25.17</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E17.png" altimg-height="25px" altimg-valign="-7px" altimg-width="205px" alttext="F\left(\phi,k\right)=R_{F}\left(x,y,z\right)," display="block"><mrow><mrow><mrow><mi href="./19.2#E4">F</mi><mo></mo><mrow><mo>(</mo><mi href="./19.1#p1.t1.r2">ϕ</mi><mo>,</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E17.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.16#E1" title="(19.16.1) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m68.png" altimg-height="23px" altimg-valign="-7px" altimg-width="101px" alttext="R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: symmetric elliptic integral of first kind</a>,
<a href="./19.2#E4" title="(19.2.4) ‣ §19.2(ii) Legendre’s Integrals ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="23px" altimg-valign="-7px" altimg-width="70px" alttext="F\left(\NVar{\phi},\NVar{k}\right)" display="inline"><mrow><mi href="./19.2#E4">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r2">ϕ</mi><mo>,</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Legendre’s incomplete elliptic integral of the first kind</a>,
<a href="./19.1#p1.t1.r2" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m110.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="\phi" display="inline"><mi href="./19.1#p1.t1.r2">ϕ</mi></math>: real or complex argument</a> and
<a href="./19.1#p1.t1.r3" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m132.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./19.1#p1.t1.r3">k</mi></math>: real or complex modulus</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">with</p>
<table id="E18" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.25.18</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E18.png" altimg-height="28px" altimg-valign="-7px" altimg-width="241px" alttext="(x,y,z)=(c-1,c-k^{2},c)," display="block"><mrow><mrow><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo stretchy="false">)</mo></mrow><mo>=</mo><mrow><mo stretchy="false">(</mo><mrow><mi>c</mi><mo>-</mo><mn>1</mn></mrow><mo>,</mo><mrow><mi>c</mi><mo>-</mo><msup><mi href="./19.1#p1.t1.r3">k</mi><mn>2</mn></msup></mrow><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E18.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./19.1#p1.t1.r3" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m132.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./19.1#p1.t1.r3">k</mi></math>: real or complex modulus</a></dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">then the five nontrivial permutations of <math class="ltx_Math" altimg="m144.png" altimg-height="16px" altimg-valign="-6px" altimg-width="54px" alttext="x,y,z" display="inline"><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi></mrow></math> that leave <math class="ltx_Math" altimg="m67.png" altimg-height="21px" altimg-valign="-5px" altimg-width="33px" alttext="R_{F}" display="inline"><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub></math>
invariant change <math class="ltx_Math" altimg="m134.png" altimg-height="20px" altimg-valign="-2px" altimg-width="24px" alttext="k^{2}" display="inline"><msup><mi href="./19.1#p1.t1.r3">k</mi><mn>2</mn></msup></math> (<math class="ltx_Math" altimg="m53.png" altimg-height="23px" altimg-valign="-7px" altimg-width="157px" alttext="=(z-y)/(z-x)" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mi>z</mi><mo>-</mo><mi>y</mi></mrow><mo stretchy="false">)</mo></mrow><mo>/</mo><mrow><mo stretchy="false">(</mo><mrow><mi>z</mi><mo>-</mo><mi>x</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></math>) into <math class="ltx_Math" altimg="m49.png" altimg-height="25px" altimg-valign="-7px" altimg-width="44px" alttext="1/k^{2}" display="inline"><mrow><mn>1</mn><mo>/</mo><msup><mi href="./19.1#p1.t1.r3">k</mi><mn>2</mn></msup></mrow></math>, <math class="ltx_Math" altimg="m164.png" altimg-height="23px" altimg-valign="-2px" altimg-width="30px" alttext="{k^{\prime}}^{2}" display="inline"><mmultiscripts><mi href="./19.1#p1.t1.r4">k</mi><none></none><mo>′</mo><none></none><mn>2</mn></mmultiscripts></math>, <math class="ltx_Math" altimg="m50.png" altimg-height="28px" altimg-valign="-7px" altimg-width="50px" alttext="1/{k^{\prime}}^{2}" display="inline"><mrow><mn>1</mn><mo>/</mo><mmultiscripts><mi href="./19.1#p1.t1.r4">k</mi><none></none><mo>′</mo><none></none><mn>2</mn></mmultiscripts></mrow></math>,
<math class="ltx_Math" altimg="m42.png" altimg-height="28px" altimg-valign="-7px" altimg-width="75px" alttext="-k^{2}/{k^{\prime}}^{2}" display="inline"><mrow><mo>-</mo><mrow><msup><mi href="./19.1#p1.t1.r3">k</mi><mn>2</mn></msup><mo>/</mo><mmultiscripts><mi href="./19.1#p1.t1.r4">k</mi><none></none><mo>′</mo><none></none><mn>2</mn></mmultiscripts></mrow></mrow></math>, <math class="ltx_Math" altimg="m43.png" altimg-height="28px" altimg-valign="-7px" altimg-width="75px" alttext="-{k^{\prime}}^{2}/k^{2}" display="inline"><mrow><mo>-</mo><mrow><mmultiscripts><mi href="./19.1#p1.t1.r4">k</mi><none></none><mo>′</mo><none></none><mn>2</mn></mmultiscripts><mo>/</mo><msup><mi href="./19.1#p1.t1.r3">k</mi><mn>2</mn></msup></mrow></mrow></math>, and <math class="ltx_Math" altimg="m119.png" altimg-height="21px" altimg-valign="-6px" altimg-width="44px" alttext="\sin\phi" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi href="./19.1#p1.t1.r2">ϕ</mi></mrow></math> (<math class="ltx_Math" altimg="m55.png" altimg-height="28px" altimg-valign="-8px" altimg-width="127px" alttext="=\sqrt{(z-x)/z}" display="inline"><mrow><mi></mi><mo>=</mo><msqrt><mrow><mrow><mo stretchy="false">(</mo><mrow><mi>z</mi><mo>-</mo><mi>x</mi></mrow><mo stretchy="false">)</mo></mrow><mo>/</mo><mi>z</mi></mrow></msqrt></mrow></math>) into
<math class="ltx_Math" altimg="m133.png" altimg-height="21px" altimg-valign="-6px" altimg-width="58px" alttext="k\sin\phi" display="inline"><mrow><mi href="./19.1#p1.t1.r3">k</mi><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi href="./19.1#p1.t1.r2">ϕ</mi></mrow></mrow></math>, <math class="ltx_Math" altimg="m39.png" altimg-height="21px" altimg-valign="-6px" altimg-width="74px" alttext="-i\tan\phi" display="inline"><mrow><mo>-</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mrow><mi href="./4.14#E4">tan</mi><mo></mo><mi href="./19.1#p1.t1.r2">ϕ</mi></mrow></mrow></mrow></math>, <math class="ltx_Math" altimg="m41.png" altimg-height="22px" altimg-valign="-6px" altimg-width="91px" alttext="-ik^{\prime}\tan\phi" display="inline"><mrow><mo>-</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><msup><mi href="./19.1#p1.t1.r4">k</mi><mo>′</mo></msup><mo></mo><mrow><mi href="./4.14#E4">tan</mi><mo></mo><mi href="./19.1#p1.t1.r2">ϕ</mi></mrow></mrow></mrow></math>,
<math class="ltx_Math" altimg="m37.png" altimg-height="29px" altimg-valign="-7px" altimg-width="209px" alttext="(k^{\prime}\sin\phi)/\sqrt{1-k^{2}{\sin^{2}}\phi}" display="inline"><mrow><mrow><mo stretchy="false">(</mo><mrow><msup><mi href="./19.1#p1.t1.r4">k</mi><mo>′</mo></msup><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi href="./19.1#p1.t1.r2">ϕ</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mo>/</mo><msqrt><mrow><mn>1</mn><mo>-</mo><mrow><msup><mi href="./19.1#p1.t1.r3">k</mi><mn>2</mn></msup><mo></mo><mrow><msup><mi href="./4.14#E1">sin</mi><mn>2</mn></msup><mo></mo><mi href="./19.1#p1.t1.r2">ϕ</mi></mrow></mrow></mrow></msqrt></mrow></math>,
<math class="ltx_Math" altimg="m40.png" altimg-height="29px" altimg-valign="-7px" altimg-width="210px" alttext="-ik\sin\phi/\sqrt{1-k^{2}{\sin^{2}}\phi}" display="inline"><mrow><mo>-</mo><mrow><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./19.1#p1.t1.r3">k</mi><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi href="./19.1#p1.t1.r2">ϕ</mi></mrow></mrow><mo>/</mo><msqrt><mrow><mn>1</mn><mo>-</mo><mrow><msup><mi href="./19.1#p1.t1.r3">k</mi><mn>2</mn></msup><mo></mo><mrow><msup><mi href="./4.14#E1">sin</mi><mn>2</mn></msup><mo></mo><mi href="./19.1#p1.t1.r2">ϕ</mi></mrow></mrow></mrow></msqrt></mrow></mrow></math>. Thus the five permutations
induce five transformations of Legendre’s integrals (and also of the Jacobian
elliptic functions).</p>
</div>
<div id="SS1.p4" class="ltx_para">
<p class="ltx_p">The three changes of parameter of <math class="ltx_Math" altimg="m78.png" altimg-height="27px" altimg-valign="-9px" altimg-width="103px" alttext="\Pi\left(\phi,\alpha^{2},k\right)" display="inline"><mrow><mi href="./19.2#E7" mathvariant="normal">Π</mi><mo></mo><mrow><mo>(</mo><mi href="./19.1#p1.t1.r2">ϕ</mi><mo>,</mo><msup><mi href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup><mo>,</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math> in
§ to convert them to
<math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="R" display="inline"><mi>R</mi></math>-functions.</dd>
</dl>
</div>
</div>
<div id="SS2.p1" class="ltx_para">
<p class="ltx_p">Let <math class="ltx_Math" altimg="m140.png" altimg-height="25px" altimg-valign="-7px" altimg-width="81px" alttext="r=1/x^{2}" display="inline"><mrow><mi href="./19.25#SS2.p1">r</mi><mo>=</mo><mrow><mn>1</mn><mo>/</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mrow></math>. Then
</p>
<table id="EGx2" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E19">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.25.19</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m28.png" altimg-height="25px" altimg-valign="-7px" altimg-width="122px" alttext="\displaystyle\mathrm{cel}\left(k_{c},p,a,b\right)" display="inline"><mrow><mi href="./19.2#E11">cel</mi><mo></mo><mrow><mo>(</mo><msub><mi href="./19.1#p1.t1.r3">k</mi><mi>c</mi></msub><mo>,</mo><mi>p</mi><mo>,</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m13.png" altimg-height="30px" altimg-valign="-9px" altimg-width="384px" alttext="\displaystyle=aR_{F}\left(0,k_{c}^{2},1\right)+\tfrac{1}{3}{(b-pa)}R_{J}\left(%
0,k_{c}^{2},1,p\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mi>a</mi><mo></mo><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><msubsup><mi href="./19.1#p1.t1.r3">k</mi><mi>c</mi><mn>2</mn></msubsup><mo>,</mo><mn>1</mn><mo>)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>b</mi><mo>-</mo><mrow><mi>p</mi><mo></mo><mi>a</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><msub><mi href="./19.16#E2">R</mi><mi href="./19.16#E2">J</mi></msub><mo></mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><msubsup><mi href="./19.1#p1.t1.r3">k</mi><mi>c</mi><mn>2</mn></msubsup><mo>,</mo><mn>1</mn><mo>,</mo><mi>p</mi><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E19.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.2#E11" title="(19.2.11) ‣ §19.2(iii) Bulirsch’s Integrals ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m89.png" altimg-height="23px" altimg-valign="-7px" altimg-width="121px" alttext="\mathrm{cel}\left(\NVar{k_{c}},\NVar{p},\NVar{a},\NVar{b}\right)" display="inline"><mrow><mi href="./19.2#E11">cel</mi><mo></mo><mrow><mo>(</mo><msub><mi class="ltx_nvar" href="./19.1#p1.t1.r3">k</mi><mi class="ltx_nvar">c</mi></msub><mo>,</mo><mi class="ltx_nvar">p</mi><mo>,</mo><mi class="ltx_nvar">a</mi><mo>,</mo><mi class="ltx_nvar">b</mi><mo>)</mo></mrow></mrow></math>: Bulirsch’s complete elliptic integral</a>,
<a href="./19.16#E1" title="(19.16.1) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m68.png" altimg-height="23px" altimg-valign="-7px" altimg-width="101px" alttext="R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: symmetric elliptic integral of first kind</a>,
<a href="./19.16#E2" title="(19.16.2) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="23px" altimg-valign="-7px" altimg-width="118px" alttext="R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)" display="inline"><mrow><msub><mi href="./19.16#E2">R</mi><mi href="./19.16#E2">J</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>,</mo><mi class="ltx_nvar">p</mi><mo>)</mo></mrow></mrow></math>: symmetric elliptic integral of third kind</a> and
<a href="./19.1#p1.t1.r3" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m132.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./19.1#p1.t1.r3">k</mi></math>: real or complex modulus</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E20">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.25.20</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m29.png" altimg-height="25px" altimg-valign="-7px" altimg-width="88px" alttext="\displaystyle\mathrm{el1}\left(x,k_{c}\right)" display="inline"><mrow><mi href="./19.2#E15">el1</mi><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><msub><mi href="./19.1#p1.t1.r3">k</mi><mi>c</mi></msub><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m5.png" altimg-height="30px" altimg-valign="-9px" altimg-width="210px" alttext="\displaystyle=R_{F}\left(r,r+k_{c}^{2},r+1\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./19.25#SS2.p1">r</mi><mo>,</mo><mrow><mi href="./19.25#SS2.p1">r</mi><mo>+</mo><msubsup><mi href="./19.1#p1.t1.r3">k</mi><mi>c</mi><mn>2</mn></msubsup></mrow><mo>,</mo><mrow><mi href="./19.25#SS2.p1">r</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E20.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.2#E15" title="(19.2.15) ‣ §19.2(iii) Bulirsch’s Integrals ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m90.png" altimg-height="23px" altimg-valign="-7px" altimg-width="86px" alttext="\mathrm{el1}\left(\NVar{x},\NVar{k_{c}}\right)" display="inline"><mrow><mi href="./19.2#E15">el1</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><msub><mi class="ltx_nvar" href="./19.1#p1.t1.r3">k</mi><mi class="ltx_nvar">c</mi></msub><mo>)</mo></mrow></mrow></math>: Bulirsch’s incomplete elliptic integral of the first kind</a>,
<a href="./19.16#E1" title="(19.16.1) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m68.png" altimg-height="23px" altimg-valign="-7px" altimg-width="101px" alttext="R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: symmetric elliptic integral of first kind</a>,
<a href="./19.1#p1.t1.r3" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m132.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./19.1#p1.t1.r3">k</mi></math>: real or complex modulus</a> and
<a href="./19.25#SS2.p1" title="§19.25(ii) Bulirsch’s Integrals as Symmetric Integrals ‣ §19.25 Relations to Other Functions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m141.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="r" display="inline"><mi href="./19.25#SS2.p1">r</mi></math>: inverse</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E21">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.25.21</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m30.png" altimg-height="25px" altimg-valign="-7px" altimg-width="125px" alttext="\displaystyle\mathrm{el2}\left(x,k_{c},a,b\right)" display="inline"><mrow><mi href="./19.2#E12">el2</mi><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><msub><mi href="./19.1#p1.t1.r3">k</mi><mi>c</mi></msub><mo>,</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m14.png" altimg-height="30px" altimg-valign="-9px" altimg-width="399px" alttext="\displaystyle=a\mathrm{el1}\left(x,k_{c}\right)+\tfrac{1}{3}{(b-a)}R_{D}\left(%
r,r+k_{c}^{2},r+1\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mi>a</mi><mo></mo><mrow><mi href="./19.2#E15">el1</mi><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><msub><mi href="./19.1#p1.t1.r3">k</mi><mi>c</mi></msub><mo>)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>b</mi><mo>-</mo><mi>a</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><msub><mi href="./19.16#E5">R</mi><mi href="./19.16#E5">D</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./19.25#SS2.p1">r</mi><mo>,</mo><mrow><mi href="./19.25#SS2.p1">r</mi><mo>+</mo><msubsup><mi href="./19.1#p1.t1.r3">k</mi><mi>c</mi><mn>2</mn></msubsup></mrow><mo>,</mo><mrow><mi href="./19.25#SS2.p1">r</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E21.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.2#E15" title="(19.2.15) ‣ §19.2(iii) Bulirsch’s Integrals ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m90.png" altimg-height="23px" altimg-valign="-7px" altimg-width="86px" alttext="\mathrm{el1}\left(\NVar{x},\NVar{k_{c}}\right)" display="inline"><mrow><mi href="./19.2#E15">el1</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><msub><mi class="ltx_nvar" href="./19.1#p1.t1.r3">k</mi><mi class="ltx_nvar">c</mi></msub><mo>)</mo></mrow></mrow></math>: Bulirsch’s incomplete elliptic integral of the first kind</a>,
<a href="./19.2#E12" title="(19.2.12) ‣ §19.2(iii) Bulirsch’s Integrals ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m91.png" altimg-height="23px" altimg-valign="-7px" altimg-width="123px" alttext="\mathrm{el2}\left(\NVar{x},\NVar{k_{c}},\NVar{a},\NVar{b}\right)" display="inline"><mrow><mi href="./19.2#E12">el2</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><msub><mi class="ltx_nvar" href="./19.1#p1.t1.r3">k</mi><mi class="ltx_nvar">c</mi></msub><mo>,</mo><mi class="ltx_nvar">a</mi><mo>,</mo><mi class="ltx_nvar">b</mi><mo>)</mo></mrow></mrow></math>: Bulirsch’s incomplete elliptic integral of the second kind</a>,
<a href="./19.16#E5" title="(19.16.5) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="23px" altimg-valign="-7px" altimg-width="102px" alttext="R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E5">R</mi><mi href="./19.16#E5">D</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: elliptic integral symmetric in only two variables</a>,
<a href="./19.1#p1.t1.r3" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m132.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./19.1#p1.t1.r3">k</mi></math>: real or complex modulus</a> and
<a href="./19.25#SS2.p1" title="§19.25(ii) Bulirsch’s Integrals as Symmetric Integrals ‣ §19.25 Relations to Other Functions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m141.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="r" display="inline"><mi href="./19.25#SS2.p1">r</mi></math>: inverse</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E22">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.25.22</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m31.png" altimg-height="25px" altimg-valign="-7px" altimg-width="107px" alttext="\displaystyle\mathrm{el3}\left(x,k_{c},p\right)" display="inline"><mrow><mi href="./19.2#E16">el3</mi><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><msub><mi href="./19.1#p1.t1.r3">k</mi><mi>c</mi></msub><mo>,</mo><mi>p</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m7.png" altimg-height="30px" altimg-valign="-9px" altimg-width="439px" alttext="\displaystyle=\mathrm{el1}\left(x,k_{c}\right)+\tfrac{1}{3}{(1-p)}R_{J}\left(r%
,r+k_{c}^{2},r+1,r+p\right)." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mi href="./19.2#E15">el1</mi><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><msub><mi href="./19.1#p1.t1.r3">k</mi><mi>c</mi></msub><mo>)</mo></mrow></mrow><mo>+</mo><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><mi>p</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><msub><mi href="./19.16#E2">R</mi><mi href="./19.16#E2">J</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./19.25#SS2.p1">r</mi><mo>,</mo><mrow><mi href="./19.25#SS2.p1">r</mi><mo>+</mo><msubsup><mi href="./19.1#p1.t1.r3">k</mi><mi>c</mi><mn>2</mn></msubsup></mrow><mo>,</mo><mrow><mi href="./19.25#SS2.p1">r</mi><mo>+</mo><mn>1</mn></mrow><mo>,</mo><mrow><mi href="./19.25#SS2.p1">r</mi><mo>+</mo><mi>p</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E22.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.2#E15" title="(19.2.15) ‣ §19.2(iii) Bulirsch’s Integrals ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m90.png" altimg-height="23px" altimg-valign="-7px" altimg-width="86px" alttext="\mathrm{el1}\left(\NVar{x},\NVar{k_{c}}\right)" display="inline"><mrow><mi href="./19.2#E15">el1</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><msub><mi class="ltx_nvar" href="./19.1#p1.t1.r3">k</mi><mi class="ltx_nvar">c</mi></msub><mo>)</mo></mrow></mrow></math>: Bulirsch’s incomplete elliptic integral of the first kind</a>,
<a href="./19.2#E16" title="(19.2.16) ‣ §19.2(iii) Bulirsch’s Integrals ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m92.png" altimg-height="23px" altimg-valign="-7px" altimg-width="105px" alttext="\mathrm{el3}\left(\NVar{x},\NVar{k_{c}},\NVar{p}\right)" display="inline"><mrow><mi href="./19.2#E16">el3</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><msub><mi class="ltx_nvar" href="./19.1#p1.t1.r3">k</mi><mi class="ltx_nvar">c</mi></msub><mo>,</mo><mi class="ltx_nvar">p</mi><mo>)</mo></mrow></mrow></math>: Bulirsch’s incomplete elliptic integral of the third kind</a>,
<a href="./19.16#E2" title="(19.16.2) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="23px" altimg-valign="-7px" altimg-width="118px" alttext="R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)" display="inline"><mrow><msub><mi href="./19.16#E2">R</mi><mi href="./19.16#E2">J</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>,</mo><mi class="ltx_nvar">p</mi><mo>)</mo></mrow></mrow></math>: symmetric elliptic integral of third kind</a>,
<a href="./19.1#p1.t1.r3" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m132.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./19.1#p1.t1.r3">k</mi></math>: real or complex modulus</a> and
<a href="./19.25#SS2.p1" title="§19.25(ii) Bulirsch’s Integrals as Symmetric Integrals ‣ §19.25 Relations to Other Functions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m141.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="r" display="inline"><mi href="./19.25#SS2.p1">r</mi></math>: inverse</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
</div>
</section>
<section id="iii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§19.25(iii) </span>Symmetric Integrals as Legendre’s Integrals</h2>
<div id="SS3.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="SS3.p1" class="ltx_para">
<p class="ltx_p">Assume <math class="ltx_Math" altimg="m45.png" altimg-height="20px" altimg-valign="-6px" altimg-width="126px" alttext="0\leq x\leq y\leq z" display="inline"><mrow><mn>0</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mi>y</mi><mo>≤</mo><mi>z</mi></mrow></math>, <math class="ltx_Math" altimg="m146.png" altimg-height="15px" altimg-valign="-3px" altimg-width="52px" alttext="x<z" display="inline"><mrow><mi>x</mi><mo><</mo><mi>z</mi></mrow></math>, and <math class="ltx_Math" altimg="m139.png" altimg-height="20px" altimg-valign="-6px" altimg-width="51px" alttext="p>0" display="inline"><mrow><mi>p</mi><mo>></mo><mn>0</mn></mrow></math>. Let</p>
<table id="E23" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex6" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="3" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">19.25.23</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m32.png" altimg-height="23px" altimg-valign="-6px" altimg-width="18px" alttext="\displaystyle\phi" display="inline"><mi href="./19.1#p1.t1.r2">ϕ</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m8.png" altimg-height="31px" altimg-valign="-7px" altimg-width="333px" alttext="\displaystyle=\operatorname{arccos}\sqrt{\ifrac{x}{z}}=\operatorname{arcsin}%
\sqrt{\ifrac{(z-x)}{z}}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./4.23#SS2.p1">arccos</mi><mo></mo><msqrt><mrow><mi>x</mi><mo>/</mo><mi>z</mi></mrow></msqrt></mrow><mo>=</mo><mrow><mi href="./4.23#SS2.p1">arcsin</mi><mo></mo><msqrt><mrow><mrow><mo stretchy="false">(</mo><mrow><mi>z</mi><mo>-</mo><mi>x</mi></mrow><mo stretchy="false">)</mo></mrow><mo>/</mo><mi>z</mi></mrow></msqrt></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex7" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m33.png" altimg-height="19px" altimg-valign="-2px" altimg-width="17px" alttext="\displaystyle k" display="inline"><mi href="./19.1#p1.t1.r3">k</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m9.png" altimg-height="54px" altimg-valign="-19px" altimg-width="103px" alttext="\displaystyle=\sqrt{\frac{z-y}{z-x}}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><msqrt><mstyle displaystyle="true"><mfrac><mrow><mi>z</mi><mo>-</mo><mi>y</mi></mrow><mrow><mi>z</mi><mo>-</mo><mi>x</mi></mrow></mfrac></mstyle></msqrt></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex8" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m26.png" altimg-height="23px" altimg-valign="-2px" altimg-width="28px" alttext="\displaystyle\alpha^{2}" display="inline"><msup><mi href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m6.png" altimg-height="46px" altimg-valign="-17px" altimg-width="83px" alttext="\displaystyle=\frac{z-p}{z-x}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mstyle displaystyle="true"><mfrac><mrow><mi>z</mi><mo>-</mo><mi>p</mi></mrow><mrow><mi>z</mi><mo>-</mo><mi>x</mi></mrow></mfrac></mstyle></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E23.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.23#SS2.p1" title="§4.23(ii) Principal Values ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m97.png" altimg-height="13px" altimg-valign="-2px" altimg-width="71px" alttext="\operatorname{arccos}\NVar{z}" display="inline"><mrow><mi href="./4.23#SS2.p1">arccos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: arccosine function</a>,
<a href="./4.23#SS2.p1" title="§4.23(ii) Principal Values ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m98.png" altimg-height="18px" altimg-valign="-2px" altimg-width="69px" alttext="\operatorname{arcsin}\NVar{z}" display="inline"><mrow><mi href="./4.23#SS2.p1">arcsin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: arcsine function</a>,
<a href="./19.1#p1.t1.r2" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m110.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="\phi" display="inline"><mi href="./19.1#p1.t1.r2">ϕ</mi></math>: real or complex argument</a>,
<a href="./19.1#p1.t1.r3" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m132.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./19.1#p1.t1.r3">k</mi></math>: real or complex modulus</a> and
<a href="./19.1#p1.t1.r5" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m81.png" altimg-height="20px" altimg-valign="-2px" altimg-width="26px" alttext="\alpha^{2}" display="inline"><msup><mi href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup></math>: real or complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">with <math class="ltx_Math" altimg="m79.png" altimg-height="21px" altimg-valign="-6px" altimg-width="54px" alttext="\alpha\neq 0" display="inline"><mrow><mi href="./19.1#p1.t1.r5">α</mi><mo>≠</mo><mn>0</mn></mrow></math>. Then</p>
<table id="E24" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.25.24</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E24.png" altimg-height="29px" altimg-valign="-7px" altimg-width="291px" alttext="(z-x)^{1/2}R_{F}\left(x,y,z\right)=F\left(\phi,k\right)," display="block"><mrow><mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mi>z</mi><mo>-</mo><mi>x</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo></mo><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mrow><mi href="./19.2#E4">F</mi><mo></mo><mrow><mo>(</mo><mi href="./19.1#p1.t1.r2">ϕ</mi><mo>,</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E24.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.16#E1" title="(19.16.1) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m68.png" altimg-height="23px" altimg-valign="-7px" altimg-width="101px" alttext="R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: symmetric elliptic integral of first kind</a>,
<a href="./19.2#E4" title="(19.2.4) ‣ §19.2(ii) Legendre’s Integrals ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="23px" altimg-valign="-7px" altimg-width="70px" alttext="F\left(\NVar{\phi},\NVar{k}\right)" display="inline"><mrow><mi href="./19.2#E4">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r2">ϕ</mi><mo>,</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Legendre’s incomplete elliptic integral of the first kind</a>,
<a href="./19.1#p1.t1.r2" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m110.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="\phi" display="inline"><mi href="./19.1#p1.t1.r2">ϕ</mi></math>: real or complex argument</a> and
<a href="./19.1#p1.t1.r3" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m132.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./19.1#p1.t1.r3">k</mi></math>: real or complex modulus</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E25" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.25.25</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E25.png" altimg-height="29px" altimg-valign="-7px" altimg-width="450px" alttext="(z-x)^{3/2}R_{D}\left(x,y,z\right)=(3/k^{2})(F\left(\phi,k\right)-E\left(\phi,%
k\right))," display="block"><mrow><mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mi>z</mi><mo>-</mo><mi>x</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mn>3</mn><mo>/</mo><mn>2</mn></mrow></msup><mo></mo><mrow><msub><mi href="./19.16#E5">R</mi><mi href="./19.16#E5">D</mi></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mn>3</mn><mo>/</mo><msup><mi href="./19.1#p1.t1.r3">k</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mi href="./19.2#E4">F</mi><mo></mo><mrow><mo>(</mo><mi href="./19.1#p1.t1.r2">ϕ</mi><mo>,</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo>-</mo><mrow><mi href="./19.2#E5">E</mi><mo></mo><mrow><mo>(</mo><mi href="./19.1#p1.t1.r2">ϕ</mi><mo>,</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E25.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.16#E5" title="(19.16.5) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="23px" altimg-valign="-7px" altimg-width="102px" alttext="R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E5">R</mi><mi href="./19.16#E5">D</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: elliptic integral symmetric in only two variables</a>,
<a href="./19.2#E4" title="(19.2.4) ‣ §19.2(ii) Legendre’s Integrals ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="23px" altimg-valign="-7px" altimg-width="70px" alttext="F\left(\NVar{\phi},\NVar{k}\right)" display="inline"><mrow><mi href="./19.2#E4">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r2">ϕ</mi><mo>,</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Legendre’s incomplete elliptic integral of the first kind</a>,
<a href="./19.2#E5" title="(19.2.5) ‣ §19.2(ii) Legendre’s Integrals ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="23px" altimg-valign="-7px" altimg-width="71px" alttext="E\left(\NVar{\phi},\NVar{k}\right)" display="inline"><mrow><mi href="./19.2#E5">E</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r2">ϕ</mi><mo>,</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Legendre’s incomplete elliptic integral of the second kind</a>,
<a href="./19.1#p1.t1.r2" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m110.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="\phi" display="inline"><mi href="./19.1#p1.t1.r2">ϕ</mi></math>: real or complex argument</a> and
<a href="./19.1#p1.t1.r3" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m132.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./19.1#p1.t1.r3">k</mi></math>: real or complex modulus</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E26" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.25.26</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E26.png" altimg-height="31px" altimg-valign="-9px" altimg-width="500px" alttext="(z-x)^{3/2}R_{J}\left(x,y,z,p\right)=(3/\alpha^{2}){(\Pi\left(\phi,\alpha^{2},%
k\right)-F\left(\phi,k\right))}," display="block"><mrow><mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mi>z</mi><mo>-</mo><mi>x</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mn>3</mn><mo>/</mo><mn>2</mn></mrow></msup><mo></mo><mrow><msub><mi href="./19.16#E2">R</mi><mi href="./19.16#E2">J</mi></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>,</mo><mi>p</mi><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mn>3</mn><mo>/</mo><msup><mi href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mi href="./19.2#E7" mathvariant="normal">Π</mi><mo></mo><mrow><mo>(</mo><mi href="./19.1#p1.t1.r2">ϕ</mi><mo>,</mo><msup><mi href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup><mo>,</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo>-</mo><mrow><mi href="./19.2#E4">F</mi><mo></mo><mrow><mo>(</mo><mi href="./19.1#p1.t1.r2">ϕ</mi><mo>,</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E26.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.16#E2" title="(19.16.2) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="23px" altimg-valign="-7px" altimg-width="118px" alttext="R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)" display="inline"><mrow><msub><mi href="./19.16#E2">R</mi><mi href="./19.16#E2">J</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>,</mo><mi class="ltx_nvar">p</mi><mo>)</mo></mrow></mrow></math>: symmetric elliptic integral of third kind</a>,
<a href="./19.2#E4" title="(19.2.4) ‣ §19.2(ii) Legendre’s Integrals ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="23px" altimg-valign="-7px" altimg-width="70px" alttext="F\left(\NVar{\phi},\NVar{k}\right)" display="inline"><mrow><mi href="./19.2#E4">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r2">ϕ</mi><mo>,</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Legendre’s incomplete elliptic integral of the first kind</a>,
<a href="./19.2#E7" title="(19.2.7) ‣ §19.2(ii) Legendre’s Integrals ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="27px" altimg-valign="-9px" altimg-width="103px" alttext="\Pi\left(\NVar{\phi},\NVar{\alpha}^{2},\NVar{k}\right)" display="inline"><mrow><mi href="./19.2#E7" mathvariant="normal">Π</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r2">ϕ</mi><mo>,</mo><msup><mi class="ltx_nvar" href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup><mo>,</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Legendre’s incomplete elliptic integral of the third kind</a>,
<a href="./19.1#p1.t1.r2" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m110.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="\phi" display="inline"><mi href="./19.1#p1.t1.r2">ϕ</mi></math>: real or complex argument</a>,
<a href="./19.1#p1.t1.r3" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m132.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./19.1#p1.t1.r3">k</mi></math>: real or complex modulus</a> and
<a href="./19.1#p1.t1.r5" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m81.png" altimg-height="20px" altimg-valign="-2px" altimg-width="26px" alttext="\alpha^{2}" display="inline"><msup><mi href="./19.1#p1.t1.r5">α</mi><mn>2</mn></msup></math>: real or complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E27" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.25.27</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E27.png" altimg-height="42px" altimg-valign="-11px" altimg-width="668px" alttext="2(z-x)^{-1/2}R_{G}\left(x,y,z\right)=E\left(\phi,k\right)+(\cot\phi)^{2}F\left%
(\phi,k\right)+(\cot\phi)\sqrt{1-k^{2}{\sin^{2}}\phi}." display="block"><mrow><mrow><mrow><mn>2</mn><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi>z</mi><mo>-</mo><mi>x</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></mrow></msup><mo></mo><mrow><msub><mi href="./19.16#E3">R</mi><mi href="./19.16#E3">G</mi></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mrow><mrow><mi href="./19.2#E5">E</mi><mo></mo><mrow><mo>(</mo><mi href="./19.1#p1.t1.r2">ϕ</mi><mo>,</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo>+</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./4.14#E7">cot</mi><mo></mo><mi href="./19.1#p1.t1.r2">ϕ</mi></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo></mo><mrow><mi href="./19.2#E4">F</mi><mo></mo><mrow><mo>(</mo><mi href="./19.1#p1.t1.r2">ϕ</mi><mo>,</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./4.14#E7">cot</mi><mo></mo><mi href="./19.1#p1.t1.r2">ϕ</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msqrt><mrow><mn>1</mn><mo>-</mo><mrow><msup><mi href="./19.1#p1.t1.r3">k</mi><mn>2</mn></msup><mo></mo><mrow><msup><mi href="./4.14#E1">sin</mi><mn>2</mn></msup><mo></mo><mi href="./19.1#p1.t1.r2">ϕ</mi></mrow></mrow></mrow></msqrt></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E27.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.16#E3" title="(19.16.3) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m71.png" altimg-height="23px" altimg-valign="-7px" altimg-width="101px" alttext="R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E3">R</mi><mi href="./19.16#E3">G</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: symmetric elliptic integral of second kind</a>,
<a href="./4.14#E7" title="(4.14.7) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="17px" altimg-valign="-2px" altimg-width="44px" alttext="\cot\NVar{z}" display="inline"><mrow><mi href="./4.14#E7">cot</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cotangent function</a>,
<a href="./19.2#E4" title="(19.2.4) ‣ §19.2(ii) Legendre’s Integrals ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="23px" altimg-valign="-7px" altimg-width="70px" alttext="F\left(\NVar{\phi},\NVar{k}\right)" display="inline"><mrow><mi href="./19.2#E4">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r2">ϕ</mi><mo>,</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Legendre’s incomplete elliptic integral of the first kind</a>,
<a href="./19.2#E5" title="(19.2.5) ‣ §19.2(ii) Legendre’s Integrals ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="23px" altimg-valign="-7px" altimg-width="71px" alttext="E\left(\NVar{\phi},\NVar{k}\right)" display="inline"><mrow><mi href="./19.2#E5">E</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r2">ϕ</mi><mo>,</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Legendre’s incomplete elliptic integral of the second kind</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m118.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./19.1#p1.t1.r2" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m110.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="\phi" display="inline"><mi href="./19.1#p1.t1.r2">ϕ</mi></math>: real or complex argument</a> and
<a href="./19.1#p1.t1.r3" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m132.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./19.1#p1.t1.r3">k</mi></math>: real or complex modulus</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="iv" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§19.25(iv) </span>Theta Functions</h2>
<div id="SS4.info" class="ltx_metadata ltx_info">
) use
<math class="ltx_Math" altimg="m35.png" altimg-height="23px" altimg-valign="-7px" altimg-width="236px" alttext="(\operatorname{cs},\operatorname{ds},\operatorname{ns})=(\operatorname{cn},%
\operatorname{dn},1)/\operatorname{sn}" display="inline"><mrow><mrow><mo stretchy="false">(</mo><mi href="./22.2#E9">cs</mi><mo>,</mo><mi href="./22.2#E7">ds</mi><mo>,</mo><mi href="./22.2#E4">ns</mi><mo stretchy="false">)</mo></mrow><mo>=</mo><mrow><mrow><mo stretchy="false">(</mo><mi href="./22.2#E5">cn</mi><mo>,</mo><mi href="./22.2#E6">dn</mi><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mo>/</mo><mi href="./22.2#E4">sn</mi></mrow></mrow></math>
(suppressing variables (u,k)).
For (), substitute <math class="ltx_Math" altimg="m152.png" altimg-height="23px" altimg-valign="-7px" altimg-width="111px" alttext="x=\operatorname{ps}\left(u,k\right)" display="inline"><mrow><mi>x</mi><mo>=</mo><mrow><mrow><mi mathvariant="normal">p</mi><mo href="./22.2#E10"></mo><mi mathvariant="normal">s</mi></mrow><mo></mo><mrow><mo>(</mo><mi>u</mi><mo>,</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></mrow></math>,
<math class="ltx_Math" altimg="m108.png" altimg-height="23px" altimg-valign="-7px" altimg-width="73px" alttext="\operatorname{sp}\left(u,k\right)" display="inline"><mrow><mrow><mi mathvariant="normal">s</mi><mo href="./22.2#E10"></mo><mi mathvariant="normal">p</mi></mrow><mo></mo><mrow><mo>(</mo><mi>u</mi><mo>,</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>, and <math class="ltx_Math" altimg="m106.png" altimg-height="23px" altimg-valign="-7px" altimg-width="77px" alttext="\operatorname{pq}\left(u,k\right)" display="inline"><mrow><mrow><mi mathvariant="normal">p</mi><mo href="./22.2#E10"></mo><mi mathvariant="normal">q</mi></mrow><mo></mo><mrow><mo>(</mo><mi>u</mi><mo>,</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>, respectively, to recover
(.</p>
</div>
<div id="SS5.p2" class="ltx_para">
<p class="ltx_p">With <math class="ltx_Math" altimg="m44.png" altimg-height="22px" altimg-valign="-5px" altimg-width="97px" alttext="0\leq k^{2}\leq 1" display="inline"><mrow><mn>0</mn><mo>≤</mo><msup><mi href="./19.1#p1.t1.r3">k</mi><mn>2</mn></msup><mo>≤</mo><mn>1</mn></mrow></math> and <math class="ltx_Math" altimg="m94.png" altimg-height="16px" altimg-valign="-6px" altimg-width="51px" alttext="\mathrm{p,q,r}" display="inline"><mrow><mi mathvariant="normal">p</mi><mo>,</mo><mi mathvariant="normal">q</mi><mo>,</mo><mi mathvariant="normal">r</mi></mrow></math> any permutation of the letters
<math class="ltx_Math" altimg="m88.png" altimg-height="21px" altimg-valign="-6px" altimg-width="53px" alttext="\mathrm{c,d,n}" display="inline"><mrow><mi mathvariant="normal">c</mi><mo>,</mo><mi mathvariant="normal">d</mi><mo>,</mo><mi mathvariant="normal">n</mi></mrow></math>, define</p>
<table id="E28" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.25.28</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E28.png" altimg-height="28px" altimg-valign="-7px" altimg-width="385px" alttext="\Delta(\mathrm{p,q})={\operatorname{ps}^{2}}\left(u,k\right)-{\operatorname{qs%
}^{2}}\left(u,k\right)=-\Delta(\mathrm{q,p})," display="block"><mrow><mrow><mrow><mi href="./19.25#E28" mathvariant="normal">Δ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi mathvariant="normal">p</mi><mo>,</mo><mi mathvariant="normal">q</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mrow><msup><mrow><mi mathvariant="normal">p</mi><mo href="./22.2#E10"></mo><mi mathvariant="normal">s</mi></mrow><mn>2</mn></msup><mo></mo><mrow><mo>(</mo><mi>u</mi><mo>,</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo>-</mo><mrow><msup><mrow><mi mathvariant="normal">q</mi><mo href="./22.2#E10"></mo><mi mathvariant="normal">s</mi></mrow><mn>2</mn></msup><mo></mo><mrow><mo>(</mo><mi>u</mi><mo>,</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mrow><mo>-</mo><mrow><mi href="./19.25#E28" mathvariant="normal">Δ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi mathvariant="normal">q</mi><mo>,</mo><mi mathvariant="normal">p</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E28.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text"><math class="ltx_Math" altimg="m75.png" altimg-height="23px" altimg-valign="-7px" altimg-width="67px" alttext="\Delta(\mathrm{p,q})" display="inline"><mrow><mi href="./19.25#E28" mathvariant="normal">Δ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi mathvariant="normal">p</mi><mo>,</mo><mi mathvariant="normal">q</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function (locally)</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./22.2#E10" title="(22.2.10) ‣ Glaisher’s Notation ‣ §22.2 Definitions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m105.png" altimg-height="23px" altimg-valign="-7px" altimg-width="75px" alttext="\operatorname{pq}\left(\NVar{z},\NVar{k}\right)" display="inline"><mrow><mrow><mi mathvariant="normal">p</mi><mo href="./22.2#E10"></mo><mi mathvariant="normal">q</mi></mrow><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r2">z</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: generic Jacobian elliptic function</a> and
<a href="./19.1#p1.t1.r3" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m132.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./19.1#p1.t1.r3">k</mi></math>: real or complex modulus</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">which implies</p>
<table id="E29" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex9" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="3" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">19.25.29</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m25.png" altimg-height="25px" altimg-valign="-7px" altimg-width="69px" alttext="\displaystyle\Delta(\mathrm{n,d})" display="inline"><mrow><mi href="./19.25#E28" mathvariant="normal">Δ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi mathvariant="normal">n</mi><mo>,</mo><mi mathvariant="normal">d</mi><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m15.png" altimg-height="27px" altimg-valign="-6px" altimg-width="53px" alttext="\displaystyle=k^{2}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><msup><mi href="./19.1#p1.t1.r3">k</mi><mn>2</mn></msup></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex10" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m23.png" altimg-height="25px" altimg-valign="-7px" altimg-width="67px" alttext="\displaystyle\Delta(\mathrm{d,c})" display="inline"><mrow><mi href="./19.25#E28" mathvariant="normal">Δ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi mathvariant="normal">d</mi><mo>,</mo><mi mathvariant="normal">c</mi><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m17.png" altimg-height="29px" altimg-valign="-6px" altimg-width="58px" alttext="\displaystyle={k^{\prime}}^{2}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mmultiscripts><mi href="./19.1#p1.t1.r4">k</mi><none></none><mo>′</mo><none></none><mn>2</mn></mmultiscripts></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex11" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m24.png" altimg-height="25px" altimg-valign="-7px" altimg-width="67px" alttext="\displaystyle\Delta(\mathrm{n,c})" display="inline"><mrow><mi href="./19.25#E28" mathvariant="normal">Δ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi mathvariant="normal">n</mi><mo>,</mo><mi mathvariant="normal">c</mi><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m1.png" altimg-height="18px" altimg-valign="-2px" altimg-width="43px" alttext="\displaystyle=1." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mn>1</mn></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E29.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.1#p1.t1.r3" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m132.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./19.1#p1.t1.r3">k</mi></math>: real or complex modulus</a>,
<a href="./19.1#p1.t1.r4" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m136.png" altimg-height="19px" altimg-valign="-2px" altimg-width="21px" alttext="k^{\prime}" display="inline"><msup><mi href="./19.1#p1.t1.r4">k</mi><mo>′</mo></msup></math>: complementary modulus</a> and
<a href="./19.25#E28" title="(19.25.28) ‣ §19.25(v) Jacobian Elliptic Functions ‣ §19.25 Relations to Other Functions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m75.png" altimg-height="23px" altimg-valign="-7px" altimg-width="67px" alttext="\Delta(\mathrm{p,q})" display="inline"><mrow><mi href="./19.25#E28" mathvariant="normal">Δ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi mathvariant="normal">p</mi><mo>,</mo><mi mathvariant="normal">q</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">If <math class="ltx_Math" altimg="m162.png" altimg-height="25px" altimg-valign="-7px" altimg-width="117px" alttext="{\operatorname{cs}^{2}}\left(u,k\right)\geq 0" display="inline"><mrow><mrow><msup><mi href="./22.2#E9">cs</mi><mn>2</mn></msup><mo></mo><mrow><mo>(</mo><mi>u</mi><mo>,</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo>≥</mo><mn>0</mn></mrow></math>, then</p>
<table id="E30" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.25.30</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E30.png" altimg-height="30px" altimg-valign="-9px" altimg-width="334px" alttext="\operatorname{am}\left(u,k\right)=R_{C}\left({\operatorname{cs}^{2}}\left(u,k%
\right),{\operatorname{ns}^{2}}\left(u,k\right)\right)," display="block"><mrow><mrow><mrow><mi href="./22.16#E1">am</mi><mo></mo><mrow><mo>(</mo><mi>u</mi><mo>,</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msub><mi href="./19.2#E17">R</mi><mi href="./19.2#E17">C</mi></msub><mo></mo><mrow><mo>(</mo><mrow><msup><mi href="./22.2#E9">cs</mi><mn>2</mn></msup><mo></mo><mrow><mo>(</mo><mi>u</mi><mo>,</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo>,</mo><mrow><msup><mi href="./22.2#E4">ns</mi><mn>2</mn></msup><mo></mo><mrow><mo>(</mo><mi>u</mi><mo>,</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E30.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.2#E17" title="(19.2.17) ‣ §19.2(iv) A Related Function: R C ( x , y ) ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m64.png" altimg-height="23px" altimg-valign="-7px" altimg-width="82px" alttext="R_{C}\left(\NVar{x},\NVar{y}\right)" display="inline"><mrow><msub><mi href="./19.2#E17">R</mi><mi href="./19.2#E17">C</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>)</mo></mrow></mrow></math>: Carlson’s combination of inverse circular and inverse hyperbolic functions</a>,
<a href="./22.16#E1" title="(22.16.1) ‣ Definition ‣ §22.16(i) Jacobi’s Amplitude ( am ) Function ‣ §22.16 Related Functions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m96.png" altimg-height="23px" altimg-valign="-7px" altimg-width="81px" alttext="\operatorname{am}\left(\NVar{x},\NVar{k}\right)" display="inline"><mrow><mi href="./22.16#E1">am</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobi’s amplitude function</a>,
<a href="./22.2#E9" title="(22.2.9) ‣ §22.2 Definitions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m101.png" altimg-height="23px" altimg-valign="-7px" altimg-width="70px" alttext="\operatorname{cs}\left(\NVar{z},\NVar{k}\right)" display="inline"><mrow><mi href="./22.2#E9">cs</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r2">z</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobian elliptic function</a>,
<a href="./22.2#E4" title="(22.2.4) ‣ §22.2 Definitions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m104.png" altimg-height="23px" altimg-valign="-7px" altimg-width="72px" alttext="\operatorname{ns}\left(\NVar{z},\NVar{k}\right)" display="inline"><mrow><mi href="./22.2#E4">ns</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r2">z</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobian elliptic function</a> and
<a href="./19.1#p1.t1.r3" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m132.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./19.1#p1.t1.r3">k</mi></math>: real or complex modulus</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E31" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.25.31</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E31.png" altimg-height="30px" altimg-valign="-9px" altimg-width="357px" alttext="u=R_{F}\left({\operatorname{ps}^{2}}\left(u,k\right),{\operatorname{qs}^{2}}%
\left(u,k\right),{\operatorname{rs}^{2}}\left(u,k\right)\right);" display="block"><mrow><mrow><mi>u</mi><mo>=</mo><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mrow><msup><mrow><mi mathvariant="normal">p</mi><mo href="./22.2#E10"></mo><mi mathvariant="normal">s</mi></mrow><mn>2</mn></msup><mo></mo><mrow><mo>(</mo><mi>u</mi><mo>,</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo>,</mo><mrow><msup><mrow><mi mathvariant="normal">q</mi><mo href="./22.2#E10"></mo><mi mathvariant="normal">s</mi></mrow><mn>2</mn></msup><mo></mo><mrow><mo>(</mo><mi>u</mi><mo>,</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo>,</mo><mrow><msup><mrow><mi mathvariant="normal">r</mi><mo href="./22.2#E10"></mo><mi mathvariant="normal">s</mi></mrow><mn>2</mn></msup><mo></mo><mrow><mo>(</mo><mi>u</mi><mo>,</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>;</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E31.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.16#E1" title="(19.16.1) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m68.png" altimg-height="23px" altimg-valign="-7px" altimg-width="101px" alttext="R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: symmetric elliptic integral of first kind</a>,
<a href="./22.2#E10" title="(22.2.10) ‣ Glaisher’s Notation ‣ §22.2 Definitions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m105.png" altimg-height="23px" altimg-valign="-7px" altimg-width="75px" alttext="\operatorname{pq}\left(\NVar{z},\NVar{k}\right)" display="inline"><mrow><mrow><mi mathvariant="normal">p</mi><mo href="./22.2#E10"></mo><mi mathvariant="normal">q</mi></mrow><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r2">z</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: generic Jacobian elliptic function</a> and
<a href="./19.1#p1.t1.r3" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m132.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./19.1#p1.t1.r3">k</mi></math>: real or complex modulus</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">compare ().</p>
<table id="E32" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.25.32</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E32.png" altimg-height="30px" altimg-valign="-9px" altimg-width="437px" alttext="\operatorname{arcps}\left(x,k\right)=R_{F}\left(x^{2},x^{2}+\Delta(\mathrm{q,p%
}),x^{2}+\Delta(\mathrm{r,p})\right)," display="block"><mrow><mrow><mrow><mrow><mrow><mi>arc</mi><mo></mo><mi mathvariant="normal">p</mi></mrow><mo></mo><mi mathvariant="normal">s</mi></mrow><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><msup><mi>x</mi><mn>2</mn></msup><mo>,</mo><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mrow><mi href="./19.25#E28" mathvariant="normal">Δ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi mathvariant="normal">q</mi><mo>,</mo><mi mathvariant="normal">p</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>,</mo><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mrow><mi href="./19.25#E28" mathvariant="normal">Δ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi mathvariant="normal">r</mi><mo>,</mo><mi mathvariant="normal">p</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E32.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.16#E1" title="(19.16.1) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m68.png" altimg-height="23px" altimg-valign="-7px" altimg-width="101px" alttext="R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: symmetric elliptic integral of first kind</a>,
<a href="./19.1#p1.t1.r3" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m132.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./19.1#p1.t1.r3">k</mi></math>: real or complex modulus</a> and
<a href="./19.25#E28" title="(19.25.28) ‣ §19.25(v) Jacobian Elliptic Functions ‣ §19.25 Relations to Other Functions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m75.png" altimg-height="23px" altimg-valign="-7px" altimg-width="67px" alttext="\Delta(\mathrm{p,q})" display="inline"><mrow><mi href="./19.25#E28" mathvariant="normal">Δ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi mathvariant="normal">p</mi><mo>,</mo><mi mathvariant="normal">q</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E33" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.25.33</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E33.png" altimg-height="30px" altimg-valign="-9px" altimg-width="458px" alttext="\operatorname{arcsp}\left(x,k\right)=xR_{F}\left(1,1+\Delta(\mathrm{q,p})x^{2}%
,1+\Delta(\mathrm{r,p})x^{2}\right)," display="block"><mrow><mrow><mrow><mrow><mrow><mi>arc</mi><mo></mo><mi mathvariant="normal">s</mi></mrow><mo></mo><mi mathvariant="normal">p</mi></mrow><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mi>x</mi><mo></mo><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mn>1</mn><mo>,</mo><mrow><mn>1</mn><mo>+</mo><mrow><mrow><mi href="./19.25#E28" mathvariant="normal">Δ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi mathvariant="normal">q</mi><mo>,</mo><mi mathvariant="normal">p</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mrow><mo>,</mo><mrow><mn>1</mn><mo>+</mo><mrow><mrow><mi href="./19.25#E28" mathvariant="normal">Δ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi mathvariant="normal">r</mi><mo>,</mo><mi mathvariant="normal">p</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E33.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.16#E1" title="(19.16.1) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m68.png" altimg-height="23px" altimg-valign="-7px" altimg-width="101px" alttext="R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: symmetric elliptic integral of first kind</a>,
<a href="./19.1#p1.t1.r3" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m132.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./19.1#p1.t1.r3">k</mi></math>: real or complex modulus</a> and
<a href="./19.25#E28" title="(19.25.28) ‣ §19.25(v) Jacobian Elliptic Functions ‣ §19.25 Relations to Other Functions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m75.png" altimg-height="23px" altimg-valign="-7px" altimg-width="67px" alttext="\Delta(\mathrm{p,q})" display="inline"><mrow><mi href="./19.25#E28" mathvariant="normal">Δ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi mathvariant="normal">p</mi><mo>,</mo><mi mathvariant="normal">q</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E34" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.25.34</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E34.png" altimg-height="30px" altimg-valign="-9px" altimg-width="378px" alttext="\operatorname{arcpq}\left(x,k\right)=\sqrt{w}R_{F}\left(x^{2},1,1+\Delta(%
\mathrm{r,q})w\right)," display="block"><mrow><mrow><mrow><mrow><mrow><mi>arc</mi><mo></mo><mi mathvariant="normal">p</mi></mrow><mo></mo><mi mathvariant="normal">q</mi></mrow><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msqrt><mi>w</mi></msqrt><mo></mo><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><msup><mi>x</mi><mn>2</mn></msup><mo>,</mo><mn>1</mn><mo>,</mo><mrow><mn>1</mn><mo>+</mo><mrow><mrow><mi href="./19.25#E28" mathvariant="normal">Δ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi mathvariant="normal">r</mi><mo>,</mo><mi mathvariant="normal">q</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mi>w</mi></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m143.png" altimg-height="27px" altimg-valign="-9px" altimg-width="195px" alttext="w=\ifrac{(1-x^{2})}{\Delta(\mathrm{q,p})}" display="inline"><mrow><mi>w</mi><mo>=</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mo>/</mo><mrow><mi href="./19.25#E28" mathvariant="normal">Δ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi mathvariant="normal">q</mi><mo>,</mo><mi mathvariant="normal">p</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E34.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.16#E1" title="(19.16.1) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m68.png" altimg-height="23px" altimg-valign="-7px" altimg-width="101px" alttext="R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: symmetric elliptic integral of first kind</a>,
<a href="./19.1#p1.t1.r3" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m132.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./19.1#p1.t1.r3">k</mi></math>: real or complex modulus</a> and
<a href="./19.25#E28" title="(19.25.28) ‣ §19.25(v) Jacobian Elliptic Functions ‣ §19.25 Relations to Other Functions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m75.png" altimg-height="23px" altimg-valign="-7px" altimg-width="67px" alttext="\Delta(\mathrm{p,q})" display="inline"><mrow><mi href="./19.25#E28" mathvariant="normal">Δ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi mathvariant="normal">p</mi><mo>,</mo><mi mathvariant="normal">q</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where we assume <math class="ltx_Math" altimg="m47.png" altimg-height="22px" altimg-valign="-5px" altimg-width="98px" alttext="0\leq x^{2}\leq 1" display="inline"><mrow><mn>0</mn><mo>≤</mo><msup><mi>x</mi><mn>2</mn></msup><mo>≤</mo><mn>1</mn></mrow></math> if <math class="ltx_Math" altimg="m154.png" altimg-height="13px" altimg-valign="-2px" altimg-width="61px" alttext="x=\operatorname{sn}" display="inline"><mrow><mi>x</mi><mo>=</mo><mi href="./22.2#E4">sn</mi></mrow></math>, <math class="ltx_Math" altimg="m100.png" altimg-height="13px" altimg-valign="-2px" altimg-width="24px" alttext="\operatorname{cn}" display="inline"><mi href="./22.2#E5">cn</mi></math>, or
<math class="ltx_Math" altimg="m99.png" altimg-height="18px" altimg-valign="-2px" altimg-width="24px" alttext="\operatorname{cd}" display="inline"><mi href="./22.2#E8">cd</mi></math>; <math class="ltx_Math" altimg="m156.png" altimg-height="22px" altimg-valign="-5px" altimg-width="61px" alttext="x^{2}\geq 1" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>≥</mo><mn>1</mn></mrow></math> if <math class="ltx_Math" altimg="m151.png" altimg-height="13px" altimg-valign="-2px" altimg-width="61px" alttext="x=\operatorname{ns}" display="inline"><mrow><mi>x</mi><mo>=</mo><mi href="./22.2#E4">ns</mi></mrow></math>, <math class="ltx_Math" altimg="m103.png" altimg-height="13px" altimg-valign="-2px" altimg-width="24px" alttext="\operatorname{nc}" display="inline"><mi href="./22.2#E5">nc</mi></math>, or <math class="ltx_Math" altimg="m102.png" altimg-height="18px" altimg-valign="-2px" altimg-width="24px" alttext="\operatorname{dc}" display="inline"><mi href="./22.2#E8">dc</mi></math>; <math class="ltx_Math" altimg="m155.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math>
real if <math class="ltx_Math" altimg="m147.png" altimg-height="13px" altimg-valign="-2px" altimg-width="59px" alttext="x=\operatorname{cs}" display="inline"><mrow><mi>x</mi><mo>=</mo><mi href="./22.2#E9">cs</mi></mrow></math> or <math class="ltx_Math" altimg="m107.png" altimg-height="13px" altimg-valign="-2px" altimg-width="21px" alttext="\operatorname{sc}" display="inline"><mi href="./22.2#E9">sc</mi></math>; <math class="ltx_Math" altimg="m137.png" altimg-height="21px" altimg-valign="-5px" altimg-width="95px" alttext="k^{\prime}\leq x\leq 1" display="inline"><mrow><msup><mi href="./19.1#p1.t1.r4">k</mi><mo>′</mo></msup><mo>≤</mo><mi>x</mi><mo>≤</mo><mn>1</mn></mrow></math> if <math class="ltx_Math" altimg="m148.png" altimg-height="18px" altimg-valign="-2px" altimg-width="64px" alttext="x=\operatorname{dn}" display="inline"><mrow><mi>x</mi><mo>=</mo><mi href="./22.2#E6">dn</mi></mrow></math>;
<math class="ltx_Math" altimg="m52.png" altimg-height="24px" altimg-valign="-7px" altimg-width="115px" alttext="1\leq x\leq 1/k^{\prime}" display="inline"><mrow><mn>1</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mrow><mn>1</mn><mo>/</mo><msup><mi href="./19.1#p1.t1.r4">k</mi><mo>′</mo></msup></mrow></mrow></math> if <math class="ltx_Math" altimg="m150.png" altimg-height="18px" altimg-valign="-2px" altimg-width="64px" alttext="x=\operatorname{nd}" display="inline"><mrow><mi>x</mi><mo>=</mo><mi href="./22.2#E6">nd</mi></mrow></math>; <math class="ltx_Math" altimg="m157.png" altimg-height="25px" altimg-valign="-5px" altimg-width="77px" alttext="x^{2}\geq{k^{\prime}}^{2}" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>≥</mo><mmultiscripts><mi href="./19.1#p1.t1.r4">k</mi><none></none><mo>′</mo><none></none><mn>2</mn></mmultiscripts></mrow></math> if <math class="ltx_Math" altimg="m149.png" altimg-height="18px" altimg-valign="-2px" altimg-width="61px" alttext="x=\operatorname{ds}" display="inline"><mrow><mi>x</mi><mo>=</mo><mi href="./22.2#E7">ds</mi></mrow></math>;
<math class="ltx_Math" altimg="m46.png" altimg-height="28px" altimg-valign="-7px" altimg-width="133px" alttext="0\leq x^{2}\leq 1/{k^{\prime}}^{2}" display="inline"><mrow><mn>0</mn><mo>≤</mo><msup><mi>x</mi><mn>2</mn></msup><mo>≤</mo><mrow><mn>1</mn><mo>/</mo><mmultiscripts><mi href="./19.1#p1.t1.r4">k</mi><none></none><mo>′</mo><none></none><mn>2</mn></mmultiscripts></mrow></mrow></math> if <math class="ltx_Math" altimg="m153.png" altimg-height="18px" altimg-valign="-2px" altimg-width="61px" alttext="x=\operatorname{sd}" display="inline"><mrow><mi>x</mi><mo>=</mo><mi href="./22.2#E7">sd</mi></mrow></math>.</p>
</div>
<div id="SS5.p3" class="ltx_para">
<p class="ltx_p">For the use of <math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="R" display="inline"><mi>R</mi></math>-functions with <math class="ltx_Math" altimg="m75.png" altimg-height="23px" altimg-valign="-7px" altimg-width="67px" alttext="\Delta(\mathrm{p,q})" display="inline"><mrow><mi href="./19.25#E28" mathvariant="normal">Δ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi mathvariant="normal">p</mi><mo>,</mo><mi mathvariant="normal">q</mi><mo stretchy="false">)</mo></mrow></mrow></math> in unifying other
properties of Jacobian elliptic functions, see <cite class="ltx_cite ltx_citemacro_citet">Carlson ()</cite>.</p>
</div>
<div id="SS5.p4" class="ltx_para">
<p class="ltx_p">Inversions of 12 elliptic integrals of the first kind, producing the 12
Jacobian elliptic functions, are combined and simplified by using the
properties of <math class="ltx_Math" altimg="m69.png" altimg-height="23px" altimg-valign="-7px" altimg-width="101px" alttext="R_{F}\left(x,y,z\right)" display="inline"><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow></mrow></math>. See (), with
<math class="ltx_Math" altimg="m159.png" altimg-height="23px" altimg-valign="-7px" altimg-width="87px" alttext="z=\wp\left(w\right)" display="inline"><mrow><mi>z</mi><mo>=</mo><mrow><mi href="./23.2#E4" mathvariant="normal">℘</mi><mo></mo><mrow><mo>(</mo><mi>w</mi><mo>)</mo></mrow></mrow></mrow></math> as prescribed in the text that follows
(), substitute
<math class="ltx_Math" altimg="m142.png" altimg-height="23px" altimg-valign="-7px" altimg-width="120px" alttext="u=t+\wp\left(w\right)" display="inline"><mrow><mi>u</mi><mo>=</mo><mrow><mi>t</mi><mo>+</mo><mrow><mi href="./23.2#E4" mathvariant="normal">℘</mi><mo></mo><mrow><mo>(</mo><mi>w</mi><mo>)</mo></mrow></mrow></mrow></mrow></math> and compare with ().
Then put <math class="ltx_Math" altimg="m158.png" altimg-height="18px" altimg-valign="-8px" altimg-width="62px" alttext="z=\omega_{j}" display="inline"><mrow><mi>z</mi><mo>=</mo><msub><mi>ω</mi><mi>j</mi></msub></mrow></math> to obtain (.</p>
<table id="E35" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.25.35</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E35.png" altimg-height="25px" altimg-valign="-7px" altimg-width="369px" alttext="z=R_{F}\left(\wp\left(z\right)-e_{1},\wp\left(z\right)-e_{2},\wp\left(z\right)%
-e_{3}\right)," display="block"><mrow><mrow><mi>z</mi><mo>=</mo><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mrow><mi href="./23.2#E4" mathvariant="normal">℘</mi><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow><mo>-</mo><msub><mi href="./4.2#E11" mathvariant="normal">e</mi><mn>1</mn></msub></mrow><mo>,</mo><mrow><mrow><mi href="./23.2#E4" mathvariant="normal">℘</mi><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow><mo>-</mo><msub><mi href="./4.2#E11" mathvariant="normal">e</mi><mn>2</mn></msub></mrow><mo>,</mo><mrow><mrow><mi href="./23.2#E4" mathvariant="normal">℘</mi><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow><mo>-</mo><msub><mi href="./4.2#E11" mathvariant="normal">e</mi><mn>3</mn></msub></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E35.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.16#E1" title="(19.16.1) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m68.png" altimg-height="23px" altimg-valign="-7px" altimg-width="101px" alttext="R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: symmetric elliptic integral of first kind</a>,
<a href="./23.2#E4" title="(23.2.4) ‣ §23.2(ii) Weierstrass Elliptic Functions ‣ §23.2 Definitions and Periodic Properties ‣ Weierstrass Elliptic Functions ‣ Chapter 23 Weierstrass Elliptic and Modular Functions" class="ltx_ref"><math class="ltx_Math" altimg="m121.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\wp\left(\NVar{z}\right)" display="inline"><mrow><mi href="./23.2#E4" mathvariant="normal">℘</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./23.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>
(= <math class="ltx_Math" altimg="m123.png" altimg-height="23px" altimg-valign="-7px" altimg-width="65px" alttext="\wp\left(z|\mathbb{L}\right)" display="inline"><mrow><mi href="./23.2#E4" mathvariant="normal">℘</mi><mo></mo><mrow><mo>(</mo><mi href="./23.1#p2.t1.r4">z</mi><mo stretchy="false">|</mo><mi href="./23.1#p2.t1.r1">𝕃</mi><mo>)</mo></mrow></mrow></math> = <math class="ltx_Math" altimg="m122.png" altimg-height="23px" altimg-valign="-7px" altimg-width="101px" alttext="\wp\left(z;g_{2},g_{3}\right)" display="inline"><mrow><mi href="./23.3#E8" mathvariant="normal">℘</mi><mo></mo><mrow><mo>(</mo><mi href="./23.1#p2.t1.r4">z</mi><mo>;</mo><msub><mi href="./23.1#p2.t1.r13">g</mi><mn>2</mn></msub><mo>,</mo><msub><mi href="./23.1#p2.t1.r13">g</mi><mn>3</mn></msub><mo>)</mo></mrow></mrow></math>): Weierstrass <math class="ltx_Math" altimg="m120.png" altimg-height="16px" altimg-valign="-6px" altimg-width="17px" alttext="\wp" display="inline"><mi href="./23.2#E4" mathvariant="normal">℘</mi></math>-function</a> and
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m93.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">provided that
</p>
<table id="E36" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.25.36</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E36.png" altimg-height="26px" altimg-valign="-8px" altimg-width="220px" alttext="\wp\left(z\right)-e_{j}\in\mathbb{C}\setminus(-\infty,0]," display="block"><mrow><mrow><mrow><mrow><mi href="./23.2#E4" mathvariant="normal">℘</mi><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow><mo>-</mo><msub><mi href="./4.2#E11" mathvariant="normal">e</mi><mi>j</mi></msub></mrow><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mrow><mi href="./front/introduction#Sx4.p1.t1.r1">ℂ</mi><mo href="./front/introduction#Sx4.p2.t1.r19">∖</mo><mrow><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">(</mo><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mo href="./front/introduction#Sx4.p2.t1.r1">,</mo><mn>0</mn><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">]</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m130.png" altimg-height="20px" altimg-valign="-6px" altimg-width="88px" alttext="j=1,2,3" display="inline"><mrow><mi>j</mi><mo>=</mo><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E36.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./23.2#E4" title="(23.2.4) ‣ §23.2(ii) Weierstrass Elliptic Functions ‣ §23.2 Definitions and Periodic Properties ‣ Weierstrass Elliptic Functions ‣ Chapter 23 Weierstrass Elliptic and Modular Functions" class="ltx_ref"><math class="ltx_Math" altimg="m121.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\wp\left(\NVar{z}\right)" display="inline"><mrow><mi href="./23.2#E4" mathvariant="normal">℘</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./23.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>
(= <math class="ltx_Math" altimg="m123.png" altimg-height="23px" altimg-valign="-7px" altimg-width="65px" alttext="\wp\left(z|\mathbb{L}\right)" display="inline"><mrow><mi href="./23.2#E4" mathvariant="normal">℘</mi><mo></mo><mrow><mo>(</mo><mi href="./23.1#p2.t1.r4">z</mi><mo stretchy="false">|</mo><mi href="./23.1#p2.t1.r1">𝕃</mi><mo>)</mo></mrow></mrow></math> = <math class="ltx_Math" altimg="m122.png" altimg-height="23px" altimg-valign="-7px" altimg-width="101px" alttext="\wp\left(z;g_{2},g_{3}\right)" display="inline"><mrow><mi href="./23.3#E8" mathvariant="normal">℘</mi><mo></mo><mrow><mo>(</mo><mi href="./23.1#p2.t1.r4">z</mi><mo>;</mo><msub><mi href="./23.1#p2.t1.r13">g</mi><mn>2</mn></msub><mo>,</mo><msub><mi href="./23.1#p2.t1.r13">g</mi><mn>3</mn></msub><mo>)</mo></mrow></mrow></math>): Weierstrass <math class="ltx_Math" altimg="m120.png" altimg-height="16px" altimg-valign="-6px" altimg-width="17px" alttext="\wp" display="inline"><mi href="./23.2#E4" mathvariant="normal">℘</mi></math>-function</a>,
<a href="./front/introduction#Sx4.p1.t1.r1" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\mathbb{C}" display="inline"><mi href="./front/introduction#Sx4.p1.t1.r1">ℂ</mi></math>: complex plane</a>,
<a href="./front/introduction#Sx4.p1.t1.r9" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m86.png" altimg-height="15px" altimg-valign="-3px" altimg-width="18px" alttext="\in" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo></math>: element of</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m93.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./front/introduction#Sx4.p2.t1.r1" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m34.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="(\NVar{a},\NVar{b}]" display="inline"><mrow><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">(</mo><mi class="ltx_nvar">a</mi><mo href="./front/introduction#Sx4.p2.t1.r1">,</mo><mi class="ltx_nvar">b</mi><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">]</mo></mrow></math>: half-closed interval</a> and
<a href="./front/introduction#Sx4.p2.t1.r19" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m113.png" altimg-height="23px" altimg-valign="-7px" altimg-width="14px" alttext="\setminus" display="inline"><mo href="./front/introduction#Sx4.p2.t1.r19">∖</mo></math>: set subtraction</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and the left-hand side does not vanish for more than one value of <math class="ltx_Math" altimg="m131.png" altimg-height="20px" altimg-valign="-6px" altimg-width="14px" alttext="j" display="inline"><mi>j</mi></math>. Also,</p>
<table id="E37" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.25.37</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E37.png" altimg-height="25px" altimg-valign="-7px" altimg-width="481px" alttext="\zeta\left(z\right)+z\wp\left(z\right)=2\!R_{G}\left(\wp\left(z\right)-e_{1},%
\wp\left(z\right)-e_{2},\wp\left(z\right)-e_{3}\right)." display="block"><mrow><mrow><mrow><mrow><mi href="./25.2#E1">ζ</mi><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow><mo>+</mo><mrow><mi>z</mi><mo></mo><mrow><mi href="./23.2#E4" mathvariant="normal">℘</mi><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>=</mo><mrow><mpadded width="-1.7pt"><mn>2</mn></mpadded><mo></mo><mrow><msub><mi href="./19.16#E3">R</mi><mi href="./19.16#E3">G</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mrow><mi href="./23.2#E4" mathvariant="normal">℘</mi><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow><mo>-</mo><msub><mi href="./4.2#E11" mathvariant="normal">e</mi><mn>1</mn></msub></mrow><mo>,</mo><mrow><mrow><mi href="./23.2#E4" mathvariant="normal">℘</mi><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow><mo>-</mo><msub><mi href="./4.2#E11" mathvariant="normal">e</mi><mn>2</mn></msub></mrow><mo>,</mo><mrow><mrow><mi href="./23.2#E4" mathvariant="normal">℘</mi><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow><mo>-</mo><msub><mi href="./4.2#E11" mathvariant="normal">e</mi><mn>3</mn></msub></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E37.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.16#E3" title="(19.16.3) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m71.png" altimg-height="23px" altimg-valign="-7px" altimg-width="101px" alttext="R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E3">R</mi><mi href="./19.16#E3">G</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: symmetric elliptic integral of second kind</a>,
<a href="./25.2#E1" title="(25.2.1) ‣ §25.2(i) Definition ‣ §25.2 Definition and Expansions ‣ Riemann Zeta Function ‣ Chapter 25 Zeta and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m124.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="\zeta\left(\NVar{s}\right)" display="inline"><mrow><mi href="./25.2#E1">ζ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./25.1#p2.t1.r5">s</mi><mo>)</mo></mrow></mrow></math>: Riemann zeta function</a>,
<a href="./23.2#E4" title="(23.2.4) ‣ §23.2(ii) Weierstrass Elliptic Functions ‣ §23.2 Definitions and Periodic Properties ‣ Weierstrass Elliptic Functions ‣ Chapter 23 Weierstrass Elliptic and Modular Functions" class="ltx_ref"><math class="ltx_Math" altimg="m121.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\wp\left(\NVar{z}\right)" display="inline"><mrow><mi href="./23.2#E4" mathvariant="normal">℘</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./23.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>
(= <math class="ltx_Math" altimg="m123.png" altimg-height="23px" altimg-valign="-7px" altimg-width="65px" alttext="\wp\left(z|\mathbb{L}\right)" display="inline"><mrow><mi href="./23.2#E4" mathvariant="normal">℘</mi><mo></mo><mrow><mo>(</mo><mi href="./23.1#p2.t1.r4">z</mi><mo stretchy="false">|</mo><mi href="./23.1#p2.t1.r1">𝕃</mi><mo>)</mo></mrow></mrow></math> = <math class="ltx_Math" altimg="m122.png" altimg-height="23px" altimg-valign="-7px" altimg-width="101px" alttext="\wp\left(z;g_{2},g_{3}\right)" display="inline"><mrow><mi href="./23.3#E8" mathvariant="normal">℘</mi><mo></mo><mrow><mo>(</mo><mi href="./23.1#p2.t1.r4">z</mi><mo>;</mo><msub><mi href="./23.1#p2.t1.r13">g</mi><mn>2</mn></msub><mo>,</mo><msub><mi href="./23.1#p2.t1.r13">g</mi><mn>3</mn></msub><mo>)</mo></mrow></mrow></math>): Weierstrass <math class="ltx_Math" altimg="m120.png" altimg-height="16px" altimg-valign="-6px" altimg-width="17px" alttext="\wp" display="inline"><mi href="./23.2#E4" mathvariant="normal">℘</mi></math>-function</a> and
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m93.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS6.p2" class="ltx_para">
<p class="ltx_p">In () <math class="ltx_Math" altimg="m129.png" altimg-height="21px" altimg-valign="-6px" altimg-width="50px" alttext="j,k,\ell" display="inline"><mrow><mi>j</mi><mo>,</mo><mi href="./19.1#p1.t1.r3">k</mi><mo>,</mo><mi mathvariant="normal">ℓ</mi></mrow></math> is any
permutation of the numbers <math class="ltx_Math" altimg="m48.png" altimg-height="20px" altimg-valign="-6px" altimg-width="52px" alttext="1,2,3" display="inline"><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn></mrow></math>.</p>
<table id="E38" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.25.38</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E38.png" altimg-height="26px" altimg-valign="-8px" altimg-width="257px" alttext="\omega_{j}=R_{F}\left(0,e_{j}-e_{k},e_{j}-e_{\ell}\right)," display="block"><mrow><mrow><msub><mi>ω</mi><mi>j</mi></msub><mo>=</mo><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mrow><msub><mi href="./4.2#E11" mathvariant="normal">e</mi><mi>j</mi></msub><mo>-</mo><msub><mi href="./4.2#E11" mathvariant="normal">e</mi><mi href="./19.1#p1.t1.r3">k</mi></msub></mrow><mo>,</mo><mrow><msub><mi href="./4.2#E11" mathvariant="normal">e</mi><mi>j</mi></msub><mo>-</mo><msub><mi href="./4.2#E11" mathvariant="normal">e</mi><mi mathvariant="normal">ℓ</mi></msub></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E38.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.16#E1" title="(19.16.1) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m68.png" altimg-height="23px" altimg-valign="-7px" altimg-width="101px" alttext="R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: symmetric elliptic integral of first kind</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m93.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a> and
<a href="./19.1#p1.t1.r3" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m132.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./19.1#p1.t1.r3">k</mi></math>: real or complex modulus</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E39" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.25.39</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E39.png" altimg-height="26px" altimg-valign="-8px" altimg-width="324px" alttext="\eta_{j}+\omega_{j}e_{j}=2\!R_{G}\left(0,e_{j}-e_{k},e_{j}-e_{\ell}\right)." display="block"><mrow><mrow><mrow><msub><mi>η</mi><mi>j</mi></msub><mo>+</mo><mrow><msub><mi>ω</mi><mi>j</mi></msub><mo></mo><msub><mi href="./4.2#E11" mathvariant="normal">e</mi><mi>j</mi></msub></mrow></mrow><mo>=</mo><mrow><mpadded width="-1.7pt"><mn>2</mn></mpadded><mo></mo><mrow><msub><mi href="./19.16#E3">R</mi><mi href="./19.16#E3">G</mi></msub><mo></mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mrow><msub><mi href="./4.2#E11" mathvariant="normal">e</mi><mi>j</mi></msub><mo>-</mo><msub><mi href="./4.2#E11" mathvariant="normal">e</mi><mi href="./19.1#p1.t1.r3">k</mi></msub></mrow><mo>,</mo><mrow><msub><mi href="./4.2#E11" mathvariant="normal">e</mi><mi>j</mi></msub><mo>-</mo><msub><mi href="./4.2#E11" mathvariant="normal">e</mi><mi mathvariant="normal">ℓ</mi></msub></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E39.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.16#E3" title="(19.16.3) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m71.png" altimg-height="23px" altimg-valign="-7px" altimg-width="101px" alttext="R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E3">R</mi><mi href="./19.16#E3">G</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: symmetric elliptic integral of second kind</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m93.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a> and
<a href="./19.1#p1.t1.r3" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m132.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./19.1#p1.t1.r3">k</mi></math>: real or complex modulus</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS6.p3" class="ltx_para">
<p class="ltx_p">Lastly,</p>
<table id="E40" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.25.40</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E40.png" altimg-height="30px" altimg-valign="-9px" altimg-width="304px" alttext="z=\sigma\left(z\right)R_{F}\left(\sigma_{1}^{2}(z),\sigma_{2}^{2}(z),\sigma_{3%
}^{2}(z)\right)," display="block"><mrow><mrow><mi>z</mi><mo>=</mo><mrow><mrow><mi href="./23.2#E6">σ</mi><mo></mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mrow><msubsup><mi href="./19.25#E41">σ</mi><mn>1</mn><mn>2</mn></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo><mrow><msubsup><mi href="./19.25#E41">σ</mi><mn>2</mn><mn>2</mn></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo><mrow><msubsup><mi href="./19.25#E41">σ</mi><mn>3</mn><mn>2</mn></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E40.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.16#E1" title="(19.16.1) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m68.png" altimg-height="23px" altimg-valign="-7px" altimg-width="101px" alttext="R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: symmetric elliptic integral of first kind</a>,
<a href="./23.2#E6" title="(23.2.6) ‣ §23.2(ii) Weierstrass Elliptic Functions ‣ §23.2 Definitions and Periodic Properties ‣ Weierstrass Elliptic Functions ‣ Chapter 23 Weierstrass Elliptic and Modular Functions" class="ltx_ref"><math class="ltx_Math" altimg="m114.png" altimg-height="23px" altimg-valign="-7px" altimg-width="45px" alttext="\sigma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./23.2#E6">σ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./23.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>
(= <math class="ltx_Math" altimg="m116.png" altimg-height="23px" altimg-valign="-7px" altimg-width="64px" alttext="\sigma\left(z|\mathbb{L}\right)" display="inline"><mrow><mi href="./23.2#E6">σ</mi><mo></mo><mrow><mo>(</mo><mi href="./23.1#p2.t1.r4">z</mi><mo stretchy="false">|</mo><mi href="./23.1#p2.t1.r1">𝕃</mi><mo>)</mo></mrow></mrow></math> = <math class="ltx_Math" altimg="m115.png" altimg-height="23px" altimg-valign="-7px" altimg-width="100px" alttext="\sigma\left(z;g_{2},g_{3}\right)" display="inline"><mrow><mi href="./23.3#SS1.p3">σ</mi><mo></mo><mrow><mo>(</mo><mi href="./23.1#p2.t1.r4">z</mi><mo>;</mo><msub><mi href="./23.1#p2.t1.r13">g</mi><mn>2</mn></msub><mo>,</mo><msub><mi href="./23.1#p2.t1.r13">g</mi><mn>3</mn></msub><mo>)</mo></mrow></mrow></math>): Weierstrass sigma function</a> and
<a href="./19.25#E41" title="(19.25.41) ‣ §19.25(vi) Weierstrass Elliptic Functions ‣ §19.25 Relations to Other Functions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m117.png" altimg-height="24px" altimg-valign="-8px" altimg-width="50px" alttext="\sigma_{j}(z)" display="inline"><mrow><msub><mi href="./19.25#E41">σ</mi><mi>j</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where</p>
<table id="E41" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.25.41</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E41.png" altimg-height="26px" altimg-valign="-8px" altimg-width="335px" alttext="\sigma_{j}(z)=\exp\left(-\eta_{j}z\right)\sigma\left(z+\omega_{j}\right)/%
\sigma\left(\omega_{j}\right)," display="block"><mrow><mrow><mrow><msub><mi href="./19.25#E41">σ</mi><mi>j</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mrow><msub><mi>η</mi><mi>j</mi></msub><mo></mo><mi>z</mi></mrow></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./23.2#E6">σ</mi><mo></mo><mrow><mo>(</mo><mrow><mi>z</mi><mo>+</mo><msub><mi>ω</mi><mi>j</mi></msub></mrow><mo>)</mo></mrow></mrow></mrow><mo>/</mo><mrow><mi href="./23.2#E6">σ</mi><mo></mo><mrow><mo>(</mo><msub><mi>ω</mi><mi>j</mi></msub><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m130.png" altimg-height="20px" altimg-valign="-6px" altimg-width="88px" alttext="j=1,2,3" display="inline"><mrow><mi>j</mi><mo>=</mo><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E41.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text"><math class="ltx_Math" altimg="m117.png" altimg-height="24px" altimg-valign="-8px" altimg-width="50px" alttext="\sigma_{j}(z)" display="inline"><mrow><msub><mi href="./19.25#E41">σ</mi><mi>j</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function (locally)</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./23.2#E6" title="(23.2.6) ‣ §23.2(ii) Weierstrass Elliptic Functions ‣ §23.2 Definitions and Periodic Properties ‣ Weierstrass Elliptic Functions ‣ Chapter 23 Weierstrass Elliptic and Modular Functions" class="ltx_ref"><math class="ltx_Math" altimg="m114.png" altimg-height="23px" altimg-valign="-7px" altimg-width="45px" alttext="\sigma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./23.2#E6">σ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./23.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>
(= <math class="ltx_Math" altimg="m116.png" altimg-height="23px" altimg-valign="-7px" altimg-width="64px" alttext="\sigma\left(z|\mathbb{L}\right)" display="inline"><mrow><mi href="./23.2#E6">σ</mi><mo></mo><mrow><mo>(</mo><mi href="./23.1#p2.t1.r4">z</mi><mo stretchy="false">|</mo><mi href="./23.1#p2.t1.r1">𝕃</mi><mo>)</mo></mrow></mrow></math> = <math class="ltx_Math" altimg="m115.png" altimg-height="23px" altimg-valign="-7px" altimg-width="100px" alttext="\sigma\left(z;g_{2},g_{3}\right)" display="inline"><mrow><mi href="./23.3#SS1.p3">σ</mi><mo></mo><mrow><mo>(</mo><mi href="./23.1#p2.t1.r4">z</mi><mo>;</mo><msub><mi href="./23.1#p2.t1.r13">g</mi><mn>2</mn></msub><mo>,</mo><msub><mi href="./23.1#p2.t1.r13">g</mi><mn>3</mn></msub><mo>)</mo></mrow></mrow></math>): Weierstrass sigma function</a> and
<a href="./4.2#E19" title="(4.2.19) ‣ §4.2(iii) The Exponential Function ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m83.png" altimg-height="16px" altimg-valign="-6px" altimg-width="48px" alttext="\exp\NVar{z}" display="inline"><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: exponential function</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="vii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§19.25(vii) </span>Hypergeometric Function</h2>
<div id="SS7.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="SS7.p1" class="ltx_para">
<table id="E42" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.25.42</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E42.png" altimg-height="25px" altimg-valign="-7px" altimg-width="342px" alttext="{{}_{2}F_{1}}\left(a,b;c;z\right)=R_{-a}\left(b,c-b;1-z,1\right)," display="block"><mrow><mrow><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mrow><mi>a</mi><mo>,</mo><mi>b</mi></mrow><mo>;</mo><mi>c</mi><mo>;</mo><mi>z</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msub><mi href="./19.16#E9">R</mi><mrow><mo>-</mo><mi>a</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mrow><mi>b</mi><mo>,</mo><mrow><mi>c</mi><mo>-</mo><mi>b</mi></mrow></mrow><mo>;</mo><mrow><mrow><mn>1</mn><mo>-</mo><mi>z</mi></mrow><mo>,</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E42.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.16#E9" title="(19.16.9) ‣ §19.16(ii) R - a ( b ; z ) ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="23px" altimg-valign="-7px" altimg-width="233px" alttext="R_{\NVar{-a}}\left(\NVar{b_{1}},\dots,\NVar{b_{n}};\NVar{z_{1}},\dots,\NVar{z_%
{n}}\right)" display="inline"><mrow><msub><mi href="./19.16#E9">R</mi><mrow><mo class="ltx_nvar">-</mo><mi class="ltx_nvar">a</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mrow><msub><mi class="ltx_nvar">b</mi><mn class="ltx_nvar">1</mn></msub><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><msub><mi class="ltx_nvar">b</mi><mi class="ltx_nvar" href="./19.1#p1.t1.r1">n</mi></msub></mrow><mo>;</mo><mrow><msub><mi class="ltx_nvar">z</mi><mn class="ltx_nvar">1</mn></msub><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><msub><mi class="ltx_nvar">z</mi><mi class="ltx_nvar" href="./19.1#p1.t1.r1">n</mi></msub></mrow><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m73.png" altimg-height="23px" altimg-valign="-7px" altimg-width="92px" alttext="R_{\NVar{-a}}\left(\NVar{\mathbf{b}};\NVar{\mathbf{z}}\right)" display="inline"><mrow><msub><mi href="./19.16#E9">R</mi><mrow><mo class="ltx_nvar">-</mo><mi class="ltx_nvar">a</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" mathvariant="bold">b</mi><mo>;</mo><mi class="ltx_nvar" mathvariant="bold">z</mi><mo>)</mo></mrow></mrow></math>: multivariate hypergeometric function</a> and
<a href="./16.2" title="§16.2 Definition and Analytic Properties ‣ Generalized Hypergeometric Functions ‣ Chapter 16 Generalized Hypergeometric Functions and Meijer G -Function" class="ltx_ref"><math class="ltx_Math" altimg="m165.png" altimg-height="23px" altimg-valign="-7px" altimg-width="118px" alttext="{{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mrow><mi class="ltx_nvar" href="./16.1#p2.t1.r5">a</mi><mo>,</mo><mi class="ltx_nvar" href="./16.1#p2.t1.r5">b</mi></mrow><mo>;</mo><mi class="ltx_nvar">c</mi><mo>;</mo><mi class="ltx_nvar" href="./16.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: <math class="ltx_Math" altimg="m54.png" altimg-height="23px" altimg-valign="-7px" altimg-width="124px" alttext="=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./16.1#p2.t1.r5">a</mi><mo>,</mo><mi class="ltx_nvar" href="./16.1#p2.t1.r5">b</mi><mo>;</mo><mi class="ltx_nvar">c</mi><mo>;</mo><mi class="ltx_nvar" href="./16.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></math> notation for Gauss’ hypergeometric function</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E43" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.25.43</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E43.png" altimg-height="29px" altimg-valign="-7px" altimg-width="487px" alttext="R_{-a}\left(b_{1},b_{2};z_{1},z_{2}\right)=z_{2}^{-a}{{}_{2}F_{1}}\left(a,b_{1%
};b_{1}+b_{2};1-(z_{1}/z_{2})\right)." display="block"><mrow><mrow><mrow><msub><mi href="./19.16#E9">R</mi><mrow><mo>-</mo><mi>a</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mrow><msub><mi>b</mi><mn>1</mn></msub><mo>,</mo><msub><mi>b</mi><mn>2</mn></msub></mrow><mo>;</mo><mrow><msub><mi>z</mi><mn>1</mn></msub><mo>,</mo><msub><mi>z</mi><mn>2</mn></msub></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msubsup><mi>z</mi><mn>2</mn><mrow><mo>-</mo><mi>a</mi></mrow></msubsup><mo></mo><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mrow><mi>a</mi><mo>,</mo><msub><mi>b</mi><mn>1</mn></msub></mrow><mo>;</mo><mrow><msub><mi>b</mi><mn>1</mn></msub><mo>+</mo><msub><mi>b</mi><mn>2</mn></msub></mrow><mo>;</mo><mrow><mn>1</mn><mo>-</mo><mrow><mo stretchy="false">(</mo><mrow><msub><mi>z</mi><mn>1</mn></msub><mo>/</mo><msub><mi>z</mi><mn>2</mn></msub></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E43.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.16#E9" title="(19.16.9) ‣ §19.16(ii) R - a ( b ; z ) ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="23px" altimg-valign="-7px" altimg-width="233px" alttext="R_{\NVar{-a}}\left(\NVar{b_{1}},\dots,\NVar{b_{n}};\NVar{z_{1}},\dots,\NVar{z_%
{n}}\right)" display="inline"><mrow><msub><mi href="./19.16#E9">R</mi><mrow><mo class="ltx_nvar">-</mo><mi class="ltx_nvar">a</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mrow><msub><mi class="ltx_nvar">b</mi><mn class="ltx_nvar">1</mn></msub><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><msub><mi class="ltx_nvar">b</mi><mi class="ltx_nvar" href="./19.1#p1.t1.r1">n</mi></msub></mrow><mo>;</mo><mrow><msub><mi class="ltx_nvar">z</mi><mn class="ltx_nvar">1</mn></msub><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><msub><mi class="ltx_nvar">z</mi><mi class="ltx_nvar" href="./19.1#p1.t1.r1">n</mi></msub></mrow><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m73.png" altimg-height="23px" altimg-valign="-7px" altimg-width="92px" alttext="R_{\NVar{-a}}\left(\NVar{\mathbf{b}};\NVar{\mathbf{z}}\right)" display="inline"><mrow><msub><mi href="./19.16#E9">R</mi><mrow><mo class="ltx_nvar">-</mo><mi class="ltx_nvar">a</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" mathvariant="bold">b</mi><mo>;</mo><mi class="ltx_nvar" mathvariant="bold">z</mi><mo>)</mo></mrow></mrow></math>: multivariate hypergeometric function</a> and
<a href="./16.2" title="§16.2 Definition and Analytic Properties ‣ Generalized Hypergeometric Functions ‣ Chapter 16 Generalized Hypergeometric Functions and Meijer G -Function" class="ltx_ref"><math class="ltx_Math" altimg="m165.png" altimg-height="23px" altimg-valign="-7px" altimg-width="118px" alttext="{{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mrow><mi class="ltx_nvar" href="./16.1#p2.t1.r5">a</mi><mo>,</mo><mi class="ltx_nvar" href="./16.1#p2.t1.r5">b</mi></mrow><mo>;</mo><mi class="ltx_nvar">c</mi><mo>;</mo><mi class="ltx_nvar" href="./16.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: <math class="ltx_Math" altimg="m54.png" altimg-height="23px" altimg-valign="-7px" altimg-width="124px" alttext="=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./16.1#p2.t1.r5">a</mi><mo>,</mo><mi class="ltx_nvar" href="./16.1#p2.t1.r5">b</mi><mo>;</mo><mi class="ltx_nvar">c</mi><mo>;</mo><mi class="ltx_nvar" href="./16.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></math> notation for Gauss’ hypergeometric function</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">For these results and extensions to the Appell function <math class="ltx_Math" altimg="m161.png" altimg-height="21px" altimg-valign="-5px" altimg-width="26px" alttext="{F_{1}}" display="inline"><msub><mi href="./16.13#E1">F</mi><mn>1</mn></msub></math>
(§) and Lauricella’s function <math class="ltx_Math" altimg="m61.png" altimg-height="21px" altimg-valign="-5px" altimg-width="32px" alttext="F_{D}" display="inline"><msub><mi href="./19.15#p1">F</mi><mi href="./19.15#p1">D</mi></msub></math> see
<cite class="ltx_cite ltx_citemacro_citet">Carlson ()</cite>. (<math class="ltx_Math" altimg="m161.png" altimg-height="21px" altimg-valign="-5px" altimg-width="26px" alttext="{F_{1}}" display="inline"><msub><mi href="./16.13#E1">F</mi><mn>1</mn></msub></math> and <math class="ltx_Math" altimg="m61.png" altimg-height="21px" altimg-valign="-5px" altimg-width="32px" alttext="F_{D}" display="inline"><msub><mi href="./19.15#p1">F</mi><mi href="./19.15#p1">D</mi></msub></math> are equivalent to
the <math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="R" display="inline"><mi>R</mi></math>-function of 3 and <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./19.1#p1.t1.r1">n</mi></math> variables, respectively, but lack full symmetry.)
</p>
</div>
</section>
</section>
</div>
<div class="ltx_page_footer">
<span></div>
</div>
</body></text>
</html>
</page>
<page>
<!DOCTYPE html><html>
<head>
<title>DLMF: 19.29 Reduction of General Elliptic Integrals</title>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
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<div class="ltx_page_navlogo">, (3.2))</cite> with <math class="ltx_Math" altimg="m51.png" altimg-height="23px" altimg-valign="-7px" altimg-width="157px" alttext="(p,q,r)=(n,d,c)" display="inline"><mrow><mrow><mo stretchy="false">(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>,</mo><mi>r</mi><mo stretchy="false">)</mo></mrow><mo>=</mo><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>d</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></mrow></math> for reduction
to <math class="ltx_Math" altimg="m90.png" altimg-height="21px" altimg-valign="-5px" altimg-width="34px" alttext="R_{D}" display="inline"><msub><mi href="./19.16#E5">R</mi><mi href="./19.16#E5">D</mi></msub></math>.</dd>
</dl>
</div>
</div>
<div id="SS1.p1" class="ltx_para">
<p class="ltx_p">These theorems reduce integrals over a real interval <math class="ltx_Math" altimg="m52.png" altimg-height="23px" altimg-valign="-7px" altimg-width="51px" alttext="(y,x)" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi>y</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi>x</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math> of certain
integrands containing the square root of a quartic or cubic polynomial to
symmetric integrals over <math class="ltx_Math" altimg="m46.png" altimg-height="23px" altimg-valign="-7px" altimg-width="59px" alttext="(0,\infty)" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mn>0</mn><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi mathvariant="normal">∞</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math> containing the square root of a cubic
polynomial (compare §). Let
</p>
<table id="E1" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex1" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="3" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">19.29.1</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m39.png" altimg-height="22px" altimg-valign="-5px" altimg-width="34px" alttext="\displaystyle X_{\alpha}" display="inline"><msub><mi href="./19.29#SS1.p1">X</mi><mi href="./19.29#SS1.p1">α</mi></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m20.png" altimg-height="30px" altimg-valign="-7px" altimg-width="130px" alttext="\displaystyle=\sqrt{a_{\alpha}+b_{\alpha}x}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><msqrt><mrow><msub><mi>a</mi><mi href="./19.29#SS1.p1">α</mi></msub><mo>+</mo><mrow><msub><mi>b</mi><mi href="./19.29#SS1.p1">α</mi></msub><mo></mo><mi>x</mi></mrow></mrow></msqrt></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex2" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m41.png" altimg-height="22px" altimg-valign="-5px" altimg-width="29px" alttext="\displaystyle Y_{\alpha}" display="inline"><msub><mi href="./19.29#SS1.p1">Y</mi><mi href="./19.29#SS1.p1">α</mi></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m21.png" altimg-height="31px" altimg-valign="-7px" altimg-width="129px" alttext="\displaystyle=\sqrt{a_{\alpha}+b_{\alpha}y}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><msqrt><mrow><msub><mi>a</mi><mi href="./19.29#SS1.p1">α</mi></msub><mo>+</mo><mrow><msub><mi>b</mi><mi href="./19.29#SS1.p1">α</mi></msub><mo></mo><mi>y</mi></mrow></mrow></msqrt></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m163.png" altimg-height="18px" altimg-valign="-6px" altimg-width="53px" alttext="x>y" display="inline"><mrow><mi>x</mi><mo>></mo><mi>y</mi></mrow></math>, <math class="ltx_Math" altimg="m65.png" altimg-height="19px" altimg-valign="-5px" altimg-width="90px" alttext="1\leq\alpha\leq 5" display="inline"><mrow><mn>1</mn><mo>≤</mo><mi href="./19.29#SS1.p1">α</mi><mo>≤</mo><mn>5</mn></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.29#SS1.p1" title="§19.29(i) Reduction Theorems ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m105.png" altimg-height="21px" altimg-valign="-5px" altimg-width="32px" alttext="X_{\alpha}" display="inline"><msub><mi href="./19.29#SS1.p1">X</mi><mi href="./19.29#SS1.p1">α</mi></msub></math></a>,
<a href="./19.29#SS1.p1" title="§19.29(i) Reduction Theorems ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m106.png" altimg-height="21px" altimg-valign="-5px" altimg-width="27px" alttext="Y_{\alpha}" display="inline"><msub><mi href="./19.29#SS1.p1">Y</mi><mi href="./19.29#SS1.p1">α</mi></msub></math></a> and
<a href="./19.29#SS1.p1" title="§19.29(i) Reduction Theorems ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m109.png" altimg-height="13px" altimg-valign="-2px" altimg-width="17px" alttext="\alpha" display="inline"><mi href="./19.29#SS1.p1">α</mi></math>: parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E2" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.29.2</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E2.png" altimg-height="25px" altimg-valign="-8px" altimg-width="176px" alttext="d_{\alpha\beta}=a_{\alpha}b_{\beta}-a_{\beta}b_{\alpha}," display="block"><mrow><mrow><msub><mi href="./19.29#SS1.p1">d</mi><mrow><mi href="./19.29#SS1.p1">α</mi><mo></mo><mi href="./19.29#SS1.p1">β</mi></mrow></msub><mo>=</mo><mrow><mrow><msub><mi>a</mi><mi href="./19.29#SS1.p1">α</mi></msub><mo></mo><msub><mi>b</mi><mi href="./19.29#SS1.p1">β</mi></msub></mrow><mo>-</mo><mrow><msub><mi>a</mi><mi href="./19.29#SS1.p1">β</mi></msub><mo></mo><msub><mi>b</mi><mi href="./19.29#SS1.p1">α</mi></msub></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m141.png" altimg-height="23px" altimg-valign="-8px" altimg-width="72px" alttext="d_{\alpha\beta}\neq 0" display="inline"><mrow><msub><mi href="./19.29#SS1.p1">d</mi><mrow><mi href="./19.29#SS1.p1">α</mi><mo></mo><mi href="./19.29#SS1.p1">β</mi></mrow></msub><mo>≠</mo><mn>0</mn></mrow></math> if <math class="ltx_Math" altimg="m110.png" altimg-height="21px" altimg-valign="-6px" altimg-width="56px" alttext="\alpha\neq\beta" display="inline"><mrow><mi href="./19.29#SS1.p1">α</mi><mo>≠</mo><mi href="./19.29#SS1.p1">β</mi></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.29#SS1.p1" title="§19.29(i) Reduction Theorems ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m140.png" altimg-height="23px" altimg-valign="-8px" altimg-width="36px" alttext="d_{\alpha\beta}" display="inline"><msub><mi href="./19.29#SS1.p1">d</mi><mrow><mi href="./19.29#SS1.p1">α</mi><mo></mo><mi href="./19.29#SS1.p1">β</mi></mrow></msub></math></a>,
<a href="./19.29#SS1.p1" title="§19.29(i) Reduction Theorems ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m109.png" altimg-height="13px" altimg-valign="-2px" altimg-width="17px" alttext="\alpha" display="inline"><mi href="./19.29#SS1.p1">α</mi></math>: parameter</a> and
<a href="./19.29#SS1.p1" title="§19.29(i) Reduction Theorems ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m111.png" altimg-height="21px" altimg-valign="-6px" altimg-width="17px" alttext="\beta" display="inline"><mi href="./19.29#SS1.p1">β</mi></math>: parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and assume that the line segment with endpoints <math class="ltx_Math" altimg="m133.png" altimg-height="21px" altimg-valign="-5px" altimg-width="82px" alttext="a_{\alpha}+b_{\alpha}x" display="inline"><mrow><msub><mi>a</mi><mi href="./19.29#SS1.p1">α</mi></msub><mo>+</mo><mrow><msub><mi>b</mi><mi href="./19.29#SS1.p1">α</mi></msub><mo></mo><mi>x</mi></mrow></mrow></math> and
<math class="ltx_Math" altimg="m134.png" altimg-height="21px" altimg-valign="-6px" altimg-width="81px" alttext="a_{\alpha}+b_{\alpha}y" display="inline"><mrow><msub><mi>a</mi><mi href="./19.29#SS1.p1">α</mi></msub><mo>+</mo><mrow><msub><mi>b</mi><mi href="./19.29#SS1.p1">α</mi></msub><mo></mo><mi>y</mi></mrow></mrow></math> lies in <math class="ltx_Math" altimg="m120.png" altimg-height="23px" altimg-valign="-7px" altimg-width="107px" alttext="\mathbb{C}\setminus(-\infty,0)" display="inline"><mrow><mi href="./front/introduction#Sx4.p1.t1.r1">ℂ</mi><mo href="./front/introduction#Sx4.p2.t1.r19">∖</mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mn>0</mn><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></mrow></math> for
<math class="ltx_Math" altimg="m64.png" altimg-height="19px" altimg-valign="-5px" altimg-width="90px" alttext="1\leq\alpha\leq 4" display="inline"><mrow><mn>1</mn><mo>≤</mo><mi href="./19.29#SS1.p1">α</mi><mo>≤</mo><mn>4</mn></mrow></math>. If
</p>
<table id="E3" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.29.3</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E3.png" altimg-height="67px" altimg-valign="-27px" altimg-width="192px" alttext="s(t)=\prod_{\alpha=1}^{4}\sqrt{a_{\alpha}+b_{\alpha}t}" display="block"><mrow><mrow><mi href="./19.29#SS1.p1">s</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∏</mo><mrow><mi href="./19.29#SS1.p1">α</mi><mo>=</mo><mn>1</mn></mrow><mn>4</mn></munderover><msqrt><mrow><msub><mi>a</mi><mi href="./19.29#SS1.p1">α</mi></msub><mo>+</mo><mrow><msub><mi>b</mi><mi href="./19.29#SS1.p1">α</mi></msub><mo></mo><mi>t</mi></mrow></mrow></msqrt></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.29#SS1.p1" title="§19.29(i) Reduction Theorems ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m159.png" altimg-height="23px" altimg-valign="-7px" altimg-width="36px" alttext="s(t)" display="inline"><mrow><mi href="./19.29#SS1.p1">s</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math>: product</a> and
<a href="./19.29#SS1.p1" title="§19.29(i) Reduction Theorems ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m109.png" altimg-height="13px" altimg-valign="-2px" altimg-width="17px" alttext="\alpha" display="inline"><mi href="./19.29#SS1.p1">α</mi></math>: parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and <math class="ltx_Math" altimg="m107.png" altimg-height="21px" altimg-valign="-6px" altimg-width="77px" alttext="\alpha,\beta,\gamma,\delta" display="inline"><mrow><mi href="./19.29#SS1.p1">α</mi><mo>,</mo><mi href="./19.29#SS1.p1">β</mi><mo>,</mo><mi href="./19.29#SS1.p1">γ</mi><mo>,</mo><mi href="./19.29#SS1.p1">δ</mi></mrow></math> is any permutation of the numbers
<math class="ltx_Math" altimg="m58.png" altimg-height="20px" altimg-valign="-6px" altimg-width="71px" alttext="1,2,3,4" display="inline"><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>4</mn></mrow></math>, then
</p>
<table id="E4" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.29.4</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E4.png" altimg-height="55px" altimg-valign="-23px" altimg-width="277px" alttext="\int_{y}^{x}\frac{\mathrm{d}t}{s(t)}=2\!R_{F}\left(U_{12}^{2},U_{13}^{2},U_{23%
}^{2}\right)," display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi>y</mi><mi>x</mi></msubsup><mfrac><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow><mrow><mi href="./19.29#SS1.p1">s</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></mfrac></mrow><mo>=</mo><mrow><mpadded width="-1.7pt"><mn>2</mn></mpadded><mo></mo><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><msubsup><mi href="./19.29#SS1.p1">U</mi><mn>12</mn><mn>2</mn></msubsup><mo>,</mo><msubsup><mi href="./19.29#SS1.p1">U</mi><mn>13</mn><mn>2</mn></msubsup><mo>,</mo><msubsup><mi href="./19.29#SS1.p1">U</mi><mn>23</mn><mn>2</mn></msubsup><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E4.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.16#E1" title="(19.16.1) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m93.png" altimg-height="23px" altimg-valign="-7px" altimg-width="101px" alttext="R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: symmetric elliptic integral of first kind</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m124.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m164.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m117.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./19.29#SS1.p1" title="§19.29(i) Reduction Theorems ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="21px" altimg-valign="-5px" altimg-width="29px" alttext="U_{n}" display="inline"><msub><mi href="./19.29#SS1.p1">U</mi><mi>n</mi></msub></math></a> and
<a href="./19.29#SS1.p1" title="§19.29(i) Reduction Theorems ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m159.png" altimg-height="23px" altimg-valign="-7px" altimg-width="36px" alttext="s(t)" display="inline"><mrow><mi href="./19.29#SS1.p1">s</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math>: product</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where
</p>
<table id="E5" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex3" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="3" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">19.29.5</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m36.png" altimg-height="25px" altimg-valign="-8px" altimg-width="41px" alttext="\displaystyle U_{\alpha\beta}" display="inline"><msub><mi href="./19.29#SS1.p1">U</mi><mrow><mi href="./19.29#SS1.p1">α</mi><mo></mo><mi href="./19.29#SS1.p1">β</mi></mrow></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m4.png" altimg-height="26px" altimg-valign="-8px" altimg-width="339px" alttext="\displaystyle=(X_{\alpha}X_{\beta}Y_{\gamma}Y_{\delta}+Y_{\alpha}Y_{\beta}X_{%
\gamma}X_{\delta})/(x-y)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><msub><mi href="./19.29#SS1.p1">X</mi><mi href="./19.29#SS1.p1">α</mi></msub><mo></mo><msub><mi href="./19.29#SS1.p1">X</mi><mi href="./19.29#SS1.p1">β</mi></msub><mo></mo><msub><mi href="./19.29#SS1.p1">Y</mi><mi href="./19.29#SS1.p1">γ</mi></msub><mo></mo><msub><mi href="./19.29#SS1.p1">Y</mi><mi href="./19.29#SS1.p1">δ</mi></msub></mrow><mo>+</mo><mrow><msub><mi href="./19.29#SS1.p1">Y</mi><mi href="./19.29#SS1.p1">α</mi></msub><mo></mo><msub><mi href="./19.29#SS1.p1">Y</mi><mi href="./19.29#SS1.p1">β</mi></msub><mo></mo><msub><mi href="./19.29#SS1.p1">X</mi><mi href="./19.29#SS1.p1">γ</mi></msub><mo></mo><msub><mi href="./19.29#SS1.p1">X</mi><mi href="./19.29#SS1.p1">δ</mi></msub></mrow></mrow><mo stretchy="false">)</mo></mrow><mo>/</mo><mrow><mo stretchy="false">(</mo><mrow><mi>x</mi><mo>-</mo><mi>y</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex4" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m36.png" altimg-height="25px" altimg-valign="-8px" altimg-width="41px" alttext="\displaystyle U_{\alpha\beta}" display="inline"><msub><mi href="./19.29#SS1.p1">U</mi><mrow><mi href="./19.29#SS1.p1">α</mi><mo></mo><mi href="./19.29#SS1.p1">β</mi></mrow></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m13.png" altimg-height="25px" altimg-valign="-8px" altimg-width="184px" alttext="\displaystyle=U_{\beta\alpha}=U_{\gamma\delta}=U_{\delta\gamma}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><msub><mi href="./19.29#SS1.p1">U</mi><mrow><mi href="./19.29#SS1.p1">β</mi><mo></mo><mi href="./19.29#SS1.p1">α</mi></mrow></msub><mo>=</mo><msub><mi href="./19.29#SS1.p1">U</mi><mrow><mi href="./19.29#SS1.p1">γ</mi><mo></mo><mi href="./19.29#SS1.p1">δ</mi></mrow></msub><mo>=</mo><msub><mi href="./19.29#SS1.p1">U</mi><mrow><mi href="./19.29#SS1.p1">δ</mi><mo></mo><mi href="./19.29#SS1.p1">γ</mi></mrow></msub></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex5" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m37.png" altimg-height="31px" altimg-valign="-10px" altimg-width="99px" alttext="\displaystyle U_{\alpha\beta}^{2}-U_{\alpha\gamma}^{2}" display="inline"><mrow><msubsup><mi href="./19.29#SS1.p1">U</mi><mrow><mi href="./19.29#SS1.p1">α</mi><mo></mo><mi href="./19.29#SS1.p1">β</mi></mrow><mn>2</mn></msubsup><mo>-</mo><msubsup><mi href="./19.29#SS1.p1">U</mi><mrow><mi href="./19.29#SS1.p1">α</mi><mo></mo><mi href="./19.29#SS1.p1">γ</mi></mrow><mn>2</mn></msubsup></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m26.png" altimg-height="25px" altimg-valign="-8px" altimg-width="92px" alttext="\displaystyle=d_{\alpha\delta}d_{\beta\gamma}." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msub><mi href="./19.29#SS1.p1">d</mi><mrow><mi href="./19.29#SS1.p1">α</mi><mo></mo><mi href="./19.29#SS1.p1">δ</mi></mrow></msub><mo></mo><msub><mi href="./19.29#SS1.p1">d</mi><mrow><mi href="./19.29#SS1.p1">β</mi><mo></mo><mi href="./19.29#SS1.p1">γ</mi></mrow></msub></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E5.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.29#SS1.p1" title="§19.29(i) Reduction Theorems ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m105.png" altimg-height="21px" altimg-valign="-5px" altimg-width="32px" alttext="X_{\alpha}" display="inline"><msub><mi href="./19.29#SS1.p1">X</mi><mi href="./19.29#SS1.p1">α</mi></msub></math></a>,
<a href="./19.29#SS1.p1" title="§19.29(i) Reduction Theorems ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="21px" altimg-valign="-5px" altimg-width="29px" alttext="U_{n}" display="inline"><msub><mi href="./19.29#SS1.p1">U</mi><mi>n</mi></msub></math></a>,
<a href="./19.29#SS1.p1" title="§19.29(i) Reduction Theorems ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m106.png" altimg-height="21px" altimg-valign="-5px" altimg-width="27px" alttext="Y_{\alpha}" display="inline"><msub><mi href="./19.29#SS1.p1">Y</mi><mi href="./19.29#SS1.p1">α</mi></msub></math></a>,
<a href="./19.29#SS1.p1" title="§19.29(i) Reduction Theorems ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m140.png" altimg-height="23px" altimg-valign="-8px" altimg-width="36px" alttext="d_{\alpha\beta}" display="inline"><msub><mi href="./19.29#SS1.p1">d</mi><mrow><mi href="./19.29#SS1.p1">α</mi><mo></mo><mi href="./19.29#SS1.p1">β</mi></mrow></msub></math></a>,
<a href="./19.29#SS1.p1" title="§19.29(i) Reduction Theorems ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m109.png" altimg-height="13px" altimg-valign="-2px" altimg-width="17px" alttext="\alpha" display="inline"><mi href="./19.29#SS1.p1">α</mi></math>: parameter</a>,
<a href="./19.29#SS1.p1" title="§19.29(i) Reduction Theorems ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m111.png" altimg-height="21px" altimg-valign="-6px" altimg-width="17px" alttext="\beta" display="inline"><mi href="./19.29#SS1.p1">β</mi></math>: parameter</a>,
<a href="./19.29#SS1.p1" title="§19.29(i) Reduction Theorems ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m114.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\gamma" display="inline"><mi href="./19.29#SS1.p1">γ</mi></math>: parameter</a> and
<a href="./19.29#SS1.p1" title="§19.29(i) Reduction Theorems ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m113.png" altimg-height="18px" altimg-valign="-2px" altimg-width="14px" alttext="\delta" display="inline"><mi href="./19.29#SS1.p1">δ</mi></math>: parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">There are only three distinct <math class="ltx_Math" altimg="m99.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="U" display="inline"><mi>U</mi></math>’s with subscripts <math class="ltx_Math" altimg="m118.png" altimg-height="19px" altimg-valign="-5px" altimg-width="35px" alttext="\leq 4" display="inline"><mrow><mi></mi><mo>≤</mo><mn>4</mn></mrow></math>, and at most one
of them can be 0 because the <math class="ltx_Math" altimg="m139.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="d" display="inline"><mi>d</mi></math>’s are nonzero. Then</p>
<table id="E6" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex6" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="4" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">19.29.6</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m45.png" altimg-height="25px" altimg-valign="-8px" altimg-width="41px" alttext="\displaystyle{U_{\alpha\beta}}" display="inline"><msub><mi href="./19.29#SS1.p1">U</mi><mrow><mi href="./19.29#SS1.p1">α</mi><mo></mo><mi href="./19.29#SS1.p1">β</mi></mrow></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m169.png" altimg-height="30px" altimg-valign="-8px" altimg-width="298px" alttext="{\displaystyle=\sqrt{b_{\alpha}}\sqrt{b_{\beta}}Y_{\gamma}Y_{\delta}+Y_{\alpha}%
Y_{\beta}\sqrt{b_{\gamma}}\sqrt{b_{\delta}},}" display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><msqrt><msub><mi>b</mi><mi href="./19.29#SS1.p1">α</mi></msub></msqrt><mo></mo><msqrt><msub><mi>b</mi><mi href="./19.29#SS1.p1">β</mi></msub></msqrt><mo></mo><msub><mi href="./19.29#SS1.p1">Y</mi><mi href="./19.29#SS1.p1">γ</mi></msub><mo></mo><msub><mi href="./19.29#SS1.p1">Y</mi><mi href="./19.29#SS1.p1">δ</mi></msub></mrow><mo>+</mo><mrow><msub><mi href="./19.29#SS1.p1">Y</mi><mi href="./19.29#SS1.p1">α</mi></msub><mo></mo><msub><mi href="./19.29#SS1.p1">Y</mi><mi href="./19.29#SS1.p1">β</mi></msub><mo></mo><msqrt><msub><mi>b</mi><mi href="./19.29#SS1.p1">γ</mi></msub></msqrt><mo></mo><msqrt><msub><mi>b</mi><mi href="./19.29#SS1.p1">δ</mi></msub></msqrt></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m162.png" altimg-height="13px" altimg-valign="-2px" altimg-width="62px" alttext="x=\infty" display="inline"><mrow><mi>x</mi><mo>=</mo><mi mathvariant="normal">∞</mi></mrow></math>,</span></td></tr>
<tr id="Ex7" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m36.png" altimg-height="25px" altimg-valign="-8px" altimg-width="41px" alttext="\displaystyle U_{\alpha\beta}" display="inline"><msub><mi href="./19.29#SS1.p1">U</mi><mrow><mi href="./19.29#SS1.p1">α</mi><mo></mo><mi href="./19.29#SS1.p1">β</mi></mrow></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m14.png" altimg-height="32px" altimg-valign="-8px" altimg-width="381px" alttext="\displaystyle=X_{\alpha}X_{\beta}\sqrt{-b_{\gamma}}\sqrt{-b_{\delta}}+\sqrt{-b%
_{\alpha}}\sqrt{-b_{\beta}}X_{\gamma}X_{\delta}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><msub><mi href="./19.29#SS1.p1">X</mi><mi href="./19.29#SS1.p1">α</mi></msub><mo></mo><msub><mi href="./19.29#SS1.p1">X</mi><mi href="./19.29#SS1.p1">β</mi></msub><mo></mo><msqrt><mrow><mo>-</mo><msub><mi>b</mi><mi href="./19.29#SS1.p1">γ</mi></msub></mrow></msqrt><mo></mo><msqrt><mrow><mo>-</mo><msub><mi>b</mi><mi href="./19.29#SS1.p1">δ</mi></msub></mrow></msqrt></mrow><mo>+</mo><mrow><msqrt><mrow><mo>-</mo><msub><mi>b</mi><mi href="./19.29#SS1.p1">α</mi></msub></mrow></msqrt><mo></mo><msqrt><mrow><mo>-</mo><msub><mi>b</mi><mi href="./19.29#SS1.p1">β</mi></msub></mrow></msqrt><mo></mo><msub><mi href="./19.29#SS1.p1">X</mi><mi href="./19.29#SS1.p1">γ</mi></msub><mo></mo><msub><mi href="./19.29#SS1.p1">X</mi><mi href="./19.29#SS1.p1">δ</mi></msub></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m168.png" altimg-height="19px" altimg-valign="-6px" altimg-width="77px" alttext="y=-\infty" display="inline"><mrow><mi>y</mi><mo>=</mo><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E6.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.29#SS1.p1" title="§19.29(i) Reduction Theorems ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m105.png" altimg-height="21px" altimg-valign="-5px" altimg-width="32px" alttext="X_{\alpha}" display="inline"><msub><mi href="./19.29#SS1.p1">X</mi><mi href="./19.29#SS1.p1">α</mi></msub></math></a>,
<a href="./19.29#SS1.p1" title="§19.29(i) Reduction Theorems ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="21px" altimg-valign="-5px" altimg-width="29px" alttext="U_{n}" display="inline"><msub><mi href="./19.29#SS1.p1">U</mi><mi>n</mi></msub></math></a>,
<a href="./19.29#SS1.p1" title="§19.29(i) Reduction Theorems ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m106.png" altimg-height="21px" altimg-valign="-5px" altimg-width="27px" alttext="Y_{\alpha}" display="inline"><msub><mi href="./19.29#SS1.p1">Y</mi><mi href="./19.29#SS1.p1">α</mi></msub></math></a>,
<a href="./19.29#SS1.p1" title="§19.29(i) Reduction Theorems ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m109.png" altimg-height="13px" altimg-valign="-2px" altimg-width="17px" alttext="\alpha" display="inline"><mi href="./19.29#SS1.p1">α</mi></math>: parameter</a>,
<a href="./19.29#SS1.p1" title="§19.29(i) Reduction Theorems ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m111.png" altimg-height="21px" altimg-valign="-6px" altimg-width="17px" alttext="\beta" display="inline"><mi href="./19.29#SS1.p1">β</mi></math>: parameter</a>,
<a href="./19.29#SS1.p1" title="§19.29(i) Reduction Theorems ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m114.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\gamma" display="inline"><mi href="./19.29#SS1.p1">γ</mi></math>: parameter</a> and
<a href="./19.29#SS1.p1" title="§19.29(i) Reduction Theorems ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m113.png" altimg-height="18px" altimg-valign="-2px" altimg-width="14px" alttext="\delta" display="inline"><mi href="./19.29#SS1.p1">δ</mi></math>: parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E7" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.29.7</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E7.png" altimg-height="55px" altimg-valign="-23px" altimg-width="539px" alttext="\int_{y}^{x}\frac{a_{\alpha}+b_{\alpha}t}{a_{\delta}+b_{\delta}t}\frac{\mathrm%
{d}t}{s(t)}=\tfrac{2}{3}d_{\alpha\beta}d_{\alpha\gamma}R_{D}\left(U_{\alpha%
\beta}^{2},U_{\alpha\gamma}^{2},U_{\alpha\delta}^{2}\right)+\frac{2X_{\alpha}Y%
_{\alpha}}{X_{\delta}Y_{\delta}U_{\alpha\delta}}," display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi>y</mi><mi>x</mi></msubsup><mrow><mfrac><mrow><msub><mi>a</mi><mi href="./19.29#SS1.p1">α</mi></msub><mo>+</mo><mrow><msub><mi>b</mi><mi href="./19.29#SS1.p1">α</mi></msub><mo></mo><mi>t</mi></mrow></mrow><mrow><msub><mi>a</mi><mi href="./19.29#SS1.p1">δ</mi></msub><mo>+</mo><mrow><msub><mi>b</mi><mi href="./19.29#SS1.p1">δ</mi></msub><mo></mo><mi>t</mi></mrow></mrow></mfrac><mo></mo><mfrac><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow><mrow><mi href="./19.29#SS1.p1">s</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></mfrac></mrow></mrow><mo>=</mo><mrow><mrow><mstyle displaystyle="false"><mfrac><mn>2</mn><mn>3</mn></mfrac></mstyle><mo></mo><msub><mi href="./19.29#SS1.p1">d</mi><mrow><mi href="./19.29#SS1.p1">α</mi><mo></mo><mi href="./19.29#SS1.p1">β</mi></mrow></msub><mo></mo><msub><mi href="./19.29#SS1.p1">d</mi><mrow><mi href="./19.29#SS1.p1">α</mi><mo></mo><mi href="./19.29#SS1.p1">γ</mi></mrow></msub><mo></mo><mrow><msub><mi href="./19.16#E5">R</mi><mi href="./19.16#E5">D</mi></msub><mo></mo><mrow><mo>(</mo><msubsup><mi href="./19.29#SS1.p1">U</mi><mrow><mi href="./19.29#SS1.p1">α</mi><mo></mo><mi href="./19.29#SS1.p1">β</mi></mrow><mn>2</mn></msubsup><mo>,</mo><msubsup><mi href="./19.29#SS1.p1">U</mi><mrow><mi href="./19.29#SS1.p1">α</mi><mo></mo><mi href="./19.29#SS1.p1">γ</mi></mrow><mn>2</mn></msubsup><mo>,</mo><msubsup><mi href="./19.29#SS1.p1">U</mi><mrow><mi href="./19.29#SS1.p1">α</mi><mo></mo><mi href="./19.29#SS1.p1">δ</mi></mrow><mn>2</mn></msubsup><mo>)</mo></mrow></mrow></mrow><mo>+</mo><mfrac><mrow><mn>2</mn><mo></mo><msub><mi href="./19.29#SS1.p1">X</mi><mi href="./19.29#SS1.p1">α</mi></msub><mo></mo><msub><mi href="./19.29#SS1.p1">Y</mi><mi href="./19.29#SS1.p1">α</mi></msub></mrow><mrow><msub><mi href="./19.29#SS1.p1">X</mi><mi href="./19.29#SS1.p1">δ</mi></msub><mo></mo><msub><mi href="./19.29#SS1.p1">Y</mi><mi href="./19.29#SS1.p1">δ</mi></msub><mo></mo><msub><mi href="./19.29#SS1.p1">U</mi><mrow><mi href="./19.29#SS1.p1">α</mi><mo></mo><mi href="./19.29#SS1.p1">δ</mi></mrow></msub></mrow></mfrac></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m102.png" altimg-height="21px" altimg-valign="-6px" altimg-width="74px" alttext="U_{\alpha\delta}\neq 0" display="inline"><mrow><msub><mi href="./19.29#SS1.p1">U</mi><mrow><mi href="./19.29#SS1.p1">α</mi><mo></mo><mi href="./19.29#SS1.p1">δ</mi></mrow></msub><mo>≠</mo><mn>0</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E7.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.16#E5" title="(19.16.5) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m91.png" altimg-height="23px" altimg-valign="-7px" altimg-width="102px" alttext="R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E5">R</mi><mi href="./19.16#E5">D</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: elliptic integral symmetric in only two variables</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m124.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m164.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m117.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./19.29#SS1.p1" title="§19.29(i) Reduction Theorems ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m105.png" altimg-height="21px" altimg-valign="-5px" altimg-width="32px" alttext="X_{\alpha}" display="inline"><msub><mi href="./19.29#SS1.p1">X</mi><mi href="./19.29#SS1.p1">α</mi></msub></math></a>,
<a href="./19.29#SS1.p1" title="§19.29(i) Reduction Theorems ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="21px" altimg-valign="-5px" altimg-width="29px" alttext="U_{n}" display="inline"><msub><mi href="./19.29#SS1.p1">U</mi><mi>n</mi></msub></math></a>,
<a href="./19.29#SS1.p1" title="§19.29(i) Reduction Theorems ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m106.png" altimg-height="21px" altimg-valign="-5px" altimg-width="27px" alttext="Y_{\alpha}" display="inline"><msub><mi href="./19.29#SS1.p1">Y</mi><mi href="./19.29#SS1.p1">α</mi></msub></math></a>,
<a href="./19.29#SS1.p1" title="§19.29(i) Reduction Theorems ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m140.png" altimg-height="23px" altimg-valign="-8px" altimg-width="36px" alttext="d_{\alpha\beta}" display="inline"><msub><mi href="./19.29#SS1.p1">d</mi><mrow><mi href="./19.29#SS1.p1">α</mi><mo></mo><mi href="./19.29#SS1.p1">β</mi></mrow></msub></math></a>,
<a href="./19.29#SS1.p1" title="§19.29(i) Reduction Theorems ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m159.png" altimg-height="23px" altimg-valign="-7px" altimg-width="36px" alttext="s(t)" display="inline"><mrow><mi href="./19.29#SS1.p1">s</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math>: product</a>,
<a href="./19.29#SS1.p1" title="§19.29(i) Reduction Theorems ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m109.png" altimg-height="13px" altimg-valign="-2px" altimg-width="17px" alttext="\alpha" display="inline"><mi href="./19.29#SS1.p1">α</mi></math>: parameter</a>,
<a href="./19.29#SS1.p1" title="§19.29(i) Reduction Theorems ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m111.png" altimg-height="21px" altimg-valign="-6px" altimg-width="17px" alttext="\beta" display="inline"><mi href="./19.29#SS1.p1">β</mi></math>: parameter</a>,
<a href="./19.29#SS1.p1" title="§19.29(i) Reduction Theorems ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m114.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\gamma" display="inline"><mi href="./19.29#SS1.p1">γ</mi></math>: parameter</a> and
<a href="./19.29#SS1.p1" title="§19.29(i) Reduction Theorems ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m113.png" altimg-height="18px" altimg-valign="-2px" altimg-width="14px" alttext="\delta" display="inline"><mi href="./19.29#SS1.p1">δ</mi></math>: parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E8" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.29.8</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E8.png" altimg-height="55px" altimg-valign="-23px" altimg-width="656px" alttext="\int_{y}^{x}\frac{a_{\alpha}+b_{\alpha}t}{a_{5}+b_{5}t}\frac{\mathrm{d}t}{s(t)%
}=\frac{2}{3}\frac{d_{\alpha\beta}d_{\alpha\gamma}d_{\alpha\delta}}{d_{\alpha 5%
}}R_{J}\left(U_{12}^{2},U_{13}^{2},U_{23}^{2},U_{\alpha 5}^{2}\right)+2\!R_{C}%
\left(S_{\alpha 5}^{2},Q_{\alpha 5}^{2}\right)," display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi>y</mi><mi>x</mi></msubsup><mrow><mfrac><mrow><msub><mi>a</mi><mi href="./19.29#SS1.p1">α</mi></msub><mo>+</mo><mrow><msub><mi>b</mi><mi href="./19.29#SS1.p1">α</mi></msub><mo></mo><mi>t</mi></mrow></mrow><mrow><msub><mi>a</mi><mn>5</mn></msub><mo>+</mo><mrow><msub><mi>b</mi><mn>5</mn></msub><mo></mo><mi>t</mi></mrow></mrow></mfrac><mo></mo><mfrac><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow><mrow><mi href="./19.29#SS1.p1">s</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></mfrac></mrow></mrow><mo>=</mo><mrow><mrow><mfrac><mn>2</mn><mn>3</mn></mfrac><mo></mo><mfrac><mrow><msub><mi href="./19.29#SS1.p1">d</mi><mrow><mi href="./19.29#SS1.p1">α</mi><mo></mo><mi href="./19.29#SS1.p1">β</mi></mrow></msub><mo></mo><msub><mi href="./19.29#SS1.p1">d</mi><mrow><mi href="./19.29#SS1.p1">α</mi><mo></mo><mi href="./19.29#SS1.p1">γ</mi></mrow></msub><mo></mo><msub><mi href="./19.29#SS1.p1">d</mi><mrow><mi href="./19.29#SS1.p1">α</mi><mo></mo><mi href="./19.29#SS1.p1">δ</mi></mrow></msub></mrow><msub><mi href="./19.29#SS1.p1">d</mi><mrow><mi href="./19.29#SS1.p1">α</mi><mo></mo><mn>5</mn></mrow></msub></mfrac><mo></mo><mrow><msub><mi href="./19.16#E2">R</mi><mi href="./19.16#E2">J</mi></msub><mo></mo><mrow><mo>(</mo><msubsup><mi href="./19.29#SS1.p1">U</mi><mn>12</mn><mn>2</mn></msubsup><mo>,</mo><msubsup><mi href="./19.29#SS1.p1">U</mi><mn>13</mn><mn>2</mn></msubsup><mo>,</mo><msubsup><mi href="./19.29#SS1.p1">U</mi><mn>23</mn><mn>2</mn></msubsup><mo>,</mo><msubsup><mi href="./19.29#SS1.p1">U</mi><mrow><mi href="./19.29#SS1.p1">α</mi><mo></mo><mn>5</mn></mrow><mn>2</mn></msubsup><mo>)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mpadded width="-1.7pt"><mn>2</mn></mpadded><mo></mo><mrow><msub><mi href="./19.2#E17">R</mi><mi href="./19.2#E17">C</mi></msub><mo></mo><mrow><mo>(</mo><msubsup><mi href="./19.29#SS1.p1">S</mi><mrow><mi href="./19.29#SS1.p1">α</mi><mo></mo><mn>5</mn></mrow><mn>2</mn></msubsup><mo>,</mo><msubsup><mi href="./19.29#SS1.p1">Q</mi><mrow><mi href="./19.29#SS1.p1">α</mi><mo></mo><mn>5</mn></mrow><mn>2</mn></msubsup><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m96.png" altimg-height="25px" altimg-valign="-7px" altimg-width="163px" alttext="S_{\alpha 5}^{2}\in\mathbb{C}\setminus(-\infty,0)" display="inline"><mrow><msubsup><mi href="./19.29#SS1.p1">S</mi><mrow><mi href="./19.29#SS1.p1">α</mi><mo></mo><mn>5</mn></mrow><mn>2</mn></msubsup><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mrow><mi href="./front/introduction#Sx4.p1.t1.r1">ℂ</mi><mo href="./front/introduction#Sx4.p2.t1.r19">∖</mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mn>0</mn><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E8.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.2#E17" title="(19.2.17) ‣ §19.2(iv) A Related Function: R C ( x , y ) ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m89.png" altimg-height="23px" altimg-valign="-7px" altimg-width="82px" alttext="R_{C}\left(\NVar{x},\NVar{y}\right)" display="inline"><mrow><msub><mi href="./19.2#E17">R</mi><mi href="./19.2#E17">C</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>)</mo></mrow></mrow></math>: Carlson’s combination of inverse circular and inverse hyperbolic functions</a>,
<a href="./19.16#E2" title="(19.16.2) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m95.png" altimg-height="23px" altimg-valign="-7px" altimg-width="118px" alttext="R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)" display="inline"><mrow><msub><mi href="./19.16#E2">R</mi><mi href="./19.16#E2">J</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>,</mo><mi class="ltx_nvar">p</mi><mo>)</mo></mrow></mrow></math>: symmetric elliptic integral of third kind</a>,
<a href="./front/introduction#Sx4.p1.t1.r1" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m119.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\mathbb{C}" display="inline"><mi href="./front/introduction#Sx4.p1.t1.r1">ℂ</mi></math>: complex plane</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m124.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m164.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./front/introduction#Sx4.p1.t1.r9" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m116.png" altimg-height="15px" altimg-valign="-3px" altimg-width="18px" alttext="\in" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo></math>: element of</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m117.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./front/introduction#Sx4.p1.t1.r28" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="(\NVar{a},\NVar{b})" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi class="ltx_nvar">a</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi class="ltx_nvar">b</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math>: open interval</a>,
<a href="./front/introduction#Sx4.p2.t1.r19" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m125.png" altimg-height="23px" altimg-valign="-7px" altimg-width="14px" alttext="\setminus" display="inline"><mo href="./front/introduction#Sx4.p2.t1.r19">∖</mo></math>: set subtraction</a>,
<a href="./19.29#SS1.p1" title="§19.29(i) Reduction Theorems ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="21px" altimg-valign="-5px" altimg-width="29px" alttext="U_{n}" display="inline"><msub><mi href="./19.29#SS1.p1">U</mi><mi>n</mi></msub></math></a>,
<a href="./19.29#SS1.p1" title="§19.29(i) Reduction Theorems ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m98.png" altimg-height="21px" altimg-valign="-5px" altimg-width="27px" alttext="S_{n}" display="inline"><msub><mi href="./19.29#SS1.p1">S</mi><mi>n</mi></msub></math></a>,
<a href="./19.29#SS1.p1" title="§19.29(i) Reduction Theorems ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="23px" altimg-valign="-8px" altimg-width="41px" alttext="Q_{\alpha\beta}" display="inline"><msub><mi href="./19.29#SS1.p1">Q</mi><mrow><mi href="./19.29#SS1.p1">α</mi><mo></mo><mi href="./19.29#SS1.p1">β</mi></mrow></msub></math></a>,
<a href="./19.29#SS1.p1" title="§19.29(i) Reduction Theorems ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m140.png" altimg-height="23px" altimg-valign="-8px" altimg-width="36px" alttext="d_{\alpha\beta}" display="inline"><msub><mi href="./19.29#SS1.p1">d</mi><mrow><mi href="./19.29#SS1.p1">α</mi><mo></mo><mi href="./19.29#SS1.p1">β</mi></mrow></msub></math></a>,
<a href="./19.29#SS1.p1" title="§19.29(i) Reduction Theorems ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m159.png" altimg-height="23px" altimg-valign="-7px" altimg-width="36px" alttext="s(t)" display="inline"><mrow><mi href="./19.29#SS1.p1">s</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math>: product</a>,
<a href="./19.29#SS1.p1" title="§19.29(i) Reduction Theorems ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m109.png" altimg-height="13px" altimg-valign="-2px" altimg-width="17px" alttext="\alpha" display="inline"><mi href="./19.29#SS1.p1">α</mi></math>: parameter</a>,
<a href="./19.29#SS1.p1" title="§19.29(i) Reduction Theorems ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m111.png" altimg-height="21px" altimg-valign="-6px" altimg-width="17px" alttext="\beta" display="inline"><mi href="./19.29#SS1.p1">β</mi></math>: parameter</a>,
<a href="./19.29#SS1.p1" title="§19.29(i) Reduction Theorems ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m114.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\gamma" display="inline"><mi href="./19.29#SS1.p1">γ</mi></math>: parameter</a> and
<a href="./19.29#SS1.p1" title="§19.29(i) Reduction Theorems ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m113.png" altimg-height="18px" altimg-valign="-2px" altimg-width="14px" alttext="\delta" display="inline"><mi href="./19.29#SS1.p1">δ</mi></math>: parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where
</p>
<table id="E9" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex8" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="4" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">19.29.9</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m35.png" altimg-height="28px" altimg-valign="-7px" altimg-width="39px" alttext="\displaystyle U_{\alpha 5}^{2}" display="inline"><msubsup><mi href="./19.29#SS1.p1">U</mi><mrow><mi href="./19.29#SS1.p1">α</mi><mo></mo><mn>5</mn></mrow><mn>2</mn></msubsup></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m12.png" altimg-height="50px" altimg-valign="-19px" altimg-width="401px" alttext="\displaystyle=U_{\alpha\beta}^{2}-\frac{d_{\alpha\gamma}d_{\alpha\delta}d_{%
\beta 5}}{d_{\alpha 5}}=U_{\beta\gamma}^{2}-\frac{d_{\alpha\beta}d_{\alpha%
\gamma}d_{\delta 5}}{d_{\alpha 5}}\neq 0," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msubsup><mi href="./19.29#SS1.p1">U</mi><mrow><mi href="./19.29#SS1.p1">α</mi><mo></mo><mi href="./19.29#SS1.p1">β</mi></mrow><mn>2</mn></msubsup><mo>-</mo><mstyle displaystyle="true"><mfrac><mrow><msub><mi href="./19.29#SS1.p1">d</mi><mrow><mi href="./19.29#SS1.p1">α</mi><mo></mo><mi href="./19.29#SS1.p1">γ</mi></mrow></msub><mo></mo><msub><mi href="./19.29#SS1.p1">d</mi><mrow><mi href="./19.29#SS1.p1">α</mi><mo></mo><mi href="./19.29#SS1.p1">δ</mi></mrow></msub><mo></mo><msub><mi href="./19.29#SS1.p1">d</mi><mrow><mi href="./19.29#SS1.p1">β</mi><mo></mo><mn>5</mn></mrow></msub></mrow><msub><mi href="./19.29#SS1.p1">d</mi><mrow><mi href="./19.29#SS1.p1">α</mi><mo></mo><mn>5</mn></mrow></msub></mfrac></mstyle></mrow><mo>=</mo><mrow><msubsup><mi href="./19.29#SS1.p1">U</mi><mrow><mi href="./19.29#SS1.p1">β</mi><mo></mo><mi href="./19.29#SS1.p1">γ</mi></mrow><mn>2</mn></msubsup><mo>-</mo><mstyle displaystyle="true"><mfrac><mrow><msub><mi href="./19.29#SS1.p1">d</mi><mrow><mi href="./19.29#SS1.p1">α</mi><mo></mo><mi href="./19.29#SS1.p1">β</mi></mrow></msub><mo></mo><msub><mi href="./19.29#SS1.p1">d</mi><mrow><mi href="./19.29#SS1.p1">α</mi><mo></mo><mi href="./19.29#SS1.p1">γ</mi></mrow></msub><mo></mo><msub><mi href="./19.29#SS1.p1">d</mi><mrow><mi href="./19.29#SS1.p1">δ</mi><mo></mo><mn>5</mn></mrow></msub></mrow><msub><mi href="./19.29#SS1.p1">d</mi><mrow><mi href="./19.29#SS1.p1">α</mi><mo></mo><mn>5</mn></mrow></msub></mfrac></mstyle></mrow><mo>≠</mo><mn>0</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex9" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m32.png" altimg-height="22px" altimg-valign="-5px" altimg-width="38px" alttext="\displaystyle S_{\alpha 5}" display="inline"><msub><mi href="./19.29#SS1.p1">S</mi><mrow><mi href="./19.29#SS1.p1">α</mi><mo></mo><mn>5</mn></mrow></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m15.png" altimg-height="53px" altimg-valign="-21px" altimg-width="349px" alttext="\displaystyle=\frac{1}{x-y}\left(\frac{X_{\beta}X_{\gamma}X_{\delta}}{X_{%
\alpha}}Y_{5}^{2}+\frac{Y_{\beta}Y_{\gamma}Y_{\delta}}{Y_{\alpha}}X_{5}^{2}%
\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><mfrac><mn>1</mn><mrow><mi>x</mi><mo>-</mo><mi>y</mi></mrow></mfrac></mstyle><mo></mo><mrow><mo>(</mo><mrow><mrow><mstyle displaystyle="true"><mfrac><mrow><msub><mi href="./19.29#SS1.p1">X</mi><mi href="./19.29#SS1.p1">β</mi></msub><mo></mo><msub><mi href="./19.29#SS1.p1">X</mi><mi href="./19.29#SS1.p1">γ</mi></msub><mo></mo><msub><mi href="./19.29#SS1.p1">X</mi><mi href="./19.29#SS1.p1">δ</mi></msub></mrow><msub><mi href="./19.29#SS1.p1">X</mi><mi href="./19.29#SS1.p1">α</mi></msub></mfrac></mstyle><mo></mo><msubsup><mi href="./19.29#SS1.p1">Y</mi><mn>5</mn><mn>2</mn></msubsup></mrow><mo>+</mo><mrow><mstyle displaystyle="true"><mfrac><mrow><msub><mi href="./19.29#SS1.p1">Y</mi><mi href="./19.29#SS1.p1">β</mi></msub><mo></mo><msub><mi href="./19.29#SS1.p1">Y</mi><mi href="./19.29#SS1.p1">γ</mi></msub><mo></mo><msub><mi href="./19.29#SS1.p1">Y</mi><mi href="./19.29#SS1.p1">δ</mi></msub></mrow><msub><mi href="./19.29#SS1.p1">Y</mi><mi href="./19.29#SS1.p1">α</mi></msub></mfrac></mstyle><mo></mo><msubsup><mi href="./19.29#SS1.p1">X</mi><mn>5</mn><mn>2</mn></msubsup></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex10" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m30.png" altimg-height="23px" altimg-valign="-6px" altimg-width="41px" alttext="\displaystyle Q_{\alpha 5}" display="inline"><msub><mi href="./19.29#SS1.p1">Q</mi><mrow><mi href="./19.29#SS1.p1">α</mi><mo></mo><mn>5</mn></mrow></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m16.png" altimg-height="49px" altimg-valign="-19px" altimg-width="158px" alttext="\displaystyle=\frac{X_{5}Y_{5}}{X_{\alpha}Y_{\alpha}}U_{\alpha 5}\neq 0," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><mfrac><mrow><msub><mi href="./19.29#SS1.p1">X</mi><mn>5</mn></msub><mo></mo><msub><mi href="./19.29#SS1.p1">Y</mi><mn>5</mn></msub></mrow><mrow><msub><mi href="./19.29#SS1.p1">X</mi><mi href="./19.29#SS1.p1">α</mi></msub><mo></mo><msub><mi href="./19.29#SS1.p1">Y</mi><mi href="./19.29#SS1.p1">α</mi></msub></mrow></mfrac></mstyle><mo></mo><msub><mi href="./19.29#SS1.p1">U</mi><mrow><mi href="./19.29#SS1.p1">α</mi><mo></mo><mn>5</mn></mrow></msub></mrow><mo>≠</mo><mn>0</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex11" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m33.png" altimg-height="28px" altimg-valign="-7px" altimg-width="97px" alttext="\displaystyle S_{\alpha 5}^{2}-Q_{\alpha 5}^{2}" display="inline"><mrow><msubsup><mi href="./19.29#SS1.p1">S</mi><mrow><mi href="./19.29#SS1.p1">α</mi><mo></mo><mn>5</mn></mrow><mn>2</mn></msubsup><mo>-</mo><msubsup><mi href="./19.29#SS1.p1">Q</mi><mrow><mi href="./19.29#SS1.p1">α</mi><mo></mo><mn>5</mn></mrow><mn>2</mn></msubsup></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m17.png" altimg-height="50px" altimg-valign="-19px" altimg-width="122px" alttext="\displaystyle=\frac{d_{\beta 5}d_{\gamma 5}d_{\delta 5}}{d_{\alpha 5}}." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mstyle displaystyle="true"><mfrac><mrow><msub><mi href="./19.29#SS1.p1">d</mi><mrow><mi href="./19.29#SS1.p1">β</mi><mo></mo><mn>5</mn></mrow></msub><mo></mo><msub><mi href="./19.29#SS1.p1">d</mi><mrow><mi href="./19.29#SS1.p1">γ</mi><mo></mo><mn>5</mn></mrow></msub><mo></mo><msub><mi href="./19.29#SS1.p1">d</mi><mrow><mi href="./19.29#SS1.p1">δ</mi><mo></mo><mn>5</mn></mrow></msub></mrow><msub><mi href="./19.29#SS1.p1">d</mi><mrow><mi href="./19.29#SS1.p1">α</mi><mo></mo><mn>5</mn></mrow></msub></mfrac></mstyle></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E9.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.29#SS1.p1" title="§19.29(i) Reduction Theorems ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m105.png" altimg-height="21px" altimg-valign="-5px" altimg-width="32px" alttext="X_{\alpha}" display="inline"><msub><mi href="./19.29#SS1.p1">X</mi><mi href="./19.29#SS1.p1">α</mi></msub></math></a>,
<a href="./19.29#SS1.p1" title="§19.29(i) Reduction Theorems ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="21px" altimg-valign="-5px" altimg-width="29px" alttext="U_{n}" display="inline"><msub><mi href="./19.29#SS1.p1">U</mi><mi>n</mi></msub></math></a>,
<a href="./19.29#SS1.p1" title="§19.29(i) Reduction Theorems ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m98.png" altimg-height="21px" altimg-valign="-5px" altimg-width="27px" alttext="S_{n}" display="inline"><msub><mi href="./19.29#SS1.p1">S</mi><mi>n</mi></msub></math></a>,
<a href="./19.29#SS1.p1" title="§19.29(i) Reduction Theorems ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="23px" altimg-valign="-8px" altimg-width="41px" alttext="Q_{\alpha\beta}" display="inline"><msub><mi href="./19.29#SS1.p1">Q</mi><mrow><mi href="./19.29#SS1.p1">α</mi><mo></mo><mi href="./19.29#SS1.p1">β</mi></mrow></msub></math></a>,
<a href="./19.29#SS1.p1" title="§19.29(i) Reduction Theorems ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m106.png" altimg-height="21px" altimg-valign="-5px" altimg-width="27px" alttext="Y_{\alpha}" display="inline"><msub><mi href="./19.29#SS1.p1">Y</mi><mi href="./19.29#SS1.p1">α</mi></msub></math></a>,
<a href="./19.29#SS1.p1" title="§19.29(i) Reduction Theorems ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m140.png" altimg-height="23px" altimg-valign="-8px" altimg-width="36px" alttext="d_{\alpha\beta}" display="inline"><msub><mi href="./19.29#SS1.p1">d</mi><mrow><mi href="./19.29#SS1.p1">α</mi><mo></mo><mi href="./19.29#SS1.p1">β</mi></mrow></msub></math></a>,
<a href="./19.29#SS1.p1" title="§19.29(i) Reduction Theorems ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m109.png" altimg-height="13px" altimg-valign="-2px" altimg-width="17px" alttext="\alpha" display="inline"><mi href="./19.29#SS1.p1">α</mi></math>: parameter</a>,
<a href="./19.29#SS1.p1" title="§19.29(i) Reduction Theorems ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m111.png" altimg-height="21px" altimg-valign="-6px" altimg-width="17px" alttext="\beta" display="inline"><mi href="./19.29#SS1.p1">β</mi></math>: parameter</a>,
<a href="./19.29#SS1.p1" title="§19.29(i) Reduction Theorems ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m114.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\gamma" display="inline"><mi href="./19.29#SS1.p1">γ</mi></math>: parameter</a> and
<a href="./19.29#SS1.p1" title="§19.29(i) Reduction Theorems ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m113.png" altimg-height="18px" altimg-valign="-2px" altimg-width="14px" alttext="\delta" display="inline"><mi href="./19.29#SS1.p1">δ</mi></math>: parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">The Cauchy principal value is taken when <math class="ltx_Math" altimg="m100.png" altimg-height="25px" altimg-valign="-7px" altimg-width="37px" alttext="U_{\alpha 5}^{2}" display="inline"><msubsup><mi href="./19.29#SS1.p1">U</mi><mrow><mi href="./19.29#SS1.p1">α</mi><mo></mo><mn>5</mn></mrow><mn>2</mn></msubsup></math> or <math class="ltx_Math" altimg="m84.png" altimg-height="25px" altimg-valign="-7px" altimg-width="39px" alttext="Q_{\alpha 5}^{2}" display="inline"><msubsup><mi href="./19.29#SS1.p1">Q</mi><mrow><mi href="./19.29#SS1.p1">α</mi><mo></mo><mn>5</mn></mrow><mn>2</mn></msubsup></math>
is real and negative. Cubic cases of these formulas are obtained by setting
one of the factors in () and its cubic case, which
replace the <math class="ltx_Math" altimg="m66.png" altimg-height="18px" altimg-valign="-4px" altimg-width="139px" alttext="8+8+12=28" display="inline"><mrow><mrow><mn>8</mn><mo>+</mo><mn>8</mn><mo>+</mo><mn>12</mn></mrow><mo>=</mo><mn>28</mn></mrow></math> formulas in
<cite class="ltx_cite ltx_citemacro_citet">Gradshteyn and Ryzhik (, 3.147, 3.131, 3.152)</cite> after taking <math class="ltx_Math" altimg="m166.png" altimg-height="20px" altimg-valign="-2px" altimg-width="25px" alttext="x^{2}" display="inline"><msup><mi>x</mi><mn>2</mn></msup></math> as the
variable of integration in 3.152. Moreover, the requirement that one limit of
integration be a branch point of the integrand is eliminated without doubling
the number of standard integrals in the result. (, 3.168)</cite>, and its cubic cases
similarly replace the <math class="ltx_Math" altimg="m60.png" altimg-height="18px" altimg-valign="-4px" altimg-width="159px" alttext="18+36+18=72" display="inline"><mrow><mrow><mn>18</mn><mo>+</mo><mn>36</mn><mo>+</mo><mn>18</mn></mrow><mo>=</mo><mn>72</mn></mrow></math> formulas in
<cite class="ltx_cite ltx_citemacro_citet">Gradshteyn and Ryzhik (, 3.133, 3.142, and 3.141(1-18))</cite>. For example,
3.142(2) is included as
</p>
<table id="E10" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.29.10</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E10.png" altimg-height="66px" altimg-valign="-26px" altimg-width="651px" alttext="\int_{u}^{b}\sqrt{\frac{a-t}{(b-t)(t-c)^{3}}}\mathrm{d}t=-\tfrac{2}{3}{(a-b)}{%
(b-u)}^{3/2}R_{D}+\frac{2}{b-c}\sqrt{\frac{(a-u)(b-u)}{u-c}}," display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi>u</mi><mi>b</mi></msubsup><mrow><msqrt><mfrac><mrow><mi>a</mi><mo>-</mo><mi>t</mi></mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mi>b</mi><mo>-</mo><mi>t</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi>t</mi><mo>-</mo><mi>c</mi></mrow><mo stretchy="false">)</mo></mrow><mn>3</mn></msup></mrow></mfrac></msqrt><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow><mo>=</mo><mrow><mrow><mo>-</mo><mrow><mstyle displaystyle="false"><mfrac><mn>2</mn><mn>3</mn></mfrac></mstyle><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>a</mi><mo>-</mo><mi>b</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi>b</mi><mo>-</mo><mi>u</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mn>3</mn><mo>/</mo><mn>2</mn></mrow></msup><mo></mo><mrow><msub><mi href="./19.16#E5">R</mi><mi href="./19.16#E5">D</mi></msub><mo></mo></mrow></mrow></mrow><mo>+</mo><mrow><mfrac><mn>2</mn><mrow><mi>b</mi><mo>-</mo><mi>c</mi></mrow></mfrac><mo></mo><msqrt><mfrac><mrow><mrow><mo stretchy="false">(</mo><mrow><mi>a</mi><mo>-</mo><mi>u</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>b</mi><mo>-</mo><mi>u</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mrow><mi>u</mi><mo>-</mo><mi>c</mi></mrow></mfrac></msqrt></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m127.png" altimg-height="18px" altimg-valign="-3px" altimg-width="123px" alttext="a>b>u>c" display="inline"><mrow><mi>a</mi><mo>></mo><mi>b</mi><mo>></mo><mi>u</mi><mo>></mo><mi>c</mi></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E10.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.16#E5" title="(19.16.5) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m91.png" altimg-height="23px" altimg-valign="-7px" altimg-width="102px" alttext="R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E5">R</mi><mi href="./19.16#E5">D</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: elliptic integral symmetric in only two variables</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m124.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m164.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m117.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where the arguments of the <math class="ltx_Math" altimg="m90.png" altimg-height="21px" altimg-valign="-5px" altimg-width="34px" alttext="R_{D}" display="inline"><msub><mi href="./19.16#E5">R</mi><mi href="./19.16#E5">D</mi></msub></math> function are, in order,
<math class="ltx_Math" altimg="m49.png" altimg-height="23px" altimg-valign="-7px" altimg-width="123px" alttext="(a-b)(u-c)" display="inline"><mrow><mrow><mo stretchy="false">(</mo><mrow><mi>a</mi><mo>-</mo><mi>b</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>u</mi><mo>-</mo><mi>c</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></math>, <math class="ltx_Math" altimg="m50.png" altimg-height="23px" altimg-valign="-7px" altimg-width="123px" alttext="(b-c)(a-u)" display="inline"><mrow><mrow><mo stretchy="false">(</mo><mrow><mi>b</mi><mo>-</mo><mi>c</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>a</mi><mo>-</mo><mi>u</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></math>, <math class="ltx_Math" altimg="m48.png" altimg-height="23px" altimg-valign="-7px" altimg-width="120px" alttext="(a-b)(b-c)" display="inline"><mrow><mrow><mo stretchy="false">(</mo><mrow><mi>a</mi><mo>-</mo><mi>b</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>b</mi><mo>-</mo><mi>c</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></math>.</p>
</div>
</section>
<section id="ii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§19.29(ii) </span>Reduction to Basic Integrals</h2>
<div id="SS2.info" class="ltx_metadata ltx_info">
) can be written
</p>
<table id="E11" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.29.11</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E11.png" altimg-height="70px" altimg-valign="-30px" altimg-width="428px" alttext="I(\mathbf{m})=\int_{y}^{x}\prod_{\alpha=1}^{h}(a_{\alpha}+b_{\alpha}t)^{-1/2}%
\prod_{j=1}^{n}(a_{j}+b_{j}t)^{m_{j}}\mathrm{d}t," display="block"><mrow><mrow><mrow><mi href="./19.29#SS2.p1">I</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi mathvariant="bold">m</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi>y</mi><mi>x</mi></msubsup><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∏</mo><mrow><mi href="./19.29#SS1.p1">α</mi><mo>=</mo><mn>1</mn></mrow><mi>h</mi></munderover><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><msub><mi>a</mi><mi href="./19.29#SS1.p1">α</mi></msub><mo>+</mo><mrow><msub><mi>b</mi><mi href="./19.29#SS1.p1">α</mi></msub><mo></mo><mi>t</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></mrow></msup><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∏</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi href="./19.1#p1.t1.r1">n</mi></munderover><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><msub><mi>a</mi><mi>j</mi></msub><mo>+</mo><mrow><msub><mi>b</mi><mi>j</mi></msub><mo></mo><mi>t</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><msub><mi href="./19.1#p1.t1.r1">m</mi><mi>j</mi></msub></msup><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E11.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m124.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m164.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m117.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./19.1#p1.t1.r1" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m155.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./19.1#p1.t1.r1">m</mi></math>: nonnegative integer</a>,
<a href="./19.1#p1.t1.r1" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m157.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./19.1#p1.t1.r1">n</mi></math>: nonnegative integer</a>,
<a href="./19.29#SS2.p1" title="§19.29(ii) Reduction to Basic Integrals ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m76.png" altimg-height="23px" altimg-valign="-7px" altimg-width="49px" alttext="I(\mathbf{m})" display="inline"><mrow><mi href="./19.29#SS2.p1">I</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi mathvariant="bold">m</mi><mo stretchy="false">)</mo></mrow></mrow></math>: integral</a> and
<a href="./19.29#SS1.p1" title="§19.29(i) Reduction Theorems ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m109.png" altimg-height="13px" altimg-valign="-2px" altimg-width="17px" alttext="\alpha" display="inline"><mi href="./19.29#SS1.p1">α</mi></math>: parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m163.png" altimg-height="18px" altimg-valign="-6px" altimg-width="53px" alttext="x>y" display="inline"><mrow><mi>x</mi><mo>></mo><mi>y</mi></mrow></math>, <math class="ltx_Math" altimg="m146.png" altimg-height="18px" altimg-valign="-2px" altimg-width="52px" alttext="h=3" display="inline"><mrow><mi>h</mi><mo>=</mo><mn>3</mn></mrow></math> or 4, <math class="ltx_Math" altimg="m158.png" altimg-height="20px" altimg-valign="-5px" altimg-width="54px" alttext="n\geq h" display="inline"><mrow><mi href="./19.1#p1.t1.r1">n</mi><mo>≥</mo><mi>h</mi></mrow></math>, and <math class="ltx_Math" altimg="m156.png" altimg-height="18px" altimg-valign="-8px" altimg-width="30px" alttext="m_{j}" display="inline"><msub><mi href="./19.1#p1.t1.r1">m</mi><mi>j</mi></msub></math> is an integer. Define</p>
<table id="E12" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.29.12</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E12.png" altimg-height="67px" altimg-valign="-30px" altimg-width="275px" alttext="\mathbf{m}=(m_{1},\dots,m_{n})=\sum_{j=1}^{n}m_{j}\mathbf{e}_{j}," display="block"><mrow><mrow><mi mathvariant="bold">m</mi><mo>=</mo><mrow><mo stretchy="false">(</mo><msub><mi href="./19.1#p1.t1.r1">m</mi><mn>1</mn></msub><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><msub><mi href="./19.1#p1.t1.r1">m</mi><mi href="./19.1#p1.t1.r1">n</mi></msub><mo stretchy="false">)</mo></mrow><mo>=</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi href="./19.1#p1.t1.r1">n</mi></munderover><mrow><msub><mi href="./19.1#p1.t1.r1">m</mi><mi>j</mi></msub><mo></mo><msub><mi mathvariant="bold">e</mi><mi>j</mi></msub></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E12.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.1#p1.t1.r1" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m155.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./19.1#p1.t1.r1">m</mi></math>: nonnegative integer</a> and
<a href="./19.1#p1.t1.r1" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m157.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./19.1#p1.t1.r1">n</mi></math>: nonnegative integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m121.png" altimg-height="18px" altimg-valign="-8px" altimg-width="23px" alttext="\mathbf{e}_{j}" display="inline"><msub><mi mathvariant="bold">e</mi><mi>j</mi></msub></math> is an <math class="ltx_Math" altimg="m157.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./19.1#p1.t1.r1">n</mi></math>-tuple with 1 in the <math class="ltx_Math" altimg="m153.png" altimg-height="20px" altimg-valign="-6px" altimg-width="14px" alttext="j" display="inline"><mi>j</mi></math>th position and 0’s
elsewhere. Define also <math class="ltx_Math" altimg="m112.png" altimg-height="23px" altimg-valign="-7px" altimg-width="122px" alttext="\boldsymbol{{0}}=(0,\dots,0)" display="inline"><mrow><mn mathvariant="bold">0</mn><mo>=</mo><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></mrow></math> and retain the notation and
conditions associated with ()
are <math class="ltx_Math" altimg="m71.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="I(\boldsymbol{{0}})" display="inline"><mrow><mi href="./19.29#SS2.p1">I</mi><mo></mo><mrow><mo stretchy="false">(</mo><mn mathvariant="bold">0</mn><mo stretchy="false">)</mo></mrow></mrow></math>, <math class="ltx_Math" altimg="m74.png" altimg-height="23px" altimg-valign="-7px" altimg-width="96px" alttext="I(\mathbf{e}_{\alpha}-\mathbf{e}_{\delta})" display="inline"><mrow><mi href="./19.29#SS2.p1">I</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><msub><mi mathvariant="bold">e</mi><mi href="./19.29#SS1.p1">α</mi></msub><mo>-</mo><msub><mi mathvariant="bold">e</mi><mi href="./19.29#SS1.p1">δ</mi></msub></mrow><mo stretchy="false">)</mo></mrow></mrow></math>, and
<math class="ltx_Math" altimg="m73.png" altimg-height="23px" altimg-valign="-7px" altimg-width="96px" alttext="I(\mathbf{e}_{\alpha}-\mathbf{e}_{5})" display="inline"><mrow><mi href="./19.29#SS2.p1">I</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><msub><mi mathvariant="bold">e</mi><mi href="./19.29#SS1.p1">α</mi></msub><mo>-</mo><msub><mi mathvariant="bold">e</mi><mn>5</mn></msub></mrow><mo stretchy="false">)</mo></mrow></mrow></math>, respectively.</p>
</div>
<div id="SS2.p2" class="ltx_para">
<p class="ltx_p">The only cases of <math class="ltx_Math" altimg="m76.png" altimg-height="23px" altimg-valign="-7px" altimg-width="49px" alttext="I(\mathbf{m})" display="inline"><mrow><mi href="./19.29#SS2.p1">I</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi mathvariant="bold">m</mi><mo stretchy="false">)</mo></mrow></mrow></math> that are integrals of the <em class="ltx_emph ltx_font_italic">first kind</em>
are the two (<math class="ltx_Math" altimg="m146.png" altimg-height="18px" altimg-valign="-2px" altimg-width="52px" alttext="h=3" display="inline"><mrow><mi>h</mi><mo>=</mo><mn>3</mn></mrow></math> or 4) with <math class="ltx_Math" altimg="m122.png" altimg-height="17px" altimg-valign="-2px" altimg-width="61px" alttext="\mathbf{m}=\boldsymbol{{0}}" display="inline"><mrow><mi mathvariant="bold">m</mi><mo>=</mo><mn mathvariant="bold">0</mn></mrow></math>. The only cases that
are integrals of the <em class="ltx_emph ltx_font_italic">third kind</em> are those in which at least one <math class="ltx_Math" altimg="m156.png" altimg-height="18px" altimg-valign="-8px" altimg-width="30px" alttext="m_{j}" display="inline"><msub><mi href="./19.1#p1.t1.r1">m</mi><mi>j</mi></msub></math>
with <math class="ltx_Math" altimg="m152.png" altimg-height="21px" altimg-valign="-6px" altimg-width="52px" alttext="j>h" display="inline"><mrow><mi>j</mi><mo>></mo><mi>h</mi></mrow></math> is a negative integer and those in which <math class="ltx_Math" altimg="m147.png" altimg-height="18px" altimg-valign="-2px" altimg-width="52px" alttext="h=4" display="inline"><mrow><mi>h</mi><mo>=</mo><mn>4</mn></mrow></math> and
<math class="ltx_Math" altimg="m126.png" altimg-height="28px" altimg-valign="-11px" altimg-width="83px" alttext="\sum_{j=1}^{n}m_{j}" display="inline"><mrow><msubsup><mo largeop="true" symmetric="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi href="./19.1#p1.t1.r1">n</mi></msubsup><msub><mi href="./19.1#p1.t1.r1">m</mi><mi>j</mi></msub></mrow></math> is a positive integer. All other cases are integrals of the
<em class="ltx_emph ltx_font_italic">second kind</em>.</p>
</div>
<div id="SS2.p3" class="ltx_para">
<p class="ltx_p"><math class="ltx_Math" altimg="m76.png" altimg-height="23px" altimg-valign="-7px" altimg-width="49px" alttext="I(\mathbf{m})" display="inline"><mrow><mi href="./19.29#SS2.p1">I</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi mathvariant="bold">m</mi><mo stretchy="false">)</mo></mrow></mrow></math> can be reduced to a linear combination of <em class="ltx_emph ltx_font_italic">basic
integrals</em> and algebraic functions. In the cubic case (<math class="ltx_Math" altimg="m146.png" altimg-height="18px" altimg-valign="-2px" altimg-width="52px" alttext="h=3" display="inline"><mrow><mi>h</mi><mo>=</mo><mn>3</mn></mrow></math>) the basic
integrals are</p>
<table id="E13" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex12" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="3" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">19.29.13</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E14a.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="I(\boldsymbol{{0}});" display="inline"><mrow><mrow><mi href="./19.29#SS2.p1">I</mi><mo></mo><mrow><mo stretchy="false">(</mo><mn mathvariant="bold">0</mn><mo stretchy="false">)</mo></mrow></mrow><mo>;</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex13" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E14b.png" altimg-height="24px" altimg-valign="-8px" altimg-width="70px" alttext="I(-\mathbf{e}_{j})," display="inline"><mrow><mrow><mi href="./19.29#SS2.p1">I</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><msub><mi mathvariant="bold">e</mi><mi>j</mi></msub></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="3"><span class="ltx_constraint"><math class="ltx_Math" altimg="m63.png" altimg-height="20px" altimg-valign="-6px" altimg-width="89px" alttext="1\leq j\leq n" display="inline"><mrow><mn>1</mn><mo>≤</mo><mi>j</mi><mo>≤</mo><mi href="./19.1#p1.t1.r1">n</mi></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E13.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.1#p1.t1.r1" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m157.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./19.1#p1.t1.r1">n</mi></math>: nonnegative integer</a> and
<a href="./19.29#SS2.p1" title="§19.29(ii) Reduction to Basic Integrals ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m76.png" altimg-height="23px" altimg-valign="-7px" altimg-width="49px" alttext="I(\mathbf{m})" display="inline"><mrow><mi href="./19.29#SS2.p1">I</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi mathvariant="bold">m</mi><mo stretchy="false">)</mo></mrow></mrow></math>: integral</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">In the quartic case (<math class="ltx_Math" altimg="m147.png" altimg-height="18px" altimg-valign="-2px" altimg-width="52px" alttext="h=4" display="inline"><mrow><mi>h</mi><mo>=</mo><mn>4</mn></mrow></math>) the basic integrals are</p>
<table id="E14" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex14" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="5" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">19.29.14</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E14a.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="I(\boldsymbol{{0}});" display="inline"><mrow><mrow><mi href="./19.29#SS2.p1">I</mi><mo></mo><mrow><mo stretchy="false">(</mo><mn mathvariant="bold">0</mn><mo stretchy="false">)</mo></mrow></mrow><mo>;</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex15" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E14b.png" altimg-height="24px" altimg-valign="-8px" altimg-width="70px" alttext="I(-\mathbf{e}_{j})," display="inline"><mrow><mrow><mi href="./19.29#SS2.p1">I</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><msub><mi mathvariant="bold">e</mi><mi>j</mi></msub></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="3"><span class="ltx_constraint"><math class="ltx_Math" altimg="m63.png" altimg-height="20px" altimg-valign="-6px" altimg-width="89px" alttext="1\leq j\leq n" display="inline"><mrow><mn>1</mn><mo>≤</mo><mi>j</mi><mo>≤</mo><mi href="./19.1#p1.t1.r1">n</mi></mrow></math>;</span></td></tr>
<tr id="Ex16" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E14c.png" altimg-height="23px" altimg-valign="-7px" altimg-width="58px" alttext="I(\mathbf{e}_{\alpha})," display="inline"><mrow><mrow><mi href="./19.29#SS2.p1">I</mi><mo></mo><mrow><mo stretchy="false">(</mo><msub><mi mathvariant="bold">e</mi><mi href="./19.29#SS1.p1">α</mi></msub><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="3"><span class="ltx_constraint"><math class="ltx_Math" altimg="m64.png" altimg-height="19px" altimg-valign="-5px" altimg-width="90px" alttext="1\leq\alpha\leq 4" display="inline"><mrow><mn>1</mn><mo>≤</mo><mi href="./19.29#SS1.p1">α</mi><mo>≤</mo><mn>4</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E14.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.1#p1.t1.r1" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m157.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./19.1#p1.t1.r1">n</mi></math>: nonnegative integer</a>,
<a href="./19.29#SS2.p1" title="§19.29(ii) Reduction to Basic Integrals ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m76.png" altimg-height="23px" altimg-valign="-7px" altimg-width="49px" alttext="I(\mathbf{m})" display="inline"><mrow><mi href="./19.29#SS2.p1">I</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi mathvariant="bold">m</mi><mo stretchy="false">)</mo></mrow></mrow></math>: integral</a> and
<a href="./19.29#SS1.p1" title="§19.29(i) Reduction Theorems ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m109.png" altimg-height="13px" altimg-valign="-2px" altimg-width="17px" alttext="\alpha" display="inline"><mi href="./19.29#SS1.p1">α</mi></math>: parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Basic integrals of type <math class="ltx_Math" altimg="m68.png" altimg-height="24px" altimg-valign="-8px" altimg-width="65px" alttext="I(-\mathbf{e}_{j})" display="inline"><mrow><mi href="./19.29#SS2.p1">I</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><msub><mi mathvariant="bold">e</mi><mi>j</mi></msub></mrow><mo stretchy="false">)</mo></mrow></mrow></math>, <math class="ltx_Math" altimg="m62.png" altimg-height="21px" altimg-valign="-6px" altimg-width="88px" alttext="1\leq j\leq h" display="inline"><mrow><mn>1</mn><mo>≤</mo><mi>j</mi><mo>≤</mo><mi>h</mi></mrow></math>, are not linearly
independent, nor are those of type <math class="ltx_Math" altimg="m75.png" altimg-height="24px" altimg-valign="-8px" altimg-width="49px" alttext="I(\mathbf{e}_{j})" display="inline"><mrow><mi href="./19.29#SS2.p1">I</mi><mo></mo><mrow><mo stretchy="false">(</mo><msub><mi mathvariant="bold">e</mi><mi>j</mi></msub><mo stretchy="false">)</mo></mrow></mrow></math>, <math class="ltx_Math" altimg="m61.png" altimg-height="20px" altimg-valign="-6px" altimg-width="87px" alttext="1\leq j\leq 4" display="inline"><mrow><mn>1</mn><mo>≤</mo><mi>j</mi><mo>≤</mo><mn>4</mn></mrow></math>.
</p>
</div>
<div id="SS2.p4" class="ltx_para">
<p class="ltx_p">The reduction of <math class="ltx_Math" altimg="m76.png" altimg-height="23px" altimg-valign="-7px" altimg-width="49px" alttext="I(\mathbf{m})" display="inline"><mrow><mi href="./19.29#SS2.p1">I</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi mathvariant="bold">m</mi><mo stretchy="false">)</mo></mrow></mrow></math> is carried out by a relation derived from
partial fractions and by use of two recurrence relations. These are given in
<cite class="ltx_cite ltx_citemacro_citet">Carlson (, (1.10), (1.7), (1.8))</cite> by means of modified
definitions. Partial fractions provide a reduction to integrals in which
<math class="ltx_Math" altimg="m123.png" altimg-height="13px" altimg-valign="-2px" altimg-width="23px" alttext="\mathbf{m}" display="inline"><mi mathvariant="bold">m</mi></math> has at most one nonzero component, and these are then reduced to
basic integrals by the recurrence relations. A special case of
<cite class="ltx_cite ltx_citemacro_citet">Carlson (, (2.19))</cite> is given by</p>
<table id="E15" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.29.15</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E15.png" altimg-height="26px" altimg-valign="-8px" altimg-width="301px" alttext="b_{j}I(\mathbf{e}_{l}-\mathbf{e}_{j})=d_{lj}I(-\mathbf{e}_{j})+b_{l}I(%
\boldsymbol{{0}})," display="block"><mrow><mrow><mrow><msub><mi>b</mi><mi>j</mi></msub><mo></mo><mrow><mi href="./19.29#SS2.p1">I</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><msub><mi mathvariant="bold">e</mi><mi href="./19.1#p1.t1.r1">l</mi></msub><mo>-</mo><msub><mi mathvariant="bold">e</mi><mi>j</mi></msub></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>=</mo><mrow><mrow><msub><mi href="./19.29#SS1.p1">d</mi><mrow><mi href="./19.1#p1.t1.r1">l</mi><mo></mo><mi>j</mi></mrow></msub><mo></mo><mrow><mi href="./19.29#SS2.p1">I</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><msub><mi mathvariant="bold">e</mi><mi>j</mi></msub></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>+</mo><mrow><msub><mi>b</mi><mi href="./19.1#p1.t1.r1">l</mi></msub><mo></mo><mrow><mi href="./19.29#SS2.p1">I</mi><mo></mo><mrow><mo stretchy="false">(</mo><mn mathvariant="bold">0</mn><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m150.png" altimg-height="21px" altimg-valign="-6px" altimg-width="139px" alttext="j,l=1,2,\dots,n" display="inline"><mrow><mrow><mrow><mi>j</mi><mo>,</mo><mi href="./19.1#p1.t1.r1">l</mi></mrow><mo>=</mo><mn>1</mn></mrow><mo>,</mo><mrow><mn>2</mn><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><mi href="./19.1#p1.t1.r1">n</mi></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E15.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.1#p1.t1.r1" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m154.png" altimg-height="18px" altimg-valign="-2px" altimg-width="11px" alttext="l" display="inline"><mi href="./19.1#p1.t1.r1">l</mi></math>: nonnegative integer</a>,
<a href="./19.1#p1.t1.r1" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m157.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./19.1#p1.t1.r1">n</mi></math>: nonnegative integer</a>,
<a href="./19.29#SS2.p1" title="§19.29(ii) Reduction to Basic Integrals ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m76.png" altimg-height="23px" altimg-valign="-7px" altimg-width="49px" alttext="I(\mathbf{m})" display="inline"><mrow><mi href="./19.29#SS2.p1">I</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi mathvariant="bold">m</mi><mo stretchy="false">)</mo></mrow></mrow></math>: integral</a> and
<a href="./19.29#SS1.p1" title="§19.29(i) Reduction Theorems ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m140.png" altimg-height="23px" altimg-valign="-8px" altimg-width="36px" alttext="d_{\alpha\beta}" display="inline"><msub><mi href="./19.29#SS1.p1">d</mi><mrow><mi href="./19.29#SS1.p1">α</mi><mo></mo><mi href="./19.29#SS1.p1">β</mi></mrow></msub></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">which shows how to express the basic integral <math class="ltx_Math" altimg="m68.png" altimg-height="24px" altimg-valign="-8px" altimg-width="65px" alttext="I(-\mathbf{e}_{j})" display="inline"><mrow><mi href="./19.29#SS2.p1">I</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><msub><mi mathvariant="bold">e</mi><mi>j</mi></msub></mrow><mo stretchy="false">)</mo></mrow></mrow></math> in terms of
symmetric integrals by using (, 3.151, 3.149, 3.137, 3.157)</cite> (after setting
<math class="ltx_Math" altimg="m165.png" altimg-height="20px" altimg-valign="-2px" altimg-width="58px" alttext="x^{2}=t" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>=</mo><mi>t</mi></mrow></math> in some cases).</p>
</div>
<div id="SS2.p5" class="ltx_para">
<p class="ltx_p">If <math class="ltx_Math" altimg="m146.png" altimg-height="18px" altimg-valign="-2px" altimg-width="52px" alttext="h=3" display="inline"><mrow><mi>h</mi><mo>=</mo><mn>3</mn></mrow></math>, then the recurrence relation (<cite class="ltx_cite ltx_citemacro_citet">Carlson (, (3.5))</cite>) has
the special case</p>
<table id="E16" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.29.16</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E16.png" altimg-height="53px" altimg-valign="-21px" altimg-width="528px" alttext="b_{\beta}b_{\gamma}I(\mathbf{e}_{\alpha})=d_{\alpha\beta}d_{\alpha\gamma}I(-%
\mathbf{e}_{\alpha})+2b_{\alpha}\left(\frac{s(x)}{a_{\alpha}+b_{\alpha}x}-%
\frac{s(y)}{a_{\alpha}+b_{\alpha}y}\right)," display="block"><mrow><mrow><mrow><msub><mi>b</mi><mi href="./19.29#SS1.p1">β</mi></msub><mo></mo><msub><mi>b</mi><mi href="./19.29#SS1.p1">γ</mi></msub><mo></mo><mrow><mi href="./19.29#SS2.p1">I</mi><mo></mo><mrow><mo stretchy="false">(</mo><msub><mi mathvariant="bold">e</mi><mi href="./19.29#SS1.p1">α</mi></msub><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>=</mo><mrow><mrow><msub><mi href="./19.29#SS1.p1">d</mi><mrow><mi href="./19.29#SS1.p1">α</mi><mo></mo><mi href="./19.29#SS1.p1">β</mi></mrow></msub><mo></mo><msub><mi href="./19.29#SS1.p1">d</mi><mrow><mi href="./19.29#SS1.p1">α</mi><mo></mo><mi href="./19.29#SS1.p1">γ</mi></mrow></msub><mo></mo><mrow><mi href="./19.29#SS2.p1">I</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><msub><mi mathvariant="bold">e</mi><mi href="./19.29#SS1.p1">α</mi></msub></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mn>2</mn><mo></mo><msub><mi>b</mi><mi href="./19.29#SS1.p1">α</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mfrac><mrow><mi href="./19.29#SS2.p5">s</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow><mrow><msub><mi>a</mi><mi href="./19.29#SS1.p1">α</mi></msub><mo>+</mo><mrow><msub><mi>b</mi><mi href="./19.29#SS1.p1">α</mi></msub><mo></mo><mi>x</mi></mrow></mrow></mfrac><mo>-</mo><mfrac><mrow><mi href="./19.29#SS2.p5">s</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow><mrow><msub><mi>a</mi><mi href="./19.29#SS1.p1">α</mi></msub><mo>+</mo><mrow><msub><mi>b</mi><mi href="./19.29#SS1.p1">α</mi></msub><mo></mo><mi>y</mi></mrow></mrow></mfrac></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E16.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.29#SS2.p1" title="§19.29(ii) Reduction to Basic Integrals ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m76.png" altimg-height="23px" altimg-valign="-7px" altimg-width="49px" alttext="I(\mathbf{m})" display="inline"><mrow><mi href="./19.29#SS2.p1">I</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi mathvariant="bold">m</mi><mo stretchy="false">)</mo></mrow></mrow></math>: integral</a>,
<a href="./19.29#SS2.p5" title="§19.29(ii) Reduction to Basic Integrals ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m159.png" altimg-height="23px" altimg-valign="-7px" altimg-width="36px" alttext="s(t)" display="inline"><mrow><mi href="./19.29#SS2.p5">s</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math>: product</a>,
<a href="./19.29#SS1.p1" title="§19.29(i) Reduction Theorems ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m140.png" altimg-height="23px" altimg-valign="-8px" altimg-width="36px" alttext="d_{\alpha\beta}" display="inline"><msub><mi href="./19.29#SS1.p1">d</mi><mrow><mi href="./19.29#SS1.p1">α</mi><mo></mo><mi href="./19.29#SS1.p1">β</mi></mrow></msub></math></a>,
<a href="./19.29#SS1.p1" title="§19.29(i) Reduction Theorems ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m109.png" altimg-height="13px" altimg-valign="-2px" altimg-width="17px" alttext="\alpha" display="inline"><mi href="./19.29#SS1.p1">α</mi></math>: parameter</a>,
<a href="./19.29#SS1.p1" title="§19.29(i) Reduction Theorems ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m111.png" altimg-height="21px" altimg-valign="-6px" altimg-width="17px" alttext="\beta" display="inline"><mi href="./19.29#SS1.p1">β</mi></math>: parameter</a> and
<a href="./19.29#SS1.p1" title="§19.29(i) Reduction Theorems ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m114.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\gamma" display="inline"><mi href="./19.29#SS1.p1">γ</mi></math>: parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m108.png" altimg-height="21px" altimg-valign="-6px" altimg-width="59px" alttext="\alpha,\beta,\gamma" display="inline"><mrow><mi href="./19.29#SS1.p1">α</mi><mo>,</mo><mi href="./19.29#SS1.p1">β</mi><mo>,</mo><mi href="./19.29#SS1.p1">γ</mi></mrow></math> is any permutation of the numbers <math class="ltx_Math" altimg="m59.png" altimg-height="20px" altimg-valign="-6px" altimg-width="52px" alttext="1,2,3" display="inline"><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn></mrow></math>, and
</p>
<table id="E17" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.29.17</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E17.png" altimg-height="67px" altimg-valign="-27px" altimg-width="197px" alttext="s(t)=\prod_{\alpha=1}^{3}\sqrt{a_{\alpha}+b_{\alpha}t}." display="block"><mrow><mrow><mrow><mi href="./19.29#SS2.p5">s</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∏</mo><mrow><mi href="./19.29#SS1.p1">α</mi><mo>=</mo><mn>1</mn></mrow><mn>3</mn></munderover><msqrt><mrow><msub><mi>a</mi><mi href="./19.29#SS1.p1">α</mi></msub><mo>+</mo><mrow><msub><mi>b</mi><mi href="./19.29#SS1.p1">α</mi></msub><mo></mo><mi>t</mi></mrow></mrow></msqrt></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E17.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.29#SS2.p5" title="§19.29(ii) Reduction to Basic Integrals ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m159.png" altimg-height="23px" altimg-valign="-7px" altimg-width="36px" alttext="s(t)" display="inline"><mrow><mi href="./19.29#SS2.p5">s</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math>: product</a> and
<a href="./19.29#SS1.p1" title="§19.29(i) Reduction Theorems ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m109.png" altimg-height="13px" altimg-valign="-2px" altimg-width="17px" alttext="\alpha" display="inline"><mi href="./19.29#SS1.p1">α</mi></math>: parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">(This shows why <math class="ltx_Math" altimg="m72.png" altimg-height="23px" altimg-valign="-7px" altimg-width="52px" alttext="I(\mathbf{e}_{\alpha})" display="inline"><mrow><mi href="./19.29#SS2.p1">I</mi><mo></mo><mrow><mo stretchy="false">(</mo><msub><mi mathvariant="bold">e</mi><mi href="./19.29#SS1.p1">α</mi></msub><mo stretchy="false">)</mo></mrow></mrow></math> is not needed as a basic integral in the
cubic case.) In the quartic case this recurrence relation has an extra term in
<math class="ltx_Math" altimg="m69.png" altimg-height="23px" altimg-valign="-7px" altimg-width="62px" alttext="I(2\mathbf{e}_{\alpha})" display="inline"><mrow><mi href="./19.29#SS2.p1">I</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mn>2</mn><mo></mo><msub><mi mathvariant="bold">e</mi><mi href="./19.29#SS1.p1">α</mi></msub></mrow><mo stretchy="false">)</mo></mrow></mrow></math>, and hence <math class="ltx_Math" altimg="m72.png" altimg-height="23px" altimg-valign="-7px" altimg-width="52px" alttext="I(\mathbf{e}_{\alpha})" display="inline"><mrow><mi href="./19.29#SS2.p1">I</mi><mo></mo><mrow><mo stretchy="false">(</mo><msub><mi mathvariant="bold">e</mi><mi href="./19.29#SS1.p1">α</mi></msub><mo stretchy="false">)</mo></mrow></mrow></math>,
<math class="ltx_Math" altimg="m64.png" altimg-height="19px" altimg-valign="-5px" altimg-width="90px" alttext="1\leq\alpha\leq 4" display="inline"><mrow><mn>1</mn><mo>≤</mo><mi href="./19.29#SS1.p1">α</mi><mo>≤</mo><mn>4</mn></mrow></math>, is a basic integral. It can be expressed in terms of
symmetric integrals by setting <math class="ltx_Math" altimg="m130.png" altimg-height="20px" altimg-valign="-5px" altimg-width="60px" alttext="a_{5}=1" display="inline"><mrow><msub><mi>a</mi><mn>5</mn></msub><mo>=</mo><mn>1</mn></mrow></math> and <math class="ltx_Math" altimg="m138.png" altimg-height="21px" altimg-valign="-5px" altimg-width="58px" alttext="b_{5}=0" display="inline"><mrow><msub><mi>b</mi><mn>5</mn></msub><mo>=</mo><mn>0</mn></mrow></math> in ().</p>
</div>
<div id="SS2.p6" class="ltx_para">
<p class="ltx_p">The other recurrence relation is</p>
<table id="E18" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.29.18</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E18.png" altimg-height="65px" altimg-valign="-27px" altimg-width="290px" alttext="b_{j}^{q}I(q\mathbf{e}_{l})=\sum_{r=0}^{q}\genfrac{(}{)}{0.0pt}{}{q}{r}b_{l}^{%
r}d_{lj}^{q-r}I(r\mathbf{e}_{j})," display="block"><mrow><mrow><mrow><msubsup><mi>b</mi><mi>j</mi><mi>q</mi></msubsup><mo></mo><mrow><mi href="./19.29#SS2.p1">I</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>q</mi><mo></mo><msub><mi mathvariant="bold">e</mi><mi href="./19.1#p1.t1.r1">l</mi></msub></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>=</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi>r</mi><mo>=</mo><mn>0</mn></mrow><mi>q</mi></munderover><mrow><mrow><mo href="./1.2#i">(</mo><mfrac linethickness="0.0pt"><mi>q</mi><mi>r</mi></mfrac><mo href="./1.2#i">)</mo></mrow><mo></mo><msubsup><mi>b</mi><mi href="./19.1#p1.t1.r1">l</mi><mi>r</mi></msubsup><mo></mo><msubsup><mi href="./19.29#SS1.p1">d</mi><mrow><mi href="./19.1#p1.t1.r1">l</mi><mo></mo><mi>j</mi></mrow><mrow><mi>q</mi><mo>-</mo><mi>r</mi></mrow></msubsup><mo></mo><mrow><mi href="./19.29#SS2.p1">I</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>r</mi><mo></mo><msub><mi mathvariant="bold">e</mi><mi>j</mi></msub></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m150.png" altimg-height="21px" altimg-valign="-6px" altimg-width="139px" alttext="j,l=1,2,\dots,n" display="inline"><mrow><mrow><mrow><mi>j</mi><mo>,</mo><mi href="./19.1#p1.t1.r1">l</mi></mrow><mo>=</mo><mn>1</mn></mrow><mo>,</mo><mrow><mn>2</mn><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><mi href="./19.1#p1.t1.r1">n</mi></mrow></mrow></math>;</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E18.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.2#i" title="§1.2(i) Binomial Coefficients ‣ §1.2 Elementary Algebra ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m115.png" altimg-height="27px" altimg-valign="-9px" altimg-width="37px" alttext="\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}" display="inline"><mrow><mo href="./1.2#i">(</mo><mfrac linethickness="0.0pt"><mi class="ltx_nvar" href="./1.1#p2.t1.r5">m</mi><mi class="ltx_nvar" href="./1.1#p2.t1.r5">n</mi></mfrac><mo href="./1.2#i">)</mo></mrow></math>: binomial coefficient</a>,
<a href="./19.1#p1.t1.r1" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m154.png" altimg-height="18px" altimg-valign="-2px" altimg-width="11px" alttext="l" display="inline"><mi href="./19.1#p1.t1.r1">l</mi></math>: nonnegative integer</a>,
<a href="./19.1#p1.t1.r1" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m157.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./19.1#p1.t1.r1">n</mi></math>: nonnegative integer</a>,
<a href="./19.29#SS2.p1" title="§19.29(ii) Reduction to Basic Integrals ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m76.png" altimg-height="23px" altimg-valign="-7px" altimg-width="49px" alttext="I(\mathbf{m})" display="inline"><mrow><mi href="./19.29#SS2.p1">I</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi mathvariant="bold">m</mi><mo stretchy="false">)</mo></mrow></mrow></math>: integral</a> and
<a href="./19.29#SS1.p1" title="§19.29(i) Reduction Theorems ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m140.png" altimg-height="23px" altimg-valign="-8px" altimg-width="36px" alttext="d_{\alpha\beta}" display="inline"><msub><mi href="./19.29#SS1.p1">d</mi><mrow><mi href="./19.29#SS1.p1">α</mi><mo></mo><mi href="./19.29#SS1.p1">β</mi></mrow></msub></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">see <cite class="ltx_cite ltx_citemacro_citet">Carlson (.</p>
</div>
<div id="SS2.p7" class="ltx_para">
<p class="ltx_p">For an implementation by James FitzSimons of the method for reducing <math class="ltx_Math" altimg="m76.png" altimg-height="23px" altimg-valign="-7px" altimg-width="49px" alttext="I(\mathbf{m})" display="inline"><mrow><mi href="./19.29#SS2.p1">I</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi mathvariant="bold">m</mi><mo stretchy="false">)</mo></mrow></mrow></math>
to basic integrals and extensive tables of such reductions,
see <cite class="ltx_cite ltx_citemacro_citet">Carlson (</dd>
</dl>
</div>
</div>
<div id="SS3.p1" class="ltx_para">
<p class="ltx_p">The first formula replaces (). Define
<math class="ltx_Math" altimg="m86.png" altimg-height="25px" altimg-valign="-8px" altimg-width="154px" alttext="Q_{j}(t)=a_{j}+b_{j}t^{2}" display="inline"><mrow><mrow><msub><mi href="./19.29#SS3.p4">Q</mi><mi>j</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><msub><mi>a</mi><mi>j</mi></msub><mo>+</mo><mrow><msub><mi>b</mi><mi>j</mi></msub><mo></mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mrow></mrow></math>, <math class="ltx_Math" altimg="m151.png" altimg-height="20px" altimg-valign="-6px" altimg-width="69px" alttext="j=1,2" display="inline"><mrow><mi>j</mi><mo>=</mo><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></mrow></math>, and assume both <math class="ltx_Math" altimg="m79.png" altimg-height="21px" altimg-valign="-6px" altimg-width="20px" alttext="Q" display="inline"><mi href="./19.29#SS3.p4">Q</mi></math>’s are positive for
<math class="ltx_Math" altimg="m56.png" altimg-height="20px" altimg-valign="-6px" altimg-width="123px" alttext="0\leq y<t<x" display="inline"><mrow><mn>0</mn><mo>≤</mo><mi>y</mi><mo><</mo><mi>t</mi><mo><</mo><mi>x</mi></mrow></math>. Then</p>
<table id="E19" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.29.19</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E19.png" altimg-height="56px" altimg-valign="-25px" altimg-width="459px" alttext="\int_{y}^{x}\frac{\mathrm{d}t}{\sqrt{Q_{1}(t)Q_{2}(t)}}=R_{F}\left(U^{2}+a_{1}%
b_{2},U^{2}+a_{2}b_{1},U^{2}\right)," display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi>y</mi><mi>x</mi></msubsup><mfrac><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow><msqrt><mrow><mrow><msub><mi href="./19.29#SS3.p4">Q</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./19.29#SS3.p4">Q</mi><mn>2</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></msqrt></mfrac></mrow><mo>=</mo><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mrow><msup><mi href="./19.29#SS3.p1">U</mi><mn>2</mn></msup><mo>+</mo><mrow><msub><mi>a</mi><mn>1</mn></msub><mo></mo><msub><mi>b</mi><mn>2</mn></msub></mrow></mrow><mo>,</mo><mrow><msup><mi href="./19.29#SS3.p1">U</mi><mn>2</mn></msup><mo>+</mo><mrow><msub><mi>a</mi><mn>2</mn></msub><mo></mo><msub><mi>b</mi><mn>1</mn></msub></mrow></mrow><mo>,</mo><msup><mi href="./19.29#SS3.p1">U</mi><mn>2</mn></msup><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E19.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.16#E1" title="(19.16.1) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m93.png" altimg-height="23px" altimg-valign="-7px" altimg-width="101px" alttext="R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: symmetric elliptic integral of first kind</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m124.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m164.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m117.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./19.29#SS3.p1" title="§19.29(iii) Examples ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m99.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="U" display="inline"><mi href="./19.29#SS3.p1">U</mi></math></a> and
<a href="./19.29#SS3.p4" title="§19.29(iii) Examples ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="25px" altimg-valign="-7px" altimg-width="52px" alttext="Q(t^{2})" display="inline"><mrow><mi href="./19.29#SS3.p4">Q</mi><mo></mo><mrow><mo stretchy="false">(</mo><msup><mi>t</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow></mrow></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E20" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.29.20</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E20.png" altimg-height="58px" altimg-valign="-25px" altimg-width="596px" alttext="\int_{y}^{x}\frac{t^{2}\mathrm{d}t}{\sqrt{Q_{1}(t)Q_{2}(t)}}=\tfrac{1}{3}a_{1}%
a_{2}R_{D}\left(U^{2}+a_{1}b_{2},U^{2}+a_{2}b_{1},U^{2}\right)+(xy/U)," display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi>y</mi><mi>x</mi></msubsup><mfrac><mrow><msup><mi>t</mi><mn>2</mn></msup><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow><msqrt><mrow><mrow><msub><mi href="./19.29#SS3.p4">Q</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./19.29#SS3.p4">Q</mi><mn>2</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></msqrt></mfrac></mrow><mo>=</mo><mrow><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>3</mn></mfrac></mstyle><mo></mo><msub><mi>a</mi><mn>1</mn></msub><mo></mo><msub><mi>a</mi><mn>2</mn></msub><mo></mo><mrow><msub><mi href="./19.16#E5">R</mi><mi href="./19.16#E5">D</mi></msub><mo></mo><mrow><mo>(</mo><mrow><msup><mi href="./19.29#SS3.p1">U</mi><mn>2</mn></msup><mo>+</mo><mrow><msub><mi>a</mi><mn>1</mn></msub><mo></mo><msub><mi>b</mi><mn>2</mn></msub></mrow></mrow><mo>,</mo><mrow><msup><mi href="./19.29#SS3.p1">U</mi><mn>2</mn></msup><mo>+</mo><mrow><msub><mi>a</mi><mn>2</mn></msub><mo></mo><msub><mi>b</mi><mn>1</mn></msub></mrow></mrow><mo>,</mo><msup><mi href="./19.29#SS3.p1">U</mi><mn>2</mn></msup><mo>)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mi>x</mi><mo></mo><mi>y</mi></mrow><mo>/</mo><mi href="./19.29#SS3.p1">U</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E20.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.16#E5" title="(19.16.5) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m91.png" altimg-height="23px" altimg-valign="-7px" altimg-width="102px" alttext="R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E5">R</mi><mi href="./19.16#E5">D</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: elliptic integral symmetric in only two variables</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m124.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m164.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m117.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./19.29#SS3.p1" title="§19.29(iii) Examples ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m99.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="U" display="inline"><mi href="./19.29#SS3.p1">U</mi></math></a> and
<a href="./19.29#SS3.p4" title="§19.29(iii) Examples ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="25px" altimg-valign="-7px" altimg-width="52px" alttext="Q(t^{2})" display="inline"><mrow><mi href="./19.29#SS3.p4">Q</mi><mo></mo><mrow><mo stretchy="false">(</mo><msup><mi>t</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow></mrow></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and
</p>
<table id="E21" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.29.21</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E21.png" altimg-height="56px" altimg-valign="-25px" altimg-width="619px" alttext="\int_{y}^{x}\frac{\mathrm{d}t}{t^{2}\sqrt{Q_{1}(t)Q_{2}(t)}}=\tfrac{1}{3}b_{1}%
b_{2}R_{D}\left(U^{2}+a_{1}b_{2},U^{2}+a_{2}b_{1},U^{2}\right)+(xyU)^{-1}," display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi>y</mi><mi>x</mi></msubsup><mfrac><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow><mrow><msup><mi>t</mi><mn>2</mn></msup><mo></mo><msqrt><mrow><mrow><msub><mi href="./19.29#SS3.p4">Q</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./19.29#SS3.p4">Q</mi><mn>2</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></msqrt></mrow></mfrac></mrow><mo>=</mo><mrow><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>3</mn></mfrac></mstyle><mo></mo><msub><mi>b</mi><mn>1</mn></msub><mo></mo><msub><mi>b</mi><mn>2</mn></msub><mo></mo><mrow><msub><mi href="./19.16#E5">R</mi><mi href="./19.16#E5">D</mi></msub><mo></mo><mrow><mo>(</mo><mrow><msup><mi href="./19.29#SS3.p1">U</mi><mn>2</mn></msup><mo>+</mo><mrow><msub><mi>a</mi><mn>1</mn></msub><mo></mo><msub><mi>b</mi><mn>2</mn></msub></mrow></mrow><mo>,</mo><mrow><msup><mi href="./19.29#SS3.p1">U</mi><mn>2</mn></msup><mo>+</mo><mrow><msub><mi>a</mi><mn>2</mn></msub><mo></mo><msub><mi>b</mi><mn>1</mn></msub></mrow></mrow><mo>,</mo><msup><mi href="./19.29#SS3.p1">U</mi><mn>2</mn></msup><mo>)</mo></mrow></mrow></mrow><mo>+</mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi>x</mi><mo></mo><mi>y</mi><mo></mo><mi href="./19.29#SS3.p1">U</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mn>1</mn></mrow></msup></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E21.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.16#E5" title="(19.16.5) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m91.png" altimg-height="23px" altimg-valign="-7px" altimg-width="102px" alttext="R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E5">R</mi><mi href="./19.16#E5">D</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: elliptic integral symmetric in only two variables</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m124.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m164.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m117.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./19.29#SS3.p1" title="§19.29(iii) Examples ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m99.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="U" display="inline"><mi href="./19.29#SS3.p1">U</mi></math></a> and
<a href="./19.29#SS3.p4" title="§19.29(iii) Examples ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="25px" altimg-valign="-7px" altimg-width="52px" alttext="Q(t^{2})" display="inline"><mrow><mi href="./19.29#SS3.p4">Q</mi><mo></mo><mrow><mo stretchy="false">(</mo><msup><mi>t</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow></mrow></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where
</p>
<table id="E22" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.29.22</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E22.png" altimg-height="31px" altimg-valign="-7px" altimg-width="424px" alttext="(x^{2}-y^{2})U=x\sqrt{Q_{1}(y)Q_{2}(y)}+y\sqrt{Q_{1}(x)Q_{2}(x)}." display="block"><mrow><mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><msup><mi>y</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi href="./19.29#SS3.p1">U</mi></mrow><mo>=</mo><mrow><mrow><mi>x</mi><mo></mo><msqrt><mrow><mrow><msub><mi href="./19.29#SS3.p4">Q</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./19.29#SS3.p4">Q</mi><mn>2</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></msqrt></mrow><mo>+</mo><mrow><mi>y</mi><mo></mo><msqrt><mrow><mrow><msub><mi href="./19.29#SS3.p4">Q</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./19.29#SS3.p4">Q</mi><mn>2</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></msqrt></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E22.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.29#SS3.p1" title="§19.29(iii) Examples ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m99.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="U" display="inline"><mi href="./19.29#SS3.p1">U</mi></math></a> and
<a href="./19.29#SS3.p4" title="§19.29(iii) Examples ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="25px" altimg-valign="-7px" altimg-width="52px" alttext="Q(t^{2})" display="inline"><mrow><mi href="./19.29#SS3.p4">Q</mi><mo></mo><mrow><mo stretchy="false">(</mo><msup><mi>t</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow></mrow></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">If both square roots in () to
evaluate the integral as <math class="ltx_Math" altimg="m94.png" altimg-height="23px" altimg-valign="-7px" altimg-width="153px" alttext="R_{G}\left(a_{1}b_{2},a_{2}b_{1},0\right)" display="inline"><mrow><msub><mi href="./19.16#E3">R</mi><mi href="./19.16#E3">G</mi></msub><mo></mo><mrow><mo>(</mo><mrow><msub><mi>a</mi><mn>1</mn></msub><mo></mo><msub><mi>b</mi><mn>2</mn></msub></mrow><mo>,</mo><mrow><msub><mi>a</mi><mn>2</mn></msub><mo></mo><msub><mi>b</mi><mn>1</mn></msub></mrow><mo>,</mo><mn>0</mn><mo>)</mo></mrow></mrow></math> multiplied either by
<math class="ltx_Math" altimg="m54.png" altimg-height="23px" altimg-valign="-7px" altimg-width="90px" alttext="-2/(b_{1}b_{2})" display="inline"><mrow><mo>-</mo><mrow><mn>2</mn><mo>/</mo><mrow><mo stretchy="false">(</mo><mrow><msub><mi>b</mi><mn>1</mn></msub><mo></mo><msub><mi>b</mi><mn>2</mn></msub></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></math> or by <math class="ltx_Math" altimg="m53.png" altimg-height="23px" altimg-valign="-7px" altimg-width="94px" alttext="-2/(a_{1}a_{2})" display="inline"><mrow><mo>-</mo><mrow><mn>2</mn><mo>/</mo><mrow><mo stretchy="false">(</mo><mrow><msub><mi>a</mi><mn>1</mn></msub><mo></mo><msub><mi>a</mi><mn>2</mn></msub></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></math> in the cases of (), respectively. If <math class="ltx_Math" altimg="m162.png" altimg-height="13px" altimg-valign="-2px" altimg-width="62px" alttext="x=\infty" display="inline"><mrow><mi>x</mi><mo>=</mo><mi mathvariant="normal">∞</mi></mrow></math>, then <math class="ltx_Math" altimg="m99.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="U" display="inline"><mi href="./19.29#SS3.p1">U</mi></math> is found by
taking the limit. For example,</p>
<table id="E23" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.29.23</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E23.png" altimg-height="56px" altimg-valign="-25px" altimg-width="467px" alttext="\int_{y}^{\infty}\frac{\mathrm{d}t}{\sqrt{(t^{2}+a^{2})(t^{2}-b^{2})}}=R_{F}%
\left(y^{2}+a^{2},y^{2}-b^{2},y^{2}\right)." display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi>y</mi><mi mathvariant="normal">∞</mi></msubsup><mfrac><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow><msqrt><mrow><mrow><mo stretchy="false">(</mo><mrow><msup><mi>t</mi><mn>2</mn></msup><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><msup><mi>t</mi><mn>2</mn></msup><mo>-</mo><msup><mi>b</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow></mrow></msqrt></mfrac></mrow><mo>=</mo><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mrow><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup></mrow><mo>,</mo><mrow><msup><mi>y</mi><mn>2</mn></msup><mo>-</mo><msup><mi>b</mi><mn>2</mn></msup></mrow><mo>,</mo><msup><mi>y</mi><mn>2</mn></msup><mo>)</mo></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E23.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.16#E1" title="(19.16.1) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m93.png" altimg-height="23px" altimg-valign="-7px" altimg-width="101px" alttext="R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: symmetric elliptic integral of first kind</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m124.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m164.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m117.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS3.p2" class="ltx_para">
<p class="ltx_p">Next, for <math class="ltx_Math" altimg="m151.png" altimg-height="20px" altimg-valign="-6px" altimg-width="69px" alttext="j=1,2" display="inline"><mrow><mi>j</mi><mo>=</mo><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></mrow></math>, define <math class="ltx_Math" altimg="m87.png" altimg-height="25px" altimg-valign="-8px" altimg-width="206px" alttext="Q_{j}(t)=f_{j}+g_{j}t+h_{j}t^{2}" display="inline"><mrow><mrow><msub><mi href="./19.29#SS3.p4">Q</mi><mi>j</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><msub><mi href="./19.29#SS3.p4">f</mi><mi>j</mi></msub><mo>+</mo><mrow><msub><mi href="./19.29#SS3.p4">g</mi><mi>j</mi></msub><mo></mo><mi>t</mi></mrow><mo>+</mo><mrow><msub><mi href="./19.29#SS3.p4">h</mi><mi>j</mi></msub><mo></mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mrow></mrow></math>, and assume both
<math class="ltx_Math" altimg="m79.png" altimg-height="21px" altimg-valign="-6px" altimg-width="20px" alttext="Q" display="inline"><mi href="./19.29#SS3.p4">Q</mi></math>’s
are positive for <math class="ltx_Math" altimg="m167.png" altimg-height="20px" altimg-valign="-6px" altimg-width="86px" alttext="y<t<x" display="inline"><mrow><mi>y</mi><mo><</mo><mi>t</mi><mo><</mo><mi>x</mi></mrow></math>. If each has real zeros, then ()
may be simpler than</p>
<table id="E24" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.29.24</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E24.png" altimg-height="56px" altimg-valign="-25px" altimg-width="504px" alttext="\int_{y}^{x}\frac{\mathrm{d}t}{\sqrt{Q_{1}(t)Q_{2}(t)}}=4\!R_{F}\left(U,U+D_{1%
2}+V,U+D_{12}-V\right)," display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi>y</mi><mi>x</mi></msubsup><mfrac><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow><msqrt><mrow><mrow><msub><mi href="./19.29#SS3.p4">Q</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./19.29#SS3.p4">Q</mi><mn>2</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></msqrt></mfrac></mrow><mo>=</mo><mrow><mpadded width="-1.7pt"><mn>4</mn></mpadded><mo></mo><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./19.29#SS3.p1">U</mi><mo>,</mo><mrow><mi href="./19.29#SS3.p1">U</mi><mo>+</mo><msub><mi href="./19.29#SS3.p2">D</mi><mn>12</mn></msub><mo>+</mo><mi href="./19.29#SS3.p2">V</mi></mrow><mo>,</mo><mrow><mrow><mi href="./19.29#SS3.p1">U</mi><mo>+</mo><msub><mi href="./19.29#SS3.p2">D</mi><mn>12</mn></msub></mrow><mo>-</mo><mi href="./19.29#SS3.p2">V</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E24.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.16#E1" title="(19.16.1) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m93.png" altimg-height="23px" altimg-valign="-7px" altimg-width="101px" alttext="R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: symmetric elliptic integral of first kind</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m124.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m164.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m117.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./19.29#SS3.p1" title="§19.29(iii) Examples ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m99.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="U" display="inline"><mi href="./19.29#SS3.p1">U</mi></math></a>,
<a href="./19.29#SS3.p2" title="§19.29(iii) Examples ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m67.png" altimg-height="23px" altimg-valign="-8px" altimg-width="34px" alttext="D_{jl}" display="inline"><msub><mi href="./19.29#SS3.p2">D</mi><mrow><mi>j</mi><mo></mo><mi href="./19.1#p1.t1.r1">l</mi></mrow></msub></math></a>,
<a href="./19.29#SS3.p2" title="§19.29(iii) Examples ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m104.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="V" display="inline"><mi href="./19.29#SS3.p2">V</mi></math></a> and
<a href="./19.29#SS3.p4" title="§19.29(iii) Examples ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="25px" altimg-valign="-7px" altimg-width="52px" alttext="Q(t^{2})" display="inline"><mrow><mi href="./19.29#SS3.p4">Q</mi><mo></mo><mrow><mo stretchy="false">(</mo><msup><mi>t</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow></mrow></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where
</p>
<table id="E25" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex17" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="4" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">19.29.25</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m1.png" altimg-height="28px" altimg-valign="-7px" altimg-width="92px" alttext="\displaystyle(x-y)^{2}U" display="inline"><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mi>x</mi><mo>-</mo><mi>y</mi></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo></mo><mi href="./19.29#SS3.p1">U</mi></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m11.png" altimg-height="23px" altimg-valign="-6px" altimg-width="75px" alttext="\displaystyle=S_{1}S_{2}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msub><mi href="./19.29#SS3.p2">S</mi><mn>1</mn></msub><mo></mo><msub><mi href="./19.29#SS3.p2">S</mi><mn>2</mn></msub></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex18" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m34.png" altimg-height="25px" altimg-valign="-8px" altimg-width="27px" alttext="\displaystyle S_{j}" display="inline"><msub><mi href="./19.29#SS3.p2">S</mi><mi>j</mi></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m18.png" altimg-height="57px" altimg-valign="-21px" altimg-width="345px" alttext="\displaystyle=\left(\sqrt{Q_{j}(x)}+\sqrt{Q_{j}(y)}\right)^{2}-h_{j}(x-y)^{2}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msup><mrow><mo>(</mo><mrow><msqrt><mrow><msub><mi href="./19.29#SS3.p4">Q</mi><mi>j</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></msqrt><mo>+</mo><msqrt><mrow><msub><mi href="./19.29#SS3.p4">Q</mi><mi>j</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></msqrt></mrow><mo>)</mo></mrow><mn>2</mn></msup><mo>-</mo><mrow><msub><mi href="./19.29#SS3.p4">h</mi><mi>j</mi></msub><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi>x</mi><mo>-</mo><mi>y</mi></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex19" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m29.png" altimg-height="25px" altimg-valign="-8px" altimg-width="36px" alttext="\displaystyle D_{jl}" display="inline"><msub><mi href="./19.29#SS3.p2">D</mi><mrow><mi>j</mi><mo></mo><mi href="./19.1#p1.t1.r1">l</mi></mrow></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m10.png" altimg-height="25px" altimg-valign="-8px" altimg-width="206px" alttext="\displaystyle=2f_{j}h_{l}+2h_{j}f_{l}-g_{j}g_{l}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mrow><mn>2</mn><mo></mo><msub><mi href="./19.29#SS3.p4">f</mi><mi>j</mi></msub><mo></mo><msub><mi href="./19.29#SS3.p4">h</mi><mi href="./19.1#p1.t1.r1">l</mi></msub></mrow><mo>+</mo><mrow><mn>2</mn><mo></mo><msub><mi href="./19.29#SS3.p4">h</mi><mi>j</mi></msub><mo></mo><msub><mi href="./19.29#SS3.p4">f</mi><mi href="./19.1#p1.t1.r1">l</mi></msub></mrow></mrow><mo>-</mo><mrow><msub><mi href="./19.29#SS3.p4">g</mi><mi>j</mi></msub><mo></mo><msub><mi href="./19.29#SS3.p4">g</mi><mi href="./19.1#p1.t1.r1">l</mi></msub></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex20" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m38.png" altimg-height="19px" altimg-valign="-2px" altimg-width="22px" alttext="\displaystyle V" display="inline"><mi href="./19.29#SS3.p2">V</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m19.png" altimg-height="42px" altimg-valign="-13px" altimg-width="177px" alttext="\displaystyle=\sqrt{D_{12}^{2}-D_{11}D_{22}}." display="inline"><mrow><mrow><mi></mi><mo>=</mo><msqrt><mrow><msubsup><mi href="./19.29#SS3.p2">D</mi><mn>12</mn><mn>2</mn></msubsup><mo>-</mo><mrow><msub><mi href="./19.29#SS3.p2">D</mi><mn>11</mn></msub><mo></mo><msub><mi href="./19.29#SS3.p2">D</mi><mn>22</mn></msub></mrow></mrow></msqrt></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E25.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.1#p1.t1.r1" title="§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m154.png" altimg-height="18px" altimg-valign="-2px" altimg-width="11px" alttext="l" display="inline"><mi href="./19.1#p1.t1.r1">l</mi></math>: nonnegative integer</a>,
<a href="./19.29#SS3.p1" title="§19.29(iii) Examples ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m99.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="U" display="inline"><mi href="./19.29#SS3.p1">U</mi></math></a>,
<a href="./19.29#SS3.p2" title="§19.29(iii) Examples ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m67.png" altimg-height="23px" altimg-valign="-8px" altimg-width="34px" alttext="D_{jl}" display="inline"><msub><mi href="./19.29#SS3.p2">D</mi><mrow><mi>j</mi><mo></mo><mi href="./19.1#p1.t1.r1">l</mi></mrow></msub></math></a>,
<a href="./19.29#SS3.p2" title="§19.29(iii) Examples ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m97.png" altimg-height="23px" altimg-valign="-8px" altimg-width="25px" alttext="S_{j}" display="inline"><msub><mi href="./19.29#SS3.p2">S</mi><mi>j</mi></msub></math></a>,
<a href="./19.29#SS3.p2" title="§19.29(iii) Examples ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m104.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="V" display="inline"><mi href="./19.29#SS3.p2">V</mi></math></a>,
<a href="./19.29#SS3.p4" title="§19.29(iii) Examples ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="25px" altimg-valign="-7px" altimg-width="52px" alttext="Q(t^{2})" display="inline"><mrow><mi href="./19.29#SS3.p4">Q</mi><mo></mo><mrow><mo stretchy="false">(</mo><msup><mi>t</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow></mrow></math></a>,
<a href="./19.29#SS3.p4" title="§19.29(iii) Examples ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m142.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi href="./19.29#SS3.p4">f</mi></math>: coefficient</a>,
<a href="./19.29#SS3.p4" title="§19.29(iii) Examples ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m144.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="g" display="inline"><mi href="./19.29#SS3.p4">g</mi></math>: coefficient</a> and
<a href="./19.29#SS3.p4" title="§19.29(iii) Examples ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m148.png" altimg-height="18px" altimg-valign="-2px" altimg-width="16px" alttext="h" display="inline"><mi href="./19.29#SS3.p4">h</mi></math>: coefficient</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">(The variables of <math class="ltx_Math" altimg="m92.png" altimg-height="21px" altimg-valign="-5px" altimg-width="33px" alttext="R_{F}" display="inline"><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub></math> are real and nonnegative unless both <math class="ltx_Math" altimg="m79.png" altimg-height="21px" altimg-valign="-6px" altimg-width="20px" alttext="Q" display="inline"><mi href="./19.29#SS3.p4">Q</mi></math>’s have
real zeros and those of <math class="ltx_Math" altimg="m81.png" altimg-height="21px" altimg-valign="-6px" altimg-width="29px" alttext="Q_{1}" display="inline"><msub><mi href="./19.29#SS3.p4">Q</mi><mn>1</mn></msub></math> interlace those of <math class="ltx_Math" altimg="m83.png" altimg-height="21px" altimg-valign="-6px" altimg-width="29px" alttext="Q_{2}" display="inline"><msub><mi href="./19.29#SS3.p4">Q</mi><mn>2</mn></msub></math>.) If
<math class="ltx_Math" altimg="m80.png" altimg-height="23px" altimg-valign="-7px" altimg-width="246px" alttext="Q_{1}(t)=(a_{1}+b_{1}t)(a_{2}+b_{2}t)" display="inline"><mrow><mrow><msub><mi href="./19.29#SS3.p4">Q</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>+</mo><mrow><msub><mi>b</mi><mn>1</mn></msub><mo></mo><mi>t</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><msub><mi>a</mi><mn>2</mn></msub><mo>+</mo><mrow><msub><mi>b</mi><mn>2</mn></msub><mo></mo><mi>t</mi></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></math>, where both linear factors are positive
for <math class="ltx_Math" altimg="m167.png" altimg-height="20px" altimg-valign="-6px" altimg-width="86px" alttext="y<t<x" display="inline"><mrow><mi>y</mi><mo><</mo><mi>t</mi><mo><</mo><mi>x</mi></mrow></math>, and <math class="ltx_Math" altimg="m82.png" altimg-height="25px" altimg-valign="-7px" altimg-width="208px" alttext="Q_{2}(t)=f_{2}+g_{2}t+h_{2}t^{2}" display="inline"><mrow><mrow><msub><mi href="./19.29#SS3.p4">Q</mi><mn>2</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><msub><mi href="./19.29#SS3.p4">f</mi><mn>2</mn></msub><mo>+</mo><mrow><msub><mi href="./19.29#SS3.p4">g</mi><mn>2</mn></msub><mo></mo><mi>t</mi></mrow><mo>+</mo><mrow><msub><mi href="./19.29#SS3.p4">h</mi><mn>2</mn></msub><mo></mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mrow></mrow></math>, then
() is modified so that</p>
<table id="E26" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex21" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="5" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">19.29.26</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m31.png" altimg-height="22px" altimg-valign="-5px" altimg-width="27px" alttext="\displaystyle S_{1}" display="inline"><msub><mi href="./19.29#SS3.p2">S</mi><mn>1</mn></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m3.png" altimg-height="28px" altimg-valign="-7px" altimg-width="173px" alttext="\displaystyle=(X_{1}Y_{2}+Y_{1}X_{2})^{2}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mrow><mrow><msub><mi href="./19.29#SS1.p1">X</mi><mn>1</mn></msub><mo></mo><msub><mi href="./19.29#SS1.p1">Y</mi><mn>2</mn></msub></mrow><mo>+</mo><mrow><msub><mi href="./19.29#SS1.p1">Y</mi><mn>1</mn></msub><mo></mo><msub><mi href="./19.29#SS1.p1">X</mi><mn>2</mn></msub></mrow></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex22" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m40.png" altimg-height="25px" altimg-valign="-8px" altimg-width="31px" alttext="\displaystyle X_{j}" display="inline"><msub><mi href="./19.29#SS1.p1">X</mi><mi>j</mi></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m22.png" altimg-height="30px" altimg-valign="-8px" altimg-width="124px" alttext="\displaystyle=\sqrt{a_{j}+b_{j}x}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><msqrt><mrow><msub><mi>a</mi><mi>j</mi></msub><mo>+</mo><mrow><msub><mi>b</mi><mi>j</mi></msub><mo></mo><mi>x</mi></mrow></mrow></msqrt></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex23" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m42.png" altimg-height="25px" altimg-valign="-8px" altimg-width="26px" alttext="\displaystyle Y_{j}" display="inline"><msub><mi href="./19.29#SS1.p1">Y</mi><mi>j</mi></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m23.png" altimg-height="30px" altimg-valign="-8px" altimg-width="123px" alttext="\displaystyle=\sqrt{a_{j}+b_{j}y}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><msqrt><mrow><msub><mi>a</mi><mi>j</mi></msub><mo>+</mo><mrow><msub><mi>b</mi><mi>j</mi></msub><mo></mo><mi>y</mi></mrow></mrow></msqrt></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex24" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m28.png" altimg-height="22px" altimg-valign="-5px" altimg-width="39px" alttext="\displaystyle D_{12}" display="inline"><msub><mi href="./19.29#SS3.p2">D</mi><mn>12</mn></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m8.png" altimg-height="25px" altimg-valign="-7px" altimg-width="346px" alttext="\displaystyle=2a_{1}a_{2}h_{2}+2b_{1}b_{2}f_{2}-(a_{1}b_{2}+a_{2}b_{1})g_{2}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mrow><mn>2</mn><mo></mo><msub><mi>a</mi><mn>1</mn></msub><mo></mo><msub><mi>a</mi><mn>2</mn></msub><mo></mo><msub><mi href="./19.29#SS3.p4">h</mi><mn>2</mn></msub></mrow><mo>+</mo><mrow><mn>2</mn><mo></mo><msub><mi>b</mi><mn>1</mn></msub><mo></mo><msub><mi>b</mi><mn>2</mn></msub><mo></mo><msub><mi href="./19.29#SS3.p4">f</mi><mn>2</mn></msub></mrow></mrow><mo>-</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><msub><mi>a</mi><mn>1</mn></msub><mo></mo><msub><mi>b</mi><mn>2</mn></msub></mrow><mo>+</mo><mrow><msub><mi>a</mi><mn>2</mn></msub><mo></mo><msub><mi>b</mi><mn>1</mn></msub></mrow></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msub><mi href="./19.29#SS3.p4">g</mi><mn>2</mn></msub></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex25" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m27.png" altimg-height="22px" altimg-valign="-5px" altimg-width="39px" alttext="\displaystyle D_{11}" display="inline"><msub><mi href="./19.29#SS3.p2">D</mi><mn>11</mn></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m6.png" altimg-height="28px" altimg-valign="-7px" altimg-width="240px" alttext="\displaystyle=-(a_{1}b_{2}-a_{2}b_{1})^{2}=-d_{12}^{2}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo>-</mo><msup><mrow><mo stretchy="false">(</mo><mrow><mrow><msub><mi>a</mi><mn>1</mn></msub><mo></mo><msub><mi>b</mi><mn>2</mn></msub></mrow><mo>-</mo><mrow><msub><mi>a</mi><mn>2</mn></msub><mo></mo><msub><mi>b</mi><mn>1</mn></msub></mrow></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow><mo>=</mo><mrow><mo>-</mo><msubsup><mi href="./19.29#SS1.p1">d</mi><mn>12</mn><mn>2</mn></msubsup></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E26.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.29#SS1.p1" title="§19.29(i) Reduction Theorems ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m105.png" altimg-height="21px" altimg-valign="-5px" altimg-width="32px" alttext="X_{\alpha}" display="inline"><msub><mi href="./19.29#SS1.p1">X</mi><mi href="./19.29#SS1.p1">α</mi></msub></math></a>,
<a href="./19.29#SS1.p1" title="§19.29(i) Reduction Theorems ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m106.png" altimg-height="21px" altimg-valign="-5px" altimg-width="27px" alttext="Y_{\alpha}" display="inline"><msub><mi href="./19.29#SS1.p1">Y</mi><mi href="./19.29#SS1.p1">α</mi></msub></math></a>,
<a href="./19.29#SS3.p2" title="§19.29(iii) Examples ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m67.png" altimg-height="23px" altimg-valign="-8px" altimg-width="34px" alttext="D_{jl}" display="inline"><msub><mi href="./19.29#SS3.p2">D</mi><mrow><mi>j</mi><mo></mo><mi href="./19.1#p1.t1.r1">l</mi></mrow></msub></math></a>,
<a href="./19.29#SS3.p2" title="§19.29(iii) Examples ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m97.png" altimg-height="23px" altimg-valign="-8px" altimg-width="25px" alttext="S_{j}" display="inline"><msub><mi href="./19.29#SS3.p2">S</mi><mi>j</mi></msub></math></a>,
<a href="./19.29#SS3.p4" title="§19.29(iii) Examples ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m142.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi href="./19.29#SS3.p4">f</mi></math>: coefficient</a>,
<a href="./19.29#SS3.p4" title="§19.29(iii) Examples ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m144.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="g" display="inline"><mi href="./19.29#SS3.p4">g</mi></math>: coefficient</a>,
<a href="./19.29#SS3.p4" title="§19.29(iii) Examples ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m148.png" altimg-height="18px" altimg-valign="-2px" altimg-width="16px" alttext="h" display="inline"><mi href="./19.29#SS3.p4">h</mi></math>: coefficient</a> and
<a href="./19.29#SS1.p1" title="§19.29(i) Reduction Theorems ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m140.png" altimg-height="23px" altimg-valign="-8px" altimg-width="36px" alttext="d_{\alpha\beta}" display="inline"><msub><mi href="./19.29#SS1.p1">d</mi><mrow><mi href="./19.29#SS1.p1">α</mi><mo></mo><mi href="./19.29#SS1.p1">β</mi></mrow></msub></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">with other quantities remaining as in (). In the cubic case,
in which <math class="ltx_Math" altimg="m129.png" altimg-height="20px" altimg-valign="-5px" altimg-width="60px" alttext="a_{2}=1" display="inline"><mrow><msub><mi>a</mi><mn>2</mn></msub><mo>=</mo><mn>1</mn></mrow></math>, <math class="ltx_Math" altimg="m137.png" altimg-height="21px" altimg-valign="-5px" altimg-width="58px" alttext="b_{2}=0" display="inline"><mrow><msub><mi>b</mi><mn>2</mn></msub><mo>=</mo><mn>0</mn></mrow></math>, () reduces further to</p>
<table id="E27" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex26" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="3" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">19.29.27</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m31.png" altimg-height="22px" altimg-valign="-5px" altimg-width="27px" alttext="\displaystyle S_{1}" display="inline"><msub><mi href="./19.29#SS3.p2">S</mi><mn>1</mn></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m2.png" altimg-height="28px" altimg-valign="-7px" altimg-width="127px" alttext="\displaystyle=(X_{1}+Y_{1})^{2}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mrow><msub><mi href="./19.29#SS1.p1">X</mi><mn>1</mn></msub><mo>+</mo><msub><mi href="./19.29#SS1.p1">Y</mi><mn>1</mn></msub></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex27" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m28.png" altimg-height="22px" altimg-valign="-5px" altimg-width="39px" alttext="\displaystyle D_{12}" display="inline"><msub><mi href="./19.29#SS3.p2">D</mi><mn>12</mn></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m9.png" altimg-height="23px" altimg-valign="-6px" altimg-width="143px" alttext="\displaystyle=2a_{1}h_{2}-b_{1}g_{2}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mn>2</mn><mo></mo><msub><mi>a</mi><mn>1</mn></msub><mo></mo><msub><mi href="./19.29#SS3.p4">h</mi><mn>2</mn></msub></mrow><mo>-</mo><mrow><msub><mi>b</mi><mn>1</mn></msub><mo></mo><msub><mi href="./19.29#SS3.p4">g</mi><mn>2</mn></msub></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex28" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m27.png" altimg-height="22px" altimg-valign="-5px" altimg-width="39px" alttext="\displaystyle D_{11}" display="inline"><msub><mi href="./19.29#SS3.p2">D</mi><mn>11</mn></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m7.png" altimg-height="28px" altimg-valign="-7px" altimg-width="66px" alttext="\displaystyle=-b_{1}^{2}." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo>-</mo><msubsup><mi>b</mi><mn>1</mn><mn>2</mn></msubsup></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E27.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.29#SS1.p1" title="§19.29(i) Reduction Theorems ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m105.png" altimg-height="21px" altimg-valign="-5px" altimg-width="32px" alttext="X_{\alpha}" display="inline"><msub><mi href="./19.29#SS1.p1">X</mi><mi href="./19.29#SS1.p1">α</mi></msub></math></a>,
<a href="./19.29#SS1.p1" title="§19.29(i) Reduction Theorems ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m106.png" altimg-height="21px" altimg-valign="-5px" altimg-width="27px" alttext="Y_{\alpha}" display="inline"><msub><mi href="./19.29#SS1.p1">Y</mi><mi href="./19.29#SS1.p1">α</mi></msub></math></a>,
<a href="./19.29#SS3.p2" title="§19.29(iii) Examples ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m67.png" altimg-height="23px" altimg-valign="-8px" altimg-width="34px" alttext="D_{jl}" display="inline"><msub><mi href="./19.29#SS3.p2">D</mi><mrow><mi>j</mi><mo></mo><mi href="./19.1#p1.t1.r1">l</mi></mrow></msub></math></a>,
<a href="./19.29#SS3.p2" title="§19.29(iii) Examples ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m97.png" altimg-height="23px" altimg-valign="-8px" altimg-width="25px" alttext="S_{j}" display="inline"><msub><mi href="./19.29#SS3.p2">S</mi><mi>j</mi></msub></math></a>,
<a href="./19.29#SS3.p4" title="§19.29(iii) Examples ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m144.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="g" display="inline"><mi href="./19.29#SS3.p4">g</mi></math>: coefficient</a> and
<a href="./19.29#SS3.p4" title="§19.29(iii) Examples ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m148.png" altimg-height="18px" altimg-valign="-2px" altimg-width="16px" alttext="h" display="inline"><mi href="./19.29#SS3.p4">h</mi></math>: coefficient</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS3.p3" class="ltx_para">
<p class="ltx_p">For example, because <math class="ltx_Math" altimg="m161.png" altimg-height="25px" altimg-valign="-7px" altimg-width="266px" alttext="t^{3}-a^{3}=(t-a)(t^{2}+at+a^{2})" display="inline"><mrow><mrow><msup><mi>t</mi><mn>3</mn></msup><mo>-</mo><msup><mi>a</mi><mn>3</mn></msup></mrow><mo>=</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mi>t</mi><mo>-</mo><mi>a</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><msup><mi>t</mi><mn>2</mn></msup><mo>+</mo><mrow><mi>a</mi><mo></mo><mi>t</mi></mrow><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></math>, we find that when
<math class="ltx_Math" altimg="m55.png" altimg-height="20px" altimg-valign="-6px" altimg-width="126px" alttext="0\leq a\leq y<x" display="inline"><mrow><mn>0</mn><mo>≤</mo><mi>a</mi><mo>≤</mo><mi>y</mi><mo><</mo><mi>x</mi></mrow></math></p>
<table id="E28" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.29.28</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E28.png" altimg-height="55px" altimg-valign="-23px" altimg-width="514px" alttext="\int_{y}^{x}\frac{\mathrm{d}t}{\sqrt{t^{3}-a^{3}}}=4\!R_{F}\left(U,U-3a+2\sqrt%
{3}a,U-3a-2\sqrt{3}a\right)," display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi>y</mi><mi>x</mi></msubsup><mfrac><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow><msqrt><mrow><msup><mi>t</mi><mn>3</mn></msup><mo>-</mo><msup><mi>a</mi><mn>3</mn></msup></mrow></msqrt></mfrac></mrow><mo>=</mo><mrow><mpadded width="-1.7pt"><mn>4</mn></mpadded><mo></mo><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./19.29#SS3.p1">U</mi><mo>,</mo><mrow><mrow><mi href="./19.29#SS3.p1">U</mi><mo>-</mo><mrow><mn>3</mn><mo></mo><mi>a</mi></mrow></mrow><mo>+</mo><mrow><mn>2</mn><mo></mo><msqrt><mn>3</mn></msqrt><mo></mo><mi>a</mi></mrow></mrow><mo>,</mo><mrow><mi href="./19.29#SS3.p1">U</mi><mo>-</mo><mrow><mn>3</mn><mo></mo><mi>a</mi></mrow><mo>-</mo><mrow><mn>2</mn><mo></mo><msqrt><mn>3</mn></msqrt><mo></mo><mi>a</mi></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E28.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.16#E1" title="(19.16.1) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m93.png" altimg-height="23px" altimg-valign="-7px" altimg-width="101px" alttext="R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: symmetric elliptic integral of first kind</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m124.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m164.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m117.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a> and
<a href="./19.29#SS3.p1" title="§19.29(iii) Examples ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m99.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="U" display="inline"><mi href="./19.29#SS3.p1">U</mi></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where</p>
<table id="E29" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex29" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="3" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">19.29.29</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m1.png" altimg-height="28px" altimg-valign="-7px" altimg-width="92px" alttext="\displaystyle(x-y)^{2}U" display="inline"><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mi>x</mi><mo>-</mo><mi>y</mi></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo></mo><mi href="./19.29#SS3.p1">U</mi></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m5.png" altimg-height="30px" altimg-valign="-9px" altimg-width="395px" alttext="\displaystyle=(\sqrt{x-a}+\sqrt{y-a})^{2}\left((\xi+\eta)^{2}-(x-y)^{2}\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><msqrt><mrow><mi>x</mi><mo>-</mo><mi>a</mi></mrow></msqrt><mo>+</mo><msqrt><mrow><mi>y</mi><mo>-</mo><mi>a</mi></mrow></msqrt></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo></mo><mrow><mo>(</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mi>ξ</mi><mo>+</mo><mi>η</mi></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>-</mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi>x</mi><mo>-</mo><mi>y</mi></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex30" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m44.png" altimg-height="23px" altimg-valign="-6px" altimg-width="16px" alttext="\displaystyle\xi" display="inline"><mi>ξ</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m24.png" altimg-height="31px" altimg-valign="-6px" altimg-width="163px" alttext="\displaystyle=\sqrt{x^{2}+ax+a^{2}}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><msqrt><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mrow><mi>a</mi><mo></mo><mi>x</mi></mrow><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup></mrow></msqrt></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex31" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m43.png" altimg-height="18px" altimg-valign="-6px" altimg-width="17px" alttext="\displaystyle\eta" display="inline"><mi>η</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m25.png" altimg-height="30px" altimg-valign="-7px" altimg-width="161px" alttext="\displaystyle=\sqrt{y^{2}+ay+a^{2}}." display="inline"><mrow><mrow><mi></mi><mo>=</mo><msqrt><mrow><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mrow><mi>a</mi><mo></mo><mi>y</mi></mrow><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup></mrow></msqrt></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E29.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./19.29#SS3.p1" title="§19.29(iii) Examples ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m99.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="U" display="inline"><mi href="./19.29#SS3.p1">U</mi></math></a></dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS3.p4" class="ltx_para">
<p class="ltx_p">Lastly, define <math class="ltx_Math" altimg="m77.png" altimg-height="25px" altimg-valign="-7px" altimg-width="193px" alttext="Q(t^{2})=f+gt^{2}+ht^{4}" display="inline"><mrow><mrow><mi href="./19.29#SS3.p4">Q</mi><mo></mo><mrow><mo stretchy="false">(</mo><msup><mi>t</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mi href="./19.29#SS3.p4">f</mi><mo>+</mo><mrow><mi href="./19.29#SS3.p4">g</mi><mo></mo><msup><mi>t</mi><mn>2</mn></msup></mrow><mo>+</mo><mrow><mi href="./19.29#SS3.p4">h</mi><mo></mo><msup><mi>t</mi><mn>4</mn></msup></mrow></mrow></mrow></math> and assume <math class="ltx_Math" altimg="m78.png" altimg-height="25px" altimg-valign="-7px" altimg-width="52px" alttext="Q(t^{2})" display="inline"><mrow><mi href="./19.29#SS3.p4">Q</mi><mo></mo><mrow><mo stretchy="false">(</mo><msup><mi>t</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow></mrow></math> is positive and
monotonic for <math class="ltx_Math" altimg="m167.png" altimg-height="20px" altimg-valign="-6px" altimg-width="86px" alttext="y<t<x" display="inline"><mrow><mi>y</mi><mo><</mo><mi>t</mi><mo><</mo><mi>x</mi></mrow></math>. Then</p>
<table id="E30" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.29.30</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E30.png" altimg-height="56px" altimg-valign="-25px" altimg-width="496px" alttext="\int_{y}^{x}\frac{\mathrm{d}t}{\sqrt{Q(t^{2})}}=2\!R_{F}\left(U,U-g+2\sqrt{fh}%
,U-g-2\sqrt{fh}\right)," display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi>y</mi><mi>x</mi></msubsup><mfrac><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow><msqrt><mrow><mi href="./19.29#SS3.p4">Q</mi><mo></mo><mrow><mo stretchy="false">(</mo><msup><mi>t</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow></mrow></msqrt></mfrac></mrow><mo>=</mo><mrow><mpadded width="-1.7pt"><mn>2</mn></mpadded><mo></mo><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./19.29#SS3.p1">U</mi><mo>,</mo><mrow><mrow><mi href="./19.29#SS3.p1">U</mi><mo>-</mo><mi href="./19.29#SS3.p4">g</mi></mrow><mo>+</mo><mrow><mn>2</mn><mo></mo><msqrt><mrow><mi href="./19.29#SS3.p4">f</mi><mo></mo><mi href="./19.29#SS3.p4">h</mi></mrow></msqrt></mrow></mrow><mo>,</mo><mrow><mi href="./19.29#SS3.p1">U</mi><mo>-</mo><mi href="./19.29#SS3.p4">g</mi><mo>-</mo><mrow><mn>2</mn><mo></mo><msqrt><mrow><mi href="./19.29#SS3.p4">f</mi><mo></mo><mi href="./19.29#SS3.p4">h</mi></mrow></msqrt></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E30.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.16#E1" title="(19.16.1) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m93.png" altimg-height="23px" altimg-valign="-7px" altimg-width="101px" alttext="R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: symmetric elliptic integral of first kind</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m124.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m164.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m117.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./19.29#SS3.p1" title="§19.29(iii) Examples ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m99.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="U" display="inline"><mi href="./19.29#SS3.p1">U</mi></math></a>,
<a href="./19.29#SS3.p4" title="§19.29(iii) Examples ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="25px" altimg-valign="-7px" altimg-width="52px" alttext="Q(t^{2})" display="inline"><mrow><mi href="./19.29#SS3.p4">Q</mi><mo></mo><mrow><mo stretchy="false">(</mo><msup><mi>t</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow></mrow></math></a>,
<a href="./19.29#SS3.p4" title="§19.29(iii) Examples ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m142.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi href="./19.29#SS3.p4">f</mi></math>: coefficient</a>,
<a href="./19.29#SS3.p4" title="§19.29(iii) Examples ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m144.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="g" display="inline"><mi href="./19.29#SS3.p4">g</mi></math>: coefficient</a> and
<a href="./19.29#SS3.p4" title="§19.29(iii) Examples ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m148.png" altimg-height="18px" altimg-valign="-2px" altimg-width="16px" alttext="h" display="inline"><mi href="./19.29#SS3.p4">h</mi></math>: coefficient</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where</p>
<table id="E31" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.29.31</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E31.png" altimg-height="45px" altimg-valign="-15px" altimg-width="444px" alttext="(x-y)^{2}U=\left(\sqrt{Q(x^{2})}+\sqrt{Q(y^{2})}\right)^{2}-h(x^{2}-y^{2})^{2}." display="block"><mrow><mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mi>x</mi><mo>-</mo><mi>y</mi></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo></mo><mi href="./19.29#SS3.p1">U</mi></mrow><mo>=</mo><mrow><msup><mrow><mo>(</mo><mrow><msqrt><mrow><mi href="./19.29#SS3.p4">Q</mi><mo></mo><mrow><mo stretchy="false">(</mo><msup><mi>x</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow></mrow></msqrt><mo>+</mo><msqrt><mrow><mi href="./19.29#SS3.p4">Q</mi><mo></mo><mrow><mo stretchy="false">(</mo><msup><mi>y</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow></mrow></msqrt></mrow><mo>)</mo></mrow><mn>2</mn></msup><mo>-</mo><mrow><mi href="./19.29#SS3.p4">h</mi><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><msup><mi>y</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E31.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.29#SS3.p1" title="§19.29(iii) Examples ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m99.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="U" display="inline"><mi href="./19.29#SS3.p1">U</mi></math></a>,
<a href="./19.29#SS3.p4" title="§19.29(iii) Examples ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="25px" altimg-valign="-7px" altimg-width="52px" alttext="Q(t^{2})" display="inline"><mrow><mi href="./19.29#SS3.p4">Q</mi><mo></mo><mrow><mo stretchy="false">(</mo><msup><mi>t</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow></mrow></math></a> and
<a href="./19.29#SS3.p4" title="§19.29(iii) Examples ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m148.png" altimg-height="18px" altimg-valign="-2px" altimg-width="16px" alttext="h" display="inline"><mi href="./19.29#SS3.p4">h</mi></math>: coefficient</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">For example, if <math class="ltx_Math" altimg="m57.png" altimg-height="20px" altimg-valign="-6px" altimg-width="89px" alttext="0\leq y\leq x" display="inline"><mrow><mn>0</mn><mo>≤</mo><mi>y</mi><mo>≤</mo><mi>x</mi></mrow></math> and <math class="ltx_Math" altimg="m128.png" altimg-height="22px" altimg-valign="-5px" altimg-width="60px" alttext="a^{4}\geq 0" display="inline"><mrow><msup><mi>a</mi><mn>4</mn></msup><mo>≥</mo><mn>0</mn></mrow></math>, then</p>
<table id="E32" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.29.32</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E32.png" altimg-height="55px" altimg-valign="-23px" altimg-width="383px" alttext="\int_{y}^{x}\frac{\mathrm{d}t}{\sqrt{t^{4}+a^{4}}}=2\!R_{F}\left(U,U+2a^{2},U-%
2a^{2}\right)," display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi>y</mi><mi>x</mi></msubsup><mfrac><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow><msqrt><mrow><msup><mi>t</mi><mn>4</mn></msup><mo>+</mo><msup><mi>a</mi><mn>4</mn></msup></mrow></msqrt></mfrac></mrow><mo>=</mo><mrow><mpadded width="-1.7pt"><mn>2</mn></mpadded><mo></mo><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./19.29#SS3.p1">U</mi><mo>,</mo><mrow><mi href="./19.29#SS3.p1">U</mi><mo>+</mo><mrow><mn>2</mn><mo></mo><msup><mi>a</mi><mn>2</mn></msup></mrow></mrow><mo>,</mo><mrow><mi href="./19.29#SS3.p1">U</mi><mo>-</mo><mrow><mn>2</mn><mo></mo><msup><mi>a</mi><mn>2</mn></msup></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E32.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.16#E1" title="(19.16.1) ‣ §19.16(i) Symmetric Integrals ‣ §19.16 Definitions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m93.png" altimg-height="23px" altimg-valign="-7px" altimg-width="101px" alttext="R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)" display="inline"><mrow><msub><mi href="./19.16#E1">R</mi><mi href="./19.16#E1">F</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>,</mo><mi class="ltx_nvar">y</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: symmetric elliptic integral of first kind</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m124.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m164.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m117.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a> and
<a href="./19.29#SS3.p1" title="§19.29(iii) Examples ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m99.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="U" display="inline"><mi href="./19.29#SS3.p1">U</mi></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where</p>
<table id="E33" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">19.29.33</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E33.png" altimg-height="45px" altimg-valign="-15px" altimg-width="458px" alttext="(x-y)^{2}U=\left(\sqrt{x^{4}+a^{4}}+\sqrt{y^{4}+a^{4}}\right)^{2}-(x^{2}-y^{2}%
)^{2}." display="block"><mrow><mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mi>x</mi><mo>-</mo><mi>y</mi></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo></mo><mi href="./19.29#SS3.p1">U</mi></mrow><mo>=</mo><mrow><msup><mrow><mo>(</mo><mrow><msqrt><mrow><msup><mi>x</mi><mn>4</mn></msup><mo>+</mo><msup><mi>a</mi><mn>4</mn></msup></mrow></msqrt><mo>+</mo><msqrt><mrow><msup><mi>y</mi><mn>4</mn></msup><mo>+</mo><msup><mi>a</mi><mn>4</mn></msup></mrow></msqrt></mrow><mo>)</mo></mrow><mn>2</mn></msup><mo>-</mo><msup><mrow><mo stretchy="false">(</mo><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><msup><mi>y</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E33.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./19.29#SS3.p1" title="§19.29(iii) Examples ‣ §19.29 Reduction of General Elliptic Integrals ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m99.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="U" display="inline"><mi href="./19.29#SS3.p1">U</mi></math></a></dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
</section>
</div>
<div class="ltx_page_footer">
<span></div>
</div>
</body></text>
</html>
</page>
<page>
<!DOCTYPE html><html>
<head>
<title>DLMF: 5.4 Special Values and Extrema</title>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
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<text><body class="color_default textfont_default titlefont_default fontsize_default navbar_default">
<div class="ltx_page_navbar">
<div class="ltx_page_navlogo"></dd>
</dl>
</div>
</div>
<div id="SS1.p1" class="ltx_para">
<table id="E1" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex1" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">5.4.1</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m14.png" altimg-height="25px" altimg-valign="-7px" altimg-width="47px" alttext="\displaystyle\Gamma\left(1\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m4.png" altimg-height="22px" altimg-valign="-6px" altimg-width="43px" alttext="\displaystyle=1," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mn>1</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex2" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m24.png" altimg-height="19px" altimg-valign="-2px" altimg-width="24px" alttext="\displaystyle n!" display="inline"><mrow><mi href="./5.1#p2.t1.r1">n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m9.png" altimg-height="25px" altimg-valign="-7px" altimg-width="113px" alttext="\displaystyle=\Gamma\left(n+1\right)." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./5.1#p2.t1.r1">n</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m26.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m71.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a> and
<a href="./5.1#p2.t1.r1" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m75.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./5.1#p2.t1.r1">n</mi></math>: nonnegative integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E2" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">5.4.2</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E2.png" altimg-height="65px" altimg-valign="-27px" altimg-width="352px" alttext="n!!=\begin{cases}2^{\frac{1}{2}n}\Gamma\left(\frac{1}{2}n+1\right),&n\text{ %
even},\\
\pi^{-\frac{1}{2}}2^{\frac{1}{2}n+\frac{1}{2}}\Gamma\left(\frac{1}{2}n+1\right%
),&n\text{ odd}.\end{cases}" display="block"><mrow><mrow><mi href="./5.1#p2.t1.r1">n</mi><mo href="./front/introduction#Sx4.p1.t1.r15" lspace="0pt" rspace="3.5pt">!!</mo></mrow><mo>=</mo><mrow><mo>{</mo><mtable columnspacing="5pt" displaystyle="true" rowspacing="0pt"><mtr><mtd columnalign="left"><mrow><mrow><msup><mn>2</mn><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./5.1#p2.t1.r1">n</mi></mrow></msup><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./5.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mrow><mi href="./5.1#p2.t1.r1">n</mi><mo></mo><mtext> even</mtext></mrow><mo>,</mo></mrow></mtd></mtr><mtr><mtd columnalign="left"><mrow><mrow><msup><mi href="./3.12#E1">π</mi><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup><mo></mo><msup><mn>2</mn><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./5.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./5.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mrow><mi href="./5.1#p2.t1.r1">n</mi><mo></mo><mtext> odd</mtext></mrow><mo>.</mo></mrow></mtd></mtr></mtable></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./front/introduction#Sx4.p1.t1.r15" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m25.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="!!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r15" lspace="0pt" rspace="3.5pt">!!</mo></math>: double factorial (as in <math class="ltx_Math" altimg="m70.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="n!!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r15" lspace="0pt" rspace="3.5pt">!!</mo></mrow></math>)</a> and
<a href="./5.1#p2.t1.r1" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m75.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./5.1#p2.t1.r1">n</mi></math>: nonnegative integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">(The second line of Formula () also applies when <math class="ltx_Math" altimg="m72.png" altimg-height="18px" altimg-valign="-4px" altimg-width="68px" alttext="n=-1" display="inline"><mrow><mi href="./5.1#p2.t1.r1">n</mi><mo>=</mo><mrow><mo>-</mo><mn>1</mn></mrow></mrow></math>.)
</p>
<table id="E3" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">5.4.3</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E3.png" altimg-height="60px" altimg-valign="-21px" altimg-width="248px" alttext="|\Gamma\left(iy\right)|=\left(\frac{\pi}{y\sinh\left(\pi y\right)}\right)^{1/2}," display="block"><mrow><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./5.1#p2.t1.r3">y</mi></mrow><mo>)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow><mo>=</mo><msup><mrow><mo>(</mo><mfrac><mi href="./3.12#E1">π</mi><mrow><mi href="./5.1#p2.t1.r3">y</mi><mo></mo><mrow><mi href="./4.28#E1">sinh</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi href="./5.1#p2.t1.r3">y</mi></mrow><mo>)</mo></mrow></mrow></mrow></mfrac><mo>)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.28#E1" title="(4.28.1) ‣ §4.28 Definitions and Periodicity ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m67.png" altimg-height="18px" altimg-valign="-2px" altimg-width="53px" alttext="\sinh\NVar{z}" display="inline"><mrow><mi href="./4.28#E1">sinh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: hyperbolic sine function</a> and
<a href="./5.1#p2.t1.r3" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./5.1#p2.t1.r3">y</mi></math>: real variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">6.1.29</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E4" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">5.4.4</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E4.png" altimg-height="46px" altimg-valign="-21px" altimg-width="436px" alttext="\Gamma\left(\tfrac{1}{2}+\mathrm{i}y\right)\Gamma\left(\tfrac{1}{2}-\mathrm{i}%
y\right)=\left|\Gamma\left(\tfrac{1}{2}+\mathrm{i}y\right)\right|^{2}=\frac{%
\pi}{\cosh\left(\pi y\right)}," display="block"><mrow><mrow><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./5.1#p2.t1.r3">y</mi></mrow></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo>-</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./5.1#p2.t1.r3">y</mi></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>=</mo><msup><mrow><mo>|</mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./5.1#p2.t1.r3">y</mi></mrow></mrow><mo>)</mo></mrow></mrow><mo>|</mo></mrow><mn>2</mn></msup><mo>=</mo><mfrac><mi href="./3.12#E1">π</mi><mrow><mi href="./4.28#E2">cosh</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi href="./5.1#p2.t1.r3">y</mi></mrow><mo>)</mo></mrow></mrow></mfrac></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E4.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.28#E2" title="(4.28.2) ‣ §4.28 Definitions and Periodicity ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m57.png" altimg-height="18px" altimg-valign="-2px" altimg-width="56px" alttext="\cosh\NVar{z}" display="inline"><mrow><mi href="./4.28#E2">cosh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: hyperbolic cosine function</a> and
<a href="./5.1#p2.t1.r3" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./5.1#p2.t1.r3">y</mi></math>: real variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">6.1.30</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E5" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">5.4.5</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E5.png" altimg-height="56px" altimg-valign="-21px" altimg-width="410px" alttext="\Gamma\left(\tfrac{1}{4}+\mathrm{i}y\right)\Gamma\left(\tfrac{3}{4}-\mathrm{i}%
y\right)=\frac{\pi\sqrt{2}}{\cosh\left(\pi y\right)+\mathrm{i}\sinh\left(\pi y%
\right)}." display="block"><mrow><mrow><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>4</mn></mfrac></mstyle><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./5.1#p2.t1.r3">y</mi></mrow></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mstyle displaystyle="false"><mfrac><mn>3</mn><mn>4</mn></mfrac></mstyle><mo>-</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./5.1#p2.t1.r3">y</mi></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mfrac><mrow><mi href="./3.12#E1">π</mi><mo></mo><msqrt><mn>2</mn></msqrt></mrow><mrow><mrow><mi href="./4.28#E2">cosh</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi href="./5.1#p2.t1.r3">y</mi></mrow><mo>)</mo></mrow></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mrow><mi href="./4.28#E1">sinh</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi href="./5.1#p2.t1.r3">y</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow></mfrac></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E5.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.28#E2" title="(4.28.2) ‣ §4.28 Definitions and Periodicity ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m57.png" altimg-height="18px" altimg-valign="-2px" altimg-width="56px" alttext="\cosh\NVar{z}" display="inline"><mrow><mi href="./4.28#E2">cosh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: hyperbolic cosine function</a>,
<a href="./4.28#E1" title="(4.28.1) ‣ §4.28 Definitions and Periodicity ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m67.png" altimg-height="18px" altimg-valign="-2px" altimg-width="53px" alttext="\sinh\NVar{z}" display="inline"><mrow><mi href="./4.28#E1">sinh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: hyperbolic sine function</a> and
<a href="./5.1#p2.t1.r3" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./5.1#p2.t1.r3">y</mi></math>: real variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">6.1.32</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="EGx1" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E6">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">5.4.6</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m15.png" altimg-height="29px" altimg-valign="-9px" altimg-width="53px" alttext="\displaystyle\Gamma\left(\tfrac{1}{2}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m10.png" altimg-height="56px" altimg-valign="-6px" altimg-width="305px" alttext="\displaystyle=\pi^{1/2}\\
=1.77245\;38509\;05516\;02729\;\dots," display="inline"><mrow><mrow><mi></mi><mo>=</mo><msup><mi href="./3.12#E1">π</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>=</mo><mrow><mpadded width="+2.8pt"><mn>1.77245 38509 05516 02729</mn></mpadded><mo></mo><mi mathvariant="normal">…</mi></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E6.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a> and
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">6.1.8</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E7">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">5.4.7</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m16.png" altimg-height="29px" altimg-valign="-9px" altimg-width="53px" alttext="\displaystyle\Gamma\left(\tfrac{1}{3}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m7.png" altimg-height="22px" altimg-valign="-6px" altimg-width="299px" alttext="\displaystyle=2.67893\;85347\;07747\;63365\;\dots," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mpadded width="+2.8pt"><mn>2.67893 85347 07747 63365</mn></mpadded><mo></mo><mi mathvariant="normal">…</mi></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E7.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a></dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">6.1.11</span> (where the value is computed to 10D.)</span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E8">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">5.4.8</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m18.png" altimg-height="29px" altimg-valign="-9px" altimg-width="53px" alttext="\displaystyle\Gamma\left(\tfrac{2}{3}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m6.png" altimg-height="22px" altimg-valign="-6px" altimg-width="299px" alttext="\displaystyle=1.35411\;79394\;26400\;41694\;\dots," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mpadded width="+2.8pt"><mn>1.35411 79394 26400 41694</mn></mpadded><mo></mo><mi mathvariant="normal">…</mi></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E8.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a></dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">6.1.13</span> (where the value is computed to 10D.)</span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E9">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">5.4.9</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m17.png" altimg-height="29px" altimg-valign="-9px" altimg-width="53px" alttext="\displaystyle\Gamma\left(\tfrac{1}{4}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m8.png" altimg-height="22px" altimg-valign="-6px" altimg-width="299px" alttext="\displaystyle=3.62560\;99082\;21908\;31193\;\dots," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mpadded width="+2.8pt"><mn>3.62560 99082 21908 31193</mn></mpadded><mo></mo><mi mathvariant="normal">…</mi></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E9.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a></dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">6.1.10</span> (where the value is computed to 10D.)</span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E10">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">5.4.10</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m19.png" altimg-height="29px" altimg-valign="-9px" altimg-width="53px" alttext="\displaystyle\Gamma\left(\tfrac{3}{4}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mfrac><mn>3</mn><mn>4</mn></mfrac><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m5.png" altimg-height="18px" altimg-valign="-2px" altimg-width="299px" alttext="\displaystyle=1.22541\;67024\;65177\;64512\;\dots." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mpadded width="+2.8pt"><mn>1.22541 67024 65177 64512</mn></mpadded><mo></mo><mi mathvariant="normal">…</mi></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E10.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a></dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">6.1.14</span> (where the value is computed to 10D.)</span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E11">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">5.4.11</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m13.png" altimg-height="26px" altimg-valign="-7px" altimg-width="53px" alttext="\displaystyle\Gamma'\left(1\right)" display="inline"><mrow><msup><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo>′</mo></msup><mo></mo><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m3.png" altimg-height="21px" altimg-valign="-6px" altimg-width="60px" alttext="\displaystyle=-\gamma." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo>-</mo><mi href="./5.2#E3">γ</mi></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E11.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a> and
<a href="./5.2#E3" title="(5.2.3) ‣ §5.2(ii) Euler’s Constant ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\gamma" display="inline"><mi href="./5.2#E3">γ</mi></math>: Euler’s constant</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">6.3.2</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
</div>
</section>
<section id="ii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§5.4(ii) </span>Psi Function</h2>
<div id="SS2.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="SS2.p1" class="ltx_para">
<table id="E12" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex3" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">5.4.12</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m22.png" altimg-height="25px" altimg-valign="-7px" altimg-width="49px" alttext="\displaystyle\psi\left(1\right)" display="inline"><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m1.png" altimg-height="21px" altimg-valign="-6px" altimg-width="60px" alttext="\displaystyle=-\gamma," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo>-</mo><mi href="./5.2#E3">γ</mi></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex4" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m20.png" altimg-height="26px" altimg-valign="-7px" altimg-width="54px" alttext="\displaystyle\psi'\left(1\right)" display="inline"><mrow><msup><mi href="./5.2#E2">ψ</mi><mo>′</mo></msup><mo></mo><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m12.png" altimg-height="30px" altimg-valign="-9px" altimg-width="66px" alttext="\displaystyle=\tfrac{1}{6}\pi^{2}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mfrac><mn>1</mn><mn>6</mn></mfrac><mo></mo><msup><mi href="./3.12#E1">π</mi><mn>2</mn></msup></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E12.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E3" title="(5.2.3) ‣ §5.2(ii) Euler’s Constant ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\gamma" display="inline"><mi href="./5.2#E3">γ</mi></math>: Euler’s constant</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a> and
<a href="./5.2#E2" title="(5.2.2) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m65.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="\psi\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: psi (or digamma) function</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">6.3.2</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E13" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex5" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">5.4.13</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m23.png" altimg-height="29px" altimg-valign="-9px" altimg-width="54px" alttext="\displaystyle\psi\left(\tfrac{1}{2}\right)" display="inline"><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m2.png" altimg-height="23px" altimg-valign="-6px" altimg-width="127px" alttext="\displaystyle=-\gamma-2\ln 2," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mo>-</mo><mi href="./5.2#E3">γ</mi></mrow><mo>-</mo><mrow><mn>2</mn><mo></mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mn>2</mn></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex6" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m21.png" altimg-height="29px" altimg-valign="-9px" altimg-width="60px" alttext="\displaystyle\psi'\left(\tfrac{1}{2}\right)" display="inline"><mrow><msup><mi href="./5.2#E2">ψ</mi><mo>′</mo></msup><mo></mo><mrow><mo>(</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m11.png" altimg-height="30px" altimg-valign="-9px" altimg-width="66px" alttext="\displaystyle=\tfrac{1}{2}\pi^{2}." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><msup><mi href="./3.12#E1">π</mi><mn>2</mn></msup></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E13.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E3" title="(5.2.3) ‣ §5.2(ii) Euler’s Constant ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\gamma" display="inline"><mi href="./5.2#E3">γ</mi></math>: Euler’s constant</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./5.2#E2" title="(5.2.2) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m65.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="\psi\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: psi (or digamma) function</a> and
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m61.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">6.3.3</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">For higher derivatives of <math class="ltx_Math" altimg="m66.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="\psi\left(z\right)" display="inline"><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mi href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math> at <math class="ltx_Math" altimg="m83.png" altimg-height="17px" altimg-valign="-2px" altimg-width="51px" alttext="z=1" display="inline"><mrow><mi href="./5.1#p2.t1.r4">z</mi><mo>=</mo><mn>1</mn></mrow></math> and <math class="ltx_Math" altimg="m84.png" altimg-height="27px" altimg-valign="-9px" altimg-width="54px" alttext="z=\frac{1}{2}" display="inline"><mrow><mi href="./5.1#p2.t1.r4">z</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math>, see §.</p>
<table id="E14" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">5.4.14</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E14.png" altimg-height="64px" altimg-valign="-28px" altimg-width="201px" alttext="\psi\left(n+1\right)=\sum_{k=1}^{n}\frac{1}{k}-\gamma," display="block"><mrow><mrow><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./5.1#p2.t1.r1">n</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./5.1#p2.t1.r2">k</mi><mo>=</mo><mn>1</mn></mrow><mi href="./5.1#p2.t1.r1">n</mi></munderover><mfrac><mn>1</mn><mi href="./5.1#p2.t1.r2">k</mi></mfrac></mrow><mo>-</mo><mi href="./5.2#E3">γ</mi></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E14.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E3" title="(5.2.3) ‣ §5.2(ii) Euler’s Constant ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\gamma" display="inline"><mi href="./5.2#E3">γ</mi></math>: Euler’s constant</a>,
<a href="./5.2#E2" title="(5.2.2) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m65.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="\psi\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: psi (or digamma) function</a>,
<a href="./5.1#p2.t1.r1" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m75.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./5.1#p2.t1.r1">n</mi></math>: nonnegative integer</a> and
<a href="./5.1#p2.t1.r2" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./5.1#p2.t1.r2">k</mi></math>: nonnegative integer</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">6.3.2</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E15" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">5.4.15</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E15.png" altimg-height="41px" altimg-valign="-15px" altimg-width="444px" alttext="\psi\left(n+\tfrac{1}{2}\right)=-\gamma-2\ln 2+2\left(1+\tfrac{1}{3}+\dots+%
\tfrac{1}{2n-1}\right)," display="block"><mrow><mrow><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./5.1#p2.t1.r1">n</mi><mo>+</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mrow><mo>-</mo><mi href="./5.2#E3">γ</mi></mrow><mo>-</mo><mrow><mn>2</mn><mo></mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mn>2</mn></mrow></mrow></mrow><mo>+</mo><mrow><mn>2</mn><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>3</mn></mfrac></mstyle><mo>+</mo><mi mathvariant="normal">…</mi><mo>+</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mrow><mrow><mn>2</mn><mo></mo><mi href="./5.1#p2.t1.r1">n</mi></mrow><mo>-</mo><mn>1</mn></mrow></mfrac></mstyle></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m74.png" altimg-height="20px" altimg-valign="-6px" altimg-width="104px" alttext="n=1,2,\dots" display="inline"><mrow><mi href="./5.1#p2.t1.r1">n</mi><mo>=</mo><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E15.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E3" title="(5.2.3) ‣ §5.2(ii) Euler’s Constant ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\gamma" display="inline"><mi href="./5.2#E3">γ</mi></math>: Euler’s constant</a>,
<a href="./5.2#E2" title="(5.2.2) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m65.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="\psi\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: psi (or digamma) function</a>,
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m61.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a> and
<a href="./5.1#p2.t1.r1" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m75.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./5.1#p2.t1.r1">n</mi></math>: nonnegative integer</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">6.3.4</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E16" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">5.4.16</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E16.png" altimg-height="49px" altimg-valign="-20px" altimg-width="254px" alttext="\Im\psi\left(iy\right)=\frac{1}{2y}+\frac{\pi}{2}\coth\left(\pi y\right)," display="block"><mrow><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℑ</mi><mo></mo><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./5.1#p2.t1.r3">y</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mrow><mfrac><mn>1</mn><mrow><mn>2</mn><mo></mo><mi href="./5.1#p2.t1.r3">y</mi></mrow></mfrac><mo>+</mo><mrow><mfrac><mi href="./3.12#E1">π</mi><mn>2</mn></mfrac><mo></mo><mrow><mi href="./4.28#E7">coth</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi href="./5.1#p2.t1.r3">y</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E16.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./5.2#E2" title="(5.2.2) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m65.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="\psi\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: psi (or digamma) function</a>,
<a href="./4.28#E7" title="(4.28.7) ‣ §4.28 Definitions and Periodicity ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="55px" alttext="\coth\NVar{z}" display="inline"><mrow><mi href="./4.28#E7">coth</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: hyperbolic cotangent function</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m55.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Im" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℑ</mi><mo></mo></mrow></math>: imaginary part</a> and
<a href="./5.1#p2.t1.r3" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./5.1#p2.t1.r3">y</mi></math>: real variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">6.3.11</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E17" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">5.4.17</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E17.png" altimg-height="41px" altimg-valign="-16px" altimg-width="247px" alttext="\Im\psi\left(\tfrac{1}{2}+iy\right)=\frac{\pi}{2}\tanh\left(\pi y\right)," display="block"><mrow><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℑ</mi><mo></mo><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./5.1#p2.t1.r3">y</mi></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mrow><mfrac><mi href="./3.12#E1">π</mi><mn>2</mn></mfrac><mo></mo><mrow><mi href="./4.28#E4">tanh</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi href="./5.1#p2.t1.r3">y</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E17.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./5.2#E2" title="(5.2.2) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m65.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="\psi\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: psi (or digamma) function</a>,
<a href="./4.28#E4" title="(4.28.4) ‣ §4.28 Definitions and Periodicity ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m68.png" altimg-height="18px" altimg-valign="-2px" altimg-width="58px" alttext="\tanh\NVar{z}" display="inline"><mrow><mi href="./4.28#E4">tanh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: hyperbolic tangent function</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m55.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Im" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℑ</mi><mo></mo></mrow></math>: imaginary part</a> and
<a href="./5.1#p2.t1.r3" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./5.1#p2.t1.r3">y</mi></math>: real variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">6.3.12</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E18" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">5.4.18</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E18.png" altimg-height="49px" altimg-valign="-20px" altimg-width="304px" alttext="\Im\psi\left(1+iy\right)=-\frac{1}{2y}+\frac{\pi}{2}\coth\left(\pi y\right)." display="block"><mrow><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℑ</mi><mo></mo><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./5.1#p2.t1.r3">y</mi></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mrow><mrow><mo>-</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mo></mo><mi href="./5.1#p2.t1.r3">y</mi></mrow></mfrac></mrow><mo>+</mo><mrow><mfrac><mi href="./3.12#E1">π</mi><mn>2</mn></mfrac><mo></mo><mrow><mi href="./4.28#E7">coth</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi href="./5.1#p2.t1.r3">y</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E18.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./5.2#E2" title="(5.2.2) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m65.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="\psi\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: psi (or digamma) function</a>,
<a href="./4.28#E7" title="(4.28.7) ‣ §4.28 Definitions and Periodicity ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="55px" alttext="\coth\NVar{z}" display="inline"><mrow><mi href="./4.28#E7">coth</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: hyperbolic cotangent function</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m55.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Im" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℑ</mi><mo></mo></mrow></math>: imaginary part</a> and
<a href="./5.1#p2.t1.r3" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./5.1#p2.t1.r3">y</mi></math>: real variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">6.3.13</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">If <math class="ltx_Math" altimg="m77.png" altimg-height="16px" altimg-valign="-6px" altimg-width="33px" alttext="p,q" display="inline"><mrow><mi>p</mi><mo>,</mo><mi href="./5.1#p2.t1.r5">q</mi></mrow></math> are integers with <math class="ltx_Math" altimg="m47.png" altimg-height="20px" altimg-valign="-6px" altimg-width="87px" alttext="0<p<q" display="inline"><mrow><mn>0</mn><mo><</mo><mi>p</mi><mo><</mo><mi href="./5.1#p2.t1.r5">q</mi></mrow></math>, then
</p>
<table id="E19" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">5.4.19</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E19.png" altimg-height="68px" altimg-valign="-28px" altimg-width="693px" alttext="\psi\left(\frac{p}{q}\right)=-\gamma-\ln q-\frac{\pi}{2}\cot\left(\frac{\pi p}%
{q}\right)+\frac{1}{2}\sum_{k=1}^{q-1}\cos\left(\frac{2\pi kp}{q}\right)\ln%
\left(2-2\cos\left(\frac{2\pi k}{q}\right)\right)." display="block"><mrow><mrow><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mfrac><mi>p</mi><mi href="./5.1#p2.t1.r5">q</mi></mfrac><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mrow><mo>-</mo><mi href="./5.2#E3">γ</mi></mrow><mo>-</mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi href="./5.1#p2.t1.r5">q</mi></mrow><mo>-</mo><mrow><mfrac><mi href="./3.12#E1">π</mi><mn>2</mn></mfrac><mo></mo><mrow><mi href="./4.14#E7">cot</mi><mo></mo><mrow><mo>(</mo><mfrac><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi>p</mi></mrow><mi href="./5.1#p2.t1.r5">q</mi></mfrac><mo>)</mo></mrow></mrow></mrow></mrow><mo>+</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./5.1#p2.t1.r2">k</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi href="./5.1#p2.t1.r5">q</mi><mo>-</mo><mn>1</mn></mrow></munderover><mrow><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mfrac><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi href="./5.1#p2.t1.r2">k</mi><mo></mo><mi>p</mi></mrow><mi href="./5.1#p2.t1.r5">q</mi></mfrac><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><mo>(</mo><mrow><mn>2</mn><mo>-</mo><mrow><mn>2</mn><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mfrac><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi href="./5.1#p2.t1.r2">k</mi></mrow><mi href="./5.1#p2.t1.r5">q</mi></mfrac><mo>)</mo></mrow></mrow></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E19.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E3" title="(5.2.3) ‣ §5.2(ii) Euler’s Constant ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\gamma" display="inline"><mi href="./5.2#E3">γ</mi></math>: Euler’s constant</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./4.14#E7" title="(4.14.7) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="17px" altimg-valign="-2px" altimg-width="44px" alttext="\cot\NVar{z}" display="inline"><mrow><mi href="./4.14#E7">cot</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cotangent function</a>,
<a href="./5.2#E2" title="(5.2.2) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m65.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="\psi\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: psi (or digamma) function</a>,
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m61.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a>,
<a href="./5.1#p2.t1.r5" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="q" display="inline"><mi href="./5.1#p2.t1.r5">q</mi></math>: real or complex variable</a> and
<a href="./5.1#p2.t1.r2" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./5.1#p2.t1.r2">k</mi></math>: nonnegative integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="iii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§5.4(iii) </span>Extrema</h2>
<div id="SS3.info" class="ltx_metadata ltx_info">
) to solve
<math class="ltx_Math" altimg="m64.png" altimg-height="23px" altimg-valign="-7px" altimg-width="193px" alttext="\psi\left(1-x\right)=\pi\cot\left(\pi x\right)" display="inline"><mrow><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mi href="./5.1#p2.t1.r3">x</mi></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mi href="./3.12#E1">π</mi><mo></mo><mrow><mi href="./4.14#E7">cot</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi href="./5.1#p2.t1.r3">x</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow></math> with <math class="ltx_Math" altimg="m79.png" altimg-height="17px" altimg-valign="-4px" altimg-width="105px" alttext="x=-n+u" display="inline"><mrow><mi href="./5.1#p2.t1.r3">x</mi><mo>=</mo><mrow><mrow><mo>-</mo><mi href="./5.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mi>u</mi></mrow></mrow></math> and <math class="ltx_Math" altimg="m75.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./5.1#p2.t1.r1">n</mi></math> large.</dd>
</dl>
</div>
</div>
<figure id="T1" class="ltx_table">
<figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_table">Table 5.4.1: </span><math class="ltx_Math" altimg="m52.png" altimg-height="24px" altimg-valign="-7px" altimg-width="181px" alttext="\Gamma'\left(x_{n}\right)=\psi\left(x_{n}\right)=0" display="inline"><mrow><mrow><msup><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo>′</mo></msup><mo></mo><mrow><mo>(</mo><msub><mi href="./5.1#p2.t1.r3">x</mi><mi href="./5.1#p2.t1.r1">n</mi></msub><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><msub><mi href="./5.1#p2.t1.r3">x</mi><mi href="./5.1#p2.t1.r1">n</mi></msub><mo>)</mo></mrow></mrow><mo>=</mo><mn>0</mn></mrow></math>.
</figcaption>
<table id="T1.t1" class="ltx_tabular ltx_centering ltx_guessed_headers ltx_align_middle">
<thead class="ltx_thead">
<tr id="T1.t1.r1" class="ltx_tr">
<th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_th_row ltx_border_tt" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m75.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./5.1#p2.t1.r1">n</mi></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m81.png" altimg-height="16px" altimg-valign="-5px" altimg-width="27px" alttext="x_{n}" display="inline"><msub><mi href="./5.1#p2.t1.r3">x</mi><mi href="./5.1#p2.t1.r1">n</mi></msub></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m54.png" altimg-height="23px" altimg-valign="-7px" altimg-width="58px" alttext="\Gamma\left(x_{n}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><msub><mi href="./5.1#p2.t1.r3">x</mi><mi href="./5.1#p2.t1.r1">n</mi></msub><mo>)</mo></mrow></mrow></math></th>
</tr>
</thead>
<tbody class="ltx_tbody">
<tr id="T1.t1.r2" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_t" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;">0</th>
<td class="ltx_td ltx_align_right ltx_border_t" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m48.png" altimg-height="17px" altimg-valign="-2px" altimg-width="125px" alttext="1.46163\;21449" display="inline"><mn>1.46163 21449</mn></math></td>
<td class="ltx_td ltx_align_right ltx_border_t" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m46.png" altimg-height="17px" altimg-valign="-2px" altimg-width="125px" alttext="0.88560\;31944" display="inline"><mn>0.88560 31944</mn></math></td>
</tr>
<tr id="T1.t1.r3" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;">1</th>
<td class="ltx_td ltx_align_right" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m30.png" altimg-height="18px" altimg-valign="-4px" altimg-width="140px" alttext="-0.50408\;30083" display="inline"><mrow><mo>-</mo><mn>0.50408 30083</mn></mrow></math></td>
<td class="ltx_td ltx_align_right" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m34.png" altimg-height="18px" altimg-valign="-4px" altimg-width="140px" alttext="-3.54464\;36112" display="inline"><mrow><mo>-</mo><mn>3.54464 36112</mn></mrow></math></td>
</tr>
<tr id="T1.t1.r4" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;">2</th>
<td class="ltx_td ltx_align_right" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m32.png" altimg-height="18px" altimg-valign="-4px" altimg-width="140px" alttext="-1.57349\;84732" display="inline"><mrow><mo>-</mo><mn>1.57349 84732</mn></mrow></math></td>
<td class="ltx_td ltx_align_right" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m49.png" altimg-height="17px" altimg-valign="-2px" altimg-width="125px" alttext="2.30240\;72583" display="inline"><mn>2.30240 72583</mn></math></td>
</tr>
<tr id="T1.t1.r5" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;">3</th>
<td class="ltx_td ltx_align_right" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m33.png" altimg-height="18px" altimg-valign="-4px" altimg-width="140px" alttext="-2.61072\;08875" display="inline"><mrow><mo>-</mo><mn>2.61072 08875</mn></mrow></math></td>
<td class="ltx_td ltx_align_right" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m31.png" altimg-height="18px" altimg-valign="-4px" altimg-width="140px" alttext="-0.88813\;63584" display="inline"><mrow><mo>-</mo><mn>0.88813 63584</mn></mrow></math></td>
</tr>
<tr id="T1.t1.r6" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_T" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;">4</th>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m35.png" altimg-height="18px" altimg-valign="-4px" altimg-width="140px" alttext="-3.63529\;33665" display="inline"><mrow><mo>-</mo><mn>3.63529 33665</mn></mrow></math></td>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m45.png" altimg-height="17px" altimg-valign="-2px" altimg-width="125px" alttext="0.24512\;75398" display="inline"><mn>0.24512 75398</mn></math></td>
</tr>
<tr id="T1.t1.r7" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;">5</th>
<td class="ltx_td ltx_align_right" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m36.png" altimg-height="18px" altimg-valign="-4px" altimg-width="140px" alttext="-4.65323\;77626" display="inline"><mrow><mo>-</mo><mn>4.65323 77626</mn></mrow></math></td>
<td class="ltx_td ltx_align_right" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m29.png" altimg-height="18px" altimg-valign="-4px" altimg-width="140px" alttext="-0.05277\;96396" display="inline"><mrow><mo>-</mo><mn>0.05277 96396</mn></mrow></math></td>
</tr>
<tr id="T1.t1.r8" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;">6</th>
<td class="ltx_td ltx_align_right" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m37.png" altimg-height="18px" altimg-valign="-4px" altimg-width="140px" alttext="-5.66716\;24513" display="inline"><mrow><mo>-</mo><mn>5.66716 24513</mn></mrow></math></td>
<td class="ltx_td ltx_align_right" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m44.png" altimg-height="17px" altimg-valign="-2px" altimg-width="125px" alttext="0.00932\;45945" display="inline"><mn>0.00932 45945</mn></math></td>
</tr>
<tr id="T1.t1.r9" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_T" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;">7</th>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m38.png" altimg-height="18px" altimg-valign="-4px" altimg-width="140px" alttext="-6.67841\;82649" display="inline"><mrow><mo>-</mo><mn>6.67841 82649</mn></mrow></math></td>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m28.png" altimg-height="18px" altimg-valign="-4px" altimg-width="140px" alttext="-0.00139\;73966" display="inline"><mrow><mo>-</mo><mn>0.00139 73966</mn></mrow></math></td>
</tr>
<tr id="T1.t1.r10" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;">8</th>
<td class="ltx_td ltx_align_right" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m39.png" altimg-height="18px" altimg-valign="-4px" altimg-width="140px" alttext="-7.68778\;83250" display="inline"><mrow><mo>-</mo><mn>7.68778 83250</mn></mrow></math></td>
<td class="ltx_td ltx_align_right" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m43.png" altimg-height="17px" altimg-valign="-2px" altimg-width="125px" alttext="0.00018\;18784" display="inline"><mn>0.00018 18784</mn></math></td>
</tr>
<tr id="T1.t1.r11" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;">9</th>
<td class="ltx_td ltx_align_right" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m40.png" altimg-height="18px" altimg-valign="-4px" altimg-width="140px" alttext="-8.69576\;41633" display="inline"><mrow><mo>-</mo><mn>8.69576 41633</mn></mrow></math></td>
<td class="ltx_td ltx_align_right" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m27.png" altimg-height="18px" altimg-valign="-4px" altimg-width="140px" alttext="-0.00002\;09253" display="inline"><mrow><mo>-</mo><mn>0.00002 09253</mn></mrow></math></td>
</tr>
<tr id="T1.t1.r12" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_b" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;">10</th>
<td class="ltx_td ltx_align_right ltx_border_b" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m41.png" altimg-height="18px" altimg-valign="-4px" altimg-width="140px" alttext="-9.70267\;25406" display="inline"><mrow><mo>-</mo><mn>9.70267 25406</mn></mrow></math></td>
<td class="ltx_td ltx_align_right ltx_border_b" style="padding-top:2.083333333333333px;padding-bottom:2.083333333333333px;"><math class="ltx_Math" altimg="m42.png" altimg-height="17px" altimg-valign="-2px" altimg-width="125px" alttext="0.00000\;21574" display="inline"><mn>0.00000 21574</mn></math></td>
</tr>
</tbody>
</table>
<div id="T1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./5.2#E2" title="(5.2.2) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m65.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="\psi\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: psi (or digamma) function</a>,
<a href="./5.1#p2.t1.r1" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m75.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./5.1#p2.t1.r1">n</mi></math>: nonnegative integer</a> and
<a href="./5.1#p2.t1.r3" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m80.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./5.1#p2.t1.r3">x</mi></math>: real variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">§6.3</span> ((Values are given to 3 decimals for <math class="ltx_Math" altimg="m73.png" altimg-height="20px" altimg-valign="-6px" altimg-width="126px" alttext="n=0,1,\dots,7" display="inline"><mrow><mi href="./5.1#p2.t1.r1">n</mi><mo>=</mo><mrow><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><mn>7</mn></mrow></mrow></math>.))</span>
</dd>
</dl>
</div>
</div>
</figure>
<div id="SS3.p1" class="ltx_para">
<p class="ltx_p">Compare Figure .</p>
</div>
<div id="SS3.p2" class="ltx_para">
<p class="ltx_p">As <math class="ltx_Math" altimg="m76.png" altimg-height="13px" altimg-valign="-2px" altimg-width="67px" alttext="n\to\infty" display="inline"><mrow><mi href="./5.1#p2.t1.r1">n</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></math>,
</p>
<table id="E20" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">5.4.20</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E20.png" altimg-height="53px" altimg-valign="-21px" altimg-width="401px" alttext="x_{n}=-n+\frac{1}{\pi}\operatorname{arctan}\left(\frac{\pi}{\ln n}\right)+O%
\left(\frac{1}{n(\ln n)^{2}}\right)." display="block"><mrow><mrow><msub><mi href="./5.1#p2.t1.r3">x</mi><mi href="./5.1#p2.t1.r1">n</mi></msub><mo>=</mo><mrow><mrow><mo>-</mo><mi href="./5.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mrow><mfrac><mn>1</mn><mi href="./3.12#E1">π</mi></mfrac><mo></mo><mrow><mi href="./4.23#SS2.p1">arctan</mi><mo></mo><mrow><mo>(</mo><mfrac><mi href="./3.12#E1">π</mi><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi href="./5.1#p2.t1.r1">n</mi></mrow></mfrac><mo>)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mi href="./2.1#E3">O</mi><mo></mo><mrow><mo>(</mo><mfrac><mn>1</mn><mrow><mi href="./5.1#p2.t1.r1">n</mi><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi href="./5.1#p2.t1.r1">n</mi></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow></mfrac><mo>)</mo></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E20.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./2.1#E3" title="(2.1.3) ‣ §2.1(i) Asymptotic and Order Symbols ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="23px" altimg-valign="-7px" altimg-width="50px" alttext="O\left(\NVar{x}\right)" display="inline"><mrow><mi href="./2.1#E3">O</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>)</mo></mrow></mrow></math>: order not exceeding</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.23#SS2.p1" title="§4.23(ii) Principal Values ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m62.png" altimg-height="17px" altimg-valign="-2px" altimg-width="73px" alttext="\operatorname{arctan}\NVar{z}" display="inline"><mrow><mi href="./4.23#SS2.p1">arctan</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: arctangent function</a>,
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m61.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a>,
<a href="./5.1#p2.t1.r1" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m75.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./5.1#p2.t1.r1">n</mi></math>: nonnegative integer</a> and
<a href="./5.1#p2.t1.r3" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m80.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./5.1#p2.t1.r3">x</mi></math>: real variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">6.3.20</span> (The error estimate has been improved.)</span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS3.p3" class="ltx_para">
<p class="ltx_p">For error bounds for this estimate see <cite class="ltx_cite ltx_citemacro_citet">Walker (</div>
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<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">7.5.1</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E1.png" altimg-height="41px" altimg-valign="-15px" altimg-width="461px" alttext="F\left(z\right)=\tfrac{1}{2}i\sqrt{\pi}\left(e^{-z^{2}}-w\left(z\right)\right)%
=-\tfrac{1}{2}i\sqrt{\pi}e^{-z^{2}}\operatorname{erf}\left(iz\right)." display="block"><mrow><mrow><mrow><mi href="./7.2#E5">F</mi><mo></mo><mrow><mo>(</mo><mi href="./7.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi mathvariant="normal">i</mi><mo></mo><msqrt><mi href="./3.12#E1">π</mi></msqrt><mo></mo><mrow><mo>(</mo><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><msup><mi href="./7.1#p1.t1.r2">z</mi><mn>2</mn></msup></mrow></msup><mo>-</mo><mrow><mi href="./7.2#E3">w</mi><mo></mo><mrow><mo>(</mo><mi href="./7.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mo>-</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi mathvariant="normal">i</mi><mo></mo><msqrt><mi href="./3.12#E1">π</mi></msqrt><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><msup><mi href="./7.1#p1.t1.r2">z</mi><mn>2</mn></msup></mrow></msup><mo></mo><mrow><mi href="./7.2#E1">erf</mi><mo></mo><mrow><mo>(</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./7.1#p1.t1.r2">z</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
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<div id="E1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./7.2#E5" title="(7.2.5) ‣ §7.2(ii) Dawson’s Integral ‣ §7.2 Definitions ‣ Properties ‣ Chapter 7 Error Functions, Dawson’s and Fresnel Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m2.png" altimg-height="23px" altimg-valign="-7px" altimg-width="49px" alttext="F\left(\NVar{z}\right)" display="inline"><mrow><mi href="./7.2#E5">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./7.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: Dawson’s integral</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m14.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./7.2#E1" title="(7.2.1) ‣ §7.2(i) Error Functions ‣ §7.2 Definitions ‣ Properties ‣ Chapter 7 Error Functions, Dawson’s and Fresnel Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m13.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\operatorname{erf}\NVar{z}" display="inline"><mrow><mi href="./7.2#E1">erf</mi><mo></mo><mi class="ltx_nvar" href="./7.1#p1.t1.r2">z</mi></mrow></math>: error function</a>,
<a href="./7.2#E3" title="(7.2.3) ‣ §7.2(i) Error Functions ‣ §7.2 Definitions ‣ Properties ‣ Chapter 7 Error Functions, Dawson’s and Fresnel Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m17.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="w\left(\NVar{z}\right)" display="inline"><mrow><mi href="./7.2#E3">w</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./7.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: Faddeeva (or Faddeyeva) function</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m9.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a> and
<a href="./7.1#p1.t1.r2" title="§7.1 Special Notation ‣ Notation ‣ Chapter 7 Error Functions, Dawson’s and Fresnel Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./7.1#p1.t1.r2">z</mi></math>: complex variable</a>
</dd>
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<table id="E2" class="ltx_equation ltx_eqn_table">
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<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">7.5.2</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E2.png" altimg-height="29px" altimg-valign="-9px" altimg-width="299px" alttext="C\left(z\right)+iS\left(z\right)=\tfrac{1}{2}(1+i)-\mathcal{F}\left(z\right)." display="block"><mrow><mrow><mrow><mrow><mi href="./7.2#E7">C</mi><mo></mo><mrow><mo>(</mo><mi href="./7.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mrow><mi href="./7.2#E8">S</mi><mo></mo><mrow><mo>(</mo><mi href="./7.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>=</mo><mrow><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>+</mo><mi mathvariant="normal">i</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>-</mo><mrow><mi class="ltx_font_mathcaligraphic" href="./7.2#E6">ℱ</mi><mo></mo><mrow><mo>(</mo><mi href="./7.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
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<tr><td class="ltx_align_right" colspan="4">
<div id="E2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./7.2#E7" title="(7.2.7) ‣ §7.2(iii) Fresnel Integrals ‣ §7.2 Definitions ‣ Properties ‣ Chapter 7 Error Functions, Dawson’s and Fresnel Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m1.png" altimg-height="23px" altimg-valign="-7px" altimg-width="49px" alttext="C\left(\NVar{z}\right)" display="inline"><mrow><mi href="./7.2#E7">C</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./7.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: Fresnel integral</a>,
<a href="./7.2#E6" title="(7.2.6) ‣ §7.2(iii) Fresnel Integrals ‣ §7.2 Definitions ‣ Properties ‣ Chapter 7 Error Functions, Dawson’s and Fresnel Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m6.png" altimg-height="23px" altimg-valign="-7px" altimg-width="50px" alttext="\mathcal{F}\left(\NVar{z}\right)" display="inline"><mrow><mi class="ltx_font_mathcaligraphic" href="./7.2#E6">ℱ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./7.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: Fresnel integral</a>,
<a href="./7.2#E8" title="(7.2.8) ‣ §7.2(iii) Fresnel Integrals ‣ §7.2 Definitions ‣ Properties ‣ Chapter 7 Error Functions, Dawson’s and Fresnel Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m4.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="S\left(\NVar{z}\right)" display="inline"><mrow><mi href="./7.2#E8">S</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./7.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: Fresnel integral</a> and
<a href="./7.1#p1.t1.r2" title="§7.1 Special Notation ‣ Notation ‣ Chapter 7 Error Functions, Dawson’s and Fresnel Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./7.1#p1.t1.r2">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
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</td></tr>
</table>
<table id="E3" class="ltx_equation ltx_eqn_table">
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<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">7.5.3</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E3.png" altimg-height="30px" altimg-valign="-9px" altimg-width="412px" alttext="C\left(z\right)=\tfrac{1}{2}+\mathrm{f}\left(z\right)\sin\left(\tfrac{1}{2}\pi
z%
^{2}\right)-\mathrm{g}\left(z\right)\cos\left(\tfrac{1}{2}\pi z^{2}\right)," display="block"><mrow><mrow><mrow><mi href="./7.2#E7">C</mi><mo></mo><mrow><mo>(</mo><mi href="./7.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo>+</mo><mrow><mrow><mi href="./7.2#E10" mathvariant="normal">f</mi><mo></mo><mrow><mo>(</mo><mi href="./7.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><msup><mi href="./7.1#p1.t1.r2">z</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>-</mo><mrow><mrow><mi href="./7.2#E11" mathvariant="normal">g</mi><mo></mo><mrow><mo>(</mo><mi href="./7.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><msup><mi href="./7.1#p1.t1.r2">z</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
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<div id="E3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./7.2#E10" title="(7.2.10) ‣ §7.2(iv) Auxiliary Functions ‣ §7.2 Definitions ‣ Properties ‣ Chapter 7 Error Functions, Dawson’s and Fresnel Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m10.png" altimg-height="23px" altimg-valign="-7px" altimg-width="41px" alttext="\mathrm{f}\left(\NVar{z}\right)" display="inline"><mrow><mi href="./7.2#E10" mathvariant="normal">f</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./7.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: auxiliary function for Fresnel integrals</a>,
<a href="./7.2#E11" title="(7.2.11) ‣ §7.2(iv) Auxiliary Functions ‣ §7.2 Definitions ‣ Properties ‣ Chapter 7 Error Functions, Dawson’s and Fresnel Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m11.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="\mathrm{g}\left(\NVar{z}\right)" display="inline"><mrow><mi href="./7.2#E11" mathvariant="normal">g</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./7.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: auxiliary function for Fresnel integrals</a>,
<a href="./7.2#E7" title="(7.2.7) ‣ §7.2(iii) Fresnel Integrals ‣ §7.2 Definitions ‣ Properties ‣ Chapter 7 Error Functions, Dawson’s and Fresnel Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m1.png" altimg-height="23px" altimg-valign="-7px" altimg-width="49px" alttext="C\left(\NVar{z}\right)" display="inline"><mrow><mi href="./7.2#E7">C</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./7.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: Fresnel integral</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m14.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m5.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m15.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a> and
<a href="./7.1#p1.t1.r2" title="§7.1 Special Notation ‣ Notation ‣ Chapter 7 Error Functions, Dawson’s and Fresnel Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./7.1#p1.t1.r2">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">7.3.9</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E4" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">7.5.4</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E4.png" altimg-height="30px" altimg-valign="-9px" altimg-width="410px" alttext="S\left(z\right)=\tfrac{1}{2}-\mathrm{f}\left(z\right)\cos\left(\tfrac{1}{2}\pi
z%
^{2}\right)-\mathrm{g}\left(z\right)\sin\left(\tfrac{1}{2}\pi z^{2}\right)." display="block"><mrow><mrow><mrow><mi href="./7.2#E8">S</mi><mo></mo><mrow><mo>(</mo><mi href="./7.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo>-</mo><mrow><mrow><mi href="./7.2#E10" mathvariant="normal">f</mi><mo></mo><mrow><mo>(</mo><mi href="./7.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><msup><mi href="./7.1#p1.t1.r2">z</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow></mrow></mrow><mo>-</mo><mrow><mrow><mi href="./7.2#E11" mathvariant="normal">g</mi><mo></mo><mrow><mo>(</mo><mi href="./7.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><msup><mi href="./7.1#p1.t1.r2">z</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E4.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./7.2#E10" title="(7.2.10) ‣ §7.2(iv) Auxiliary Functions ‣ §7.2 Definitions ‣ Properties ‣ Chapter 7 Error Functions, Dawson’s and Fresnel Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m10.png" altimg-height="23px" altimg-valign="-7px" altimg-width="41px" alttext="\mathrm{f}\left(\NVar{z}\right)" display="inline"><mrow><mi href="./7.2#E10" mathvariant="normal">f</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./7.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: auxiliary function for Fresnel integrals</a>,
<a href="./7.2#E11" title="(7.2.11) ‣ §7.2(iv) Auxiliary Functions ‣ §7.2 Definitions ‣ Properties ‣ Chapter 7 Error Functions, Dawson’s and Fresnel Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m11.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="\mathrm{g}\left(\NVar{z}\right)" display="inline"><mrow><mi href="./7.2#E11" mathvariant="normal">g</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./7.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: auxiliary function for Fresnel integrals</a>,
<a href="./7.2#E8" title="(7.2.8) ‣ §7.2(iii) Fresnel Integrals ‣ §7.2 Definitions ‣ Properties ‣ Chapter 7 Error Functions, Dawson’s and Fresnel Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m4.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="S\left(\NVar{z}\right)" display="inline"><mrow><mi href="./7.2#E8">S</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./7.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: Fresnel integral</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m14.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m5.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m15.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a> and
<a href="./7.1#p1.t1.r2" title="§7.1 Special Notation ‣ Notation ‣ Chapter 7 Error Functions, Dawson’s and Fresnel Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./7.1#p1.t1.r2">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">7.3.10</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E5" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">7.5.5</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E5.png" altimg-height="31px" altimg-valign="-7px" altimg-width="259px" alttext="e^{-\frac{1}{2}\pi iz^{2}}\mathcal{F}\left(z\right)=\mathrm{g}\left(z\right)+i%
\mathrm{f}\left(z\right)." display="block"><mrow><mrow><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi><mo></mo><msup><mi href="./7.1#p1.t1.r2">z</mi><mn>2</mn></msup></mrow></mrow></msup><mo></mo><mrow><mi class="ltx_font_mathcaligraphic" href="./7.2#E6">ℱ</mi><mo></mo><mrow><mo>(</mo><mi href="./7.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mrow><mrow><mi href="./7.2#E11" mathvariant="normal">g</mi><mo></mo><mrow><mo>(</mo><mi href="./7.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mrow><mi href="./7.2#E10" mathvariant="normal">f</mi><mo></mo><mrow><mo>(</mo><mi href="./7.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E5.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./7.2#E10" title="(7.2.10) ‣ §7.2(iv) Auxiliary Functions ‣ §7.2 Definitions ‣ Properties ‣ Chapter 7 Error Functions, Dawson’s and Fresnel Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m10.png" altimg-height="23px" altimg-valign="-7px" altimg-width="41px" alttext="\mathrm{f}\left(\NVar{z}\right)" display="inline"><mrow><mi href="./7.2#E10" mathvariant="normal">f</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./7.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: auxiliary function for Fresnel integrals</a>,
<a href="./7.2#E11" title="(7.2.11) ‣ §7.2(iv) Auxiliary Functions ‣ §7.2 Definitions ‣ Properties ‣ Chapter 7 Error Functions, Dawson’s and Fresnel Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m11.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="\mathrm{g}\left(\NVar{z}\right)" display="inline"><mrow><mi href="./7.2#E11" mathvariant="normal">g</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./7.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: auxiliary function for Fresnel integrals</a>,
<a href="./7.2#E6" title="(7.2.6) ‣ §7.2(iii) Fresnel Integrals ‣ §7.2 Definitions ‣ Properties ‣ Chapter 7 Error Functions, Dawson’s and Fresnel Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m6.png" altimg-height="23px" altimg-valign="-7px" altimg-width="50px" alttext="\mathcal{F}\left(\NVar{z}\right)" display="inline"><mrow><mi class="ltx_font_mathcaligraphic" href="./7.2#E6">ℱ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./7.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: Fresnel integral</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m14.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m9.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a> and
<a href="./7.1#p1.t1.r2" title="§7.1 Special Notation ‣ Notation ‣ Chapter 7 Error Functions, Dawson’s and Fresnel Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./7.1#p1.t1.r2">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E6" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">7.5.6</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E6.png" altimg-height="33px" altimg-valign="-9px" altimg-width="454px" alttext="e^{\pm\frac{1}{2}\pi iz^{2}}(\mathrm{g}\left(z\right)\pm i\mathrm{f}\left(z%
\right))=\tfrac{1}{2}(1\pm i)-(C\left(z\right)\pm iS\left(z\right))." display="block"><mrow><mrow><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>±</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi><mo></mo><msup><mi href="./7.1#p1.t1.r2">z</mi><mn>2</mn></msup></mrow></mrow></msup><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mi href="./7.2#E11" mathvariant="normal">g</mi><mo></mo><mrow><mo>(</mo><mi href="./7.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow><mo>±</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mrow><mi href="./7.2#E10" mathvariant="normal">f</mi><mo></mo><mrow><mo>(</mo><mi href="./7.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>±</mo><mi mathvariant="normal">i</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>-</mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mi href="./7.2#E7">C</mi><mo></mo><mrow><mo>(</mo><mi href="./7.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow><mo>±</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mrow><mi href="./7.2#E8">S</mi><mo></mo><mrow><mo>(</mo><mi href="./7.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E6.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./7.2#E10" title="(7.2.10) ‣ §7.2(iv) Auxiliary Functions ‣ §7.2 Definitions ‣ Properties ‣ Chapter 7 Error Functions, Dawson’s and Fresnel Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m10.png" altimg-height="23px" altimg-valign="-7px" altimg-width="41px" alttext="\mathrm{f}\left(\NVar{z}\right)" display="inline"><mrow><mi href="./7.2#E10" mathvariant="normal">f</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./7.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: auxiliary function for Fresnel integrals</a>,
<a href="./7.2#E11" title="(7.2.11) ‣ §7.2(iv) Auxiliary Functions ‣ §7.2 Definitions ‣ Properties ‣ Chapter 7 Error Functions, Dawson’s and Fresnel Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m11.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="\mathrm{g}\left(\NVar{z}\right)" display="inline"><mrow><mi href="./7.2#E11" mathvariant="normal">g</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./7.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: auxiliary function for Fresnel integrals</a>,
<a href="./7.2#E7" title="(7.2.7) ‣ §7.2(iii) Fresnel Integrals ‣ §7.2 Definitions ‣ Properties ‣ Chapter 7 Error Functions, Dawson’s and Fresnel Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m1.png" altimg-height="23px" altimg-valign="-7px" altimg-width="49px" alttext="C\left(\NVar{z}\right)" display="inline"><mrow><mi href="./7.2#E7">C</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./7.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: Fresnel integral</a>,
<a href="./7.2#E8" title="(7.2.8) ‣ §7.2(iii) Fresnel Integrals ‣ §7.2 Definitions ‣ Properties ‣ Chapter 7 Error Functions, Dawson’s and Fresnel Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m4.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="S\left(\NVar{z}\right)" display="inline"><mrow><mi href="./7.2#E8">S</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./7.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: Fresnel integral</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m14.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m9.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a> and
<a href="./7.1#p1.t1.r2" title="§7.1 Special Notation ‣ Notation ‣ Chapter 7 Error Functions, Dawson’s and Fresnel Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./7.1#p1.t1.r2">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="p2" class="ltx_para">
<p class="ltx_p">In ()</p>
<table id="E7" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">7.5.7</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E7.png" altimg-height="29px" altimg-valign="-9px" altimg-width="157px" alttext="\zeta=\tfrac{1}{2}\sqrt{\pi}(1\mp i)z," display="block"><mrow><mrow><mi href="./7.5#E7">ζ</mi><mo>=</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><msqrt><mi href="./3.12#E1">π</mi></msqrt><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>∓</mo><mi mathvariant="normal">i</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi href="./7.1#p1.t1.r2">z</mi></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E7.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text"><math class="ltx_Math" altimg="m16.png" altimg-height="21px" altimg-valign="-6px" altimg-width="14px" alttext="\zeta" display="inline"><mi href="./7.5#E7">ζ</mi></math>: change of variable (locally)</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m14.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a> and
<a href="./7.1#p1.t1.r2" title="§7.1 Special Notation ‣ Notation ‣ Chapter 7 Error Functions, Dawson’s and Fresnel Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./7.1#p1.t1.r2">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and either all upper signs or all lower signs are taken throughout.
</p>
<table id="E8" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">7.5.8</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E8.png" altimg-height="29px" altimg-valign="-9px" altimg-width="267px" alttext="C\left(z\right)\pm iS\left(z\right)=\tfrac{1}{2}(1\pm i)\operatorname{erf}\zeta." display="block"><mrow><mrow><mrow><mrow><mi href="./7.2#E7">C</mi><mo></mo><mrow><mo>(</mo><mi href="./7.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow><mo>±</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mrow><mi href="./7.2#E8">S</mi><mo></mo><mrow><mo>(</mo><mi href="./7.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>=</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>±</mo><mi mathvariant="normal">i</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mi href="./7.2#E1">erf</mi><mo></mo><mi href="./7.5#E7">ζ</mi></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E8.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./7.2#E7" title="(7.2.7) ‣ §7.2(iii) Fresnel Integrals ‣ §7.2 Definitions ‣ Properties ‣ Chapter 7 Error Functions, Dawson’s and Fresnel Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m1.png" altimg-height="23px" altimg-valign="-7px" altimg-width="49px" alttext="C\left(\NVar{z}\right)" display="inline"><mrow><mi href="./7.2#E7">C</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./7.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: Fresnel integral</a>,
<a href="./7.2#E8" title="(7.2.8) ‣ §7.2(iii) Fresnel Integrals ‣ §7.2 Definitions ‣ Properties ‣ Chapter 7 Error Functions, Dawson’s and Fresnel Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m4.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="S\left(\NVar{z}\right)" display="inline"><mrow><mi href="./7.2#E8">S</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./7.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: Fresnel integral</a>,
<a href="./7.2#E1" title="(7.2.1) ‣ §7.2(i) Error Functions ‣ §7.2 Definitions ‣ Properties ‣ Chapter 7 Error Functions, Dawson’s and Fresnel Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m13.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\operatorname{erf}\NVar{z}" display="inline"><mrow><mi href="./7.2#E1">erf</mi><mo></mo><mi class="ltx_nvar" href="./7.1#p1.t1.r2">z</mi></mrow></math>: error function</a>,
<a href="./7.1#p1.t1.r2" title="§7.1 Special Notation ‣ Notation ‣ Chapter 7 Error Functions, Dawson’s and Fresnel Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./7.1#p1.t1.r2">z</mi></math>: complex variable</a> and
<a href="./7.5#E7" title="(7.5.7) ‣ §7.5 Interrelations ‣ Properties ‣ Chapter 7 Error Functions, Dawson’s and Fresnel Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m16.png" altimg-height="21px" altimg-valign="-6px" altimg-width="14px" alttext="\zeta" display="inline"><mi href="./7.5#E7">ζ</mi></math>: change of variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E9" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">7.5.9</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E9.png" altimg-height="41px" altimg-valign="-15px" altimg-width="407px" alttext="C\left(z\right)\pm iS\left(z\right)=\tfrac{1}{2}(1\pm i)\left(1-e^{\pm\frac{1}%
{2}\pi iz^{2}}w\left(i\zeta\right)\right)." display="block"><mrow><mrow><mrow><mrow><mi href="./7.2#E7">C</mi><mo></mo><mrow><mo>(</mo><mi href="./7.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow><mo>±</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mrow><mi href="./7.2#E8">S</mi><mo></mo><mrow><mo>(</mo><mi href="./7.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>=</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>±</mo><mi mathvariant="normal">i</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>±</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi><mo></mo><msup><mi href="./7.1#p1.t1.r2">z</mi><mn>2</mn></msup></mrow></mrow></msup><mo></mo><mrow><mi href="./7.2#E3">w</mi><mo></mo><mrow><mo>(</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./7.5#E7">ζ</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E9.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./7.2#E7" title="(7.2.7) ‣ §7.2(iii) Fresnel Integrals ‣ §7.2 Definitions ‣ Properties ‣ Chapter 7 Error Functions, Dawson’s and Fresnel Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m1.png" altimg-height="23px" altimg-valign="-7px" altimg-width="49px" alttext="C\left(\NVar{z}\right)" display="inline"><mrow><mi href="./7.2#E7">C</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./7.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: Fresnel integral</a>,
<a href="./7.2#E8" title="(7.2.8) ‣ §7.2(iii) Fresnel Integrals ‣ §7.2 Definitions ‣ Properties ‣ Chapter 7 Error Functions, Dawson’s and Fresnel Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m4.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="S\left(\NVar{z}\right)" display="inline"><mrow><mi href="./7.2#E8">S</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./7.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: Fresnel integral</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m14.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./7.2#E3" title="(7.2.3) ‣ §7.2(i) Error Functions ‣ §7.2 Definitions ‣ Properties ‣ Chapter 7 Error Functions, Dawson’s and Fresnel Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m17.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="w\left(\NVar{z}\right)" display="inline"><mrow><mi href="./7.2#E3">w</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./7.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: Faddeeva (or Faddeyeva) function</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m9.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./7.1#p1.t1.r2" title="§7.1 Special Notation ‣ Notation ‣ Chapter 7 Error Functions, Dawson’s and Fresnel Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./7.1#p1.t1.r2">z</mi></math>: complex variable</a> and
<a href="./7.5#E7" title="(7.5.7) ‣ §7.5 Interrelations ‣ Properties ‣ Chapter 7 Error Functions, Dawson’s and Fresnel Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m16.png" altimg-height="21px" altimg-valign="-6px" altimg-width="14px" alttext="\zeta" display="inline"><mi href="./7.5#E7">ζ</mi></math>: change of variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E10" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">7.5.10</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E10.png" altimg-height="33px" altimg-valign="-9px" altimg-width="289px" alttext="\mathrm{g}\left(z\right)\pm i\mathrm{f}\left(z\right)=\tfrac{1}{2}(1\pm i)e^{%
\zeta^{2}}\operatorname{erfc}\zeta." display="block"><mrow><mrow><mrow><mrow><mi href="./7.2#E11" mathvariant="normal">g</mi><mo></mo><mrow><mo>(</mo><mi href="./7.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow><mo>±</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mrow><mi href="./7.2#E10" mathvariant="normal">f</mi><mo></mo><mrow><mo>(</mo><mi href="./7.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>=</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>±</mo><mi mathvariant="normal">i</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><msup><mi href="./7.5#E7">ζ</mi><mn>2</mn></msup></msup><mo></mo><mrow><mi href="./7.2#E2">erfc</mi><mo></mo><mi href="./7.5#E7">ζ</mi></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E10.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./7.2#E10" title="(7.2.10) ‣ §7.2(iv) Auxiliary Functions ‣ §7.2 Definitions ‣ Properties ‣ Chapter 7 Error Functions, Dawson’s and Fresnel Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m10.png" altimg-height="23px" altimg-valign="-7px" altimg-width="41px" alttext="\mathrm{f}\left(\NVar{z}\right)" display="inline"><mrow><mi href="./7.2#E10" mathvariant="normal">f</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./7.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: auxiliary function for Fresnel integrals</a>,
<a href="./7.2#E11" title="(7.2.11) ‣ §7.2(iv) Auxiliary Functions ‣ §7.2 Definitions ‣ Properties ‣ Chapter 7 Error Functions, Dawson’s and Fresnel Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m11.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="\mathrm{g}\left(\NVar{z}\right)" display="inline"><mrow><mi href="./7.2#E11" mathvariant="normal">g</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./7.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: auxiliary function for Fresnel integrals</a>,
<a href="./7.2#E2" title="(7.2.2) ‣ §7.2(i) Error Functions ‣ §7.2 Definitions ‣ Properties ‣ Chapter 7 Error Functions, Dawson’s and Fresnel Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m12.png" altimg-height="18px" altimg-valign="-2px" altimg-width="49px" alttext="\operatorname{erfc}\NVar{z}" display="inline"><mrow><mi href="./7.2#E2">erfc</mi><mo></mo><mi class="ltx_nvar" href="./7.1#p1.t1.r2">z</mi></mrow></math>: complementary error function</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m9.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./7.1#p1.t1.r2" title="§7.1 Special Notation ‣ Notation ‣ Chapter 7 Error Functions, Dawson’s and Fresnel Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./7.1#p1.t1.r2">z</mi></math>: complex variable</a> and
<a href="./7.5#E7" title="(7.5.7) ‣ §7.5 Interrelations ‣ Properties ‣ Chapter 7 Error Functions, Dawson’s and Fresnel Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m16.png" altimg-height="21px" altimg-valign="-6px" altimg-width="14px" alttext="\zeta" display="inline"><mi href="./7.5#E7">ζ</mi></math>: change of variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">7.3.22</span> (in different form)</span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E11" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">7.5.11</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E11.png" altimg-height="28px" altimg-valign="-7px" altimg-width="232px" alttext="|\mathcal{F}\left(x\right)|^{2}={\mathrm{f}^{2}}\left(x\right)+{\mathrm{g}^{2}%
}\left(x\right)," display="block"><mrow><mrow><msup><mrow><mo stretchy="false">|</mo><mrow><mi class="ltx_font_mathcaligraphic" href="./7.2#E6">ℱ</mi><mo></mo><mrow><mo>(</mo><mi href="./7.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow><mn>2</mn></msup><mo>=</mo><mrow><mrow><msup><mi href="./7.2#E10" mathvariant="normal">f</mi><mn>2</mn></msup><mo></mo><mrow><mo>(</mo><mi href="./7.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow><mo>+</mo><mrow><msup><mi href="./7.2#E11" mathvariant="normal">g</mi><mn>2</mn></msup><mo></mo><mrow><mo>(</mo><mi href="./7.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m20.png" altimg-height="19px" altimg-valign="-5px" altimg-width="52px" alttext="x\geq 0" display="inline"><mrow><mi href="./7.1#p1.t1.r1">x</mi><mo>≥</mo><mn>0</mn></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E11.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./7.2#E10" title="(7.2.10) ‣ §7.2(iv) Auxiliary Functions ‣ §7.2 Definitions ‣ Properties ‣ Chapter 7 Error Functions, Dawson’s and Fresnel Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m10.png" altimg-height="23px" altimg-valign="-7px" altimg-width="41px" alttext="\mathrm{f}\left(\NVar{z}\right)" display="inline"><mrow><mi href="./7.2#E10" mathvariant="normal">f</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./7.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: auxiliary function for Fresnel integrals</a>,
<a href="./7.2#E11" title="(7.2.11) ‣ §7.2(iv) Auxiliary Functions ‣ §7.2 Definitions ‣ Properties ‣ Chapter 7 Error Functions, Dawson’s and Fresnel Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m11.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="\mathrm{g}\left(\NVar{z}\right)" display="inline"><mrow><mi href="./7.2#E11" mathvariant="normal">g</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./7.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: auxiliary function for Fresnel integrals</a>,
<a href="./7.2#E6" title="(7.2.6) ‣ §7.2(iii) Fresnel Integrals ‣ §7.2 Definitions ‣ Properties ‣ Chapter 7 Error Functions, Dawson’s and Fresnel Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m6.png" altimg-height="23px" altimg-valign="-7px" altimg-width="50px" alttext="\mathcal{F}\left(\NVar{z}\right)" display="inline"><mrow><mi class="ltx_font_mathcaligraphic" href="./7.2#E6">ℱ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./7.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: Fresnel integral</a> and
<a href="./7.1#p1.t1.r1" title="§7.1 Special Notation ‣ Notation ‣ Chapter 7 Error Functions, Dawson’s and Fresnel Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./7.1#p1.t1.r1">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E12" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">7.5.12</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_Math ltx_eqn_cell ltx_align_center"><math class="ltx_Math" alttext="|\mathcal{F}\left(x\right)|^{2}=2+{\mathrm{f}^{2}}\left(-x\right)+{\mathrm{g}^%
{2}}\left(-x\right)-2\sqrt{2}\cos\left(\tfrac{1}{4}\pi+\tfrac{1}{2}\pi x^{2}%
\right)\mathrm{f}\left(-x\right)-2\sqrt{2}\cos\left(\tfrac{1}{4}\pi-\tfrac{1}{%
2}\pi x^{2}\right)\mathrm{g}\left(-x\right)," display="block"><mrow><msup><mrow><mo stretchy="false">|</mo><mrow><mi class="ltx_font_mathcaligraphic" href="./7.2#E6">ℱ</mi><mo></mo><mrow><mo>(</mo><mi href="./7.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow><mn>2</mn></msup><mo>=</mo><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mrow><mn>2</mn><mo>+</mo><mrow><msup><mi href="./7.2#E10" mathvariant="normal">f</mi><mn>2</mn></msup><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mi href="./7.1#p1.t1.r1">x</mi></mrow><mo>)</mo></mrow></mrow><mo>+</mo><mrow><msup><mi href="./7.2#E11" mathvariant="normal">g</mi><mn>2</mn></msup><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mi href="./7.1#p1.t1.r1">x</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>-</mo><mrow><mn>2</mn><mo></mo><msqrt><mn>2</mn></msqrt><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>4</mn></mfrac></mstyle><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo>+</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><msup><mi href="./7.1#p1.t1.r1">x</mi><mn>2</mn></msup></mrow></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./7.2#E10" mathvariant="normal">f</mi><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mi href="./7.1#p1.t1.r1">x</mi></mrow><mo>)</mo></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>-</mo><mrow><mn>2</mn><mo></mo><msqrt><mn>2</mn></msqrt><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>4</mn></mfrac></mstyle><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo>-</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><msup><mi href="./7.1#p1.t1.r1">x</mi><mn>2</mn></msup></mrow></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./7.2#E11" mathvariant="normal">g</mi><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mi href="./7.1#p1.t1.r1">x</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mtd></mtr></mtable></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m21.png" altimg-height="19px" altimg-valign="-5px" altimg-width="52px" alttext="x\leq 0" display="inline"><mrow><mi href="./7.1#p1.t1.r1">x</mi><mo>≤</mo><mn>0</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E12.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./7.2#E10" title="(7.2.10) ‣ §7.2(iv) Auxiliary Functions ‣ §7.2 Definitions ‣ Properties ‣ Chapter 7 Error Functions, Dawson’s and Fresnel Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m10.png" altimg-height="23px" altimg-valign="-7px" altimg-width="41px" alttext="\mathrm{f}\left(\NVar{z}\right)" display="inline"><mrow><mi href="./7.2#E10" mathvariant="normal">f</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./7.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: auxiliary function for Fresnel integrals</a>,
<a href="./7.2#E11" title="(7.2.11) ‣ §7.2(iv) Auxiliary Functions ‣ §7.2 Definitions ‣ Properties ‣ Chapter 7 Error Functions, Dawson’s and Fresnel Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m11.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="\mathrm{g}\left(\NVar{z}\right)" display="inline"><mrow><mi href="./7.2#E11" mathvariant="normal">g</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./7.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: auxiliary function for Fresnel integrals</a>,
<a href="./7.2#E6" title="(7.2.6) ‣ §7.2(iii) Fresnel Integrals ‣ §7.2 Definitions ‣ Properties ‣ Chapter 7 Error Functions, Dawson’s and Fresnel Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m6.png" altimg-height="23px" altimg-valign="-7px" altimg-width="50px" alttext="\mathcal{F}\left(\NVar{z}\right)" display="inline"><mrow><mi class="ltx_font_mathcaligraphic" href="./7.2#E6">ℱ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./7.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: Fresnel integral</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m14.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m5.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a> and
<a href="./7.1#p1.t1.r1" title="§7.1 Special Notation ‣ Notation ‣ Chapter 7 Error Functions, Dawson’s and Fresnel Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./7.1#p1.t1.r1">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">See Figure .
</p>
<table id="E13" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">7.5.13</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E13.png" altimg-height="33px" altimg-valign="-9px" altimg-width="299px" alttext="G\left(x\right)=\sqrt{\pi}F\left(x\right)-\tfrac{1}{2}e^{-x^{2}}\mathrm{Ei}%
\left(x^{2}\right)," display="block"><mrow><mrow><mrow><mi href="./7.2#E12">G</mi><mo></mo><mrow><mo>(</mo><mi href="./7.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mrow><msqrt><mi href="./3.12#E1">π</mi></msqrt><mo></mo><mrow><mi href="./7.2#E5">F</mi><mo></mo><mrow><mo>(</mo><mi href="./7.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow></mrow><mo>-</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><msup><mi href="./7.1#p1.t1.r1">x</mi><mn>2</mn></msup></mrow></msup><mo></mo><mrow><mi href="./6.2#SS1.p3">Ei</mi><mo></mo><mrow><mo>(</mo><msup><mi href="./7.1#p1.t1.r1">x</mi><mn>2</mn></msup><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m18.png" altimg-height="17px" altimg-valign="-3px" altimg-width="52px" alttext="x>0" display="inline"><mrow><mi href="./7.1#p1.t1.r1">x</mi><mo>></mo><mn>0</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E13.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./7.2#E5" title="(7.2.5) ‣ §7.2(ii) Dawson’s Integral ‣ §7.2 Definitions ‣ Properties ‣ Chapter 7 Error Functions, Dawson’s and Fresnel Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m2.png" altimg-height="23px" altimg-valign="-7px" altimg-width="49px" alttext="F\left(\NVar{z}\right)" display="inline"><mrow><mi href="./7.2#E5">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./7.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: Dawson’s integral</a>,
<a href="./7.2#E12" title="(7.2.12) ‣ §7.2(v) Goodwin–Staton Integral ‣ §7.2 Definitions ‣ Properties ‣ Chapter 7 Error Functions, Dawson’s and Fresnel Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m3.png" altimg-height="23px" altimg-valign="-7px" altimg-width="49px" alttext="G\left(\NVar{z}\right)" display="inline"><mrow><mi href="./7.2#E12">G</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./7.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: Goodwin–Staton integral</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m14.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m9.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./6.2#SS1.p3" title="§6.2(i) Exponential and Logarithmic Integrals ‣ §6.2 Definitions and Interrelations ‣ Properties ‣ Chapter 6 Exponential, Logarithmic, Sine, and Cosine Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m7.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="\mathrm{Ei}\left(\NVar{x}\right)" display="inline"><mrow><mi href="./6.2#SS1.p3">Ei</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./6.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow></math>: exponential integral</a> and
<a href="./7.1#p1.t1.r1" title="§7.1 Special Notation ‣ Notation ‣ Chapter 7 Error Functions, Dawson’s and Fresnel Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./7.1#p1.t1.r1">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">For <math class="ltx_Math" altimg="m8.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="\mathrm{Ei}\left(x\right)" display="inline"><mrow><mi href="./6.2#SS1.p3">Ei</mi><mo></mo><mrow><mo>(</mo><mi href="./7.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow></math> see §</div>
</div>
</body></text>
</html>
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<title>DLMF: 5.5 Functional Relations</title>
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<div class="ltx_page_navlogo"></dd>
</dl>
</div>
</div>
<div id="SS1.p1" class="ltx_para">
<table id="E1" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">5.5.1</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E1.png" altimg-height="25px" altimg-valign="-7px" altimg-width="169px" alttext="\Gamma\left(z+1\right)=z\Gamma\left(z\right)," display="block"><mrow><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./5.1#p2.t1.r4">z</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mi href="./5.1#p2.t1.r4">z</mi><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m4.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a> and
<a href="./5.1#p2.t1.r4" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./5.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">6.1.15</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E2" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">5.5.2</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E2.png" altimg-height="46px" altimg-valign="-16px" altimg-width="197px" alttext="\psi\left(z+1\right)=\psi\left(z\right)+\frac{1}{z}." display="block"><mrow><mrow><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./5.1#p2.t1.r4">z</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mi href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow><mo>+</mo><mfrac><mn>1</mn><mi href="./5.1#p2.t1.r4">z</mi></mfrac></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E2" title="(5.2.2) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m8.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="\psi\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: psi (or digamma) function</a> and
<a href="./5.1#p2.t1.r4" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./5.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">6.3.5</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="ii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§5.5(ii) </span>Reflection</h2>
<div id="SS2.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="SS2.p1" class="ltx_para">
<table id="E3" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">5.5.3</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E3.png" altimg-height="25px" altimg-valign="-7px" altimg-width="253px" alttext="\Gamma\left(z\right)\Gamma\left(1-z\right)=\pi/\sin\left(\pi z\right)," display="block"><mrow><mrow><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mi href="./5.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mrow><mi href="./3.12#E1">π</mi><mo>/</mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi href="./5.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m20.png" altimg-height="21px" altimg-valign="-6px" altimg-width="117px" alttext="z\neq 0,\pm 1,\dots" display="inline"><mrow><mi href="./5.1#p2.t1.r4">z</mi><mo>≠</mo><mrow><mn>0</mn><mo>,</mo><mrow><mo>±</mo><mn>1</mn></mrow><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m4.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m7.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m9.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a> and
<a href="./5.1#p2.t1.r4" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./5.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">6.1.17</span> (without the condition on <math class="ltx_Math" altimg="m19.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./5.1#p2.t1.r4">z</mi></math>.)</span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E4" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">5.5.4</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E4.png" altimg-height="25px" altimg-valign="-7px" altimg-width="296px" alttext="\psi\left(z\right)-\psi\left(1-z\right)=-\pi/\tan\left(\pi z\right)," display="block"><mrow><mrow><mrow><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mi href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow><mo>-</mo><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mi href="./5.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mrow><mo>-</mo><mrow><mi href="./3.12#E1">π</mi><mo>/</mo><mrow><mi href="./4.14#E4">tan</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi href="./5.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m20.png" altimg-height="21px" altimg-valign="-6px" altimg-width="117px" alttext="z\neq 0,\pm 1,\dots" display="inline"><mrow><mi href="./5.1#p2.t1.r4">z</mi><mo>≠</mo><mrow><mn>0</mn><mo>,</mo><mrow><mo>±</mo><mn>1</mn></mrow><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E4.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m7.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./5.2#E2" title="(5.2.2) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m8.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="\psi\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: psi (or digamma) function</a>,
<a href="./4.14#E4" title="(4.14.4) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m10.png" altimg-height="17px" altimg-valign="-2px" altimg-width="47px" alttext="\tan\NVar{z}" display="inline"><mrow><mi href="./4.14#E4">tan</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: tangent function</a> and
<a href="./5.1#p2.t1.r4" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./5.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">6.3.7</span> (without the condition on <math class="ltx_Math" altimg="m19.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./5.1#p2.t1.r4">z</mi></math>.)</span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="iii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§5.5(iii) </span>Multiplication</h2>
<div id="SS3.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px1.p1" class="ltx_para">
<p class="ltx_p">For <math class="ltx_Math" altimg="m2.png" altimg-height="21px" altimg-valign="-6px" altimg-width="162px" alttext="2z\neq 0,-1,-2,\dots" display="inline"><mrow><mrow><mn>2</mn><mo></mo><mi href="./5.1#p2.t1.r4">z</mi></mrow><mo>≠</mo><mrow><mn>0</mn><mo>,</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo>,</mo><mrow><mo>-</mo><mn>2</mn></mrow><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>,
</p>
<table id="E5" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">5.5.5</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E5.png" altimg-height="31px" altimg-valign="-9px" altimg-width="316px" alttext="\Gamma\left(2z\right)=\pi^{-1/2}2^{2z-1}\Gamma\left(z\right)\Gamma\left(z+%
\tfrac{1}{2}\right)." display="block"><mrow><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><mi href="./5.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msup><mi href="./3.12#E1">π</mi><mrow><mo>-</mo><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></mrow></msup><mo></mo><msup><mn>2</mn><mrow><mrow><mn>2</mn><mo></mo><mi href="./5.1#p2.t1.r4">z</mi></mrow><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./5.1#p2.t1.r4">z</mi><mo>+</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E5.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m4.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m7.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a> and
<a href="./5.1#p2.t1.r4" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./5.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">6.1.18</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px2" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Gauss’s Multiplication Formula</h3>
<div id="Px2.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px2.p1" class="ltx_para">
<p class="ltx_p">For <math class="ltx_Math" altimg="m17.png" altimg-height="21px" altimg-valign="-6px" altimg-width="164px" alttext="nz\neq 0,-1,-2,\dots" display="inline"><mrow><mrow><mi href="./5.1#p2.t1.r1">n</mi><mo></mo><mi href="./5.1#p2.t1.r4">z</mi></mrow><mo>≠</mo><mrow><mn>0</mn><mo>,</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo>,</mo><mrow><mo>-</mo><mn>2</mn></mrow><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>,</p>
<table id="E6" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">5.5.6</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E6.png" altimg-height="67px" altimg-valign="-28px" altimg-width="405px" alttext="\Gamma\left(nz\right)=(2\pi)^{(1-n)/2}n^{nz-(1/2)}\prod_{k=0}^{n-1}\Gamma\left%
(z+\frac{k}{n}\right)." display="block"><mrow><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./5.1#p2.t1.r1">n</mi><mo></mo><mi href="./5.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><mi href="./5.1#p2.t1.r1">n</mi></mrow><mo stretchy="false">)</mo></mrow><mo>/</mo><mn>2</mn></mrow></msup><mo></mo><msup><mi href="./5.1#p2.t1.r1">n</mi><mrow><mrow><mi href="./5.1#p2.t1.r1">n</mi><mo></mo><mi href="./5.1#p2.t1.r4">z</mi></mrow><mo>-</mo><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow><mo stretchy="false">)</mo></mrow></mrow></msup><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∏</mo><mrow><mi href="./5.1#p2.t1.r2">k</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi href="./5.1#p2.t1.r1">n</mi><mo>-</mo><mn>1</mn></mrow></munderover><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./5.1#p2.t1.r4">z</mi><mo>+</mo><mfrac><mi href="./5.1#p2.t1.r2">k</mi><mi href="./5.1#p2.t1.r1">n</mi></mfrac></mrow><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E6.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m4.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m7.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./5.1#p2.t1.r1" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m16.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./5.1#p2.t1.r1">n</mi></math>: nonnegative integer</a>,
<a href="./5.1#p2.t1.r2" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m15.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./5.1#p2.t1.r2">k</mi></math>: nonnegative integer</a> and
<a href="./5.1#p2.t1.r4" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./5.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">6.1.20</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E7" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">5.5.7</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E7.png" altimg-height="67px" altimg-valign="-28px" altimg-width="284px" alttext="\prod_{k=1}^{n-1}\Gamma\left(\frac{k}{n}\right)=(2\pi)^{(n-1)/2}n^{-1/2}." display="block"><mrow><mrow><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∏</mo><mrow><mi href="./5.1#p2.t1.r2">k</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi href="./5.1#p2.t1.r1">n</mi><mo>-</mo><mn>1</mn></mrow></munderover><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mfrac><mi href="./5.1#p2.t1.r2">k</mi><mi href="./5.1#p2.t1.r1">n</mi></mfrac><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./5.1#p2.t1.r1">n</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo>/</mo><mn>2</mn></mrow></msup><mo></mo><msup><mi href="./5.1#p2.t1.r1">n</mi><mrow><mo>-</mo><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></mrow></msup></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E7.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m4.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m7.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./5.1#p2.t1.r1" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m16.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./5.1#p2.t1.r1">n</mi></math>: nonnegative integer</a> and
<a href="./5.1#p2.t1.r2" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m15.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./5.1#p2.t1.r2">k</mi></math>: nonnegative integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E8" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">5.5.8</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E8.png" altimg-height="29px" altimg-valign="-9px" altimg-width="329px" alttext="\psi\left(2z\right)=\tfrac{1}{2}\left(\psi\left(z\right)+\psi\left(z+\tfrac{1}%
{2}\right)\right)+\ln 2," display="block"><mrow><mrow><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><mi href="./5.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mrow><mo>(</mo><mrow><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mi href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow><mo>+</mo><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./5.1#p2.t1.r4">z</mi><mo>+</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle></mrow><mo>)</mo></mrow></mrow></mrow><mo>)</mo></mrow></mrow><mo>+</mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mn>2</mn></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E8.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E2" title="(5.2.2) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m8.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="\psi\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: psi (or digamma) function</a>,
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m6.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a> and
<a href="./5.1#p2.t1.r4" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./5.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">6.3.8</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E9" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">5.5.9</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E9.png" altimg-height="67px" altimg-valign="-28px" altimg-width="300px" alttext="\psi\left(nz\right)=\frac{1}{n}\sum_{k=0}^{n-1}\psi\left(z+\frac{k}{n}\right)+%
\ln n." display="block"><mrow><mrow><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./5.1#p2.t1.r1">n</mi><mo></mo><mi href="./5.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mfrac><mn>1</mn><mi href="./5.1#p2.t1.r1">n</mi></mfrac><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./5.1#p2.t1.r2">k</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi href="./5.1#p2.t1.r1">n</mi><mo>-</mo><mn>1</mn></mrow></munderover><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./5.1#p2.t1.r4">z</mi><mo>+</mo><mfrac><mi href="./5.1#p2.t1.r2">k</mi><mi href="./5.1#p2.t1.r1">n</mi></mfrac></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>+</mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi href="./5.1#p2.t1.r1">n</mi></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E9.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E2" title="(5.2.2) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m8.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="\psi\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: psi (or digamma) function</a>,
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m6.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a>,
<a href="./5.1#p2.t1.r1" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m16.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./5.1#p2.t1.r1">n</mi></math>: nonnegative integer</a>,
<a href="./5.1#p2.t1.r2" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m15.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./5.1#p2.t1.r2">k</mi></math>: nonnegative integer</a> and
<a href="./5.1#p2.t1.r4" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./5.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="Px2.p2" class="ltx_para">
<p class="ltx_p">See also <cite class="ltx_cite ltx_citemacro_citet">Sándor and Tóth (</dd>
</dl>
</div>
</div>
<div id="SS4.p1" class="ltx_para">
<p class="ltx_p">If a positive function <math class="ltx_Math" altimg="m13.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="f(x)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./5.1#p2.t1.r3">x</mi><mo stretchy="false">)</mo></mrow></mrow></math> on <math class="ltx_Math" altimg="m1.png" altimg-height="23px" altimg-valign="-7px" altimg-width="59px" alttext="(0,\infty)" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mn>0</mn><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi mathvariant="normal">∞</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math> satisfies <math class="ltx_Math" altimg="m14.png" altimg-height="23px" altimg-valign="-7px" altimg-width="154px" alttext="f(x+1)=xf(x)" display="inline"><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./5.1#p2.t1.r3">x</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mi href="./5.1#p2.t1.r3">x</mi><mo></mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./5.1#p2.t1.r3">x</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow></math>,
<math class="ltx_Math" altimg="m11.png" altimg-height="23px" altimg-valign="-7px" altimg-width="78px" alttext="f(1)=1" display="inline"><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mn>1</mn></mrow></math>, and <math class="ltx_Math" altimg="m5.png" altimg-height="23px" altimg-valign="-7px" altimg-width="63px" alttext="\ln f(x)" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./5.1#p2.t1.r3">x</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></math> is convex (see §), then
<math class="ltx_Math" altimg="m12.png" altimg-height="23px" altimg-valign="-7px" altimg-width="112px" alttext="f(x)=\Gamma\left(x\right)" display="inline"><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./5.1#p2.t1.r3">x</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi href="./5.1#p2.t1.r3">x</mi><mo>)</mo></mrow></mrow></mrow></math>.</p>
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<title>DLMF: 5.7 Series Expansions</title>
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<div id="SS1.p1" class="ltx_para">
<p class="ltx_p">Throughout this subsection <math class="ltx_Math" altimg="m16.png" altimg-height="23px" altimg-valign="-7px" altimg-width="44px" alttext="\zeta\left(k\right)" display="inline"><mrow><mi href="./25.2#E1">ζ</mi><mo></mo><mrow><mo>(</mo><mi href="./5.1#p2.t1.r2">k</mi><mo>)</mo></mrow></mrow></math> is as in Chapter .</p>
<table id="E1" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">5.7.1</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E1.png" altimg-height="64px" altimg-valign="-28px" altimg-width="155px" alttext="\frac{1}{\Gamma\left(z\right)}=\sum_{k=1}^{\infty}c_{k}z^{k}," display="block"><mrow><mrow><mfrac><mn>1</mn><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></mfrac><mo>=</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./5.1#p2.t1.r2">k</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><msub><mi href="./5.7#SS1.p1">c</mi><mi href="./5.1#p2.t1.r2">k</mi></msub><mo></mo><msup><mi href="./5.1#p2.t1.r4">z</mi><mi href="./5.1#p2.t1.r2">k</mi></msup></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
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<div id="E1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m7.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./5.1#p2.t1.r2" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./5.1#p2.t1.r2">k</mi></math>: nonnegative integer</a>,
<a href="./5.1#p2.t1.r4" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m24.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./5.1#p2.t1.r4">z</mi></math>: complex variable</a> and
<a href="./5.7#SS1.p1" title="§5.7(i) Maclaurin and Taylor Series ‣ §5.7 Series Expansions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="16px" altimg-valign="-5px" altimg-width="23px" alttext="c_{k}" display="inline"><msub><mi href="./5.7#SS1.p1">c</mi><mi href="./5.1#p2.t1.r2">k</mi></msub></math>: coefficient</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">6.1.34</span></span>
</dd>
</dl>
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</td></tr>
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<p class="ltx_p">where <math class="ltx_Math" altimg="m17.png" altimg-height="20px" altimg-valign="-5px" altimg-width="58px" alttext="c_{1}=1" display="inline"><mrow><msub><mi href="./5.7#SS1.p1">c</mi><mn>1</mn></msub><mo>=</mo><mn>1</mn></mrow></math>, <math class="ltx_Math" altimg="m18.png" altimg-height="16px" altimg-valign="-6px" altimg-width="60px" alttext="c_{2}=\gamma" display="inline"><mrow><msub><mi href="./5.7#SS1.p1">c</mi><mn>2</mn></msub><mo>=</mo><mi href="./5.2#E3">γ</mi></mrow></math>, and</p>
<table id="E2" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">5.7.2</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E2.png" altimg-height="28px" altimg-valign="-7px" altimg-width="596px" alttext="(k-1)c_{k}=\gamma c_{k-1}-\zeta\left(2\right)c_{k-2}+\zeta\left(3\right)c_{k-3%
}-\dots+(-1)^{k}\zeta\left(k-1\right)c_{1}," display="block"><mrow><mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./5.1#p2.t1.r2">k</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msub><mi href="./5.7#SS1.p1">c</mi><mi href="./5.1#p2.t1.r2">k</mi></msub></mrow><mo>=</mo><mrow><mrow><mrow><mrow><mrow><mi href="./5.2#E3">γ</mi><mo></mo><msub><mi href="./5.7#SS1.p1">c</mi><mrow><mi href="./5.1#p2.t1.r2">k</mi><mo>-</mo><mn>1</mn></mrow></msub></mrow><mo>-</mo><mrow><mrow><mi href="./25.2#E1">ζ</mi><mo></mo><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow></mrow><mo></mo><msub><mi href="./5.7#SS1.p1">c</mi><mrow><mi href="./5.1#p2.t1.r2">k</mi><mo>-</mo><mn>2</mn></mrow></msub></mrow></mrow><mo>+</mo><mrow><mrow><mi href="./25.2#E1">ζ</mi><mo></mo><mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow></mrow><mo></mo><msub><mi href="./5.7#SS1.p1">c</mi><mrow><mi href="./5.1#p2.t1.r2">k</mi><mo>-</mo><mn>3</mn></mrow></msub></mrow></mrow><mo>-</mo><mi mathvariant="normal">…</mi></mrow><mo>+</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./5.1#p2.t1.r2">k</mi></msup><mo></mo><mrow><mi href="./25.2#E1">ζ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./5.1#p2.t1.r2">k</mi><mo>-</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mo></mo><msub><mi href="./5.7#SS1.p1">c</mi><mn>1</mn></msub></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
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<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m21.png" altimg-height="20px" altimg-valign="-5px" altimg-width="52px" alttext="k\geq 3" display="inline"><mrow><mi href="./5.1#p2.t1.r2">k</mi><mo>≥</mo><mn>3</mn></mrow></math>.</span></td></tr>
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<div id="E2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E3" title="(5.2.3) ‣ §5.2(ii) Euler’s Constant ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m11.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\gamma" display="inline"><mi href="./5.2#E3">γ</mi></math>: Euler’s constant</a>,
<a href="./25.2#E1" title="(25.2.1) ‣ §25.2(i) Definition ‣ §25.2 Definition and Expansions ‣ Riemann Zeta Function ‣ Chapter 25 Zeta and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m15.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="\zeta\left(\NVar{s}\right)" display="inline"><mrow><mi href="./25.2#E1">ζ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./25.1#p2.t1.r5">s</mi><mo>)</mo></mrow></mrow></math>: Riemann zeta function</a>,
<a href="./5.1#p2.t1.r2" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./5.1#p2.t1.r2">k</mi></math>: nonnegative integer</a> and
<a href="./5.7#SS1.p1" title="§5.7(i) Maclaurin and Taylor Series ‣ §5.7 Series Expansions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="16px" altimg-valign="-5px" altimg-width="23px" alttext="c_{k}" display="inline"><msub><mi href="./5.7#SS1.p1">c</mi><mi href="./5.1#p2.t1.r2">k</mi></msub></math>: coefficient</a>
</dd>
</dl>
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</td></tr>
</table>
<p class="ltx_p">For 15D numerical values of <math class="ltx_Math" altimg="m19.png" altimg-height="16px" altimg-valign="-5px" altimg-width="23px" alttext="c_{k}" display="inline"><msub><mi href="./5.7#SS1.p1">c</mi><mi href="./5.1#p2.t1.r2">k</mi></msub></math> see <cite class="ltx_cite ltx_citemacro_citet">Abramowitz and Stegun ()</cite>.</p>
<table id="EGx1" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E3">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">5.7.3</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m4.png" altimg-height="25px" altimg-valign="-7px" altimg-width="102px" alttext="\displaystyle\ln\Gamma\left(1+z\right)" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mi href="./5.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m2.png" altimg-height="64px" altimg-valign="-28px" altimg-width="446px" alttext="\displaystyle=-\ln\left(1+z\right)+z(1-\gamma)+\sum_{k=2}^{\infty}(-1)^{k}(%
\zeta\left(k\right)-1)\frac{z^{k}}{k}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mo>-</mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mi href="./5.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mi href="./5.1#p2.t1.r4">z</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><mi href="./5.2#E3">γ</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>+</mo><mrow><mstyle displaystyle="true"><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./5.1#p2.t1.r2">k</mi><mo>=</mo><mn>2</mn></mrow><mi mathvariant="normal">∞</mi></munderover></mstyle><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./5.1#p2.t1.r2">k</mi></msup><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mi href="./25.2#E1">ζ</mi><mo></mo><mrow><mo>(</mo><mi href="./5.1#p2.t1.r2">k</mi><mo>)</mo></mrow></mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mstyle displaystyle="true"><mfrac><msup><mi href="./5.1#p2.t1.r4">z</mi><mi href="./5.1#p2.t1.r2">k</mi></msup><mi href="./5.1#p2.t1.r2">k</mi></mfrac></mstyle></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m28.png" altimg-height="23px" altimg-valign="-7px" altimg-width="62px" alttext="|z|<2" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mi href="./5.1#p2.t1.r4">z</mi><mo stretchy="false">|</mo></mrow><mo><</mo><mn>2</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m7.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./5.2#E3" title="(5.2.3) ‣ §5.2(ii) Euler’s Constant ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m11.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\gamma" display="inline"><mi href="./5.2#E3">γ</mi></math>: Euler’s constant</a>,
<a href="./25.2#E1" title="(25.2.1) ‣ §25.2(i) Definition ‣ §25.2 Definition and Expansions ‣ Riemann Zeta Function ‣ Chapter 25 Zeta and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m15.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="\zeta\left(\NVar{s}\right)" display="inline"><mrow><mi href="./25.2#E1">ζ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./25.1#p2.t1.r5">s</mi><mo>)</mo></mrow></mrow></math>: Riemann zeta function</a>,
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m12.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a>,
<a href="./5.1#p2.t1.r2" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./5.1#p2.t1.r2">k</mi></math>: nonnegative integer</a> and
<a href="./5.1#p2.t1.r4" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m24.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./5.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">6.1.33</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E4">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">5.7.4</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m5.png" altimg-height="25px" altimg-valign="-7px" altimg-width="83px" alttext="\displaystyle\psi\left(1+z\right)" display="inline"><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mi href="./5.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m1.png" altimg-height="64px" altimg-valign="-28px" altimg-width="247px" alttext="\displaystyle=-\gamma+\sum_{k=2}^{\infty}(-1)^{k}\zeta\left(k\right)z^{k-1}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mo>-</mo><mi href="./5.2#E3">γ</mi></mrow><mo>+</mo><mrow><mstyle displaystyle="true"><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./5.1#p2.t1.r2">k</mi><mo>=</mo><mn>2</mn></mrow><mi mathvariant="normal">∞</mi></munderover></mstyle><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./5.1#p2.t1.r2">k</mi></msup><mo></mo><mrow><mi href="./25.2#E1">ζ</mi><mo></mo><mrow><mo>(</mo><mi href="./5.1#p2.t1.r2">k</mi><mo>)</mo></mrow></mrow><mo></mo><msup><mi href="./5.1#p2.t1.r4">z</mi><mrow><mi href="./5.1#p2.t1.r2">k</mi><mo>-</mo><mn>1</mn></mrow></msup></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m27.png" altimg-height="23px" altimg-valign="-7px" altimg-width="62px" alttext="|z|<1" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mi href="./5.1#p2.t1.r4">z</mi><mo stretchy="false">|</mo></mrow><mo><</mo><mn>1</mn></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E4.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E3" title="(5.2.3) ‣ §5.2(ii) Euler’s Constant ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m11.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\gamma" display="inline"><mi href="./5.2#E3">γ</mi></math>: Euler’s constant</a>,
<a href="./25.2#E1" title="(25.2.1) ‣ §25.2(i) Definition ‣ §25.2 Definition and Expansions ‣ Riemann Zeta Function ‣ Chapter 25 Zeta and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m15.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="\zeta\left(\NVar{s}\right)" display="inline"><mrow><mi href="./25.2#E1">ζ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./25.1#p2.t1.r5">s</mi><mo>)</mo></mrow></mrow></math>: Riemann zeta function</a>,
<a href="./5.2#E2" title="(5.2.2) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m14.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="\psi\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: psi (or digamma) function</a>,
<a href="./5.1#p2.t1.r2" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./5.1#p2.t1.r2">k</mi></math>: nonnegative integer</a> and
<a href="./5.1#p2.t1.r4" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m24.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./5.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">6.3.14</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E5">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">5.7.5</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m5.png" altimg-height="25px" altimg-valign="-7px" altimg-width="83px" alttext="\displaystyle\psi\left(1+z\right)" display="inline"><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mi href="./5.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m3.png" altimg-height="64px" altimg-valign="-28px" altimg-width="538px" alttext="\displaystyle=\frac{1}{2z}-\frac{\pi}{2}\cot\left(\pi z\right)+\frac{1}{z^{2}-%
1}+1-\gamma-\sum_{k=1}^{\infty}(\zeta\left(2k+1\right)-1)z^{2k}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mrow><mstyle displaystyle="true"><mfrac><mn>1</mn><mrow><mn>2</mn><mo></mo><mi href="./5.1#p2.t1.r4">z</mi></mrow></mfrac></mstyle><mo>-</mo><mrow><mstyle displaystyle="true"><mfrac><mi href="./3.12#E1">π</mi><mn>2</mn></mfrac></mstyle><mo></mo><mrow><mi href="./4.14#E7">cot</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi href="./5.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>+</mo><mstyle displaystyle="true"><mfrac><mn>1</mn><mrow><msup><mi href="./5.1#p2.t1.r4">z</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow></mfrac></mstyle><mo>+</mo><mn>1</mn></mrow><mo>-</mo><mi href="./5.2#E3">γ</mi><mo>-</mo><mrow><mstyle displaystyle="true"><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./5.1#p2.t1.r2">k</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover></mstyle><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mi href="./25.2#E1">ζ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mn>2</mn><mo></mo><mi href="./5.1#p2.t1.r2">k</mi></mrow><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msup><mi href="./5.1#p2.t1.r4">z</mi><mrow><mn>2</mn><mo></mo><mi href="./5.1#p2.t1.r2">k</mi></mrow></msup></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m28.png" altimg-height="23px" altimg-valign="-7px" altimg-width="62px" alttext="|z|<2" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mi href="./5.1#p2.t1.r4">z</mi><mo stretchy="false">|</mo></mrow><mo><</mo><mn>2</mn></mrow></math>, <math class="ltx_Math" altimg="m26.png" altimg-height="21px" altimg-valign="-6px" altimg-width="85px" alttext="z\neq 0,\pm 1" display="inline"><mrow><mi href="./5.1#p2.t1.r4">z</mi><mo>≠</mo><mrow><mn>0</mn><mo>,</mo><mrow><mo>±</mo><mn>1</mn></mrow></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E5.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E3" title="(5.2.3) ‣ §5.2(ii) Euler’s Constant ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m11.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\gamma" display="inline"><mi href="./5.2#E3">γ</mi></math>: Euler’s constant</a>,
<a href="./25.2#E1" title="(25.2.1) ‣ §25.2(i) Definition ‣ §25.2 Definition and Expansions ‣ Riemann Zeta Function ‣ Chapter 25 Zeta and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m15.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="\zeta\left(\NVar{s}\right)" display="inline"><mrow><mi href="./25.2#E1">ζ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./25.1#p2.t1.r5">s</mi><mo>)</mo></mrow></mrow></math>: Riemann zeta function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m13.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.14#E7" title="(4.14.7) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m10.png" altimg-height="17px" altimg-valign="-2px" altimg-width="44px" alttext="\cot\NVar{z}" display="inline"><mrow><mi href="./4.14#E7">cot</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cotangent function</a>,
<a href="./5.2#E2" title="(5.2.2) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m14.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="\psi\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: psi (or digamma) function</a>,
<a href="./5.1#p2.t1.r2" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./5.1#p2.t1.r2">k</mi></math>: nonnegative integer</a> and
<a href="./5.1#p2.t1.r4" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m24.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./5.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">6.3.15</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
</div>
<div id="SS1.p2" class="ltx_para">
<p class="ltx_p">For 20D numerical values of the coefficients of the Maclaurin series for
<math class="ltx_Math" altimg="m8.png" altimg-height="23px" altimg-valign="-7px" altimg-width="80px" alttext="\Gamma\left(z+3\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./5.1#p2.t1.r4">z</mi><mo>+</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow></math> see <cite class="ltx_cite ltx_citemacro_citet">Luke (</dd>
</dl>
</div>
</div>
<div id="SS2.p1" class="ltx_para">
<p class="ltx_p">When <math class="ltx_Math" altimg="m25.png" altimg-height="21px" altimg-valign="-6px" altimg-width="152px" alttext="z\neq 0,-1,-2,\dots" display="inline"><mrow><mi href="./5.1#p2.t1.r4">z</mi><mo>≠</mo><mrow><mn>0</mn><mo>,</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo>,</mo><mrow><mo>-</mo><mn>2</mn></mrow><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>,</p>
<table id="E6" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">5.7.6</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E6.png" altimg-height="64px" altimg-valign="-28px" altimg-width="548px" alttext="\psi\left(z\right)=-\gamma-\frac{1}{z}+\sum_{k=1}^{\infty}\frac{z}{k(k+z)}=-%
\gamma+\sum_{k=0}^{\infty}\left(\frac{1}{k+1}-\frac{1}{k+z}\right)," display="block"><mrow><mrow><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mi href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mrow><mo>-</mo><mi href="./5.2#E3">γ</mi></mrow><mo>-</mo><mfrac><mn>1</mn><mi href="./5.1#p2.t1.r4">z</mi></mfrac></mrow><mo>+</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./5.1#p2.t1.r2">k</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mfrac><mi href="./5.1#p2.t1.r4">z</mi><mrow><mi href="./5.1#p2.t1.r2">k</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./5.1#p2.t1.r2">k</mi><mo>+</mo><mi href="./5.1#p2.t1.r4">z</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mfrac></mrow></mrow><mo>=</mo><mrow><mrow><mo>-</mo><mi href="./5.2#E3">γ</mi></mrow><mo>+</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./5.1#p2.t1.r2">k</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mrow><mi href="./5.1#p2.t1.r2">k</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mo>-</mo><mfrac><mn>1</mn><mrow><mi href="./5.1#p2.t1.r2">k</mi><mo>+</mo><mi href="./5.1#p2.t1.r4">z</mi></mrow></mfrac></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E6.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E3" title="(5.2.3) ‣ §5.2(ii) Euler’s Constant ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m11.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\gamma" display="inline"><mi href="./5.2#E3">γ</mi></math>: Euler’s constant</a>,
<a href="./5.2#E2" title="(5.2.2) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m14.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="\psi\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: psi (or digamma) function</a>,
<a href="./5.1#p2.t1.r2" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./5.1#p2.t1.r2">k</mi></math>: nonnegative integer</a> and
<a href="./5.1#p2.t1.r4" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m24.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./5.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and
</p>
<table id="E7" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">5.7.7</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E7.png" altimg-height="64px" altimg-valign="-28px" altimg-width="315px" alttext="\psi\left(\frac{z+1}{2}\right)-\psi\left(\frac{z}{2}\right)=2\sum_{k=0}^{%
\infty}\frac{(-1)^{k}}{k+z}." display="block"><mrow><mrow><mrow><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mfrac><mrow><mi href="./5.1#p2.t1.r4">z</mi><mo>+</mo><mn>1</mn></mrow><mn>2</mn></mfrac><mo>)</mo></mrow></mrow><mo>-</mo><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mfrac><mi href="./5.1#p2.t1.r4">z</mi><mn>2</mn></mfrac><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mrow><mn>2</mn><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./5.1#p2.t1.r2">k</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mfrac><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./5.1#p2.t1.r2">k</mi></msup><mrow><mi href="./5.1#p2.t1.r2">k</mi><mo>+</mo><mi href="./5.1#p2.t1.r4">z</mi></mrow></mfrac></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E7.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E2" title="(5.2.2) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m14.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="\psi\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: psi (or digamma) function</a>,
<a href="./5.1#p2.t1.r2" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./5.1#p2.t1.r2">k</mi></math>: nonnegative integer</a> and
<a href="./5.1#p2.t1.r4" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m24.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./5.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Also,</p>
<table id="E8" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">5.7.8</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E8.png" altimg-height="64px" altimg-valign="-28px" altimg-width="236px" alttext="\Im\psi\left(1+\mathrm{i}y\right)=\sum_{k=1}^{\infty}\frac{y}{k^{2}+y^{2}}." display="block"><mrow><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℑ</mi><mo></mo><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./5.1#p2.t1.r3">y</mi></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./5.1#p2.t1.r2">k</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mfrac><mi href="./5.1#p2.t1.r3">y</mi><mrow><msup><mi href="./5.1#p2.t1.r2">k</mi><mn>2</mn></msup><mo>+</mo><msup><mi href="./5.1#p2.t1.r3">y</mi><mn>2</mn></msup></mrow></mfrac></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E8.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E2" title="(5.2.2) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m14.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="\psi\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: psi (or digamma) function</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m9.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Im" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℑ</mi><mo></mo></mrow></math>: imaginary part</a>,
<a href="./5.1#p2.t1.r2" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./5.1#p2.t1.r2">k</mi></math>: nonnegative integer</a> and
<a href="./5.1#p2.t1.r3" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m23.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./5.1#p2.t1.r3">y</mi></math>: real variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">6.3.13</span></span>
</dd>
</dl>
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<title>DLMF: 5.10 Continued Fractions</title>
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<div id="p1" class="ltx_para">
<p class="ltx_p">For <math class="ltx_Math" altimg="m17.png" altimg-height="18px" altimg-valign="-3px" altimg-width="65px" alttext="\Re z>0" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./5.1#p2.t1.r4">z</mi></mrow><mo>></mo><mn>0</mn></mrow></math>,</p>
<table id="E1" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">5.10.1</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E1.png" altimg-height="53px" altimg-valign="-17px" altimg-width="646px" alttext="\operatorname{Ln}\Gamma\left(z\right)+z-\left(z-\tfrac{1}{2}\right)\ln z-%
\tfrac{1}{2}\ln\left(2\pi\right)=\cfrac{a_{0}}{z+\cfrac{a_{1}}{z+\cfrac{a_{2}}%
{z+\cfrac{a_{3}}{z+\cfrac{a_{4}}{z+\cfrac{a_{5}}{z+}}}}}}\cdots," display="block"><mrow><mrow><mrow><mrow><mrow><mi href="./4.2#E1">Ln</mi><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow><mo>+</mo><mi href="./5.1#p2.t1.r4">z</mi></mrow><mo>-</mo><mrow><mrow><mo>(</mo><mrow><mi href="./5.1#p2.t1.r4">z</mi><mo>-</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle></mrow><mo>)</mo></mrow><mo></mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi href="./5.1#p2.t1.r4">z</mi></mrow></mrow><mo>-</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>=</mo><mrow><mrow><mfrac><msub><mi href="./5.10#p1">a</mi><mn>0</mn></msub><mrow><mi href="./5.1#p2.t1.r4">z</mi><mo>+</mo></mrow></mfrac><mfrac><msub><mi href="./5.10#p1">a</mi><mn>1</mn></msub><mrow><mi href="./5.1#p2.t1.r4">z</mi><mo>+</mo></mrow></mfrac><mfrac><msub><mi href="./5.10#p1">a</mi><mn>2</mn></msub><mrow><mi href="./5.1#p2.t1.r4">z</mi><mo>+</mo></mrow></mfrac><mfrac><msub><mi href="./5.10#p1">a</mi><mn>3</mn></msub><mrow><mi href="./5.1#p2.t1.r4">z</mi><mo>+</mo></mrow></mfrac><mfrac><msub><mi href="./5.10#p1">a</mi><mn>4</mn></msub><mrow><mi href="./5.1#p2.t1.r4">z</mi><mo>+</mo></mrow></mfrac><mfrac><msub><mi href="./5.10#p1">a</mi><mn>5</mn></msub><mrow><mi href="./5.1#p2.t1.r4">z</mi><mo>+</mo></mrow></mfrac></mrow><mo></mo><mi mathvariant="normal">⋯</mi></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m16.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m24.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.2#E1" title="(4.2.1) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="18px" altimg-valign="-2px" altimg-width="41px" alttext="\operatorname{Ln}\NVar{z}" display="inline"><mrow><mi href="./4.2#E1">Ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: general logarithm function</a>,
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a>,
<a href="./5.1#p2.t1.r4" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m31.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./5.1#p2.t1.r4">z</mi></math>: complex variable</a> and
<a href="./5.10#p1" title="§5.10 Continued Fractions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="16px" altimg-valign="-5px" altimg-width="25px" alttext="a_{k}" display="inline"><msub><mi href="./5.10#p1">a</mi><mi href="./5.1#p2.t1.r2">k</mi></msub></math>: coefficient</a>
</dd>
<dt>Addition (effective with 1.0.10):</dt>
<dd>
To increase the region of validity of this equation, the logarithm of the
gamma function that appears on its left-hand side has been changed to
<math class="ltx_Math" altimg="m21.png" altimg-height="23px" altimg-valign="-7px" altimg-width="73px" alttext="\operatorname{Ln}\Gamma\left(z\right)" display="inline"><mrow><mi href="./4.2#E1">Ln</mi><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></math>, where <math class="ltx_Math" altimg="m23.png" altimg-height="18px" altimg-valign="-2px" altimg-width="28px" alttext="\operatorname{Ln}" display="inline"><mi href="./4.2#E1">Ln</mi></math> is the <span class="ltx_text ltx_font_italic">general logarithm</span>.
Originally <math class="ltx_Math" altimg="m18.png" altimg-height="23px" altimg-valign="-7px" altimg-width="66px" alttext="\ln\Gamma\left(z\right)" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></math> was used, where <math class="ltx_Math" altimg="m20.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="\ln" display="inline"><mi href="./4.2#E2">ln</mi></math>
is the <span class="ltx_text ltx_font_italic">principal branch</span> of the logarithm.
<p><span class="ltx_font_italic">Suggested 2015-02-13 by Philippe Spindel</span></p>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where
</p>
<table id="E2" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex1" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="7" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">5.10.2</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m8.png" altimg-height="17px" altimg-valign="-5px" altimg-width="26px" alttext="\displaystyle a_{0}" display="inline"><msub><mi href="./5.10#p1">a</mi><mn>0</mn></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m3.png" altimg-height="29px" altimg-valign="-9px" altimg-width="53px" alttext="\displaystyle=\tfrac{1}{12}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mfrac><mn>1</mn><mn>12</mn></mfrac></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex2" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m9.png" altimg-height="17px" altimg-valign="-5px" altimg-width="26px" alttext="\displaystyle a_{1}" display="inline"><msub><mi href="./5.10#p1">a</mi><mn>1</mn></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m4.png" altimg-height="29px" altimg-valign="-9px" altimg-width="53px" alttext="\displaystyle=\tfrac{1}{30}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mfrac><mn>1</mn><mn>30</mn></mfrac></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex3" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m10.png" altimg-height="17px" altimg-valign="-5px" altimg-width="26px" alttext="\displaystyle a_{2}" display="inline"><msub><mi href="./5.10#p1">a</mi><mn>2</mn></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m7.png" altimg-height="29px" altimg-valign="-9px" altimg-width="61px" alttext="\displaystyle=\tfrac{53}{210}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mfrac><mn>53</mn><mn>210</mn></mfrac></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex4" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m11.png" altimg-height="17px" altimg-valign="-5px" altimg-width="26px" alttext="\displaystyle a_{3}" display="inline"><msub><mi href="./5.10#p1">a</mi><mn>3</mn></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m2.png" altimg-height="29px" altimg-valign="-9px" altimg-width="61px" alttext="\displaystyle=\tfrac{195}{371}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mfrac><mn>195</mn><mn>371</mn></mfrac></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex5" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m12.png" altimg-height="17px" altimg-valign="-5px" altimg-width="26px" alttext="\displaystyle a_{4}" display="inline"><msub><mi href="./5.10#p1">a</mi><mn>4</mn></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m5.png" altimg-height="29px" altimg-valign="-9px" altimg-width="77px" alttext="\displaystyle=\tfrac{22999}{22737}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mfrac><mn>22999</mn><mn>22737</mn></mfrac></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex6" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m13.png" altimg-height="17px" altimg-valign="-5px" altimg-width="26px" alttext="\displaystyle a_{5}" display="inline"><msub><mi href="./5.10#p1">a</mi><mn>5</mn></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m6.png" altimg-height="29px" altimg-valign="-9px" altimg-width="105px" alttext="\displaystyle=\tfrac{299\;44523}{197\;33142}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mfrac><mn>299 44523</mn><mn>197 33142</mn></mfrac></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex7" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m14.png" altimg-height="17px" altimg-valign="-5px" altimg-width="26px" alttext="\displaystyle a_{6}" display="inline"><msub><mi href="./5.10#p1">a</mi><mn>6</mn></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m1.png" altimg-height="29px" altimg-valign="-9px" altimg-width="142px" alttext="\displaystyle=\tfrac{10\;95352\;41009}{4\;82642\;75462}." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mfrac><mn>10 95352 41009</mn><mn>4 82642 75462</mn></mfrac></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./5.10#p1" title="§5.10 Continued Fractions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="16px" altimg-valign="-5px" altimg-width="25px" alttext="a_{k}" display="inline"><msub><mi href="./5.10#p1">a</mi><mi href="./5.1#p2.t1.r2">k</mi></msub></math>: coefficient</a></dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">6.1.48</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">For exact values of <math class="ltx_Math" altimg="m28.png" altimg-height="16px" altimg-valign="-5px" altimg-width="24px" alttext="a_{7}" display="inline"><msub><mi href="./5.10#p1">a</mi><mn>7</mn></msub></math> to <math class="ltx_Math" altimg="m26.png" altimg-height="16px" altimg-valign="-5px" altimg-width="32px" alttext="a_{11}" display="inline"><msub><mi href="./5.10#p1">a</mi><mn>11</mn></msub></math> and 40S values of <math class="ltx_Math" altimg="m25.png" altimg-height="16px" altimg-valign="-5px" altimg-width="24px" alttext="a_{0}" display="inline"><msub><mi href="./5.10#p1">a</mi><mn>0</mn></msub></math> to <math class="ltx_Math" altimg="m27.png" altimg-height="16px" altimg-valign="-5px" altimg-width="32px" alttext="a_{40}" display="inline"><msub><mi href="./5.10#p1">a</mi><mn>40</mn></msub></math>,
see <cite class="ltx_cite ltx_citemacro_citet">Char (</div>
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<title>DLMF: 5.9 Integral Representations</title>
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<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">5.9.1</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E1.png" altimg-height="53px" altimg-valign="-21px" altimg-width="355px" alttext="\frac{1}{\mu}\Gamma\left(\frac{\nu}{\mu}\right)\frac{1}{z^{\nu/\mu}}=\int_{0}^%
{\infty}\exp\left(-zt^{\mu}\right)t^{\nu-1}\mathrm{d}t," display="block"><mrow><mrow><mrow><mfrac><mn>1</mn><mi>μ</mi></mfrac><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mfrac><mi>ν</mi><mi>μ</mi></mfrac><mo>)</mo></mrow></mrow><mo></mo><mfrac><mn>1</mn><msup><mi href="./5.1#p2.t1.r4">z</mi><mrow><mi>ν</mi><mo>/</mo><mi>μ</mi></mrow></msup></mfrac></mrow><mo>=</mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><mrow><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mrow><mi href="./5.1#p2.t1.r4">z</mi><mo></mo><msup><mi>t</mi><mi>μ</mi></msup></mrow></mrow><mo>)</mo></mrow></mrow><mo></mo><msup><mi>t</mi><mrow><mi>ν</mi><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m13.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m27.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m52.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E19" title="(4.2.19) ‣ §4.2(iii) The Exponential Function ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m18.png" altimg-height="16px" altimg-valign="-6px" altimg-width="48px" alttext="\exp\NVar{z}" display="inline"><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: exponential function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a> and
<a href="./5.1#p2.t1.r4" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./5.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p"><math class="ltx_Math" altimg="m16.png" altimg-height="18px" altimg-valign="-3px" altimg-width="66px" alttext="\Re\nu>0" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>ν</mi></mrow><mo>></mo><mn>0</mn></mrow></math>, <math class="ltx_Math" altimg="m29.png" altimg-height="20px" altimg-valign="-6px" altimg-width="53px" alttext="\mu>0" display="inline"><mrow><mi>μ</mi><mo>></mo><mn>0</mn></mrow></math>, and <math class="ltx_Math" altimg="m14.png" altimg-height="18px" altimg-valign="-3px" altimg-width="65px" alttext="\Re z>0" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./5.1#p2.t1.r4">z</mi></mrow><mo>></mo><mn>0</mn></mrow></math>. (The fractional powers
have their principal values.)</p>
</div>
<section id="Px1" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Hankel’s Loop Integral</h3>
<div id="Px1.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px1.p1" class="ltx_para">
<table id="E2" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">5.9.2</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E2.png" altimg-height="58px" altimg-valign="-22px" altimg-width="242px" alttext="\frac{1}{\Gamma\left(z\right)}=\frac{1}{2\pi i}\int_{-\infty}^{(0+)}e^{t}t^{-z%
}\mathrm{d}t," display="block"><mrow><mrow><mfrac><mn>1</mn><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></mfrac><mo>=</mo><mrow><mfrac><mn>1</mn><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi></mrow></mfrac><mo></mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mo stretchy="false">(</mo><mrow><mn>0</mn><mo>+</mo></mrow><mo stretchy="false">)</mo></mrow></msubsup><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mi>t</mi></msup><mo></mo><msup><mi>t</mi><mrow><mo>-</mo><mi href="./5.1#p2.t1.r4">z</mi></mrow></msup><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m13.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m35.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m27.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m52.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m28.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a> and
<a href="./5.1#p2.t1.r4" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./5.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">6.1.4</span> (in a slightly different form.)</span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where the contour begins at <math class="ltx_Math" altimg="m7.png" altimg-height="17px" altimg-valign="-4px" altimg-width="40px" alttext="-\infty" display="inline"><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow></math>, circles the origin once in the positive
direction, and returns to <math class="ltx_Math" altimg="m7.png" altimg-height="17px" altimg-valign="-4px" altimg-width="40px" alttext="-\infty" display="inline"><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow></math>. <math class="ltx_Math" altimg="m51.png" altimg-height="19px" altimg-valign="-2px" altimg-width="33px" alttext="t^{-z}" display="inline"><msup><mi>t</mi><mrow><mo>-</mo><mi href="./5.1#p2.t1.r4">z</mi></mrow></msup></math> has its principal value where <math class="ltx_Math" altimg="m50.png" altimg-height="17px" altimg-valign="-2px" altimg-width="11px" alttext="t" display="inline"><mi>t</mi></math>
crosses the positive real axis, and is continuous. See Figure
<figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 5.9.1: </span><math class="ltx_Math" altimg="m50.png" altimg-height="17px" altimg-valign="-2px" altimg-width="11px" alttext="t" display="inline"><mi>t</mi></math>-plane. Contour for Hankel’s loop integral.
</dd>
</dl>
</div>
</div>
</figure>
<div id="Px1.p2" class="ltx_para">
<table id="E3" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">5.9.3</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E3.png" altimg-height="54px" altimg-valign="-22px" altimg-width="271px" alttext="c^{-z}\Gamma\left(z\right)=\int_{-\infty}^{\infty}|t|^{2z-1}e^{-ct^{2}}\mathrm%
{d}t," display="block"><mrow><mrow><mrow><msup><mi>c</mi><mrow><mo>-</mo><mi href="./5.1#p2.t1.r4">z</mi></mrow></msup><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mi mathvariant="normal">∞</mi></msubsup><mrow><msup><mrow><mo stretchy="false">|</mo><mi>t</mi><mo stretchy="false">|</mo></mrow><mrow><mrow><mn>2</mn><mo></mo><mi href="./5.1#p2.t1.r4">z</mi></mrow><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mi>c</mi><mo></mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mrow></msup><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m42.png" altimg-height="17px" altimg-valign="-3px" altimg-width="49px" alttext="c>0" display="inline"><mrow><mi>c</mi><mo>></mo><mn>0</mn></mrow></math>, <math class="ltx_Math" altimg="m14.png" altimg-height="18px" altimg-valign="-3px" altimg-width="65px" alttext="\Re z>0" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./5.1#p2.t1.r4">z</mi></mrow><mo>></mo><mn>0</mn></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m13.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m27.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m52.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m28.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m15.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a> and
<a href="./5.1#p2.t1.r4" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./5.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where the path is the real axis.</p>
<table id="E4" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">5.9.4</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E4.png" altimg-height="64px" altimg-valign="-28px" altimg-width="342px" alttext="\Gamma\left(z\right)=\int_{1}^{\infty}t^{z-1}e^{-t}\mathrm{d}t+\sum_{k=0}^{%
\infty}\frac{(-1)^{k}}{(z+k)k!}," display="block"><mrow><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>1</mn><mi mathvariant="normal">∞</mi></msubsup><mrow><msup><mi>t</mi><mrow><mi href="./5.1#p2.t1.r4">z</mi><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mi>t</mi></mrow></msup><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow><mo>+</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./5.1#p2.t1.r2">k</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mfrac><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./5.1#p2.t1.r2">k</mi></msup><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./5.1#p2.t1.r4">z</mi><mo>+</mo><mi href="./5.1#p2.t1.r2">k</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mi href="./5.1#p2.t1.r2">k</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mrow></mfrac></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m54.png" altimg-height="21px" altimg-valign="-6px" altimg-width="152px" alttext="z\neq 0,-1,-2,\dots" display="inline"><mrow><mi href="./5.1#p2.t1.r4">z</mi><mo>≠</mo><mrow><mn>0</mn><mo>,</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo>,</mo><mrow><mo>-</mo><mn>2</mn></mrow><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E4.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m13.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m27.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m52.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m28.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m5.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m44.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./5.1#p2.t1.r2" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./5.1#p2.t1.r2">k</mi></math>: nonnegative integer</a> and
<a href="./5.1#p2.t1.r4" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./5.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E5" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">5.9.5</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E5.png" altimg-height="66px" altimg-valign="-28px" altimg-width="371px" alttext="\Gamma\left(z\right)=\int_{0}^{\infty}t^{z-1}\left(e^{-t}-\sum_{k=0}^{n}\frac{%
(-1)^{k}t^{k}}{k!}\right)\mathrm{d}t," display="block"><mrow><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><mrow><msup><mi>t</mi><mrow><mi href="./5.1#p2.t1.r4">z</mi><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mrow><mo>(</mo><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mi>t</mi></mrow></msup><mo>-</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./5.1#p2.t1.r2">k</mi><mo>=</mo><mn>0</mn></mrow><mi href="./5.1#p2.t1.r1">n</mi></munderover><mfrac><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./5.1#p2.t1.r2">k</mi></msup><mo></mo><msup><mi>t</mi><mi href="./5.1#p2.t1.r2">k</mi></msup></mrow><mrow><mi href="./5.1#p2.t1.r2">k</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mfrac></mrow></mrow><mo>)</mo></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m8.png" altimg-height="19px" altimg-valign="-4px" altimg-width="171px" alttext="-n-1<\Re z<-n" display="inline"><mrow><mrow><mrow><mo>-</mo><mi href="./5.1#p2.t1.r1">n</mi></mrow><mo>-</mo><mn>1</mn></mrow><mo><</mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./5.1#p2.t1.r4">z</mi></mrow><mo><</mo><mrow><mo>-</mo><mi href="./5.1#p2.t1.r1">n</mi></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E5.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m13.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m27.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m52.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m28.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m5.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m44.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m15.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./5.1#p2.t1.r1" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m46.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./5.1#p2.t1.r1">n</mi></math>: nonnegative integer</a>,
<a href="./5.1#p2.t1.r2" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./5.1#p2.t1.r2">k</mi></math>: nonnegative integer</a> and
<a href="./5.1#p2.t1.r4" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./5.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="EGx1" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E6">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">5.9.6</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m3.png" altimg-height="29px" altimg-valign="-9px" altimg-width="134px" alttext="\displaystyle\Gamma\left(z\right)\cos\left(\tfrac{1}{2}\pi z\right)" display="inline"><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi href="./5.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m1.png" altimg-height="53px" altimg-valign="-20px" altimg-width="168px" alttext="\displaystyle=\int_{0}^{\infty}t^{z-1}\cos t\mathrm{d}t," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup></mstyle><mrow><msup><mi>t</mi><mrow><mi href="./5.1#p2.t1.r4">z</mi><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi>t</mi></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m9.png" altimg-height="18px" altimg-valign="-3px" altimg-width="102px" alttext="0<\Re z<1" display="inline"><mrow><mn>0</mn><mo><</mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./5.1#p2.t1.r4">z</mi></mrow><mo><</mo><mn>1</mn></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E6.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m13.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m35.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m17.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m27.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m52.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m15.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a> and
<a href="./5.1#p2.t1.r4" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./5.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E7">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">5.9.7</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m4.png" altimg-height="29px" altimg-valign="-9px" altimg-width="132px" alttext="\displaystyle\Gamma\left(z\right)\sin\left(\tfrac{1}{2}\pi z\right)" display="inline"><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi href="./5.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m2.png" altimg-height="53px" altimg-valign="-20px" altimg-width="166px" alttext="\displaystyle=\int_{0}^{\infty}t^{z-1}\sin t\mathrm{d}t," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup></mstyle><mrow><msup><mi>t</mi><mrow><mi href="./5.1#p2.t1.r4">z</mi><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi>t</mi></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m6.png" altimg-height="19px" altimg-valign="-4px" altimg-width="117px" alttext="-1<\Re z<1" display="inline"><mrow><mrow><mo>-</mo><mn>1</mn></mrow><mo><</mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./5.1#p2.t1.r4">z</mi></mrow><mo><</mo><mn>1</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E7.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m13.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m35.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m27.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m52.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m15.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a> and
<a href="./5.1#p2.t1.r4" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./5.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
<table id="E8" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">5.9.8</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E8.png" altimg-height="53px" altimg-valign="-21px" altimg-width="343px" alttext="\Gamma\left(1+\frac{1}{n}\right)\cos\left(\frac{\pi}{2n}\right)=\int_{0}^{%
\infty}\cos\left(t^{n}\right)\mathrm{d}t," display="block"><mrow><mrow><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mfrac><mn>1</mn><mi href="./5.1#p2.t1.r1">n</mi></mfrac></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mfrac><mi href="./3.12#E1">π</mi><mrow><mn>2</mn><mo></mo><mi href="./5.1#p2.t1.r1">n</mi></mrow></mfrac><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><mrow><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><msup><mi>t</mi><mi href="./5.1#p2.t1.r1">n</mi></msup><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m45.png" altimg-height="20px" altimg-valign="-6px" altimg-width="123px" alttext="n=2,3,4,\dots" display="inline"><mrow><mi href="./5.1#p2.t1.r1">n</mi><mo>=</mo><mrow><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E8.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m13.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m35.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m17.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m27.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m52.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a> and
<a href="./5.1#p2.t1.r1" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m46.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./5.1#p2.t1.r1">n</mi></math>: nonnegative integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E9" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">5.9.9</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E9.png" altimg-height="53px" altimg-valign="-21px" altimg-width="339px" alttext="\Gamma\left(1+\frac{1}{n}\right)\sin\left(\frac{\pi}{2n}\right)=\int_{0}^{%
\infty}\sin\left(t^{n}\right)\mathrm{d}t," display="block"><mrow><mrow><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mfrac><mn>1</mn><mi href="./5.1#p2.t1.r1">n</mi></mfrac></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mfrac><mi href="./3.12#E1">π</mi><mrow><mn>2</mn><mo></mo><mi href="./5.1#p2.t1.r1">n</mi></mrow></mfrac><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><mrow><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><msup><mi>t</mi><mi href="./5.1#p2.t1.r1">n</mi></msup><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m45.png" altimg-height="20px" altimg-valign="-6px" altimg-width="123px" alttext="n=2,3,4,\dots" display="inline"><mrow><mi href="./5.1#p2.t1.r1">n</mi><mo>=</mo><mrow><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E9.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m13.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m35.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m27.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m52.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a> and
<a href="./5.1#p2.t1.r1" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m46.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./5.1#p2.t1.r1">n</mi></math>: nonnegative integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px2" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Binet’s Formula</h3>
<div id="Px2.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px2.p1" class="ltx_para">
<table id="E10" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">5.9.10</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E10.png" altimg-height="52px" altimg-valign="-20px" altimg-width="540px" alttext="\operatorname{Ln}\Gamma\left(z\right)=\left(z-\tfrac{1}{2}\right)\ln z-z+%
\tfrac{1}{2}\ln\left(2\pi\right)+2\int_{0}^{\infty}\frac{\operatorname{arctan}%
\left(t/z\right)}{e^{2\pi t}-1}\mathrm{d}t," display="block"><mrow><mrow><mrow><mi href="./4.2#E1">Ln</mi><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mrow><mrow><mrow><mrow><mo>(</mo><mrow><mi href="./5.1#p2.t1.r4">z</mi><mo>-</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle></mrow><mo>)</mo></mrow><mo></mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi href="./5.1#p2.t1.r4">z</mi></mrow></mrow><mo>-</mo><mi href="./5.1#p2.t1.r4">z</mi></mrow><mo>+</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mn>2</mn><mo></mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><mrow><mfrac><mrow><mi href="./4.23#SS2.p1">arctan</mi><mo></mo><mrow><mo>(</mo><mrow><mi>t</mi><mo>/</mo><mi href="./5.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi>t</mi></mrow></msup><mo>-</mo><mn>1</mn></mrow></mfrac><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E10.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m13.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m35.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m27.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m52.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m28.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./4.23#SS2.p1" title="§4.23(ii) Principal Values ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m34.png" altimg-height="17px" altimg-valign="-2px" altimg-width="73px" alttext="\operatorname{arctan}\NVar{z}" display="inline"><mrow><mi href="./4.23#SS2.p1">arctan</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: arctangent function</a>,
<a href="./4.2#E1" title="(4.2.1) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m32.png" altimg-height="18px" altimg-valign="-2px" altimg-width="41px" alttext="\operatorname{Ln}\NVar{z}" display="inline"><mrow><mi href="./4.2#E1">Ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: general logarithm function</a>,
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m24.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a> and
<a href="./5.1#p2.t1.r4" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./5.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">6.1.50</span></span>
</dd>
<dt>Addition (effective with 1.0.10):</dt>
<dd>
To increase the region of validity of this equation, the logarithm of the
gamma function that appears on its left-hand side has been changed to
<math class="ltx_Math" altimg="m31.png" altimg-height="23px" altimg-valign="-7px" altimg-width="73px" alttext="\operatorname{Ln}\Gamma\left(z\right)" display="inline"><mrow><mi href="./4.2#E1">Ln</mi><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></math>, where <math class="ltx_Math" altimg="m33.png" altimg-height="18px" altimg-valign="-2px" altimg-width="28px" alttext="\operatorname{Ln}" display="inline"><mi href="./4.2#E1">Ln</mi></math> is the <span class="ltx_text ltx_font_italic">general logarithm</span>.
Originally <math class="ltx_Math" altimg="m23.png" altimg-height="23px" altimg-valign="-7px" altimg-width="66px" alttext="\ln\Gamma\left(z\right)" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></math> was used, where <math class="ltx_Math" altimg="m25.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="\ln" display="inline"><mi href="./4.2#E2">ln</mi></math>
is the <span class="ltx_text ltx_font_italic">principal branch</span> of the logarithm.
<p><span class="ltx_font_italic">Suggested 2015-02-13 by Philippe Spindel</span></p>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m55.png" altimg-height="23px" altimg-valign="-7px" altimg-width="113px" alttext="|\operatorname{ph}z|<\pi/2" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mi href="./5.1#p2.t1.r4">z</mi></mrow><mo stretchy="false">|</mo></mrow><mo><</mo><mrow><mi href="./3.12#E1">π</mi><mo>/</mo><mn>2</mn></mrow></mrow></math> and the inverse tangent has its principal value.</p>
<table id="E11" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">5.9.11</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E11.png" altimg-height="57px" altimg-valign="-22px" altimg-width="476px" alttext="\operatorname{Ln}\Gamma\left(z+1\right)=-\gamma z-\frac{1}{2\pi i}\int_{-c-%
\infty i}^{-c+\infty i}\frac{\pi z^{-s}}{s\sin\left(\pi s\right)}\zeta\left(-s%
\right)\mathrm{d}s," display="block"><mrow><mrow><mrow><mi href="./4.2#E1">Ln</mi><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./5.1#p2.t1.r4">z</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mrow><mrow><mo>-</mo><mrow><mi href="./5.2#E3">γ</mi><mo></mo><mi href="./5.1#p2.t1.r4">z</mi></mrow></mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi></mrow></mfrac><mo></mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><mrow><mo>-</mo><mi>c</mi></mrow><mo>-</mo><mrow><mi mathvariant="normal">∞</mi><mo></mo><mi mathvariant="normal">i</mi></mrow></mrow><mrow><mrow><mo>-</mo><mi>c</mi></mrow><mo>+</mo><mrow><mi mathvariant="normal">∞</mi><mo></mo><mi mathvariant="normal">i</mi></mrow></mrow></msubsup><mrow><mfrac><mrow><mi href="./3.12#E1">π</mi><mo></mo><msup><mi href="./5.1#p2.t1.r4">z</mi><mrow><mo>-</mo><mi href="./5.1#p2.t1.r5">s</mi></mrow></msup></mrow><mrow><mi href="./5.1#p2.t1.r5">s</mi><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi href="./5.1#p2.t1.r5">s</mi></mrow><mo>)</mo></mrow></mrow></mrow></mfrac><mo></mo><mrow><mi href="./25.2#E1">ζ</mi><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mi href="./5.1#p2.t1.r5">s</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./5.1#p2.t1.r5">s</mi></mrow></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E11.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m13.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./5.2#E3" title="(5.2.3) ‣ §5.2(ii) Euler’s Constant ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\gamma" display="inline"><mi href="./5.2#E3">γ</mi></math>: Euler’s constant</a>,
<a href="./25.2#E1" title="(25.2.1) ‣ §25.2(i) Definition ‣ §25.2 Definition and Expansions ‣ Riemann Zeta Function ‣ Chapter 25 Zeta and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="\zeta\left(\NVar{s}\right)" display="inline"><mrow><mi href="./25.2#E1">ζ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./25.1#p2.t1.r5">s</mi><mo>)</mo></mrow></mrow></math>: Riemann zeta function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m35.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m27.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m52.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./4.2#E1" title="(4.2.1) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m32.png" altimg-height="18px" altimg-valign="-2px" altimg-width="41px" alttext="\operatorname{Ln}\NVar{z}" display="inline"><mrow><mi href="./4.2#E1">Ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: general logarithm function</a>,
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m24.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./5.1#p2.t1.r5" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="s" display="inline"><mi href="./5.1#p2.t1.r5">s</mi></math>: real or complex variable</a> and
<a href="./5.1#p2.t1.r4" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./5.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>Addition (effective with 1.0.10):</dt>
<dd>
To increase the region of validity of this equation, the logarithm of the
gamma function that appears on its left-hand side has been changed to
<math class="ltx_Math" altimg="m30.png" altimg-height="23px" altimg-valign="-7px" altimg-width="107px" alttext="\operatorname{Ln}\Gamma\left(z+1\right)" display="inline"><mrow><mi href="./4.2#E1">Ln</mi><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./5.1#p2.t1.r4">z</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mrow></math>, where <math class="ltx_Math" altimg="m33.png" altimg-height="18px" altimg-valign="-2px" altimg-width="28px" alttext="\operatorname{Ln}" display="inline"><mi href="./4.2#E1">Ln</mi></math> is the <span class="ltx_text ltx_font_italic">general logarithm</span>.
Originally <math class="ltx_Math" altimg="m22.png" altimg-height="23px" altimg-valign="-7px" altimg-width="100px" alttext="\ln\Gamma\left(z+1\right)" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./5.1#p2.t1.r4">z</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mrow></math> was used, where <math class="ltx_Math" altimg="m25.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="\ln" display="inline"><mi href="./4.2#E2">ln</mi></math>
is the <span class="ltx_text ltx_font_italic">principal branch</span> of the logarithm.
<p><span class="ltx_font_italic">Suggested 2015-02-13 by Philippe Spindel</span></p>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m57.png" altimg-height="23px" altimg-valign="-7px" altimg-width="127px" alttext="|\operatorname{ph}z|\leq\pi-\delta" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mi href="./5.1#p2.t1.r4">z</mi></mrow><mo stretchy="false">|</mo></mrow><mo>≤</mo><mrow><mi href="./3.12#E1">π</mi><mo>-</mo><mi href="./5.1#p2.t1.r6">δ</mi></mrow></mrow></math> (<math class="ltx_Math" altimg="m11.png" altimg-height="15px" altimg-valign="-3px" altimg-width="37px" alttext="<\pi" display="inline"><mrow><mi></mi><mo><</mo><mi href="./3.12#E1">π</mi></mrow></math>), <math class="ltx_Math" altimg="m10.png" altimg-height="17px" altimg-valign="-3px" altimg-width="86px" alttext="1<c<2" display="inline"><mrow><mn>1</mn><mo><</mo><mi>c</mi><mo><</mo><mn>2</mn></mrow></math>, and
<math class="ltx_Math" altimg="m39.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="\zeta\left(s\right)" display="inline"><mrow><mi href="./25.2#E1">ζ</mi><mo></mo><mrow><mo>(</mo><mi href="./5.1#p2.t1.r5">s</mi><mo>)</mo></mrow></mrow></math> is as in Chapter </dd>
</dl>
</div>
</div>
<div id="SS2.p1" class="ltx_para">
<p class="ltx_p">For <math class="ltx_Math" altimg="m14.png" altimg-height="18px" altimg-valign="-3px" altimg-width="65px" alttext="\Re z>0" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./5.1#p2.t1.r4">z</mi></mrow><mo>></mo><mn>0</mn></mrow></math>,</p>
<table id="E12" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">5.9.12</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E12.png" altimg-height="55px" altimg-valign="-21px" altimg-width="298px" alttext="\psi\left(z\right)=\int_{0}^{\infty}\left(\frac{e^{-t}}{t}-\frac{e^{-zt}}{1-e^%
{-t}}\right)\mathrm{d}t," display="block"><mrow><mrow><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mi href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><mrow><mrow><mo>(</mo><mrow><mfrac><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mi>t</mi></mrow></msup><mi>t</mi></mfrac><mo>-</mo><mfrac><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mi href="./5.1#p2.t1.r4">z</mi><mo></mo><mi>t</mi></mrow></mrow></msup><mrow><mn>1</mn><mo>-</mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mi>t</mi></mrow></msup></mrow></mfrac></mrow><mo>)</mo></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E12.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m27.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m52.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./5.2#E2" title="(5.2.2) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m36.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="\psi\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: psi (or digamma) function</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m28.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a> and
<a href="./5.1#p2.t1.r4" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./5.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">6.3.21</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E13" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">5.9.13</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E13.png" altimg-height="53px" altimg-valign="-21px" altimg-width="370px" alttext="\psi\left(z\right)=\ln z+\int_{0}^{\infty}\left(\frac{1}{t}-\frac{1}{1-e^{-t}}%
\right)e^{-tz}\mathrm{d}t," display="block"><mrow><mrow><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mi href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi href="./5.1#p2.t1.r4">z</mi></mrow><mo>+</mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><mrow><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mi>t</mi></mfrac><mo>-</mo><mfrac><mn>1</mn><mrow><mn>1</mn><mo>-</mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mi>t</mi></mrow></msup></mrow></mfrac></mrow><mo>)</mo></mrow><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mi>t</mi><mo></mo><mi href="./5.1#p2.t1.r4">z</mi></mrow></mrow></msup><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E13.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m27.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m52.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./5.2#E2" title="(5.2.2) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m36.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="\psi\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: psi (or digamma) function</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m28.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m24.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a> and
<a href="./5.1#p2.t1.r4" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./5.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E14" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">5.9.14</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E14.png" altimg-height="53px" altimg-valign="-21px" altimg-width="301px" alttext="\psi\left(z\right)=\int_{0}^{\infty}\left(e^{-t}-\frac{1}{(1+t)^{z}}\right)%
\frac{\mathrm{d}t}{t}," display="block"><mrow><mrow><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mi href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><mrow><mrow><mo>(</mo><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mi>t</mi></mrow></msup><mo>-</mo><mfrac><mn>1</mn><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>+</mo><mi>t</mi></mrow><mo stretchy="false">)</mo></mrow><mi href="./5.1#p2.t1.r4">z</mi></msup></mfrac></mrow><mo>)</mo></mrow><mo></mo><mfrac><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow><mi>t</mi></mfrac></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E14.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m27.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m52.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./5.2#E2" title="(5.2.2) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m36.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="\psi\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: psi (or digamma) function</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m28.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a> and
<a href="./5.1#p2.t1.r4" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./5.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">6.3.21</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E15" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">5.9.15</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E15.png" altimg-height="52px" altimg-valign="-21px" altimg-width="402px" alttext="\psi\left(z\right)=\ln z-\frac{1}{2z}-2\int_{0}^{\infty}\frac{t\mathrm{d}t}{(t%
^{2}+z^{2})(e^{2\pi t}-1)}." display="block"><mrow><mrow><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mi href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi href="./5.1#p2.t1.r4">z</mi></mrow><mo>-</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mo></mo><mi href="./5.1#p2.t1.r4">z</mi></mrow></mfrac><mo>-</mo><mrow><mn>2</mn><mo></mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><mfrac><mrow><mi>t</mi><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><msup><mi>t</mi><mn>2</mn></msup><mo>+</mo><msup><mi href="./5.1#p2.t1.r4">z</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi>t</mi></mrow></msup><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow></mfrac></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E15.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m35.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m27.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m52.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./5.2#E2" title="(5.2.2) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m36.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="\psi\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: psi (or digamma) function</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m28.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m24.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a> and
<a href="./5.1#p2.t1.r4" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./5.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">6.3.21</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E16" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">5.9.16</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E16.png" altimg-height="56px" altimg-valign="-20px" altimg-width="423px" alttext="\psi\left(z\right)+\gamma=\int_{0}^{\infty}\frac{e^{-t}-e^{-zt}}{1-e^{-t}}%
\mathrm{d}t=\int_{0}^{1}\frac{1-t^{z-1}}{1-t}\mathrm{d}t." display="block"><mrow><mrow><mrow><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mi href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow><mo>+</mo><mi href="./5.2#E3">γ</mi></mrow><mo>=</mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><mrow><mfrac><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mi>t</mi></mrow></msup><mo>-</mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mi href="./5.1#p2.t1.r4">z</mi><mo></mo><mi>t</mi></mrow></mrow></msup></mrow><mrow><mn>1</mn><mo>-</mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mi>t</mi></mrow></msup></mrow></mfrac><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow><mo>=</mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mn>1</mn></msubsup><mrow><mfrac><mrow><mn>1</mn><mo>-</mo><msup><mi>t</mi><mrow><mi href="./5.1#p2.t1.r4">z</mi><mo>-</mo><mn>1</mn></mrow></msup></mrow><mrow><mn>1</mn><mo>-</mo><mi>t</mi></mrow></mfrac><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E16.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E3" title="(5.2.3) ‣ §5.2(ii) Euler’s Constant ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\gamma" display="inline"><mi href="./5.2#E3">γ</mi></math>: Euler’s constant</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m27.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m52.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./5.2#E2" title="(5.2.2) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m36.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="\psi\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: psi (or digamma) function</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m28.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a> and
<a href="./5.1#p2.t1.r4" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./5.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">6.3.22</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E17" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">5.9.17</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E17.png" altimg-height="57px" altimg-valign="-22px" altimg-width="428px" alttext="\psi\left(z+1\right)=-\gamma+\frac{1}{2\pi i}\int_{-c-\infty i}^{-c+\infty i}%
\frac{\pi z^{-s-1}}{\sin\left(\pi s\right)}\zeta\left(-s\right)\mathrm{d}s," display="block"><mrow><mrow><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./5.1#p2.t1.r4">z</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mo>-</mo><mi href="./5.2#E3">γ</mi></mrow><mo>+</mo><mrow><mfrac><mn>1</mn><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi></mrow></mfrac><mo></mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><mrow><mo>-</mo><mi>c</mi></mrow><mo>-</mo><mrow><mi mathvariant="normal">∞</mi><mo></mo><mi mathvariant="normal">i</mi></mrow></mrow><mrow><mrow><mo>-</mo><mi>c</mi></mrow><mo>+</mo><mrow><mi mathvariant="normal">∞</mi><mo></mo><mi mathvariant="normal">i</mi></mrow></mrow></msubsup><mrow><mfrac><mrow><mi href="./3.12#E1">π</mi><mo></mo><msup><mi href="./5.1#p2.t1.r4">z</mi><mrow><mrow><mo>-</mo><mi href="./5.1#p2.t1.r5">s</mi></mrow><mo>-</mo><mn>1</mn></mrow></msup></mrow><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi href="./5.1#p2.t1.r5">s</mi></mrow><mo>)</mo></mrow></mrow></mfrac><mo></mo><mrow><mi href="./25.2#E1">ζ</mi><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mi href="./5.1#p2.t1.r5">s</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./5.1#p2.t1.r5">s</mi></mrow></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E17.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E3" title="(5.2.3) ‣ §5.2(ii) Euler’s Constant ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\gamma" display="inline"><mi href="./5.2#E3">γ</mi></math>: Euler’s constant</a>,
<a href="./25.2#E1" title="(25.2.1) ‣ §25.2(i) Definition ‣ §25.2 Definition and Expansions ‣ Riemann Zeta Function ‣ Chapter 25 Zeta and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="\zeta\left(\NVar{s}\right)" display="inline"><mrow><mi href="./25.2#E1">ζ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./25.1#p2.t1.r5">s</mi><mo>)</mo></mrow></mrow></math>: Riemann zeta function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m35.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m27.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m52.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./5.2#E2" title="(5.2.2) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m36.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="\psi\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: psi (or digamma) function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./5.1#p2.t1.r5" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="s" display="inline"><mi href="./5.1#p2.t1.r5">s</mi></math>: real or complex variable</a> and
<a href="./5.1#p2.t1.r4" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./5.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m56.png" altimg-height="23px" altimg-valign="-7px" altimg-width="175px" alttext="|\operatorname{ph}z|\leq\pi-\delta(<\pi)" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mi href="./5.1#p2.t1.r4">z</mi></mrow><mo stretchy="false">|</mo></mrow><mo>≤</mo><mrow><mrow><mi href="./3.12#E1">π</mi><mo>-</mo><mi href="./5.1#p2.t1.r6">δ</mi></mrow><mspace width="veryverythickmathspace"></mspace><mrow><mo stretchy="false">(</mo><mrow><mi></mi><mo><</mo><mi href="./3.12#E1">π</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></math> and <math class="ltx_Math" altimg="m10.png" altimg-height="17px" altimg-valign="-3px" altimg-width="86px" alttext="1<c<2" display="inline"><mrow><mn>1</mn><mo><</mo><mi>c</mi><mo><</mo><mn>2</mn></mrow></math>.
</p>
<table id="E18" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">5.9.18</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E18.png" altimg-height="56px" altimg-valign="-21px" altimg-width="907px" alttext="\gamma=-\int_{0}^{\infty}e^{-t}\ln t\mathrm{d}t=\int_{0}^{\infty}\left(\frac{1%
}{1+t}-e^{-t}\right)\frac{\mathrm{d}t}{t}=\int_{0}^{1}(1-e^{-t})\frac{\mathrm{%
d}t}{t}-\int_{1}^{\infty}e^{-t}\frac{\mathrm{d}t}{t}=\int_{0}^{\infty}\left(%
\frac{e^{-t}}{1-e^{-t}}-\frac{e^{-t}}{t}\right)\mathrm{d}t." display="block"><mrow><mrow><mi href="./5.2#E3">γ</mi><mo>=</mo><mrow><mo>-</mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mi>t</mi></mrow></msup><mo></mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi>t</mi></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow><mo>=</mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><mrow><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mrow><mn>1</mn><mo>+</mo><mi>t</mi></mrow></mfrac><mo>-</mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mi>t</mi></mrow></msup></mrow><mo>)</mo></mrow><mo></mo><mfrac><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow><mi>t</mi></mfrac></mrow></mrow><mo>=</mo><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mn>1</mn></msubsup><mrow><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mi>t</mi></mrow></msup></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mfrac><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow><mi>t</mi></mfrac></mrow></mrow><mo>-</mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>1</mn><mi mathvariant="normal">∞</mi></msubsup><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mi>t</mi></mrow></msup><mo></mo><mfrac><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow><mi>t</mi></mfrac></mrow></mrow></mrow><mo>=</mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><mrow><mrow><mo>(</mo><mrow><mfrac><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mi>t</mi></mrow></msup><mrow><mn>1</mn><mo>-</mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mi>t</mi></mrow></msup></mrow></mfrac><mo>-</mo><mfrac><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mi>t</mi></mrow></msup><mi>t</mi></mfrac></mrow><mo>)</mo></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E18.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E3" title="(5.2.3) ‣ §5.2(ii) Euler’s Constant ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\gamma" display="inline"><mi href="./5.2#E3">γ</mi></math>: Euler’s constant</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m27.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m52.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m28.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a> and
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m24.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">6.3.22</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E19" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">5.9.19</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E19.png" altimg-height="53px" altimg-valign="-20px" altimg-width="277px" alttext="{\Gamma^{(n)}}\left(z\right)=\int_{0}^{\infty}(\ln t)^{n}e^{-t}t^{z-1}\mathrm{%
d}t," display="block"><mrow><mrow><mrow><msup><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mrow><mo stretchy="false">(</mo><mi href="./5.1#p2.t1.r1">n</mi><mo stretchy="false">)</mo></mrow></msup><mo></mo><mrow><mo>(</mo><mi href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi>t</mi></mrow><mo stretchy="false">)</mo></mrow><mi href="./5.1#p2.t1.r1">n</mi></msup><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mi>t</mi></mrow></msup><mo></mo><msup><mi>t</mi><mrow><mi href="./5.1#p2.t1.r4">z</mi><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m47.png" altimg-height="19px" altimg-valign="-5px" altimg-width="53px" alttext="n\geq 0" display="inline"><mrow><mi href="./5.1#p2.t1.r1">n</mi><mo>≥</mo><mn>0</mn></mrow></math>, <math class="ltx_Math" altimg="m14.png" altimg-height="18px" altimg-valign="-3px" altimg-width="65px" alttext="\Re z>0" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./5.1#p2.t1.r4">z</mi></mrow><mo>></mo><mn>0</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E19.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m13.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m27.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m52.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m28.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m24.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m15.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./5.1#p2.t1.r1" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m46.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./5.1#p2.t1.r1">n</mi></math>: nonnegative integer</a> and
<a href="./5.1#p2.t1.r4" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./5.1#p2.t1.r4">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
</section>
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<span></div>
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<title>DLMF: 5.20 Physical Applications</title>
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<div id="Px1.p1" class="ltx_para">
<p class="ltx_p">In nonrelativistic quantum mechanics, collisions between two charged particles
are described with the aid of the Coulomb phase shift
<math class="ltx_Math" altimg="m16.png" altimg-height="23px" altimg-valign="-7px" altimg-width="144px" alttext="\operatorname{ph}\Gamma\left(\ell+1+\mathrm{i}\eta\right)" display="inline"><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi mathvariant="normal">ℓ</mi><mo>+</mo><mn>1</mn><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi>η</mi></mrow></mrow><mo>)</mo></mrow></mrow></mrow></math>; see (</dd>
</dl>
</div>
</div>
<div id="Px2.p1" class="ltx_para">
<p class="ltx_p">Suppose the potential energy of a gas of <math class="ltx_Math" altimg="m21.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./5.1#p2.t1.r1">n</mi></math> point charges with positions
<math class="ltx_Math" altimg="m24.png" altimg-height="16px" altimg-valign="-6px" altimg-width="120px" alttext="x_{1},x_{2},\dots,x_{n}" display="inline"><mrow><msub><mi href="./5.1#p2.t1.r3">x</mi><mn>1</mn></msub><mo>,</mo><msub><mi href="./5.1#p2.t1.r3">x</mi><mn>2</mn></msub><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><msub><mi href="./5.1#p2.t1.r3">x</mi><mi href="./5.1#p2.t1.r1">n</mi></msub></mrow></math> and free to move on the infinite line
<math class="ltx_Math" altimg="m1.png" altimg-height="17px" altimg-valign="-4px" altimg-width="124px" alttext="-\infty<x<\infty" display="inline"><mrow><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mo><</mo><mi href="./5.1#p2.t1.r3">x</mi><mo><</mo><mi mathvariant="normal">∞</mi></mrow></math>, is given by
</p>
<table id="E1" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">5.20.1</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E1.png" altimg-height="67px" altimg-valign="-31px" altimg-width="321px" alttext="W=\frac{1}{2}\sum_{\ell=1}^{n}x_{\ell}^{2}-\sum_{1\leq\ell<j\leq n}\ln|x_{\ell%
}-x_{j}|." display="block"><mrow><mrow><mi href="./5.20#Px2.p1">W</mi><mo>=</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi mathvariant="normal">ℓ</mi><mo>=</mo><mn>1</mn></mrow><mi href="./5.1#p2.t1.r1">n</mi></munderover><msubsup><mi href="./5.1#p2.t1.r3">x</mi><mi mathvariant="normal">ℓ</mi><mn>2</mn></msubsup></mrow></mrow><mo>-</mo><mrow><munder><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mn>1</mn><mo>≤</mo><mi mathvariant="normal">ℓ</mi><mo><</mo><mi href="./5.1#p2.t1.r1">j</mi><mo>≤</mo><mi href="./5.1#p2.t1.r1">n</mi></mrow></munder><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><mo stretchy="false">|</mo><mrow><msub><mi href="./5.1#p2.t1.r3">x</mi><mi mathvariant="normal">ℓ</mi></msub><mo>-</mo><msub><mi href="./5.1#p2.t1.r3">x</mi><mi href="./5.1#p2.t1.r1">j</mi></msub></mrow><mo stretchy="false">|</mo></mrow></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m12.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a>,
<a href="./5.1#p2.t1.r1" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="20px" altimg-valign="-6px" altimg-width="14px" alttext="j" display="inline"><mi href="./5.1#p2.t1.r1">j</mi></math>: nonnegative integer</a>,
<a href="./5.1#p2.t1.r1" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m21.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./5.1#p2.t1.r1">n</mi></math>: nonnegative integer</a>,
<a href="./5.1#p2.t1.r3" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m23.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./5.1#p2.t1.r3">x</mi></math>: real variable</a> and
<a href="./5.20#Px2.p1" title="Solvable Models of Statistical Mechanics ‣ §5.20 Physical Applications ‣ Applications ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m6.png" altimg-height="18px" altimg-valign="-2px" altimg-width="26px" alttext="W" display="inline"><mi href="./5.20#Px2.p1">W</mi></math>: potential energy</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">The probability density of the positions when the gas is in thermodynamic
equilibrium is:</p>
<table id="E2" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">5.20.2</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E2.png" altimg-height="25px" altimg-valign="-7px" altimg-width="313px" alttext="P(x_{1},\dots,x_{n})=C\exp\left(-W/(kT)\right)," display="block"><mrow><mrow><mrow><mi>P</mi><mo></mo><mrow><mo stretchy="false">(</mo><msub><mi href="./5.1#p2.t1.r3">x</mi><mn>1</mn></msub><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><msub><mi href="./5.1#p2.t1.r3">x</mi><mi href="./5.1#p2.t1.r1">n</mi></msub><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mi href="./5.20#Px2.p1">C</mi><mo></mo><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mrow><mi href="./5.20#Px2.p1">W</mi><mo>/</mo><mrow><mo stretchy="false">(</mo><mrow><mi>k</mi><mo></mo><mi href="./5.20#Px2.p1">T</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.2#E19" title="(4.2.19) ‣ §4.2(iii) The Exponential Function ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m10.png" altimg-height="16px" altimg-valign="-6px" altimg-width="48px" alttext="\exp\NVar{z}" display="inline"><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: exponential function</a>,
<a href="./5.1#p2.t1.r1" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m21.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./5.1#p2.t1.r1">n</mi></math>: nonnegative integer</a>,
<a href="./5.1#p2.t1.r3" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m23.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./5.1#p2.t1.r3">x</mi></math>: real variable</a>,
<a href="./5.20#Px2.p1" title="Solvable Models of Statistical Mechanics ‣ §5.20 Physical Applications ‣ Applications ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m6.png" altimg-height="18px" altimg-valign="-2px" altimg-width="26px" alttext="W" display="inline"><mi href="./5.20#Px2.p1">W</mi></math>: potential energy</a>,
<a href="./5.20#Px2.p1" title="Solvable Models of Statistical Mechanics ‣ §5.20 Physical Applications ‣ Applications ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m5.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="T" display="inline"><mi href="./5.20#Px2.p1">T</mi></math>: temperature</a> and
<a href="./5.20#Px2.p1" title="Solvable Models of Statistical Mechanics ‣ §5.20 Physical Applications ‣ Applications ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m3.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./5.20#Px2.p1">C</mi></math>: constant</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m20.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi>k</mi></math> is the Boltzmann constant, <math class="ltx_Math" altimg="m5.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="T" display="inline"><mi href="./5.20#Px2.p1">T</mi></math> the temperature and <math class="ltx_Math" altimg="m3.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./5.20#Px2.p1">C</mi></math> a constant.
Then the partition function (with <math class="ltx_Math" altimg="m9.png" altimg-height="23px" altimg-valign="-7px" altimg-width="104px" alttext="\beta=1/(kT)" display="inline"><mrow><mi href="./5.20#Px2.p1">β</mi><mo>=</mo><mrow><mn>1</mn><mo>/</mo><mrow><mo stretchy="false">(</mo><mrow><mi>k</mi><mo></mo><mi href="./5.20#Px2.p1">T</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></math>) is given by
</p>
<table id="E3" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">5.20.3</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_Math ltx_eqn_cell ltx_align_center"><math class="ltx_Math" alttext="\psi_{n}(\beta)=\int_{{\mathbb{R}^{n}}}e^{-\beta W}\mathrm{d}x\\
=(2\pi)^{n/2}\beta^{-(n/2)-(\beta n(n-1)/4)}\*\left(\Gamma\left(1+\tfrac{1}{2}%
\beta\right)\right)^{-n}\prod_{j=1}^{n}\Gamma\left(1+\tfrac{1}{2}j\beta\right)." display="block"><mrow><mtable align="baseline 1" columnalign="left" columnspacing="0.1em" displaystyle="true"><mtr><mtd><mrow><msub><mi href="./5.20#Px2.p1">ψ</mi><mi href="./5.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./5.20#Px2.p1">β</mi><mo stretchy="false">)</mo></mrow></mrow></mtd><mtd><mo>=</mo><mrow><msub><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><msup><mi href="./front/introduction#Sx4.p2.t1.r14">ℝ</mi><mi href="./5.1#p2.t1.r1">n</mi></msup></msub><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mi href="./5.20#Px2.p1">β</mi><mo></mo><mi href="./5.20#Px2.p1">W</mi></mrow></mrow></msup><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./5.1#p2.t1.r3">x</mi></mrow></mrow></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mi href="./5.1#p2.t1.r1">n</mi><mo>/</mo><mn>2</mn></mrow></msup><mo></mo><msup><mi href="./5.20#Px2.p1">β</mi><mrow><mrow><mo>-</mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./5.1#p2.t1.r1">n</mi><mo>/</mo><mn>2</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>-</mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mi href="./5.20#Px2.p1">β</mi><mo></mo><mi href="./5.1#p2.t1.r1">n</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./5.1#p2.t1.r1">n</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>/</mo><mn>4</mn></mrow><mo stretchy="false">)</mo></mrow></mrow></msup><mo></mo><msup><mrow><mo>(</mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./5.20#Px2.p1">β</mi></mrow></mrow><mo>)</mo></mrow></mrow><mo>)</mo></mrow><mrow><mo>-</mo><mi href="./5.1#p2.t1.r1">n</mi></mrow></msup><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∏</mo><mrow><mi href="./5.1#p2.t1.r1">j</mi><mo>=</mo><mn>1</mn></mrow><mi href="./5.1#p2.t1.r1">n</mi></munderover><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./5.1#p2.t1.r1">j</mi><mo></mo><mi href="./5.20#Px2.p1">β</mi></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>.</mo></mtd></mtr></mtable></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m8.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m17.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m14.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m23.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m15.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m11.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./front/introduction#Sx4.p2.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m13.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\mathbb{R}" display="inline"><mi href="./front/introduction#Sx4.p2.t1.r14">ℝ</mi></math>: real line</a>,
<a href="./5.1#p2.t1.r1" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="20px" altimg-valign="-6px" altimg-width="14px" alttext="j" display="inline"><mi href="./5.1#p2.t1.r1">j</mi></math>: nonnegative integer</a>,
<a href="./5.1#p2.t1.r1" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m21.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./5.1#p2.t1.r1">n</mi></math>: nonnegative integer</a>,
<a href="./5.1#p2.t1.r3" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m23.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./5.1#p2.t1.r3">x</mi></math>: real variable</a>,
<a href="./5.20#Px2.p1" title="Solvable Models of Statistical Mechanics ‣ §5.20 Physical Applications ‣ Applications ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m6.png" altimg-height="18px" altimg-valign="-2px" altimg-width="26px" alttext="W" display="inline"><mi href="./5.20#Px2.p1">W</mi></math>: potential energy</a>,
<a href="./5.20#Px2.p1" title="Solvable Models of Statistical Mechanics ‣ §5.20 Physical Applications ‣ Applications ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m18.png" altimg-height="23px" altimg-valign="-7px" altimg-width="56px" alttext="\psi_{n}(\beta)" display="inline"><mrow><msub><mi href="./5.20#Px2.p1">ψ</mi><mi href="./5.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./5.20#Px2.p1">β</mi><mo stretchy="false">)</mo></mrow></mrow></math>: partition function</a> and
<a href="./5.20#Px2.p1" title="Solvable Models of Statistical Mechanics ‣ §5.20 Physical Applications ‣ Applications ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m9.png" altimg-height="23px" altimg-valign="-7px" altimg-width="104px" alttext="\beta=1/(kT)" display="inline"><mrow><mi href="./5.20#Px2.p1">β</mi><mo>=</mo><mrow><mn>1</mn><mo>/</mo><mrow><mo stretchy="false">(</mo><mrow><mi>k</mi><mo></mo><mi href="./5.20#Px2.p1">T</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">See ().</p>
</div>
<div id="Px2.p2" class="ltx_para">
<p class="ltx_p">For <math class="ltx_Math" altimg="m21.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./5.1#p2.t1.r1">n</mi></math> charges free to move on a circular wire of radius <math class="ltx_Math" altimg="m2.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="1" display="inline"><mn>1</mn></math>,</p>
<table id="E4" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">5.20.4</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E4.png" altimg-height="55px" altimg-valign="-31px" altimg-width="268px" alttext="W=-\sum_{1\leq\ell<j\leq n}\ln|e^{i\theta_{\ell}}-e^{i\theta_{j}}|," display="block"><mrow><mrow><mi href="./5.20#Px2.p1">W</mi><mo>=</mo><mrow><mo>-</mo><mrow><munder><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mn>1</mn><mo>≤</mo><mi mathvariant="normal">ℓ</mi><mo><</mo><mi href="./5.1#p2.t1.r1">j</mi><mo>≤</mo><mi href="./5.1#p2.t1.r1">n</mi></mrow></munder><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><mo stretchy="false">|</mo><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mo></mo><msub><mi>θ</mi><mi mathvariant="normal">ℓ</mi></msub></mrow></msup><mo>-</mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mo></mo><msub><mi>θ</mi><mi href="./5.1#p2.t1.r1">j</mi></msub></mrow></msup></mrow><mo stretchy="false">|</mo></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E4.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m15.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m12.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a>,
<a href="./5.1#p2.t1.r1" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="20px" altimg-valign="-6px" altimg-width="14px" alttext="j" display="inline"><mi href="./5.1#p2.t1.r1">j</mi></math>: nonnegative integer</a>,
<a href="./5.1#p2.t1.r1" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m21.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./5.1#p2.t1.r1">n</mi></math>: nonnegative integer</a> and
<a href="./5.20#Px2.p1" title="Solvable Models of Statistical Mechanics ‣ §5.20 Physical Applications ‣ Applications ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m6.png" altimg-height="18px" altimg-valign="-2px" altimg-width="26px" alttext="W" display="inline"><mi href="./5.20#Px2.p1">W</mi></math>: potential energy</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and the partition function is given by</p>
<table id="E5" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">5.20.5</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E5.png" altimg-height="54px" altimg-valign="-24px" altimg-width="625px" alttext="\psi_{n}(\beta)=\frac{1}{(2\pi)^{n}}\int_{[-\pi,\pi]^{n}}e^{-\beta W}\mathrm{d%
}\theta_{1}\cdots\mathrm{d}\theta_{n}=\Gamma\left(1+\tfrac{1}{2}n\beta\right)(%
\Gamma\left(1+\tfrac{1}{2}\beta\right))^{-n}." display="block"><mrow><mrow><mrow><msub><mi href="./5.20#Px2.p1">ψ</mi><mi href="./5.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./5.20#Px2.p1">β</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mfrac><mn>1</mn><msup><mrow><mo stretchy="false">(</mo><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo stretchy="false">)</mo></mrow><mi href="./5.1#p2.t1.r1">n</mi></msup></mfrac><mo></mo><mrow><msub><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><msup><mrow><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">[</mo><mrow><mo>-</mo><mi href="./3.12#E1">π</mi></mrow><mo href="./front/introduction#Sx4.p1.t1.r29">,</mo><mi href="./3.12#E1">π</mi><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">]</mo></mrow><mi href="./5.1#p2.t1.r1">n</mi></msup></msub><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mi href="./5.20#Px2.p1">β</mi><mo></mo><mi href="./5.20#Px2.p1">W</mi></mrow></mrow></msup><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><msub><mi>θ</mi><mn>1</mn></msub></mrow><mo></mo><mi mathvariant="normal">⋯</mi><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><msub><mi>θ</mi><mi href="./5.1#p2.t1.r1">n</mi></msub></mrow></mrow></mrow></mrow><mo>=</mo><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./5.1#p2.t1.r1">n</mi><mo></mo><mi href="./5.20#Px2.p1">β</mi></mrow></mrow><mo>)</mo></mrow></mrow><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./5.20#Px2.p1">β</mi></mrow></mrow><mo>)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mi href="./5.1#p2.t1.r1">n</mi></mrow></msup></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E5.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m8.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m17.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./front/introduction#Sx4.p1.t1.r29" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m7.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="[\NVar{a},\NVar{b}]" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">[</mo><mi class="ltx_nvar">a</mi><mo href="./front/introduction#Sx4.p1.t1.r29">,</mo><mi class="ltx_nvar">b</mi><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">]</mo></mrow></math>: closed interval</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m14.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m23.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m15.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m11.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./5.1#p2.t1.r1" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m21.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./5.1#p2.t1.r1">n</mi></math>: nonnegative integer</a>,
<a href="./5.20#Px2.p1" title="Solvable Models of Statistical Mechanics ‣ §5.20 Physical Applications ‣ Applications ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m6.png" altimg-height="18px" altimg-valign="-2px" altimg-width="26px" alttext="W" display="inline"><mi href="./5.20#Px2.p1">W</mi></math>: potential energy</a>,
<a href="./5.20#Px2.p1" title="Solvable Models of Statistical Mechanics ‣ §5.20 Physical Applications ‣ Applications ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m18.png" altimg-height="23px" altimg-valign="-7px" altimg-width="56px" alttext="\psi_{n}(\beta)" display="inline"><mrow><msub><mi href="./5.20#Px2.p1">ψ</mi><mi href="./5.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./5.20#Px2.p1">β</mi><mo stretchy="false">)</mo></mrow></mrow></math>: partition function</a> and
<a href="./5.20#Px2.p1" title="Solvable Models of Statistical Mechanics ‣ §5.20 Physical Applications ‣ Applications ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m9.png" altimg-height="23px" altimg-valign="-7px" altimg-width="104px" alttext="\beta=1/(kT)" display="inline"><mrow><mi href="./5.20#Px2.p1">β</mi><mo>=</mo><mrow><mn>1</mn><mo>/</mo><mrow><mo stretchy="false">(</mo><mrow><mi>k</mi><mo></mo><mi href="./5.20#Px2.p1">T</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">See (</div>
</div>
</body></text>
</html>
</page>
<page>
<!DOCTYPE html><html>
<head>
<title>DLMF: 5.12 Beta Function</title>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<!--GOOGLE BOOTSTRAP--></head>
<text><body class="color_default textfont_default titlefont_default fontsize_default navbar_default">
<div class="ltx_page_navbar">
<div class="ltx_page_navlogo">) it is assumed
<math class="ltx_Math" altimg="m9.png" altimg-height="18px" altimg-valign="-3px" altimg-width="66px" alttext="\Re a>0" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./5.1#p2.t1.r5">a</mi></mrow><mo>></mo><mn>0</mn></mrow></math> and <math class="ltx_Math" altimg="m11.png" altimg-height="18px" altimg-valign="-3px" altimg-width="64px" alttext="\Re b>0" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./5.1#p2.t1.r5">b</mi></mrow><mo>></mo><mn>0</mn></mrow></math>.</p>
</div>
<section id="Px1" class="ltx_paragraph">
<h2 class="ltx_title ltx_title_paragraph">Euler’s Beta Integral</h2>
<div id="Px1.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px1.p1" class="ltx_para">
<table id="E1" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">5.12.1</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E1.png" altimg-height="55px" altimg-valign="-21px" altimg-width="388px" alttext="\mathrm{B}\left(a,b\right)=\int_{0}^{1}t^{a-1}(1-t)^{b-1}\mathrm{d}t=\frac{%
\Gamma\left(a\right)\Gamma\left(b\right)}{\Gamma\left(a+b\right)}." display="block"><mrow><mrow><mrow><mi href="./5.12#E1" mathvariant="normal">B</mi><mo></mo><mrow><mo>(</mo><mi href="./5.1#p2.t1.r5">a</mi><mo>,</mo><mi href="./5.1#p2.t1.r5">b</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mn>1</mn></msubsup><mrow><msup><mi>t</mi><mrow><mi href="./5.1#p2.t1.r5">a</mi><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><mi>t</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mi href="./5.1#p2.t1.r5">b</mi><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow><mo>=</mo><mfrac><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi href="./5.1#p2.t1.r5">a</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi href="./5.1#p2.t1.r5">b</mi><mo>)</mo></mrow></mrow></mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./5.1#p2.t1.r5">a</mi><mo>+</mo><mi href="./5.1#p2.t1.r5">b</mi></mrow><mo>)</mo></mrow></mrow></mfrac></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m21.png" altimg-height="23px" altimg-valign="-7px" altimg-width="65px" alttext="\mathrm{B}\left(\NVar{a},\NVar{b}\right)" display="inline"><mrow><mi href="./5.12#E1" mathvariant="normal">B</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r5">a</mi><mo>,</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r5">b</mi><mo>)</mo></mrow></mrow></math>: beta function</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m8.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m34.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./5.1#p2.t1.r5" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m28.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./5.1#p2.t1.r5">a</mi></math>: real or complex variable</a> and
<a href="./5.1#p2.t1.r5" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="b" display="inline"><mi href="./5.1#p2.t1.r5">b</mi></math>: real or complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">6.2.1 and 6.2.2</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E2" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">5.12.2</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E2.png" altimg-height="56px" altimg-valign="-20px" altimg-width="331px" alttext="\int_{0}^{\pi/2}{\sin^{2a-1}}\theta{\cos^{2b-1}}\theta\mathrm{d}\theta=\tfrac{%
1}{2}\mathrm{B}\left(a,b\right)." display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mrow><mi href="./3.12#E1">π</mi><mo>/</mo><mn>2</mn></mrow></msubsup><mrow><mrow><msup><mi href="./4.14#E1">sin</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./5.1#p2.t1.r5">a</mi></mrow><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mi>θ</mi></mrow><mo></mo><mrow><msup><mi href="./4.14#E2">cos</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./5.1#p2.t1.r5">b</mi></mrow><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mi>θ</mi></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>θ</mi></mrow></mrow></mrow><mo>=</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mrow><mi href="./5.12#E1" mathvariant="normal">B</mi><mo></mo><mrow><mo>(</mo><mi href="./5.1#p2.t1.r5">a</mi><mo>,</mo><mi href="./5.1#p2.t1.r5">b</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.12#E1" title="(5.12.1) ‣ Euler’s Beta Integral ‣ §5.12 Beta Function ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m21.png" altimg-height="23px" altimg-valign="-7px" altimg-width="65px" alttext="\mathrm{B}\left(\NVar{a},\NVar{b}\right)" display="inline"><mrow><mi href="./5.12#E1" mathvariant="normal">B</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r5">a</mi><mo>,</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r5">b</mi><mo>)</mo></mrow></mrow></math>: beta function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m25.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m17.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m34.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m26.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./5.1#p2.t1.r5" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m28.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./5.1#p2.t1.r5">a</mi></math>: real or complex variable</a> and
<a href="./5.1#p2.t1.r5" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="b" display="inline"><mi href="./5.1#p2.t1.r5">b</mi></math>: real or complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">6.2.1 and 6.2.2</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E3" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">5.12.3</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E3.png" altimg-height="55px" altimg-valign="-21px" altimg-width="233px" alttext="\int_{0}^{\infty}\frac{t^{a-1}\mathrm{d}t}{(1+t)^{a+b}}=\mathrm{B}\left(a,b%
\right)." display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><mfrac><mrow><msup><mi>t</mi><mrow><mi href="./5.1#p2.t1.r5">a</mi><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>+</mo><mi>t</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mi href="./5.1#p2.t1.r5">a</mi><mo>+</mo><mi href="./5.1#p2.t1.r5">b</mi></mrow></msup></mfrac></mrow><mo>=</mo><mrow><mi href="./5.12#E1" mathvariant="normal">B</mi><mo></mo><mrow><mo>(</mo><mi href="./5.1#p2.t1.r5">a</mi><mo>,</mo><mi href="./5.1#p2.t1.r5">b</mi><mo>)</mo></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.12#E1" title="(5.12.1) ‣ Euler’s Beta Integral ‣ §5.12 Beta Function ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m21.png" altimg-height="23px" altimg-valign="-7px" altimg-width="65px" alttext="\mathrm{B}\left(\NVar{a},\NVar{b}\right)" display="inline"><mrow><mi href="./5.12#E1" mathvariant="normal">B</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r5">a</mi><mo>,</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r5">b</mi><mo>)</mo></mrow></mrow></math>: beta function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m34.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./5.1#p2.t1.r5" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m28.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./5.1#p2.t1.r5">a</mi></math>: real or complex variable</a> and
<a href="./5.1#p2.t1.r5" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="b" display="inline"><mi href="./5.1#p2.t1.r5">b</mi></math>: real or complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">6.2.1 and 6.2.2</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E4" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">5.12.4</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E4.png" altimg-height="55px" altimg-valign="-21px" altimg-width="393px" alttext="\int_{0}^{1}\frac{t^{a-1}(1-t)^{b-1}}{(t+z)^{a+b}}\mathrm{d}t=\mathrm{B}\left(%
a,b\right)(1+z)^{-a}z^{-b}," display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mn>1</mn></msubsup><mrow><mfrac><mrow><msup><mi>t</mi><mrow><mi href="./5.1#p2.t1.r5">a</mi><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><mi>t</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mi href="./5.1#p2.t1.r5">b</mi><mo>-</mo><mn>1</mn></mrow></msup></mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mi>t</mi><mo>+</mo><mi href="./5.1#p2.t1.r4">z</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mi href="./5.1#p2.t1.r5">a</mi><mo>+</mo><mi href="./5.1#p2.t1.r5">b</mi></mrow></msup></mfrac><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow><mo>=</mo><mrow><mrow><mi href="./5.12#E1" mathvariant="normal">B</mi><mo></mo><mrow><mo>(</mo><mi href="./5.1#p2.t1.r5">a</mi><mo>,</mo><mi href="./5.1#p2.t1.r5">b</mi><mo>)</mo></mrow></mrow><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>+</mo><mi href="./5.1#p2.t1.r4">z</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mi href="./5.1#p2.t1.r5">a</mi></mrow></msup><mo></mo><msup><mi href="./5.1#p2.t1.r4">z</mi><mrow><mo>-</mo><mi href="./5.1#p2.t1.r5">b</mi></mrow></msup></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m37.png" altimg-height="23px" altimg-valign="-7px" altimg-width="93px" alttext="|\operatorname{ph}z|<\pi" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mi href="./5.1#p2.t1.r4">z</mi></mrow><mo stretchy="false">|</mo></mrow><mo><</mo><mi href="./3.12#E1">π</mi></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E4.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.12#E1" title="(5.12.1) ‣ Euler’s Beta Integral ‣ §5.12 Beta Function ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m21.png" altimg-height="23px" altimg-valign="-7px" altimg-width="65px" alttext="\mathrm{B}\left(\NVar{a},\NVar{b}\right)" display="inline"><mrow><mi href="./5.12#E1" mathvariant="normal">B</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r5">a</mi><mo>,</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r5">b</mi><mo>)</mo></mrow></mrow></math>: beta function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m25.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m34.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.9#E7" title="(1.9.7) ‣ Modulus and Phase ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m24.png" altimg-height="21px" altimg-valign="-6px" altimg-width="26px" alttext="\operatorname{ph}" display="inline"><mi href="./1.9#E7">ph</mi></math>: phase</a>,
<a href="./5.1#p2.t1.r4" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m36.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./5.1#p2.t1.r4">z</mi></math>: complex variable</a>,
<a href="./5.1#p2.t1.r5" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m28.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./5.1#p2.t1.r5">a</mi></math>: real or complex variable</a> and
<a href="./5.1#p2.t1.r5" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="b" display="inline"><mi href="./5.1#p2.t1.r5">b</mi></math>: real or complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E5" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">5.12.5</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E5.png" altimg-height="60px" altimg-valign="-24px" altimg-width="550px" alttext="\int_{0}^{\pi/2}(\cos t)^{a-1}\cos\left(bt\right)\mathrm{d}t=\frac{\pi}{2^{a}}%
\frac{1}{a\mathrm{B}\left(\frac{1}{2}(a+b+1),\frac{1}{2}(a-b+1)\right)}," display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mrow><mi href="./3.12#E1">π</mi><mo>/</mo><mn>2</mn></mrow></msubsup><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi>t</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mi href="./5.1#p2.t1.r5">a</mi><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./5.1#p2.t1.r5">b</mi><mo></mo><mi>t</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow><mo>=</mo><mrow><mfrac><mi href="./3.12#E1">π</mi><msup><mn>2</mn><mi href="./5.1#p2.t1.r5">a</mi></msup></mfrac><mo></mo><mfrac><mn>1</mn><mrow><mi href="./5.1#p2.t1.r5">a</mi><mo></mo><mrow><mi href="./5.12#E1" mathvariant="normal">B</mi><mo></mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./5.1#p2.t1.r5">a</mi><mo>+</mo><mi href="./5.1#p2.t1.r5">b</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mi href="./5.1#p2.t1.r5">a</mi><mo>-</mo><mi href="./5.1#p2.t1.r5">b</mi></mrow><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mfrac></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m9.png" altimg-height="18px" altimg-valign="-3px" altimg-width="66px" alttext="\Re a>0" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./5.1#p2.t1.r5">a</mi></mrow><mo>></mo><mn>0</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E5.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.12#E1" title="(5.12.1) ‣ Euler’s Beta Integral ‣ §5.12 Beta Function ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m21.png" altimg-height="23px" altimg-valign="-7px" altimg-width="65px" alttext="\mathrm{B}\left(\NVar{a},\NVar{b}\right)" display="inline"><mrow><mi href="./5.12#E1" mathvariant="normal">B</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r5">a</mi><mo>,</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r5">b</mi><mo>)</mo></mrow></mrow></math>: beta function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m25.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m17.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m34.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m16.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./5.1#p2.t1.r5" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m28.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./5.1#p2.t1.r5">a</mi></math>: real or complex variable</a> and
<a href="./5.1#p2.t1.r5" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="b" display="inline"><mi href="./5.1#p2.t1.r5">b</mi></math>: real or complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E6" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">5.12.6</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E6.png" altimg-height="58px" altimg-valign="-24px" altimg-width="513px" alttext="\int_{0}^{\pi}(\sin t)^{a-1}e^{ibt}\mathrm{d}t=\frac{\pi}{2^{a-1}}\frac{e^{i%
\pi b/2}}{a\mathrm{B}\left(\frac{1}{2}(a+b+1),\frac{1}{2}(a-b+1)\right)}," display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi href="./3.12#E1">π</mi></msubsup><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi>t</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mi href="./5.1#p2.t1.r5">a</mi><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./5.1#p2.t1.r5">b</mi><mo></mo><mi>t</mi></mrow></msup><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow><mo>=</mo><mrow><mfrac><mi href="./3.12#E1">π</mi><msup><mn>2</mn><mrow><mi href="./5.1#p2.t1.r5">a</mi><mo>-</mo><mn>1</mn></mrow></msup></mfrac><mo></mo><mfrac><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi href="./5.1#p2.t1.r5">b</mi></mrow><mo>/</mo><mn>2</mn></mrow></msup><mrow><mi href="./5.1#p2.t1.r5">a</mi><mo></mo><mrow><mi href="./5.12#E1" mathvariant="normal">B</mi><mo></mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./5.1#p2.t1.r5">a</mi><mo>+</mo><mi href="./5.1#p2.t1.r5">b</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mi href="./5.1#p2.t1.r5">a</mi><mo>-</mo><mi href="./5.1#p2.t1.r5">b</mi></mrow><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mfrac></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m9.png" altimg-height="18px" altimg-valign="-3px" altimg-width="66px" alttext="\Re a>0" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./5.1#p2.t1.r5">a</mi></mrow><mo>></mo><mn>0</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E6.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.12#E1" title="(5.12.1) ‣ Euler’s Beta Integral ‣ §5.12 Beta Function ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m21.png" altimg-height="23px" altimg-valign="-7px" altimg-width="65px" alttext="\mathrm{B}\left(\NVar{a},\NVar{b}\right)" display="inline"><mrow><mi href="./5.12#E1" mathvariant="normal">B</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r5">a</mi><mo>,</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r5">b</mi><mo>)</mo></mrow></mrow></math>: beta function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m25.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m34.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m23.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m16.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m26.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./5.1#p2.t1.r5" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m28.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./5.1#p2.t1.r5">a</mi></math>: real or complex variable</a> and
<a href="./5.1#p2.t1.r5" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="b" display="inline"><mi href="./5.1#p2.t1.r5">b</mi></math>: real or complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E7" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">5.12.7</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E7.png" altimg-height="53px" altimg-valign="-21px" altimg-width="356px" alttext="\int_{0}^{\infty}\frac{\cosh\left(2bt\right)}{(\cosh t)^{2a}}\mathrm{d}t=4^{a-%
1}\mathrm{B}\left(a+b,a-b\right)," display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><mrow><mfrac><mrow><mi href="./4.28#E2">cosh</mi><mo></mo><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><mi href="./5.1#p2.t1.r5">b</mi><mo></mo><mi>t</mi></mrow><mo>)</mo></mrow></mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./4.28#E2">cosh</mi><mo></mo><mi>t</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mn>2</mn><mo></mo><mi href="./5.1#p2.t1.r5">a</mi></mrow></msup></mfrac><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow><mo>=</mo><mrow><msup><mn>4</mn><mrow><mi href="./5.1#p2.t1.r5">a</mi><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mrow><mi href="./5.12#E1" mathvariant="normal">B</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./5.1#p2.t1.r5">a</mi><mo>+</mo><mi href="./5.1#p2.t1.r5">b</mi></mrow><mo>,</mo><mrow><mi href="./5.1#p2.t1.r5">a</mi><mo>-</mo><mi href="./5.1#p2.t1.r5">b</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m10.png" altimg-height="23px" altimg-valign="-7px" altimg-width="90px" alttext="\Re a>|\Re b|" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./5.1#p2.t1.r5">a</mi></mrow><mo>></mo><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./5.1#p2.t1.r5">b</mi></mrow><mo stretchy="false">|</mo></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E7.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.12#E1" title="(5.12.1) ‣ Euler’s Beta Integral ‣ §5.12 Beta Function ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m21.png" altimg-height="23px" altimg-valign="-7px" altimg-width="65px" alttext="\mathrm{B}\left(\NVar{a},\NVar{b}\right)" display="inline"><mrow><mi href="./5.12#E1" mathvariant="normal">B</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r5">a</mi><mo>,</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r5">b</mi><mo>)</mo></mrow></mrow></math>: beta function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m34.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.28#E2" title="(4.28.2) ‣ §4.28 Definitions and Periodicity ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m18.png" altimg-height="18px" altimg-valign="-2px" altimg-width="56px" alttext="\cosh\NVar{z}" display="inline"><mrow><mi href="./4.28#E2">cosh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: hyperbolic cosine function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m16.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./5.1#p2.t1.r5" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m28.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./5.1#p2.t1.r5">a</mi></math>: real or complex variable</a> and
<a href="./5.1#p2.t1.r5" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="b" display="inline"><mi href="./5.1#p2.t1.r5">b</mi></math>: real or complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E8" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">5.12.8</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E8.png" altimg-height="56px" altimg-valign="-22px" altimg-width="432px" alttext="{\frac{1}{2\pi}\int_{-\infty}^{\infty}\frac{\mathrm{d}t}{(w+it)^{a}(z-it)^{b}}%
=\frac{(w+z)^{1-a-b}}{(a+b-1)\mathrm{B}\left(a,b\right)}}," display="block"><mrow><mrow><mrow><mfrac><mn>1</mn><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi></mrow></mfrac><mo></mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mi mathvariant="normal">∞</mi></msubsup><mfrac><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./5.1#p2.t1.r5">w</mi><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi>t</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mi href="./5.1#p2.t1.r5">a</mi></msup><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./5.1#p2.t1.r4">z</mi><mo>-</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi>t</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mi href="./5.1#p2.t1.r5">b</mi></msup></mrow></mfrac></mrow></mrow><mo>=</mo><mfrac><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./5.1#p2.t1.r5">w</mi><mo>+</mo><mi href="./5.1#p2.t1.r4">z</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>-</mo><mi href="./5.1#p2.t1.r5">a</mi><mo>-</mo><mi href="./5.1#p2.t1.r5">b</mi></mrow></msup><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mi href="./5.1#p2.t1.r5">a</mi><mo>+</mo><mi href="./5.1#p2.t1.r5">b</mi></mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mi href="./5.12#E1" mathvariant="normal">B</mi><mo></mo><mrow><mo>(</mo><mi href="./5.1#p2.t1.r5">a</mi><mo>,</mo><mi href="./5.1#p2.t1.r5">b</mi><mo>)</mo></mrow></mrow></mrow></mfrac></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m15.png" altimg-height="23px" altimg-valign="-7px" altimg-width="114px" alttext="\Re(a+b)>1" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./5.1#p2.t1.r5">a</mi><mo>+</mo><mi href="./5.1#p2.t1.r5">b</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>></mo><mn>1</mn></mrow></math>, <math class="ltx_Math" altimg="m12.png" altimg-height="18px" altimg-valign="-3px" altimg-width="70px" alttext="\Re w>0" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./5.1#p2.t1.r5">w</mi></mrow><mo>></mo><mn>0</mn></mrow></math>, <math class="ltx_Math" altimg="m13.png" altimg-height="18px" altimg-valign="-3px" altimg-width="65px" alttext="\Re z>0" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./5.1#p2.t1.r4">z</mi></mrow><mo>></mo><mn>0</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E8.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.12#E1" title="(5.12.1) ‣ Euler’s Beta Integral ‣ §5.12 Beta Function ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m21.png" altimg-height="23px" altimg-valign="-7px" altimg-width="65px" alttext="\mathrm{B}\left(\NVar{a},\NVar{b}\right)" display="inline"><mrow><mi href="./5.12#E1" mathvariant="normal">B</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r5">a</mi><mo>,</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r5">b</mi><mo>)</mo></mrow></mrow></math>: beta function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m25.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m34.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m16.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./5.1#p2.t1.r5" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m33.png" altimg-height="13px" altimg-valign="-2px" altimg-width="19px" alttext="w" display="inline"><mi href="./5.1#p2.t1.r5">w</mi></math>: real or complex variable</a>,
<a href="./5.1#p2.t1.r4" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m36.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./5.1#p2.t1.r4">z</mi></math>: complex variable</a>,
<a href="./5.1#p2.t1.r5" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m28.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./5.1#p2.t1.r5">a</mi></math>: real or complex variable</a> and
<a href="./5.1#p2.t1.r5" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="b" display="inline"><mi href="./5.1#p2.t1.r5">b</mi></math>: real or complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">In () the fractional powers have their principal values when
<math class="ltx_Math" altimg="m32.png" altimg-height="17px" altimg-valign="-3px" altimg-width="56px" alttext="w>0" display="inline"><mrow><mi href="./5.1#p2.t1.r5">w</mi><mo>></mo><mn>0</mn></mrow></math> and <math class="ltx_Math" altimg="m35.png" altimg-height="17px" altimg-valign="-3px" altimg-width="51px" alttext="z>0" display="inline"><mrow><mi href="./5.1#p2.t1.r4">z</mi><mo>></mo><mn>0</mn></mrow></math>, and are continued via continuity.</p>
<table id="E9" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">5.12.9</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E9.png" altimg-height="57px" altimg-valign="-22px" altimg-width="365px" alttext="{\frac{1}{2\pi i}\int_{c-\infty i}^{c+\infty i}t^{-a}(1-t)^{-1-b}\mathrm{d}t=%
\frac{1}{b\mathrm{B}\left(a,b\right)}}," display="block"><mrow><mrow><mrow><mfrac><mn>1</mn><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi></mrow></mfrac><mo></mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><mi>c</mi><mo>-</mo><mrow><mi mathvariant="normal">∞</mi><mo></mo><mi mathvariant="normal">i</mi></mrow></mrow><mrow><mi>c</mi><mo>+</mo><mrow><mi mathvariant="normal">∞</mi><mo></mo><mi mathvariant="normal">i</mi></mrow></mrow></msubsup><mrow><msup><mi>t</mi><mrow><mo>-</mo><mi href="./5.1#p2.t1.r5">a</mi></mrow></msup><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><mi>t</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mrow><mo>-</mo><mn>1</mn></mrow><mo>-</mo><mi href="./5.1#p2.t1.r5">b</mi></mrow></msup><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow><mo>=</mo><mfrac><mn>1</mn><mrow><mi href="./5.1#p2.t1.r5">b</mi><mo></mo><mrow><mi href="./5.12#E1" mathvariant="normal">B</mi><mo></mo><mrow><mo>(</mo><mi href="./5.1#p2.t1.r5">a</mi><mo>,</mo><mi href="./5.1#p2.t1.r5">b</mi><mo>)</mo></mrow></mrow></mrow></mfrac></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m3.png" altimg-height="17px" altimg-valign="-3px" altimg-width="86px" alttext="0<c<1" display="inline"><mrow><mn>0</mn><mo><</mo><mi>c</mi><mo><</mo><mn>1</mn></mrow></math>, <math class="ltx_Math" altimg="m14.png" altimg-height="23px" altimg-valign="-7px" altimg-width="114px" alttext="\Re(a+b)>0" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./5.1#p2.t1.r5">a</mi><mo>+</mo><mi href="./5.1#p2.t1.r5">b</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>></mo><mn>0</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E9.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.12#E1" title="(5.12.1) ‣ Euler’s Beta Integral ‣ §5.12 Beta Function ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m21.png" altimg-height="23px" altimg-valign="-7px" altimg-width="65px" alttext="\mathrm{B}\left(\NVar{a},\NVar{b}\right)" display="inline"><mrow><mi href="./5.12#E1" mathvariant="normal">B</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r5">a</mi><mo>,</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r5">b</mi><mo>)</mo></mrow></mrow></math>: beta function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m25.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m34.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m16.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./5.1#p2.t1.r5" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m28.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./5.1#p2.t1.r5">a</mi></math>: real or complex variable</a> and
<a href="./5.1#p2.t1.r5" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="b" display="inline"><mi href="./5.1#p2.t1.r5">b</mi></math>: real or complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E10" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">5.12.10</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E10.png" altimg-height="56px" altimg-valign="-20px" altimg-width="408px" alttext="{\frac{1}{2\pi i}\int_{0}^{(1+)}t^{a-1}(t-1)^{b-1}\mathrm{d}t=\frac{\sin\left(%
\pi b\right)}{\pi}\mathrm{B}\left(a,b\right)}," display="block"><mrow><mrow><mrow><mfrac><mn>1</mn><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi></mrow></mfrac><mo></mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>+</mo></mrow><mo stretchy="false">)</mo></mrow></msubsup><mrow><msup><mi>t</mi><mrow><mi href="./5.1#p2.t1.r5">a</mi><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi>t</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mrow><mi href="./5.1#p2.t1.r5">b</mi><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow><mo>=</mo><mrow><mfrac><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi href="./5.1#p2.t1.r5">b</mi></mrow><mo>)</mo></mrow></mrow><mi href="./3.12#E1">π</mi></mfrac><mo></mo><mrow><mi href="./5.12#E1" mathvariant="normal">B</mi><mo></mo><mrow><mo>(</mo><mi href="./5.1#p2.t1.r5">a</mi><mo>,</mo><mi href="./5.1#p2.t1.r5">b</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m9.png" altimg-height="18px" altimg-valign="-3px" altimg-width="66px" alttext="\Re a>0" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./5.1#p2.t1.r5">a</mi></mrow><mo>></mo><mn>0</mn></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E10.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.12#E1" title="(5.12.1) ‣ Euler’s Beta Integral ‣ §5.12 Beta Function ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m21.png" altimg-height="23px" altimg-valign="-7px" altimg-width="65px" alttext="\mathrm{B}\left(\NVar{a},\NVar{b}\right)" display="inline"><mrow><mi href="./5.12#E1" mathvariant="normal">B</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r5">a</mi><mo>,</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r5">b</mi><mo>)</mo></mrow></mrow></math>: beta function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m25.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m34.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m16.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m26.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./5.1#p2.t1.r5" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m28.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./5.1#p2.t1.r5">a</mi></math>: real or complex variable</a> and
<a href="./5.1#p2.t1.r5" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="b" display="inline"><mi href="./5.1#p2.t1.r5">b</mi></math>: real or complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">with the contour as shown in Figure <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 5.12.1: </span><math class="ltx_Math" altimg="m31.png" altimg-height="17px" altimg-valign="-2px" altimg-width="11px" alttext="t" display="inline"><mi>t</mi></math>-plane. Contour for first loop integral for the beta function.
) the fractional powers are
continuous on the integration paths and take their principal values at the
beginning.</p>
<table id="E11" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">5.12.11</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E11.png" altimg-height="56px" altimg-valign="-20px" altimg-width="398px" alttext="\frac{1}{e^{2\pi ia}-1}\int_{\infty}^{(0+)}t^{a-1}(1+t)^{-a-b}\mathrm{d}t=%
\mathrm{B}\left(a,b\right)," display="block"><mrow><mrow><mrow><mfrac><mn>1</mn><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi><mo></mo><mi href="./5.1#p2.t1.r5">a</mi></mrow></msup><mo>-</mo><mn>1</mn></mrow></mfrac><mo></mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi mathvariant="normal">∞</mi><mrow><mo stretchy="false">(</mo><mrow><mn>0</mn><mo>+</mo></mrow><mo stretchy="false">)</mo></mrow></msubsup><mrow><msup><mi>t</mi><mrow><mi href="./5.1#p2.t1.r5">a</mi><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>+</mo><mi>t</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mrow><mo>-</mo><mi href="./5.1#p2.t1.r5">a</mi></mrow><mo>-</mo><mi href="./5.1#p2.t1.r5">b</mi></mrow></msup><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow><mo>=</mo><mrow><mi href="./5.12#E1" mathvariant="normal">B</mi><mo></mo><mrow><mo>(</mo><mi href="./5.1#p2.t1.r5">a</mi><mo>,</mo><mi href="./5.1#p2.t1.r5">b</mi><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E11.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.12#E1" title="(5.12.1) ‣ Euler’s Beta Integral ‣ §5.12 Beta Function ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m21.png" altimg-height="23px" altimg-valign="-7px" altimg-width="65px" alttext="\mathrm{B}\left(\NVar{a},\NVar{b}\right)" display="inline"><mrow><mi href="./5.12#E1" mathvariant="normal">B</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r5">a</mi><mo>,</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r5">b</mi><mo>)</mo></mrow></mrow></math>: beta function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m25.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m34.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m23.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./5.1#p2.t1.r5" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m28.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./5.1#p2.t1.r5">a</mi></math>: real or complex variable</a> and
<a href="./5.1#p2.t1.r5" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="b" display="inline"><mi href="./5.1#p2.t1.r5">b</mi></math>: real or complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">when <math class="ltx_Math" altimg="m11.png" altimg-height="18px" altimg-valign="-3px" altimg-width="64px" alttext="\Re b>0" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./5.1#p2.t1.r5">b</mi></mrow><mo>></mo><mn>0</mn></mrow></math>, <math class="ltx_Math" altimg="m28.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./5.1#p2.t1.r5">a</mi></math> is not an integer and the contour cuts the real
axis between <math class="ltx_Math" altimg="m2.png" altimg-height="18px" altimg-valign="-4px" altimg-width="30px" alttext="-1" display="inline"><mrow><mo>-</mo><mn>1</mn></mrow></math> and the origin. See Figure <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 5.12.2: </span><math class="ltx_Math" altimg="m31.png" altimg-height="17px" altimg-valign="-2px" altimg-width="11px" alttext="t" display="inline"><mi>t</mi></math>-plane. Contour for second loop integral for the beta function.
</dd>
</dl>
</div>
</div>
<div id="Px2.p1" class="ltx_para">
<p class="ltx_p">When <math class="ltx_Math" altimg="m27.png" altimg-height="21px" altimg-valign="-6px" altimg-width="71px" alttext="a,b\in\mathbb{C}" display="inline"><mrow><mrow><mi href="./5.1#p2.t1.r5">a</mi><mo>,</mo><mi href="./5.1#p2.t1.r5">b</mi></mrow><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mi href="./front/introduction#Sx4.p1.t1.r1">ℂ</mi></mrow></math>
</p>
<table id="E12" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">5.12.12</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E12.png" altimg-height="56px" altimg-valign="-20px" altimg-width="607px" alttext="\int_{P}^{(1+,0+,1-,0-)}t^{a-1}(1-t)^{b-1}\mathrm{d}t=-4e^{\pi i(a+b)}\sin%
\left(\pi a\right)\sin\left(\pi b\right)\mathrm{B}\left(a,b\right)," display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi href="./5.12#Px2.p1">P</mi><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>+</mo></mrow><mo>,</mo><mrow><mn>0</mn><mo>+</mo></mrow><mo>,</mo><mrow><mn>1</mn><mo>-</mo></mrow><mo>,</mo><mrow><mn>0</mn><mo>-</mo></mrow><mo stretchy="false">)</mo></mrow></msubsup><mrow><msup><mi>t</mi><mrow><mi href="./5.1#p2.t1.r5">a</mi><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><mi>t</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mi href="./5.1#p2.t1.r5">b</mi><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow><mo>=</mo><mrow><mo>-</mo><mrow><mn>4</mn><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./5.1#p2.t1.r5">a</mi><mo>+</mo><mi href="./5.1#p2.t1.r5">b</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></msup><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi href="./5.1#p2.t1.r5">a</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi href="./5.1#p2.t1.r5">b</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.12#E1" mathvariant="normal">B</mi><mo></mo><mrow><mo>(</mo><mi href="./5.1#p2.t1.r5">a</mi><mo>,</mo><mi href="./5.1#p2.t1.r5">b</mi><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E12.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.12#E1" title="(5.12.1) ‣ Euler’s Beta Integral ‣ §5.12 Beta Function ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m21.png" altimg-height="23px" altimg-valign="-7px" altimg-width="65px" alttext="\mathrm{B}\left(\NVar{a},\NVar{b}\right)" display="inline"><mrow><mi href="./5.12#E1" mathvariant="normal">B</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r5">a</mi><mo>,</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r5">b</mi><mo>)</mo></mrow></mrow></math>: beta function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m25.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m34.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m23.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m26.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./5.1#p2.t1.r5" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m28.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./5.1#p2.t1.r5">a</mi></math>: real or complex variable</a>,
<a href="./5.1#p2.t1.r5" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="b" display="inline"><mi href="./5.1#p2.t1.r5">b</mi></math>: real or complex variable</a> and
<a href="./5.12#Px2.p1" title="Pochhammer’s Integral ‣ §5.12 Beta Function ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m7.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="P" display="inline"><mi href="./5.12#Px2.p1">P</mi></math>: point</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where the contour starts from an arbitrary point <math class="ltx_Math" altimg="m7.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="P" display="inline"><mi href="./5.12#Px2.p1">P</mi></math> in the interval <math class="ltx_Math" altimg="m1.png" altimg-height="23px" altimg-valign="-7px" altimg-width="49px" alttext="(0,1)" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mn>0</mn><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mn>1</mn><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math>,
circles <math class="ltx_Math" altimg="m5.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="1" display="inline"><mn>1</mn></math> and then <math class="ltx_Math" altimg="m4.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math> in the positive sense, circles <math class="ltx_Math" altimg="m5.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="1" display="inline"><mn>1</mn></math> and then <math class="ltx_Math" altimg="m4.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math> in the
negative sense, and returns to <math class="ltx_Math" altimg="m7.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="P" display="inline"><mi href="./5.12#Px2.p1">P</mi></math>. It can always be deformed into the contour
shown in Figure <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 5.12.3: </span><math class="ltx_Math" altimg="m31.png" altimg-height="17px" altimg-valign="-2px" altimg-width="11px" alttext="t" display="inline"><mi>t</mi></math>-plane. Contour for Pochhammer’s integral.
</div>
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<title>DLMF: 31.11 Expansions in Series of Hypergeometric Functions</title>
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<div class="ltx_page_navlogo"></dd>
</dl>
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</div>
<div id="SS2.p1" class="ltx_para">
<p class="ltx_p">Let <math class="ltx_Math" altimg="m46.png" altimg-height="23px" altimg-valign="-7px" altimg-width="45px" alttext="w(z)" display="inline"><mrow><mi>w</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./31.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> be any Fuchs–Frobenius solution of Heun’s equation. Expand</p>
<table id="E1" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">31.11.1</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E1.png" altimg-height="67px" altimg-valign="-30px" altimg-width="149px" alttext="w(z)=\sum_{j=0}^{\infty}c_{j}P_{j}," display="block"><mrow><mrow><mrow><mi>w</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./31.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./31.1#p2.t1.r3">j</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><msub><mi href="./31.11#SS2.p1">c</mi><mi href="./31.1#p2.t1.r3">j</mi></msub><mo></mo><msub><mi href="./31.11#E2">P</mi><mi href="./31.1#p2.t1.r3">j</mi></msub></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./31.1#p2.t1.r2" title="§31.1 Special Notation ‣ Notation ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./31.1#p2.t1.r2">z</mi></math>: complex variable</a>,
<a href="./31.1#p2.t1.r3" title="§31.1 Special Notation ‣ Notation ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="20px" altimg-valign="-6px" altimg-width="14px" alttext="j" display="inline"><mi href="./31.1#p2.t1.r3">j</mi></math>: nonnegative integer</a>,
<a href="./31.11#E2" title="(31.11.2) ‣ §31.11(ii) General Form ‣ §31.11 Expansions in Series of Hypergeometric Functions ‣ Properties ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m27.png" altimg-height="23px" altimg-valign="-8px" altimg-width="25px" alttext="P_{j}" display="inline"><msub><mi href="./31.11#E2">P</mi><mi href="./31.1#p2.t1.r3">j</mi></msub></math></a> and
<a href="./31.11#SS2.p1" title="§31.11(ii) General Form ‣ §31.11 Expansions in Series of Hypergeometric Functions ‣ Properties ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m41.png" altimg-height="18px" altimg-valign="-8px" altimg-width="21px" alttext="c_{j}" display="inline"><msub><mi href="./31.11#SS2.p1">c</mi><mi href="./31.1#p2.t1.r3">j</mi></msub></math>: coefficients</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where (§)</p>
<table id="E2" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">31.11.2</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E2.png" altimg-height="77px" altimg-valign="-33px" altimg-width="322px" alttext="P_{j}=P\begin{Bmatrix}0&1&\infty&\\
0&0&\lambda+j&z\\
1-\gamma&1-\delta&\mu-j&\end{Bmatrix}," display="block"><mrow><mrow><msub><mi href="./31.11#E2">P</mi><mi href="./31.1#p2.t1.r3">j</mi></msub><mo>=</mo><mrow><mi href="./15.11#E3">P</mi><mo></mo><mrow><mo>{</mo><mtable columnspacing="5pt" displaystyle="true" rowspacing="0pt"><mtr><mtd columnalign="center"><mn>0</mn></mtd><mtd columnalign="center"><mn>1</mn></mtd><mtd columnalign="center"><mi mathvariant="normal">∞</mi></mtd><mtd columnalign="center"><mi></mi></mtd></mtr><mtr><mtd columnalign="center"><mn>0</mn></mtd><mtd columnalign="center"><mn>0</mn></mtd><mtd columnalign="center"><mrow><mi href="./31.11#E3">λ</mi><mo>+</mo><mi href="./31.1#p2.t1.r3">j</mi></mrow></mtd><mtd columnalign="center"><mi href="./31.1#p2.t1.r2">z</mi></mtd></mtr><mtr><mtd columnalign="center"><mrow><mn>1</mn><mo>-</mo><mi href="./31.1#p2.t1.r5">γ</mi></mrow></mtd><mtd columnalign="center"><mrow><mn>1</mn><mo>-</mo><mi href="./31.1#p2.t1.r5">δ</mi></mrow></mtd><mtd columnalign="center"><mrow><mi href="./31.11#E3">μ</mi><mo>-</mo><mi href="./31.1#p2.t1.r3">j</mi></mrow></mtd><mtd columnalign="center"><mi></mi></mtd></mtr></mtable><mo>}</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text"><math class="ltx_Math" altimg="m27.png" altimg-height="23px" altimg-valign="-8px" altimg-width="25px" alttext="P_{j}" display="inline"><msub><mi href="./31.11#E2">P</mi><mi href="./31.1#p2.t1.r3">j</mi></msub></math> (locally)</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./15.11#E3" title="(15.11.3) ‣ §15.11(i) Equations with Three Singularities ‣ §15.11 Riemann’s Differential Equation ‣ Properties ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m25.png" altimg-height="75px" altimg-valign="-33px" altimg-width="180px" alttext="P\NVar{\begin{Bmatrix}\alpha&\beta&\gamma&\\
a_{1}&b_{1}&c_{1}&z\\
a_{2}&b_{2}&c_{2}&\end{Bmatrix}}" display="inline"><mrow><mi href="./15.11#E3">P</mi><mo></mo><mrow class="ltx_nvar"><mo class="ltx_nvar">{</mo><mtable class="ltx_nvar" columnspacing="5pt" rowspacing="0pt"><mtr><mtd class="ltx_nvar" columnalign="center"><mi class="ltx_nvar" href="./15.11#SS1.p1">α</mi></mtd><mtd class="ltx_nvar" columnalign="center"><mi class="ltx_nvar" href="./15.11#SS1.p1">β</mi></mtd><mtd class="ltx_nvar" columnalign="center"><mi class="ltx_nvar" href="./15.11#SS1.p1">γ</mi></mtd><mtd class="ltx_nvar" columnalign="center"><mi class="ltx_nvar"></mi></mtd></mtr><mtr><mtd class="ltx_nvar" columnalign="center"><msub class="ltx_nvar"><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mn class="ltx_nvar">1</mn></msub></mtd><mtd class="ltx_nvar" columnalign="center"><msub class="ltx_nvar"><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi><mn class="ltx_nvar">1</mn></msub></mtd><mtd class="ltx_nvar" columnalign="center"><msub class="ltx_nvar"><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mn class="ltx_nvar">1</mn></msub></mtd><mtd class="ltx_nvar" columnalign="center"><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi></mtd></mtr><mtr><mtd class="ltx_nvar" columnalign="center"><msub class="ltx_nvar"><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mn class="ltx_nvar">2</mn></msub></mtd><mtd class="ltx_nvar" columnalign="center"><msub class="ltx_nvar"><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi><mn class="ltx_nvar">2</mn></msub></mtd><mtd class="ltx_nvar" columnalign="center"><msub class="ltx_nvar"><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mn class="ltx_nvar">2</mn></msub></mtd><mtd class="ltx_nvar" columnalign="center"><mi class="ltx_nvar"></mi></mtd></mtr></mtable><mo class="ltx_nvar">}</mo></mrow></mrow></math>: Riemann’s <math class="ltx_Math" altimg="m26.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="P" display="inline"><mi href="./15.11#E3">P</mi></math>-symbol for solutions of the generalized hypergeometric differential equation</a>,
<a href="./31.1#p2.t1.r2" title="§31.1 Special Notation ‣ Notation ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./31.1#p2.t1.r2">z</mi></math>: complex variable</a>,
<a href="./31.1#p2.t1.r5" title="§31.1 Special Notation ‣ Notation ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m34.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\gamma" display="inline"><mi href="./31.1#p2.t1.r5">γ</mi></math>: real or complex parameter</a>,
<a href="./31.1#p2.t1.r5" title="§31.1 Special Notation ‣ Notation ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m32.png" altimg-height="18px" altimg-valign="-2px" altimg-width="14px" alttext="\delta" display="inline"><mi href="./31.1#p2.t1.r5">δ</mi></math>: real or complex parameter</a>,
<a href="./31.1#p2.t1.r3" title="§31.1 Special Notation ‣ Notation ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="20px" altimg-valign="-6px" altimg-width="14px" alttext="j" display="inline"><mi href="./31.1#p2.t1.r3">j</mi></math>: nonnegative integer</a>,
<a href="./31.11#E3" title="(31.11.3) ‣ §31.11(ii) General Form ‣ §31.11 Expansions in Series of Hypergeometric Functions ‣ Properties ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m36.png" altimg-height="18px" altimg-valign="-2px" altimg-width="16px" alttext="\lambda" display="inline"><mi href="./31.11#E3">λ</mi></math></a> and
<a href="./31.11#E3" title="(31.11.3) ‣ §31.11(ii) General Form ‣ §31.11 Expansions in Series of Hypergeometric Functions ‣ Properties ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\mu" display="inline"><mi href="./31.11#E3">μ</mi></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">with</p>
<table id="E3" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">31.11.3</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E3.png" altimg-height="23px" altimg-valign="-6px" altimg-width="274px" alttext="\lambda+\mu=\gamma+\delta-1=\alpha+\beta-\epsilon." display="block"><mrow><mrow><mrow><mi href="./31.11#E3">λ</mi><mo>+</mo><mi href="./31.11#E3">μ</mi></mrow><mo>=</mo><mrow><mrow><mi href="./31.1#p2.t1.r5">γ</mi><mo>+</mo><mi href="./31.1#p2.t1.r5">δ</mi></mrow><mo>-</mo><mn>1</mn></mrow><mo>=</mo><mrow><mrow><mi href="./31.1#p2.t1.r5">α</mi><mo>+</mo><mi href="./31.1#p2.t1.r5">β</mi></mrow><mo>-</mo><mi href="./31.1#p2.t1.r5">ϵ</mi></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd>
<span class="ltx_text"><math class="ltx_Math" altimg="m36.png" altimg-height="18px" altimg-valign="-2px" altimg-width="16px" alttext="\lambda" display="inline"><mi href="./31.11#E3">λ</mi></math> (locally)</span> and
<span class="ltx_text"><math class="ltx_Math" altimg="m39.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\mu" display="inline"><mi href="./31.11#E3">μ</mi></math> (locally)</span>
</dd>
<dt>Symbols:</dt>
<dd>
<a href="./31.1#p2.t1.r5" title="§31.1 Special Notation ‣ Notation ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m34.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\gamma" display="inline"><mi href="./31.1#p2.t1.r5">γ</mi></math>: real or complex parameter</a>,
<a href="./31.1#p2.t1.r5" title="§31.1 Special Notation ‣ Notation ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m32.png" altimg-height="18px" altimg-valign="-2px" altimg-width="14px" alttext="\delta" display="inline"><mi href="./31.1#p2.t1.r5">δ</mi></math>: real or complex parameter</a>,
<a href="./31.1#p2.t1.r5" title="§31.1 Special Notation ‣ Notation ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m33.png" altimg-height="13px" altimg-valign="-2px" altimg-width="12px" alttext="\epsilon" display="inline"><mi href="./31.1#p2.t1.r5">ϵ</mi></math>: real or complex parameter</a>,
<a href="./31.1#p2.t1.r5" title="§31.1 Special Notation ‣ Notation ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m30.png" altimg-height="13px" altimg-valign="-2px" altimg-width="17px" alttext="\alpha" display="inline"><mi href="./31.1#p2.t1.r5">α</mi></math>: real or complex parameter</a> and
<a href="./31.1#p2.t1.r5" title="§31.1 Special Notation ‣ Notation ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m31.png" altimg-height="21px" altimg-valign="-6px" altimg-width="17px" alttext="\beta" display="inline"><mi href="./31.1#p2.t1.r5">β</mi></math>: real or complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">The coefficients <math class="ltx_Math" altimg="m41.png" altimg-height="18px" altimg-valign="-8px" altimg-width="21px" alttext="c_{j}" display="inline"><msub><mi href="./31.11#SS2.p1">c</mi><mi href="./31.1#p2.t1.r3">j</mi></msub></math> satisfy the equations
</p>
<table id="E4" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">31.11.4</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E4.png" altimg-height="23px" altimg-valign="-6px" altimg-width="158px" alttext="L_{0}c_{0}+M_{0}c_{1}=0," display="block"><mrow><mrow><mrow><mrow><msub><mi href="./31.11#EGx1">L</mi><mn>0</mn></msub><mo></mo><msub><mi href="./31.11#SS2.p1">c</mi><mn>0</mn></msub></mrow><mo>+</mo><mrow><msub><mi href="./31.11#EGx1">M</mi><mn>0</mn></msub><mo></mo><msub><mi href="./31.11#SS2.p1">c</mi><mn>1</mn></msub></mrow></mrow><mo>=</mo><mn>0</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E4.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./31.11#SS2.p1" title="§31.11(ii) General Form ‣ §31.11 Expansions in Series of Hypergeometric Functions ‣ Properties ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m41.png" altimg-height="18px" altimg-valign="-8px" altimg-width="21px" alttext="c_{j}" display="inline"><msub><mi href="./31.11#SS2.p1">c</mi><mi href="./31.1#p2.t1.r3">j</mi></msub></math>: coefficients</a>,
<a href="./31.11#EGx1" title="§31.11(ii) General Form ‣ §31.11 Expansions in Series of Hypergeometric Functions ‣ Properties ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m23.png" altimg-height="23px" altimg-valign="-8px" altimg-width="26px" alttext="L_{j}" display="inline"><msub><mi href="./31.11#EGx1">L</mi><mi href="./31.1#p2.t1.r3">j</mi></msub></math>: coefficients</a> and
<a href="./31.11#EGx1" title="§31.11(ii) General Form ‣ §31.11 Expansions in Series of Hypergeometric Functions ‣ Properties ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m24.png" altimg-height="23px" altimg-valign="-8px" altimg-width="32px" alttext="M_{j}" display="inline"><msub><mi href="./31.11#EGx1">M</mi><mi href="./31.1#p2.t1.r3">j</mi></msub></math>: coefficients</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E5" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">31.11.5</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E5.png" altimg-height="25px" altimg-valign="-8px" altimg-width="263px" alttext="K_{j}c_{j-1}+L_{j}c_{j}+M_{j}c_{j+1}=0," display="block"><mrow><mrow><mrow><mrow><msub><mi href="./31.11#EGx1">K</mi><mi href="./31.1#p2.t1.r3">j</mi></msub><mo></mo><msub><mi href="./31.11#SS2.p1">c</mi><mrow><mi href="./31.1#p2.t1.r3">j</mi><mo>-</mo><mn>1</mn></mrow></msub></mrow><mo>+</mo><mrow><msub><mi href="./31.11#EGx1">L</mi><mi href="./31.1#p2.t1.r3">j</mi></msub><mo></mo><msub><mi href="./31.11#SS2.p1">c</mi><mi href="./31.1#p2.t1.r3">j</mi></msub></mrow><mo>+</mo><mrow><msub><mi href="./31.11#EGx1">M</mi><mi href="./31.1#p2.t1.r3">j</mi></msub><mo></mo><msub><mi href="./31.11#SS2.p1">c</mi><mrow><mi href="./31.1#p2.t1.r3">j</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></mrow><mo>=</mo><mn>0</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m42.png" altimg-height="20px" altimg-valign="-6px" altimg-width="101px" alttext="j=1,2,\dots" display="inline"><mrow><mi href="./31.1#p2.t1.r3">j</mi><mo>=</mo><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E5.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./31.1#p2.t1.r3" title="§31.1 Special Notation ‣ Notation ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="20px" altimg-valign="-6px" altimg-width="14px" alttext="j" display="inline"><mi href="./31.1#p2.t1.r3">j</mi></math>: nonnegative integer</a>,
<a href="./31.11#SS2.p1" title="§31.11(ii) General Form ‣ §31.11 Expansions in Series of Hypergeometric Functions ‣ Properties ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m41.png" altimg-height="18px" altimg-valign="-8px" altimg-width="21px" alttext="c_{j}" display="inline"><msub><mi href="./31.11#SS2.p1">c</mi><mi href="./31.1#p2.t1.r3">j</mi></msub></math>: coefficients</a>,
<a href="./31.11#EGx1" title="§31.11(ii) General Form ‣ §31.11 Expansions in Series of Hypergeometric Functions ‣ Properties ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="23px" altimg-valign="-8px" altimg-width="30px" alttext="K_{j}" display="inline"><msub><mi href="./31.11#EGx1">K</mi><mi href="./31.1#p2.t1.r3">j</mi></msub></math>: coefficients</a>,
<a href="./31.11#EGx1" title="§31.11(ii) General Form ‣ §31.11 Expansions in Series of Hypergeometric Functions ‣ Properties ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m23.png" altimg-height="23px" altimg-valign="-8px" altimg-width="26px" alttext="L_{j}" display="inline"><msub><mi href="./31.11#EGx1">L</mi><mi href="./31.1#p2.t1.r3">j</mi></msub></math>: coefficients</a> and
<a href="./31.11#EGx1" title="§31.11(ii) General Form ‣ §31.11 Expansions in Series of Hypergeometric Functions ‣ Properties ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m24.png" altimg-height="23px" altimg-valign="-8px" altimg-width="32px" alttext="M_{j}" display="inline"><msub><mi href="./31.11#EGx1">M</mi><mi href="./31.1#p2.t1.r3">j</mi></msub></math>: coefficients</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where
</p>
</div>
<div id="SS2.p2" class="ltx_para">
<table id="EGx1" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E6">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">31.11.6</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m16.png" altimg-height="25px" altimg-valign="-8px" altimg-width="31px" alttext="\displaystyle K_{j}" display="inline"><msub><mi href="./31.11#EGx1">K</mi><mi href="./31.1#p2.t1.r3">j</mi></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m1.png" altimg-height="53px" altimg-valign="-21px" altimg-width="544px" alttext="\displaystyle=-\frac{(j+\alpha-\mu-1)(j+\beta-\mu-1)(j+\gamma-\mu-1)(j+\lambda%
-1)}{(2j+\lambda-\mu-1)(2j+\lambda-\mu-2)}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo>-</mo><mstyle displaystyle="true"><mfrac><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mi href="./31.1#p2.t1.r3">j</mi><mo>+</mo><mi href="./31.1#p2.t1.r5">α</mi></mrow><mo>-</mo><mi href="./31.11#E3">μ</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mi href="./31.1#p2.t1.r3">j</mi><mo>+</mo><mi href="./31.1#p2.t1.r5">β</mi></mrow><mo>-</mo><mi href="./31.11#E3">μ</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mi href="./31.1#p2.t1.r3">j</mi><mo>+</mo><mi href="./31.1#p2.t1.r5">γ</mi></mrow><mo>-</mo><mi href="./31.11#E3">μ</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mi href="./31.1#p2.t1.r3">j</mi><mo>+</mo><mi href="./31.11#E3">λ</mi></mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mrow><mn>2</mn><mo></mo><mi href="./31.1#p2.t1.r3">j</mi></mrow><mo>+</mo><mi href="./31.11#E3">λ</mi></mrow><mo>-</mo><mi href="./31.11#E3">μ</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mrow><mn>2</mn><mo></mo><mi href="./31.1#p2.t1.r3">j</mi></mrow><mo>+</mo><mi href="./31.11#E3">λ</mi></mrow><mo>-</mo><mi href="./31.11#E3">μ</mi><mo>-</mo><mn>2</mn></mrow><mo stretchy="false">)</mo></mrow></mrow></mfrac></mstyle></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E6.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./31.1#p2.t1.r5" title="§31.1 Special Notation ‣ Notation ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m34.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\gamma" display="inline"><mi href="./31.1#p2.t1.r5">γ</mi></math>: real or complex parameter</a>,
<a href="./31.1#p2.t1.r3" title="§31.1 Special Notation ‣ Notation ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="20px" altimg-valign="-6px" altimg-width="14px" alttext="j" display="inline"><mi href="./31.1#p2.t1.r3">j</mi></math>: nonnegative integer</a>,
<a href="./31.1#p2.t1.r5" title="§31.1 Special Notation ‣ Notation ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m30.png" altimg-height="13px" altimg-valign="-2px" altimg-width="17px" alttext="\alpha" display="inline"><mi href="./31.1#p2.t1.r5">α</mi></math>: real or complex parameter</a>,
<a href="./31.1#p2.t1.r5" title="§31.1 Special Notation ‣ Notation ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m31.png" altimg-height="21px" altimg-valign="-6px" altimg-width="17px" alttext="\beta" display="inline"><mi href="./31.1#p2.t1.r5">β</mi></math>: real or complex parameter</a>,
<a href="./31.11#E3" title="(31.11.3) ‣ §31.11(ii) General Form ‣ §31.11 Expansions in Series of Hypergeometric Functions ‣ Properties ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m36.png" altimg-height="18px" altimg-valign="-2px" altimg-width="16px" alttext="\lambda" display="inline"><mi href="./31.11#E3">λ</mi></math></a>,
<a href="./31.11#E3" title="(31.11.3) ‣ §31.11(ii) General Form ‣ §31.11 Expansions in Series of Hypergeometric Functions ‣ Properties ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\mu" display="inline"><mi href="./31.11#E3">μ</mi></math></a> and
<a href="./31.11#EGx1" title="§31.11(ii) General Form ‣ §31.11 Expansions in Series of Hypergeometric Functions ‣ Properties ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="23px" altimg-valign="-8px" altimg-width="30px" alttext="K_{j}" display="inline"><msub><mi href="./31.11#EGx1">K</mi><mi href="./31.1#p2.t1.r3">j</mi></msub></math>: coefficients</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E7">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">31.11.7</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m17.png" altimg-height="25px" altimg-valign="-8px" altimg-width="28px" alttext="\displaystyle L_{j}" display="inline"><msub><mi href="./31.11#EGx1">L</mi><mi href="./31.1#p2.t1.r3">j</mi></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m15.png" altimg-height="106px" altimg-valign="-21px" altimg-width="577px" alttext="\displaystyle=a(\lambda+j)(\mu-j)-q+\frac{(j+\alpha-\mu)(j+\beta-\mu)(j+\gamma%
-\mu)(j+\lambda)}{(2j+\lambda-\mu)(2j+\lambda-\mu+1)}+\frac{(j-\alpha+\lambda)%
(j-\beta+\lambda)(j-\gamma+\lambda)(j-\mu)}{(2j+\lambda-\mu)(2j+\lambda-\mu-1)}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mrow><mi href="./31.1#p2.t1.r4">a</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./31.11#E3">λ</mi><mo>+</mo><mi href="./31.1#p2.t1.r3">j</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./31.11#E3">μ</mi><mo>-</mo><mi href="./31.1#p2.t1.r3">j</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>-</mo><mi href="./31.1#p2.t1.r5">q</mi></mrow><mo>+</mo><mstyle displaystyle="true"><mfrac><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mi href="./31.1#p2.t1.r3">j</mi><mo>+</mo><mi href="./31.1#p2.t1.r5">α</mi></mrow><mo>-</mo><mi href="./31.11#E3">μ</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mi href="./31.1#p2.t1.r3">j</mi><mo>+</mo><mi href="./31.1#p2.t1.r5">β</mi></mrow><mo>-</mo><mi href="./31.11#E3">μ</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mi href="./31.1#p2.t1.r3">j</mi><mo>+</mo><mi href="./31.1#p2.t1.r5">γ</mi></mrow><mo>-</mo><mi href="./31.11#E3">μ</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./31.1#p2.t1.r3">j</mi><mo>+</mo><mi href="./31.11#E3">λ</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mrow><mn>2</mn><mo></mo><mi href="./31.1#p2.t1.r3">j</mi></mrow><mo>+</mo><mi href="./31.11#E3">λ</mi></mrow><mo>-</mo><mi href="./31.11#E3">μ</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mrow><mrow><mn>2</mn><mo></mo><mi href="./31.1#p2.t1.r3">j</mi></mrow><mo>+</mo><mi href="./31.11#E3">λ</mi></mrow><mo>-</mo><mi href="./31.11#E3">μ</mi></mrow><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow></mfrac></mstyle><mo>+</mo><mstyle displaystyle="true"><mfrac><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mi href="./31.1#p2.t1.r3">j</mi><mo>-</mo><mi href="./31.1#p2.t1.r5">α</mi></mrow><mo>+</mo><mi href="./31.11#E3">λ</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mi href="./31.1#p2.t1.r3">j</mi><mo>-</mo><mi href="./31.1#p2.t1.r5">β</mi></mrow><mo>+</mo><mi href="./31.11#E3">λ</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mi href="./31.1#p2.t1.r3">j</mi><mo>-</mo><mi href="./31.1#p2.t1.r5">γ</mi></mrow><mo>+</mo><mi href="./31.11#E3">λ</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./31.1#p2.t1.r3">j</mi><mo>-</mo><mi href="./31.11#E3">μ</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mrow><mn>2</mn><mo></mo><mi href="./31.1#p2.t1.r3">j</mi></mrow><mo>+</mo><mi href="./31.11#E3">λ</mi></mrow><mo>-</mo><mi href="./31.11#E3">μ</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mrow><mn>2</mn><mo></mo><mi href="./31.1#p2.t1.r3">j</mi></mrow><mo>+</mo><mi href="./31.11#E3">λ</mi></mrow><mo>-</mo><mi href="./31.11#E3">μ</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow></mfrac></mstyle></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E7.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./31.1#p2.t1.r5" title="§31.1 Special Notation ‣ Notation ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m34.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\gamma" display="inline"><mi href="./31.1#p2.t1.r5">γ</mi></math>: real or complex parameter</a>,
<a href="./31.1#p2.t1.r3" title="§31.1 Special Notation ‣ Notation ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="20px" altimg-valign="-6px" altimg-width="14px" alttext="j" display="inline"><mi href="./31.1#p2.t1.r3">j</mi></math>: nonnegative integer</a>,
<a href="./31.1#p2.t1.r4" title="§31.1 Special Notation ‣ Notation ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m40.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./31.1#p2.t1.r4">a</mi></math>: complex parameter</a>,
<a href="./31.1#p2.t1.r5" title="§31.1 Special Notation ‣ Notation ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m45.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="q" display="inline"><mi href="./31.1#p2.t1.r5">q</mi></math>: real or complex parameter</a>,
<a href="./31.1#p2.t1.r5" title="§31.1 Special Notation ‣ Notation ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m30.png" altimg-height="13px" altimg-valign="-2px" altimg-width="17px" alttext="\alpha" display="inline"><mi href="./31.1#p2.t1.r5">α</mi></math>: real or complex parameter</a>,
<a href="./31.1#p2.t1.r5" title="§31.1 Special Notation ‣ Notation ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m31.png" altimg-height="21px" altimg-valign="-6px" altimg-width="17px" alttext="\beta" display="inline"><mi href="./31.1#p2.t1.r5">β</mi></math>: real or complex parameter</a>,
<a href="./31.11#E3" title="(31.11.3) ‣ §31.11(ii) General Form ‣ §31.11 Expansions in Series of Hypergeometric Functions ‣ Properties ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m36.png" altimg-height="18px" altimg-valign="-2px" altimg-width="16px" alttext="\lambda" display="inline"><mi href="./31.11#E3">λ</mi></math></a>,
<a href="./31.11#E3" title="(31.11.3) ‣ §31.11(ii) General Form ‣ §31.11 Expansions in Series of Hypergeometric Functions ‣ Properties ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\mu" display="inline"><mi href="./31.11#E3">μ</mi></math></a> and
<a href="./31.11#EGx1" title="§31.11(ii) General Form ‣ §31.11 Expansions in Series of Hypergeometric Functions ‣ Properties ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m23.png" altimg-height="23px" altimg-valign="-8px" altimg-width="26px" alttext="L_{j}" display="inline"><msub><mi href="./31.11#EGx1">L</mi><mi href="./31.1#p2.t1.r3">j</mi></msub></math>: coefficients</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E8">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">31.11.8</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m18.png" altimg-height="25px" altimg-valign="-8px" altimg-width="34px" alttext="\displaystyle M_{j}" display="inline"><msub><mi href="./31.11#EGx1">M</mi><mi href="./31.1#p2.t1.r3">j</mi></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m2.png" altimg-height="53px" altimg-valign="-21px" altimg-width="543px" alttext="\displaystyle=-\frac{(j-\alpha+\lambda+1)(j-\beta+\lambda+1)(j-\gamma+\lambda+%
1)(j-\mu+1)}{(2j+\lambda-\mu+1)(2j+\lambda-\mu+2)}." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo>-</mo><mstyle displaystyle="true"><mfrac><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mi href="./31.1#p2.t1.r3">j</mi><mo>-</mo><mi href="./31.1#p2.t1.r5">α</mi></mrow><mo>+</mo><mi href="./31.11#E3">λ</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mi href="./31.1#p2.t1.r3">j</mi><mo>-</mo><mi href="./31.1#p2.t1.r5">β</mi></mrow><mo>+</mo><mi href="./31.11#E3">λ</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mi href="./31.1#p2.t1.r3">j</mi><mo>-</mo><mi href="./31.1#p2.t1.r5">γ</mi></mrow><mo>+</mo><mi href="./31.11#E3">λ</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mi href="./31.1#p2.t1.r3">j</mi><mo>-</mo><mi href="./31.11#E3">μ</mi></mrow><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mrow><mrow><mn>2</mn><mo></mo><mi href="./31.1#p2.t1.r3">j</mi></mrow><mo>+</mo><mi href="./31.11#E3">λ</mi></mrow><mo>-</mo><mi href="./31.11#E3">μ</mi></mrow><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mrow><mrow><mn>2</mn><mo></mo><mi href="./31.1#p2.t1.r3">j</mi></mrow><mo>+</mo><mi href="./31.11#E3">λ</mi></mrow><mo>-</mo><mi href="./31.11#E3">μ</mi></mrow><mo>+</mo><mn>2</mn></mrow><mo stretchy="false">)</mo></mrow></mrow></mfrac></mstyle></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E8.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./31.1#p2.t1.r5" title="§31.1 Special Notation ‣ Notation ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m34.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\gamma" display="inline"><mi href="./31.1#p2.t1.r5">γ</mi></math>: real or complex parameter</a>,
<a href="./31.1#p2.t1.r3" title="§31.1 Special Notation ‣ Notation ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="20px" altimg-valign="-6px" altimg-width="14px" alttext="j" display="inline"><mi href="./31.1#p2.t1.r3">j</mi></math>: nonnegative integer</a>,
<a href="./31.1#p2.t1.r5" title="§31.1 Special Notation ‣ Notation ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m30.png" altimg-height="13px" altimg-valign="-2px" altimg-width="17px" alttext="\alpha" display="inline"><mi href="./31.1#p2.t1.r5">α</mi></math>: real or complex parameter</a>,
<a href="./31.1#p2.t1.r5" title="§31.1 Special Notation ‣ Notation ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m31.png" altimg-height="21px" altimg-valign="-6px" altimg-width="17px" alttext="\beta" display="inline"><mi href="./31.1#p2.t1.r5">β</mi></math>: real or complex parameter</a>,
<a href="./31.11#E3" title="(31.11.3) ‣ §31.11(ii) General Form ‣ §31.11 Expansions in Series of Hypergeometric Functions ‣ Properties ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m36.png" altimg-height="18px" altimg-valign="-2px" altimg-width="16px" alttext="\lambda" display="inline"><mi href="./31.11#E3">λ</mi></math></a>,
<a href="./31.11#E3" title="(31.11.3) ‣ §31.11(ii) General Form ‣ §31.11 Expansions in Series of Hypergeometric Functions ‣ Properties ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\mu" display="inline"><mi href="./31.11#E3">μ</mi></math></a> and
<a href="./31.11#EGx1" title="§31.11(ii) General Form ‣ §31.11 Expansions in Series of Hypergeometric Functions ‣ Properties ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m24.png" altimg-height="23px" altimg-valign="-8px" altimg-width="32px" alttext="M_{j}" display="inline"><msub><mi href="./31.11#EGx1">M</mi><mi href="./31.1#p2.t1.r3">j</mi></msub></math>: coefficients</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
</div>
<div id="SS2.p3" class="ltx_para">
<p class="ltx_p"><math class="ltx_Math" altimg="m36.png" altimg-height="18px" altimg-valign="-2px" altimg-width="16px" alttext="\lambda" display="inline"><mi href="./31.11#E3">λ</mi></math>, <math class="ltx_Math" altimg="m39.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\mu" display="inline"><mi href="./31.11#E3">μ</mi></math> must also satisfy the condition</p>
<table id="E9" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">31.11.9</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E9.png" altimg-height="24px" altimg-valign="-7px" altimg-width="123px" alttext="M_{-1}P_{-1}=0." display="block"><mrow><mrow><mrow><msub><mi href="./31.11#EGx1">M</mi><mrow><mo>-</mo><mn>1</mn></mrow></msub><mo></mo><msub><mi href="./31.11#E2">P</mi><mrow><mo>-</mo><mn>1</mn></mrow></msub></mrow><mo>=</mo><mn>0</mn></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E9.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./31.11#E2" title="(31.11.2) ‣ §31.11(ii) General Form ‣ §31.11 Expansions in Series of Hypergeometric Functions ‣ Properties ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m27.png" altimg-height="23px" altimg-valign="-8px" altimg-width="25px" alttext="P_{j}" display="inline"><msub><mi href="./31.11#E2">P</mi><mi href="./31.1#p2.t1.r3">j</mi></msub></math></a> and
<a href="./31.11#EGx1" title="§31.11(ii) General Form ‣ §31.11 Expansions in Series of Hypergeometric Functions ‣ Properties ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m24.png" altimg-height="23px" altimg-valign="-8px" altimg-width="32px" alttext="M_{j}" display="inline"><msub><mi href="./31.11#EGx1">M</mi><mi href="./31.1#p2.t1.r3">j</mi></msub></math>: coefficients</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="iii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§31.11(iii) </span>Type I</h2>
<div id="SS3.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="SS3.p1" class="ltx_para">
<p class="ltx_p">Here
</p>
<table id="E10" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex1" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">31.11.10</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m19.png" altimg-height="19px" altimg-valign="-2px" altimg-width="18px" alttext="\displaystyle\lambda" display="inline"><mi href="./31.11#SS3.p1">λ</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m5.png" altimg-height="18px" altimg-valign="-6px" altimg-width="45px" alttext="\displaystyle=\alpha," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mi href="./31.1#p2.t1.r5">α</mi></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex2" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m20.png" altimg-height="18px" altimg-valign="-6px" altimg-width="18px" alttext="\displaystyle\mu" display="inline"><mi href="./31.11#SS3.p1">μ</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m8.png" altimg-height="23px" altimg-valign="-6px" altimg-width="77px" alttext="\displaystyle=\beta-\epsilon," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./31.1#p2.t1.r5">β</mi><mo>-</mo><mi href="./31.1#p2.t1.r5">ϵ</mi></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E10.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./31.1#p2.t1.r5" title="§31.1 Special Notation ‣ Notation ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m33.png" altimg-height="13px" altimg-valign="-2px" altimg-width="12px" alttext="\epsilon" display="inline"><mi href="./31.1#p2.t1.r5">ϵ</mi></math>: real or complex parameter</a>,
<a href="./31.1#p2.t1.r5" title="§31.1 Special Notation ‣ Notation ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m30.png" altimg-height="13px" altimg-valign="-2px" altimg-width="17px" alttext="\alpha" display="inline"><mi href="./31.1#p2.t1.r5">α</mi></math>: real or complex parameter</a>,
<a href="./31.1#p2.t1.r5" title="§31.1 Special Notation ‣ Notation ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m31.png" altimg-height="21px" altimg-valign="-6px" altimg-width="17px" alttext="\beta" display="inline"><mi href="./31.1#p2.t1.r5">β</mi></math>: real or complex parameter</a>,
<a href="./31.11#SS3.p1" title="§31.11(iii) Type I ‣ §31.11 Expansions in Series of Hypergeometric Functions ‣ Properties ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m36.png" altimg-height="18px" altimg-valign="-2px" altimg-width="16px" alttext="\lambda" display="inline"><mi href="./31.11#SS3.p1">λ</mi></math></a> and
<a href="./31.11#SS3.p1" title="§31.11(iii) Type I ‣ §31.11 Expansions in Series of Hypergeometric Functions ‣ Properties ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\mu" display="inline"><mi href="./31.11#SS3.p1">μ</mi></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">or</p>
<table id="E11" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex3" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">31.11.11</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m19.png" altimg-height="19px" altimg-valign="-2px" altimg-width="18px" alttext="\displaystyle\lambda" display="inline"><mi href="./31.11#SS3.p1">λ</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m7.png" altimg-height="23px" altimg-valign="-6px" altimg-width="45px" alttext="\displaystyle=\beta," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mi href="./31.1#p2.t1.r5">β</mi></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex4" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m20.png" altimg-height="18px" altimg-valign="-6px" altimg-width="18px" alttext="\displaystyle\mu" display="inline"><mi href="./31.11#SS3.p1">μ</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m6.png" altimg-height="19px" altimg-valign="-4px" altimg-width="78px" alttext="\displaystyle=\alpha-\epsilon." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./31.1#p2.t1.r5">α</mi><mo>-</mo><mi href="./31.1#p2.t1.r5">ϵ</mi></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E11.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./31.1#p2.t1.r5" title="§31.1 Special Notation ‣ Notation ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m33.png" altimg-height="13px" altimg-valign="-2px" altimg-width="12px" alttext="\epsilon" display="inline"><mi href="./31.1#p2.t1.r5">ϵ</mi></math>: real or complex parameter</a>,
<a href="./31.1#p2.t1.r5" title="§31.1 Special Notation ‣ Notation ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m30.png" altimg-height="13px" altimg-valign="-2px" altimg-width="17px" alttext="\alpha" display="inline"><mi href="./31.1#p2.t1.r5">α</mi></math>: real or complex parameter</a>,
<a href="./31.1#p2.t1.r5" title="§31.1 Special Notation ‣ Notation ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m31.png" altimg-height="21px" altimg-valign="-6px" altimg-width="17px" alttext="\beta" display="inline"><mi href="./31.1#p2.t1.r5">β</mi></math>: real or complex parameter</a>,
<a href="./31.11#SS3.p1" title="§31.11(iii) Type I ‣ §31.11 Expansions in Series of Hypergeometric Functions ‣ Properties ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m36.png" altimg-height="18px" altimg-valign="-2px" altimg-width="16px" alttext="\lambda" display="inline"><mi href="./31.11#SS3.p1">λ</mi></math></a> and
<a href="./31.11#SS3.p1" title="§31.11(iii) Type I ‣ §31.11 Expansions in Series of Hypergeometric Functions ‣ Properties ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\mu" display="inline"><mi href="./31.11#SS3.p1">μ</mi></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Then condition ().
Then the Fuchs–Frobenius solution at <math class="ltx_Math" altimg="m35.png" altimg-height="13px" altimg-valign="-2px" altimg-width="24px" alttext="\infty" display="inline"><mi mathvariant="normal">∞</mi></math> belonging to the exponent
<math class="ltx_Math" altimg="m30.png" altimg-height="13px" altimg-valign="-2px" altimg-width="17px" alttext="\alpha" display="inline"><mi href="./31.1#p2.t1.r5">α</mi></math> has the expansion () with</p>
<table id="E12" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">31.11.12</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E12.png" altimg-height="53px" altimg-valign="-21px" altimg-width="616px" alttext="P_{j}=\frac{\Gamma\left(\alpha+j\right)\Gamma\left(1-\gamma+\alpha+j\right)}{%
\Gamma\left(1+\alpha-\beta+\epsilon+2j\right)}z^{-\alpha-j}\*{{}_{2}F_{1}}%
\left({\alpha+j,1-\gamma+\alpha+j\atop 1+\alpha-\beta+\epsilon+2j};\frac{1}{z}%
\right)," display="block"><mrow><mrow><msub><mi href="./31.11#E2">P</mi><mi href="./31.1#p2.t1.r3">j</mi></msub><mo>=</mo><mrow><mfrac><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./31.1#p2.t1.r5">α</mi><mo>+</mo><mi href="./31.1#p2.t1.r3">j</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mn>1</mn><mo>-</mo><mi href="./31.1#p2.t1.r5">γ</mi></mrow><mo>+</mo><mi href="./31.1#p2.t1.r5">α</mi><mo>+</mo><mi href="./31.1#p2.t1.r3">j</mi></mrow><mo>)</mo></mrow></mrow></mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mrow><mn>1</mn><mo>+</mo><mi href="./31.1#p2.t1.r5">α</mi></mrow><mo>-</mo><mi href="./31.1#p2.t1.r5">β</mi></mrow><mo>+</mo><mi href="./31.1#p2.t1.r5">ϵ</mi><mo>+</mo><mrow><mn>2</mn><mo></mo><mi href="./31.1#p2.t1.r3">j</mi></mrow></mrow><mo>)</mo></mrow></mrow></mfrac><mo></mo><msup><mi href="./31.1#p2.t1.r2">z</mi><mrow><mrow><mo>-</mo><mi href="./31.1#p2.t1.r5">α</mi></mrow><mo>-</mo><mi href="./31.1#p2.t1.r3">j</mi></mrow></msup><mo></mo><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mrow><mi href="./31.1#p2.t1.r5">α</mi><mo>+</mo><mi href="./31.1#p2.t1.r3">j</mi></mrow><mo>,</mo><mrow><mrow><mn>1</mn><mo>-</mo><mi href="./31.1#p2.t1.r5">γ</mi></mrow><mo>+</mo><mi href="./31.1#p2.t1.r5">α</mi><mo>+</mo><mi href="./31.1#p2.t1.r3">j</mi></mrow></mrow><mrow><mrow><mrow><mn>1</mn><mo>+</mo><mi href="./31.1#p2.t1.r5">α</mi></mrow><mo>-</mo><mi href="./31.1#p2.t1.r5">β</mi></mrow><mo>+</mo><mi href="./31.1#p2.t1.r5">ϵ</mi><mo>+</mo><mrow><mn>2</mn><mo></mo><mi href="./31.1#p2.t1.r3">j</mi></mrow></mrow></mfrac><mo>;</mo><mfrac><mn>1</mn><mi href="./31.1#p2.t1.r2">z</mi></mfrac><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E12.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m28.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./16.2" title="§16.2 Definition and Analytic Properties ‣ Generalized Hypergeometric Functions ‣ Chapter 16 Generalized Hypergeometric Functions and Meijer G -Function" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="23px" altimg-valign="-7px" altimg-width="118px" alttext="{{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mrow><mi class="ltx_nvar" href="./16.1#p2.t1.r5">a</mi><mo>,</mo><mi class="ltx_nvar" href="./16.1#p2.t1.r5">b</mi></mrow><mo>;</mo><mi class="ltx_nvar">c</mi><mo>;</mo><mi class="ltx_nvar" href="./16.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: <math class="ltx_Math" altimg="m21.png" altimg-height="23px" altimg-valign="-7px" altimg-width="124px" alttext="=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./16.1#p2.t1.r5">a</mi><mo>,</mo><mi class="ltx_nvar" href="./16.1#p2.t1.r5">b</mi><mo>;</mo><mi class="ltx_nvar">c</mi><mo>;</mo><mi class="ltx_nvar" href="./16.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></math> notation for Gauss’ hypergeometric function</a>,
<a href="./31.1#p2.t1.r2" title="§31.1 Special Notation ‣ Notation ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./31.1#p2.t1.r2">z</mi></math>: complex variable</a>,
<a href="./31.1#p2.t1.r5" title="§31.1 Special Notation ‣ Notation ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m34.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\gamma" display="inline"><mi href="./31.1#p2.t1.r5">γ</mi></math>: real or complex parameter</a>,
<a href="./31.1#p2.t1.r5" title="§31.1 Special Notation ‣ Notation ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m33.png" altimg-height="13px" altimg-valign="-2px" altimg-width="12px" alttext="\epsilon" display="inline"><mi href="./31.1#p2.t1.r5">ϵ</mi></math>: real or complex parameter</a>,
<a href="./31.1#p2.t1.r3" title="§31.1 Special Notation ‣ Notation ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="20px" altimg-valign="-6px" altimg-width="14px" alttext="j" display="inline"><mi href="./31.1#p2.t1.r3">j</mi></math>: nonnegative integer</a>,
<a href="./31.1#p2.t1.r5" title="§31.1 Special Notation ‣ Notation ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m30.png" altimg-height="13px" altimg-valign="-2px" altimg-width="17px" alttext="\alpha" display="inline"><mi href="./31.1#p2.t1.r5">α</mi></math>: real or complex parameter</a>,
<a href="./31.1#p2.t1.r5" title="§31.1 Special Notation ‣ Notation ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m31.png" altimg-height="21px" altimg-valign="-6px" altimg-width="17px" alttext="\beta" display="inline"><mi href="./31.1#p2.t1.r5">β</mi></math>: real or complex parameter</a> and
<a href="./31.11#E2" title="(31.11.2) ‣ §31.11(ii) General Form ‣ §31.11 Expansions in Series of Hypergeometric Functions ‣ Properties ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m27.png" altimg-height="23px" altimg-valign="-8px" altimg-width="25px" alttext="P_{j}" display="inline"><msub><mi href="./31.11#E2">P</mi><mi href="./31.1#p2.t1.r3">j</mi></msub></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and () converges outside the ellipse <math class="ltx_Math" altimg="m37.png" altimg-height="18px" altimg-valign="-2px" altimg-width="17px" alttext="\mathcal{E}" display="inline"><mi class="ltx_font_mathcaligraphic" href="./31.11#SS3.p2">ℰ</mi></math> in the
<math class="ltx_Math" altimg="m50.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./31.1#p2.t1.r2">z</mi></math>-plane with foci at 0, 1, and passing through the third finite singularity at
<math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="52px" alttext="z=a" display="inline"><mrow><mi href="./31.1#p2.t1.r2">z</mi><mo>=</mo><mi href="./31.1#p2.t1.r4">a</mi></mrow></math>.</p>
</div>
<div id="SS3.p3" class="ltx_para">
<p class="ltx_p">Every Heun function (§) can be represented by a series of Type
I convergent in the whole plane cut along a line joining the two singularities
of the Heun function.</p>
</div>
<div id="SS3.p4" class="ltx_para">
<p class="ltx_p">For example, consider the Heun function which is analytic at <math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="52px" alttext="z=a" display="inline"><mrow><mi href="./31.1#p2.t1.r2">z</mi><mo>=</mo><mi href="./31.1#p2.t1.r4">a</mi></mrow></math> and has
exponent <math class="ltx_Math" altimg="m30.png" altimg-height="13px" altimg-valign="-2px" altimg-width="17px" alttext="\alpha" display="inline"><mi href="./31.1#p2.t1.r5">α</mi></math> at <math class="ltx_Math" altimg="m35.png" altimg-height="13px" altimg-valign="-2px" altimg-width="24px" alttext="\infty" display="inline"><mi mathvariant="normal">∞</mi></math>. The expansion () is convergent in the plane cut along the line joining the
two singularities <math class="ltx_Math" altimg="m47.png" altimg-height="17px" altimg-valign="-2px" altimg-width="51px" alttext="z=0" display="inline"><mrow><mi href="./31.1#p2.t1.r2">z</mi><mo>=</mo><mn>0</mn></mrow></math> and <math class="ltx_Math" altimg="m48.png" altimg-height="17px" altimg-valign="-2px" altimg-width="51px" alttext="z=1" display="inline"><mrow><mi href="./31.1#p2.t1.r2">z</mi><mo>=</mo><mn>1</mn></mrow></math>. In this case the accessory parameter <math class="ltx_Math" altimg="m45.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="q" display="inline"><mi href="./31.1#p2.t1.r5">q</mi></math>
is a root of the continued-fraction equation</p>
<table id="E13" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">31.11.13</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E13.png" altimg-height="57px" altimg-valign="-21px" altimg-width="363px" alttext="\left(L_{0}/M_{0}\right)-\cfrac{K_{1}/M_{1}}{L_{1}/M_{1}-\cfrac{K_{2}/M_{2}}{L%
_{2}/M_{2}-\cdots}}=0." display="block"><mrow><mrow><mrow><mrow><mo>(</mo><mrow><msub><mi href="./31.11#EGx1">L</mi><mn>0</mn></msub><mo>/</mo><msub><mi href="./31.11#EGx1">M</mi><mn>0</mn></msub></mrow><mo>)</mo></mrow><mo>-</mo><mrow><mfrac><mrow><msub><mi href="./31.11#EGx1">K</mi><mn>1</mn></msub><mo>/</mo><msub><mi href="./31.11#EGx1">M</mi><mn>1</mn></msub></mrow><mrow><mrow><msub><mi href="./31.11#EGx1">L</mi><mn>1</mn></msub><mo>/</mo><msub><mi href="./31.11#EGx1">M</mi><mn>1</mn></msub></mrow><mo>-</mo></mrow></mfrac><mfrac><mrow><msub><mi href="./31.11#EGx1">K</mi><mn>2</mn></msub><mo>/</mo><msub><mi href="./31.11#EGx1">M</mi><mn>2</mn></msub></mrow><mrow><mrow><msub><mi href="./31.11#EGx1">L</mi><mn>2</mn></msub><mo>/</mo><msub><mi href="./31.11#EGx1">M</mi><mn>2</mn></msub></mrow><mo>-</mo></mrow></mfrac><mi mathvariant="normal">⋯</mi></mrow></mrow><mo>=</mo><mn>0</mn></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E13.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./31.11#EGx1" title="§31.11(ii) General Form ‣ §31.11 Expansions in Series of Hypergeometric Functions ‣ Properties ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="23px" altimg-valign="-8px" altimg-width="30px" alttext="K_{j}" display="inline"><msub><mi href="./31.11#EGx1">K</mi><mi href="./31.1#p2.t1.r3">j</mi></msub></math>: coefficients</a>,
<a href="./31.11#EGx1" title="§31.11(ii) General Form ‣ §31.11 Expansions in Series of Hypergeometric Functions ‣ Properties ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m23.png" altimg-height="23px" altimg-valign="-8px" altimg-width="26px" alttext="L_{j}" display="inline"><msub><mi href="./31.11#EGx1">L</mi><mi href="./31.1#p2.t1.r3">j</mi></msub></math>: coefficients</a> and
<a href="./31.11#EGx1" title="§31.11(ii) General Form ‣ §31.11 Expansions in Series of Hypergeometric Functions ‣ Properties ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m24.png" altimg-height="23px" altimg-valign="-8px" altimg-width="32px" alttext="M_{j}" display="inline"><msub><mi href="./31.11#EGx1">M</mi><mi href="./31.1#p2.t1.r3">j</mi></msub></math>: coefficients</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">The case <math class="ltx_Math" altimg="m29.png" altimg-height="17px" altimg-valign="-4px" altimg-width="71px" alttext="\alpha=-n" display="inline"><mrow><mi href="./31.1#p2.t1.r5">α</mi><mo>=</mo><mrow><mo>-</mo><mi href="./31.1#p2.t1.r3">n</mi></mrow></mrow></math> for nonnegative integer <math class="ltx_Math" altimg="m44.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./31.1#p2.t1.r3">n</mi></math> corresponds to the Heun
polynomial <math class="ltx_Math" altimg="m38.png" altimg-height="26px" altimg-valign="-10px" altimg-width="89px" alttext="\mathit{Hp}_{n,m}\left(z\right)" display="inline"><mrow><msub><mi href="./31.5#E2" mathvariant="italic">Hp</mi><mrow><mi href="./31.1#p2.t1.r3">n</mi><mo href="./31.5#E2">,</mo><mi href="./31.1#p2.t1.r3">m</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./31.1#p2.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>.</p>
</div>
<div id="SS3.p5" class="ltx_para">
<p class="ltx_p">The expansion ()—is convergent inside the ellipse
<math class="ltx_Math" altimg="m37.png" altimg-height="18px" altimg-valign="-2px" altimg-width="17px" alttext="\mathcal{E}" display="inline"><mi class="ltx_font_mathcaligraphic" href="./31.11#SS3.p2">ℰ</mi></math>.</p>
</div>
</section>
<section id="iv" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§31.11(iv) </span>Type II</h2>
<div id="SS4.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="SS4.p1" class="ltx_para">
<p class="ltx_p">Here one of the following four pairs of conditions is satisfied:</p>
<table id="EGx2" class="ltx_equationgroup ltx_eqn_table">
<tr id="E14" class="ltx_eqn_row"><td class="ltx_eqn_cell"></td></tr>
<tr id="Ex5" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">31.11.14</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m19.png" altimg-height="19px" altimg-valign="-2px" altimg-width="18px" alttext="\displaystyle\lambda" display="inline"><mi href="./31.11#EGx2">λ</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m11.png" altimg-height="23px" altimg-valign="-6px" altimg-width="112px" alttext="\displaystyle=\gamma+\delta-1," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mi href="./31.1#p2.t1.r5">γ</mi><mo>+</mo><mi href="./31.1#p2.t1.r5">δ</mi></mrow><mo>-</mo><mn>1</mn></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex6" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m20.png" altimg-height="18px" altimg-valign="-6px" altimg-width="18px" alttext="\displaystyle\mu" display="inline"><mi href="./31.11#EGx2">μ</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m3.png" altimg-height="22px" altimg-valign="-6px" altimg-width="43px" alttext="\displaystyle=0," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mn>0</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E14.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./31.1#p2.t1.r5" title="§31.1 Special Notation ‣ Notation ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m34.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\gamma" display="inline"><mi href="./31.1#p2.t1.r5">γ</mi></math>: real or complex parameter</a>,
<a href="./31.1#p2.t1.r5" title="§31.1 Special Notation ‣ Notation ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m32.png" altimg-height="18px" altimg-valign="-2px" altimg-width="14px" alttext="\delta" display="inline"><mi href="./31.1#p2.t1.r5">δ</mi></math>: real or complex parameter</a>,
<a href="./31.11#EGx2" title="§31.11(iv) Type II ‣ §31.11 Expansions in Series of Hypergeometric Functions ‣ Properties ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m36.png" altimg-height="18px" altimg-valign="-2px" altimg-width="16px" alttext="\lambda" display="inline"><mi href="./31.11#EGx2">λ</mi></math></a> and
<a href="./31.11#EGx2" title="§31.11(iv) Type II ‣ §31.11 Expansions in Series of Hypergeometric Functions ‣ Properties ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\mu" display="inline"><mi href="./31.11#EGx2">μ</mi></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
<tr id="E15" class="ltx_eqn_row"><td class="ltx_eqn_cell"></td></tr>
<tr id="Ex7" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">31.11.15</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m19.png" altimg-height="19px" altimg-valign="-2px" altimg-width="18px" alttext="\displaystyle\lambda" display="inline"><mi href="./31.11#EGx2">λ</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m13.png" altimg-height="18px" altimg-valign="-6px" altimg-width="44px" alttext="\displaystyle=\gamma," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mi href="./31.1#p2.t1.r5">γ</mi></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex8" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m20.png" altimg-height="18px" altimg-valign="-6px" altimg-width="18px" alttext="\displaystyle\mu" display="inline"><mi href="./31.11#EGx2">μ</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m10.png" altimg-height="23px" altimg-valign="-6px" altimg-width="77px" alttext="\displaystyle=\delta-1," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./31.1#p2.t1.r5">δ</mi><mo>-</mo><mn>1</mn></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E15.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./31.1#p2.t1.r5" title="§31.1 Special Notation ‣ Notation ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m34.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\gamma" display="inline"><mi href="./31.1#p2.t1.r5">γ</mi></math>: real or complex parameter</a>,
<a href="./31.1#p2.t1.r5" title="§31.1 Special Notation ‣ Notation ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m32.png" altimg-height="18px" altimg-valign="-2px" altimg-width="14px" alttext="\delta" display="inline"><mi href="./31.1#p2.t1.r5">δ</mi></math>: real or complex parameter</a>,
<a href="./31.11#EGx2" title="§31.11(iv) Type II ‣ §31.11 Expansions in Series of Hypergeometric Functions ‣ Properties ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m36.png" altimg-height="18px" altimg-valign="-2px" altimg-width="16px" alttext="\lambda" display="inline"><mi href="./31.11#EGx2">λ</mi></math></a> and
<a href="./31.11#EGx2" title="§31.11(iv) Type II ‣ §31.11 Expansions in Series of Hypergeometric Functions ‣ Properties ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\mu" display="inline"><mi href="./31.11#EGx2">μ</mi></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
<tr id="E16" class="ltx_eqn_row"><td class="ltx_eqn_cell"></td></tr>
<tr id="Ex9" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">31.11.16</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m19.png" altimg-height="19px" altimg-valign="-2px" altimg-width="18px" alttext="\displaystyle\lambda" display="inline"><mi href="./31.11#EGx2">λ</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m9.png" altimg-height="23px" altimg-valign="-6px" altimg-width="41px" alttext="\displaystyle=\delta," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mi href="./31.1#p2.t1.r5">δ</mi></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex10" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m20.png" altimg-height="18px" altimg-valign="-6px" altimg-width="18px" alttext="\displaystyle\mu" display="inline"><mi href="./31.11#EGx2">μ</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m14.png" altimg-height="22px" altimg-valign="-6px" altimg-width="78px" alttext="\displaystyle=\gamma-1," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./31.1#p2.t1.r5">γ</mi><mo>-</mo><mn>1</mn></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E16.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./31.1#p2.t1.r5" title="§31.1 Special Notation ‣ Notation ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m34.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\gamma" display="inline"><mi href="./31.1#p2.t1.r5">γ</mi></math>: real or complex parameter</a>,
<a href="./31.1#p2.t1.r5" title="§31.1 Special Notation ‣ Notation ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m32.png" altimg-height="18px" altimg-valign="-2px" altimg-width="14px" alttext="\delta" display="inline"><mi href="./31.1#p2.t1.r5">δ</mi></math>: real or complex parameter</a>,
<a href="./31.11#EGx2" title="§31.11(iv) Type II ‣ §31.11 Expansions in Series of Hypergeometric Functions ‣ Properties ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m36.png" altimg-height="18px" altimg-valign="-2px" altimg-width="16px" alttext="\lambda" display="inline"><mi href="./31.11#EGx2">λ</mi></math></a> and
<a href="./31.11#EGx2" title="§31.11(iv) Type II ‣ §31.11 Expansions in Series of Hypergeometric Functions ‣ Properties ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\mu" display="inline"><mi href="./31.11#EGx2">μ</mi></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
<tr id="E17" class="ltx_eqn_row"><td class="ltx_eqn_cell"></td></tr>
<tr id="Ex11" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">31.11.17</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m19.png" altimg-height="19px" altimg-valign="-2px" altimg-width="18px" alttext="\displaystyle\lambda" display="inline"><mi href="./31.11#EGx2">λ</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m4.png" altimg-height="22px" altimg-valign="-6px" altimg-width="43px" alttext="\displaystyle=1," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mn>1</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex12" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m20.png" altimg-height="18px" altimg-valign="-6px" altimg-width="18px" alttext="\displaystyle\mu" display="inline"><mi href="./31.11#EGx2">μ</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m12.png" altimg-height="23px" altimg-valign="-6px" altimg-width="112px" alttext="\displaystyle=\gamma+\delta-2." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mi href="./31.1#p2.t1.r5">γ</mi><mo>+</mo><mi href="./31.1#p2.t1.r5">δ</mi></mrow><mo>-</mo><mn>2</mn></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E17.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./31.1#p2.t1.r5" title="§31.1 Special Notation ‣ Notation ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m34.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\gamma" display="inline"><mi href="./31.1#p2.t1.r5">γ</mi></math>: real or complex parameter</a>,
<a href="./31.1#p2.t1.r5" title="§31.1 Special Notation ‣ Notation ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m32.png" altimg-height="18px" altimg-valign="-2px" altimg-width="14px" alttext="\delta" display="inline"><mi href="./31.1#p2.t1.r5">δ</mi></math>: real or complex parameter</a>,
<a href="./31.11#EGx2" title="§31.11(iv) Type II ‣ §31.11 Expansions in Series of Hypergeometric Functions ‣ Properties ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m36.png" altimg-height="18px" altimg-valign="-2px" altimg-width="16px" alttext="\lambda" display="inline"><mi href="./31.11#EGx2">λ</mi></math></a> and
<a href="./31.11#EGx2" title="§31.11(iv) Type II ‣ §31.11 Expansions in Series of Hypergeometric Functions ‣ Properties ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\mu" display="inline"><mi href="./31.11#EGx2">μ</mi></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">In each case <math class="ltx_Math" altimg="m27.png" altimg-height="23px" altimg-valign="-8px" altimg-width="25px" alttext="P_{j}" display="inline"><msub><mi href="./31.11#E2">P</mi><mi href="./31.1#p2.t1.r3">j</mi></msub></math> can be expressed in terms of a Jacobi polynomial
(§). Such series diverge for Fuchs–Frobenius solutions. For
Heun functions they are convergent inside the ellipse <math class="ltx_Math" altimg="m37.png" altimg-height="18px" altimg-valign="-2px" altimg-width="17px" alttext="\mathcal{E}" display="inline"><mi class="ltx_font_mathcaligraphic" href="./31.11#SS3.p2">ℰ</mi></math>. Every
Heun function can be represented by a series of Type II.</p>
</div>
</section>
<section id="v" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§31.11(v) </span>Doubly-Infinite Series</h2>
<div id="SS5.info" class="ltx_metadata ltx_info">
</div>
</div>
</body></text>
</html>
</page>
<page>
<!DOCTYPE html><html>
<head>
<title>DLMF: 31.16 Mathematical Applications</title>
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<div class="ltx_page_navlogo">)</cite>
from the viewpoint of interrelation between two bases in a Hilbert space:
</p>
<table id="E1" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">31.16.1</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_Math ltx_eqn_cell ltx_align_center"><math class="ltx_Math" alttext="\mathit{Hp}_{n,m}\left(x\right)\mathit{Hp}_{n,m}\left(y\right)=\sum_{j=0}^{n}A%
_{j}{\sin^{2j}}\theta\*P_{n-j}^{(\gamma+\delta+2j-1,\epsilon-1)}(\cos 2\theta)%
P_{j}^{(\delta-1,\gamma-1)}(\cos 2\phi)," display="block"><mrow><mrow><mrow><msub><mi href="./31.5#E2" mathvariant="italic">Hp</mi><mrow><mi href="./31.1#p2.t1.r3">n</mi><mo href="./31.5#E2">,</mo><mi href="./31.1#p2.t1.r3">m</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./31.16#E2">x</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./31.5#E2" mathvariant="italic">Hp</mi><mrow><mi href="./31.1#p2.t1.r3">n</mi><mo href="./31.5#E2">,</mo><mi href="./31.1#p2.t1.r3">m</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./31.16#E2">y</mi><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./31.1#p2.t1.r3">j</mi><mo>=</mo><mn>0</mn></mrow><mi href="./31.1#p2.t1.r3">n</mi></munderover><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><msub><mi href="./31.16#SS2.p1">A</mi><mi href="./31.1#p2.t1.r3">j</mi></msub><mo></mo><mrow><msup><mi href="./4.14#E1">sin</mi><mrow><mn>2</mn><mo></mo><mi href="./31.1#p2.t1.r3">j</mi></mrow></msup><mo></mo><mi href="./31.16#E2">θ</mi></mrow><mo></mo><msubsup><mi href="./31.16#E5">P</mi><mrow><mi href="./31.1#p2.t1.r3">n</mi><mo>-</mo><mi href="./31.1#p2.t1.r3">j</mi></mrow><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mrow><mrow><mi href="./31.1#p2.t1.r5">γ</mi><mo>+</mo><mi href="./31.1#p2.t1.r5">δ</mi><mo>+</mo><mrow><mn>2</mn><mo></mo><mi href="./31.1#p2.t1.r3">j</mi></mrow></mrow><mo>-</mo><mn>1</mn></mrow><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mrow><mi href="./31.1#p2.t1.r5">ϵ</mi><mo>-</mo><mn>1</mn></mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></msubsup></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>×</mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mn>2</mn><mo></mo><mi href="./31.16#E2">θ</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msubsup><mi href="./31.16#E5">P</mi><mi href="./31.1#p2.t1.r3">j</mi><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mrow><mi href="./31.1#p2.t1.r5">δ</mi><mo>-</mo><mn>1</mn></mrow><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mrow><mi href="./31.1#p2.t1.r5">γ</mi><mo>-</mo><mn>1</mn></mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mn>2</mn><mo></mo><mi href="./31.16#E2">ϕ</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mo>,</mo></mtd></mtr></mtable></mrow></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./31.5#E2" title="(31.5.2) ‣ §31.5 Solutions Analytic at Three Singularities: Heun Polynomials ‣ Properties ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m21.png" altimg-height="26px" altimg-valign="-10px" altimg-width="252px" alttext="\mathit{Hp}_{\NVar{n},\NVar{m}}\left(\NVar{a},\NVar{q_{n,m}};\NVar{-n},\NVar{%
\beta},\NVar{\gamma},\NVar{\delta};\NVar{z}\right)" display="inline"><mrow><msub><mi href="./31.5#E2" mathvariant="italic">Hp</mi><mrow><mi class="ltx_nvar" href="./31.1#p2.t1.r3">n</mi><mo href="./31.5#E2">,</mo><mi class="ltx_nvar" href="./31.1#p2.t1.r3">m</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./31.1#p2.t1.r4">a</mi><mo>,</mo><msub><mi class="ltx_nvar" href="./31.1#p2.t1.r5">q</mi><mrow class="ltx_nvar"><mi class="ltx_nvar" href="./31.1#p2.t1.r3">n</mi><mo class="ltx_nvar">,</mo><mi class="ltx_nvar" href="./31.1#p2.t1.r3">m</mi></mrow></msub><mo>;</mo><mrow><mo class="ltx_nvar">-</mo><mi class="ltx_nvar" href="./31.1#p2.t1.r3">n</mi></mrow><mo>,</mo><mi class="ltx_nvar" href="./31.1#p2.t1.r5">β</mi><mo>,</mo><mi class="ltx_nvar" href="./31.1#p2.t1.r5">γ</mi><mo>,</mo><mi class="ltx_nvar" href="./31.1#p2.t1.r5">δ</mi><mo>;</mo><mi class="ltx_nvar" href="./31.1#p2.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: Heun polynomials</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m17.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./front/introduction#Sx4.p1.t1.r28" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m11.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="(\NVar{a},\NVar{b})" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi class="ltx_nvar">a</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi class="ltx_nvar">b</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math>: open interval</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m23.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./31.1#p2.t1.r5" title="§31.1 Special Notation ‣ Notation ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\gamma" display="inline"><mi href="./31.1#p2.t1.r5">γ</mi></math>: real or complex parameter</a>,
<a href="./31.1#p2.t1.r5" title="§31.1 Special Notation ‣ Notation ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m18.png" altimg-height="18px" altimg-valign="-2px" altimg-width="14px" alttext="\delta" display="inline"><mi href="./31.1#p2.t1.r5">δ</mi></math>: real or complex parameter</a>,
<a href="./31.1#p2.t1.r5" title="§31.1 Special Notation ‣ Notation ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="13px" altimg-valign="-2px" altimg-width="12px" alttext="\epsilon" display="inline"><mi href="./31.1#p2.t1.r5">ϵ</mi></math>: real or complex parameter</a>,
<a href="./31.1#p2.t1.r3" title="§31.1 Special Notation ‣ Notation ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m27.png" altimg-height="20px" altimg-valign="-6px" altimg-width="14px" alttext="j" display="inline"><mi href="./31.1#p2.t1.r3">j</mi></math>: nonnegative integer</a>,
<a href="./31.1#p2.t1.r3" title="§31.1 Special Notation ‣ Notation ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./31.1#p2.t1.r3">m</mi></math>: nonnegative integer</a>,
<a href="./31.1#p2.t1.r3" title="§31.1 Special Notation ‣ Notation ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m31.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./31.1#p2.t1.r3">n</mi></math>: nonnegative integer</a>,
<a href="./31.1#p2.t1.r4" title="§31.1 Special Notation ‣ Notation ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m25.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./31.1#p2.t1.r4">a</mi></math>: complex parameter</a>,
<a href="./31.1#p2.t1.r5" title="§31.1 Special Notation ‣ Notation ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m32.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="q" display="inline"><mi href="./31.1#p2.t1.r5">q</mi></math>: real or complex parameter</a>,
<a href="./31.1#p2.t1.r5" title="§31.1 Special Notation ‣ Notation ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m16.png" altimg-height="21px" altimg-valign="-6px" altimg-width="17px" alttext="\beta" display="inline"><mi href="./31.1#p2.t1.r5">β</mi></math>: real or complex parameter</a>,
<a href="./31.16#E2" title="(31.16.2) ‣ §31.16(ii) Heun Polynomial Products ‣ §31.16 Mathematical Applications ‣ Applications ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m24.png" altimg-height="18px" altimg-valign="-2px" altimg-width="14px" alttext="\theta" display="inline"><mi href="./31.16#E2">θ</mi></math>: angle</a>,
<a href="./31.16#E2" title="(31.16.2) ‣ §31.16(ii) Heun Polynomial Products ‣ §31.16 Mathematical Applications ‣ Applications ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="\phi" display="inline"><mi href="./31.16#E2">ϕ</mi></math>: angle</a>,
<a href="./31.16#E2" title="(31.16.2) ‣ §31.16(ii) Heun Polynomial Products ‣ §31.16 Mathematical Applications ‣ Applications ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m33.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./31.16#E2">x</mi></math>: variable</a>,
<a href="./31.16#E2" title="(31.16.2) ‣ §31.16(ii) Heun Polynomial Products ‣ §31.16 Mathematical Applications ‣ Applications ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m34.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./31.16#E2">y</mi></math>: variable</a>,
<a href="./31.16#SS2.p1" title="§31.16(ii) Heun Polynomial Products ‣ §31.16 Mathematical Applications ‣ Applications ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m12.png" altimg-height="23px" altimg-valign="-8px" altimg-width="28px" alttext="A_{j}" display="inline"><msub><mi href="./31.16#SS2.p1">A</mi><mi href="./31.1#p2.t1.r3">j</mi></msub></math>: coefficients</a> and
<a href="./31.16#E5" title="(31.16.5) ‣ §31.16(ii) Heun Polynomial Products ‣ §31.16 Mathematical Applications ‣ Applications ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m13.png" altimg-height="23px" altimg-valign="-8px" altimg-width="25px" alttext="P_{j}" display="inline"><msub><mi href="./31.16#E5">P</mi><mi href="./31.1#p2.t1.r3">j</mi></msub></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m30.png" altimg-height="20px" altimg-valign="-6px" altimg-width="104px" alttext="n=0,1,\dots" display="inline"><mrow><mi href="./31.1#p2.t1.r3">n</mi><mo>=</mo><mrow><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>, <math class="ltx_Math" altimg="m28.png" altimg-height="20px" altimg-valign="-6px" altimg-width="133px" alttext="m=0,1,\dots,n" display="inline"><mrow><mi href="./31.1#p2.t1.r3">m</mi><mo>=</mo><mrow><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><mi href="./31.1#p2.t1.r3">n</mi></mrow></mrow></math>, and</p>
<table id="E2" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex1" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">31.16.2</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m9.png" altimg-height="14px" altimg-valign="-2px" altimg-width="17px" alttext="\displaystyle x" display="inline"><mi href="./31.16#E2">x</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m4.png" altimg-height="27px" altimg-valign="-6px" altimg-width="123px" alttext="\displaystyle={\sin^{2}}\theta{\cos^{2}}\phi," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><msup><mi href="./4.14#E1">sin</mi><mn>2</mn></msup><mo></mo><mi href="./31.16#E2">θ</mi></mrow><mo></mo><mrow><msup><mi href="./4.14#E2">cos</mi><mn>2</mn></msup><mo></mo><mi href="./31.16#E2">ϕ</mi></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex2" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m10.png" altimg-height="18px" altimg-valign="-6px" altimg-width="17px" alttext="\displaystyle y" display="inline"><mi href="./31.16#E2">y</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m5.png" altimg-height="27px" altimg-valign="-6px" altimg-width="121px" alttext="\displaystyle={\sin^{2}}\theta{\sin^{2}}\phi." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><msup><mi href="./4.14#E1">sin</mi><mn>2</mn></msup><mo></mo><mi href="./31.16#E2">θ</mi></mrow><mo></mo><mrow><msup><mi href="./4.14#E1">sin</mi><mn>2</mn></msup><mo></mo><mi href="./31.16#E2">ϕ</mi></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd>
<span class="ltx_text"><math class="ltx_Math" altimg="m24.png" altimg-height="18px" altimg-valign="-2px" altimg-width="14px" alttext="\theta" display="inline"><mi href="./31.16#E2">θ</mi></math>: angle (locally)</span>,
<span class="ltx_text"><math class="ltx_Math" altimg="m22.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="\phi" display="inline"><mi href="./31.16#E2">ϕ</mi></math>: angle (locally)</span>,
<span class="ltx_text"><math class="ltx_Math" altimg="m33.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./31.16#E2">x</mi></math>: variable (locally)</span> and
<span class="ltx_text"><math class="ltx_Math" altimg="m34.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./31.16#E2">y</mi></math>: variable (locally)</span>
</dd>
<dt>Symbols:</dt>
<dd>
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m17.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a> and
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m23.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">The coefficients <math class="ltx_Math" altimg="m12.png" altimg-height="23px" altimg-valign="-8px" altimg-width="28px" alttext="A_{j}" display="inline"><msub><mi href="./31.16#SS2.p1">A</mi><mi href="./31.1#p2.t1.r3">j</mi></msub></math> satisfy the relations:
</p>
<table id="E3" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">31.16.3</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E3.png" altimg-height="23px" altimg-valign="-6px" altimg-width="169px" alttext="Q_{0}A_{0}+R_{0}A_{1}=0," display="block"><mrow><mrow><mrow><mrow><msub><mi href="./31.16#E6">Q</mi><mn>0</mn></msub><mo></mo><msub><mi href="./31.16#SS2.p1">A</mi><mn>0</mn></msub></mrow><mo>+</mo><mrow><msub><mi href="./31.16#E7">R</mi><mn>0</mn></msub><mo></mo><msub><mi href="./31.16#SS2.p1">A</mi><mn>1</mn></msub></mrow></mrow><mo>=</mo><mn>0</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./31.16#SS2.p1" title="§31.16(ii) Heun Polynomial Products ‣ §31.16 Mathematical Applications ‣ Applications ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m12.png" altimg-height="23px" altimg-valign="-8px" altimg-width="28px" alttext="A_{j}" display="inline"><msub><mi href="./31.16#SS2.p1">A</mi><mi href="./31.1#p2.t1.r3">j</mi></msub></math>: coefficients</a>,
<a href="./31.16#E6" title="(31.16.6) ‣ §31.16(ii) Heun Polynomial Products ‣ §31.16 Mathematical Applications ‣ Applications ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m14.png" altimg-height="23px" altimg-valign="-8px" altimg-width="28px" alttext="Q_{j}" display="inline"><msub><mi href="./31.16#E6">Q</mi><mi href="./31.1#p2.t1.r3">j</mi></msub></math></a> and
<a href="./31.16#E7" title="(31.16.7) ‣ §31.16(ii) Heun Polynomial Products ‣ §31.16 Mathematical Applications ‣ Applications ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m15.png" altimg-height="23px" altimg-valign="-8px" altimg-width="28px" alttext="R_{j}" display="inline"><msub><mi href="./31.16#E7">R</mi><mi href="./31.1#p2.t1.r3">j</mi></msub></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E4" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">31.16.4</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E4.png" altimg-height="25px" altimg-valign="-8px" altimg-width="276px" alttext="P_{j}A_{j-1}+Q_{j}A_{j}+R_{j}A_{j+1}=0," display="block"><mrow><mrow><mrow><mrow><msub><mi href="./31.16#E5">P</mi><mi href="./31.1#p2.t1.r3">j</mi></msub><mo></mo><msub><mi href="./31.16#SS2.p1">A</mi><mrow><mi href="./31.1#p2.t1.r3">j</mi><mo>-</mo><mn>1</mn></mrow></msub></mrow><mo>+</mo><mrow><msub><mi href="./31.16#E6">Q</mi><mi href="./31.1#p2.t1.r3">j</mi></msub><mo></mo><msub><mi href="./31.16#SS2.p1">A</mi><mi href="./31.1#p2.t1.r3">j</mi></msub></mrow><mo>+</mo><mrow><msub><mi href="./31.16#E7">R</mi><mi href="./31.1#p2.t1.r3">j</mi></msub><mo></mo><msub><mi href="./31.16#SS2.p1">A</mi><mrow><mi href="./31.1#p2.t1.r3">j</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></mrow><mo>=</mo><mn>0</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m26.png" altimg-height="20px" altimg-valign="-6px" altimg-width="125px" alttext="j=1,2,\dots,n" display="inline"><mrow><mi href="./31.1#p2.t1.r3">j</mi><mo>=</mo><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><mi href="./31.1#p2.t1.r3">n</mi></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E4.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./31.1#p2.t1.r3" title="§31.1 Special Notation ‣ Notation ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m27.png" altimg-height="20px" altimg-valign="-6px" altimg-width="14px" alttext="j" display="inline"><mi href="./31.1#p2.t1.r3">j</mi></math>: nonnegative integer</a>,
<a href="./31.1#p2.t1.r3" title="§31.1 Special Notation ‣ Notation ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m31.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./31.1#p2.t1.r3">n</mi></math>: nonnegative integer</a>,
<a href="./31.16#SS2.p1" title="§31.16(ii) Heun Polynomial Products ‣ §31.16 Mathematical Applications ‣ Applications ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m12.png" altimg-height="23px" altimg-valign="-8px" altimg-width="28px" alttext="A_{j}" display="inline"><msub><mi href="./31.16#SS2.p1">A</mi><mi href="./31.1#p2.t1.r3">j</mi></msub></math>: coefficients</a>,
<a href="./31.16#E5" title="(31.16.5) ‣ §31.16(ii) Heun Polynomial Products ‣ §31.16 Mathematical Applications ‣ Applications ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m13.png" altimg-height="23px" altimg-valign="-8px" altimg-width="25px" alttext="P_{j}" display="inline"><msub><mi href="./31.16#E5">P</mi><mi href="./31.1#p2.t1.r3">j</mi></msub></math></a>,
<a href="./31.16#E6" title="(31.16.6) ‣ §31.16(ii) Heun Polynomial Products ‣ §31.16 Mathematical Applications ‣ Applications ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m14.png" altimg-height="23px" altimg-valign="-8px" altimg-width="28px" alttext="Q_{j}" display="inline"><msub><mi href="./31.16#E6">Q</mi><mi href="./31.1#p2.t1.r3">j</mi></msub></math></a> and
<a href="./31.16#E7" title="(31.16.7) ‣ §31.16(ii) Heun Polynomial Products ‣ §31.16 Mathematical Applications ‣ Applications ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m15.png" altimg-height="23px" altimg-valign="-8px" altimg-width="28px" alttext="R_{j}" display="inline"><msub><mi href="./31.16#E7">R</mi><mi href="./31.1#p2.t1.r3">j</mi></msub></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where</p>
</div>
<div id="SS2.p2" class="ltx_para">
<table id="EGx1" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E5">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">31.16.5</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m6.png" altimg-height="25px" altimg-valign="-8px" altimg-width="27px" alttext="\displaystyle P_{j}" display="inline"><msub><mi href="./31.16#E5">P</mi><mi href="./31.1#p2.t1.r3">j</mi></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m2.png" altimg-height="53px" altimg-valign="-21px" altimg-width="365px" alttext="\displaystyle=\frac{(\epsilon-j+n)j(\beta+j-1)(\gamma+\delta+j-2)}{(\gamma+%
\delta+2j-3)(\gamma+\delta+2j-2)}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mstyle displaystyle="true"><mfrac><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mi href="./31.1#p2.t1.r5">ϵ</mi><mo>-</mo><mi href="./31.1#p2.t1.r3">j</mi></mrow><mo>+</mo><mi href="./31.1#p2.t1.r3">n</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi href="./31.1#p2.t1.r3">j</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mi href="./31.1#p2.t1.r5">β</mi><mo>+</mo><mi href="./31.1#p2.t1.r3">j</mi></mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mi href="./31.1#p2.t1.r5">γ</mi><mo>+</mo><mi href="./31.1#p2.t1.r5">δ</mi><mo>+</mo><mi href="./31.1#p2.t1.r3">j</mi></mrow><mo>-</mo><mn>2</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mi href="./31.1#p2.t1.r5">γ</mi><mo>+</mo><mi href="./31.1#p2.t1.r5">δ</mi><mo>+</mo><mrow><mn>2</mn><mo></mo><mi href="./31.1#p2.t1.r3">j</mi></mrow></mrow><mo>-</mo><mn>3</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mi href="./31.1#p2.t1.r5">γ</mi><mo>+</mo><mi href="./31.1#p2.t1.r5">δ</mi><mo>+</mo><mrow><mn>2</mn><mo></mo><mi href="./31.1#p2.t1.r3">j</mi></mrow></mrow><mo>-</mo><mn>2</mn></mrow><mo stretchy="false">)</mo></mrow></mrow></mfrac></mstyle></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E5.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text"><math class="ltx_Math" altimg="m13.png" altimg-height="23px" altimg-valign="-8px" altimg-width="25px" alttext="P_{j}" display="inline"><msub><mi href="./31.16#E5">P</mi><mi href="./31.1#p2.t1.r3">j</mi></msub></math> (locally)</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./31.1#p2.t1.r5" title="§31.1 Special Notation ‣ Notation ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\gamma" display="inline"><mi href="./31.1#p2.t1.r5">γ</mi></math>: real or complex parameter</a>,
<a href="./31.1#p2.t1.r5" title="§31.1 Special Notation ‣ Notation ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m18.png" altimg-height="18px" altimg-valign="-2px" altimg-width="14px" alttext="\delta" display="inline"><mi href="./31.1#p2.t1.r5">δ</mi></math>: real or complex parameter</a>,
<a href="./31.1#p2.t1.r5" title="§31.1 Special Notation ‣ Notation ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="13px" altimg-valign="-2px" altimg-width="12px" alttext="\epsilon" display="inline"><mi href="./31.1#p2.t1.r5">ϵ</mi></math>: real or complex parameter</a>,
<a href="./31.1#p2.t1.r3" title="§31.1 Special Notation ‣ Notation ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m27.png" altimg-height="20px" altimg-valign="-6px" altimg-width="14px" alttext="j" display="inline"><mi href="./31.1#p2.t1.r3">j</mi></math>: nonnegative integer</a>,
<a href="./31.1#p2.t1.r3" title="§31.1 Special Notation ‣ Notation ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m31.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./31.1#p2.t1.r3">n</mi></math>: nonnegative integer</a> and
<a href="./31.1#p2.t1.r5" title="§31.1 Special Notation ‣ Notation ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m16.png" altimg-height="21px" altimg-valign="-6px" altimg-width="17px" alttext="\beta" display="inline"><mi href="./31.1#p2.t1.r5">β</mi></math>: real or complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E6">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">31.16.6</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m7.png" altimg-height="25px" altimg-valign="-8px" altimg-width="30px" alttext="\displaystyle Q_{j}" display="inline"><msub><mi href="./31.16#E6">Q</mi><mi href="./31.1#p2.t1.r3">j</mi></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m1.png" altimg-height="106px" altimg-valign="-21px" altimg-width="567px" alttext="\displaystyle=-aj(j+\gamma+\delta-1)-q+\frac{(j-n)(j+\beta)(j+\gamma)(j+\gamma%
+\delta-1)}{(2j+\gamma+\delta)(2j+\gamma+\delta-1)}+\frac{(j+n+\gamma+\delta-1%
)j(j+\delta-1)(j-\beta+\gamma+\delta-1)}{(2j+\gamma+\delta-1)(2j+\gamma+\delta%
-2)}," display="inline"><mrow><mi></mi><mo>=</mo><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mrow><mrow><mo>-</mo><mrow><mi href="./31.1#p2.t1.r4">a</mi><mo></mo><mi href="./31.1#p2.t1.r3">j</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mi href="./31.1#p2.t1.r3">j</mi><mo>+</mo><mi href="./31.1#p2.t1.r5">γ</mi><mo>+</mo><mi href="./31.1#p2.t1.r5">δ</mi></mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>-</mo><mi href="./31.1#p2.t1.r5">q</mi></mrow><mo>+</mo><mstyle displaystyle="true"><mfrac><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./31.1#p2.t1.r3">j</mi><mo>-</mo><mi href="./31.1#p2.t1.r3">n</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./31.1#p2.t1.r3">j</mi><mo>+</mo><mi href="./31.1#p2.t1.r5">β</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./31.1#p2.t1.r3">j</mi><mo>+</mo><mi href="./31.1#p2.t1.r5">γ</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mi href="./31.1#p2.t1.r3">j</mi><mo>+</mo><mi href="./31.1#p2.t1.r5">γ</mi><mo>+</mo><mi href="./31.1#p2.t1.r5">δ</mi></mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>2</mn><mo></mo><mi href="./31.1#p2.t1.r3">j</mi></mrow><mo>+</mo><mi href="./31.1#p2.t1.r5">γ</mi><mo>+</mo><mi href="./31.1#p2.t1.r5">δ</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mrow><mn>2</mn><mo></mo><mi href="./31.1#p2.t1.r3">j</mi></mrow><mo>+</mo><mi href="./31.1#p2.t1.r5">γ</mi><mo>+</mo><mi href="./31.1#p2.t1.r5">δ</mi></mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow></mfrac></mstyle></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>+</mo><mstyle displaystyle="true"><mfrac><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mi href="./31.1#p2.t1.r3">j</mi><mo>+</mo><mi href="./31.1#p2.t1.r3">n</mi><mo>+</mo><mi href="./31.1#p2.t1.r5">γ</mi><mo>+</mo><mi href="./31.1#p2.t1.r5">δ</mi></mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi href="./31.1#p2.t1.r3">j</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mi href="./31.1#p2.t1.r3">j</mi><mo>+</mo><mi href="./31.1#p2.t1.r5">δ</mi></mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mrow><mi href="./31.1#p2.t1.r3">j</mi><mo>-</mo><mi href="./31.1#p2.t1.r5">β</mi></mrow><mo>+</mo><mi href="./31.1#p2.t1.r5">γ</mi><mo>+</mo><mi href="./31.1#p2.t1.r5">δ</mi></mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mrow><mn>2</mn><mo></mo><mi href="./31.1#p2.t1.r3">j</mi></mrow><mo>+</mo><mi href="./31.1#p2.t1.r5">γ</mi><mo>+</mo><mi href="./31.1#p2.t1.r5">δ</mi></mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mrow><mn>2</mn><mo></mo><mi href="./31.1#p2.t1.r3">j</mi></mrow><mo>+</mo><mi href="./31.1#p2.t1.r5">γ</mi><mo>+</mo><mi href="./31.1#p2.t1.r5">δ</mi></mrow><mo>-</mo><mn>2</mn></mrow><mo stretchy="false">)</mo></mrow></mrow></mfrac></mstyle><mo>,</mo></mtd></mtr></mtable></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E6.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text"><math class="ltx_Math" altimg="m14.png" altimg-height="23px" altimg-valign="-8px" altimg-width="28px" alttext="Q_{j}" display="inline"><msub><mi href="./31.16#E6">Q</mi><mi href="./31.1#p2.t1.r3">j</mi></msub></math> (locally)</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./31.1#p2.t1.r5" title="§31.1 Special Notation ‣ Notation ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\gamma" display="inline"><mi href="./31.1#p2.t1.r5">γ</mi></math>: real or complex parameter</a>,
<a href="./31.1#p2.t1.r5" title="§31.1 Special Notation ‣ Notation ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m18.png" altimg-height="18px" altimg-valign="-2px" altimg-width="14px" alttext="\delta" display="inline"><mi href="./31.1#p2.t1.r5">δ</mi></math>: real or complex parameter</a>,
<a href="./31.1#p2.t1.r3" title="§31.1 Special Notation ‣ Notation ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m27.png" altimg-height="20px" altimg-valign="-6px" altimg-width="14px" alttext="j" display="inline"><mi href="./31.1#p2.t1.r3">j</mi></math>: nonnegative integer</a>,
<a href="./31.1#p2.t1.r3" title="§31.1 Special Notation ‣ Notation ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m31.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./31.1#p2.t1.r3">n</mi></math>: nonnegative integer</a>,
<a href="./31.1#p2.t1.r4" title="§31.1 Special Notation ‣ Notation ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m25.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./31.1#p2.t1.r4">a</mi></math>: complex parameter</a>,
<a href="./31.1#p2.t1.r5" title="§31.1 Special Notation ‣ Notation ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m32.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="q" display="inline"><mi href="./31.1#p2.t1.r5">q</mi></math>: real or complex parameter</a> and
<a href="./31.1#p2.t1.r5" title="§31.1 Special Notation ‣ Notation ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m16.png" altimg-height="21px" altimg-valign="-6px" altimg-width="17px" alttext="\beta" display="inline"><mi href="./31.1#p2.t1.r5">β</mi></math>: real or complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E7">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">31.16.7</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m8.png" altimg-height="25px" altimg-valign="-8px" altimg-width="30px" alttext="\displaystyle R_{j}" display="inline"><msub><mi href="./31.16#E7">R</mi><mi href="./31.1#p2.t1.r3">j</mi></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m3.png" altimg-height="53px" altimg-valign="-21px" altimg-width="349px" alttext="\displaystyle=\frac{(n-j)(j+n+\gamma+\delta)(j+\gamma)(j+\delta)}{(\gamma+%
\delta+2j)(\gamma+\delta+2j+1)}." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mstyle displaystyle="true"><mfrac><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./31.1#p2.t1.r3">n</mi><mo>-</mo><mi href="./31.1#p2.t1.r3">j</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./31.1#p2.t1.r3">j</mi><mo>+</mo><mi href="./31.1#p2.t1.r3">n</mi><mo>+</mo><mi href="./31.1#p2.t1.r5">γ</mi><mo>+</mo><mi href="./31.1#p2.t1.r5">δ</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./31.1#p2.t1.r3">j</mi><mo>+</mo><mi href="./31.1#p2.t1.r5">γ</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./31.1#p2.t1.r3">j</mi><mo>+</mo><mi href="./31.1#p2.t1.r5">δ</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./31.1#p2.t1.r5">γ</mi><mo>+</mo><mi href="./31.1#p2.t1.r5">δ</mi><mo>+</mo><mrow><mn>2</mn><mo></mo><mi href="./31.1#p2.t1.r3">j</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./31.1#p2.t1.r5">γ</mi><mo>+</mo><mi href="./31.1#p2.t1.r5">δ</mi><mo>+</mo><mrow><mn>2</mn><mo></mo><mi href="./31.1#p2.t1.r3">j</mi></mrow><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow></mfrac></mstyle></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E7.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text"><math class="ltx_Math" altimg="m15.png" altimg-height="23px" altimg-valign="-8px" altimg-width="28px" alttext="R_{j}" display="inline"><msub><mi href="./31.16#E7">R</mi><mi href="./31.1#p2.t1.r3">j</mi></msub></math> (locally)</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./31.1#p2.t1.r5" title="§31.1 Special Notation ‣ Notation ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\gamma" display="inline"><mi href="./31.1#p2.t1.r5">γ</mi></math>: real or complex parameter</a>,
<a href="./31.1#p2.t1.r5" title="§31.1 Special Notation ‣ Notation ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m18.png" altimg-height="18px" altimg-valign="-2px" altimg-width="14px" alttext="\delta" display="inline"><mi href="./31.1#p2.t1.r5">δ</mi></math>: real or complex parameter</a>,
<a href="./31.1#p2.t1.r3" title="§31.1 Special Notation ‣ Notation ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m27.png" altimg-height="20px" altimg-valign="-6px" altimg-width="14px" alttext="j" display="inline"><mi href="./31.1#p2.t1.r3">j</mi></math>: nonnegative integer</a> and
<a href="./31.1#p2.t1.r3" title="§31.1 Special Notation ‣ Notation ‣ Chapter 31 Heun Functions" class="ltx_ref"><math class="ltx_Math" altimg="m31.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./31.1#p2.t1.r3">n</mi></math>: nonnegative integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
</div>
<div id="SS2.p3" class="ltx_para">
<p class="ltx_p">By specifying either <math class="ltx_Math" altimg="m24.png" altimg-height="18px" altimg-valign="-2px" altimg-width="14px" alttext="\theta" display="inline"><mi href="./31.16#E2">θ</mi></math> or <math class="ltx_Math" altimg="m22.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="\phi" display="inline"><mi href="./31.16#E2">ϕ</mi></math> in (</div>
</div>
</body></text>
</html>
</page>
<page>
<!DOCTYPE html><html>
<head>
<title>DLMF: 15.8 Transformations of Variable</title>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<!--GOOGLE BOOTSTRAP--></head>
<text><body class="color_default textfont_default titlefont_default fontsize_default navbar_default">
<div class="ltx_page_navbar">
<div class="ltx_page_navlogo"> assume their
principal values.</p>
<table id="E1" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">15.8.1</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_Math ltx_eqn_cell ltx_align_center"><math class="ltx_Math" alttext="\mathbf{F}\left({a,b\atop c};z\right)=(1-z)^{-a}\mathbf{F}\left({a,c-b\atop c}%
;\frac{z}{z-1}\right)=(1-z)^{-b}\mathbf{F}\left({c-a,b\atop c};\frac{z}{z-1}%
\right)=(1-z)^{c-a-b}\mathbf{F}\left({c-a,c-b\atop c};z\right)," display="block"><mrow><mtable align="baseline 1" columnalign="left" columnspacing="0.1em" displaystyle="true"><mtr><mtd><mrow><mi href="./15.2#E2" mathvariant="bold">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mi href="./15.1#p1.t1.r3">c</mi></mfrac><mo>;</mo><mi href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mtd><mtd><mo>=</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi></mrow></msup><mo></mo><mrow><mi href="./15.2#E2" mathvariant="bold">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mrow><mi href="./15.1#p1.t1.r3">c</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow></mrow><mi href="./15.1#p1.t1.r3">c</mi></mfrac><mo>;</mo><mfrac><mi href="./15.1#p1.t1.r2">z</mi><mrow><mi href="./15.1#p1.t1.r2">z</mi><mo>-</mo><mn>1</mn></mrow></mfrac><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow></msup><mo></mo><mrow><mi href="./15.2#E2" mathvariant="bold">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mrow><mi href="./15.1#p1.t1.r3">c</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>,</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mi href="./15.1#p1.t1.r3">c</mi></mfrac><mo>;</mo><mfrac><mi href="./15.1#p1.t1.r2">z</mi><mrow><mi href="./15.1#p1.t1.r2">z</mi><mo>-</mo><mn>1</mn></mrow></mfrac><mo>)</mo></mrow></mrow></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mi href="./15.1#p1.t1.r3">c</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow></msup><mo></mo><mrow><mi href="./15.2#E2" mathvariant="bold">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mrow><mi href="./15.1#p1.t1.r3">c</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>,</mo><mrow><mi href="./15.1#p1.t1.r3">c</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow></mrow><mi href="./15.1#p1.t1.r3">c</mi></mfrac><mo>;</mo><mi href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mtd></mtr></mtable></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m85.png" altimg-height="23px" altimg-valign="-7px" altimg-width="146px" alttext="|\operatorname{ph}\left(1-z\right)|<\pi" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo>)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow><mo><</mo><mi href="./3.12#E1">π</mi></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.9#E7" title="(1.9.7) ‣ Modulus and Phase ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="21px" altimg-valign="-6px" altimg-width="26px" alttext="\operatorname{ph}" display="inline"><mi href="./1.9#E7">ph</mi></math>: phase</a>,
<a href="./15.2#E2" title="(15.2.2) ‣ §15.2(i) Gauss Series ‣ §15.2 Definitions and Analytical Properties ‣ Properties ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="23px" altimg-valign="-7px" altimg-width="102px" alttext="\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E2" mathvariant="bold">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m45.png" altimg-height="39px" altimg-valign="-15px" altimg-width="90px" alttext="\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E2" mathvariant="bold">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi></mfrac><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: Olver’s hypergeometric function</a>,
<a href="./15.1#p1.t1.r2" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./15.1#p1.t1.r2">z</mi></math>: complex variable</a>,
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./15.1#p1.t1.r3">a</mi></math>: real or complex parameter</a>,
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="b" display="inline"><mi href="./15.1#p1.t1.r3">b</mi></math>: real or complex parameter</a> and
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m76.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="c" display="inline"><mi href="./15.1#p1.t1.r3">c</mi></math>: real or complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="EGx1" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E2">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">15.8.2</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m23.png" altimg-height="53px" altimg-valign="-21px" altimg-width="224px" alttext="\displaystyle\frac{\sin\left(\pi(b-a)\right)}{\pi}\mathbf{F}\left({a,b\atop c}%
;z\right)" display="inline"><mrow><mstyle displaystyle="true"><mfrac><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./3.12#E1">π</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./15.1#p1.t1.r3">b</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>)</mo></mrow></mrow><mi href="./3.12#E1">π</mi></mfrac></mstyle><mo></mo><mrow><mi href="./15.2#E2" mathvariant="bold">F</mi><mo></mo><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac linethickness="0pt"><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mi href="./15.1#p1.t1.r3">c</mi></mfrac></mstyle><mo>;</mo><mi href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m7.png" altimg-height="56px" altimg-valign="-21px" altimg-width="647px" alttext="\displaystyle=\frac{(-z)^{-a}}{\Gamma\left(b\right)\Gamma\left(c-a\right)}%
\mathbf{F}\left({a,a-c+1\atop a-b+1};\frac{1}{z}\right)-\frac{(-z)^{-b}}{%
\Gamma\left(a\right)\Gamma\left(c-b\right)}\mathbf{F}\left({b,b-c+1\atop b-a+1%
};\frac{1}{z}\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mstyle displaystyle="true"><mfrac><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi></mrow></msup><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi href="./15.1#p1.t1.r3">b</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./15.1#p1.t1.r3">c</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>)</mo></mrow></mrow></mrow></mfrac></mstyle><mo></mo><mrow><mi href="./15.2#E2" mathvariant="bold">F</mi><mo></mo><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac linethickness="0pt"><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mrow><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">c</mi></mrow><mo>+</mo><mn>1</mn></mrow></mrow><mrow><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mo>+</mo><mn>1</mn></mrow></mfrac></mstyle><mo>;</mo><mstyle displaystyle="true"><mfrac><mn>1</mn><mi href="./15.1#p1.t1.r2">z</mi></mfrac></mstyle><mo>)</mo></mrow></mrow></mrow><mo>-</mo><mrow><mstyle displaystyle="true"><mfrac><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow></msup><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi href="./15.1#p1.t1.r3">a</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./15.1#p1.t1.r3">c</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mo>)</mo></mrow></mrow></mrow></mfrac></mstyle><mo></mo><mrow><mi href="./15.2#E2" mathvariant="bold">F</mi><mo></mo><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac linethickness="0pt"><mrow><mi href="./15.1#p1.t1.r3">b</mi><mo>,</mo><mrow><mrow><mi href="./15.1#p1.t1.r3">b</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">c</mi></mrow><mo>+</mo><mn>1</mn></mrow></mrow><mrow><mrow><mi href="./15.1#p1.t1.r3">b</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>+</mo><mn>1</mn></mrow></mfrac></mstyle><mo>;</mo><mstyle displaystyle="true"><mfrac><mn>1</mn><mi href="./15.1#p1.t1.r2">z</mi></mfrac></mstyle><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m84.png" altimg-height="23px" altimg-valign="-7px" altimg-width="127px" alttext="|\operatorname{ph}\left(-z\right)|<\pi" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo>)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow><mo><</mo><mi href="./3.12#E1">π</mi></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m31.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.9#E7" title="(1.9.7) ‣ Modulus and Phase ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="21px" altimg-valign="-6px" altimg-width="26px" alttext="\operatorname{ph}" display="inline"><mi href="./1.9#E7">ph</mi></math>: phase</a>,
<a href="./15.2#E2" title="(15.2.2) ‣ §15.2(i) Gauss Series ‣ §15.2 Definitions and Analytical Properties ‣ Properties ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="23px" altimg-valign="-7px" altimg-width="102px" alttext="\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E2" mathvariant="bold">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m45.png" altimg-height="39px" altimg-valign="-15px" altimg-width="90px" alttext="\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E2" mathvariant="bold">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi></mfrac><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: Olver’s hypergeometric function</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./15.1#p1.t1.r2" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./15.1#p1.t1.r2">z</mi></math>: complex variable</a>,
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./15.1#p1.t1.r3">a</mi></math>: real or complex parameter</a>,
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="b" display="inline"><mi href="./15.1#p1.t1.r3">b</mi></math>: real or complex parameter</a> and
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m76.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="c" display="inline"><mi href="./15.1#p1.t1.r3">c</mi></math>: real or complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E3">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">15.8.3</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m23.png" altimg-height="53px" altimg-valign="-21px" altimg-width="224px" alttext="\displaystyle\frac{\sin\left(\pi(b-a)\right)}{\pi}\mathbf{F}\left({a,b\atop c}%
;z\right)" display="inline"><mrow><mstyle displaystyle="true"><mfrac><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./3.12#E1">π</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./15.1#p1.t1.r3">b</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>)</mo></mrow></mrow><mi href="./3.12#E1">π</mi></mfrac></mstyle><mo></mo><mrow><mi href="./15.2#E2" mathvariant="bold">F</mi><mo></mo><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac linethickness="0pt"><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mi href="./15.1#p1.t1.r3">c</mi></mfrac></mstyle><mo>;</mo><mi href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m8.png" altimg-height="56px" altimg-valign="-21px" altimg-width="681px" alttext="\displaystyle=\frac{(1-z)^{-a}}{\Gamma\left(b\right)\Gamma\left(c-a\right)}%
\mathbf{F}\left({a,c-b\atop a-b+1};\frac{1}{1-z}\right)-\frac{(1-z)^{-b}}{%
\Gamma\left(a\right)\Gamma\left(c-b\right)}\mathbf{F}\left({b,c-a\atop b-a+1};%
\frac{1}{1-z}\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mstyle displaystyle="true"><mfrac><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi></mrow></msup><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi href="./15.1#p1.t1.r3">b</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./15.1#p1.t1.r3">c</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>)</mo></mrow></mrow></mrow></mfrac></mstyle><mo></mo><mrow><mi href="./15.2#E2" mathvariant="bold">F</mi><mo></mo><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac linethickness="0pt"><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mrow><mi href="./15.1#p1.t1.r3">c</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow></mrow><mrow><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mo>+</mo><mn>1</mn></mrow></mfrac></mstyle><mo>;</mo><mstyle displaystyle="true"><mfrac><mn>1</mn><mrow><mn>1</mn><mo>-</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow></mfrac></mstyle><mo>)</mo></mrow></mrow></mrow><mo>-</mo><mrow><mstyle displaystyle="true"><mfrac><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow></msup><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi href="./15.1#p1.t1.r3">a</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./15.1#p1.t1.r3">c</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mo>)</mo></mrow></mrow></mrow></mfrac></mstyle><mo></mo><mrow><mi href="./15.2#E2" mathvariant="bold">F</mi><mo></mo><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac linethickness="0pt"><mrow><mi href="./15.1#p1.t1.r3">b</mi><mo>,</mo><mrow><mi href="./15.1#p1.t1.r3">c</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi></mrow></mrow><mrow><mrow><mi href="./15.1#p1.t1.r3">b</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>+</mo><mn>1</mn></mrow></mfrac></mstyle><mo>;</mo><mstyle displaystyle="true"><mfrac><mn>1</mn><mrow><mn>1</mn><mo>-</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow></mfrac></mstyle><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m84.png" altimg-height="23px" altimg-valign="-7px" altimg-width="127px" alttext="|\operatorname{ph}\left(-z\right)|<\pi" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo>)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow><mo><</mo><mi href="./3.12#E1">π</mi></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m31.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.9#E7" title="(1.9.7) ‣ Modulus and Phase ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="21px" altimg-valign="-6px" altimg-width="26px" alttext="\operatorname{ph}" display="inline"><mi href="./1.9#E7">ph</mi></math>: phase</a>,
<a href="./15.2#E2" title="(15.2.2) ‣ §15.2(i) Gauss Series ‣ §15.2 Definitions and Analytical Properties ‣ Properties ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="23px" altimg-valign="-7px" altimg-width="102px" alttext="\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E2" mathvariant="bold">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m45.png" altimg-height="39px" altimg-valign="-15px" altimg-width="90px" alttext="\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E2" mathvariant="bold">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi></mfrac><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: Olver’s hypergeometric function</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./15.1#p1.t1.r2" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./15.1#p1.t1.r2">z</mi></math>: complex variable</a>,
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./15.1#p1.t1.r3">a</mi></math>: real or complex parameter</a>,
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="b" display="inline"><mi href="./15.1#p1.t1.r3">b</mi></math>: real or complex parameter</a> and
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m76.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="c" display="inline"><mi href="./15.1#p1.t1.r3">c</mi></math>: real or complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E4">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">15.8.4</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m24.png" altimg-height="53px" altimg-valign="-21px" altimg-width="257px" alttext="\displaystyle\frac{\sin\left(\pi(c-a-b)\right)}{\pi}\mathbf{F}\left({a,b\atop c%
};z\right)" display="inline"><mrow><mstyle displaystyle="true"><mfrac><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./3.12#E1">π</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./15.1#p1.t1.r3">c</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>)</mo></mrow></mrow><mi href="./3.12#E1">π</mi></mfrac></mstyle><mo></mo><mrow><mi href="./15.2#E2" mathvariant="bold">F</mi><mo></mo><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac linethickness="0pt"><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mi href="./15.1#p1.t1.r3">c</mi></mfrac></mstyle><mo>;</mo><mi href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m11.png" altimg-height="56px" altimg-valign="-21px" altimg-width="761px" alttext="\displaystyle=\frac{1}{\Gamma\left(c-a\right)\Gamma\left(c-b\right)}\mathbf{F}%
\left({a,b\atop a+b-c+1};1-z\right)-\frac{(1-z)^{c-a-b}}{\Gamma\left(a\right)%
\Gamma\left(b\right)}\mathbf{F}\left({c-a,c-b\atop c-a-b+1};1-z\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mstyle displaystyle="true"><mfrac><mn>1</mn><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./15.1#p1.t1.r3">c</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./15.1#p1.t1.r3">c</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mo>)</mo></mrow></mrow></mrow></mfrac></mstyle><mo></mo><mrow><mi href="./15.2#E2" mathvariant="bold">F</mi><mo></mo><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac linethickness="0pt"><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mrow><mrow><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>+</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mo>-</mo><mi href="./15.1#p1.t1.r3">c</mi></mrow><mo>+</mo><mn>1</mn></mrow></mfrac></mstyle><mo>;</mo><mrow><mn>1</mn><mo>-</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>-</mo><mrow><mstyle displaystyle="true"><mfrac><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mi href="./15.1#p1.t1.r3">c</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow></msup><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi href="./15.1#p1.t1.r3">a</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi href="./15.1#p1.t1.r3">b</mi><mo>)</mo></mrow></mrow></mrow></mfrac></mstyle><mo></mo><mrow><mi href="./15.2#E2" mathvariant="bold">F</mi><mo></mo><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac linethickness="0pt"><mrow><mrow><mi href="./15.1#p1.t1.r3">c</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>,</mo><mrow><mi href="./15.1#p1.t1.r3">c</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow></mrow><mrow><mrow><mi href="./15.1#p1.t1.r3">c</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mo>+</mo><mn>1</mn></mrow></mfrac></mstyle><mo>;</mo><mrow><mn>1</mn><mo>-</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m86.png" altimg-height="23px" altimg-valign="-7px" altimg-width="93px" alttext="|\operatorname{ph}z|<\pi" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo stretchy="false">|</mo></mrow><mo><</mo><mi href="./3.12#E1">π</mi></mrow></math>, <math class="ltx_Math" altimg="m85.png" altimg-height="23px" altimg-valign="-7px" altimg-width="146px" alttext="|\operatorname{ph}\left(1-z\right)|<\pi" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo>)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow><mo><</mo><mi href="./3.12#E1">π</mi></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E4.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m31.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.9#E7" title="(1.9.7) ‣ Modulus and Phase ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="21px" altimg-valign="-6px" altimg-width="26px" alttext="\operatorname{ph}" display="inline"><mi href="./1.9#E7">ph</mi></math>: phase</a>,
<a href="./15.2#E2" title="(15.2.2) ‣ §15.2(i) Gauss Series ‣ §15.2 Definitions and Analytical Properties ‣ Properties ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="23px" altimg-valign="-7px" altimg-width="102px" alttext="\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E2" mathvariant="bold">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m45.png" altimg-height="39px" altimg-valign="-15px" altimg-width="90px" alttext="\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E2" mathvariant="bold">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi></mfrac><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: Olver’s hypergeometric function</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./15.1#p1.t1.r2" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./15.1#p1.t1.r2">z</mi></math>: complex variable</a>,
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./15.1#p1.t1.r3">a</mi></math>: real or complex parameter</a>,
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="b" display="inline"><mi href="./15.1#p1.t1.r3">b</mi></math>: real or complex parameter</a> and
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m76.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="c" display="inline"><mi href="./15.1#p1.t1.r3">c</mi></math>: real or complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E5">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">15.8.5</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m24.png" altimg-height="53px" altimg-valign="-21px" altimg-width="257px" alttext="\displaystyle\frac{\sin\left(\pi(c-a-b)\right)}{\pi}\mathbf{F}\left({a,b\atop c%
};z\right)" display="inline"><mrow><mstyle displaystyle="true"><mfrac><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./3.12#E1">π</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./15.1#p1.t1.r3">c</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>)</mo></mrow></mrow><mi href="./3.12#E1">π</mi></mfrac></mstyle><mo></mo><mrow><mi href="./15.2#E2" mathvariant="bold">F</mi><mo></mo><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac linethickness="0pt"><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mi href="./15.1#p1.t1.r3">c</mi></mfrac></mstyle><mo>;</mo><mi href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m12.png" altimg-height="56px" altimg-valign="-21px" altimg-width="810px" alttext="\displaystyle=\frac{z^{-a}}{\Gamma\left(c-a\right)\Gamma\left(c-b\right)}%
\mathbf{F}\left({a,a-c+1\atop a+b-c+1};1-\frac{1}{z}\right)-\frac{(1-z)^{c-a-b%
}z^{a-c}}{\Gamma\left(a\right)\Gamma\left(b\right)}\mathbf{F}\left({c-a,1-a%
\atop c-a-b+1};1-\frac{1}{z}\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mstyle displaystyle="true"><mfrac><msup><mi href="./15.1#p1.t1.r2">z</mi><mrow><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi></mrow></msup><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./15.1#p1.t1.r3">c</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./15.1#p1.t1.r3">c</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mo>)</mo></mrow></mrow></mrow></mfrac></mstyle><mo></mo><mrow><mi href="./15.2#E2" mathvariant="bold">F</mi><mo></mo><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac linethickness="0pt"><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mrow><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">c</mi></mrow><mo>+</mo><mn>1</mn></mrow></mrow><mrow><mrow><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>+</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mo>-</mo><mi href="./15.1#p1.t1.r3">c</mi></mrow><mo>+</mo><mn>1</mn></mrow></mfrac></mstyle><mo>;</mo><mrow><mn>1</mn><mo>-</mo><mstyle displaystyle="true"><mfrac><mn>1</mn><mi href="./15.1#p1.t1.r2">z</mi></mfrac></mstyle></mrow><mo>)</mo></mrow></mrow></mrow><mo>-</mo><mrow><mstyle displaystyle="true"><mfrac><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mi href="./15.1#p1.t1.r3">c</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow></msup><mo></mo><msup><mi href="./15.1#p1.t1.r2">z</mi><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">c</mi></mrow></msup></mrow><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi href="./15.1#p1.t1.r3">a</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi href="./15.1#p1.t1.r3">b</mi><mo>)</mo></mrow></mrow></mrow></mfrac></mstyle><mo></mo><mrow><mi href="./15.2#E2" mathvariant="bold">F</mi><mo></mo><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac linethickness="0pt"><mrow><mrow><mi href="./15.1#p1.t1.r3">c</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>,</mo><mrow><mn>1</mn><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi></mrow></mrow><mrow><mrow><mi href="./15.1#p1.t1.r3">c</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mo>+</mo><mn>1</mn></mrow></mfrac></mstyle><mo>;</mo><mrow><mn>1</mn><mo>-</mo><mstyle displaystyle="true"><mfrac><mn>1</mn><mi href="./15.1#p1.t1.r2">z</mi></mfrac></mstyle></mrow><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m86.png" altimg-height="23px" altimg-valign="-7px" altimg-width="93px" alttext="|\operatorname{ph}z|<\pi" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo stretchy="false">|</mo></mrow><mo><</mo><mi href="./3.12#E1">π</mi></mrow></math>, <math class="ltx_Math" altimg="m85.png" altimg-height="23px" altimg-valign="-7px" altimg-width="146px" alttext="|\operatorname{ph}\left(1-z\right)|<\pi" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo>)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow><mo><</mo><mi href="./3.12#E1">π</mi></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E5.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m31.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.9#E7" title="(1.9.7) ‣ Modulus and Phase ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="21px" altimg-valign="-6px" altimg-width="26px" alttext="\operatorname{ph}" display="inline"><mi href="./1.9#E7">ph</mi></math>: phase</a>,
<a href="./15.2#E2" title="(15.2.2) ‣ §15.2(i) Gauss Series ‣ §15.2 Definitions and Analytical Properties ‣ Properties ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="23px" altimg-valign="-7px" altimg-width="102px" alttext="\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E2" mathvariant="bold">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m45.png" altimg-height="39px" altimg-valign="-15px" altimg-width="90px" alttext="\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E2" mathvariant="bold">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi></mfrac><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: Olver’s hypergeometric function</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./15.1#p1.t1.r2" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./15.1#p1.t1.r2">z</mi></math>: complex variable</a>,
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./15.1#p1.t1.r3">a</mi></math>: real or complex parameter</a>,
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="b" display="inline"><mi href="./15.1#p1.t1.r3">b</mi></math>: real or complex parameter</a> and
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m76.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="c" display="inline"><mi href="./15.1#p1.t1.r3">c</mi></math>: real or complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
</div>
<div id="SS1.p2" class="ltx_para">
<p class="ltx_p">For an alternative version of the transformations () as <math class="ltx_Math" altimg="m57.png" altimg-height="17px" altimg-valign="-4px" altimg-width="79px" alttext="a\to-m" display="inline"><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>→</mo><mrow><mo>-</mo><mi href="./15.1#p1.t1.r4">m</mi></mrow></mrow></math>, together with
(</dd>
</dl>
</div>
</div>
<div id="SS2.p1" class="ltx_para">
<p class="ltx_p">With <math class="ltx_Math" altimg="m78.png" altimg-height="20px" altimg-valign="-6px" altimg-width="128px" alttext="m=0,1,2,\dots" display="inline"><mrow><mi href="./15.1#p1.t1.r4">m</mi><mo>=</mo><mrow><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>, polynomial cases of () are given by</p>
<table id="EGx2" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E6">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">15.8.6</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m17.png" altimg-height="53px" altimg-valign="-21px" altimg-width="128px" alttext="\displaystyle F\left({-m,b\atop c};z\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac linethickness="0pt"><mrow><mrow><mo>-</mo><mi href="./15.1#p1.t1.r4">m</mi></mrow><mo>,</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mi href="./15.1#p1.t1.r3">c</mi></mfrac></mstyle><mo>;</mo><mi href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m9.png" altimg-height="53px" altimg-valign="-21px" altimg-width="681px" alttext="\displaystyle=\frac{(b)_{m}}{(c)_{m}}(-z)^{m}F\left({-m,1-c-m\atop 1-b-m};%
\frac{1}{z}\right)=\frac{(b)_{m}}{(c)_{m}}(1-z)^{m}F\left({-m,c-b\atop 1-b-m};%
\frac{1}{1-z}\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><mfrac><msub><mrow><mo stretchy="false">(</mo><mi href="./15.1#p1.t1.r3">b</mi><mo stretchy="false">)</mo></mrow><mi href="./15.1#p1.t1.r4">m</mi></msub><msub><mrow><mo stretchy="false">(</mo><mi href="./15.1#p1.t1.r3">c</mi><mo stretchy="false">)</mo></mrow><mi href="./15.1#p1.t1.r4">m</mi></msub></mfrac></mstyle><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo stretchy="false">)</mo></mrow><mi href="./15.1#p1.t1.r4">m</mi></msup><mo></mo><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac linethickness="0pt"><mrow><mrow><mo>-</mo><mi href="./15.1#p1.t1.r4">m</mi></mrow><mo>,</mo><mrow><mn>1</mn><mo>-</mo><mi href="./15.1#p1.t1.r3">c</mi><mo>-</mo><mi href="./15.1#p1.t1.r4">m</mi></mrow></mrow><mrow><mn>1</mn><mo>-</mo><mi href="./15.1#p1.t1.r3">b</mi><mo>-</mo><mi href="./15.1#p1.t1.r4">m</mi></mrow></mfrac></mstyle><mo>;</mo><mstyle displaystyle="true"><mfrac><mn>1</mn><mi href="./15.1#p1.t1.r2">z</mi></mfrac></mstyle><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mrow><mstyle displaystyle="true"><mfrac><msub><mrow><mo stretchy="false">(</mo><mi href="./15.1#p1.t1.r3">b</mi><mo stretchy="false">)</mo></mrow><mi href="./15.1#p1.t1.r4">m</mi></msub><msub><mrow><mo stretchy="false">(</mo><mi href="./15.1#p1.t1.r3">c</mi><mo stretchy="false">)</mo></mrow><mi href="./15.1#p1.t1.r4">m</mi></msub></mfrac></mstyle><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo stretchy="false">)</mo></mrow><mi href="./15.1#p1.t1.r4">m</mi></msup><mo></mo><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac linethickness="0pt"><mrow><mrow><mo>-</mo><mi href="./15.1#p1.t1.r4">m</mi></mrow><mo>,</mo><mrow><mi href="./15.1#p1.t1.r3">c</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow></mrow><mrow><mn>1</mn><mo>-</mo><mi href="./15.1#p1.t1.r3">b</mi><mo>-</mo><mi href="./15.1#p1.t1.r4">m</mi></mrow></mfrac></mstyle><mo>;</mo><mstyle displaystyle="true"><mfrac><mn>1</mn><mrow><mn>1</mn><mo>-</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow></mfrac></mstyle><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E6.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./15.2#E1" title="(15.2.1) ‣ §15.2(i) Gauss Series ‣ §15.2 Definitions and Analytical Properties ‣ Properties ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="23px" altimg-valign="-7px" altimg-width="103px" alttext="F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m30.png" altimg-height="39px" altimg-valign="-15px" altimg-width="91px" alttext="F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi></mfrac><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: <math class="ltx_Math" altimg="m28.png" altimg-height="23px" altimg-valign="-7px" altimg-width="139px" alttext="={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow></math>
Gauss’ hypergeometric function</a>,
<a href="./15.1#p1.t1.r2" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./15.1#p1.t1.r2">z</mi></math>: complex variable</a>,
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="b" display="inline"><mi href="./15.1#p1.t1.r3">b</mi></math>: real or complex parameter</a>,
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m76.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="c" display="inline"><mi href="./15.1#p1.t1.r3">c</mi></math>: real or complex parameter</a> and
<a href="./15.1#p1.t1.r4" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m79.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./15.1#p1.t1.r4">m</mi></math>: integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E7">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">15.8.7</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m17.png" altimg-height="53px" altimg-valign="-21px" altimg-width="128px" alttext="\displaystyle F\left({-m,b\atop c};z\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac linethickness="0pt"><mrow><mrow><mo>-</mo><mi href="./15.1#p1.t1.r4">m</mi></mrow><mo>,</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mi href="./15.1#p1.t1.r3">c</mi></mfrac></mstyle><mo>;</mo><mi href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m10.png" altimg-height="53px" altimg-valign="-21px" altimg-width="703px" alttext="\displaystyle=\frac{(c-b)_{m}}{(c)_{m}}F\left({-m,b\atop b-c-m+1};1-z\right)=%
\frac{(c-b)_{m}}{(c)_{m}}z^{m}F\left({-m,1-c-m\atop b-c-m+1};1-\frac{1}{z}%
\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><mfrac><msub><mrow><mo stretchy="false">(</mo><mrow><mi href="./15.1#p1.t1.r3">c</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mo stretchy="false">)</mo></mrow><mi href="./15.1#p1.t1.r4">m</mi></msub><msub><mrow><mo stretchy="false">(</mo><mi href="./15.1#p1.t1.r3">c</mi><mo stretchy="false">)</mo></mrow><mi href="./15.1#p1.t1.r4">m</mi></msub></mfrac></mstyle><mo></mo><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac linethickness="0pt"><mrow><mrow><mo>-</mo><mi href="./15.1#p1.t1.r4">m</mi></mrow><mo>,</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mrow><mrow><mi href="./15.1#p1.t1.r3">b</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">c</mi><mo>-</mo><mi href="./15.1#p1.t1.r4">m</mi></mrow><mo>+</mo><mn>1</mn></mrow></mfrac></mstyle><mo>;</mo><mrow><mn>1</mn><mo>-</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mrow><mstyle displaystyle="true"><mfrac><msub><mrow><mo stretchy="false">(</mo><mrow><mi href="./15.1#p1.t1.r3">c</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mo stretchy="false">)</mo></mrow><mi href="./15.1#p1.t1.r4">m</mi></msub><msub><mrow><mo stretchy="false">(</mo><mi href="./15.1#p1.t1.r3">c</mi><mo stretchy="false">)</mo></mrow><mi href="./15.1#p1.t1.r4">m</mi></msub></mfrac></mstyle><mo></mo><msup><mi href="./15.1#p1.t1.r2">z</mi><mi href="./15.1#p1.t1.r4">m</mi></msup><mo></mo><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac linethickness="0pt"><mrow><mrow><mo>-</mo><mi href="./15.1#p1.t1.r4">m</mi></mrow><mo>,</mo><mrow><mn>1</mn><mo>-</mo><mi href="./15.1#p1.t1.r3">c</mi><mo>-</mo><mi href="./15.1#p1.t1.r4">m</mi></mrow></mrow><mrow><mrow><mi href="./15.1#p1.t1.r3">b</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">c</mi><mo>-</mo><mi href="./15.1#p1.t1.r4">m</mi></mrow><mo>+</mo><mn>1</mn></mrow></mfrac></mstyle><mo>;</mo><mrow><mn>1</mn><mo>-</mo><mstyle displaystyle="true"><mfrac><mn>1</mn><mi href="./15.1#p1.t1.r2">z</mi></mfrac></mstyle></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E7.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./15.2#E1" title="(15.2.1) ‣ §15.2(i) Gauss Series ‣ §15.2 Definitions and Analytical Properties ‣ Properties ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="23px" altimg-valign="-7px" altimg-width="103px" alttext="F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m30.png" altimg-height="39px" altimg-valign="-15px" altimg-width="91px" alttext="F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi></mfrac><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: <math class="ltx_Math" altimg="m28.png" altimg-height="23px" altimg-valign="-7px" altimg-width="139px" alttext="={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow></math>
Gauss’ hypergeometric function</a>,
<a href="./15.1#p1.t1.r2" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./15.1#p1.t1.r2">z</mi></math>: complex variable</a>,
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="b" display="inline"><mi href="./15.1#p1.t1.r3">b</mi></math>: real or complex parameter</a>,
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m76.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="c" display="inline"><mi href="./15.1#p1.t1.r3">c</mi></math>: real or complex parameter</a> and
<a href="./15.1#p1.t1.r4" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m79.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./15.1#p1.t1.r4">m</mi></math>: integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
<p class="ltx_p">with the understanding that if <math class="ltx_Math" altimg="m60.png" altimg-height="19px" altimg-valign="-4px" altimg-width="63px" alttext="b=-\ell" display="inline"><mrow><mi href="./15.1#p1.t1.r3">b</mi><mo>=</mo><mrow><mo>-</mo><mi href="./15.1#p1.t1.r4" mathvariant="normal">ℓ</mi></mrow></mrow></math>, <math class="ltx_Math" altimg="m36.png" altimg-height="21px" altimg-valign="-6px" altimg-width="119px" alttext="\ell=0,1,2,\dots" display="inline"><mrow><mi href="./15.1#p1.t1.r4" mathvariant="normal">ℓ</mi><mo>=</mo><mrow><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>, then
<math class="ltx_Math" altimg="m80.png" altimg-height="20px" altimg-valign="-5px" altimg-width="57px" alttext="m\leq\ell" display="inline"><mrow><mi href="./15.1#p1.t1.r4">m</mi><mo>≤</mo><mi href="./15.1#p1.t1.r4" mathvariant="normal">ℓ</mi></mrow></math>.</p>
</div>
<div id="SS2.p2" class="ltx_para">
<p class="ltx_p">When <math class="ltx_Math" altimg="m58.png" altimg-height="19px" altimg-valign="-4px" altimg-width="48px" alttext="b-a" display="inline"><mrow><mi href="./15.1#p1.t1.r3">b</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi></mrow></math> is an integer limits are taken in () as follows.</p>
</div>
<div id="SS2.p3" class="ltx_para">
<p class="ltx_p">If <math class="ltx_Math" altimg="m58.png" altimg-height="19px" altimg-valign="-4px" altimg-width="48px" alttext="b-a" display="inline"><mrow><mi href="./15.1#p1.t1.r3">b</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi></mrow></math> is a nonnegative integer, then
</p>
<table id="E8" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">15.8.8</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_Math ltx_eqn_cell ltx_align_center"><math class="ltx_Math" alttext="\mathbf{F}\left({a,a+m\atop c};z\right)=\frac{(-z)^{-a}}{\Gamma\left(a+m\right%
)}\sum_{k=0}^{m-1}\frac{(a)_{k}(m-k-1)!}{k!\Gamma\left(c-a-k\right)}z^{-k}+%
\frac{(-z)^{-a}}{\Gamma\left(a\right)}\sum_{k=0}^{\infty}\frac{(a+m)_{k}}{k!(k%
+m)!\Gamma\left(c-a-k-m\right)}(-1)^{k}z^{-k-m}\*\left(\ln(-z)+\psi\left(k+1%
\right)+\psi\left(k+m+1\right)-\psi\left(a+k+m\right)-\psi\left(c-a-k-m\right)%
\right)," display="block"><mrow><mrow><mi href="./15.2#E2" mathvariant="bold">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>+</mo><mi href="./15.1#p1.t1.r4">m</mi></mrow></mrow><mi href="./15.1#p1.t1.r3">c</mi></mfrac><mo>;</mo><mi href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mrow><mfrac><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi></mrow></msup><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>+</mo><mi href="./15.1#p1.t1.r4">m</mi></mrow><mo>)</mo></mrow></mrow></mfrac><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./15.1#p1.t1.r4">k</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi href="./15.1#p1.t1.r4">m</mi><mo>-</mo><mn>1</mn></mrow></munderover><mrow><mfrac><mrow><msub><mrow><mo stretchy="false">(</mo><mi href="./15.1#p1.t1.r3">a</mi><mo stretchy="false">)</mo></mrow><mi href="./15.1#p1.t1.r4">k</mi></msub><mo></mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./15.1#p1.t1.r4">m</mi><mo>-</mo><mi href="./15.1#p1.t1.r4">k</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mrow><mrow><mrow><mi href="./15.1#p1.t1.r4">k</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./15.1#p1.t1.r3">c</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi><mo>-</mo><mi href="./15.1#p1.t1.r4">k</mi></mrow><mo>)</mo></mrow></mrow></mrow></mfrac><mo></mo><msup><mi href="./15.1#p1.t1.r2">z</mi><mrow><mo>-</mo><mi href="./15.1#p1.t1.r4">k</mi></mrow></msup></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>+</mo><mrow><mfrac><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi></mrow></msup><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi href="./15.1#p1.t1.r3">a</mi><mo>)</mo></mrow></mrow></mfrac><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./15.1#p1.t1.r4">k</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mfrac><msub><mrow><mo stretchy="false">(</mo><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>+</mo><mi href="./15.1#p1.t1.r4">m</mi></mrow><mo stretchy="false">)</mo></mrow><mi href="./15.1#p1.t1.r4">k</mi></msub><mrow><mrow><mi href="./15.1#p1.t1.r4">k</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow><mo></mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./15.1#p1.t1.r4">k</mi><mo>+</mo><mi href="./15.1#p1.t1.r4">m</mi></mrow><mo stretchy="false">)</mo></mrow><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./15.1#p1.t1.r3">c</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi><mo>-</mo><mi href="./15.1#p1.t1.r4">k</mi><mo>-</mo><mi href="./15.1#p1.t1.r4">m</mi></mrow><mo>)</mo></mrow></mrow></mrow></mfrac><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./15.1#p1.t1.r4">k</mi></msup><mo></mo><msup><mi href="./15.1#p1.t1.r2">z</mi><mrow><mrow><mo>-</mo><mi href="./15.1#p1.t1.r4">k</mi></mrow><mo>-</mo><mi href="./15.1#p1.t1.r4">m</mi></mrow></msup></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>×</mo><mrow><mo maxsize="1.296em" minsize="1.296em">(</mo><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mrow><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>+</mo><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./15.1#p1.t1.r4">k</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mo>+</mo><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./15.1#p1.t1.r4">k</mi><mo>+</mo><mi href="./15.1#p1.t1.r4">m</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>-</mo><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>+</mo><mi href="./15.1#p1.t1.r4">k</mi><mo>+</mo><mi href="./15.1#p1.t1.r4">m</mi></mrow><mo>)</mo></mrow></mrow><mo>-</mo><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./15.1#p1.t1.r3">c</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi><mo>-</mo><mi href="./15.1#p1.t1.r4">k</mi><mo>-</mo><mi href="./15.1#p1.t1.r4">m</mi></mrow><mo>)</mo></mrow></mrow><mo maxsize="1.296em" minsize="1.296em">)</mo><mo>,</mo></mtd></mtr></mtable></mrow></mrow></mtd></mtr></mtable></mrow></mrow></mrow></mtd></mtr></mtable></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m90.png" altimg-height="23px" altimg-valign="-7px" altimg-width="194px" alttext="|z|>1,|\operatorname{ph}\left(-z\right)|<\pi" display="inline"><mrow><mrow><mrow><mo stretchy="false">|</mo><mi href="./15.1#p1.t1.r2">z</mi><mo stretchy="false">|</mo></mrow><mo>></mo><mn>1</mn></mrow><mo>,</mo><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo>)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow><mo><</mo><mi href="./3.12#E1">π</mi></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E8.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m31.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./5.2#E2" title="(5.2.2) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="\psi\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: psi (or digamma) function</a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m25.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m81.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m41.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a>,
<a href="./1.9#E7" title="(1.9.7) ‣ Modulus and Phase ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="21px" altimg-valign="-6px" altimg-width="26px" alttext="\operatorname{ph}" display="inline"><mi href="./1.9#E7">ph</mi></math>: phase</a>,
<a href="./15.2#E2" title="(15.2.2) ‣ §15.2(i) Gauss Series ‣ §15.2 Definitions and Analytical Properties ‣ Properties ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="23px" altimg-valign="-7px" altimg-width="102px" alttext="\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E2" mathvariant="bold">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m45.png" altimg-height="39px" altimg-valign="-15px" altimg-width="90px" alttext="\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E2" mathvariant="bold">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi></mfrac><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: Olver’s hypergeometric function</a>,
<a href="./15.1#p1.t1.r2" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./15.1#p1.t1.r2">z</mi></math>: complex variable</a>,
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./15.1#p1.t1.r3">a</mi></math>: real or complex parameter</a>,
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m76.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="c" display="inline"><mi href="./15.1#p1.t1.r3">c</mi></math>: real or complex parameter</a>,
<a href="./15.1#p1.t1.r4" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./15.1#p1.t1.r4">k</mi></math>: integer</a> and
<a href="./15.1#p1.t1.r4" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m79.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./15.1#p1.t1.r4">m</mi></math>: integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E9" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">15.8.9</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_Math ltx_eqn_cell ltx_align_center"><math class="ltx_Math" alttext="\mathbf{F}\left({a,a+m\atop c};z\right)=\frac{(1-z)^{-a}}{\Gamma\left(a+m%
\right)\Gamma\left(c-a\right)}\sum_{k=0}^{m-1}\frac{(a)_{k}(c-a-m)_{k}(m-k-1)!%
}{k!}(z-1)^{-k}+\frac{(-1)^{m}(1-z)^{-a-m}}{\Gamma\left(a\right)\Gamma\left(c-%
a-m\right)}\sum_{k=0}^{\infty}\frac{(a+m)_{k}(c-a)_{k}}{k!(k+m)!}(1-z)^{-k}\*(%
\ln\left(1-z\right)+\psi\left(k+1\right)+\psi\left(k+m+1\right)-\psi\left(a+k+%
m\right)-\psi\left(c-a+k\right))," display="block"><mrow><mrow><mi href="./15.2#E2" mathvariant="bold">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>+</mo><mi href="./15.1#p1.t1.r4">m</mi></mrow></mrow><mi href="./15.1#p1.t1.r3">c</mi></mfrac><mo>;</mo><mi href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mrow><mfrac><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi></mrow></msup><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>+</mo><mi href="./15.1#p1.t1.r4">m</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./15.1#p1.t1.r3">c</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>)</mo></mrow></mrow></mrow></mfrac><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./15.1#p1.t1.r4">k</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi href="./15.1#p1.t1.r4">m</mi><mo>-</mo><mn>1</mn></mrow></munderover><mrow><mfrac><mrow><msub><mrow><mo stretchy="false">(</mo><mi href="./15.1#p1.t1.r3">a</mi><mo stretchy="false">)</mo></mrow><mi href="./15.1#p1.t1.r4">k</mi></msub><mo></mo><msub><mrow><mo stretchy="false">(</mo><mrow><mi href="./15.1#p1.t1.r3">c</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi><mo>-</mo><mi href="./15.1#p1.t1.r4">m</mi></mrow><mo stretchy="false">)</mo></mrow><mi href="./15.1#p1.t1.r4">k</mi></msub><mo></mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./15.1#p1.t1.r4">m</mi><mo>-</mo><mi href="./15.1#p1.t1.r4">k</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mrow><mrow><mi href="./15.1#p1.t1.r4">k</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mfrac><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./15.1#p1.t1.r2">z</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mi href="./15.1#p1.t1.r4">k</mi></mrow></msup></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>+</mo><mrow><mfrac><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./15.1#p1.t1.r4">m</mi></msup><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mrow><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>-</mo><mi href="./15.1#p1.t1.r4">m</mi></mrow></msup></mrow><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi href="./15.1#p1.t1.r3">a</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./15.1#p1.t1.r3">c</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi><mo>-</mo><mi href="./15.1#p1.t1.r4">m</mi></mrow><mo>)</mo></mrow></mrow></mrow></mfrac><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./15.1#p1.t1.r4">k</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mfrac><mrow><msub><mrow><mo stretchy="false">(</mo><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>+</mo><mi href="./15.1#p1.t1.r4">m</mi></mrow><mo stretchy="false">)</mo></mrow><mi href="./15.1#p1.t1.r4">k</mi></msub><mo></mo><msub><mrow><mo stretchy="false">(</mo><mrow><mi href="./15.1#p1.t1.r3">c</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo stretchy="false">)</mo></mrow><mi href="./15.1#p1.t1.r4">k</mi></msub></mrow><mrow><mrow><mi href="./15.1#p1.t1.r4">k</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow><mo></mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./15.1#p1.t1.r4">k</mi><mo>+</mo><mi href="./15.1#p1.t1.r4">m</mi></mrow><mo stretchy="false">)</mo></mrow><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mrow></mfrac><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mi href="./15.1#p1.t1.r4">k</mi></mrow></msup></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>×</mo><mrow><mo maxsize="1.296em" minsize="1.296em">(</mo><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mrow><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo>)</mo></mrow></mrow><mo>+</mo><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./15.1#p1.t1.r4">k</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mo>+</mo><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./15.1#p1.t1.r4">k</mi><mo>+</mo><mi href="./15.1#p1.t1.r4">m</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>-</mo><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>+</mo><mi href="./15.1#p1.t1.r4">k</mi><mo>+</mo><mi href="./15.1#p1.t1.r4">m</mi></mrow><mo>)</mo></mrow></mrow><mo>-</mo><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mi href="./15.1#p1.t1.r3">c</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>+</mo><mi href="./15.1#p1.t1.r4">k</mi></mrow><mo>)</mo></mrow></mrow><mo maxsize="1.296em" minsize="1.296em">)</mo><mo>,</mo></mtd></mtr></mtable></mrow></mrow></mtd></mtr></mtable></mrow></mrow></mrow></mtd></mtr></mtable></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m88.png" altimg-height="23px" altimg-valign="-7px" altimg-width="247px" alttext="|z-1|>1,|\operatorname{ph}\left(1-z\right)|<\pi" display="inline"><mrow><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./15.1#p1.t1.r2">z</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">|</mo></mrow><mo>></mo><mn>1</mn></mrow><mo>,</mo><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo>)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow><mo><</mo><mi href="./3.12#E1">π</mi></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E9.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m31.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./5.2#E2" title="(5.2.2) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="\psi\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: psi (or digamma) function</a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m25.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m81.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m41.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a>,
<a href="./1.9#E7" title="(1.9.7) ‣ Modulus and Phase ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="21px" altimg-valign="-6px" altimg-width="26px" alttext="\operatorname{ph}" display="inline"><mi href="./1.9#E7">ph</mi></math>: phase</a>,
<a href="./15.2#E2" title="(15.2.2) ‣ §15.2(i) Gauss Series ‣ §15.2 Definitions and Analytical Properties ‣ Properties ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="23px" altimg-valign="-7px" altimg-width="102px" alttext="\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E2" mathvariant="bold">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m45.png" altimg-height="39px" altimg-valign="-15px" altimg-width="90px" alttext="\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E2" mathvariant="bold">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi></mfrac><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: Olver’s hypergeometric function</a>,
<a href="./15.1#p1.t1.r2" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./15.1#p1.t1.r2">z</mi></math>: complex variable</a>,
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./15.1#p1.t1.r3">a</mi></math>: real or complex parameter</a>,
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m76.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="c" display="inline"><mi href="./15.1#p1.t1.r3">c</mi></math>: real or complex parameter</a>,
<a href="./15.1#p1.t1.r4" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./15.1#p1.t1.r4">k</mi></math>: integer</a> and
<a href="./15.1#p1.t1.r4" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m79.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./15.1#p1.t1.r4">m</mi></math>: integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">In () when <math class="ltx_Math" altimg="m65.png" altimg-height="19px" altimg-valign="-4px" altimg-width="125px" alttext="c-a-k-m" display="inline"><mrow><mi href="./15.1#p1.t1.r3">c</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi><mo>-</mo><mi href="./15.1#p1.t1.r4">k</mi><mo>-</mo><mi href="./15.1#p1.t1.r4">m</mi></mrow></math> is a nonpositive integer
<math class="ltx_Math" altimg="m38.png" altimg-height="23px" altimg-valign="-7px" altimg-width="324px" alttext="\ifrac{\psi\left(c-a-k-m\right)}{\Gamma\left(c-a-k-m\right)}" display="inline"><mrow><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./15.1#p1.t1.r3">c</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi><mo>-</mo><mi href="./15.1#p1.t1.r4">k</mi><mo>-</mo><mi href="./15.1#p1.t1.r4">m</mi></mrow><mo>)</mo></mrow></mrow><mo>/</mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./15.1#p1.t1.r3">c</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi><mo>-</mo><mi href="./15.1#p1.t1.r4">k</mi><mo>-</mo><mi href="./15.1#p1.t1.r4">m</mi></mrow><mo>)</mo></mrow></mrow></mrow></math> is interpreted as
<math class="ltx_Math" altimg="m27.png" altimg-height="25px" altimg-valign="-7px" altimg-width="284px" alttext="(-1)^{m+k+a-c+1}(m+k+a-c)!" display="inline"><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mrow><mrow><mrow><mi href="./15.1#p1.t1.r4">m</mi><mo>+</mo><mi href="./15.1#p1.t1.r4">k</mi><mo>+</mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>-</mo><mi href="./15.1#p1.t1.r3">c</mi></mrow><mo>+</mo><mn>1</mn></mrow></msup><mo></mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mi href="./15.1#p1.t1.r4">m</mi><mo>+</mo><mi href="./15.1#p1.t1.r4">k</mi><mo>+</mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>-</mo><mi href="./15.1#p1.t1.r3">c</mi></mrow><mo stretchy="false">)</mo></mrow><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mrow></math>. Also, if <math class="ltx_Math" altimg="m56.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./15.1#p1.t1.r3">a</mi></math> is a nonpositive integer, then
() applies.</p>
</div>
<div id="SS2.p4" class="ltx_para">
<p class="ltx_p">Alternatively, if <math class="ltx_Math" altimg="m58.png" altimg-height="19px" altimg-valign="-4px" altimg-width="48px" alttext="b-a" display="inline"><mrow><mi href="./15.1#p1.t1.r3">b</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi></mrow></math> is a negative integer, then we interchange <math class="ltx_Math" altimg="m56.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./15.1#p1.t1.r3">a</mi></math> and <math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="b" display="inline"><mi href="./15.1#p1.t1.r3">b</mi></math>
in <math class="ltx_Math" altimg="m44.png" altimg-height="23px" altimg-valign="-7px" altimg-width="102px" alttext="\mathbf{F}\left(a,b;c;z\right)" display="inline"><mrow><mi href="./15.2#E2" mathvariant="bold">F</mi><mo></mo><mrow><mo>(</mo><mi href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>.</p>
</div>
<div id="SS2.p5" class="ltx_para">
<p class="ltx_p">In a similar way, when <math class="ltx_Math" altimg="m64.png" altimg-height="19px" altimg-valign="-4px" altimg-width="81px" alttext="c-a-b" display="inline"><mrow><mi href="./15.1#p1.t1.r3">c</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow></math> is an integer limits are taken in
() as follows.</p>
</div>
<div id="SS2.p6" class="ltx_para">
<p class="ltx_p">If <math class="ltx_Math" altimg="m64.png" altimg-height="19px" altimg-valign="-4px" altimg-width="81px" alttext="c-a-b" display="inline"><mrow><mi href="./15.1#p1.t1.r3">c</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow></math> is a nonnegative integer, then
</p>
<table id="E10" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">15.8.10</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_Math ltx_eqn_cell ltx_align_center"><math class="ltx_Math" alttext="\mathbf{F}\left({a,b\atop a+b+m};z\right)=\frac{1}{\Gamma\left(a+m\right)%
\Gamma\left(b+m\right)}\sum_{k=0}^{m-1}\frac{(a)_{k}(b)_{k}(m-k-1)!}{k!}(z-1)^%
{k}-\frac{(z-1)^{m}}{\Gamma\left(a\right)\Gamma\left(b\right)}\sum_{k=0}^{%
\infty}\frac{(a+m)_{k}(b+m)_{k}}{k!(k+m)!}(1-z)^{k}\*\left(\ln(1-z)-\psi\left(%
k+1\right)-\psi\left(k+m+1\right)+\psi\left(a+k+m\right)+\psi\left(b+k+m\right%
)\right)," display="block"><mrow><mrow><mi href="./15.2#E2" mathvariant="bold">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>+</mo><mi href="./15.1#p1.t1.r3">b</mi><mo>+</mo><mi href="./15.1#p1.t1.r4">m</mi></mrow></mfrac><mo>;</mo><mi href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mrow><mfrac><mn>1</mn><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>+</mo><mi href="./15.1#p1.t1.r4">m</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./15.1#p1.t1.r3">b</mi><mo>+</mo><mi href="./15.1#p1.t1.r4">m</mi></mrow><mo>)</mo></mrow></mrow></mrow></mfrac><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./15.1#p1.t1.r4">k</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi href="./15.1#p1.t1.r4">m</mi><mo>-</mo><mn>1</mn></mrow></munderover><mrow><mfrac><mrow><msub><mrow><mo stretchy="false">(</mo><mi href="./15.1#p1.t1.r3">a</mi><mo stretchy="false">)</mo></mrow><mi href="./15.1#p1.t1.r4">k</mi></msub><mo></mo><msub><mrow><mo stretchy="false">(</mo><mi href="./15.1#p1.t1.r3">b</mi><mo stretchy="false">)</mo></mrow><mi href="./15.1#p1.t1.r4">k</mi></msub><mo></mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./15.1#p1.t1.r4">m</mi><mo>-</mo><mi href="./15.1#p1.t1.r4">k</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mrow><mrow><mi href="./15.1#p1.t1.r4">k</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mfrac><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./15.1#p1.t1.r2">z</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./15.1#p1.t1.r4">k</mi></msup></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>-</mo><mrow><mfrac><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./15.1#p1.t1.r2">z</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./15.1#p1.t1.r4">m</mi></msup><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi href="./15.1#p1.t1.r3">a</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi href="./15.1#p1.t1.r3">b</mi><mo>)</mo></mrow></mrow></mrow></mfrac><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./15.1#p1.t1.r4">k</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mfrac><mrow><msub><mrow><mo stretchy="false">(</mo><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>+</mo><mi href="./15.1#p1.t1.r4">m</mi></mrow><mo stretchy="false">)</mo></mrow><mi href="./15.1#p1.t1.r4">k</mi></msub><mo></mo><msub><mrow><mo stretchy="false">(</mo><mrow><mi href="./15.1#p1.t1.r3">b</mi><mo>+</mo><mi href="./15.1#p1.t1.r4">m</mi></mrow><mo stretchy="false">)</mo></mrow><mi href="./15.1#p1.t1.r4">k</mi></msub></mrow><mrow><mrow><mi href="./15.1#p1.t1.r4">k</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow><mo></mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./15.1#p1.t1.r4">k</mi><mo>+</mo><mi href="./15.1#p1.t1.r4">m</mi></mrow><mo stretchy="false">)</mo></mrow><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mrow></mfrac><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo stretchy="false">)</mo></mrow><mi href="./15.1#p1.t1.r4">k</mi></msup></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>×</mo><mrow><mo maxsize="1.296em" minsize="1.296em">(</mo><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mrow><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>-</mo><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./15.1#p1.t1.r4">k</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mo>-</mo><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./15.1#p1.t1.r4">k</mi><mo>+</mo><mi href="./15.1#p1.t1.r4">m</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>+</mo><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>+</mo><mi href="./15.1#p1.t1.r4">k</mi><mo>+</mo><mi href="./15.1#p1.t1.r4">m</mi></mrow><mo>)</mo></mrow></mrow><mo>+</mo><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./15.1#p1.t1.r3">b</mi><mo>+</mo><mi href="./15.1#p1.t1.r4">k</mi><mo>+</mo><mi href="./15.1#p1.t1.r4">m</mi></mrow><mo>)</mo></mrow></mrow><mo maxsize="1.296em" minsize="1.296em">)</mo><mo>,</mo></mtd></mtr></mtable></mrow></mrow></mtd></mtr></mtable></mrow></mrow></mrow></mtd></mtr></mtable></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m87.png" altimg-height="23px" altimg-valign="-7px" altimg-width="247px" alttext="|z-1|<1,|\operatorname{ph}\left(1-z\right)|<\pi" display="inline"><mrow><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./15.1#p1.t1.r2">z</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">|</mo></mrow><mo><</mo><mn>1</mn></mrow><mo>,</mo><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo>)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow><mo><</mo><mi href="./3.12#E1">π</mi></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E10.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m31.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./5.2#E2" title="(5.2.2) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="\psi\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: psi (or digamma) function</a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m25.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m81.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m41.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a>,
<a href="./1.9#E7" title="(1.9.7) ‣ Modulus and Phase ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="21px" altimg-valign="-6px" altimg-width="26px" alttext="\operatorname{ph}" display="inline"><mi href="./1.9#E7">ph</mi></math>: phase</a>,
<a href="./15.2#E2" title="(15.2.2) ‣ §15.2(i) Gauss Series ‣ §15.2 Definitions and Analytical Properties ‣ Properties ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="23px" altimg-valign="-7px" altimg-width="102px" alttext="\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E2" mathvariant="bold">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m45.png" altimg-height="39px" altimg-valign="-15px" altimg-width="90px" alttext="\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E2" mathvariant="bold">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi></mfrac><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: Olver’s hypergeometric function</a>,
<a href="./15.1#p1.t1.r2" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./15.1#p1.t1.r2">z</mi></math>: complex variable</a>,
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./15.1#p1.t1.r3">a</mi></math>: real or complex parameter</a>,
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="b" display="inline"><mi href="./15.1#p1.t1.r3">b</mi></math>: real or complex parameter</a>,
<a href="./15.1#p1.t1.r4" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./15.1#p1.t1.r4">k</mi></math>: integer</a> and
<a href="./15.1#p1.t1.r4" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m79.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./15.1#p1.t1.r4">m</mi></math>: integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E11" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">15.8.11</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_Math ltx_eqn_cell ltx_align_center"><math class="ltx_Math" alttext="\mathbf{F}\left({a,b\atop a+b+m};z\right)=\frac{z^{-a}}{\Gamma\left(a+m\right)%
}\sum_{k=0}^{m-1}\frac{(a)_{k}(m-k-1)!}{k!\Gamma\left(b+m-k\right)}\left(1-%
\frac{1}{z}\right)^{k}-\frac{z^{-a}}{\Gamma\left(a\right)}\sum_{k=0}^{\infty}%
\frac{(a+m)_{k}}{k!(k+m)!\Gamma\left(b-k\right)}(-1)^{k}\left(1-\frac{1}{z}%
\right)^{k+m}\*\left(\ln\left(\frac{1-z}{z}\right)-\psi\left(k+1\right)-\psi%
\left(k+m+1\right)+\psi\left(a+k+m\right)+\psi\left(b-k\right)\right)," display="block"><mrow><mrow><mi href="./15.2#E2" mathvariant="bold">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>+</mo><mi href="./15.1#p1.t1.r3">b</mi><mo>+</mo><mi href="./15.1#p1.t1.r4">m</mi></mrow></mfrac><mo>;</mo><mi href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mrow><mfrac><msup><mi href="./15.1#p1.t1.r2">z</mi><mrow><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi></mrow></msup><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>+</mo><mi href="./15.1#p1.t1.r4">m</mi></mrow><mo>)</mo></mrow></mrow></mfrac><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./15.1#p1.t1.r4">k</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi href="./15.1#p1.t1.r4">m</mi><mo>-</mo><mn>1</mn></mrow></munderover><mrow><mfrac><mrow><msub><mrow><mo stretchy="false">(</mo><mi href="./15.1#p1.t1.r3">a</mi><mo stretchy="false">)</mo></mrow><mi href="./15.1#p1.t1.r4">k</mi></msub><mo></mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./15.1#p1.t1.r4">m</mi><mo>-</mo><mi href="./15.1#p1.t1.r4">k</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mrow><mrow><mrow><mi href="./15.1#p1.t1.r4">k</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mi href="./15.1#p1.t1.r3">b</mi><mo>+</mo><mi href="./15.1#p1.t1.r4">m</mi></mrow><mo>-</mo><mi href="./15.1#p1.t1.r4">k</mi></mrow><mo>)</mo></mrow></mrow></mrow></mfrac><mo></mo><msup><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mfrac><mn>1</mn><mi href="./15.1#p1.t1.r2">z</mi></mfrac></mrow><mo>)</mo></mrow><mi href="./15.1#p1.t1.r4">k</mi></msup></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>-</mo><mrow><mfrac><msup><mi href="./15.1#p1.t1.r2">z</mi><mrow><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi></mrow></msup><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi href="./15.1#p1.t1.r3">a</mi><mo>)</mo></mrow></mrow></mfrac><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./15.1#p1.t1.r4">k</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mfrac><msub><mrow><mo stretchy="false">(</mo><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>+</mo><mi href="./15.1#p1.t1.r4">m</mi></mrow><mo stretchy="false">)</mo></mrow><mi href="./15.1#p1.t1.r4">k</mi></msub><mrow><mrow><mi href="./15.1#p1.t1.r4">k</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow><mo></mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./15.1#p1.t1.r4">k</mi><mo>+</mo><mi href="./15.1#p1.t1.r4">m</mi></mrow><mo stretchy="false">)</mo></mrow><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./15.1#p1.t1.r3">b</mi><mo>-</mo><mi href="./15.1#p1.t1.r4">k</mi></mrow><mo>)</mo></mrow></mrow></mrow></mfrac><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./15.1#p1.t1.r4">k</mi></msup><mo></mo><msup><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mfrac><mn>1</mn><mi href="./15.1#p1.t1.r2">z</mi></mfrac></mrow><mo>)</mo></mrow><mrow><mi href="./15.1#p1.t1.r4">k</mi><mo>+</mo><mi href="./15.1#p1.t1.r4">m</mi></mrow></msup></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>×</mo><mrow><mo maxsize="2.736em" minsize="2.736em">(</mo><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mrow><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><mo>(</mo><mfrac><mrow><mn>1</mn><mo>-</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mi href="./15.1#p1.t1.r2">z</mi></mfrac><mo>)</mo></mrow></mrow><mo>-</mo><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./15.1#p1.t1.r4">k</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mo>-</mo><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./15.1#p1.t1.r4">k</mi><mo>+</mo><mi href="./15.1#p1.t1.r4">m</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>+</mo><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>+</mo><mi href="./15.1#p1.t1.r4">k</mi><mo>+</mo><mi href="./15.1#p1.t1.r4">m</mi></mrow><mo>)</mo></mrow></mrow><mo>+</mo><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./15.1#p1.t1.r3">b</mi><mo>-</mo><mi href="./15.1#p1.t1.r4">k</mi></mrow><mo>)</mo></mrow></mrow><mo maxsize="2.736em" minsize="2.736em">)</mo><mo>,</mo></mtd></mtr></mtable></mrow></mrow></mtd></mtr></mtable></mrow></mrow></mrow></mtd></mtr></mtable></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m34.png" altimg-height="27px" altimg-valign="-9px" altimg-width="316px" alttext="\Re z>\tfrac{1}{2},|\operatorname{ph}z|<\pi,|\operatorname{ph}\left(1-z\right)%
|<\pi" display="inline"><mrow><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo>></mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>,</mo><mrow><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo stretchy="false">|</mo></mrow><mo><</mo><mi href="./3.12#E1">π</mi></mrow><mo>,</mo><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo>)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow><mo><</mo><mi href="./3.12#E1">π</mi></mrow></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E11.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m31.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./5.2#E2" title="(5.2.2) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="\psi\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: psi (or digamma) function</a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m25.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m81.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m41.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a>,
<a href="./1.9#E7" title="(1.9.7) ‣ Modulus and Phase ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="21px" altimg-valign="-6px" altimg-width="26px" alttext="\operatorname{ph}" display="inline"><mi href="./1.9#E7">ph</mi></math>: phase</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m35.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./15.2#E2" title="(15.2.2) ‣ §15.2(i) Gauss Series ‣ §15.2 Definitions and Analytical Properties ‣ Properties ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="23px" altimg-valign="-7px" altimg-width="102px" alttext="\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E2" mathvariant="bold">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m45.png" altimg-height="39px" altimg-valign="-15px" altimg-width="90px" alttext="\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E2" mathvariant="bold">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi></mfrac><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: Olver’s hypergeometric function</a>,
<a href="./15.1#p1.t1.r2" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./15.1#p1.t1.r2">z</mi></math>: complex variable</a>,
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./15.1#p1.t1.r3">a</mi></math>: real or complex parameter</a>,
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="b" display="inline"><mi href="./15.1#p1.t1.r3">b</mi></math>: real or complex parameter</a>,
<a href="./15.1#p1.t1.r4" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./15.1#p1.t1.r4">k</mi></math>: integer</a> and
<a href="./15.1#p1.t1.r4" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m79.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./15.1#p1.t1.r4">m</mi></math>: integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS2.p7" class="ltx_para">
<p class="ltx_p">In () when <math class="ltx_Math" altimg="m59.png" altimg-height="19px" altimg-valign="-4px" altimg-width="48px" alttext="b-k" display="inline"><mrow><mi href="./15.1#p1.t1.r3">b</mi><mo>-</mo><mi href="./15.1#p1.t1.r4">k</mi></mrow></math> is a nonpositive integer,
<math class="ltx_Math" altimg="m37.png" altimg-height="23px" altimg-valign="-7px" altimg-width="171px" alttext="\ifrac{\psi\left(b-k\right)}{\Gamma\left(b-k\right)}" display="inline"><mrow><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./15.1#p1.t1.r3">b</mi><mo>-</mo><mi href="./15.1#p1.t1.r4">k</mi></mrow><mo>)</mo></mrow></mrow><mo>/</mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./15.1#p1.t1.r3">b</mi><mo>-</mo><mi href="./15.1#p1.t1.r4">k</mi></mrow><mo>)</mo></mrow></mrow></mrow></math> is interpreted as
<math class="ltx_Math" altimg="m26.png" altimg-height="25px" altimg-valign="-7px" altimg-width="160px" alttext="(-1)^{k-b+1}(k-b)!" display="inline"><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mrow><mrow><mi href="./15.1#p1.t1.r4">k</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mo>+</mo><mn>1</mn></mrow></msup><mo></mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./15.1#p1.t1.r4">k</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mo stretchy="false">)</mo></mrow><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mrow></math>. Also, if <math class="ltx_Math" altimg="m56.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./15.1#p1.t1.r3">a</mi></math> or <math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="b" display="inline"><mi href="./15.1#p1.t1.r3">b</mi></math> or both are nonpositive
integers, then () applies.</p>
</div>
<div id="SS2.p8" class="ltx_para">
<p class="ltx_p">Lastly, if <math class="ltx_Math" altimg="m64.png" altimg-height="19px" altimg-valign="-4px" altimg-width="81px" alttext="c-a-b" display="inline"><mrow><mi href="./15.1#p1.t1.r3">c</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow></math> is a negative integer, then we first apply the
transformation</p>
<table id="E12" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">15.8.12</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E12.png" altimg-height="41px" altimg-valign="-15px" altimg-width="486px" alttext="\mathbf{F}\left(a,b;a+b-m;z\right)=(1-z)^{-m}\mathbf{F}\left(\tilde{a},\tilde{%
b};\tilde{a}+\tilde{b}+m;z\right)," display="block"><mrow><mrow><mrow><mi href="./15.2#E2" mathvariant="bold">F</mi><mo></mo><mrow><mo>(</mo><mi href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mrow><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>+</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mo>-</mo><mi href="./15.1#p1.t1.r4">m</mi></mrow><mo>;</mo><mi href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mi href="./15.1#p1.t1.r4">m</mi></mrow></msup><mo></mo><mrow><mi href="./15.2#E2" mathvariant="bold">F</mi><mo></mo><mrow><mo>(</mo><mover accent="true"><mi href="./15.1#p1.t1.r3">a</mi><mo stretchy="false">~</mo></mover><mo>,</mo><mover accent="true"><mi href="./15.1#p1.t1.r3">b</mi><mo stretchy="false">~</mo></mover><mo>;</mo><mrow><mover accent="true"><mi href="./15.1#p1.t1.r3">a</mi><mo stretchy="false">~</mo></mover><mo>+</mo><mover accent="true"><mi href="./15.1#p1.t1.r3">b</mi><mo stretchy="false">~</mo></mover><mo>+</mo><mi href="./15.1#p1.t1.r4">m</mi></mrow><mo>;</mo><mi href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m51.png" altimg-height="26px" altimg-valign="-6px" altimg-width="188px" alttext="\tilde{a}=a-m,\tilde{b}=b-m" display="inline"><mrow><mrow><mover accent="true"><mi href="./15.1#p1.t1.r3">a</mi><mo stretchy="false">~</mo></mover><mo>=</mo><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>-</mo><mi href="./15.1#p1.t1.r4">m</mi></mrow></mrow><mo>,</mo><mrow><mover accent="true"><mi href="./15.1#p1.t1.r3">b</mi><mo stretchy="false">~</mo></mover><mo>=</mo><mrow><mi href="./15.1#p1.t1.r3">b</mi><mo>-</mo><mi href="./15.1#p1.t1.r4">m</mi></mrow></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E12.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./15.2#E2" title="(15.2.2) ‣ §15.2(i) Gauss Series ‣ §15.2 Definitions and Analytical Properties ‣ Properties ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="23px" altimg-valign="-7px" altimg-width="102px" alttext="\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E2" mathvariant="bold">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m45.png" altimg-height="39px" altimg-valign="-15px" altimg-width="90px" alttext="\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E2" mathvariant="bold">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi></mfrac><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: Olver’s hypergeometric function</a>,
<a href="./15.1#p1.t1.r2" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./15.1#p1.t1.r2">z</mi></math>: complex variable</a>,
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./15.1#p1.t1.r3">a</mi></math>: real or complex parameter</a>,
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="b" display="inline"><mi href="./15.1#p1.t1.r3">b</mi></math>: real or complex parameter</a> and
<a href="./15.1#p1.t1.r4" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m79.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./15.1#p1.t1.r4">m</mi></math>: integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="iii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§15.8(iii) </span>Quadratic Transformations</h2>
<div id="SS3.info" class="ltx_metadata ltx_info">
is satisfied.</p>
</div>
<figure id="T1" class="ltx_table">
<figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_table">Table 15.8.1: </span>Quadratic transformations of the hypergeometric function.
</figcaption>
<table id="T1.t1" class="ltx_tabular ltx_centering ltx_guessed_headers ltx_align_middle">
<thead class="ltx_thead">
<tr id="T1.t1.r1" class="ltx_tr">
<th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" style="padding:2.083333333333333px 3.0pt;">Group 1</th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" style="padding:2.083333333333333px 3.0pt;">Group 2</th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" style="padding:2.083333333333333px 3.0pt;">Group 3</th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" style="padding:2.083333333333333px 3.0pt;">Group 4</th>
</tr>
</thead>
<tbody class="ltx_tbody">
<tr id="T1.t1.r2" class="ltx_tr">
<td class="ltx_td ltx_border_t" style="padding:2.083333333333333px 3.0pt;"></td>
<td class="ltx_td ltx_align_center ltx_border_t" style="padding:2.083333333333333px 3.0pt;"><math class="ltx_Math" altimg="m74.png" altimg-height="19px" altimg-valign="-4px" altimg-width="117px" alttext="c=a-b+1" display="inline"><mrow><mi href="./15.1#p1.t1.r3">c</mi><mo>=</mo><mrow><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mo>+</mo><mn>1</mn></mrow></mrow></math></td>
<td class="ltx_td ltx_align_center ltx_border_t" style="padding:2.083333333333333px 3.0pt;"><math class="ltx_Math" altimg="m55.png" altimg-height="27px" altimg-valign="-9px" altimg-width="87px" alttext="a=b+\frac{1}{2}" display="inline"><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>=</mo><mrow><mi href="./15.1#p1.t1.r3">b</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow></math></td>
<td class="ltx_td ltx_border_t" style="padding:2.083333333333333px 3.0pt;"></td>
</tr>
<tr id="T1.t1.r3" class="ltx_tr">
<td class="ltx_td ltx_align_center" style="padding:2.083333333333333px 3.0pt;"><math class="ltx_Math" altimg="m66.png" altimg-height="17px" altimg-valign="-2px" altimg-width="60px" alttext="c=2a" display="inline"><mrow><mi href="./15.1#p1.t1.r3">c</mi><mo>=</mo><mrow><mn>2</mn><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow></mrow></math></td>
<td class="ltx_td ltx_align_center" style="padding:2.083333333333333px 3.0pt;"><math class="ltx_Math" altimg="m75.png" altimg-height="19px" altimg-valign="-4px" altimg-width="117px" alttext="c=b-a+1" display="inline"><mrow><mi href="./15.1#p1.t1.r3">c</mi><mo>=</mo><mrow><mrow><mi href="./15.1#p1.t1.r3">b</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>+</mo><mn>1</mn></mrow></mrow></math></td>
<td class="ltx_td ltx_align_center" style="padding:2.083333333333333px 3.0pt;"><math class="ltx_Math" altimg="m62.png" altimg-height="27px" altimg-valign="-9px" altimg-width="87px" alttext="b=a+\frac{1}{2}" display="inline"><mrow><mi href="./15.1#p1.t1.r3">b</mi><mo>=</mo><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow></math></td>
<td class="ltx_td ltx_align_center" style="padding:2.083333333333333px 3.0pt;"><math class="ltx_Math" altimg="m70.png" altimg-height="27px" altimg-valign="-9px" altimg-width="52px" alttext="c=\frac{1}{2}" display="inline"><mrow><mi href="./15.1#p1.t1.r3">c</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math></td>
</tr>
<tr id="T1.t1.r4" class="ltx_tr">
<td class="ltx_td ltx_align_center" style="padding:2.083333333333333px 3.0pt;"><math class="ltx_Math" altimg="m68.png" altimg-height="18px" altimg-valign="-2px" altimg-width="58px" alttext="c=2b" display="inline"><mrow><mi href="./15.1#p1.t1.r3">c</mi><mo>=</mo><mrow><mn>2</mn><mo></mo><mi href="./15.1#p1.t1.r3">b</mi></mrow></mrow></math></td>
<td class="ltx_td ltx_align_center" style="padding:2.083333333333333px 3.0pt;"><math class="ltx_Math" altimg="m69.png" altimg-height="27px" altimg-valign="-9px" altimg-width="145px" alttext="c=\frac{1}{2}(a+b+1)" display="inline"><mrow><mi href="./15.1#p1.t1.r3">c</mi><mo>=</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>+</mo><mi href="./15.1#p1.t1.r3">b</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_center" style="padding:2.083333333333333px 3.0pt;"><math class="ltx_Math" altimg="m72.png" altimg-height="27px" altimg-valign="-9px" altimg-width="120px" alttext="c=a+b+\frac{1}{2}" display="inline"><mrow><mi href="./15.1#p1.t1.r3">c</mi><mo>=</mo><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>+</mo><mi href="./15.1#p1.t1.r3">b</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow></math></td>
<td class="ltx_td ltx_align_center" style="padding:2.083333333333333px 3.0pt;"><math class="ltx_Math" altimg="m71.png" altimg-height="27px" altimg-valign="-9px" altimg-width="52px" alttext="c=\frac{3}{2}" display="inline"><mrow><mi href="./15.1#p1.t1.r3">c</mi><mo>=</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow></math></td>
</tr>
<tr id="T1.t1.r5" class="ltx_tr">
<td class="ltx_td ltx_border_b" style="padding:2.083333333333333px 3.0pt;"></td>
<td class="ltx_td ltx_align_center ltx_border_b" style="padding:2.083333333333333px 3.0pt;"><math class="ltx_Math" altimg="m54.png" altimg-height="19px" altimg-valign="-4px" altimg-width="84px" alttext="a+b=1" display="inline"><mrow><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>+</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mo>=</mo><mn>1</mn></mrow></math></td>
<td class="ltx_td ltx_align_center ltx_border_b" style="padding:2.083333333333333px 3.0pt;"><math class="ltx_Math" altimg="m73.png" altimg-height="27px" altimg-valign="-9px" altimg-width="120px" alttext="c=a+b-\frac{1}{2}" display="inline"><mrow><mi href="./15.1#p1.t1.r3">c</mi><mo>=</mo><mrow><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>+</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow></math></td>
<td class="ltx_td ltx_border_b" style="padding:2.083333333333333px 3.0pt;"></td>
</tr>
</tbody>
</table>
<div id="T1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./15.1#p1.t1.r3">a</mi></math>: real or complex parameter</a>,
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="b" display="inline"><mi href="./15.1#p1.t1.r3">b</mi></math>: real or complex parameter</a> and
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m76.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="c" display="inline"><mi href="./15.1#p1.t1.r3">c</mi></math>: real or complex parameter</a>
</dd>
</dl>
</div>
</div>
</figure>
<div id="SS3.p3" class="ltx_para">
<p class="ltx_p">The hypergeometric functions that correspond to Groups 1 and 2 have <math class="ltx_Math" altimg="m82.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./15.1#p1.t1.r2">z</mi></math> as
variable. The hypergeometric functions that correspond to Groups 3 and 4 have
a nonlinear function of <math class="ltx_Math" altimg="m82.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./15.1#p1.t1.r2">z</mi></math> as variable. The transformation formulas between
two hypergeometric functions in Group 2, or two hypergeometric functions in
Group 3, are the linear transformations ().</p>
</div>
<div id="SS3.p4" class="ltx_para">
<p class="ltx_p">In the equations that follow in this subsection all functions take their
principal values.</p>
</div>
<section id="Px1" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Group 1 <math class="ltx_Math" altimg="m42.png" altimg-height="12px" altimg-valign="-2px" altimg-width="36px" alttext="\longrightarrow" display="inline"><mo>⟶</mo></math> Group 3</h3>
<div id="Px1.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px1.p1" class="ltx_para">
<table id="EGx3" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E13">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">15.8.13</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m19.png" altimg-height="53px" altimg-valign="-21px" altimg-width="106px" alttext="\displaystyle F\left({a,b\atop 2b};z\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac linethickness="0pt"><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mrow><mn>2</mn><mo></mo><mi href="./15.1#p1.t1.r3">b</mi></mrow></mfrac></mstyle><mo>;</mo><mi href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m14.png" altimg-height="65px" altimg-valign="-27px" altimg-width="372px" alttext="\displaystyle=\left(1-\tfrac{1}{2}z\right)^{-a}F\left({\tfrac{1}{2}a,\tfrac{1}%
{2}a+\tfrac{1}{2}\atop b+\tfrac{1}{2}};\left(\frac{z}{2-z}\right)^{2}\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msup><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./15.1#p1.t1.r2">z</mi></mrow></mrow><mo>)</mo></mrow><mrow><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi></mrow></msup><mo></mo><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac linethickness="0pt"><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>,</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow><mrow><mi href="./15.1#p1.t1.r3">b</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mfrac></mstyle><mo>;</mo><msup><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac><mi href="./15.1#p1.t1.r2">z</mi><mrow><mn>2</mn><mo>-</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow></mfrac></mstyle><mo>)</mo></mrow><mn>2</mn></msup><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m85.png" altimg-height="23px" altimg-valign="-7px" altimg-width="146px" alttext="|\operatorname{ph}\left(1-z\right)|<\pi" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo>)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow><mo><</mo><mi href="./3.12#E1">π</mi></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E13.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./15.2#E1" title="(15.2.1) ‣ §15.2(i) Gauss Series ‣ §15.2 Definitions and Analytical Properties ‣ Properties ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="23px" altimg-valign="-7px" altimg-width="103px" alttext="F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m30.png" altimg-height="39px" altimg-valign="-15px" altimg-width="91px" alttext="F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi></mfrac><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: <math class="ltx_Math" altimg="m28.png" altimg-height="23px" altimg-valign="-7px" altimg-width="139px" alttext="={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow></math>
Gauss’ hypergeometric function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.9#E7" title="(1.9.7) ‣ Modulus and Phase ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="21px" altimg-valign="-6px" altimg-width="26px" alttext="\operatorname{ph}" display="inline"><mi href="./1.9#E7">ph</mi></math>: phase</a>,
<a href="./15.1#p1.t1.r2" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./15.1#p1.t1.r2">z</mi></math>: complex variable</a>,
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./15.1#p1.t1.r3">a</mi></math>: real or complex parameter</a> and
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="b" display="inline"><mi href="./15.1#p1.t1.r3">b</mi></math>: real or complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E14">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">15.8.14</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m19.png" altimg-height="53px" altimg-valign="-21px" altimg-width="106px" alttext="\displaystyle F\left({a,b\atop 2b};z\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac linethickness="0pt"><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mrow><mn>2</mn><mo></mo><mi href="./15.1#p1.t1.r3">b</mi></mrow></mfrac></mstyle><mo>;</mo><mi href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m15.png" altimg-height="65px" altimg-valign="-27px" altimg-width="352px" alttext="\displaystyle=\left(1-z\right)^{-\ifrac{a}{2}}F\left({\tfrac{1}{2}a,b-\tfrac{1%
}{2}a\atop b+\tfrac{1}{2}};\frac{z^{2}}{4z-4}\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msup><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo>)</mo></mrow><mrow><mo>-</mo><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>/</mo><mn>2</mn></mrow></mrow></msup><mo></mo><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac linethickness="0pt"><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>,</mo><mrow><mi href="./15.1#p1.t1.r3">b</mi><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow></mrow></mrow><mrow><mi href="./15.1#p1.t1.r3">b</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mfrac></mstyle><mo>;</mo><mstyle displaystyle="true"><mfrac><msup><mi href="./15.1#p1.t1.r2">z</mi><mn>2</mn></msup><mrow><mrow><mn>4</mn><mo></mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo>-</mo><mn>4</mn></mrow></mfrac></mstyle><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m85.png" altimg-height="23px" altimg-valign="-7px" altimg-width="146px" alttext="|\operatorname{ph}\left(1-z\right)|<\pi" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo>)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow><mo><</mo><mi href="./3.12#E1">π</mi></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E14.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./15.2#E1" title="(15.2.1) ‣ §15.2(i) Gauss Series ‣ §15.2 Definitions and Analytical Properties ‣ Properties ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="23px" altimg-valign="-7px" altimg-width="103px" alttext="F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m30.png" altimg-height="39px" altimg-valign="-15px" altimg-width="91px" alttext="F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi></mfrac><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: <math class="ltx_Math" altimg="m28.png" altimg-height="23px" altimg-valign="-7px" altimg-width="139px" alttext="={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow></math>
Gauss’ hypergeometric function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.9#E7" title="(1.9.7) ‣ Modulus and Phase ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="21px" altimg-valign="-6px" altimg-width="26px" alttext="\operatorname{ph}" display="inline"><mi href="./1.9#E7">ph</mi></math>: phase</a>,
<a href="./15.1#p1.t1.r2" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./15.1#p1.t1.r2">z</mi></math>: complex variable</a>,
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./15.1#p1.t1.r3">a</mi></math>: real or complex parameter</a> and
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="b" display="inline"><mi href="./15.1#p1.t1.r3">b</mi></math>: real or complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
</div>
</section>
<section id="Px2" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Group 2 <math class="ltx_Math" altimg="m42.png" altimg-height="12px" altimg-valign="-2px" altimg-width="36px" alttext="\longrightarrow" display="inline"><mo>⟶</mo></math> Group 3</h3>
<div id="Px2.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px2.p1" class="ltx_para">
<table id="EGx4" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E15">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">15.8.15</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m20.png" altimg-height="53px" altimg-valign="-21px" altimg-width="156px" alttext="\displaystyle F\left({a,b\atop a-b+1};z\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac linethickness="0pt"><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mrow><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mo>+</mo><mn>1</mn></mrow></mfrac></mstyle><mo>;</mo><mi href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m1.png" altimg-height="55px" altimg-valign="-21px" altimg-width="346px" alttext="\displaystyle=(1+z)^{-a}F\left({\frac{1}{2}a,\frac{1}{2}a+\frac{1}{2}\atop a-b%
+1};\frac{4z}{(1+z)^{2}}\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>+</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi></mrow></msup><mo></mo><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac linethickness="0pt"><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>,</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow><mrow><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mo>+</mo><mn>1</mn></mrow></mfrac></mstyle><mo>;</mo><mstyle displaystyle="true"><mfrac><mrow><mn>4</mn><mo></mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>+</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mfrac></mstyle><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m89.png" altimg-height="23px" altimg-valign="-7px" altimg-width="62px" alttext="|z|<1" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mi href="./15.1#p1.t1.r2">z</mi><mo stretchy="false">|</mo></mrow><mo><</mo><mn>1</mn></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E15.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./15.2#E1" title="(15.2.1) ‣ §15.2(i) Gauss Series ‣ §15.2 Definitions and Analytical Properties ‣ Properties ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="23px" altimg-valign="-7px" altimg-width="103px" alttext="F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m30.png" altimg-height="39px" altimg-valign="-15px" altimg-width="91px" alttext="F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi></mfrac><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: <math class="ltx_Math" altimg="m28.png" altimg-height="23px" altimg-valign="-7px" altimg-width="139px" alttext="={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow></math>
Gauss’ hypergeometric function</a>,
<a href="./15.1#p1.t1.r2" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./15.1#p1.t1.r2">z</mi></math>: complex variable</a>,
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./15.1#p1.t1.r3">a</mi></math>: real or complex parameter</a> and
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="b" display="inline"><mi href="./15.1#p1.t1.r3">b</mi></math>: real or complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E16">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">15.8.16</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m20.png" altimg-height="53px" altimg-valign="-21px" altimg-width="156px" alttext="\displaystyle F\left({a,b\atop a-b+1};z\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac linethickness="0pt"><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mrow><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mo>+</mo><mn>1</mn></mrow></mfrac></mstyle><mo>;</mo><mi href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m4.png" altimg-height="55px" altimg-valign="-21px" altimg-width="379px" alttext="\displaystyle=(1-z)^{-a}F\left({\frac{1}{2}a,\frac{1}{2}a-b+\frac{1}{2}\atop a%
-b+1};\frac{-4z}{(1-z)^{2}}\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi></mrow></msup><mo></mo><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac linethickness="0pt"><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>,</mo><mrow><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>-</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow><mrow><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mo>+</mo><mn>1</mn></mrow></mfrac></mstyle><mo>;</mo><mstyle displaystyle="true"><mfrac><mrow><mo>-</mo><mrow><mn>4</mn><mo></mo><mi href="./15.1#p1.t1.r2">z</mi></mrow></mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mfrac></mstyle><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m89.png" altimg-height="23px" altimg-valign="-7px" altimg-width="62px" alttext="|z|<1" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mi href="./15.1#p1.t1.r2">z</mi><mo stretchy="false">|</mo></mrow><mo><</mo><mn>1</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E16.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./15.2#E1" title="(15.2.1) ‣ §15.2(i) Gauss Series ‣ §15.2 Definitions and Analytical Properties ‣ Properties ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="23px" altimg-valign="-7px" altimg-width="103px" alttext="F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m30.png" altimg-height="39px" altimg-valign="-15px" altimg-width="91px" alttext="F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi></mfrac><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: <math class="ltx_Math" altimg="m28.png" altimg-height="23px" altimg-valign="-7px" altimg-width="139px" alttext="={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow></math>
Gauss’ hypergeometric function</a>,
<a href="./15.1#p1.t1.r2" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./15.1#p1.t1.r2">z</mi></math>: complex variable</a>,
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./15.1#p1.t1.r3">a</mi></math>: real or complex parameter</a> and
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="b" display="inline"><mi href="./15.1#p1.t1.r3">b</mi></math>: real or complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
<table id="EGx5" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E17">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">15.8.17</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m21.png" altimg-height="55px" altimg-valign="-23px" altimg-width="184px" alttext="\displaystyle F\left({a,b\atop\frac{1}{2}(a+b+1)};z\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac linethickness="0pt"><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>+</mo><mi href="./15.1#p1.t1.r3">b</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow></mfrac></mstyle><mo>;</mo><mi href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m2.png" altimg-height="65px" altimg-valign="-27px" altimg-width="383px" alttext="\displaystyle=(1-2z)^{-a}F\left({\frac{1}{2}a,\frac{1}{2}a+\frac{1}{2}\atop%
\frac{1}{2}(a+b+1)};\frac{4z(z-1)}{(1-2z)^{2}}\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><mrow><mn>2</mn><mo></mo><mi href="./15.1#p1.t1.r2">z</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi></mrow></msup><mo></mo><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac linethickness="0pt"><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>,</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>+</mo><mi href="./15.1#p1.t1.r3">b</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow></mfrac></mstyle><mo>;</mo><mstyle displaystyle="true"><mfrac><mrow><mn>4</mn><mo></mo><mi href="./15.1#p1.t1.r2">z</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./15.1#p1.t1.r2">z</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><mrow><mn>2</mn><mo></mo><mi href="./15.1#p1.t1.r2">z</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mfrac></mstyle><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m32.png" altimg-height="27px" altimg-valign="-9px" altimg-width="68px" alttext="\Re z<\frac{1}{2}" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo><</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E17.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./15.2#E1" title="(15.2.1) ‣ §15.2(i) Gauss Series ‣ §15.2 Definitions and Analytical Properties ‣ Properties ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="23px" altimg-valign="-7px" altimg-width="103px" alttext="F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m30.png" altimg-height="39px" altimg-valign="-15px" altimg-width="91px" alttext="F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi></mfrac><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: <math class="ltx_Math" altimg="m28.png" altimg-height="23px" altimg-valign="-7px" altimg-width="139px" alttext="={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow></math>
Gauss’ hypergeometric function</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m35.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./15.1#p1.t1.r2" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./15.1#p1.t1.r2">z</mi></math>: complex variable</a>,
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./15.1#p1.t1.r3">a</mi></math>: real or complex parameter</a> and
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="b" display="inline"><mi href="./15.1#p1.t1.r3">b</mi></math>: real or complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E18">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">15.8.18</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m21.png" altimg-height="55px" altimg-valign="-23px" altimg-width="184px" alttext="\displaystyle F\left({a,b\atop\frac{1}{2}(a+b+1)};z\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac linethickness="0pt"><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>+</mo><mi href="./15.1#p1.t1.r3">b</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow></mfrac></mstyle><mo>;</mo><mi href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m6.png" altimg-height="65px" altimg-valign="-27px" altimg-width="286px" alttext="\displaystyle=F\left({\frac{1}{2}a,\frac{1}{2}b\atop\frac{1}{2}(a+b+1)};4z(1-z%
)\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac linethickness="0pt"><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>,</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./15.1#p1.t1.r3">b</mi></mrow></mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>+</mo><mi href="./15.1#p1.t1.r3">b</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow></mfrac></mstyle><mo>;</mo><mrow><mn>4</mn><mo></mo><mi href="./15.1#p1.t1.r2">z</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m32.png" altimg-height="27px" altimg-valign="-9px" altimg-width="68px" alttext="\Re z<\frac{1}{2}" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo><</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E18.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./15.2#E1" title="(15.2.1) ‣ §15.2(i) Gauss Series ‣ §15.2 Definitions and Analytical Properties ‣ Properties ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="23px" altimg-valign="-7px" altimg-width="103px" alttext="F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m30.png" altimg-height="39px" altimg-valign="-15px" altimg-width="91px" alttext="F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi></mfrac><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: <math class="ltx_Math" altimg="m28.png" altimg-height="23px" altimg-valign="-7px" altimg-width="139px" alttext="={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow></math>
Gauss’ hypergeometric function</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m35.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./15.1#p1.t1.r2" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./15.1#p1.t1.r2">z</mi></math>: complex variable</a>,
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./15.1#p1.t1.r3">a</mi></math>: real or complex parameter</a> and
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="b" display="inline"><mi href="./15.1#p1.t1.r3">b</mi></math>: real or complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
<table id="EGx6" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E19">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">15.8.19</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m18.png" altimg-height="53px" altimg-valign="-21px" altimg-width="142px" alttext="\displaystyle F\left({a,1-a\atop c};z\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac linethickness="0pt"><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mrow><mn>1</mn><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi></mrow></mrow><mi href="./15.1#p1.t1.r3">c</mi></mfrac></mstyle><mo>;</mo><mi href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m3.png" altimg-height="55px" altimg-valign="-21px" altimg-width="577px" alttext="\displaystyle=(1-2z)^{1-a-c}(1-z)^{c-1}F\left({\frac{1}{2}(a+c),\frac{1}{2}(a+%
c-1)\atop c};\frac{4z(z-1)}{(1-2z)^{2}}\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><mrow><mn>2</mn><mo></mo><mi href="./15.1#p1.t1.r2">z</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">c</mi></mrow></msup><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mi href="./15.1#p1.t1.r3">c</mi><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac linethickness="0pt"><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>+</mo><mi href="./15.1#p1.t1.r3">c</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>+</mo><mi href="./15.1#p1.t1.r3">c</mi></mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mi href="./15.1#p1.t1.r3">c</mi></mfrac></mstyle><mo>;</mo><mstyle displaystyle="true"><mfrac><mrow><mn>4</mn><mo></mo><mi href="./15.1#p1.t1.r2">z</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./15.1#p1.t1.r2">z</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><mrow><mn>2</mn><mo></mo><mi href="./15.1#p1.t1.r2">z</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mfrac></mstyle><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m32.png" altimg-height="27px" altimg-valign="-9px" altimg-width="68px" alttext="\Re z<\frac{1}{2}" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo><</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E19.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./15.2#E1" title="(15.2.1) ‣ §15.2(i) Gauss Series ‣ §15.2 Definitions and Analytical Properties ‣ Properties ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="23px" altimg-valign="-7px" altimg-width="103px" alttext="F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m30.png" altimg-height="39px" altimg-valign="-15px" altimg-width="91px" alttext="F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi></mfrac><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: <math class="ltx_Math" altimg="m28.png" altimg-height="23px" altimg-valign="-7px" altimg-width="139px" alttext="={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow></math>
Gauss’ hypergeometric function</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m35.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./15.1#p1.t1.r2" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./15.1#p1.t1.r2">z</mi></math>: complex variable</a>,
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./15.1#p1.t1.r3">a</mi></math>: real or complex parameter</a> and
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m76.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="c" display="inline"><mi href="./15.1#p1.t1.r3">c</mi></math>: real or complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E20">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">15.8.20</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m18.png" altimg-height="53px" altimg-valign="-21px" altimg-width="142px" alttext="\displaystyle F\left({a,1-a\atop c};z\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac linethickness="0pt"><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mrow><mn>1</mn><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi></mrow></mrow><mi href="./15.1#p1.t1.r3">c</mi></mfrac></mstyle><mo>;</mo><mi href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m5.png" altimg-height="55px" altimg-valign="-21px" altimg-width="453px" alttext="\displaystyle=(1-z)^{c-1}F\left({\frac{1}{2}(c-a),\frac{1}{2}(a+c-1)\atop c};4%
z(1-z)\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mi href="./15.1#p1.t1.r3">c</mi><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac linethickness="0pt"><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./15.1#p1.t1.r3">c</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>+</mo><mi href="./15.1#p1.t1.r3">c</mi></mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mi href="./15.1#p1.t1.r3">c</mi></mfrac></mstyle><mo>;</mo><mrow><mn>4</mn><mo></mo><mi href="./15.1#p1.t1.r2">z</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m32.png" altimg-height="27px" altimg-valign="-9px" altimg-width="68px" alttext="\Re z<\frac{1}{2}" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo><</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E20.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./15.2#E1" title="(15.2.1) ‣ §15.2(i) Gauss Series ‣ §15.2 Definitions and Analytical Properties ‣ Properties ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="23px" altimg-valign="-7px" altimg-width="103px" alttext="F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m30.png" altimg-height="39px" altimg-valign="-15px" altimg-width="91px" alttext="F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi></mfrac><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: <math class="ltx_Math" altimg="m28.png" altimg-height="23px" altimg-valign="-7px" altimg-width="139px" alttext="={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow></math>
Gauss’ hypergeometric function</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m35.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./15.1#p1.t1.r2" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./15.1#p1.t1.r2">z</mi></math>: complex variable</a>,
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./15.1#p1.t1.r3">a</mi></math>: real or complex parameter</a> and
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m76.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="c" display="inline"><mi href="./15.1#p1.t1.r3">c</mi></math>: real or complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
</div>
</section>
<section id="Px3" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Group 2 <math class="ltx_Math" altimg="m42.png" altimg-height="12px" altimg-valign="-2px" altimg-width="36px" alttext="\longrightarrow" display="inline"><mo>⟶</mo></math> Group 1</h3>
<div id="Px3.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px3.p1" class="ltx_para">
<table id="EGx7" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E21">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">15.8.21</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m20.png" altimg-height="53px" altimg-valign="-21px" altimg-width="156px" alttext="\displaystyle F\left({a,b\atop a-b+1};z\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac linethickness="0pt"><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mrow><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mo>+</mo><mn>1</mn></mrow></mfrac></mstyle><mo>;</mo><mi href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m13.png" altimg-height="55px" altimg-valign="-21px" altimg-width="398px" alttext="\displaystyle=\left(1+\sqrt{z}\right)^{-2a}F\left({a,a-b+\tfrac{1}{2}\atop 2a-%
2b+1};\frac{4\sqrt{z}}{(1+\sqrt{z})^{2}}\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msup><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><msqrt><mi href="./15.1#p1.t1.r2">z</mi></msqrt></mrow><mo>)</mo></mrow><mrow><mo>-</mo><mrow><mn>2</mn><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow></mrow></msup><mo></mo><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac linethickness="0pt"><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mrow><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow><mrow><mrow><mrow><mn>2</mn><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>-</mo><mrow><mn>2</mn><mo></mo><mi href="./15.1#p1.t1.r3">b</mi></mrow></mrow><mo>+</mo><mn>1</mn></mrow></mfrac></mstyle><mo>;</mo><mstyle displaystyle="true"><mfrac><mrow><mn>4</mn><mo></mo><msqrt><mi href="./15.1#p1.t1.r2">z</mi></msqrt></mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>+</mo><msqrt><mi href="./15.1#p1.t1.r2">z</mi></msqrt></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mfrac></mstyle><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m86.png" altimg-height="23px" altimg-valign="-7px" altimg-width="93px" alttext="|\operatorname{ph}z|<\pi" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo stretchy="false">|</mo></mrow><mo><</mo><mi href="./3.12#E1">π</mi></mrow></math>, <math class="ltx_Math" altimg="m89.png" altimg-height="23px" altimg-valign="-7px" altimg-width="62px" alttext="|z|<1" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mi href="./15.1#p1.t1.r2">z</mi><mo stretchy="false">|</mo></mrow><mo><</mo><mn>1</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E21.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./15.2#E1" title="(15.2.1) ‣ §15.2(i) Gauss Series ‣ §15.2 Definitions and Analytical Properties ‣ Properties ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="23px" altimg-valign="-7px" altimg-width="103px" alttext="F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m30.png" altimg-height="39px" altimg-valign="-15px" altimg-width="91px" alttext="F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi></mfrac><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: <math class="ltx_Math" altimg="m28.png" altimg-height="23px" altimg-valign="-7px" altimg-width="139px" alttext="={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow></math>
Gauss’ hypergeometric function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.9#E7" title="(1.9.7) ‣ Modulus and Phase ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="21px" altimg-valign="-6px" altimg-width="26px" alttext="\operatorname{ph}" display="inline"><mi href="./1.9#E7">ph</mi></math>: phase</a>,
<a href="./15.1#p1.t1.r2" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./15.1#p1.t1.r2">z</mi></math>: complex variable</a>,
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./15.1#p1.t1.r3">a</mi></math>: real or complex parameter</a> and
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="b" display="inline"><mi href="./15.1#p1.t1.r3">b</mi></math>: real or complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E22">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">15.8.22</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m22.png" altimg-height="55px" altimg-valign="-23px" altimg-width="184px" alttext="\displaystyle F\left({a,b\atop\tfrac{1}{2}(a+b+1)};z\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac linethickness="0pt"><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>+</mo><mi href="./15.1#p1.t1.r3">b</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow></mfrac></mstyle><mo>;</mo><mi href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m16.png" altimg-height="69px" altimg-valign="-29px" altimg-width="499px" alttext="\displaystyle=\left(\frac{\sqrt{1-z^{-1}}-1}{\sqrt{1-z^{-1}}+1}\right)^{a}F%
\left({a,\tfrac{1}{2}(a+b)\atop a+b};\frac{4\sqrt{1-z^{-1}}}{\left(\sqrt{1-z^{%
-1}}+1\right)^{2}}\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msup><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac><mrow><msqrt><mrow><mn>1</mn><mo>-</mo><msup><mi href="./15.1#p1.t1.r2">z</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></mrow></msqrt><mo>-</mo><mn>1</mn></mrow><mrow><msqrt><mrow><mn>1</mn><mo>-</mo><msup><mi href="./15.1#p1.t1.r2">z</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></mrow></msqrt><mo>+</mo><mn>1</mn></mrow></mfrac></mstyle><mo>)</mo></mrow><mi href="./15.1#p1.t1.r3">a</mi></msup><mo></mo><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac linethickness="0pt"><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>+</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>+</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow></mfrac></mstyle><mo>;</mo><mstyle displaystyle="true"><mfrac><mrow><mn>4</mn><mo></mo><msqrt><mrow><mn>1</mn><mo>-</mo><msup><mi href="./15.1#p1.t1.r2">z</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></mrow></msqrt></mrow><msup><mrow><mo>(</mo><mrow><msqrt><mrow><mn>1</mn><mo>-</mo><msup><mi href="./15.1#p1.t1.r2">z</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></mrow></msqrt><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mn>2</mn></msup></mfrac></mstyle><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m84.png" altimg-height="23px" altimg-valign="-7px" altimg-width="127px" alttext="|\operatorname{ph}\left(-z\right)|<\pi" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo>)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow><mo><</mo><mi href="./3.12#E1">π</mi></mrow></math>, <math class="ltx_Math" altimg="m32.png" altimg-height="27px" altimg-valign="-9px" altimg-width="68px" alttext="\Re z<\frac{1}{2}" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo><</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E22.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./15.2#E1" title="(15.2.1) ‣ §15.2(i) Gauss Series ‣ §15.2 Definitions and Analytical Properties ‣ Properties ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="23px" altimg-valign="-7px" altimg-width="103px" alttext="F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m30.png" altimg-height="39px" altimg-valign="-15px" altimg-width="91px" alttext="F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi></mfrac><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: <math class="ltx_Math" altimg="m28.png" altimg-height="23px" altimg-valign="-7px" altimg-width="139px" alttext="={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow></math>
Gauss’ hypergeometric function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.9#E7" title="(1.9.7) ‣ Modulus and Phase ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="21px" altimg-valign="-6px" altimg-width="26px" alttext="\operatorname{ph}" display="inline"><mi href="./1.9#E7">ph</mi></math>: phase</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m35.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./15.1#p1.t1.r2" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./15.1#p1.t1.r2">z</mi></math>: complex variable</a>,
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./15.1#p1.t1.r3">a</mi></math>: real or complex parameter</a> and
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="b" display="inline"><mi href="./15.1#p1.t1.r3">b</mi></math>: real or complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
<table id="E23" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">15.8.23</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E23.png" altimg-height="121px" altimg-valign="-29px" altimg-width="581px" alttext="F\left({a,1-a\atop c};z\right)=\left(\sqrt{1-z^{-1}}-1\right)^{1-a}\left(\sqrt%
{1-z^{-1}}+1\right)^{a-2c+1}\left(1-z^{-1}\right)^{c-1}F\left({c-a,c-\tfrac{1}%
{2}\atop 2c-1};\frac{4\sqrt{1-z^{-1}}}{\left(\sqrt{1-z^{-1}}+1\right)^{2}}%
\right)," display="block"><mrow><mrow><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mrow><mn>1</mn><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi></mrow></mrow><mi href="./15.1#p1.t1.r3">c</mi></mfrac><mo>;</mo><mi href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msup><mrow><mo>(</mo><mrow><msqrt><mrow><mn>1</mn><mo>-</mo><msup><mi href="./15.1#p1.t1.r2">z</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></mrow></msqrt><mo>-</mo><mn>1</mn></mrow><mo>)</mo></mrow><mrow><mn>1</mn><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi></mrow></msup><mo></mo><msup><mrow><mo>(</mo><mrow><msqrt><mrow><mn>1</mn><mo>-</mo><msup><mi href="./15.1#p1.t1.r2">z</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></mrow></msqrt><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mrow><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>-</mo><mrow><mn>2</mn><mo></mo><mi href="./15.1#p1.t1.r3">c</mi></mrow></mrow><mo>+</mo><mn>1</mn></mrow></msup><mo></mo><msup><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><msup><mi href="./15.1#p1.t1.r2">z</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></mrow><mo>)</mo></mrow><mrow><mi href="./15.1#p1.t1.r3">c</mi><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mrow><mi href="./15.1#p1.t1.r3">c</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>,</mo><mrow><mi href="./15.1#p1.t1.r3">c</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow><mrow><mrow><mn>2</mn><mo></mo><mi href="./15.1#p1.t1.r3">c</mi></mrow><mo>-</mo><mn>1</mn></mrow></mfrac><mo>;</mo><mfrac><mrow><mn>4</mn><mo></mo><msqrt><mrow><mn>1</mn><mo>-</mo><msup><mi href="./15.1#p1.t1.r2">z</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></mrow></msqrt></mrow><msup><mrow><mo>(</mo><mrow><msqrt><mrow><mn>1</mn><mo>-</mo><msup><mi href="./15.1#p1.t1.r2">z</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></mrow></msqrt><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mn>2</mn></msup></mfrac><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m84.png" altimg-height="23px" altimg-valign="-7px" altimg-width="127px" alttext="|\operatorname{ph}\left(-z\right)|<\pi" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo>)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow><mo><</mo><mi href="./3.12#E1">π</mi></mrow></math>, <math class="ltx_Math" altimg="m32.png" altimg-height="27px" altimg-valign="-9px" altimg-width="68px" alttext="\Re z<\frac{1}{2}" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo><</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E23.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./15.2#E1" title="(15.2.1) ‣ §15.2(i) Gauss Series ‣ §15.2 Definitions and Analytical Properties ‣ Properties ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="23px" altimg-valign="-7px" altimg-width="103px" alttext="F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m30.png" altimg-height="39px" altimg-valign="-15px" altimg-width="91px" alttext="F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi></mfrac><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: <math class="ltx_Math" altimg="m28.png" altimg-height="23px" altimg-valign="-7px" altimg-width="139px" alttext="={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow></math>
Gauss’ hypergeometric function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.9#E7" title="(1.9.7) ‣ Modulus and Phase ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="21px" altimg-valign="-6px" altimg-width="26px" alttext="\operatorname{ph}" display="inline"><mi href="./1.9#E7">ph</mi></math>: phase</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m35.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./15.1#p1.t1.r2" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./15.1#p1.t1.r2">z</mi></math>: complex variable</a>,
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./15.1#p1.t1.r3">a</mi></math>: real or complex parameter</a> and
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m76.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="c" display="inline"><mi href="./15.1#p1.t1.r3">c</mi></math>: real or complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px4" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Group 2 <math class="ltx_Math" altimg="m42.png" altimg-height="12px" altimg-valign="-2px" altimg-width="36px" alttext="\longrightarrow" display="inline"><mo>⟶</mo></math> Group 4</h3>
<div id="Px4.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px4.p1" class="ltx_para">
<table id="E24" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">15.8.24</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_Math ltx_eqn_cell ltx_align_center"><math class="ltx_Math" alttext="F\left({a,b\atop a-b+1};z\right)=(1-z)^{-a}\frac{\Gamma\left(a-b+1\right)%
\Gamma\left(\tfrac{1}{2}\right)}{\Gamma\left(\tfrac{1}{2}a+\tfrac{1}{2}\right)%
\Gamma\left(\tfrac{1}{2}a-b+1\right)}F\left({\tfrac{1}{2}a,\tfrac{1}{2}a-b+%
\tfrac{1}{2}\atop\tfrac{1}{2}};\left(\frac{z+1}{z-1}\right)^{2}\right)+(1+z)(1%
-z)^{-a-1}\frac{\Gamma\left(a-b+1\right)\Gamma\left(-\tfrac{1}{2}\right)}{%
\Gamma\left(\tfrac{1}{2}a\right)\Gamma\left(\tfrac{1}{2}a-b+\tfrac{1}{2}\right%
)}F\left({\tfrac{1}{2}a+\tfrac{1}{2},\tfrac{1}{2}a-b+1\atop\tfrac{3}{2}};\left%
(\frac{z+1}{z-1}\right)^{2}\right)," display="block"><mrow><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mrow><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mo>+</mo><mn>1</mn></mrow></mfrac><mo>;</mo><mi href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi></mrow></msup><mo></mo><mfrac><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>)</mo></mrow></mrow></mrow><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>-</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mrow></mfrac><mo></mo><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>,</mo><mrow><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>-</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></mfrac><mo>;</mo><msup><mrow><mo>(</mo><mfrac><mrow><mi href="./15.1#p1.t1.r2">z</mi><mo>+</mo><mn>1</mn></mrow><mrow><mi href="./15.1#p1.t1.r2">z</mi><mo>-</mo><mn>1</mn></mrow></mfrac><mo>)</mo></mrow><mn>2</mn></msup><mo>)</mo></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>+</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>+</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mrow><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mfrac><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow></mrow><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>-</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow></mrow></mfrac><mo></mo><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>,</mo><mrow><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>-</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mo>+</mo><mn>1</mn></mrow></mrow><mfrac><mn>3</mn><mn>2</mn></mfrac></mfrac><mo>;</mo><msup><mrow><mo>(</mo><mfrac><mrow><mi href="./15.1#p1.t1.r2">z</mi><mo>+</mo><mn>1</mn></mrow><mrow><mi href="./15.1#p1.t1.r2">z</mi><mo>-</mo><mn>1</mn></mrow></mfrac><mo>)</mo></mrow><mn>2</mn></msup><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mtd></mtr></mtable></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m84.png" altimg-height="23px" altimg-valign="-7px" altimg-width="127px" alttext="|\operatorname{ph}\left(-z\right)|<\pi" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo>)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow><mo><</mo><mi href="./3.12#E1">π</mi></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E24.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m31.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./15.2#E1" title="(15.2.1) ‣ §15.2(i) Gauss Series ‣ §15.2 Definitions and Analytical Properties ‣ Properties ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="23px" altimg-valign="-7px" altimg-width="103px" alttext="F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m30.png" altimg-height="39px" altimg-valign="-15px" altimg-width="91px" alttext="F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi></mfrac><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: <math class="ltx_Math" altimg="m28.png" altimg-height="23px" altimg-valign="-7px" altimg-width="139px" alttext="={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow></math>
Gauss’ hypergeometric function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.9#E7" title="(1.9.7) ‣ Modulus and Phase ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="21px" altimg-valign="-6px" altimg-width="26px" alttext="\operatorname{ph}" display="inline"><mi href="./1.9#E7">ph</mi></math>: phase</a>,
<a href="./15.1#p1.t1.r2" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./15.1#p1.t1.r2">z</mi></math>: complex variable</a>,
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./15.1#p1.t1.r3">a</mi></math>: real or complex parameter</a> and
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="b" display="inline"><mi href="./15.1#p1.t1.r3">b</mi></math>: real or complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E25" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">15.8.25</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_Math ltx_eqn_cell ltx_align_center"><math class="ltx_Math" alttext="F\left({a,b\atop\tfrac{1}{2}(a+b+1)};z\right)=\frac{\Gamma\left(\tfrac{1}{2}(a%
+b+1)\right)\Gamma\left(\tfrac{1}{2}\right)}{\Gamma\left(\tfrac{1}{2}a+\tfrac{%
1}{2}\right)\Gamma\left(\tfrac{1}{2}b+\tfrac{1}{2}\right)}F\left({\tfrac{1}{2}%
a,\tfrac{1}{2}b\atop\tfrac{1}{2}};(1-2z)^{2}\right)+(1-2z)\frac{\Gamma\left(%
\tfrac{1}{2}(a+b+1)\right)\Gamma\left(-\tfrac{1}{2}\right)}{\Gamma\left(\tfrac%
{1}{2}a\right)\Gamma\left(\tfrac{1}{2}b\right)}F\left({\tfrac{1}{2}a+\tfrac{1}%
{2},\tfrac{1}{2}b+\tfrac{1}{2}\atop\tfrac{3}{2}};(1-2z)^{2}\right)," display="block"><mrow><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>+</mo><mi href="./15.1#p1.t1.r3">b</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow></mfrac><mo>;</mo><mi href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mrow><mfrac><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>+</mo><mi href="./15.1#p1.t1.r3">b</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>)</mo></mrow></mrow></mrow><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow></mrow></mfrac><mo></mo><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>,</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./15.1#p1.t1.r3">b</mi></mrow></mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></mfrac><mo>;</mo><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><mrow><mn>2</mn><mo></mo><mi href="./15.1#p1.t1.r2">z</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>)</mo></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>+</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><mrow><mn>2</mn><mo></mo><mi href="./15.1#p1.t1.r2">z</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mfrac><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>+</mo><mi href="./15.1#p1.t1.r3">b</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow></mrow><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mo>)</mo></mrow></mrow></mrow></mfrac><mo></mo><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>,</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow><mfrac><mn>3</mn><mn>2</mn></mfrac></mfrac><mo>;</mo><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><mrow><mn>2</mn><mo></mo><mi href="./15.1#p1.t1.r2">z</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mtd></mtr></mtable></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m86.png" altimg-height="23px" altimg-valign="-7px" altimg-width="93px" alttext="|\operatorname{ph}z|<\pi" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo stretchy="false">|</mo></mrow><mo><</mo><mi href="./3.12#E1">π</mi></mrow></math>, <math class="ltx_Math" altimg="m85.png" altimg-height="23px" altimg-valign="-7px" altimg-width="146px" alttext="|\operatorname{ph}\left(1-z\right)|<\pi" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo>)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow><mo><</mo><mi href="./3.12#E1">π</mi></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E25.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m31.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./15.2#E1" title="(15.2.1) ‣ §15.2(i) Gauss Series ‣ §15.2 Definitions and Analytical Properties ‣ Properties ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="23px" altimg-valign="-7px" altimg-width="103px" alttext="F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m30.png" altimg-height="39px" altimg-valign="-15px" altimg-width="91px" alttext="F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi></mfrac><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: <math class="ltx_Math" altimg="m28.png" altimg-height="23px" altimg-valign="-7px" altimg-width="139px" alttext="={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow></math>
Gauss’ hypergeometric function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.9#E7" title="(1.9.7) ‣ Modulus and Phase ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="21px" altimg-valign="-6px" altimg-width="26px" alttext="\operatorname{ph}" display="inline"><mi href="./1.9#E7">ph</mi></math>: phase</a>,
<a href="./15.1#p1.t1.r2" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./15.1#p1.t1.r2">z</mi></math>: complex variable</a>,
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./15.1#p1.t1.r3">a</mi></math>: real or complex parameter</a> and
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="b" display="inline"><mi href="./15.1#p1.t1.r3">b</mi></math>: real or complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E26" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">15.8.26</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_Math ltx_eqn_cell ltx_align_center"><math class="ltx_Math" alttext="F\left({a,1-a\atop c};z\right)=(1-z)^{c-1}\frac{\Gamma\left(c\right)\Gamma%
\left(\tfrac{1}{2}\right)}{\Gamma\left(\tfrac{1}{2}(c-a+1)\right)\Gamma\left(%
\tfrac{1}{2}c+\tfrac{1}{2}a\right)}F\left({\tfrac{1}{2}c-\tfrac{1}{2}a,\tfrac{%
1}{2}c+\tfrac{1}{2}a-\tfrac{1}{2}\atop\tfrac{1}{2}};(1-2z)^{2}\right)+(1-2z)(1%
-z)^{c-1}\frac{\Gamma\left(c\right)\Gamma\left(-\tfrac{1}{2}\right)}{\Gamma%
\left(\tfrac{1}{2}c-\tfrac{1}{2}a\right)\Gamma\left(\tfrac{1}{2}(c+a-1)\right)%
}F\left({\tfrac{1}{2}c-\tfrac{1}{2}a+\tfrac{1}{2},\tfrac{1}{2}c+\tfrac{1}{2}a%
\atop\tfrac{3}{2}};(1-2z)^{2}\right)," display="block"><mrow><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mrow><mn>1</mn><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi></mrow></mrow><mi href="./15.1#p1.t1.r3">c</mi></mfrac><mo>;</mo><mi href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mi href="./15.1#p1.t1.r3">c</mi><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mfrac><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi href="./15.1#p1.t1.r3">c</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>)</mo></mrow></mrow></mrow><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mi href="./15.1#p1.t1.r3">c</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./15.1#p1.t1.r3">c</mi></mrow><mo>+</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mfrac><mo></mo><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./15.1#p1.t1.r3">c</mi></mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow></mrow><mo>,</mo><mrow><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./15.1#p1.t1.r3">c</mi></mrow><mo>+</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow></mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></mfrac><mo>;</mo><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><mrow><mn>2</mn><mo></mo><mi href="./15.1#p1.t1.r2">z</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>)</mo></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>+</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><mrow><mn>2</mn><mo></mo><mi href="./15.1#p1.t1.r2">z</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mi href="./15.1#p1.t1.r3">c</mi><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mfrac><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi href="./15.1#p1.t1.r3">c</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow></mrow><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./15.1#p1.t1.r3">c</mi></mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mi href="./15.1#p1.t1.r3">c</mi><mo>+</mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mfrac><mo></mo><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mrow><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./15.1#p1.t1.r3">c</mi></mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow></mrow><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>,</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./15.1#p1.t1.r3">c</mi></mrow><mo>+</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow></mrow></mrow><mfrac><mn>3</mn><mn>2</mn></mfrac></mfrac><mo>;</mo><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><mrow><mn>2</mn><mo></mo><mi href="./15.1#p1.t1.r2">z</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mtd></mtr></mtable></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m86.png" altimg-height="23px" altimg-valign="-7px" altimg-width="93px" alttext="|\operatorname{ph}z|<\pi" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo stretchy="false">|</mo></mrow><mo><</mo><mi href="./3.12#E1">π</mi></mrow></math>, <math class="ltx_Math" altimg="m85.png" altimg-height="23px" altimg-valign="-7px" altimg-width="146px" alttext="|\operatorname{ph}\left(1-z\right)|<\pi" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo>)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow><mo><</mo><mi href="./3.12#E1">π</mi></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E26.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m31.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./15.2#E1" title="(15.2.1) ‣ §15.2(i) Gauss Series ‣ §15.2 Definitions and Analytical Properties ‣ Properties ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="23px" altimg-valign="-7px" altimg-width="103px" alttext="F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m30.png" altimg-height="39px" altimg-valign="-15px" altimg-width="91px" alttext="F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi></mfrac><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: <math class="ltx_Math" altimg="m28.png" altimg-height="23px" altimg-valign="-7px" altimg-width="139px" alttext="={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow></math>
Gauss’ hypergeometric function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.9#E7" title="(1.9.7) ‣ Modulus and Phase ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="21px" altimg-valign="-6px" altimg-width="26px" alttext="\operatorname{ph}" display="inline"><mi href="./1.9#E7">ph</mi></math>: phase</a>,
<a href="./15.1#p1.t1.r2" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./15.1#p1.t1.r2">z</mi></math>: complex variable</a>,
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./15.1#p1.t1.r3">a</mi></math>: real or complex parameter</a> and
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m76.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="c" display="inline"><mi href="./15.1#p1.t1.r3">c</mi></math>: real or complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px5" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Group 4 <math class="ltx_Math" altimg="m42.png" altimg-height="12px" altimg-valign="-2px" altimg-width="36px" alttext="\longrightarrow" display="inline"><mo>⟶</mo></math> Group 2</h3>
<div id="Px5.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px5.p1" class="ltx_para">
<table id="E27" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">15.8.27</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_Math ltx_eqn_cell ltx_align_center"><math class="ltx_Math" alttext="\frac{2\Gamma\left(\tfrac{1}{2}\right)\Gamma\left(a+b+\tfrac{1}{2}\right)}{%
\Gamma\left(a+\tfrac{1}{2}\right)\Gamma\left(b+\tfrac{1}{2}\right)}F\left(a,b;%
\tfrac{1}{2};z\right)=F\left(2a,2b;a+b+\tfrac{1}{2};\tfrac{1}{2}-\tfrac{1}{2}%
\sqrt{z}\right)+F\left(2a,2b;a+b+\tfrac{1}{2};\tfrac{1}{2}+\tfrac{1}{2}\sqrt{z%
}\right)," display="block"><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mrow><mfrac><mrow><mn>2</mn><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>+</mo><mi href="./15.1#p1.t1.r3">b</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow></mrow><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./15.1#p1.t1.r3">b</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow></mrow></mfrac><mo></mo><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo>;</mo><mi href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>=</mo><mrow><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>,</mo><mrow><mn>2</mn><mo></mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mo>;</mo><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>+</mo><mi href="./15.1#p1.t1.r3">b</mi><mo>+</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle></mrow><mo>;</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo>-</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><msqrt><mi href="./15.1#p1.t1.r2">z</mi></msqrt></mrow></mrow><mo>)</mo></mrow></mrow><mo>+</mo><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>,</mo><mrow><mn>2</mn><mo></mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mo>;</mo><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>+</mo><mi href="./15.1#p1.t1.r3">b</mi><mo>+</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle></mrow><mo>;</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo>+</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><msqrt><mi href="./15.1#p1.t1.r2">z</mi></msqrt></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mtd></mtr></mtable></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m86.png" altimg-height="23px" altimg-valign="-7px" altimg-width="93px" alttext="|\operatorname{ph}z|<\pi" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo stretchy="false">|</mo></mrow><mo><</mo><mi href="./3.12#E1">π</mi></mrow></math>, <math class="ltx_Math" altimg="m85.png" altimg-height="23px" altimg-valign="-7px" altimg-width="146px" alttext="|\operatorname{ph}\left(1-z\right)|<\pi" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo>)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow><mo><</mo><mi href="./3.12#E1">π</mi></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E27.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m31.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./15.2#E1" title="(15.2.1) ‣ §15.2(i) Gauss Series ‣ §15.2 Definitions and Analytical Properties ‣ Properties ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="23px" altimg-valign="-7px" altimg-width="103px" alttext="F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m30.png" altimg-height="39px" altimg-valign="-15px" altimg-width="91px" alttext="F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi></mfrac><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: <math class="ltx_Math" altimg="m28.png" altimg-height="23px" altimg-valign="-7px" altimg-width="139px" alttext="={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow></math>
Gauss’ hypergeometric function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.9#E7" title="(1.9.7) ‣ Modulus and Phase ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="21px" altimg-valign="-6px" altimg-width="26px" alttext="\operatorname{ph}" display="inline"><mi href="./1.9#E7">ph</mi></math>: phase</a>,
<a href="./15.1#p1.t1.r2" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./15.1#p1.t1.r2">z</mi></math>: complex variable</a>,
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./15.1#p1.t1.r3">a</mi></math>: real or complex parameter</a> and
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="b" display="inline"><mi href="./15.1#p1.t1.r3">b</mi></math>: real or complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E28" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">15.8.28</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_Math ltx_eqn_cell ltx_align_center"><math class="ltx_Math" alttext="\frac{2\sqrt{z}\Gamma\left(-\tfrac{1}{2}\right)\Gamma\left(a+b-\tfrac{1}{2}%
\right)}{\Gamma\left(a-\tfrac{1}{2}\right)\Gamma\left(b-\tfrac{1}{2}\right)}F%
\left(a,b;\tfrac{3}{2};z\right)=F\left(2a-1,2b-1;a+b-\tfrac{1}{2};\tfrac{1}{2}%
-\tfrac{1}{2}\sqrt{z}\right)-F\left(2a-1,2b-1;a+b-\tfrac{1}{2};\tfrac{1}{2}+%
\tfrac{1}{2}\sqrt{z}\right)," display="block"><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mrow><mfrac><mrow><mn>2</mn><mo></mo><msqrt><mi href="./15.1#p1.t1.r2">z</mi></msqrt><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>+</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow></mrow><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./15.1#p1.t1.r3">b</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow></mrow></mfrac><mo></mo><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mstyle displaystyle="false"><mfrac><mn>3</mn><mn>2</mn></mfrac></mstyle><mo>;</mo><mi href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>=</mo><mrow><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mn>2</mn><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>-</mo><mn>1</mn></mrow><mo>,</mo><mrow><mrow><mn>2</mn><mo></mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mo>-</mo><mn>1</mn></mrow><mo>;</mo><mrow><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>+</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mo>-</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle></mrow><mo>;</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo>-</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><msqrt><mi href="./15.1#p1.t1.r2">z</mi></msqrt></mrow></mrow><mo>)</mo></mrow></mrow><mo>-</mo><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mn>2</mn><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>-</mo><mn>1</mn></mrow><mo>,</mo><mrow><mrow><mn>2</mn><mo></mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mo>-</mo><mn>1</mn></mrow><mo>;</mo><mrow><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>+</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mo>-</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle></mrow><mo>;</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo>+</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><msqrt><mi href="./15.1#p1.t1.r2">z</mi></msqrt></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mtd></mtr></mtable></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m86.png" altimg-height="23px" altimg-valign="-7px" altimg-width="93px" alttext="|\operatorname{ph}z|<\pi" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo stretchy="false">|</mo></mrow><mo><</mo><mi href="./3.12#E1">π</mi></mrow></math>, <math class="ltx_Math" altimg="m85.png" altimg-height="23px" altimg-valign="-7px" altimg-width="146px" alttext="|\operatorname{ph}\left(1-z\right)|<\pi" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo>)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow><mo><</mo><mi href="./3.12#E1">π</mi></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E28.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m31.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./15.2#E1" title="(15.2.1) ‣ §15.2(i) Gauss Series ‣ §15.2 Definitions and Analytical Properties ‣ Properties ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="23px" altimg-valign="-7px" altimg-width="103px" alttext="F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m30.png" altimg-height="39px" altimg-valign="-15px" altimg-width="91px" alttext="F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi></mfrac><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: <math class="ltx_Math" altimg="m28.png" altimg-height="23px" altimg-valign="-7px" altimg-width="139px" alttext="={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow></math>
Gauss’ hypergeometric function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.9#E7" title="(1.9.7) ‣ Modulus and Phase ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="21px" altimg-valign="-6px" altimg-width="26px" alttext="\operatorname{ph}" display="inline"><mi href="./1.9#E7">ph</mi></math>: phase</a>,
<a href="./15.1#p1.t1.r2" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./15.1#p1.t1.r2">z</mi></math>: complex variable</a>,
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./15.1#p1.t1.r3">a</mi></math>: real or complex parameter</a> and
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="b" display="inline"><mi href="./15.1#p1.t1.r3">b</mi></math>: real or complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
</section>
<section id="iv" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§15.8(iv) </span>Quadratic Transformations (Continued)</h2>
<div id="SS4.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px6.p1" class="ltx_para">
<p class="ltx_p"><math class="ltx_Math" altimg="m61.png" altimg-height="27px" altimg-valign="-9px" altimg-width="100px" alttext="b=\tfrac{1}{3}a+\tfrac{1}{3}" display="inline"><mrow><mi href="./15.1#p1.t1.r3">b</mi><mo>=</mo><mrow><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>+</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></mrow></mrow></math>, <math class="ltx_Math" altimg="m67.png" altimg-height="19px" altimg-valign="-4px" altimg-width="162px" alttext="c=2b=a-b+1" display="inline"><mrow><mi href="./15.1#p1.t1.r3">c</mi><mo>=</mo><mrow><mn>2</mn><mo></mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mo>=</mo><mrow><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mo>+</mo><mn>1</mn></mrow></mrow></math> in Groups 1 and 2.</p>
</div>
<div id="Px6.p2" class="ltx_para">
<p class="ltx_p">() becomes</p>
<table id="E29" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">15.8.29</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E29.png" altimg-height="65px" altimg-valign="-27px" altimg-width="543px" alttext="F\left({a,\tfrac{1}{3}a+\tfrac{1}{3}\atop\tfrac{2}{3}a+\tfrac{2}{3}};z\right)=%
\left(1+\sqrt{z}\right)^{-2a}\*F\left({a,\tfrac{2}{3}a+\tfrac{1}{6}\atop\tfrac%
{4}{3}a+\tfrac{1}{3}};\frac{4\sqrt{z}}{(1+\sqrt{z})^{2}}\right)." display="block"><mrow><mrow><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mrow><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>+</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></mrow></mrow><mrow><mrow><mfrac><mn>2</mn><mn>3</mn></mfrac><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>+</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></mrow></mfrac><mo>;</mo><mi href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msup><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><msqrt><mi href="./15.1#p1.t1.r2">z</mi></msqrt></mrow><mo>)</mo></mrow><mrow><mo>-</mo><mrow><mn>2</mn><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow></mrow></msup><mo></mo><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mrow><mrow><mfrac><mn>2</mn><mn>3</mn></mfrac><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>+</mo><mfrac><mn>1</mn><mn>6</mn></mfrac></mrow></mrow><mrow><mrow><mfrac><mn>4</mn><mn>3</mn></mfrac><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>+</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></mrow></mfrac><mo>;</mo><mfrac><mrow><mn>4</mn><mo></mo><msqrt><mi href="./15.1#p1.t1.r2">z</mi></msqrt></mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>+</mo><msqrt><mi href="./15.1#p1.t1.r2">z</mi></msqrt></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mfrac><mo>)</mo></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E29.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./15.2#E1" title="(15.2.1) ‣ §15.2(i) Gauss Series ‣ §15.2 Definitions and Analytical Properties ‣ Properties ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="23px" altimg-valign="-7px" altimg-width="103px" alttext="F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m30.png" altimg-height="39px" altimg-valign="-15px" altimg-width="91px" alttext="F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi></mfrac><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: <math class="ltx_Math" altimg="m28.png" altimg-height="23px" altimg-valign="-7px" altimg-width="139px" alttext="={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow></math>
Gauss’ hypergeometric function</a>,
<a href="./15.1#p1.t1.r2" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./15.1#p1.t1.r2">z</mi></math>: complex variable</a> and
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./15.1#p1.t1.r3">a</mi></math>: real or complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">This is a quadratic transformation between two cases in Group 1.</p>
</div>
<div id="Px6.p3" class="ltx_para">
<p class="ltx_p">We can also use (), and obtain</p>
<table id="E30" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">15.8.30</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E30.png" altimg-height="65px" altimg-valign="-27px" altimg-width="871px" alttext="\left(1-\tfrac{1}{2}z\right)^{-a}F\left({\tfrac{1}{2}a,\tfrac{1}{2}a+\tfrac{1}%
{2}\atop\tfrac{1}{3}a+\tfrac{5}{6}};\left(\frac{z}{2-z}\right)^{2}\right)=F%
\left({a,\tfrac{1}{3}a+\tfrac{1}{3}\atop\tfrac{2}{3}a+\tfrac{2}{3}};z\right)=(%
1+z)^{-a}F\left({\tfrac{1}{2}a,\tfrac{1}{2}a+\tfrac{1}{2}\atop\tfrac{2}{3}a+%
\tfrac{2}{3}};\frac{4z}{(1+z)^{2}}\right)," display="block"><mrow><mrow><mrow><msup><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./15.1#p1.t1.r2">z</mi></mrow></mrow><mo>)</mo></mrow><mrow><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi></mrow></msup><mo></mo><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>,</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow><mrow><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>+</mo><mfrac><mn>5</mn><mn>6</mn></mfrac></mrow></mfrac><mo>;</mo><msup><mrow><mo>(</mo><mfrac><mi href="./15.1#p1.t1.r2">z</mi><mrow><mn>2</mn><mo>-</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow></mfrac><mo>)</mo></mrow><mn>2</mn></msup><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mrow><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>+</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></mrow></mrow><mrow><mrow><mfrac><mn>2</mn><mn>3</mn></mfrac><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>+</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></mrow></mfrac><mo>;</mo><mi href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>+</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi></mrow></msup><mo></mo><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>,</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow><mrow><mrow><mfrac><mn>2</mn><mn>3</mn></mfrac><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>+</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></mrow></mfrac><mo>;</mo><mfrac><mrow><mn>4</mn><mo></mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>+</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mfrac><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E30.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./15.2#E1" title="(15.2.1) ‣ §15.2(i) Gauss Series ‣ §15.2 Definitions and Analytical Properties ‣ Properties ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="23px" altimg-valign="-7px" altimg-width="103px" alttext="F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m30.png" altimg-height="39px" altimg-valign="-15px" altimg-width="91px" alttext="F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi></mfrac><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: <math class="ltx_Math" altimg="m28.png" altimg-height="23px" altimg-valign="-7px" altimg-width="139px" alttext="={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow></math>
Gauss’ hypergeometric function</a>,
<a href="./15.1#p1.t1.r2" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./15.1#p1.t1.r2">z</mi></math>: complex variable</a> and
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./15.1#p1.t1.r3">a</mi></math>: real or complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">which is a quadratic transformation between two cases in Group 3.</p>
</div>
<div id="Px6.p4" class="ltx_para">
<p class="ltx_p">For further examples see <cite class="ltx_cite ltx_citemacro_citet">Andrews<span class="ltx_text ltx_bib_etal"> et al.</span> (</dd>
</dl>
</div>
</div>
<div id="Px7.p1" class="ltx_para">
<table id="E31" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">15.8.31</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E31.png" altimg-height="65px" altimg-valign="-27px" altimg-width="547px" alttext="F\left({3a,3a+\frac{1}{2}\atop 4a+\frac{2}{3}};z\right)=\left(1-\tfrac{9}{8}z%
\right)^{-2a}\*F\left({a,a+\frac{1}{2}\atop 2a+\frac{5}{6}};\frac{27z^{2}(z-1)%
}{(9z-8)^{2}}\right)," display="block"><mrow><mrow><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mrow><mn>3</mn><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>,</mo><mrow><mrow><mn>3</mn><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow><mrow><mrow><mn>4</mn><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>+</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></mrow></mfrac><mo>;</mo><mi href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msup><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mrow><mstyle displaystyle="false"><mfrac><mn>9</mn><mn>8</mn></mfrac></mstyle><mo></mo><mi href="./15.1#p1.t1.r2">z</mi></mrow></mrow><mo>)</mo></mrow><mrow><mo>-</mo><mrow><mn>2</mn><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow></mrow></msup><mo></mo><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow><mrow><mrow><mn>2</mn><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>+</mo><mfrac><mn>5</mn><mn>6</mn></mfrac></mrow></mfrac><mo>;</mo><mfrac><mrow><mn>27</mn><mo></mo><msup><mi href="./15.1#p1.t1.r2">z</mi><mn>2</mn></msup><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./15.1#p1.t1.r2">z</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>9</mn><mo></mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo>-</mo><mn>8</mn></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mfrac><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m33.png" altimg-height="27px" altimg-valign="-9px" altimg-width="68px" alttext="\Re z<\frac{8}{9}" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo><</mo><mfrac><mn>8</mn><mn>9</mn></mfrac></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E31.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./15.2#E1" title="(15.2.1) ‣ §15.2(i) Gauss Series ‣ §15.2 Definitions and Analytical Properties ‣ Properties ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="23px" altimg-valign="-7px" altimg-width="103px" alttext="F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m30.png" altimg-height="39px" altimg-valign="-15px" altimg-width="91px" alttext="F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi></mfrac><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: <math class="ltx_Math" altimg="m28.png" altimg-height="23px" altimg-valign="-7px" altimg-width="139px" alttext="={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow></math>
Gauss’ hypergeometric function</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m35.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./15.1#p1.t1.r2" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./15.1#p1.t1.r2">z</mi></math>: complex variable</a> and
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./15.1#p1.t1.r3">a</mi></math>: real or complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="Px7.p2" class="ltx_para">
<p class="ltx_p">With <math class="ltx_Math" altimg="m52.png" altimg-height="28px" altimg-valign="-9px" altimg-width="273px" alttext="\zeta=e^{\ifrac{2\pi\mathrm{i}}{3}}(1-z)/\left(z-e^{\ifrac{4\pi\mathrm{i}}{3}}\right)" display="inline"><mrow><mi href="./15.8#Px7.p2">ζ</mi><mo>=</mo><mrow><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi></mrow><mo>/</mo><mn>3</mn></mrow></msup><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>/</mo><mrow><mo>(</mo><mrow><mi href="./15.1#p1.t1.r2">z</mi><mo>-</mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mrow><mn>4</mn><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi></mrow><mo>/</mo><mn>3</mn></mrow></msup></mrow><mo>)</mo></mrow></mrow></mrow></math>
</p>
<table id="E32" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">15.8.32</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_Math ltx_eqn_cell ltx_align_center"><math class="ltx_Math" alttext="\frac{\left(1-z^{3}\right)^{a}}{\left(-z\right)^{3a}}\left(\frac{1}{\Gamma%
\left(a+\frac{2}{3}\right)\Gamma\left(\frac{2}{3}\right)}F\left({a,a+\frac{1}{%
3}\atop\frac{2}{3}};z^{-3}\right)+\frac{e^{\frac{1}{3}\pi\mathrm{i}}}{z\Gamma%
\left(a\right)\Gamma\left(\frac{4}{3}\right)}F\left({a+\frac{1}{3},a+\frac{2}{%
3}\atop\frac{4}{3}};z^{-3}\right)\right)=\frac{3^{\frac{3}{2}a+\frac{1}{2}}e^{%
\frac{1}{2}a\pi\mathrm{i}}\Gamma\left(a+\frac{1}{3}\right)(1-\zeta)^{a}}{2\pi%
\Gamma\left(2a+\frac{2}{3}\right)(-\zeta)^{2a}}F\left({a+\frac{1}{3},3a\atop 2%
a+\frac{2}{3}};\zeta^{-1}\right)," display="block"><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mrow><mfrac><msup><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><msup><mi href="./15.1#p1.t1.r2">z</mi><mn>3</mn></msup></mrow><mo>)</mo></mrow><mi href="./15.1#p1.t1.r3">a</mi></msup><msup><mrow><mo>(</mo><mrow><mo>-</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo>)</mo></mrow><mrow><mn>3</mn><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow></msup></mfrac><mo></mo><mrow><mo>(</mo><mrow><mrow><mfrac><mn>1</mn><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>+</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><mo>)</mo></mrow></mrow></mrow></mfrac><mo></mo><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>+</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></mrow></mrow><mfrac><mn>2</mn><mn>3</mn></mfrac></mfrac><mo>;</mo><msup><mi href="./15.1#p1.t1.r2">z</mi><mrow><mo>-</mo><mn>3</mn></mrow></msup><mo>)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mfrac><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi></mrow></msup><mrow><mi href="./15.1#p1.t1.r2">z</mi><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi href="./15.1#p1.t1.r3">a</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mfrac><mn>4</mn><mn>3</mn></mfrac><mo>)</mo></mrow></mrow></mrow></mfrac><mo></mo><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>+</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></mrow><mo>,</mo><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>+</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></mrow></mrow><mfrac><mn>4</mn><mn>3</mn></mfrac></mfrac><mo>;</mo><msup><mi href="./15.1#p1.t1.r2">z</mi><mrow><mo>-</mo><mn>3</mn></mrow></msup><mo>)</mo></mrow></mrow></mrow></mrow><mo>)</mo></mrow></mrow></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>=</mo><mrow><mfrac><mrow><msup><mn>3</mn><mrow><mrow><mfrac><mn>3</mn><mn>2</mn></mfrac><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./15.1#p1.t1.r3">a</mi><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi></mrow></msup><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>+</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><mi href="./15.8#Px7.p2">ζ</mi></mrow><mo stretchy="false">)</mo></mrow><mi href="./15.1#p1.t1.r3">a</mi></msup></mrow><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mn>2</mn><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>+</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mi href="./15.8#Px7.p2">ζ</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mn>2</mn><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow></msup></mrow></mfrac><mo></mo><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>+</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></mrow><mo>,</mo><mrow><mn>3</mn><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow></mrow><mrow><mrow><mn>2</mn><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>+</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></mrow></mfrac><mo>;</mo><msup><mi href="./15.8#Px7.p2">ζ</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mtd></mtr></mtable></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m91.png" altimg-height="23px" altimg-valign="-7px" altimg-width="62px" alttext="|z|>1" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mi href="./15.1#p1.t1.r2">z</mi><mo stretchy="false">|</mo></mrow><mo>></mo><mn>1</mn></mrow></math>, <math class="ltx_Math" altimg="m83.png" altimg-height="27px" altimg-valign="-9px" altimg-width="140px" alttext="|\operatorname{ph}\left(-z\right)|<\frac{1}{3}\pi" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo>)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow><mo><</mo><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E32.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m31.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./15.2#E1" title="(15.2.1) ‣ §15.2(i) Gauss Series ‣ §15.2 Definitions and Analytical Properties ‣ Properties ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="23px" altimg-valign="-7px" altimg-width="103px" alttext="F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m30.png" altimg-height="39px" altimg-valign="-15px" altimg-width="91px" alttext="F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi></mfrac><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: <math class="ltx_Math" altimg="m28.png" altimg-height="23px" altimg-valign="-7px" altimg-width="139px" alttext="={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow></math>
Gauss’ hypergeometric function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m46.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./1.9#E7" title="(1.9.7) ‣ Modulus and Phase ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="21px" altimg-valign="-6px" altimg-width="26px" alttext="\operatorname{ph}" display="inline"><mi href="./1.9#E7">ph</mi></math>: phase</a>,
<a href="./15.1#p1.t1.r2" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./15.1#p1.t1.r2">z</mi></math>: complex variable</a>,
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./15.1#p1.t1.r3">a</mi></math>: real or complex parameter</a> and
<a href="./15.8#Px7.p2" title="Examples ‣ §15.8(v) Cubic Transformations ‣ §15.8 Transformations of Variable ‣ Properties ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="21px" altimg-valign="-6px" altimg-width="14px" alttext="\zeta" display="inline"><mi href="./15.8#Px7.p2">ζ</mi></math>: change of variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px8" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Ramanujan’s Cubic Transformation</h3>
<div id="Px8.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px8.p1" class="ltx_para">
<table id="E33" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">15.8.33</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E33.png" altimg-height="65px" altimg-valign="-27px" altimg-width="451px" alttext="F\left({\frac{1}{3},\frac{2}{3}\atop 1};1-\left(\frac{1-z}{1+2z}\right)^{3}%
\right)=(1+2z)F\left({\frac{1}{3},\frac{2}{3}\atop 1};z^{3}\right)," display="block"><mrow><mrow><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac><mo>,</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></mrow><mn>1</mn></mfrac><mo>;</mo><mrow><mn>1</mn><mo>-</mo><msup><mrow><mo>(</mo><mfrac><mrow><mn>1</mn><mo>-</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mrow><mn>1</mn><mo>+</mo><mrow><mn>2</mn><mo></mo><mi href="./15.1#p1.t1.r2">z</mi></mrow></mrow></mfrac><mo>)</mo></mrow><mn>3</mn></msup></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>+</mo><mrow><mn>2</mn><mo></mo><mi href="./15.1#p1.t1.r2">z</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac><mo>,</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></mrow><mn>1</mn></mfrac><mo>;</mo><msup><mi href="./15.1#p1.t1.r2">z</mi><mn>3</mn></msup><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E33.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./15.2#E1" title="(15.2.1) ‣ §15.2(i) Gauss Series ‣ §15.2 Definitions and Analytical Properties ‣ Properties ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="23px" altimg-valign="-7px" altimg-width="103px" alttext="F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m30.png" altimg-height="39px" altimg-valign="-15px" altimg-width="91px" alttext="F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi></mfrac><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: <math class="ltx_Math" altimg="m28.png" altimg-height="23px" altimg-valign="-7px" altimg-width="139px" alttext="={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow></math>
Gauss’ hypergeometric function</a> and
<a href="./15.1#p1.t1.r2" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./15.1#p1.t1.r2">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">provided that <math class="ltx_Math" altimg="m82.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./15.1#p1.t1.r2">z</mi></math> lies in the intersection of the open disks
<math class="ltx_Math" altimg="m40.png" altimg-height="29px" altimg-valign="-9px" altimg-width="200px" alttext="\left|z-\frac{1}{4}\pm\frac{1}{4}\sqrt{3}\mathrm{i}\right|<\frac{1}{2}\sqrt{3}" display="inline"><mrow><mrow><mo>|</mo><mrow><mrow><mi href="./15.1#p1.t1.r2">z</mi><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow><mo>±</mo><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac><mo></mo><msqrt><mn>3</mn></msqrt><mo></mo><mi mathvariant="normal">i</mi></mrow></mrow><mo>|</mo></mrow><mo><</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><msqrt><mn>3</mn></msqrt></mrow></mrow></math>,
or equivalently, <math class="ltx_Math" altimg="m39.png" altimg-height="23px" altimg-valign="-7px" altimg-width="260px" alttext="\left|\operatorname{ph}\left(\ifrac{(1-z)}{(1+2z)}\right)\right|<\pi/3" display="inline"><mrow><mrow><mo>|</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo stretchy="false">)</mo></mrow><mo>/</mo><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>+</mo><mrow><mn>2</mn><mo></mo><mi href="./15.1#p1.t1.r2">z</mi></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>)</mo></mrow></mrow><mo>|</mo></mrow><mo><</mo><mrow><mi href="./3.12#E1">π</mi><mo>/</mo><mn>3</mn></mrow></mrow></math>. This is used in a
cubic analog of the arithmetic-geometric mean. See <cite class="ltx_cite ltx_citemacro_citet">Borwein and Borwein (</div>
</div>
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<page>
<!DOCTYPE html><html>
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<title>DLMF: 15.4 Special Cases</title>
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<div class="ltx_page_navlogo"></dd>
</dl>
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<div id="SS1.p1" class="ltx_para">
<p class="ltx_p">The following results hold for principal branches when <math class="ltx_Math" altimg="m47.png" altimg-height="23px" altimg-valign="-7px" altimg-width="62px" alttext="|z|<1" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mi href="./15.1#p1.t1.r2">z</mi><mo stretchy="false">|</mo></mrow><mo><</mo><mn>1</mn></mrow></math>, and by
analytic continuation elsewhere. Exceptions are (), that hold for <math class="ltx_Math" altimg="m49.png" altimg-height="23px" altimg-valign="-7px" altimg-width="89px" alttext="|z|<\ifrac{\pi}{4}" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mi href="./15.1#p1.t1.r2">z</mi><mo stretchy="false">|</mo></mrow><mo><</mo><mrow><mi href="./3.12#E1">π</mi><mo>/</mo><mn>4</mn></mrow></mrow></math>, and
(), that hold
for <math class="ltx_Math" altimg="m48.png" altimg-height="23px" altimg-valign="-7px" altimg-width="89px" alttext="|z|<\ifrac{\pi}{2}" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mi href="./15.1#p1.t1.r2">z</mi><mo stretchy="false">|</mo></mrow><mo><</mo><mrow><mi href="./3.12#E1">π</mi><mo>/</mo><mn>2</mn></mrow></mrow></math>.</p>
<table id="EGx1" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E1">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">15.4.1</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m8.png" altimg-height="25px" altimg-valign="-7px" altimg-width="107px" alttext="\displaystyle F\left(1,1;2;z\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>;</mo><mn>2</mn><mo>;</mo><mi href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m3.png" altimg-height="28px" altimg-valign="-7px" altimg-width="166px" alttext="\displaystyle=-z^{-1}\ln\left(1-z\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo>-</mo><mrow><msup><mi href="./15.1#p1.t1.r2">z</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./15.2#E1" title="(15.2.1) ‣ §15.2(i) Gauss Series ‣ §15.2 Definitions and Analytical Properties ‣ Properties ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m16.png" altimg-height="23px" altimg-valign="-7px" altimg-width="103px" alttext="F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m18.png" altimg-height="39px" altimg-valign="-15px" altimg-width="91px" alttext="F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi></mfrac><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: <math class="ltx_Math" altimg="m15.png" altimg-height="23px" altimg-valign="-7px" altimg-width="139px" alttext="={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow></math>
Gauss’ hypergeometric function</a>,
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m25.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a> and
<a href="./15.1#p1.t1.r2" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./15.1#p1.t1.r2">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E2">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">15.4.2</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m10.png" altimg-height="30px" altimg-valign="-9px" altimg-width="124px" alttext="\displaystyle F\left(\tfrac{1}{2},1;\tfrac{3}{2};z^{2}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>,</mo><mn>1</mn><mo>;</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mo>;</mo><msup><mi href="./15.1#p1.t1.r2">z</mi><mn>2</mn></msup><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m4.png" altimg-height="53px" altimg-valign="-21px" altimg-width="163px" alttext="\displaystyle=\frac{1}{2z}\ln\left(\frac{1+z}{1-z}\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><mfrac><mn>1</mn><mrow><mn>2</mn><mo></mo><mi href="./15.1#p1.t1.r2">z</mi></mrow></mfrac></mstyle><mo></mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac><mrow><mn>1</mn><mo>+</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mrow><mn>1</mn><mo>-</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow></mfrac></mstyle><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./15.2#E1" title="(15.2.1) ‣ §15.2(i) Gauss Series ‣ §15.2 Definitions and Analytical Properties ‣ Properties ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m16.png" altimg-height="23px" altimg-valign="-7px" altimg-width="103px" alttext="F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m18.png" altimg-height="39px" altimg-valign="-15px" altimg-width="91px" alttext="F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi></mfrac><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: <math class="ltx_Math" altimg="m15.png" altimg-height="23px" altimg-valign="-7px" altimg-width="139px" alttext="={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow></math>
Gauss’ hypergeometric function</a>,
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m25.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a> and
<a href="./15.1#p1.t1.r2" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./15.1#p1.t1.r2">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E3">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">15.4.3</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m9.png" altimg-height="30px" altimg-valign="-9px" altimg-width="140px" alttext="\displaystyle F\left(\tfrac{1}{2},1;\tfrac{3}{2};-z^{2}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>,</mo><mn>1</mn><mo>;</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mo>;</mo><mrow><mo>-</mo><msup><mi href="./15.1#p1.t1.r2">z</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m7.png" altimg-height="27px" altimg-valign="-6px" altimg-width="136px" alttext="\displaystyle=z^{-1}\operatorname{arctan}z," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msup><mi href="./15.1#p1.t1.r2">z</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mrow><mi href="./4.23#SS2.p1">arctan</mi><mo></mo><mi href="./15.1#p1.t1.r2">z</mi></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./15.2#E1" title="(15.2.1) ‣ §15.2(i) Gauss Series ‣ §15.2 Definitions and Analytical Properties ‣ Properties ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m16.png" altimg-height="23px" altimg-valign="-7px" altimg-width="103px" alttext="F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m18.png" altimg-height="39px" altimg-valign="-15px" altimg-width="91px" alttext="F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi></mfrac><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: <math class="ltx_Math" altimg="m15.png" altimg-height="23px" altimg-valign="-7px" altimg-width="139px" alttext="={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow></math>
Gauss’ hypergeometric function</a>,
<a href="./4.23#SS2.p1" title="§4.23(ii) Principal Values ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m28.png" altimg-height="17px" altimg-valign="-2px" altimg-width="73px" alttext="\operatorname{arctan}\NVar{z}" display="inline"><mrow><mi href="./4.23#SS2.p1">arctan</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: arctangent function</a> and
<a href="./15.1#p1.t1.r2" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./15.1#p1.t1.r2">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E4">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">15.4.4</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m12.png" altimg-height="30px" altimg-valign="-9px" altimg-width="127px" alttext="\displaystyle F\left(\tfrac{1}{2},\tfrac{1}{2};\tfrac{3}{2};z^{2}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>,</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>;</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mo>;</mo><msup><mi href="./15.1#p1.t1.r2">z</mi><mn>2</mn></msup><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m6.png" altimg-height="27px" altimg-valign="-6px" altimg-width="132px" alttext="\displaystyle=z^{-1}\operatorname{arcsin}z," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msup><mi href="./15.1#p1.t1.r2">z</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mrow><mi href="./4.23#SS2.p1">arcsin</mi><mo></mo><mi href="./15.1#p1.t1.r2">z</mi></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E4.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./15.2#E1" title="(15.2.1) ‣ §15.2(i) Gauss Series ‣ §15.2 Definitions and Analytical Properties ‣ Properties ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m16.png" altimg-height="23px" altimg-valign="-7px" altimg-width="103px" alttext="F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m18.png" altimg-height="39px" altimg-valign="-15px" altimg-width="91px" alttext="F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi></mfrac><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: <math class="ltx_Math" altimg="m15.png" altimg-height="23px" altimg-valign="-7px" altimg-width="139px" alttext="={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow></math>
Gauss’ hypergeometric function</a>,
<a href="./4.23#SS2.p1" title="§4.23(ii) Principal Values ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m27.png" altimg-height="18px" altimg-valign="-2px" altimg-width="69px" alttext="\operatorname{arcsin}\NVar{z}" display="inline"><mrow><mi href="./4.23#SS2.p1">arcsin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: arcsine function</a> and
<a href="./15.1#p1.t1.r2" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./15.1#p1.t1.r2">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E5">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">15.4.5</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m11.png" altimg-height="30px" altimg-valign="-9px" altimg-width="143px" alttext="\displaystyle F\left(\tfrac{1}{2},\tfrac{1}{2};\tfrac{3}{2};-z^{2}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>,</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>;</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mo>;</mo><mrow><mo>-</mo><msup><mi href="./15.1#p1.t1.r2">z</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m5.png" altimg-height="41px" altimg-valign="-15px" altimg-width="222px" alttext="\displaystyle=z^{-1}\ln\left(z+\sqrt{1+z^{2}}\right)." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msup><mi href="./15.1#p1.t1.r2">z</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./15.1#p1.t1.r2">z</mi><mo>+</mo><msqrt><mrow><mn>1</mn><mo>+</mo><msup><mi href="./15.1#p1.t1.r2">z</mi><mn>2</mn></msup></mrow></msqrt></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E5.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./15.2#E1" title="(15.2.1) ‣ §15.2(i) Gauss Series ‣ §15.2 Definitions and Analytical Properties ‣ Properties ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m16.png" altimg-height="23px" altimg-valign="-7px" altimg-width="103px" alttext="F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m18.png" altimg-height="39px" altimg-valign="-15px" altimg-width="91px" alttext="F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi></mfrac><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: <math class="ltx_Math" altimg="m15.png" altimg-height="23px" altimg-valign="-7px" altimg-width="139px" alttext="={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow></math>
Gauss’ hypergeometric function</a>,
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m25.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a> and
<a href="./15.1#p1.t1.r2" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./15.1#p1.t1.r2">z</mi></math>: complex variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
<table id="E6" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex1" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">15.4.6</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m13.png" altimg-height="25px" altimg-valign="-7px" altimg-width="107px" alttext="\displaystyle F\left(a,b;a;z\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi href="./15.1#p1.t1.r3">a</mi><mo>;</mo><mi href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m2.png" altimg-height="28px" altimg-valign="-7px" altimg-width="113px" alttext="\displaystyle=(1-z)^{-b}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow></msup></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex2" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m14.png" altimg-height="25px" altimg-valign="-7px" altimg-width="105px" alttext="\displaystyle F\left(a,b;b;z\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m1.png" altimg-height="27px" altimg-valign="-7px" altimg-width="115px" alttext="\displaystyle=(1-z)^{-a}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi></mrow></msup></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E6.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./15.2#E1" title="(15.2.1) ‣ §15.2(i) Gauss Series ‣ §15.2 Definitions and Analytical Properties ‣ Properties ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m16.png" altimg-height="23px" altimg-valign="-7px" altimg-width="103px" alttext="F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m18.png" altimg-height="39px" altimg-valign="-15px" altimg-width="91px" alttext="F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi></mfrac><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: <math class="ltx_Math" altimg="m15.png" altimg-height="23px" altimg-valign="-7px" altimg-width="139px" alttext="={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow></math>
Gauss’ hypergeometric function</a>,
<a href="./15.1#p1.t1.r2" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./15.1#p1.t1.r2">z</mi></math>: complex variable</a>,
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m36.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./15.1#p1.t1.r3">a</mi></math>: real or complex parameter</a> and
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="b" display="inline"><mi href="./15.1#p1.t1.r3">b</mi></math>: real or complex parameter</a>
</dd>
<dt>Addition (effective with 1.0.16):</dt>
<dd>
This equation was expanded by adding the formula <math class="ltx_Math" altimg="m17.png" altimg-height="25px" altimg-valign="-7px" altimg-width="212px" alttext="F\left(a,b;a;z\right)=(1-z)^{-b}" display="inline"><mrow><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi href="./15.1#p1.t1.r3">a</mi><mo>;</mo><mi href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow></msup></mrow></math>.
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where the limit interpretation (.
</p>
<table id="E7" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">15.4.7</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E7.png" altimg-height="30px" altimg-valign="-9px" altimg-width="434px" alttext="F\left(a,\tfrac{1}{2}+a;\tfrac{1}{2};z^{2}\right)=\tfrac{1}{2}\left((1+z)^{-2a%
}+(1-z)^{-2a}\right)," display="block"><mrow><mrow><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo>+</mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>;</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo>;</mo><msup><mi href="./15.1#p1.t1.r2">z</mi><mn>2</mn></msup><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mrow><mo>(</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>+</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mrow><mn>2</mn><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow></mrow></msup><mo>+</mo><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mrow><mn>2</mn><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow></mrow></msup></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E7.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./15.2#E1" title="(15.2.1) ‣ §15.2(i) Gauss Series ‣ §15.2 Definitions and Analytical Properties ‣ Properties ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m16.png" altimg-height="23px" altimg-valign="-7px" altimg-width="103px" alttext="F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m18.png" altimg-height="39px" altimg-valign="-15px" altimg-width="91px" alttext="F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi></mfrac><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: <math class="ltx_Math" altimg="m15.png" altimg-height="23px" altimg-valign="-7px" altimg-width="139px" alttext="={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow></math>
Gauss’ hypergeometric function</a>,
<a href="./15.1#p1.t1.r2" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./15.1#p1.t1.r2">z</mi></math>: complex variable</a> and
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m36.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./15.1#p1.t1.r3">a</mi></math>: real or complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E8" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">15.4.8</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E8.png" altimg-height="30px" altimg-valign="-9px" altimg-width="392px" alttext="F\left(a,\tfrac{1}{2}+a;\tfrac{1}{2};-{\tan^{2}}z\right)=(\cos z)^{2a}\cos%
\left(2az\right)." display="block"><mrow><mrow><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo>+</mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>;</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo>;</mo><mrow><mo>-</mo><mrow><msup><mi href="./4.14#E4">tan</mi><mn>2</mn></msup><mo></mo><mi href="./15.1#p1.t1.r2">z</mi></mrow></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mn>2</mn><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow></msup><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><mi href="./15.1#p1.t1.r3">a</mi><mo></mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E8.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./15.2#E1" title="(15.2.1) ‣ §15.2(i) Gauss Series ‣ §15.2 Definitions and Analytical Properties ‣ Properties ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m16.png" altimg-height="23px" altimg-valign="-7px" altimg-width="103px" alttext="F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m18.png" altimg-height="39px" altimg-valign="-15px" altimg-width="91px" alttext="F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi></mfrac><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: <math class="ltx_Math" altimg="m15.png" altimg-height="23px" altimg-valign="-7px" altimg-width="139px" alttext="={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow></math>
Gauss’ hypergeometric function</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m24.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./4.14#E4" title="(4.14.4) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m33.png" altimg-height="17px" altimg-valign="-2px" altimg-width="47px" alttext="\tan\NVar{z}" display="inline"><mrow><mi href="./4.14#E4">tan</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: tangent function</a>,
<a href="./15.1#p1.t1.r2" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./15.1#p1.t1.r2">z</mi></math>: complex variable</a> and
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m36.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./15.1#p1.t1.r3">a</mi></math>: real or complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E9" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">15.4.9</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E9.png" altimg-height="51px" altimg-valign="-21px" altimg-width="522px" alttext="F\left(a,\tfrac{1}{2}+a;\tfrac{3}{2};z^{2}\right)=\frac{1}{(2-4a)z}\left((1+z)%
^{1-2a}-(1-z)^{1-2a}\right)," display="block"><mrow><mrow><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo>+</mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>;</mo><mstyle displaystyle="false"><mfrac><mn>3</mn><mn>2</mn></mfrac></mstyle><mo>;</mo><msup><mi href="./15.1#p1.t1.r2">z</mi><mn>2</mn></msup><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mfrac><mn>1</mn><mrow><mrow><mo stretchy="false">(</mo><mrow><mn>2</mn><mo>-</mo><mrow><mn>4</mn><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi href="./15.1#p1.t1.r2">z</mi></mrow></mfrac><mo></mo><mrow><mo>(</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>+</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>-</mo><mrow><mn>2</mn><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow></mrow></msup><mo>-</mo><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>-</mo><mrow><mn>2</mn><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow></mrow></msup></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E9.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./15.2#E1" title="(15.2.1) ‣ §15.2(i) Gauss Series ‣ §15.2 Definitions and Analytical Properties ‣ Properties ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m16.png" altimg-height="23px" altimg-valign="-7px" altimg-width="103px" alttext="F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m18.png" altimg-height="39px" altimg-valign="-15px" altimg-width="91px" alttext="F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi></mfrac><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: <math class="ltx_Math" altimg="m15.png" altimg-height="23px" altimg-valign="-7px" altimg-width="139px" alttext="={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow></math>
Gauss’ hypergeometric function</a>,
<a href="./15.1#p1.t1.r2" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./15.1#p1.t1.r2">z</mi></math>: complex variable</a> and
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m36.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./15.1#p1.t1.r3">a</mi></math>: real or complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E10" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">15.4.10</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E10.png" altimg-height="53px" altimg-valign="-21px" altimg-width="438px" alttext="F\left(a,\tfrac{1}{2}+a;\tfrac{3}{2};-{\tan^{2}}z\right)=(\cos z)^{2a}\frac{%
\sin\left((1-2a)z\right)}{(1-2a)\sin z}." display="block"><mrow><mrow><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo>+</mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>;</mo><mstyle displaystyle="false"><mfrac><mn>3</mn><mn>2</mn></mfrac></mstyle><mo>;</mo><mrow><mo>-</mo><mrow><msup><mi href="./4.14#E4">tan</mi><mn>2</mn></msup><mo></mo><mi href="./15.1#p1.t1.r2">z</mi></mrow></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mn>2</mn><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow></msup><mo></mo><mfrac><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><mrow><mn>2</mn><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo>)</mo></mrow></mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><mrow><mn>2</mn><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi href="./15.1#p1.t1.r2">z</mi></mrow></mrow></mfrac></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E10.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./15.2#E1" title="(15.2.1) ‣ §15.2(i) Gauss Series ‣ §15.2 Definitions and Analytical Properties ‣ Properties ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m16.png" altimg-height="23px" altimg-valign="-7px" altimg-width="103px" alttext="F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m18.png" altimg-height="39px" altimg-valign="-15px" altimg-width="91px" alttext="F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi></mfrac><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: <math class="ltx_Math" altimg="m15.png" altimg-height="23px" altimg-valign="-7px" altimg-width="139px" alttext="={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow></math>
Gauss’ hypergeometric function</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m24.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m32.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./4.14#E4" title="(4.14.4) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m33.png" altimg-height="17px" altimg-valign="-2px" altimg-width="47px" alttext="\tan\NVar{z}" display="inline"><mrow><mi href="./4.14#E4">tan</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: tangent function</a>,
<a href="./15.1#p1.t1.r2" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./15.1#p1.t1.r2">z</mi></math>: complex variable</a> and
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m36.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./15.1#p1.t1.r3">a</mi></math>: real or complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E11" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">15.4.11</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E11.png" altimg-height="53px" altimg-valign="-21px" altimg-width="549px" alttext="F\left(-a,a;\tfrac{1}{2};-z^{2}\right)=\tfrac{1}{2}\left(\left(\sqrt{1+z^{2}}+%
z\right)^{2a}+\left(\sqrt{1+z^{2}}-z\right)^{2a}\right)," display="block"><mrow><mrow><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>,</mo><mi href="./15.1#p1.t1.r3">a</mi><mo>;</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo>;</mo><mrow><mo>-</mo><msup><mi href="./15.1#p1.t1.r2">z</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mrow><mo>(</mo><mrow><msup><mrow><mo>(</mo><mrow><msqrt><mrow><mn>1</mn><mo>+</mo><msup><mi href="./15.1#p1.t1.r2">z</mi><mn>2</mn></msup></mrow></msqrt><mo>+</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo>)</mo></mrow><mrow><mn>2</mn><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow></msup><mo>+</mo><msup><mrow><mo>(</mo><mrow><msqrt><mrow><mn>1</mn><mo>+</mo><msup><mi href="./15.1#p1.t1.r2">z</mi><mn>2</mn></msup></mrow></msqrt><mo>-</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo>)</mo></mrow><mrow><mn>2</mn><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow></msup></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E11.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./15.2#E1" title="(15.2.1) ‣ §15.2(i) Gauss Series ‣ §15.2 Definitions and Analytical Properties ‣ Properties ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m16.png" altimg-height="23px" altimg-valign="-7px" altimg-width="103px" alttext="F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m18.png" altimg-height="39px" altimg-valign="-15px" altimg-width="91px" alttext="F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi></mfrac><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: <math class="ltx_Math" altimg="m15.png" altimg-height="23px" altimg-valign="-7px" altimg-width="139px" alttext="={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow></math>
Gauss’ hypergeometric function</a>,
<a href="./15.1#p1.t1.r2" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./15.1#p1.t1.r2">z</mi></math>: complex variable</a> and
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m36.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./15.1#p1.t1.r3">a</mi></math>: real or complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E12" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">15.4.12</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E12.png" altimg-height="30px" altimg-valign="-9px" altimg-width="274px" alttext="F\left(-a,a;\tfrac{1}{2};{\sin^{2}}z\right)=\cos\left(2az\right)." display="block"><mrow><mrow><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>,</mo><mi href="./15.1#p1.t1.r3">a</mi><mo>;</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo>;</mo><mrow><msup><mi href="./4.14#E1">sin</mi><mn>2</mn></msup><mo></mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><mi href="./15.1#p1.t1.r3">a</mi><mo></mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E12.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./15.2#E1" title="(15.2.1) ‣ §15.2(i) Gauss Series ‣ §15.2 Definitions and Analytical Properties ‣ Properties ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m16.png" altimg-height="23px" altimg-valign="-7px" altimg-width="103px" alttext="F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m18.png" altimg-height="39px" altimg-valign="-15px" altimg-width="91px" alttext="F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi></mfrac><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: <math class="ltx_Math" altimg="m15.png" altimg-height="23px" altimg-valign="-7px" altimg-width="139px" alttext="={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow></math>
Gauss’ hypergeometric function</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m24.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m32.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./15.1#p1.t1.r2" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./15.1#p1.t1.r2">z</mi></math>: complex variable</a> and
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m36.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./15.1#p1.t1.r3">a</mi></math>: real or complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E13" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">15.4.13</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E13.png" altimg-height="53px" altimg-valign="-21px" altimg-width="680px" alttext="F\left(a,1-a;\tfrac{1}{2};-z^{2}\right)=\frac{1}{2\sqrt{1+z^{2}}}\left(\left(%
\sqrt{1+z^{2}}+z\right)^{2a-1}+\left(\sqrt{1+z^{2}}-z\right)^{2a-1}\right)," display="block"><mrow><mrow><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mrow><mn>1</mn><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>;</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo>;</mo><mrow><mo>-</mo><msup><mi href="./15.1#p1.t1.r2">z</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mfrac><mn>1</mn><mrow><mn>2</mn><mo></mo><msqrt><mrow><mn>1</mn><mo>+</mo><msup><mi href="./15.1#p1.t1.r2">z</mi><mn>2</mn></msup></mrow></msqrt></mrow></mfrac><mo></mo><mrow><mo>(</mo><mrow><msup><mrow><mo>(</mo><mrow><msqrt><mrow><mn>1</mn><mo>+</mo><msup><mi href="./15.1#p1.t1.r2">z</mi><mn>2</mn></msup></mrow></msqrt><mo>+</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo>)</mo></mrow><mrow><mrow><mn>2</mn><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>-</mo><mn>1</mn></mrow></msup><mo>+</mo><msup><mrow><mo>(</mo><mrow><msqrt><mrow><mn>1</mn><mo>+</mo><msup><mi href="./15.1#p1.t1.r2">z</mi><mn>2</mn></msup></mrow></msqrt><mo>-</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo>)</mo></mrow><mrow><mrow><mn>2</mn><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>-</mo><mn>1</mn></mrow></msup></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E13.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./15.2#E1" title="(15.2.1) ‣ §15.2(i) Gauss Series ‣ §15.2 Definitions and Analytical Properties ‣ Properties ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m16.png" altimg-height="23px" altimg-valign="-7px" altimg-width="103px" alttext="F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m18.png" altimg-height="39px" altimg-valign="-15px" altimg-width="91px" alttext="F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi></mfrac><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: <math class="ltx_Math" altimg="m15.png" altimg-height="23px" altimg-valign="-7px" altimg-width="139px" alttext="={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow></math>
Gauss’ hypergeometric function</a>,
<a href="./15.1#p1.t1.r2" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./15.1#p1.t1.r2">z</mi></math>: complex variable</a> and
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m36.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./15.1#p1.t1.r3">a</mi></math>: real or complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E14" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">15.4.14</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E14.png" altimg-height="48px" altimg-valign="-16px" altimg-width="344px" alttext="F\left(a,1-a;\tfrac{1}{2};{\sin^{2}}z\right)=\frac{\cos\left((2a-1)z\right)}{%
\cos z}." display="block"><mrow><mrow><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mrow><mn>1</mn><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>;</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo>;</mo><mrow><msup><mi href="./4.14#E1">sin</mi><mn>2</mn></msup><mo></mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mfrac><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>2</mn><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo>)</mo></mrow></mrow><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi href="./15.1#p1.t1.r2">z</mi></mrow></mfrac></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E14.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./15.2#E1" title="(15.2.1) ‣ §15.2(i) Gauss Series ‣ §15.2 Definitions and Analytical Properties ‣ Properties ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m16.png" altimg-height="23px" altimg-valign="-7px" altimg-width="103px" alttext="F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m18.png" altimg-height="39px" altimg-valign="-15px" altimg-width="91px" alttext="F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi></mfrac><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: <math class="ltx_Math" altimg="m15.png" altimg-height="23px" altimg-valign="-7px" altimg-width="139px" alttext="={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow></math>
Gauss’ hypergeometric function</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m24.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m32.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./15.1#p1.t1.r2" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./15.1#p1.t1.r2">z</mi></math>: complex variable</a> and
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m36.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./15.1#p1.t1.r3">a</mi></math>: real or complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E15" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">15.4.15</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E15.png" altimg-height="53px" altimg-valign="-21px" altimg-width="681px" alttext="F\left(a,1-a;\tfrac{3}{2};-z^{2}\right)=\frac{1}{(2-4a)z}\left(\left(\sqrt{1+z%
^{2}}+z\right)^{1-2a}-\left(\sqrt{1+z^{2}}-z\right)^{1-2a}\right)," display="block"><mrow><mrow><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mrow><mn>1</mn><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>;</mo><mstyle displaystyle="false"><mfrac><mn>3</mn><mn>2</mn></mfrac></mstyle><mo>;</mo><mrow><mo>-</mo><msup><mi href="./15.1#p1.t1.r2">z</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mfrac><mn>1</mn><mrow><mrow><mo stretchy="false">(</mo><mrow><mn>2</mn><mo>-</mo><mrow><mn>4</mn><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi href="./15.1#p1.t1.r2">z</mi></mrow></mfrac><mo></mo><mrow><mo>(</mo><mrow><msup><mrow><mo>(</mo><mrow><msqrt><mrow><mn>1</mn><mo>+</mo><msup><mi href="./15.1#p1.t1.r2">z</mi><mn>2</mn></msup></mrow></msqrt><mo>+</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo>)</mo></mrow><mrow><mn>1</mn><mo>-</mo><mrow><mn>2</mn><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow></mrow></msup><mo>-</mo><msup><mrow><mo>(</mo><mrow><msqrt><mrow><mn>1</mn><mo>+</mo><msup><mi href="./15.1#p1.t1.r2">z</mi><mn>2</mn></msup></mrow></msqrt><mo>-</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo>)</mo></mrow><mrow><mn>1</mn><mo>-</mo><mrow><mn>2</mn><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow></mrow></msup></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E15.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./15.2#E1" title="(15.2.1) ‣ §15.2(i) Gauss Series ‣ §15.2 Definitions and Analytical Properties ‣ Properties ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m16.png" altimg-height="23px" altimg-valign="-7px" altimg-width="103px" alttext="F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m18.png" altimg-height="39px" altimg-valign="-15px" altimg-width="91px" alttext="F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi></mfrac><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: <math class="ltx_Math" altimg="m15.png" altimg-height="23px" altimg-valign="-7px" altimg-width="139px" alttext="={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow></math>
Gauss’ hypergeometric function</a>,
<a href="./15.1#p1.t1.r2" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./15.1#p1.t1.r2">z</mi></math>: complex variable</a> and
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m36.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./15.1#p1.t1.r3">a</mi></math>: real or complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E16" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">15.4.16</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E16.png" altimg-height="53px" altimg-valign="-21px" altimg-width="342px" alttext="F\left(a,1-a;\tfrac{3}{2};{\sin^{2}}z\right)=\frac{\sin\left((2a-1)z\right)}{(%
2a-1)\sin z}." display="block"><mrow><mrow><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mrow><mn>1</mn><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>;</mo><mstyle displaystyle="false"><mfrac><mn>3</mn><mn>2</mn></mfrac></mstyle><mo>;</mo><mrow><msup><mi href="./4.14#E1">sin</mi><mn>2</mn></msup><mo></mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mfrac><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>2</mn><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo>)</mo></mrow></mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>2</mn><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi href="./15.1#p1.t1.r2">z</mi></mrow></mrow></mfrac></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E16.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./15.2#E1" title="(15.2.1) ‣ §15.2(i) Gauss Series ‣ §15.2 Definitions and Analytical Properties ‣ Properties ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m16.png" altimg-height="23px" altimg-valign="-7px" altimg-width="103px" alttext="F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m18.png" altimg-height="39px" altimg-valign="-15px" altimg-width="91px" alttext="F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi></mfrac><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: <math class="ltx_Math" altimg="m15.png" altimg-height="23px" altimg-valign="-7px" altimg-width="139px" alttext="={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow></math>
Gauss’ hypergeometric function</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m32.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./15.1#p1.t1.r2" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./15.1#p1.t1.r2">z</mi></math>: complex variable</a> and
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m36.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./15.1#p1.t1.r3">a</mi></math>: real or complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E17" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">15.4.17</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E17.png" altimg-height="36px" altimg-valign="-9px" altimg-width="385px" alttext="F\left(a,\tfrac{1}{2}+a;1+2a;z\right)=\left(\tfrac{1}{2}+\tfrac{1}{2}\sqrt{1-z%
}\right)^{-2a}," display="block"><mrow><mrow><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo>+</mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>;</mo><mrow><mn>1</mn><mo>+</mo><mrow><mn>2</mn><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow></mrow><mo>;</mo><mi href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow><mo>=</mo><msup><mrow><mo>(</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo>+</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><msqrt><mrow><mn>1</mn><mo>-</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow></msqrt></mrow></mrow><mo>)</mo></mrow><mrow><mo>-</mo><mrow><mn>2</mn><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow></mrow></msup></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E17.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./15.2#E1" title="(15.2.1) ‣ §15.2(i) Gauss Series ‣ §15.2 Definitions and Analytical Properties ‣ Properties ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m16.png" altimg-height="23px" altimg-valign="-7px" altimg-width="103px" alttext="F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m18.png" altimg-height="39px" altimg-valign="-15px" altimg-width="91px" alttext="F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi></mfrac><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: <math class="ltx_Math" altimg="m15.png" altimg-height="23px" altimg-valign="-7px" altimg-width="139px" alttext="={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow></math>
Gauss’ hypergeometric function</a>,
<a href="./15.1#p1.t1.r2" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./15.1#p1.t1.r2">z</mi></math>: complex variable</a> and
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m36.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./15.1#p1.t1.r3">a</mi></math>: real or complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E18" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">15.4.18</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E18.png" altimg-height="50px" altimg-valign="-21px" altimg-width="427px" alttext="F\left(a,\tfrac{1}{2}+a;2a;z\right)=\frac{1}{\sqrt{1-z}}\left(\tfrac{1}{2}+%
\tfrac{1}{2}\sqrt{1-z}\right)^{1-2a}," display="block"><mrow><mrow><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo>+</mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>;</mo><mrow><mn>2</mn><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>;</mo><mi href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mfrac><mn>1</mn><msqrt><mrow><mn>1</mn><mo>-</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow></msqrt></mfrac><mo></mo><msup><mrow><mo>(</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo>+</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><msqrt><mrow><mn>1</mn><mo>-</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow></msqrt></mrow></mrow><mo>)</mo></mrow><mrow><mn>1</mn><mo>-</mo><mrow><mn>2</mn><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow></mrow></msup></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E18.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./15.2#E1" title="(15.2.1) ‣ §15.2(i) Gauss Series ‣ §15.2 Definitions and Analytical Properties ‣ Properties ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m16.png" altimg-height="23px" altimg-valign="-7px" altimg-width="103px" alttext="F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m18.png" altimg-height="39px" altimg-valign="-15px" altimg-width="91px" alttext="F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi></mfrac><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: <math class="ltx_Math" altimg="m15.png" altimg-height="23px" altimg-valign="-7px" altimg-width="139px" alttext="={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow></math>
Gauss’ hypergeometric function</a>,
<a href="./15.1#p1.t1.r2" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./15.1#p1.t1.r2">z</mi></math>: complex variable</a> and
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m36.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./15.1#p1.t1.r3">a</mi></math>: real or complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E19" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">15.4.19</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E19.png" altimg-height="28px" altimg-valign="-7px" altimg-width="436px" alttext="F\left(a+1,b;a;z\right)=\left(1-(1-(\ifrac{b}{a}))z\right)(1-z)^{-1-b}." display="block"><mrow><mrow><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>+</mo><mn>1</mn></mrow><mo>,</mo><mi href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi href="./15.1#p1.t1.r3">a</mi><mo>;</mo><mi href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./15.1#p1.t1.r3">b</mi><mo>/</mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi href="./15.1#p1.t1.r2">z</mi></mrow></mrow><mo>)</mo></mrow><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mrow><mo>-</mo><mn>1</mn></mrow><mo>-</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow></msup></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E19.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./15.2#E1" title="(15.2.1) ‣ §15.2(i) Gauss Series ‣ §15.2 Definitions and Analytical Properties ‣ Properties ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m16.png" altimg-height="23px" altimg-valign="-7px" altimg-width="103px" alttext="F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m18.png" altimg-height="39px" altimg-valign="-15px" altimg-width="91px" alttext="F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi></mfrac><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: <math class="ltx_Math" altimg="m15.png" altimg-height="23px" altimg-valign="-7px" altimg-width="139px" alttext="={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow></math>
Gauss’ hypergeometric function</a>,
<a href="./15.1#p1.t1.r2" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./15.1#p1.t1.r2">z</mi></math>: complex variable</a>,
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m36.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./15.1#p1.t1.r3">a</mi></math>: real or complex parameter</a> and
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="b" display="inline"><mi href="./15.1#p1.t1.r3">b</mi></math>: real or complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">In (</dd>
</dl>
</div>
</div>
<div id="SS2.p1" class="ltx_para">
<p class="ltx_p">If <math class="ltx_Math" altimg="m23.png" altimg-height="23px" altimg-valign="-7px" altimg-width="147px" alttext="\Re(c-a-b)>0" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./15.1#p1.t1.r3">c</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>></mo><mn>0</mn></mrow></math>, then</p>
<table id="E20" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">15.4.20</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E20.png" altimg-height="53px" altimg-valign="-21px" altimg-width="292px" alttext="F\left(a,b;c;1\right)=\frac{\Gamma\left(c\right)\Gamma\left(c-a-b\right)}{%
\Gamma\left(c-a\right)\Gamma\left(c-b\right)}." display="block"><mrow><mrow><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mn>1</mn><mo>)</mo></mrow></mrow><mo>=</mo><mfrac><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi href="./15.1#p1.t1.r3">c</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./15.1#p1.t1.r3">c</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mo>)</mo></mrow></mrow></mrow><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./15.1#p1.t1.r3">c</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./15.1#p1.t1.r3">c</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mo>)</mo></mrow></mrow></mrow></mfrac></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E20.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./15.2#E1" title="(15.2.1) ‣ §15.2(i) Gauss Series ‣ §15.2 Definitions and Analytical Properties ‣ Properties ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m16.png" altimg-height="23px" altimg-valign="-7px" altimg-width="103px" alttext="F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m18.png" altimg-height="39px" altimg-valign="-15px" altimg-width="91px" alttext="F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi></mfrac><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: <math class="ltx_Math" altimg="m15.png" altimg-height="23px" altimg-valign="-7px" altimg-width="139px" alttext="={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow></math>
Gauss’ hypergeometric function</a>,
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m36.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./15.1#p1.t1.r3">a</mi></math>: real or complex parameter</a>,
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="b" display="inline"><mi href="./15.1#p1.t1.r3">b</mi></math>: real or complex parameter</a> and
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="c" display="inline"><mi href="./15.1#p1.t1.r3">c</mi></math>: real or complex parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">15.1.20</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS2.p2" class="ltx_para">
<p class="ltx_p">If <math class="ltx_Math" altimg="m38.png" altimg-height="19px" altimg-valign="-4px" altimg-width="83px" alttext="c=a+b" display="inline"><mrow><mi href="./15.1#p1.t1.r3">c</mi><mo>=</mo><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>+</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow></mrow></math>, then</p>
<table id="E21" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">15.4.21</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E21.png" altimg-height="53px" altimg-valign="-21px" altimg-width="319px" alttext="\lim_{z\to 1-}\frac{F\left(a,b;a+b;z\right)}{-\ln\left(1-z\right)}=\frac{%
\Gamma\left(a+b\right)}{\Gamma\left(a\right)\Gamma\left(b\right)}." display="block"><mrow><mrow><mrow><munder><mo movablelimits="false">lim</mo><mrow><mi href="./15.1#p1.t1.r2">z</mi><mo>→</mo><mrow><mn>1</mn><mo>-</mo></mrow></mrow></munder><mo></mo><mfrac><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>+</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mo>;</mo><mi href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow><mrow><mo>-</mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo>)</mo></mrow></mrow></mrow></mfrac></mrow><mo>=</mo><mfrac><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>+</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mo>)</mo></mrow></mrow><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi href="./15.1#p1.t1.r3">a</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi href="./15.1#p1.t1.r3">b</mi><mo>)</mo></mrow></mrow></mrow></mfrac></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E21.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./15.2#E1" title="(15.2.1) ‣ §15.2(i) Gauss Series ‣ §15.2 Definitions and Analytical Properties ‣ Properties ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m16.png" altimg-height="23px" altimg-valign="-7px" altimg-width="103px" alttext="F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m18.png" altimg-height="39px" altimg-valign="-15px" altimg-width="91px" alttext="F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi></mfrac><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: <math class="ltx_Math" altimg="m15.png" altimg-height="23px" altimg-valign="-7px" altimg-width="139px" alttext="={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow></math>
Gauss’ hypergeometric function</a>,
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m25.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a>,
<a href="./15.1#p1.t1.r2" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./15.1#p1.t1.r2">z</mi></math>: complex variable</a>,
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m36.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./15.1#p1.t1.r3">a</mi></math>: real or complex parameter</a> and
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="b" display="inline"><mi href="./15.1#p1.t1.r3">b</mi></math>: real or complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS2.p3" class="ltx_para">
<p class="ltx_p">If <math class="ltx_Math" altimg="m22.png" altimg-height="23px" altimg-valign="-7px" altimg-width="147px" alttext="\Re(c-a-b)=0" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./15.1#p1.t1.r3">c</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mn>0</mn></mrow></math> and <math class="ltx_Math" altimg="m40.png" altimg-height="21px" altimg-valign="-6px" altimg-width="83px" alttext="c\neq a+b" display="inline"><mrow><mi href="./15.1#p1.t1.r3">c</mi><mo>≠</mo><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>+</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow></mrow></math>, then</p>
<table id="E22" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">15.4.22</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E22.png" altimg-height="53px" altimg-valign="-21px" altimg-width="663px" alttext="\lim_{z\to 1-}(1-z)^{a+b-c}\left(F\left(a,b;c;z\right)-\frac{\Gamma\left(c%
\right)\Gamma\left(c-a-b\right)}{\Gamma\left(c-a\right)\Gamma\left(c-b\right)}%
\right)=\frac{\Gamma\left(c\right)\Gamma\left(a+b-c\right)}{\Gamma\left(a%
\right)\Gamma\left(b\right)}." display="block"><mrow><mrow><mrow><munder><mo movablelimits="false">lim</mo><mrow><mi href="./15.1#p1.t1.r2">z</mi><mo>→</mo><mrow><mn>1</mn><mo>-</mo></mrow></mrow></munder><mo></mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>+</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mo>-</mo><mi href="./15.1#p1.t1.r3">c</mi></mrow></msup><mo></mo><mrow><mo>(</mo><mrow><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow><mo>-</mo><mfrac><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi href="./15.1#p1.t1.r3">c</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./15.1#p1.t1.r3">c</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mo>)</mo></mrow></mrow></mrow><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./15.1#p1.t1.r3">c</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./15.1#p1.t1.r3">c</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mo>)</mo></mrow></mrow></mrow></mfrac></mrow><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mfrac><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi href="./15.1#p1.t1.r3">c</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>+</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mo>-</mo><mi href="./15.1#p1.t1.r3">c</mi></mrow><mo>)</mo></mrow></mrow></mrow><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi href="./15.1#p1.t1.r3">a</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi href="./15.1#p1.t1.r3">b</mi><mo>)</mo></mrow></mrow></mrow></mfrac></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E22.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./15.2#E1" title="(15.2.1) ‣ §15.2(i) Gauss Series ‣ §15.2 Definitions and Analytical Properties ‣ Properties ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m16.png" altimg-height="23px" altimg-valign="-7px" altimg-width="103px" alttext="F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m18.png" altimg-height="39px" altimg-valign="-15px" altimg-width="91px" alttext="F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi></mfrac><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: <math class="ltx_Math" altimg="m15.png" altimg-height="23px" altimg-valign="-7px" altimg-width="139px" alttext="={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow></math>
Gauss’ hypergeometric function</a>,
<a href="./15.1#p1.t1.r2" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./15.1#p1.t1.r2">z</mi></math>: complex variable</a>,
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m36.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./15.1#p1.t1.r3">a</mi></math>: real or complex parameter</a>,
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="b" display="inline"><mi href="./15.1#p1.t1.r3">b</mi></math>: real or complex parameter</a> and
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="c" display="inline"><mi href="./15.1#p1.t1.r3">c</mi></math>: real or complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS2.p4" class="ltx_para">
<p class="ltx_p">If <math class="ltx_Math" altimg="m21.png" altimg-height="23px" altimg-valign="-7px" altimg-width="147px" alttext="\Re(c-a-b)<0" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./15.1#p1.t1.r3">c</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo><</mo><mn>0</mn></mrow></math>, then</p>
<table id="E23" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">15.4.23</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E23.png" altimg-height="53px" altimg-valign="-21px" altimg-width="360px" alttext="\lim_{z\to 1-}\frac{F\left(a,b;c;z\right)}{(1-z)^{c-a-b}}=\frac{\Gamma\left(c%
\right)\Gamma\left(a+b-c\right)}{\Gamma\left(a\right)\Gamma\left(b\right)}." display="block"><mrow><mrow><mrow><munder><mo movablelimits="false">lim</mo><mrow><mi href="./15.1#p1.t1.r2">z</mi><mo>→</mo><mrow><mn>1</mn><mo>-</mo></mrow></mrow></munder><mo></mo><mfrac><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><mi href="./15.1#p1.t1.r2">z</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mi href="./15.1#p1.t1.r3">c</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow></msup></mfrac></mrow><mo>=</mo><mfrac><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi href="./15.1#p1.t1.r3">c</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>+</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mo>-</mo><mi href="./15.1#p1.t1.r3">c</mi></mrow><mo>)</mo></mrow></mrow></mrow><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi href="./15.1#p1.t1.r3">a</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi href="./15.1#p1.t1.r3">b</mi><mo>)</mo></mrow></mrow></mrow></mfrac></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E23.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./15.2#E1" title="(15.2.1) ‣ §15.2(i) Gauss Series ‣ §15.2 Definitions and Analytical Properties ‣ Properties ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m16.png" altimg-height="23px" altimg-valign="-7px" altimg-width="103px" alttext="F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m18.png" altimg-height="39px" altimg-valign="-15px" altimg-width="91px" alttext="F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi></mfrac><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: <math class="ltx_Math" altimg="m15.png" altimg-height="23px" altimg-valign="-7px" altimg-width="139px" alttext="={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow></math>
Gauss’ hypergeometric function</a>,
<a href="./15.1#p1.t1.r2" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./15.1#p1.t1.r2">z</mi></math>: complex variable</a>,
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m36.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./15.1#p1.t1.r3">a</mi></math>: real or complex parameter</a>,
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="b" display="inline"><mi href="./15.1#p1.t1.r3">b</mi></math>: real or complex parameter</a> and
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="c" display="inline"><mi href="./15.1#p1.t1.r3">c</mi></math>: real or complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<section id="Px1" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Chu–Vandermonde Identity</h3>
<div id="Px1.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px1.p1" class="ltx_para">
<table id="E24" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">15.4.24</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E24.png" altimg-height="55px" altimg-valign="-22px" altimg-width="226px" alttext="F\left(-n,b;c;1\right)=\frac{{\left(c-b\right)_{n}}}{{\left(c\right)_{n}}}," display="block"><mrow><mrow><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mi href="./15.1#p1.t1.r4">n</mi></mrow><mo>,</mo><mi href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mn>1</mn><mo>)</mo></mrow></mrow><mo>=</mo><mfrac><msub><mrow><mo href="./5.2#iii">(</mo><mrow><mi href="./15.1#p1.t1.r3">c</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mo href="./5.2#iii">)</mo></mrow><mi href="./15.1#p1.t1.r4">n</mi></msub><msub><mrow><mo href="./5.2#iii">(</mo><mi href="./15.1#p1.t1.r3">c</mi><mo href="./5.2#iii">)</mo></mrow><mi href="./15.1#p1.t1.r4">n</mi></msub></mfrac></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m42.png" altimg-height="20px" altimg-valign="-6px" altimg-width="123px" alttext="n=0,1,2,\dots" display="inline"><mrow><mi href="./15.1#p1.t1.r4">n</mi><mo>=</mo><mrow><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E24.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./15.2#E1" title="(15.2.1) ‣ §15.2(i) Gauss Series ‣ §15.2 Definitions and Analytical Properties ‣ Properties ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m16.png" altimg-height="23px" altimg-valign="-7px" altimg-width="103px" alttext="F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m18.png" altimg-height="39px" altimg-valign="-15px" altimg-width="91px" alttext="F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi></mfrac><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: <math class="ltx_Math" altimg="m15.png" altimg-height="23px" altimg-valign="-7px" altimg-width="139px" alttext="={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow></math>
Gauss’ hypergeometric function</a>,
<a href="./5.2#iii" title="§5.2(iii) Pochhammer’s Symbol ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m46.png" altimg-height="24px" altimg-valign="-8px" altimg-width="41px" alttext="{\left(\NVar{a}\right)_{\NVar{n}}}" display="inline"><msub><mrow><mo href="./5.2#iii">(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r5">a</mi><mo href="./5.2#iii">)</mo></mrow><mi class="ltx_nvar" href="./5.1#p2.t1.r1">n</mi></msub></math>: Pochhammer’s symbol (or shifted factorial)</a>,
<a href="./15.1#p1.t1.r4" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./15.1#p1.t1.r4">n</mi></math>: integer</a>,
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="b" display="inline"><mi href="./15.1#p1.t1.r3">b</mi></math>: real or complex parameter</a> and
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="c" display="inline"><mi href="./15.1#p1.t1.r3">c</mi></math>: real or complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px2" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Dougall’s Bilateral Sum</h3>
<div id="Px2.info" class="ltx_metadata ltx_info">
). If <math class="ltx_Math" altimg="m34.png" altimg-height="21px" altimg-valign="-6px" altimg-width="32px" alttext="a,b" display="inline"><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow></math> are not integers and
<math class="ltx_Math" altimg="m20.png" altimg-height="23px" altimg-valign="-7px" altimg-width="182px" alttext="\Re(c+d-a-b)>1" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mi href="./15.1#p1.t1.r3">c</mi><mo>+</mo><mi>d</mi></mrow><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>></mo><mn>1</mn></mrow></math>, then</p>
<table id="E25" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">15.4.25</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E25.png" altimg-height="65px" altimg-valign="-29px" altimg-width="713px" alttext="\sum_{n=-\infty}^{\infty}\frac{\Gamma\left(a+n\right)\Gamma\left(b+n\right)}{%
\Gamma\left(c+n\right)\Gamma\left(d+n\right)}=\frac{\pi^{2}}{\sin\left(\pi a%
\right)\sin\left(\pi b\right)}\*\frac{\Gamma\left(c+d-a-b-1\right)}{\Gamma%
\left(c-a\right)\Gamma\left(d-a\right)\Gamma\left(c-b\right)\Gamma\left(d-b%
\right)}." display="block"><mrow><mrow><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./15.1#p1.t1.r4">n</mi><mo>=</mo><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow></mrow><mi mathvariant="normal">∞</mi></munderover><mfrac><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>+</mo><mi href="./15.1#p1.t1.r4">n</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./15.1#p1.t1.r3">b</mi><mo>+</mo><mi href="./15.1#p1.t1.r4">n</mi></mrow><mo>)</mo></mrow></mrow></mrow><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./15.1#p1.t1.r3">c</mi><mo>+</mo><mi href="./15.1#p1.t1.r4">n</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi>d</mi><mo>+</mo><mi href="./15.1#p1.t1.r4">n</mi></mrow><mo>)</mo></mrow></mrow></mrow></mfrac></mrow><mo>=</mo><mrow><mfrac><msup><mi href="./3.12#E1">π</mi><mn>2</mn></msup><mrow><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mo>)</mo></mrow></mrow></mrow></mfrac><mo></mo><mfrac><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mi href="./15.1#p1.t1.r3">c</mi><mo>+</mo><mi>d</mi></mrow><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">b</mi><mo>-</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./15.1#p1.t1.r3">c</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi>d</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./15.1#p1.t1.r3">c</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi>d</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mo>)</mo></mrow></mrow></mrow></mfrac></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E25.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m32.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./15.1#p1.t1.r4" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./15.1#p1.t1.r4">n</mi></math>: integer</a>,
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m36.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./15.1#p1.t1.r3">a</mi></math>: real or complex parameter</a>,
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="b" display="inline"><mi href="./15.1#p1.t1.r3">b</mi></math>: real or complex parameter</a> and
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="c" display="inline"><mi href="./15.1#p1.t1.r3">c</mi></math>: real or complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
</section>
<section id="iii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§15.4(iii) </span>Other Arguments</h2>
<div id="SS3.info" class="ltx_metadata ltx_info">
) let <math class="ltx_Math" altimg="m45.png" altimg-height="17px" altimg-valign="-4px" altimg-width="81px" alttext="z\to-\infty" display="inline"><mrow><mi href="./15.1#p1.t1.r2">z</mi><mo>→</mo><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow></mrow></math> in (</dd>
</dl>
</div>
</div>
<div id="SS3.p1" class="ltx_para">
<table id="E26" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">15.4.26</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E26.png" altimg-height="59px" altimg-valign="-24px" altimg-width="430px" alttext="F\left(a,b;a-b+1;-1\right)=\frac{\Gamma\left(a-b+1\right)\Gamma\left(\tfrac{1}%
{2}a+1\right)}{\Gamma\left(a+1\right)\Gamma\left(\tfrac{1}{2}a-b+1\right)}." display="block"><mrow><mrow><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mrow><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mo>+</mo><mn>1</mn></mrow><mo>;</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mfrac><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mrow><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>-</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mrow></mfrac></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E26.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./15.2#E1" title="(15.2.1) ‣ §15.2(i) Gauss Series ‣ §15.2 Definitions and Analytical Properties ‣ Properties ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m16.png" altimg-height="23px" altimg-valign="-7px" altimg-width="103px" alttext="F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m18.png" altimg-height="39px" altimg-valign="-15px" altimg-width="91px" alttext="F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi></mfrac><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: <math class="ltx_Math" altimg="m15.png" altimg-height="23px" altimg-valign="-7px" altimg-width="139px" alttext="={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow></math>
Gauss’ hypergeometric function</a>,
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m36.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./15.1#p1.t1.r3">a</mi></math>: real or complex parameter</a> and
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="b" display="inline"><mi href="./15.1#p1.t1.r3">b</mi></math>: real or complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E27" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">15.4.27</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E27.png" altimg-height="29px" altimg-valign="-9px" altimg-width="417px" alttext="F\left(1,a;a+1;-1\right)=\tfrac{1}{2}a\left(\psi\left(\tfrac{1}{2}a+\tfrac{1}{%
2}\right)-\psi\left(\tfrac{1}{2}a\right)\right)." display="block"><mrow><mrow><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mn>1</mn><mo>,</mo><mi href="./15.1#p1.t1.r3">a</mi><mo>;</mo><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>+</mo><mn>1</mn></mrow><mo>;</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./15.1#p1.t1.r3">a</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>+</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle></mrow><mo>)</mo></mrow></mrow><mo>-</mo><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E27.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./15.2#E1" title="(15.2.1) ‣ §15.2(i) Gauss Series ‣ §15.2 Definitions and Analytical Properties ‣ Properties ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m16.png" altimg-height="23px" altimg-valign="-7px" altimg-width="103px" alttext="F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m18.png" altimg-height="39px" altimg-valign="-15px" altimg-width="91px" alttext="F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi></mfrac><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: <math class="ltx_Math" altimg="m15.png" altimg-height="23px" altimg-valign="-7px" altimg-width="139px" alttext="={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow></math>
Gauss’ hypergeometric function</a>,
<a href="./5.2#E2" title="(5.2.2) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m31.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="\psi\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: psi (or digamma) function</a> and
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m36.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./15.1#p1.t1.r3">a</mi></math>: real or complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E28" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">15.4.28</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E28.png" altimg-height="59px" altimg-valign="-24px" altimg-width="463px" alttext="F\left(a,b;\tfrac{1}{2}a+\tfrac{1}{2}b+\tfrac{1}{2};\tfrac{1}{2}\right)=\sqrt{%
\pi}\frac{\Gamma\left(\tfrac{1}{2}a+\tfrac{1}{2}b+\tfrac{1}{2}\right)}{\Gamma%
\left(\tfrac{1}{2}a+\tfrac{1}{2}\right)\Gamma\left(\tfrac{1}{2}b+\tfrac{1}{2}%
\right)}." display="block"><mrow><mrow><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mrow><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>+</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mo>+</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle></mrow><mo>;</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msqrt><mi href="./3.12#E1">π</mi></msqrt><mo></mo><mfrac><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>+</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow></mrow></mfrac></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E28.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./15.2#E1" title="(15.2.1) ‣ §15.2(i) Gauss Series ‣ §15.2 Definitions and Analytical Properties ‣ Properties ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m16.png" altimg-height="23px" altimg-valign="-7px" altimg-width="103px" alttext="F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m18.png" altimg-height="39px" altimg-valign="-15px" altimg-width="91px" alttext="F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi></mfrac><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: <math class="ltx_Math" altimg="m15.png" altimg-height="23px" altimg-valign="-7px" altimg-width="139px" alttext="={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow></math>
Gauss’ hypergeometric function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m36.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./15.1#p1.t1.r3">a</mi></math>: real or complex parameter</a> and
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="b" display="inline"><mi href="./15.1#p1.t1.r3">b</mi></math>: real or complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E29" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">15.4.29</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E29.png" altimg-height="65px" altimg-valign="-27px" altimg-width="804px" alttext="F\left(a,b;\tfrac{1}{2}a+\tfrac{1}{2}b+1;\tfrac{1}{2}\right)=\frac{2\sqrt{\pi}%
}{a-b}\Gamma\left(\tfrac{1}{2}a+\tfrac{1}{2}b+1\right)\*\left(\frac{1}{\Gamma%
\left(\tfrac{1}{2}a\right)\Gamma\left(\tfrac{1}{2}b+\tfrac{1}{2}\right)}-\frac%
{1}{\Gamma\left(\tfrac{1}{2}a+\tfrac{1}{2}\right)\Gamma\left(\tfrac{1}{2}b%
\right)}\right)." display="block"><mrow><mrow><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mrow><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>+</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mo>+</mo><mn>1</mn></mrow><mo>;</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mfrac><mrow><mn>2</mn><mo></mo><msqrt><mi href="./3.12#E1">π</mi></msqrt></mrow><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>-</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow></mfrac><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>+</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow></mrow></mfrac><mo>-</mo><mfrac><mn>1</mn><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mo>)</mo></mrow></mrow></mrow></mfrac></mrow><mo>)</mo></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E29.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./15.2#E1" title="(15.2.1) ‣ §15.2(i) Gauss Series ‣ §15.2 Definitions and Analytical Properties ‣ Properties ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m16.png" altimg-height="23px" altimg-valign="-7px" altimg-width="103px" alttext="F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m18.png" altimg-height="39px" altimg-valign="-15px" altimg-width="91px" alttext="F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi></mfrac><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: <math class="ltx_Math" altimg="m15.png" altimg-height="23px" altimg-valign="-7px" altimg-width="139px" alttext="={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow></math>
Gauss’ hypergeometric function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m36.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./15.1#p1.t1.r3">a</mi></math>: real or complex parameter</a> and
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="b" display="inline"><mi href="./15.1#p1.t1.r3">b</mi></math>: real or complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E30" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">15.4.30</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E30.png" altimg-height="58px" altimg-valign="-24px" altimg-width="430px" alttext="F\left(a,1-a;b;\tfrac{1}{2}\right)=\frac{2^{1-b}\sqrt{\pi}\Gamma\left(b\right)%
}{\Gamma\left(\tfrac{1}{2}a+\tfrac{1}{2}b\right)\Gamma\left(\tfrac{1}{2}b-%
\tfrac{1}{2}a+\tfrac{1}{2}\right)}." display="block"><mrow><mrow><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mrow><mn>1</mn><mo>-</mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>;</mo><mi href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo>)</mo></mrow></mrow><mo>=</mo><mfrac><mrow><msup><mn>2</mn><mrow><mn>1</mn><mo>-</mo><mi href="./15.1#p1.t1.r3">b</mi></mrow></msup><mo></mo><msqrt><mi href="./3.12#E1">π</mi></msqrt><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi href="./15.1#p1.t1.r3">b</mi><mo>)</mo></mrow></mrow></mrow><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>+</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./15.1#p1.t1.r3">b</mi></mrow></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./15.1#p1.t1.r3">b</mi></mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow></mrow><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow></mrow></mfrac></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E30.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./15.2#E1" title="(15.2.1) ‣ §15.2(i) Gauss Series ‣ §15.2 Definitions and Analytical Properties ‣ Properties ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m16.png" altimg-height="23px" altimg-valign="-7px" altimg-width="103px" alttext="F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m18.png" altimg-height="39px" altimg-valign="-15px" altimg-width="91px" alttext="F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi></mfrac><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: <math class="ltx_Math" altimg="m15.png" altimg-height="23px" altimg-valign="-7px" altimg-width="139px" alttext="={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow></math>
Gauss’ hypergeometric function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m36.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./15.1#p1.t1.r3">a</mi></math>: real or complex parameter</a> and
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="b" display="inline"><mi href="./15.1#p1.t1.r3">b</mi></math>: real or complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E31" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">15.4.31</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E31.png" altimg-height="60px" altimg-valign="-24px" altimg-width="463px" alttext="F\left(a,\tfrac{1}{2}+a;\tfrac{3}{2}-2a;-\tfrac{1}{3}\right)=\left(\frac{8}{9}%
\right)^{-2a}\frac{\Gamma\left(\tfrac{4}{3}\right)\Gamma\left(\tfrac{3}{2}-2a%
\right)}{\Gamma\left(\tfrac{3}{2}\right)\Gamma\left(\tfrac{4}{3}-2a\right)}." display="block"><mrow><mrow><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo>+</mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>;</mo><mrow><mstyle displaystyle="false"><mfrac><mn>3</mn><mn>2</mn></mfrac></mstyle><mo>-</mo><mrow><mn>2</mn><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow></mrow><mo>;</mo><mrow><mo>-</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>3</mn></mfrac></mstyle></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msup><mrow><mo>(</mo><mfrac><mn>8</mn><mn>9</mn></mfrac><mo>)</mo></mrow><mrow><mo>-</mo><mrow><mn>2</mn><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow></mrow></msup><mo></mo><mfrac><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mfrac><mn>4</mn><mn>3</mn></mfrac><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mfrac><mn>3</mn><mn>2</mn></mfrac><mo>-</mo><mrow><mn>2</mn><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mfrac><mn>4</mn><mn>3</mn></mfrac><mo>-</mo><mrow><mn>2</mn><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mfrac></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E31.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./15.2#E1" title="(15.2.1) ‣ §15.2(i) Gauss Series ‣ §15.2 Definitions and Analytical Properties ‣ Properties ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m16.png" altimg-height="23px" altimg-valign="-7px" altimg-width="103px" alttext="F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m18.png" altimg-height="39px" altimg-valign="-15px" altimg-width="91px" alttext="F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi></mfrac><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: <math class="ltx_Math" altimg="m15.png" altimg-height="23px" altimg-valign="-7px" altimg-width="139px" alttext="={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow></math>
Gauss’ hypergeometric function</a> and
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m36.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./15.1#p1.t1.r3">a</mi></math>: real or complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E32" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">15.4.32</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E32.png" altimg-height="59px" altimg-valign="-24px" altimg-width="513px" alttext="F\left(a,\tfrac{1}{2}+a;\tfrac{5}{6}+\tfrac{2}{3}a;\tfrac{1}{9}\right)=\sqrt{%
\pi}\left(\frac{3}{4}\right)^{a}\frac{\Gamma\left(\tfrac{5}{6}+\tfrac{2}{3}a%
\right)}{\Gamma\left(\tfrac{1}{2}+\tfrac{1}{3}a\right)\Gamma\left(\tfrac{5}{6}%
+\tfrac{1}{3}a\right)}." display="block"><mrow><mrow><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo>+</mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>;</mo><mrow><mstyle displaystyle="false"><mfrac><mn>5</mn><mn>6</mn></mfrac></mstyle><mo>+</mo><mrow><mstyle displaystyle="false"><mfrac><mn>2</mn><mn>3</mn></mfrac></mstyle><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow></mrow><mo>;</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>9</mn></mfrac></mstyle><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msqrt><mi href="./3.12#E1">π</mi></msqrt><mo></mo><msup><mrow><mo>(</mo><mfrac><mn>3</mn><mn>4</mn></mfrac><mo>)</mo></mrow><mi href="./15.1#p1.t1.r3">a</mi></msup><mo></mo><mfrac><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mfrac><mn>5</mn><mn>6</mn></mfrac><mo>+</mo><mrow><mfrac><mn>2</mn><mn>3</mn></mfrac><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow></mrow><mo>)</mo></mrow></mrow><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>+</mo><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mfrac><mn>5</mn><mn>6</mn></mfrac><mo>+</mo><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mfrac></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E32.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./15.2#E1" title="(15.2.1) ‣ §15.2(i) Gauss Series ‣ §15.2 Definitions and Analytical Properties ‣ Properties ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m16.png" altimg-height="23px" altimg-valign="-7px" altimg-width="103px" alttext="F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m18.png" altimg-height="39px" altimg-valign="-15px" altimg-width="91px" alttext="F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi></mfrac><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: <math class="ltx_Math" altimg="m15.png" altimg-height="23px" altimg-valign="-7px" altimg-width="139px" alttext="={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow></math>
Gauss’ hypergeometric function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a> and
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m36.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./15.1#p1.t1.r3">a</mi></math>: real or complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E33" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">15.4.33</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E33.png" altimg-height="62px" altimg-valign="-24px" altimg-width="618px" alttext="F\left(3a,\tfrac{1}{3}+a;\tfrac{2}{3}+2a;e^{\ifrac{\mathrm{i}\pi}{3}}\right)=%
\sqrt{\pi}e^{\ifrac{\mathrm{i}\pi a}{2}}\left(\frac{16}{27}\right)^{(3a+1)/6}%
\frac{\Gamma\left(\frac{5}{6}+a\right)}{\Gamma\left(\frac{2}{3}+a\right)\Gamma%
\left(\frac{2}{3}\right)}," display="block"><mrow><mrow><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mrow><mn>3</mn><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>,</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>3</mn></mfrac></mstyle><mo>+</mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>;</mo><mrow><mstyle displaystyle="false"><mfrac><mn>2</mn><mn>3</mn></mfrac></mstyle><mo>+</mo><mrow><mn>2</mn><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow></mrow><mo>;</mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo>/</mo><mn>3</mn></mrow></msup><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msqrt><mi href="./3.12#E1">π</mi></msqrt><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>/</mo><mn>2</mn></mrow></msup><mo></mo><msup><mrow><mo>(</mo><mfrac><mn>16</mn><mn>27</mn></mfrac><mo>)</mo></mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>3</mn><mo></mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo>/</mo><mn>6</mn></mrow></msup><mo></mo><mfrac><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mfrac><mn>5</mn><mn>6</mn></mfrac><mo>+</mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>)</mo></mrow></mrow><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mfrac><mn>2</mn><mn>3</mn></mfrac><mo>+</mo><mi href="./15.1#p1.t1.r3">a</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><mo>)</mo></mrow></mrow></mrow></mfrac></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E33.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./15.2#E1" title="(15.2.1) ‣ §15.2(i) Gauss Series ‣ §15.2 Definitions and Analytical Properties ‣ Properties ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m16.png" altimg-height="23px" altimg-valign="-7px" altimg-width="103px" alttext="F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m18.png" altimg-height="39px" altimg-valign="-15px" altimg-width="91px" alttext="F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi></mfrac><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: <math class="ltx_Math" altimg="m15.png" altimg-height="23px" altimg-valign="-7px" altimg-width="139px" alttext="={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow></math>
Gauss’ hypergeometric function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m26.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a> and
<a href="./15.1#p1.t1.r3" title="§15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m36.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./15.1#p1.t1.r3">a</mi></math>: real or complex parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where the limit interpretation (), has to be taken when
<math class="ltx_Math" altimg="m35.png" altimg-height="27px" altimg-valign="-9px" altimg-width="176px" alttext="a=-\frac{1}{3},-\frac{4}{3},-\frac{7}{3},\dots" display="inline"><mrow><mi href="./15.1#p1.t1.r3">a</mi><mo>=</mo><mrow><mrow><mo>-</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></mrow><mo>,</mo><mrow><mo>-</mo><mfrac><mn>4</mn><mn>3</mn></mfrac></mrow><mo>,</mo><mrow><mo>-</mo><mfrac><mn>7</mn><mn>3</mn></mfrac></mrow><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>.
Compare the final paragraph in §</div>
</div>
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</html>
</page>
<page>
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<head>
<title>DLMF: 36.2 Catastrophes and Canonical Integrals</title>
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<div class="ltx_page_navlogo">) shift the <math class="ltx_Math" altimg="m72.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="s" display="inline"><mi href="./36.1#p2.t1.r2">s</mi></math> variable in ()) to remove the quadratic term, integrate, and
then deform the contour of the remaining <math class="ltx_Math" altimg="m73.png" altimg-height="17px" altimg-valign="-2px" altimg-width="11px" alttext="t" display="inline"><mi href="./36.1#p2.t1.r2">t</mi></math> integration.
For ()) with respect to <math class="ltx_Math" altimg="m73.png" altimg-height="17px" altimg-valign="-2px" altimg-width="11px" alttext="t" display="inline"><mi href="./36.1#p2.t1.r2">t</mi></math>.</dd>
</dl>
</div>
</div>
<section id="Px1" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Normal Forms Associated with Canonical Integrals:
Cuspoid Catastrophe with Codimension <math class="ltx_Math" altimg="m33.png" altimg-height="18px" altimg-valign="-2px" altimg-width="23px" alttext="K" display="inline"><mi href="./36.1#p2.t1.r3">K</mi></math>
</h3>
<div id="Px1.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px1.p1" class="ltx_para">
<table id="E1" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">36.2.1</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E1.png" altimg-height="67px" altimg-valign="-27px" altimg-width="268px" alttext="\Phi_{K}\left(t;\mathbf{x}\right)=t^{K+2}+\sum_{m=1}^{K}x_{m}t^{m}." display="block"><mrow><mrow><mrow><msub><mi href="./36.2#E1" mathvariant="normal">Φ</mi><mi href="./36.1#p2.t1.r3">K</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./36.1#p2.t1.r2">t</mi><mo>;</mo><mi mathvariant="bold">x</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msup><mi href="./36.1#p2.t1.r2">t</mi><mrow><mi href="./36.1#p2.t1.r3">K</mi><mo>+</mo><mn>2</mn></mrow></msup><mo>+</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./36.1#p2.t1.r1">m</mi><mo>=</mo><mn>1</mn></mrow><mi href="./36.1#p2.t1.r3">K</mi></munderover><mrow><msub><mi href="./36.1#p2.t1.r4">x</mi><mi href="./36.1#p2.t1.r1">m</mi></msub><mo></mo><msup><mi href="./36.1#p2.t1.r2">t</mi><mi href="./36.1#p2.t1.r1">m</mi></msup></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m38.png" altimg-height="23px" altimg-valign="-7px" altimg-width="81px" alttext="\Phi_{\NVar{K}}\left(\NVar{t};\NVar{\mathbf{x}}\right)" display="inline"><mrow><msub><mi href="./36.2#E1" mathvariant="normal">Φ</mi><mi class="ltx_nvar" href="./36.1#p2.t1.r3">K</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./36.1#p2.t1.r2">t</mi><mo>;</mo><mi class="ltx_nvar" mathvariant="bold">x</mi><mo>)</mo></mrow></mrow></math>: cuspoid catastrophe of codimension <math class="ltx_Math" altimg="m33.png" altimg-height="18px" altimg-valign="-2px" altimg-width="23px" alttext="K" display="inline"><mi href="./36.1#p2.t1.r3">K</mi></math></span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./36.1#p2.t1.r1" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m71.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./36.1#p2.t1.r1">n</mi></math>: integer</a>,
<a href="./36.1#p2.t1.r2" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m73.png" altimg-height="17px" altimg-valign="-2px" altimg-width="11px" alttext="t" display="inline"><mi href="./36.1#p2.t1.r2">t</mi></math>: variable</a>,
<a href="./36.1#p2.t1.r3" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m33.png" altimg-height="18px" altimg-valign="-2px" altimg-width="23px" alttext="K" display="inline"><mi href="./36.1#p2.t1.r3">K</mi></math>: codimension</a> and
<a href="./36.1#p2.t1.r4" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m81.png" altimg-height="16px" altimg-valign="-5px" altimg-width="22px" alttext="x_{i}" display="inline"><msub><mi href="./36.1#p2.t1.r4">x</mi><mi mathvariant="normal">i</mi></msub></math>: real parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Special cases: <math class="ltx_Math" altimg="m30.png" altimg-height="18px" altimg-valign="-2px" altimg-width="59px" alttext="K=1" display="inline"><mrow><mi href="./36.1#p2.t1.r3">K</mi><mo>=</mo><mn>1</mn></mrow></math>, <em class="ltx_emph ltx_font_italic">fold catastrophe</em>; <math class="ltx_Math" altimg="m31.png" altimg-height="18px" altimg-valign="-2px" altimg-width="59px" alttext="K=2" display="inline"><mrow><mi href="./36.1#p2.t1.r3">K</mi><mo>=</mo><mn>2</mn></mrow></math>, <em class="ltx_emph ltx_font_italic">cusp catastrophe</em>;
<math class="ltx_Math" altimg="m32.png" altimg-height="18px" altimg-valign="-2px" altimg-width="59px" alttext="K=3" display="inline"><mrow><mi href="./36.1#p2.t1.r3">K</mi><mo>=</mo><mn>3</mn></mrow></math>, <em class="ltx_emph ltx_font_italic">swallowtail catastrophe</em>.</p>
</div>
</section>
<section id="Px2" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Normal Forms for Umbilic Catastrophes with Codimension <math class="ltx_Math" altimg="m32.png" altimg-height="18px" altimg-valign="-2px" altimg-width="59px" alttext="K=3" display="inline"><mrow><mi href="./36.1#p2.t1.r3">K</mi><mo>=</mo><mn>3</mn></mrow></math>
</h3>
<div id="Px2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m37.png" altimg-height="26px" altimg-valign="-7px" altimg-width="109px" alttext="\Phi^{(\mathrm{U})}\left(\NVar{s},\NVar{t};\NVar{\mathbf{x}}\right)" display="inline"><mrow><msup><mi href="./36.2#Px2" mathvariant="normal">Φ</mi><mrow><mo href="./36.2#Px2" stretchy="false">(</mo><mi href="./36.2#Px2" mathvariant="normal">U</mi><mo href="./36.2#Px2" stretchy="false">)</mo></mrow></msup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./36.1#p2.t1.r2">s</mi><mo>,</mo><mi class="ltx_nvar" href="./36.1#p2.t1.r2">t</mi><mo>;</mo><mi class="ltx_nvar" mathvariant="bold">x</mi><mo>)</mo></mrow></mrow></math>: elliptic umbilic catastrophe for <math class="ltx_Math" altimg="m62.png" altimg-height="18px" altimg-valign="-2px" altimg-width="106px" alttext="\mathrm{U}=\mathrm{E}\mbox{ or }\mathrm{K}" display="inline"><mrow><mi mathvariant="normal">U</mi><mo>=</mo><mrow><mi mathvariant="normal">E</mi><mo></mo><mtext> or </mtext><mo></mo><mi mathvariant="normal">K</mi></mrow></mrow></math></span></dd>
<dt>Keywords:</dt>
<dd>
</dd>
</dl>
</div>
</div>
<div id="Px2.p1" class="ltx_para">
<table id="EGx1" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E2">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">36.2.2</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m14.png" altimg-height="29px" altimg-valign="-7px" altimg-width="110px" alttext="\displaystyle\Phi^{(\mathrm{E})}\left(s,t;\mathbf{x}\right)" display="inline"><mrow><msup><mi href="./36.2#E2" mathvariant="normal">Φ</mi><mrow><mo href="./36.2#E2" stretchy="false">(</mo><mi href="./36.2#E2" mathvariant="normal">E</mi><mo href="./36.2#E2" stretchy="false">)</mo></mrow></msup><mo></mo><mrow><mo>(</mo><mi href="./36.1#p2.t1.r2">s</mi><mo>,</mo><mi href="./36.1#p2.t1.r2">t</mi><mo>;</mo><mi mathvariant="bold">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m13.png" altimg-height="28px" altimg-valign="-7px" altimg-width="307px" alttext="\displaystyle=s^{3}-3st^{2}+z(s^{2}+t^{2})+yt+xs," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><msup><mi href="./36.1#p2.t1.r2">s</mi><mn>3</mn></msup><mo>-</mo><mrow><mn>3</mn><mo></mo><mi href="./36.1#p2.t1.r2">s</mi><mo></mo><msup><mi href="./36.1#p2.t1.r2">t</mi><mn>2</mn></msup></mrow></mrow><mo>+</mo><mrow><mi href="./36.1#p2.t1.r4">z</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><msup><mi href="./36.1#p2.t1.r2">s</mi><mn>2</mn></msup><mo>+</mo><msup><mi href="./36.1#p2.t1.r2">t</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>+</mo><mrow><mi href="./36.1#p2.t1.r4">y</mi><mo></mo><mi href="./36.1#p2.t1.r2">t</mi></mrow><mo>+</mo><mrow><mi href="./36.1#p2.t1.r4">x</mi><mo></mo><mi href="./36.1#p2.t1.r2">s</mi></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m57.png" altimg-height="23px" altimg-valign="-7px" altimg-width="113px" alttext="\mathbf{x}=\{x,y,z\}" display="inline"><mrow><mi mathvariant="bold">x</mi><mo>=</mo><mrow><mo stretchy="false">{</mo><mi href="./36.1#p2.t1.r4">x</mi><mo>,</mo><mi href="./36.1#p2.t1.r4">y</mi><mo>,</mo><mi href="./36.1#p2.t1.r4">z</mi><mo stretchy="false">}</mo></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m35.png" altimg-height="26px" altimg-valign="-7px" altimg-width="108px" alttext="\Phi^{(\mathrm{E})}\left(\NVar{s},\NVar{t};\NVar{\mathbf{x}}\right)" display="inline"><mrow><msup><mi href="./36.2#E2" mathvariant="normal">Φ</mi><mrow><mo href="./36.2#E2" stretchy="false">(</mo><mi href="./36.2#E2" mathvariant="normal">E</mi><mo href="./36.2#E2" stretchy="false">)</mo></mrow></msup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./36.1#p2.t1.r2">s</mi><mo>,</mo><mi class="ltx_nvar" href="./36.1#p2.t1.r2">t</mi><mo>;</mo><mi class="ltx_nvar" mathvariant="bold">x</mi><mo>)</mo></mrow></mrow></math>: elliptic umbilic catastrophe</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./36.1#p2.t1.r4" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./36.1#p2.t1.r4">y</mi></math>: real parameter</a>,
<a href="./36.1#p2.t1.r4" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m83.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./36.1#p2.t1.r4">z</mi></math>: real parameter</a>,
<a href="./36.1#p2.t1.r2" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m73.png" altimg-height="17px" altimg-valign="-2px" altimg-width="11px" alttext="t" display="inline"><mi href="./36.1#p2.t1.r2">t</mi></math>: variable</a>,
<a href="./36.1#p2.t1.r2" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="s" display="inline"><mi href="./36.1#p2.t1.r2">s</mi></math>: variable</a> and
<a href="./36.1#p2.t1.r4" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m76.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./36.1#p2.t1.r4">x</mi></math>: real parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tr class="ltx_eqn_row ltx_align_baseline"><td class="ltx_eqn_cell ltx_align_left" style="white-space:normal;" colspan="5">(elliptic umbilic).</td></tr>
<tbody id="E3">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">36.2.3</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m15.png" altimg-height="29px" altimg-valign="-7px" altimg-width="111px" alttext="\displaystyle\Phi^{(\mathrm{H})}\left(s,t;\mathbf{x}\right)" display="inline"><mrow><msup><mi href="./36.2#E3" mathvariant="normal">Φ</mi><mrow><mo href="./36.2#E3" stretchy="false">(</mo><mi href="./36.2#E3" mathvariant="normal">H</mi><mo href="./36.2#E3" stretchy="false">)</mo></mrow></msup><mo></mo><mrow><mo>(</mo><mi href="./36.1#p2.t1.r2">s</mi><mo>,</mo><mi href="./36.1#p2.t1.r2">t</mi><mo>;</mo><mi mathvariant="bold">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m12.png" altimg-height="27px" altimg-valign="-6px" altimg-width="230px" alttext="\displaystyle=s^{3}+t^{3}+zst+yt+xs," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msup><mi href="./36.1#p2.t1.r2">s</mi><mn>3</mn></msup><mo>+</mo><msup><mi href="./36.1#p2.t1.r2">t</mi><mn>3</mn></msup><mo>+</mo><mrow><mi href="./36.1#p2.t1.r4">z</mi><mo></mo><mi href="./36.1#p2.t1.r2">s</mi><mo></mo><mi href="./36.1#p2.t1.r2">t</mi></mrow><mo>+</mo><mrow><mi href="./36.1#p2.t1.r4">y</mi><mo></mo><mi href="./36.1#p2.t1.r2">t</mi></mrow><mo>+</mo><mrow><mi href="./36.1#p2.t1.r4">x</mi><mo></mo><mi href="./36.1#p2.t1.r2">s</mi></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m57.png" altimg-height="23px" altimg-valign="-7px" altimg-width="113px" alttext="\mathbf{x}=\{x,y,z\}" display="inline"><mrow><mi mathvariant="bold">x</mi><mo>=</mo><mrow><mo stretchy="false">{</mo><mi href="./36.1#p2.t1.r4">x</mi><mo>,</mo><mi href="./36.1#p2.t1.r4">y</mi><mo>,</mo><mi href="./36.1#p2.t1.r4">z</mi><mo stretchy="false">}</mo></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m36.png" altimg-height="26px" altimg-valign="-7px" altimg-width="109px" alttext="\Phi^{(\mathrm{H})}\left(\NVar{s},\NVar{t};\NVar{\mathbf{x}}\right)" display="inline"><mrow><msup><mi href="./36.2#E3" mathvariant="normal">Φ</mi><mrow><mo href="./36.2#E3" stretchy="false">(</mo><mi href="./36.2#E3" mathvariant="normal">H</mi><mo href="./36.2#E3" stretchy="false">)</mo></mrow></msup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./36.1#p2.t1.r2">s</mi><mo>,</mo><mi class="ltx_nvar" href="./36.1#p2.t1.r2">t</mi><mo>;</mo><mi class="ltx_nvar" mathvariant="bold">x</mi><mo>)</mo></mrow></mrow></math>: hyperbolic umbilic catastrophe</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./36.1#p2.t1.r4" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./36.1#p2.t1.r4">y</mi></math>: real parameter</a>,
<a href="./36.1#p2.t1.r4" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m83.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./36.1#p2.t1.r4">z</mi></math>: real parameter</a>,
<a href="./36.1#p2.t1.r2" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m73.png" altimg-height="17px" altimg-valign="-2px" altimg-width="11px" alttext="t" display="inline"><mi href="./36.1#p2.t1.r2">t</mi></math>: variable</a>,
<a href="./36.1#p2.t1.r2" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="s" display="inline"><mi href="./36.1#p2.t1.r2">s</mi></math>: variable</a> and
<a href="./36.1#p2.t1.r4" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m76.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./36.1#p2.t1.r4">x</mi></math>: real parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tr class="ltx_eqn_row ltx_align_baseline"><td class="ltx_eqn_cell ltx_align_left" style="white-space:normal;" colspan="5">(hyperbolic umbilic).</td></tr>
</table>
</div>
</section>
<section id="Px3" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Canonical Integrals</h3>
<div id="Px3.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px3.p1" class="ltx_para">
<table id="E4" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">36.2.4</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E4.png" altimg-height="54px" altimg-valign="-22px" altimg-width="298px" alttext="\Psi_{K}\left(\mathbf{x}\right)=\int_{-\infty}^{\infty}\exp\left(i\Phi_{K}%
\left(t;\mathbf{x}\right)\right)\mathrm{d}t." display="block"><mrow><mrow><mrow><msub><mi href="./36.2#E4" mathvariant="normal">Ψ</mi><mi href="./36.1#p2.t1.r3">K</mi></msub><mo></mo><mrow><mo>(</mo><mi mathvariant="bold">x</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mi mathvariant="normal">∞</mi></msubsup><mrow><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mrow><mo>(</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mrow><msub><mi href="./36.2#E1" mathvariant="normal">Φ</mi><mi href="./36.1#p2.t1.r3">K</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./36.1#p2.t1.r2">t</mi><mo>;</mo><mi mathvariant="bold">x</mi><mo>)</mo></mrow></mrow></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./36.1#p2.t1.r2">t</mi></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E4.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m51.png" altimg-height="23px" altimg-valign="-7px" altimg-width="66px" alttext="\Psi_{\NVar{K}}\left(\NVar{\mathbf{x}}\right)" display="inline"><mrow><msub><mi href="./36.2#E4" mathvariant="normal">Ψ</mi><mi class="ltx_nvar" href="./36.1#p2.t1.r3">K</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" mathvariant="bold">x</mi><mo>)</mo></mrow></mrow></math>: canonical integral function</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./36.2#E1" title="(36.2.1) ‣ Normal Forms Associated with Canonical Integrals: Cuspoid Catastrophe with Codimension K ‣ §36.2(i) Definitions ‣ §36.2 Catastrophes and Canonical Integrals ‣ Properties ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="23px" altimg-valign="-7px" altimg-width="81px" alttext="\Phi_{\NVar{K}}\left(\NVar{t};\NVar{\mathbf{x}}\right)" display="inline"><mrow><msub><mi href="./36.2#E1" mathvariant="normal">Φ</mi><mi class="ltx_nvar" href="./36.1#p2.t1.r3">K</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./36.1#p2.t1.r2">t</mi><mo>;</mo><mi class="ltx_nvar" mathvariant="bold">x</mi><mo>)</mo></mrow></mrow></math>: cuspoid catastrophe of codimension <math class="ltx_Math" altimg="m33.png" altimg-height="18px" altimg-valign="-2px" altimg-width="23px" alttext="K" display="inline"><mi href="./36.1#p2.t1.r3">K</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m76.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E19" title="(4.2.19) ‣ §4.2(iii) The Exponential Function ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="16px" altimg-valign="-6px" altimg-width="48px" alttext="\exp\NVar{z}" display="inline"><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: exponential function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./36.1#p2.t1.r2" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m73.png" altimg-height="17px" altimg-valign="-2px" altimg-width="11px" alttext="t" display="inline"><mi href="./36.1#p2.t1.r2">t</mi></math>: variable</a> and
<a href="./36.1#p2.t1.r3" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m33.png" altimg-height="18px" altimg-valign="-2px" altimg-width="23px" alttext="K" display="inline"><mi href="./36.1#p2.t1.r3">K</mi></math>: codimension</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E5" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">36.2.5</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E5.png" altimg-height="54px" altimg-valign="-22px" altimg-width="408px" alttext="\Psi^{(\mathrm{U})}\left(\mathbf{x}\right)=\int_{-\infty}^{\infty}\int_{-%
\infty}^{\infty}\exp\left(i\Phi^{(\mathrm{U})}\left(s,t;\mathbf{x}\right)%
\right)\mathrm{d}s\mathrm{d}t," display="block"><mrow><mrow><mrow><msup><mi href="./36.2#E5" mathvariant="normal">Ψ</mi><mrow><mo href="./36.2#E5" stretchy="false">(</mo><mi href="./36.2#E5" mathvariant="normal">U</mi><mo href="./36.2#E5" stretchy="false">)</mo></mrow></msup><mo></mo><mrow><mo>(</mo><mi mathvariant="bold">x</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mi mathvariant="normal">∞</mi></msubsup><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mi mathvariant="normal">∞</mi></msubsup><mrow><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mrow><mo>(</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mrow><msup><mi href="./36.2#Px2" mathvariant="normal">Φ</mi><mrow><mo href="./36.2#Px2" stretchy="false">(</mo><mi href="./36.2#Px2" mathvariant="normal">U</mi><mo href="./36.2#Px2" stretchy="false">)</mo></mrow></msup><mo></mo><mrow><mo>(</mo><mi href="./36.1#p2.t1.r2">s</mi><mo>,</mo><mi href="./36.1#p2.t1.r2">t</mi><mo>;</mo><mi mathvariant="bold">x</mi><mo>)</mo></mrow></mrow></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./36.1#p2.t1.r2">s</mi></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./36.1#p2.t1.r2">t</mi></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m61.png" altimg-height="21px" altimg-valign="-6px" altimg-width="83px" alttext="\mathrm{U}=\mathrm{E},\mathrm{H}" display="inline"><mrow><mi mathvariant="normal">U</mi><mo>=</mo><mrow><mi mathvariant="normal">E</mi><mo>,</mo><mi mathvariant="normal">H</mi></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E5.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd>
<span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m41.png" altimg-height="26px" altimg-valign="-7px" altimg-width="75px" alttext="\Psi^{(\mathrm{E})}\left(\NVar{\mathbf{x}}\right)" display="inline"><mrow><msup><mi href="./36.2#E5" mathvariant="normal">Ψ</mi><mrow><mo href="./36.2#E5" stretchy="false">(</mo><mi href="./36.2#E5" mathvariant="normal">E</mi><mo href="./36.2#E5" stretchy="false">)</mo></mrow></msup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" mathvariant="bold">x</mi><mo>)</mo></mrow></mrow></math>: elliptic umbilic canonical integral function</span>,
<span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m44.png" altimg-height="26px" altimg-valign="-7px" altimg-width="76px" alttext="\Psi^{(\mathrm{H})}\left(\NVar{\mathbf{x}}\right)" display="inline"><mrow><msup><mi href="./36.2#E5" mathvariant="normal">Ψ</mi><mrow><mo href="./36.2#E5" stretchy="false">(</mo><mi href="./36.2#E5" mathvariant="normal">H</mi><mo href="./36.2#E5" stretchy="false">)</mo></mrow></msup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" mathvariant="bold">x</mi><mo>)</mo></mrow></mrow></math>: hyperbolic umbilic canonical integral function</span> and
<span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m46.png" altimg-height="26px" altimg-valign="-7px" altimg-width="76px" alttext="\Psi^{(\mathrm{U})}\left(\NVar{\mathbf{x}}\right)" display="inline"><mrow><msup><mi href="./36.2#E5" mathvariant="normal">Ψ</mi><mrow><mo href="./36.2#E5" stretchy="false">(</mo><mi href="./36.2#E5" mathvariant="normal">U</mi><mo href="./36.2#E5" stretchy="false">)</mo></mrow></msup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" mathvariant="bold">x</mi><mo>)</mo></mrow></mrow></math>: umbilic canonical integral function</span>
</dd>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m76.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E19" title="(4.2.19) ‣ §4.2(iii) The Exponential Function ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="16px" altimg-valign="-6px" altimg-width="48px" alttext="\exp\NVar{z}" display="inline"><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: exponential function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./36.2#Px2" title="Normal Forms for Umbilic Catastrophes with Codimension = K 3 ‣ §36.2(i) Definitions ‣ §36.2 Catastrophes and Canonical Integrals ‣ Properties ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="26px" altimg-valign="-7px" altimg-width="109px" alttext="\Phi^{(\mathrm{U})}\left(\NVar{s},\NVar{t};\NVar{\mathbf{x}}\right)" display="inline"><mrow><msup><mi href="./36.2#Px2" mathvariant="normal">Φ</mi><mrow><mo href="./36.2#Px2" stretchy="false">(</mo><mi href="./36.2#Px2" mathvariant="normal">U</mi><mo href="./36.2#Px2" stretchy="false">)</mo></mrow></msup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./36.1#p2.t1.r2">s</mi><mo>,</mo><mi class="ltx_nvar" href="./36.1#p2.t1.r2">t</mi><mo>;</mo><mi class="ltx_nvar" mathvariant="bold">x</mi><mo>)</mo></mrow></mrow></math>: elliptic umbilic catastrophe for <math class="ltx_Math" altimg="m62.png" altimg-height="18px" altimg-valign="-2px" altimg-width="106px" alttext="\mathrm{U}=\mathrm{E}\mbox{ or }\mathrm{K}" display="inline"><mrow><mi mathvariant="normal">U</mi><mo>=</mo><mrow><mi mathvariant="normal">E</mi><mo></mo><mtext> or </mtext><mo></mo><mi mathvariant="normal">K</mi></mrow></mrow></math></a>,
<a href="./36.1#p2.t1.r2" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m73.png" altimg-height="17px" altimg-valign="-2px" altimg-width="11px" alttext="t" display="inline"><mi href="./36.1#p2.t1.r2">t</mi></math>: variable</a> and
<a href="./36.1#p2.t1.r2" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="s" display="inline"><mi href="./36.1#p2.t1.r2">s</mi></math>: variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E6" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">36.2.6</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_Math ltx_eqn_cell ltx_align_center"><math class="ltx_Math" alttext="\Psi^{(\mathrm{E})}\left(\mathbf{x}\right)=2\sqrt{\ifrac{\pi}{3}}\,\exp\left(i%
\left(\tfrac{4}{27}z^{3}+\tfrac{1}{3}xz-\tfrac{1}{4}\pi\right)\right)\int_{%
\infty\exp\left(-7\pi i/12\right)}^{\infty\exp\left(\pi i/12\right)}\exp\left(%
i\left(u^{6}+2zu^{4}+(z^{2}+x)u^{2}+\frac{y^{2}}{12u^{2}}\right)\right)\mathrm%
{d}u," display="block"><mrow><mrow><msup><mi href="./36.2#E5" mathvariant="normal">Ψ</mi><mrow><mo href="./36.2#E5" stretchy="false">(</mo><mi href="./36.2#E5" mathvariant="normal">E</mi><mo href="./36.2#E5" stretchy="false">)</mo></mrow></msup><mo></mo><mrow><mo>(</mo><mi mathvariant="bold">x</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mn>2</mn><mo></mo><mpadded width="+1.7pt"><msqrt><mrow><mi href="./3.12#E1">π</mi><mo>/</mo><mn>3</mn></mrow></msqrt></mpadded><mo></mo><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mrow><mo>(</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mrow><mstyle displaystyle="false"><mfrac><mn>4</mn><mn>27</mn></mfrac></mstyle><mo></mo><msup><mi href="./36.1#p2.t1.r4">z</mi><mn>3</mn></msup></mrow><mo>+</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>3</mn></mfrac></mstyle><mo></mo><mi href="./36.1#p2.t1.r4">x</mi><mo></mo><mi href="./36.1#p2.t1.r4">z</mi></mrow></mrow><mo>-</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>4</mn></mfrac></mstyle><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>×</mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><mi mathvariant="normal">∞</mi><mo></mo><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mrow><mrow><mn>7</mn><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi></mrow><mo>/</mo><mn>12</mn></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mrow><mi mathvariant="normal">∞</mi><mo></mo><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi></mrow><mo>/</mo><mn>12</mn></mrow><mo>)</mo></mrow></mrow></mrow></msubsup><mrow><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mrow><mo>(</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mrow><mo>(</mo><mrow><msup><mi>u</mi><mn>6</mn></msup><mo>+</mo><mrow><mn>2</mn><mo></mo><mi href="./36.1#p2.t1.r4">z</mi><mo></mo><msup><mi>u</mi><mn>4</mn></msup></mrow><mo>+</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><msup><mi href="./36.1#p2.t1.r4">z</mi><mn>2</mn></msup><mo>+</mo><mi href="./36.1#p2.t1.r4">x</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msup><mi>u</mi><mn>2</mn></msup></mrow><mo>+</mo><mfrac><msup><mi href="./36.1#p2.t1.r4">y</mi><mn>2</mn></msup><mrow><mn>12</mn><mo></mo><msup><mi>u</mi><mn>2</mn></msup></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>u</mi></mrow></mrow></mrow><mo>,</mo></mtd></mtr></mtable></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E6.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m76.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./36.2#E5" title="(36.2.5) ‣ Canonical Integrals ‣ §36.2(i) Definitions ‣ §36.2 Catastrophes and Canonical Integrals ‣ Properties ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m41.png" altimg-height="26px" altimg-valign="-7px" altimg-width="75px" alttext="\Psi^{(\mathrm{E})}\left(\NVar{\mathbf{x}}\right)" display="inline"><mrow><msup><mi href="./36.2#E5" mathvariant="normal">Ψ</mi><mrow><mo href="./36.2#E5" stretchy="false">(</mo><mi href="./36.2#E5" mathvariant="normal">E</mi><mo href="./36.2#E5" stretchy="false">)</mo></mrow></msup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" mathvariant="bold">x</mi><mo>)</mo></mrow></mrow></math>: elliptic umbilic canonical integral function</a>,
<a href="./4.2#E19" title="(4.2.19) ‣ §4.2(iii) The Exponential Function ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="16px" altimg-valign="-6px" altimg-width="48px" alttext="\exp\NVar{z}" display="inline"><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: exponential function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./36.1#p2.t1.r4" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./36.1#p2.t1.r4">y</mi></math>: real parameter</a>,
<a href="./36.1#p2.t1.r4" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m83.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./36.1#p2.t1.r4">z</mi></math>: real parameter</a> and
<a href="./36.1#p2.t1.r4" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m76.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./36.1#p2.t1.r4">x</mi></math>: real parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">with the contour passing to the lower right of <math class="ltx_Math" altimg="m74.png" altimg-height="17px" altimg-valign="-2px" altimg-width="52px" alttext="u=0" display="inline"><mrow><mi>u</mi><mo>=</mo><mn>0</mn></mrow></math>.</p>
<table id="E7" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex1" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">36.2.7</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m17.png" altimg-height="29px" altimg-valign="-7px" altimg-width="77px" alttext="\displaystyle\Psi^{(\mathrm{E})}\left(\mathbf{x}\right)" display="inline"><mrow><msup><mi href="./36.2#E5" mathvariant="normal">Ψ</mi><mrow><mo href="./36.2#E5" stretchy="false">(</mo><mi href="./36.2#E5" mathvariant="normal">E</mi><mo href="./36.2#E5" stretchy="false">)</mo></mrow></msup><mo></mo><mrow><mo>(</mo><mi mathvariant="bold">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m6.png" altimg-height="47px" altimg-valign="-16px" altimg-width="602px" alttext="\displaystyle=\dfrac{4\pi}{3^{1/3}}\exp\left(i\left(\tfrac{2}{27}z^{3}-\tfrac{%
1}{3}xz\right)\right)\left(\exp\left(-i\dfrac{\pi}{6}\right)\mathrm{F}_{+}(%
\mathbf{x})+\exp\left(i\dfrac{\pi}{6}\right)\mathrm{F}_{-}(\mathbf{x})\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><mfrac><mrow><mn>4</mn><mo></mo><mi href="./3.12#E1">π</mi></mrow><msup><mn>3</mn><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup></mfrac></mstyle><mo></mo><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mrow><mo>(</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mfrac><mn>2</mn><mn>27</mn></mfrac><mo></mo><msup><mi href="./36.1#p2.t1.r4">z</mi><mn>3</mn></msup></mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac><mo></mo><mi href="./36.1#p2.t1.r4">x</mi><mo></mo><mi href="./36.1#p2.t1.r4">z</mi></mrow></mrow><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mo>(</mo><mrow><mrow><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mstyle displaystyle="true"><mfrac><mi href="./3.12#E1">π</mi><mn>6</mn></mfrac></mstyle></mrow></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi mathvariant="normal">F</mi><mo>+</mo></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi mathvariant="bold">x</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mrow><mo>(</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mstyle displaystyle="true"><mfrac><mi href="./3.12#E1">π</mi><mn>6</mn></mfrac></mstyle></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi mathvariant="normal">F</mi><mo>-</mo></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi mathvariant="bold">x</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex2" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m26.png" altimg-height="25px" altimg-valign="-7px" altimg-width="60px" alttext="\displaystyle\mathrm{F}_{\pm}(\mathbf{x})" display="inline"><mrow><msub><mi mathvariant="normal">F</mi><mo>±</mo></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi mathvariant="bold">x</mi><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m9.png" altimg-height="53px" altimg-valign="-20px" altimg-width="841px" alttext="\displaystyle=\int_{0}^{\infty}\cos\left(ry\exp\left(\pm i\dfrac{\pi}{6}\right%
)\right)\exp\left(2ir^{2}z\exp\left(\pm i\dfrac{\pi}{3}\right)\right)\mathrm{%
Ai}\left(3^{2/3}r^{2}+3^{-1/3}\exp\left(\mp i\dfrac{\pi}{3}\right)\left(\tfrac%
{1}{3}z^{2}-x\right)\right)\mathrm{d}r." display="inline"><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup></mstyle><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mi>r</mi><mo></mo><mi href="./36.1#p2.t1.r4">y</mi><mo></mo><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mrow><mo>(</mo><mrow><mo>±</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mstyle displaystyle="true"><mfrac><mi href="./3.12#E1">π</mi><mn>6</mn></mfrac></mstyle></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><mi mathvariant="normal">i</mi><mo></mo><msup><mi>r</mi><mn>2</mn></msup><mo></mo><mi href="./36.1#p2.t1.r4">z</mi><mo></mo><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mrow><mo>(</mo><mrow><mo>±</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mstyle displaystyle="true"><mfrac><mi href="./3.12#E1">π</mi><mn>3</mn></mfrac></mstyle></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>)</mo></mrow></mrow></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>×</mo><mrow><mi href="./9.2#i">Ai</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><msup><mn>3</mn><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup><mo></mo><msup><mi>r</mi><mn>2</mn></msup></mrow><mo>+</mo><mrow><msup><mn>3</mn><mrow><mo>-</mo><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></mrow></msup><mo></mo><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mrow><mo>(</mo><mrow><mo>∓</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mstyle displaystyle="true"><mfrac><mi href="./3.12#E1">π</mi><mn>3</mn></mfrac></mstyle></mrow></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mo>(</mo><mrow><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac><mo></mo><msup><mi href="./36.1#p2.t1.r4">z</mi><mn>2</mn></msup></mrow><mo>-</mo><mi href="./36.1#p2.t1.r4">x</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>r</mi></mrow><mo>.</mo></mtd></mtr></mtable></mrow></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E7.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./9.2#i" title="§9.2(i) Airy’s Equation ‣ §9.2 Differential Equation ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="\mathrm{Ai}\left(\NVar{z}\right)" display="inline"><mrow><mi href="./9.2#i">Ai</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./9.1#p2.t1.r3">z</mi><mo>)</mo></mrow></mrow></math>: Airy function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m76.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./36.2#E5" title="(36.2.5) ‣ Canonical Integrals ‣ §36.2(i) Definitions ‣ §36.2 Catastrophes and Canonical Integrals ‣ Properties ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m41.png" altimg-height="26px" altimg-valign="-7px" altimg-width="75px" alttext="\Psi^{(\mathrm{E})}\left(\NVar{\mathbf{x}}\right)" display="inline"><mrow><msup><mi href="./36.2#E5" mathvariant="normal">Ψ</mi><mrow><mo href="./36.2#E5" stretchy="false">(</mo><mi href="./36.2#E5" mathvariant="normal">E</mi><mo href="./36.2#E5" stretchy="false">)</mo></mrow></msup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" mathvariant="bold">x</mi><mo>)</mo></mrow></mrow></math>: elliptic umbilic canonical integral function</a>,
<a href="./4.2#E19" title="(4.2.19) ‣ §4.2(iii) The Exponential Function ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="16px" altimg-valign="-6px" altimg-width="48px" alttext="\exp\NVar{z}" display="inline"><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: exponential function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./36.1#p2.t1.r4" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./36.1#p2.t1.r4">y</mi></math>: real parameter</a>,
<a href="./36.1#p2.t1.r4" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m83.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./36.1#p2.t1.r4">z</mi></math>: real parameter</a> and
<a href="./36.1#p2.t1.r4" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m76.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./36.1#p2.t1.r4">x</mi></math>: real parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E8" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">36.2.8</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_Math ltx_eqn_cell ltx_align_center"><math class="ltx_Math" alttext="\Psi^{(\mathrm{H})}\left(\mathbf{x}\right)=4\sqrt{\ifrac{\pi}{6}}\,\exp\left(i%
\left(\tfrac{1}{27}z^{3}+\tfrac{1}{6}z(y+x)+\tfrac{1}{4}\pi\right)\right)\*%
\int_{\infty\exp\left(5\pi i/12\right)}^{\infty\exp\left(\pi i/12\right)}\exp%
\left(i\left(2u^{6}+2zu^{4}+\left(\tfrac{1}{2}z^{2}+x+y\right)u^{2}-\frac{(y-x%
)^{2}}{24u^{2}}\right)\right)\mathrm{d}u," display="block"><mrow><mrow><msup><mi href="./36.2#E5" mathvariant="normal">Ψ</mi><mrow><mo href="./36.2#E5" stretchy="false">(</mo><mi href="./36.2#E5" mathvariant="normal">H</mi><mo href="./36.2#E5" stretchy="false">)</mo></mrow></msup><mo></mo><mrow><mo>(</mo><mi mathvariant="bold">x</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mn>4</mn><mo></mo><mpadded width="+1.7pt"><msqrt><mrow><mi href="./3.12#E1">π</mi><mo>/</mo><mn>6</mn></mrow></msqrt></mpadded><mo></mo><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mrow><mo>(</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>27</mn></mfrac></mstyle><mo></mo><msup><mi href="./36.1#p2.t1.r4">z</mi><mn>3</mn></msup></mrow><mo>+</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>6</mn></mfrac></mstyle><mo></mo><mi href="./36.1#p2.t1.r4">z</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./36.1#p2.t1.r4">y</mi><mo>+</mo><mi href="./36.1#p2.t1.r4">x</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>+</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>4</mn></mfrac></mstyle><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>×</mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><mi mathvariant="normal">∞</mi><mo></mo><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mn>5</mn><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi></mrow><mo>/</mo><mn>12</mn></mrow><mo>)</mo></mrow></mrow></mrow><mrow><mi mathvariant="normal">∞</mi><mo></mo><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi></mrow><mo>/</mo><mn>12</mn></mrow><mo>)</mo></mrow></mrow></mrow></msubsup><mrow><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mrow><mo>(</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mrow><mn>2</mn><mo></mo><msup><mi>u</mi><mn>6</mn></msup></mrow><mo>+</mo><mrow><mn>2</mn><mo></mo><mi href="./36.1#p2.t1.r4">z</mi><mo></mo><msup><mi>u</mi><mn>4</mn></msup></mrow><mo>+</mo><mrow><mrow><mo>(</mo><mrow><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><msup><mi href="./36.1#p2.t1.r4">z</mi><mn>2</mn></msup></mrow><mo>+</mo><mi href="./36.1#p2.t1.r4">x</mi><mo>+</mo><mi href="./36.1#p2.t1.r4">y</mi></mrow><mo>)</mo></mrow><mo></mo><msup><mi>u</mi><mn>2</mn></msup></mrow></mrow><mo>-</mo><mfrac><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./36.1#p2.t1.r4">y</mi><mo>-</mo><mi href="./36.1#p2.t1.r4">x</mi></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mrow><mn>24</mn><mo></mo><msup><mi>u</mi><mn>2</mn></msup></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>u</mi></mrow></mrow></mrow><mo>,</mo></mtd></mtr></mtable></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E8.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m76.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E19" title="(4.2.19) ‣ §4.2(iii) The Exponential Function ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="16px" altimg-valign="-6px" altimg-width="48px" alttext="\exp\NVar{z}" display="inline"><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: exponential function</a>,
<a href="./36.2#E5" title="(36.2.5) ‣ Canonical Integrals ‣ §36.2(i) Definitions ‣ §36.2 Catastrophes and Canonical Integrals ‣ Properties ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="26px" altimg-valign="-7px" altimg-width="76px" alttext="\Psi^{(\mathrm{H})}\left(\NVar{\mathbf{x}}\right)" display="inline"><mrow><msup><mi href="./36.2#E5" mathvariant="normal">Ψ</mi><mrow><mo href="./36.2#E5" stretchy="false">(</mo><mi href="./36.2#E5" mathvariant="normal">H</mi><mo href="./36.2#E5" stretchy="false">)</mo></mrow></msup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" mathvariant="bold">x</mi><mo>)</mo></mrow></mrow></math>: hyperbolic umbilic canonical integral function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./36.1#p2.t1.r4" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./36.1#p2.t1.r4">y</mi></math>: real parameter</a>,
<a href="./36.1#p2.t1.r4" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m83.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./36.1#p2.t1.r4">z</mi></math>: real parameter</a> and
<a href="./36.1#p2.t1.r4" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m76.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./36.1#p2.t1.r4">x</mi></math>: real parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">with the contour passing to the upper right of <math class="ltx_Math" altimg="m74.png" altimg-height="17px" altimg-valign="-2px" altimg-width="52px" alttext="u=0" display="inline"><mrow><mi>u</mi><mo>=</mo><mn>0</mn></mrow></math>.</p>
<table id="E9" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">36.2.9</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E9.png" altimg-height="61px" altimg-valign="-24px" altimg-width="541px" alttext="\Psi^{(\mathrm{H})}\left(\mathbf{x}\right)=\frac{2\pi}{3^{1/3}}\int_{\infty%
\exp\left(5\pi i/6\right)}^{\infty\exp\left(\pi i/6\right)}\exp\left(i(s^{3}+%
xs)\right)\mathrm{Ai}\left(\frac{zs+y}{3^{1/3}}\right)\mathrm{d}s." display="block"><mrow><mrow><mrow><msup><mi href="./36.2#E5" mathvariant="normal">Ψ</mi><mrow><mo href="./36.2#E5" stretchy="false">(</mo><mi href="./36.2#E5" mathvariant="normal">H</mi><mo href="./36.2#E5" stretchy="false">)</mo></mrow></msup><mo></mo><mrow><mo>(</mo><mi mathvariant="bold">x</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mfrac><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi></mrow><msup><mn>3</mn><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup></mfrac><mo></mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><mi mathvariant="normal">∞</mi><mo></mo><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mn>5</mn><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi></mrow><mo>/</mo><mn>6</mn></mrow><mo>)</mo></mrow></mrow></mrow><mrow><mi mathvariant="normal">∞</mi><mo></mo><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi></mrow><mo>/</mo><mn>6</mn></mrow><mo>)</mo></mrow></mrow></mrow></msubsup><mrow><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mrow><mo>(</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><msup><mi href="./36.1#p2.t1.r2">s</mi><mn>3</mn></msup><mo>+</mo><mrow><mi href="./36.1#p2.t1.r4">x</mi><mo></mo><mi href="./36.1#p2.t1.r2">s</mi></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./9.2#i">Ai</mi><mo></mo><mrow><mo>(</mo><mfrac><mrow><mrow><mi href="./36.1#p2.t1.r4">z</mi><mo></mo><mi href="./36.1#p2.t1.r2">s</mi></mrow><mo>+</mo><mi href="./36.1#p2.t1.r4">y</mi></mrow><msup><mn>3</mn><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup></mfrac><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./36.1#p2.t1.r2">s</mi></mrow></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E9.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./9.2#i" title="§9.2(i) Airy’s Equation ‣ §9.2 Differential Equation ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="\mathrm{Ai}\left(\NVar{z}\right)" display="inline"><mrow><mi href="./9.2#i">Ai</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./9.1#p2.t1.r3">z</mi><mo>)</mo></mrow></mrow></math>: Airy function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m76.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E19" title="(4.2.19) ‣ §4.2(iii) The Exponential Function ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="16px" altimg-valign="-6px" altimg-width="48px" alttext="\exp\NVar{z}" display="inline"><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: exponential function</a>,
<a href="./36.2#E5" title="(36.2.5) ‣ Canonical Integrals ‣ §36.2(i) Definitions ‣ §36.2 Catastrophes and Canonical Integrals ‣ Properties ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="26px" altimg-valign="-7px" altimg-width="76px" alttext="\Psi^{(\mathrm{H})}\left(\NVar{\mathbf{x}}\right)" display="inline"><mrow><msup><mi href="./36.2#E5" mathvariant="normal">Ψ</mi><mrow><mo href="./36.2#E5" stretchy="false">(</mo><mi href="./36.2#E5" mathvariant="normal">H</mi><mo href="./36.2#E5" stretchy="false">)</mo></mrow></msup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" mathvariant="bold">x</mi><mo>)</mo></mrow></mrow></math>: hyperbolic umbilic canonical integral function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./36.1#p2.t1.r4" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./36.1#p2.t1.r4">y</mi></math>: real parameter</a>,
<a href="./36.1#p2.t1.r4" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m83.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./36.1#p2.t1.r4">z</mi></math>: real parameter</a>,
<a href="./36.1#p2.t1.r2" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="s" display="inline"><mi href="./36.1#p2.t1.r2">s</mi></math>: variable</a> and
<a href="./36.1#p2.t1.r4" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m76.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./36.1#p2.t1.r4">x</mi></math>: real parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px4" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Diffraction Catastrophes</h3>
<div id="Px4.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px4.p1" class="ltx_para">
<table id="E10" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">36.2.10</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E10.png" altimg-height="54px" altimg-valign="-22px" altimg-width="357px" alttext="\Psi_{K}(\mathbf{x};k)=\sqrt{k}\int_{-\infty}^{\infty}\exp\left(ik\Phi_{K}%
\left(t;\mathbf{x}\right)\right)\mathrm{d}t," display="block"><mrow><mrow><mrow><msub><mi href="./36.2#E10" mathvariant="normal">Ψ</mi><mi href="./36.1#p2.t1.r3">K</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi mathvariant="bold">x</mi><mo>;</mo><mi href="./36.1#p2.t1.r2">k</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><msqrt><mi href="./36.1#p2.t1.r2">k</mi></msqrt><mo></mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mi mathvariant="normal">∞</mi></msubsup><mrow><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mrow><mo>(</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./36.1#p2.t1.r2">k</mi><mo></mo><mrow><msub><mi href="./36.2#E1" mathvariant="normal">Φ</mi><mi href="./36.1#p2.t1.r3">K</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./36.1#p2.t1.r2">t</mi><mo>;</mo><mi mathvariant="bold">x</mi><mo>)</mo></mrow></mrow></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./36.1#p2.t1.r2">t</mi></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m69.png" altimg-height="18px" altimg-valign="-3px" altimg-width="52px" alttext="k>0" display="inline"><mrow><mi href="./36.1#p2.t1.r2">k</mi><mo>></mo><mn>0</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E10.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m50.png" altimg-height="23px" altimg-valign="-7px" altimg-width="83px" alttext="\Psi_{\NVar{K}}(\NVar{\mathbf{x}};k)" display="inline"><mrow><msub><mi href="./36.2#E10" mathvariant="normal">Ψ</mi><mi class="ltx_nvar" href="./36.1#p2.t1.r3">K</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi class="ltx_nvar" mathvariant="bold">x</mi><mo>;</mo><mi href="./36.1#p2.t1.r2">k</mi><mo stretchy="false">)</mo></mrow></mrow></math>: diffraction catastrophe</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./36.2#E1" title="(36.2.1) ‣ Normal Forms Associated with Canonical Integrals: Cuspoid Catastrophe with Codimension K ‣ §36.2(i) Definitions ‣ §36.2 Catastrophes and Canonical Integrals ‣ Properties ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="23px" altimg-valign="-7px" altimg-width="81px" alttext="\Phi_{\NVar{K}}\left(\NVar{t};\NVar{\mathbf{x}}\right)" display="inline"><mrow><msub><mi href="./36.2#E1" mathvariant="normal">Φ</mi><mi class="ltx_nvar" href="./36.1#p2.t1.r3">K</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./36.1#p2.t1.r2">t</mi><mo>;</mo><mi class="ltx_nvar" mathvariant="bold">x</mi><mo>)</mo></mrow></mrow></math>: cuspoid catastrophe of codimension <math class="ltx_Math" altimg="m33.png" altimg-height="18px" altimg-valign="-2px" altimg-width="23px" alttext="K" display="inline"><mi href="./36.1#p2.t1.r3">K</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m76.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E19" title="(4.2.19) ‣ §4.2(iii) The Exponential Function ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="16px" altimg-valign="-6px" altimg-width="48px" alttext="\exp\NVar{z}" display="inline"><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: exponential function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./36.1#p2.t1.r2" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./36.1#p2.t1.r2">k</mi></math>: variable</a>,
<a href="./36.1#p2.t1.r2" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m73.png" altimg-height="17px" altimg-valign="-2px" altimg-width="11px" alttext="t" display="inline"><mi href="./36.1#p2.t1.r2">t</mi></math>: variable</a> and
<a href="./36.1#p2.t1.r3" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m33.png" altimg-height="18px" altimg-valign="-2px" altimg-width="23px" alttext="K" display="inline"><mi href="./36.1#p2.t1.r3">K</mi></math>: codimension</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E11" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">36.2.11</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E11.png" altimg-height="54px" altimg-valign="-22px" altimg-width="450px" alttext="\Psi^{(\mathrm{U})}(\mathbf{x};k)=k\int_{-\infty}^{\infty}\int_{-\infty}^{%
\infty}\exp\left(ik\Phi^{(\mathrm{U})}\left(s,t;\mathbf{x}\right)\right)%
\mathrm{d}s\mathrm{d}t," display="block"><mrow><mrow><mrow><msup><mi href="./36.2#E11" mathvariant="normal">Ψ</mi><mrow><mo href="./36.2#E11" stretchy="false">(</mo><mi href="./36.2#E11" mathvariant="normal">U</mi><mo href="./36.2#E11" stretchy="false">)</mo></mrow></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi mathvariant="bold">x</mi><mo>;</mo><mi href="./36.1#p2.t1.r2">k</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mi href="./36.1#p2.t1.r2">k</mi><mo></mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mi mathvariant="normal">∞</mi></msubsup><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mi mathvariant="normal">∞</mi></msubsup><mrow><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mrow><mo>(</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./36.1#p2.t1.r2">k</mi><mo></mo><mrow><msup><mi href="./36.2#Px2" mathvariant="normal">Φ</mi><mrow><mo href="./36.2#Px2" stretchy="false">(</mo><mi href="./36.2#Px2" mathvariant="normal">U</mi><mo href="./36.2#Px2" stretchy="false">)</mo></mrow></msup><mo></mo><mrow><mo>(</mo><mi href="./36.1#p2.t1.r2">s</mi><mo>,</mo><mi href="./36.1#p2.t1.r2">t</mi><mo>;</mo><mi mathvariant="bold">x</mi><mo>)</mo></mrow></mrow></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./36.1#p2.t1.r2">s</mi></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./36.1#p2.t1.r2">t</mi></mrow></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m60.png" altimg-height="21px" altimg-valign="-6px" altimg-width="83px" alttext="\mathrm{U=E,H}" display="inline"><mrow><mi mathvariant="normal">U</mi><mo>=</mo><mrow><mi mathvariant="normal">E</mi><mo>,</mo><mi mathvariant="normal">H</mi></mrow></mrow></math>; <math class="ltx_Math" altimg="m69.png" altimg-height="18px" altimg-valign="-3px" altimg-width="52px" alttext="k>0" display="inline"><mrow><mi href="./36.1#p2.t1.r2">k</mi><mo>></mo><mn>0</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E11.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd>
<span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m39.png" altimg-height="26px" altimg-valign="-7px" altimg-width="91px" alttext="\Psi^{(\mathrm{E})}(\NVar{\mathbf{x}};\NVar{k})" display="inline"><mrow><msup><mi href="./36.2#E11" mathvariant="normal">Ψ</mi><mrow><mo href="./36.2#E11" stretchy="false">(</mo><mi href="./36.2#E11" mathvariant="normal">E</mi><mo href="./36.2#E11" stretchy="false">)</mo></mrow></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi class="ltx_nvar" mathvariant="bold">x</mi><mo>;</mo><mi class="ltx_nvar" href="./36.1#p2.t1.r2">k</mi><mo stretchy="false">)</mo></mrow></mrow></math>: elliptic umbilic canonical integral function</span>,
<span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m42.png" altimg-height="26px" altimg-valign="-7px" altimg-width="92px" alttext="\Psi^{(\mathrm{H})}(\NVar{\mathbf{x}};\NVar{k})" display="inline"><mrow><msup><mi href="./36.2#E11" mathvariant="normal">Ψ</mi><mrow><mo href="./36.2#E11" stretchy="false">(</mo><mi href="./36.2#E11" mathvariant="normal">H</mi><mo href="./36.2#E11" stretchy="false">)</mo></mrow></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi class="ltx_nvar" mathvariant="bold">x</mi><mo>;</mo><mi class="ltx_nvar" href="./36.1#p2.t1.r2">k</mi><mo stretchy="false">)</mo></mrow></mrow></math>: hyperbolic umbilic canonical integral function</span> and
<span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m45.png" altimg-height="26px" altimg-valign="-7px" altimg-width="92px" alttext="\Psi^{(\mathrm{U})}(\NVar{\mathbf{x}};\NVar{k})" display="inline"><mrow><msup><mi href="./36.2#E11" mathvariant="normal">Ψ</mi><mrow><mo href="./36.2#E11" stretchy="false">(</mo><mi href="./36.2#E11" mathvariant="normal">U</mi><mo href="./36.2#E11" stretchy="false">)</mo></mrow></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi class="ltx_nvar" mathvariant="bold">x</mi><mo>;</mo><mi class="ltx_nvar" href="./36.1#p2.t1.r2">k</mi><mo stretchy="false">)</mo></mrow></mrow></math>: umbilic canonical integral function</span>
</dd>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m76.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E19" title="(4.2.19) ‣ §4.2(iii) The Exponential Function ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="16px" altimg-valign="-6px" altimg-width="48px" alttext="\exp\NVar{z}" display="inline"><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: exponential function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./36.2#Px2" title="Normal Forms for Umbilic Catastrophes with Codimension = K 3 ‣ §36.2(i) Definitions ‣ §36.2 Catastrophes and Canonical Integrals ‣ Properties ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="26px" altimg-valign="-7px" altimg-width="109px" alttext="\Phi^{(\mathrm{U})}\left(\NVar{s},\NVar{t};\NVar{\mathbf{x}}\right)" display="inline"><mrow><msup><mi href="./36.2#Px2" mathvariant="normal">Φ</mi><mrow><mo href="./36.2#Px2" stretchy="false">(</mo><mi href="./36.2#Px2" mathvariant="normal">U</mi><mo href="./36.2#Px2" stretchy="false">)</mo></mrow></msup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./36.1#p2.t1.r2">s</mi><mo>,</mo><mi class="ltx_nvar" href="./36.1#p2.t1.r2">t</mi><mo>;</mo><mi class="ltx_nvar" mathvariant="bold">x</mi><mo>)</mo></mrow></mrow></math>: elliptic umbilic catastrophe for <math class="ltx_Math" altimg="m62.png" altimg-height="18px" altimg-valign="-2px" altimg-width="106px" alttext="\mathrm{U}=\mathrm{E}\mbox{ or }\mathrm{K}" display="inline"><mrow><mi mathvariant="normal">U</mi><mo>=</mo><mrow><mi mathvariant="normal">E</mi><mo></mo><mtext> or </mtext><mo></mo><mi mathvariant="normal">K</mi></mrow></mrow></math></a>,
<a href="./36.1#p2.t1.r2" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./36.1#p2.t1.r2">k</mi></math>: variable</a>,
<a href="./36.1#p2.t1.r2" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m73.png" altimg-height="17px" altimg-valign="-2px" altimg-width="11px" alttext="t" display="inline"><mi href="./36.1#p2.t1.r2">t</mi></math>: variable</a> and
<a href="./36.1#p2.t1.r2" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="s" display="inline"><mi href="./36.1#p2.t1.r2">s</mi></math>: variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="Px4.p2" class="ltx_para">
<p class="ltx_p">For more extensive lists of normal forms of catastrophes (umbilic and beyond)
involving two variables (“corank two”) see <cite class="ltx_cite ltx_citemacro_citet">Arnol’d (</dd>
</dl>
</div>
</div>
<div id="SS2.p1" class="ltx_para">
<table id="E12" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">36.2.12</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E12.png" altimg-height="42px" altimg-valign="-16px" altimg-width="179px" alttext="\Psi_{0}=\sqrt{\pi}\exp\left(i\frac{\pi}{4}\right)." display="block"><mrow><mrow><msub><mi href="./36.2#E4" mathvariant="normal">Ψ</mi><mn>0</mn></msub><mo>=</mo><mrow><msqrt><mi href="./3.12#E1">π</mi></msqrt><mo></mo><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mrow><mo>(</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mfrac><mi href="./3.12#E1">π</mi><mn>4</mn></mfrac></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E12.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./36.2#E4" title="(36.2.4) ‣ Canonical Integrals ‣ §36.2(i) Definitions ‣ §36.2 Catastrophes and Canonical Integrals ‣ Properties ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="23px" altimg-valign="-7px" altimg-width="66px" alttext="\Psi_{\NVar{K}}\left(\NVar{\mathbf{x}}\right)" display="inline"><mrow><msub><mi href="./36.2#E4" mathvariant="normal">Ψ</mi><mi class="ltx_nvar" href="./36.1#p2.t1.r3">K</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" mathvariant="bold">x</mi><mo>)</mo></mrow></mrow></math>: canonical integral function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a> and
<a href="./4.2#E19" title="(4.2.19) ‣ §4.2(iii) The Exponential Function ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="16px" altimg-valign="-6px" altimg-width="48px" alttext="\exp\NVar{z}" display="inline"><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: exponential function</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p"><math class="ltx_Math" altimg="m47.png" altimg-height="21px" altimg-valign="-5px" altimg-width="29px" alttext="\Psi_{1}" display="inline"><msub><mi href="./36.2#E4" mathvariant="normal">Ψ</mi><mn>1</mn></msub></math> is related to the Airy function (§):
</p>
<table id="E13" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">36.2.13</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E13.png" altimg-height="47px" altimg-valign="-16px" altimg-width="223px" alttext="\Psi_{1}\left(x\right)=\frac{2\pi}{3^{1/3}}\mathrm{Ai}\left(\frac{x}{3^{1/3}}%
\right)." display="block"><mrow><mrow><mrow><msub><mi href="./36.2#E4" mathvariant="normal">Ψ</mi><mn>1</mn></msub><mo></mo><mrow><mo>(</mo><mi href="./36.1#p2.t1.r4">x</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mfrac><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi></mrow><msup><mn>3</mn><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup></mfrac><mo></mo><mrow><mi href="./9.2#i">Ai</mi><mo></mo><mrow><mo>(</mo><mfrac><mi href="./36.1#p2.t1.r4">x</mi><msup><mn>3</mn><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup></mfrac><mo>)</mo></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E13.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./9.2#i" title="§9.2(i) Airy’s Equation ‣ §9.2 Differential Equation ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="\mathrm{Ai}\left(\NVar{z}\right)" display="inline"><mrow><mi href="./9.2#i">Ai</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./9.1#p2.t1.r3">z</mi><mo>)</mo></mrow></mrow></math>: Airy function</a>,
<a href="./36.2#E4" title="(36.2.4) ‣ Canonical Integrals ‣ §36.2(i) Definitions ‣ §36.2 Catastrophes and Canonical Integrals ‣ Properties ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="23px" altimg-valign="-7px" altimg-width="66px" alttext="\Psi_{\NVar{K}}\left(\NVar{\mathbf{x}}\right)" display="inline"><mrow><msub><mi href="./36.2#E4" mathvariant="normal">Ψ</mi><mi class="ltx_nvar" href="./36.1#p2.t1.r3">K</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" mathvariant="bold">x</mi><mo>)</mo></mrow></mrow></math>: canonical integral function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a> and
<a href="./36.1#p2.t1.r4" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m76.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./36.1#p2.t1.r4">x</mi></math>: real parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p"><math class="ltx_Math" altimg="m48.png" altimg-height="21px" altimg-valign="-5px" altimg-width="29px" alttext="\Psi_{2}" display="inline"><msub><mi href="./36.2#E4" mathvariant="normal">Ψ</mi><mn>2</mn></msub></math> is the Pearcey integral (<cite class="ltx_cite ltx_citemacro_citet">Pearcey ()</cite>):
</p>
<table id="E14" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">36.2.14</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E14.png" altimg-height="54px" altimg-valign="-22px" altimg-width="468px" alttext="\Psi_{2}\left(\mathbf{x}\right)=P(x_{2},x_{1})=\int_{-\infty}^{\infty}\exp%
\left(\mathrm{i}(t^{4}+x_{2}t^{2}+x_{1}t)\right)\mathrm{d}t." display="block"><mrow><mrow><mrow><msub><mi href="./36.2#E4" mathvariant="normal">Ψ</mi><mn>2</mn></msub><mo></mo><mrow><mo>(</mo><mi mathvariant="bold">x</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mi>P</mi><mo></mo><mrow><mo stretchy="false">(</mo><msub><mi href="./36.1#p2.t1.r4">x</mi><mn>2</mn></msub><mo>,</mo><msub><mi href="./36.1#p2.t1.r4">x</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mi mathvariant="normal">∞</mi></msubsup><mrow><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mrow><mo>(</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><msup><mi href="./36.1#p2.t1.r2">t</mi><mn>4</mn></msup><mo>+</mo><mrow><msub><mi href="./36.1#p2.t1.r4">x</mi><mn>2</mn></msub><mo></mo><msup><mi href="./36.1#p2.t1.r2">t</mi><mn>2</mn></msup></mrow><mo>+</mo><mrow><msub><mi href="./36.1#p2.t1.r4">x</mi><mn>1</mn></msub><mo></mo><mi href="./36.1#p2.t1.r2">t</mi></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./36.1#p2.t1.r2">t</mi></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E14.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./36.2#E4" title="(36.2.4) ‣ Canonical Integrals ‣ §36.2(i) Definitions ‣ §36.2 Catastrophes and Canonical Integrals ‣ Properties ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="23px" altimg-valign="-7px" altimg-width="66px" alttext="\Psi_{\NVar{K}}\left(\NVar{\mathbf{x}}\right)" display="inline"><mrow><msub><mi href="./36.2#E4" mathvariant="normal">Ψ</mi><mi class="ltx_nvar" href="./36.1#p2.t1.r3">K</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" mathvariant="bold">x</mi><mo>)</mo></mrow></mrow></math>: canonical integral function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m76.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E19" title="(4.2.19) ‣ §4.2(iii) The Exponential Function ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="16px" altimg-valign="-6px" altimg-width="48px" alttext="\exp\NVar{z}" display="inline"><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: exponential function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./36.1#p2.t1.r2" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m73.png" altimg-height="17px" altimg-valign="-2px" altimg-width="11px" alttext="t" display="inline"><mi href="./36.1#p2.t1.r2">t</mi></math>: variable</a> and
<a href="./36.1#p2.t1.r4" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m81.png" altimg-height="16px" altimg-valign="-5px" altimg-width="22px" alttext="x_{i}" display="inline"><msub><mi href="./36.1#p2.t1.r4">x</mi><mi mathvariant="normal">i</mi></msub></math>: real parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">(Other notations also appear in the literature.)
</p>
<table id="E15" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">36.2.15</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E15.png" altimg-height="101px" altimg-valign="-45px" altimg-width="531px" alttext="\Psi_{K}\left(\boldsymbol{{0}}\right)=\frac{2}{K+2}\Gamma\left(\frac{1}{K+2}%
\right)\*\begin{cases}\exp\left(i\dfrac{\pi}{2(K+2)}\right),&K\text{ even,}\\
\cos\left(\dfrac{\pi}{2(K+2)}\right),&K\text{ odd}.\end{cases}" display="block"><mrow><mrow><msub><mi href="./36.2#E4" mathvariant="normal">Ψ</mi><mi href="./36.1#p2.t1.r3">K</mi></msub><mo></mo><mrow><mo>(</mo><mn mathvariant="bold">0</mn><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mfrac><mn>2</mn><mrow><mi href="./36.1#p2.t1.r3">K</mi><mo>+</mo><mn>2</mn></mrow></mfrac><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mfrac><mn>1</mn><mrow><mi href="./36.1#p2.t1.r3">K</mi><mo>+</mo><mn>2</mn></mrow></mfrac><mo>)</mo></mrow></mrow><mo></mo><mrow><mo>{</mo><mtable columnspacing="5pt" displaystyle="true" rowspacing="0pt"><mtr><mtd columnalign="left"><mrow><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mrow><mo>(</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mfrac><mi href="./3.12#E1">π</mi><mrow><mn>2</mn><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./36.1#p2.t1.r3">K</mi><mo>+</mo><mn>2</mn></mrow><mo stretchy="false">)</mo></mrow></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mi href="./36.1#p2.t1.r3">K</mi><mo></mo><mtext> even,</mtext></mrow></mtd></mtr><mtr><mtd columnalign="left"><mrow><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mfrac><mi href="./3.12#E1">π</mi><mrow><mn>2</mn><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./36.1#p2.t1.r3">K</mi><mo>+</mo><mn>2</mn></mrow><mo stretchy="false">)</mo></mrow></mrow></mfrac><mo>)</mo></mrow></mrow><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mrow><mi href="./36.1#p2.t1.r3">K</mi><mo></mo><mtext> odd</mtext></mrow><mo>.</mo></mrow></mtd></mtr></mtable></mrow></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E15.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m34.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./36.2#E4" title="(36.2.4) ‣ Canonical Integrals ‣ §36.2(i) Definitions ‣ §36.2 Catastrophes and Canonical Integrals ‣ Properties ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="23px" altimg-valign="-7px" altimg-width="66px" alttext="\Psi_{\NVar{K}}\left(\NVar{\mathbf{x}}\right)" display="inline"><mrow><msub><mi href="./36.2#E4" mathvariant="normal">Ψ</mi><mi class="ltx_nvar" href="./36.1#p2.t1.r3">K</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" mathvariant="bold">x</mi><mo>)</mo></mrow></mrow></math>: canonical integral function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./4.2#E19" title="(4.2.19) ‣ §4.2(iii) The Exponential Function ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="16px" altimg-valign="-6px" altimg-width="48px" alttext="\exp\NVar{z}" display="inline"><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: exponential function</a> and
<a href="./36.1#p2.t1.r3" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m33.png" altimg-height="18px" altimg-valign="-2px" altimg-width="23px" alttext="K" display="inline"><mi href="./36.1#p2.t1.r3">K</mi></math>: codimension</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E16" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex3" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="4" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">36.2.16</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m19.png" altimg-height="25px" altimg-valign="-7px" altimg-width="61px" alttext="\displaystyle\Psi_{1}\left(\boldsymbol{{0}}\right)" display="inline"><mrow><msub><mi href="./36.2#E4" mathvariant="normal">Ψ</mi><mn>1</mn></msub><mo></mo><mrow><mo>(</mo><mn mathvariant="bold">0</mn><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m2.png" altimg-height="22px" altimg-valign="-6px" altimg-width="98px" alttext="\displaystyle=1.54669," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mn>1.54669</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex4" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m20.png" altimg-height="25px" altimg-valign="-7px" altimg-width="61px" alttext="\displaystyle\Psi_{2}\left(\boldsymbol{{0}}\right)" display="inline"><mrow><msub><mi href="./36.2#E4" mathvariant="normal">Ψ</mi><mn>2</mn></msub><mo></mo><mrow><mo>(</mo><mn mathvariant="bold">0</mn><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m3.png" altimg-height="21px" altimg-valign="-4px" altimg-width="191px" alttext="\displaystyle=1.67481+\mathrm{i}\,0.69373" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mn>1.67481</mn><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mn> 0.69373</mn></mrow></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex5" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m21.png" altimg-height="25px" altimg-valign="-7px" altimg-width="61px" alttext="\displaystyle\Psi_{3}\left(\boldsymbol{{0}}\right)" display="inline"><mrow><msub><mi href="./36.2#E4" mathvariant="normal">Ψ</mi><mn>3</mn></msub><mo></mo><mrow><mo>(</mo><mn mathvariant="bold">0</mn><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m4.png" altimg-height="22px" altimg-valign="-6px" altimg-width="98px" alttext="\displaystyle=1.74646," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mn>1.74646</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex6" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m22.png" altimg-height="25px" altimg-valign="-7px" altimg-width="61px" alttext="\displaystyle\Psi_{4}\left(\boldsymbol{{0}}\right)" display="inline"><mrow><msub><mi href="./36.2#E4" mathvariant="normal">Ψ</mi><mn>4</mn></msub><mo></mo><mrow><mo>(</mo><mn mathvariant="bold">0</mn><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m5.png" altimg-height="21px" altimg-valign="-4px" altimg-width="196px" alttext="\displaystyle=1.79222+\mathrm{i}\,0.48022." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mn>1.79222</mn><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mn> 0.48022</mn></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E16.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./36.2#E4" title="(36.2.4) ‣ Canonical Integrals ‣ §36.2(i) Definitions ‣ §36.2 Catastrophes and Canonical Integrals ‣ Properties ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="23px" altimg-valign="-7px" altimg-width="66px" alttext="\Psi_{\NVar{K}}\left(\NVar{\mathbf{x}}\right)" display="inline"><mrow><msub><mi href="./36.2#E4" mathvariant="normal">Ψ</mi><mi class="ltx_nvar" href="./36.1#p2.t1.r3">K</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" mathvariant="bold">x</mi><mo>)</mo></mrow></mrow></math>: canonical integral function</a></dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS2.p2" class="ltx_para">
<table id="E17" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex7" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="6" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">36.2.17</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m25.png" altimg-height="50px" altimg-valign="-19px" altimg-width="113px" alttext="\displaystyle\frac{{\partial}^{p}}{{\partial x_{1}}^{p}}\Psi_{K}\left(%
\boldsymbol{{0}}\right)" display="inline"><mrow><mstyle displaystyle="true"><mfrac><mpadded width="-1.7pt"><msup><mo href="./1.5#E3">∂</mo><mi>p</mi></msup></mpadded><msup><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><msub><mi href="./36.1#p2.t1.r4">x</mi><mn>1</mn></msub></mrow><mi href="./1.5#E3">p</mi></msup></mfrac></mstyle><mo></mo><mrow><msub><mi href="./36.2#E4" mathvariant="normal">Ψ</mi><mi href="./36.1#p2.t1.r3">K</mi></msub><mo></mo><mrow><mo>(</mo><mn mathvariant="bold">0</mn><mo>)</mo></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m8.png" altimg-height="53px" altimg-valign="-21px" altimg-width="400px" alttext="\displaystyle=\frac{2}{K+2}\Gamma\left(\frac{p+1}{K+2}\right)\cos\left(\frac{%
\pi}{2}\left(\frac{p+1}{K+2}+p\right)\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><mfrac><mn>2</mn><mrow><mi href="./36.1#p2.t1.r3">K</mi><mo>+</mo><mn>2</mn></mrow></mfrac></mstyle><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac><mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow><mrow><mi href="./36.1#p2.t1.r3">K</mi><mo>+</mo><mn>2</mn></mrow></mfrac></mstyle><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mstyle displaystyle="true"><mfrac><mi href="./3.12#E1">π</mi><mn>2</mn></mfrac></mstyle><mo></mo><mrow><mo>(</mo><mrow><mstyle displaystyle="true"><mfrac><mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow><mrow><mi href="./36.1#p2.t1.r3">K</mi><mo>+</mo><mn>2</mn></mrow></mfrac></mstyle><mo>+</mo><mi>p</mi></mrow><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m33.png" altimg-height="18px" altimg-valign="-2px" altimg-width="23px" alttext="K" display="inline"><mi href="./36.1#p2.t1.r3">K</mi></math> odd,</span></td></tr>
<tr id="Ex8" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m23.png" altimg-height="54px" altimg-valign="-21px" altimg-width="141px" alttext="\displaystyle\frac{{\partial}^{2q+1}}{{\partial x_{1}}^{2q+1}}\Psi_{K}\left(%
\boldsymbol{{0}}\right)" display="inline"><mrow><mstyle displaystyle="true"><mfrac><mpadded width="-1.7pt"><msup><mo href="./1.5#E3">∂</mo><mrow><mrow><mn>2</mn><mo></mo><mi>q</mi></mrow><mo>+</mo><mn>1</mn></mrow></msup></mpadded><msup><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><msub><mi href="./36.1#p2.t1.r4">x</mi><mn>1</mn></msub></mrow><mrow><mrow><mn>2</mn><mo href="./1.5#E3"></mo><mi href="./1.5#E3">q</mi></mrow><mo href="./1.5#E3">+</mo><mn>1</mn></mrow></msup></mfrac></mstyle><mo></mo><mrow><msub><mi href="./36.2#E4" mathvariant="normal">Ψ</mi><mi href="./36.1#p2.t1.r3">K</mi></msub><mo></mo><mrow><mo>(</mo><mn mathvariant="bold">0</mn><mo>)</mo></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m1.png" altimg-height="22px" altimg-valign="-6px" altimg-width="43px" alttext="\displaystyle=0," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mn>0</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m33.png" altimg-height="18px" altimg-valign="-2px" altimg-width="23px" alttext="K" display="inline"><mi href="./36.1#p2.t1.r3">K</mi></math> even,</span></td></tr>
<tr id="Ex9" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m24.png" altimg-height="54px" altimg-valign="-21px" altimg-width="121px" alttext="\displaystyle\frac{{\partial}^{2q}}{{\partial x_{1}}^{2q}}\Psi_{K}\left(%
\boldsymbol{{0}}\right)" display="inline"><mrow><mstyle displaystyle="true"><mfrac><mpadded width="-1.7pt"><msup><mo href="./1.5#E3">∂</mo><mrow><mn>2</mn><mo></mo><mi>q</mi></mrow></msup></mpadded><msup><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><msub><mi href="./36.1#p2.t1.r4">x</mi><mn>1</mn></msub></mrow><mrow><mn>2</mn><mo href="./1.5#E3"></mo><mi href="./1.5#E3">q</mi></mrow></msup></mfrac></mstyle><mo></mo><mrow><msub><mi href="./36.2#E4" mathvariant="normal">Ψ</mi><mi href="./36.1#p2.t1.r3">K</mi></msub><mo></mo><mrow><mo>(</mo><mn mathvariant="bold">0</mn><mo>)</mo></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m7.png" altimg-height="53px" altimg-valign="-21px" altimg-width="423px" alttext="\displaystyle=\frac{2}{K+2}\Gamma\left(\frac{2q+1}{K+2}\right)\exp\left(i\frac%
{\pi}{2}\left(\frac{2q+1}{K+2}+2q\right)\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><mfrac><mn>2</mn><mrow><mi href="./36.1#p2.t1.r3">K</mi><mo>+</mo><mn>2</mn></mrow></mfrac></mstyle><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac><mrow><mrow><mn>2</mn><mo></mo><mi>q</mi></mrow><mo>+</mo><mn>1</mn></mrow><mrow><mi href="./36.1#p2.t1.r3">K</mi><mo>+</mo><mn>2</mn></mrow></mfrac></mstyle><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mrow><mo>(</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mstyle displaystyle="true"><mfrac><mi href="./3.12#E1">π</mi><mn>2</mn></mfrac></mstyle><mo></mo><mrow><mo>(</mo><mrow><mstyle displaystyle="true"><mfrac><mrow><mrow><mn>2</mn><mo></mo><mi>q</mi></mrow><mo>+</mo><mn>1</mn></mrow><mrow><mi href="./36.1#p2.t1.r3">K</mi><mo>+</mo><mn>2</mn></mrow></mfrac></mstyle><mo>+</mo><mrow><mn>2</mn><mo></mo><mi>q</mi></mrow></mrow><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m33.png" altimg-height="18px" altimg-valign="-2px" altimg-width="23px" alttext="K" display="inline"><mi href="./36.1#p2.t1.r3">K</mi></math> even.</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E17.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m34.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./36.2#E4" title="(36.2.4) ‣ Canonical Integrals ‣ §36.2(i) Definitions ‣ §36.2 Catastrophes and Canonical Integrals ‣ Properties ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="23px" altimg-valign="-7px" altimg-width="66px" alttext="\Psi_{\NVar{K}}\left(\NVar{\mathbf{x}}\right)" display="inline"><mrow><msub><mi href="./36.2#E4" mathvariant="normal">Ψ</mi><mi class="ltx_nvar" href="./36.1#p2.t1.r3">K</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" mathvariant="bold">x</mi><mo>)</mo></mrow></mrow></math>: canonical integral function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./4.2#E19" title="(4.2.19) ‣ §4.2(iii) The Exponential Function ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="16px" altimg-valign="-6px" altimg-width="48px" alttext="\exp\NVar{z}" display="inline"><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: exponential function</a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m55.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m68.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m76.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m65.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m76.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./36.1#p2.t1.r3" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m33.png" altimg-height="18px" altimg-valign="-2px" altimg-width="23px" alttext="K" display="inline"><mi href="./36.1#p2.t1.r3">K</mi></math>: codimension</a> and
<a href="./36.1#p2.t1.r4" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m81.png" altimg-height="16px" altimg-valign="-5px" altimg-width="22px" alttext="x_{i}" display="inline"><msub><mi href="./36.1#p2.t1.r4">x</mi><mi mathvariant="normal">i</mi></msub></math>: real parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS2.p3" class="ltx_para">
<table id="E18" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex10" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">36.2.18</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m16.png" altimg-height="29px" altimg-valign="-7px" altimg-width="76px" alttext="\displaystyle\Psi^{(\mathrm{E})}\left(\boldsymbol{{0}}\right)" display="inline"><mrow><msup><mi href="./36.2#E5" mathvariant="normal">Ψ</mi><mrow><mo href="./36.2#E5" stretchy="false">(</mo><mi href="./36.2#E5" mathvariant="normal">E</mi><mo href="./36.2#E5" stretchy="false">)</mo></mrow></msup><mo></mo><mrow><mo>(</mo><mn mathvariant="bold">0</mn><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m10.png" altimg-height="29px" altimg-valign="-9px" altimg-width="213px" alttext="\displaystyle=\tfrac{1}{3}\sqrt{\pi}\Gamma\left(\tfrac{1}{6}\right)=3.28868," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac><mo></mo><msqrt><mi href="./3.12#E1">π</mi></msqrt><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mfrac><mn>1</mn><mn>6</mn></mfrac><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mn>3.28868</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex11" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m18.png" altimg-height="29px" altimg-valign="-7px" altimg-width="76px" alttext="\displaystyle\Psi^{(\mathrm{H})}\left(0\right)" display="inline"><mrow><msup><mi href="./36.2#E5" mathvariant="normal">Ψ</mi><mrow><mo href="./36.2#E5" stretchy="false">(</mo><mi href="./36.2#E5" mathvariant="normal">H</mi><mo href="./36.2#E5" stretchy="false">)</mo></mrow></msup><mo></mo><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m11.png" altimg-height="30px" altimg-valign="-9px" altimg-width="193px" alttext="\displaystyle=\tfrac{1}{3}{\Gamma^{2}}\left(\tfrac{1}{3}\right)=2.39224." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac><mo></mo><mrow><msup><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mn>2</mn></msup><mo></mo><mrow><mo>(</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mn>2.39224</mn></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E18.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m34.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./36.2#E5" title="(36.2.5) ‣ Canonical Integrals ‣ §36.2(i) Definitions ‣ §36.2 Catastrophes and Canonical Integrals ‣ Properties ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m41.png" altimg-height="26px" altimg-valign="-7px" altimg-width="75px" alttext="\Psi^{(\mathrm{E})}\left(\NVar{\mathbf{x}}\right)" display="inline"><mrow><msup><mi href="./36.2#E5" mathvariant="normal">Ψ</mi><mrow><mo href="./36.2#E5" stretchy="false">(</mo><mi href="./36.2#E5" mathvariant="normal">E</mi><mo href="./36.2#E5" stretchy="false">)</mo></mrow></msup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" mathvariant="bold">x</mi><mo>)</mo></mrow></mrow></math>: elliptic umbilic canonical integral function</a> and
<a href="./36.2#E5" title="(36.2.5) ‣ Canonical Integrals ‣ §36.2(i) Definitions ‣ §36.2 Catastrophes and Canonical Integrals ‣ Properties ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="26px" altimg-valign="-7px" altimg-width="76px" alttext="\Psi^{(\mathrm{H})}\left(\NVar{\mathbf{x}}\right)" display="inline"><mrow><msup><mi href="./36.2#E5" mathvariant="normal">Ψ</mi><mrow><mo href="./36.2#E5" stretchy="false">(</mo><mi href="./36.2#E5" mathvariant="normal">H</mi><mo href="./36.2#E5" stretchy="false">)</mo></mrow></msup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" mathvariant="bold">x</mi><mo>)</mo></mrow></mrow></math>: hyperbolic umbilic canonical integral function</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E19" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">36.2.19</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_Math ltx_eqn_cell ltx_align_center"><math class="ltx_Math" alttext="\Psi_{2}\left(0,y\right)=\frac{\pi}{2}\sqrt{\frac{|y|}{2}}\exp\left(-i\frac{y^%
{2}}{8}\right)\left(\exp\left(i\frac{\pi}{8}\right)J_{-\ifrac{1}{4}}\left(%
\frac{y^{2}}{8}\right)-\operatorname{sign}\left(y\right)\exp\left(-i\frac{\pi}%
{8}\right)J_{\ifrac{1}{4}}\left(\frac{y^{2}}{8}\right)\right)." display="block"><mrow><mrow><msub><mi href="./36.2#E4" mathvariant="normal">Ψ</mi><mn>2</mn></msub><mo></mo><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mi href="./36.1#p2.t1.r4">y</mi></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mfrac><mi href="./3.12#E1">π</mi><mn>2</mn></mfrac><mo></mo><msqrt><mfrac><mrow><mo stretchy="false">|</mo><mi href="./36.1#p2.t1.r4">y</mi><mo stretchy="false">|</mo></mrow><mn>2</mn></mfrac></msqrt><mo></mo><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mfrac><msup><mi href="./36.1#p2.t1.r4">y</mi><mn>2</mn></msup><mn>8</mn></mfrac></mrow></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mo maxsize="3.588em" minsize="3.588em">(</mo><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mrow><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mrow><mo>(</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mfrac><mi href="./3.12#E1">π</mi><mn>8</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mrow><mo>-</mo><mrow><mn>1</mn><mo>/</mo><mn>4</mn></mrow></mrow></msub><mo></mo><mrow><mo>(</mo><mfrac><msup><mi href="./36.1#p2.t1.r4">y</mi><mn>2</mn></msup><mn>8</mn></mfrac><mo>)</mo></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>-</mo><mrow><mrow><mi href="./front/introduction#Sx4.p2.t1.r18">sign</mi><mo></mo><mrow><mo>(</mo><mi href="./36.1#p2.t1.r4">y</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mfrac><mi href="./3.12#E1">π</mi><mn>8</mn></mfrac></mrow></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mrow><mn>1</mn><mo>/</mo><mn>4</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mfrac><msup><mi href="./36.1#p2.t1.r4">y</mi><mn>2</mn></msup><mn>8</mn></mfrac><mo>)</mo></mrow></mrow></mrow><mo maxsize="3.588em" minsize="3.588em">)</mo><mo>.</mo></mtd></mtr></mtable></mrow></mrow></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E19.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./36.2#E4" title="(36.2.4) ‣ Canonical Integrals ‣ §36.2(i) Definitions ‣ §36.2 Catastrophes and Canonical Integrals ‣ Properties ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="23px" altimg-valign="-7px" altimg-width="66px" alttext="\Psi_{\NVar{K}}\left(\NVar{\mathbf{x}}\right)" display="inline"><mrow><msub><mi href="./36.2#E4" mathvariant="normal">Ψ</mi><mi class="ltx_nvar" href="./36.1#p2.t1.r3">K</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" mathvariant="bold">x</mi><mo>)</mo></mrow></mrow></math>: canonical integral function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.2#E19" title="(4.2.19) ‣ §4.2(iii) The Exponential Function ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="16px" altimg-valign="-6px" altimg-width="48px" alttext="\exp\NVar{z}" display="inline"><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: exponential function</a>,
<a href="./front/introduction#Sx4.p2.t1.r18" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m64.png" altimg-height="21px" altimg-valign="-6px" altimg-width="53px" alttext="\operatorname{sign}\NVar{x}" display="inline"><mrow><mi href="./front/introduction#Sx4.p2.t1.r18">sign</mi><mo></mo><mi class="ltx_nvar">x</mi></mrow></math>: sign of <math class="ltx_Math" altimg="m76.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./36.1#p2.t1.r4" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./36.1#p2.t1.r4">y</mi></math>: real parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">For the Bessel function <math class="ltx_Math" altimg="m28.png" altimg-height="18px" altimg-valign="-2px" altimg-width="17px" alttext="J" display="inline"><mi href="./10.2#E2">J</mi></math> see §.
</p>
<table id="E20" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">36.2.20</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E20.png" altimg-height="53px" altimg-valign="-21px" altimg-width="505px" alttext="\Psi^{(\mathrm{E})}\left(x,y,0\right)=2\pi^{2}(\tfrac{2}{3})^{2/3}\Re\left(%
\mathrm{Ai}\left(\frac{x+iy}{12^{1/3}}\right)\mathrm{Bi}\left(\frac{x-iy}{12^{%
1/3}}\right)\right)," display="block"><mrow><mrow><mrow><msup><mi href="./36.2#E5" mathvariant="normal">Ψ</mi><mrow><mo href="./36.2#E5" stretchy="false">(</mo><mi href="./36.2#E5" mathvariant="normal">E</mi><mo href="./36.2#E5" stretchy="false">)</mo></mrow></msup><mo></mo><mrow><mo>(</mo><mrow><mi href="./36.1#p2.t1.r4">x</mi><mo>,</mo><mi href="./36.1#p2.t1.r4">y</mi><mo>,</mo><mn>0</mn></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mn>2</mn><mo></mo><msup><mi href="./3.12#E1">π</mi><mn>2</mn></msup><mo></mo><msup><mrow><mo stretchy="false">(</mo><mstyle displaystyle="false"><mfrac><mn>2</mn><mn>3</mn></mfrac></mstyle><mo stretchy="false">)</mo></mrow><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup><mo></mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mi href="./9.2#i">Ai</mi><mo></mo><mrow><mo>(</mo><mfrac><mrow><mi href="./36.1#p2.t1.r4">x</mi><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./36.1#p2.t1.r4">y</mi></mrow></mrow><msup><mn>12</mn><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup></mfrac><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./9.2#i">Bi</mi><mo></mo><mrow><mo>(</mo><mfrac><mrow><mi href="./36.1#p2.t1.r4">x</mi><mo>-</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./36.1#p2.t1.r4">y</mi></mrow></mrow><msup><mn>12</mn><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup></mfrac><mo>)</mo></mrow></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E20.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./9.2#i" title="§9.2(i) Airy’s Equation ‣ §9.2 Differential Equation ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="\mathrm{Ai}\left(\NVar{z}\right)" display="inline"><mrow><mi href="./9.2#i">Ai</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./9.1#p2.t1.r3">z</mi><mo>)</mo></mrow></mrow></math>: Airy function</a>,
<a href="./9.2#i" title="§9.2(i) Airy’s Equation ‣ §9.2 Differential Equation ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="23px" altimg-valign="-7px" altimg-width="53px" alttext="\mathrm{Bi}\left(\NVar{z}\right)" display="inline"><mrow><mi href="./9.2#i">Bi</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./9.1#p2.t1.r3">z</mi><mo>)</mo></mrow></mrow></math>: Airy function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./36.2#E5" title="(36.2.5) ‣ Canonical Integrals ‣ §36.2(i) Definitions ‣ §36.2 Catastrophes and Canonical Integrals ‣ Properties ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m41.png" altimg-height="26px" altimg-valign="-7px" altimg-width="75px" alttext="\Psi^{(\mathrm{E})}\left(\NVar{\mathbf{x}}\right)" display="inline"><mrow><msup><mi href="./36.2#E5" mathvariant="normal">Ψ</mi><mrow><mo href="./36.2#E5" stretchy="false">(</mo><mi href="./36.2#E5" mathvariant="normal">E</mi><mo href="./36.2#E5" stretchy="false">)</mo></mrow></msup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" mathvariant="bold">x</mi><mo>)</mo></mrow></mrow></math>: elliptic umbilic canonical integral function</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./36.1#p2.t1.r4" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./36.1#p2.t1.r4">y</mi></math>: real parameter</a> and
<a href="./36.1#p2.t1.r4" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m76.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./36.1#p2.t1.r4">x</mi></math>: real parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E21" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">36.2.21</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E21.png" altimg-height="49px" altimg-valign="-16px" altimg-width="368px" alttext="\Psi^{(\mathrm{H})}\left(x,y,0\right)=\frac{4\pi^{2}}{3^{2/3}}\mathrm{Ai}\left%
(\frac{x}{3^{1/3}}\right)\mathrm{Ai}\left(\frac{y}{3^{1/3}}\right)." display="block"><mrow><mrow><mrow><msup><mi href="./36.2#E5" mathvariant="normal">Ψ</mi><mrow><mo href="./36.2#E5" stretchy="false">(</mo><mi href="./36.2#E5" mathvariant="normal">H</mi><mo href="./36.2#E5" stretchy="false">)</mo></mrow></msup><mo></mo><mrow><mo>(</mo><mrow><mi href="./36.1#p2.t1.r4">x</mi><mo>,</mo><mi href="./36.1#p2.t1.r4">y</mi><mo>,</mo><mn>0</mn></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mfrac><mrow><mn>4</mn><mo></mo><msup><mi href="./3.12#E1">π</mi><mn>2</mn></msup></mrow><msup><mn>3</mn><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup></mfrac><mo></mo><mrow><mi href="./9.2#i">Ai</mi><mo></mo><mrow><mo>(</mo><mfrac><mi href="./36.1#p2.t1.r4">x</mi><msup><mn>3</mn><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup></mfrac><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./9.2#i">Ai</mi><mo></mo><mrow><mo>(</mo><mfrac><mi href="./36.1#p2.t1.r4">y</mi><msup><mn>3</mn><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup></mfrac><mo>)</mo></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E21.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./9.2#i" title="§9.2(i) Airy’s Equation ‣ §9.2 Differential Equation ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="\mathrm{Ai}\left(\NVar{z}\right)" display="inline"><mrow><mi href="./9.2#i">Ai</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./9.1#p2.t1.r3">z</mi><mo>)</mo></mrow></mrow></math>: Airy function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./36.2#E5" title="(36.2.5) ‣ Canonical Integrals ‣ §36.2(i) Definitions ‣ §36.2 Catastrophes and Canonical Integrals ‣ Properties ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="26px" altimg-valign="-7px" altimg-width="76px" alttext="\Psi^{(\mathrm{H})}\left(\NVar{\mathbf{x}}\right)" display="inline"><mrow><msup><mi href="./36.2#E5" mathvariant="normal">Ψ</mi><mrow><mo href="./36.2#E5" stretchy="false">(</mo><mi href="./36.2#E5" mathvariant="normal">H</mi><mo href="./36.2#E5" stretchy="false">)</mo></mrow></msup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" mathvariant="bold">x</mi><mo>)</mo></mrow></mrow></math>: hyperbolic umbilic canonical integral function</a>,
<a href="./36.1#p2.t1.r4" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./36.1#p2.t1.r4">y</mi></math>: real parameter</a> and
<a href="./36.1#p2.t1.r4" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m76.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./36.1#p2.t1.r4">x</mi></math>: real parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS2.p4" class="ltx_para">
<p class="ltx_p">Addendum: For further special cases see §</dd>
</dl>
</div>
</div>
<div id="SS3.p1" class="ltx_para">
<table id="E22" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">36.2.22</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E22.png" altimg-height="26px" altimg-valign="-7px" altimg-width="186px" alttext="\Psi_{2K}\left(\mathbf{x}^{\prime}\right)=\Psi_{2K}\left(\mathbf{x}\right)," display="block"><mrow><mrow><mrow><msub><mi href="./36.2#E4" mathvariant="normal">Ψ</mi><mrow><mn>2</mn><mo></mo><mi href="./36.1#p2.t1.r3">K</mi></mrow></msub><mo></mo><mrow><mo>(</mo><msup><mi mathvariant="bold">x</mi><mo>′</mo></msup><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msub><mi href="./36.2#E4" mathvariant="normal">Ψ</mi><mrow><mn>2</mn><mo></mo><mi href="./36.1#p2.t1.r3">K</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi mathvariant="bold">x</mi><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m77.png" altimg-height="25px" altimg-valign="-9px" altimg-width="156px" alttext="x_{2m+1}^{\prime}=-x_{2m+1}" display="inline"><mrow><msubsup><mi href="./36.1#p2.t1.r4">x</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./36.1#p2.t1.r1">m</mi></mrow><mo>+</mo><mn>1</mn></mrow><mo>′</mo></msubsup><mo>=</mo><mrow><mo>-</mo><msub><mi href="./36.1#p2.t1.r4">x</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./36.1#p2.t1.r1">m</mi></mrow><mo>+</mo><mn>1</mn></mrow></msub></mrow></mrow></math>, <math class="ltx_Math" altimg="m80.png" altimg-height="23px" altimg-valign="-7px" altimg-width="100px" alttext="x_{2m}^{\prime}=x_{2m}" display="inline"><mrow><msubsup><mi href="./36.1#p2.t1.r4">x</mi><mrow><mn>2</mn><mo></mo><mi href="./36.1#p2.t1.r1">m</mi></mrow><mo>′</mo></msubsup><mo>=</mo><msub><mi href="./36.1#p2.t1.r4">x</mi><mrow><mn>2</mn><mo></mo><mi href="./36.1#p2.t1.r1">m</mi></mrow></msub></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E22.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./36.2#E4" title="(36.2.4) ‣ Canonical Integrals ‣ §36.2(i) Definitions ‣ §36.2 Catastrophes and Canonical Integrals ‣ Properties ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="23px" altimg-valign="-7px" altimg-width="66px" alttext="\Psi_{\NVar{K}}\left(\NVar{\mathbf{x}}\right)" display="inline"><mrow><msub><mi href="./36.2#E4" mathvariant="normal">Ψ</mi><mi class="ltx_nvar" href="./36.1#p2.t1.r3">K</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" mathvariant="bold">x</mi><mo>)</mo></mrow></mrow></math>: canonical integral function</a>,
<a href="./36.1#p2.t1.r1" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m71.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./36.1#p2.t1.r1">n</mi></math>: integer</a>,
<a href="./36.1#p2.t1.r3" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m33.png" altimg-height="18px" altimg-valign="-2px" altimg-width="23px" alttext="K" display="inline"><mi href="./36.1#p2.t1.r3">K</mi></math>: codimension</a> and
<a href="./36.1#p2.t1.r4" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m81.png" altimg-height="16px" altimg-valign="-5px" altimg-width="22px" alttext="x_{i}" display="inline"><msub><mi href="./36.1#p2.t1.r4">x</mi><mi mathvariant="normal">i</mi></msub></math>: real parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E23" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">36.2.23</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E23.png" altimg-height="28px" altimg-valign="-9px" altimg-width="227px" alttext="\Psi_{2K+1}\left(\mathbf{x}^{\prime}\right)={\Psi_{2K+1}^{\ast}}\left(\mathbf{%
x}\right)," display="block"><mrow><mrow><mrow><msub><mi href="./36.2#E4" mathvariant="normal">Ψ</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./36.1#p2.t1.r3">K</mi></mrow><mo>+</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><msup><mi mathvariant="bold">x</mi><mo>′</mo></msup><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msubsup><mi href="./36.2#E4" mathvariant="normal">Ψ</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./36.1#p2.t1.r3">K</mi></mrow><mo>+</mo><mn>1</mn></mrow><mo>∗</mo></msubsup><mo></mo><mrow><mo>(</mo><mi mathvariant="bold">x</mi><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m78.png" altimg-height="25px" altimg-valign="-9px" altimg-width="140px" alttext="x_{2m+1}^{\prime}=x_{2m+1}" display="inline"><mrow><msubsup><mi href="./36.1#p2.t1.r4">x</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./36.1#p2.t1.r1">m</mi></mrow><mo>+</mo><mn>1</mn></mrow><mo>′</mo></msubsup><mo>=</mo><msub><mi href="./36.1#p2.t1.r4">x</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./36.1#p2.t1.r1">m</mi></mrow><mo>+</mo><mn>1</mn></mrow></msub></mrow></math>, <math class="ltx_Math" altimg="m79.png" altimg-height="23px" altimg-valign="-7px" altimg-width="115px" alttext="x_{2m}^{\prime}=-x_{2m}" display="inline"><mrow><msubsup><mi href="./36.1#p2.t1.r4">x</mi><mrow><mn>2</mn><mo></mo><mi href="./36.1#p2.t1.r1">m</mi></mrow><mo>′</mo></msubsup><mo>=</mo><mrow><mo>-</mo><msub><mi href="./36.1#p2.t1.r4">x</mi><mrow><mn>2</mn><mo></mo><mi href="./36.1#p2.t1.r1">m</mi></mrow></msub></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E23.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./36.2#E4" title="(36.2.4) ‣ Canonical Integrals ‣ §36.2(i) Definitions ‣ §36.2 Catastrophes and Canonical Integrals ‣ Properties ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="23px" altimg-valign="-7px" altimg-width="66px" alttext="\Psi_{\NVar{K}}\left(\NVar{\mathbf{x}}\right)" display="inline"><mrow><msub><mi href="./36.2#E4" mathvariant="normal">Ψ</mi><mi class="ltx_nvar" href="./36.1#p2.t1.r3">K</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" mathvariant="bold">x</mi><mo>)</mo></mrow></mrow></math>: canonical integral function</a>,
<a href="./36.1#p2.t1.r1" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m71.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./36.1#p2.t1.r1">n</mi></math>: integer</a>,
<a href="./36.1#p2.t1.r3" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m33.png" altimg-height="18px" altimg-valign="-2px" altimg-width="23px" alttext="K" display="inline"><mi href="./36.1#p2.t1.r3">K</mi></math>: codimension</a> and
<a href="./36.1#p2.t1.r4" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m81.png" altimg-height="16px" altimg-valign="-5px" altimg-width="22px" alttext="x_{i}" display="inline"><msub><mi href="./36.1#p2.t1.r4">x</mi><mi mathvariant="normal">i</mi></msub></math>: real parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E24" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">36.2.24</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E24.png" altimg-height="31px" altimg-valign="-7px" altimg-width="278px" alttext="\Psi^{(\mathrm{U})}\left(x,y,z\right)={\Psi^{\ast}}^{(\mathrm{U})}(x,y,-z)," display="block"><mrow><mrow><mrow><msup><mi href="./36.2#E5" mathvariant="normal">Ψ</mi><mrow><mo href="./36.2#E5" stretchy="false">(</mo><mi href="./36.2#E5" mathvariant="normal">U</mi><mo href="./36.2#E5" stretchy="false">)</mo></mrow></msup><mo></mo><mrow><mo>(</mo><mrow><mi href="./36.1#p2.t1.r4">x</mi><mo>,</mo><mi href="./36.1#p2.t1.r4">y</mi><mo>,</mo><mi href="./36.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mmultiscripts><mi mathvariant="normal">Ψ</mi><none></none><mo>∗</mo><none></none><mrow><mo stretchy="false">(</mo><mi mathvariant="normal">U</mi><mo stretchy="false">)</mo></mrow></mmultiscripts><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./36.1#p2.t1.r4">x</mi><mo>,</mo><mi href="./36.1#p2.t1.r4">y</mi><mo>,</mo><mrow><mo>-</mo><mi href="./36.1#p2.t1.r4">z</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m60.png" altimg-height="21px" altimg-valign="-6px" altimg-width="83px" alttext="\mathrm{U=E,H}" display="inline"><mrow><mi mathvariant="normal">U</mi><mo>=</mo><mrow><mi mathvariant="normal">E</mi><mo>,</mo><mi mathvariant="normal">H</mi></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E24.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./36.2#E5" title="(36.2.5) ‣ Canonical Integrals ‣ §36.2(i) Definitions ‣ §36.2 Catastrophes and Canonical Integrals ‣ Properties ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m46.png" altimg-height="26px" altimg-valign="-7px" altimg-width="76px" alttext="\Psi^{(\mathrm{U})}\left(\NVar{\mathbf{x}}\right)" display="inline"><mrow><msup><mi href="./36.2#E5" mathvariant="normal">Ψ</mi><mrow><mo href="./36.2#E5" stretchy="false">(</mo><mi href="./36.2#E5" mathvariant="normal">U</mi><mo href="./36.2#E5" stretchy="false">)</mo></mrow></msup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" mathvariant="bold">x</mi><mo>)</mo></mrow></mrow></math>: umbilic canonical integral function</a>,
<a href="./36.1#p2.t1.r4" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./36.1#p2.t1.r4">y</mi></math>: real parameter</a>,
<a href="./36.1#p2.t1.r4" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m83.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./36.1#p2.t1.r4">z</mi></math>: real parameter</a> and
<a href="./36.1#p2.t1.r4" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m76.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./36.1#p2.t1.r4">x</mi></math>: real parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E25" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">36.2.25</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E25.png" altimg-height="29px" altimg-valign="-7px" altimg-width="273px" alttext="\Psi^{(\mathrm{E})}\left(x,-y,z\right)=\Psi^{(\mathrm{E})}\left(x,y,z\right)." display="block"><mrow><mrow><mrow><msup><mi href="./36.2#E5" mathvariant="normal">Ψ</mi><mrow><mo href="./36.2#E5" stretchy="false">(</mo><mi href="./36.2#E5" mathvariant="normal">E</mi><mo href="./36.2#E5" stretchy="false">)</mo></mrow></msup><mo></mo><mrow><mo>(</mo><mrow><mi href="./36.1#p2.t1.r4">x</mi><mo>,</mo><mrow><mo>-</mo><mi href="./36.1#p2.t1.r4">y</mi></mrow><mo>,</mo><mi href="./36.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msup><mi href="./36.2#E5" mathvariant="normal">Ψ</mi><mrow><mo href="./36.2#E5" stretchy="false">(</mo><mi href="./36.2#E5" mathvariant="normal">E</mi><mo href="./36.2#E5" stretchy="false">)</mo></mrow></msup><mo></mo><mrow><mo>(</mo><mrow><mi href="./36.1#p2.t1.r4">x</mi><mo>,</mo><mi href="./36.1#p2.t1.r4">y</mi><mo>,</mo><mi href="./36.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E25.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./36.2#E5" title="(36.2.5) ‣ Canonical Integrals ‣ §36.2(i) Definitions ‣ §36.2 Catastrophes and Canonical Integrals ‣ Properties ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m41.png" altimg-height="26px" altimg-valign="-7px" altimg-width="75px" alttext="\Psi^{(\mathrm{E})}\left(\NVar{\mathbf{x}}\right)" display="inline"><mrow><msup><mi href="./36.2#E5" mathvariant="normal">Ψ</mi><mrow><mo href="./36.2#E5" stretchy="false">(</mo><mi href="./36.2#E5" mathvariant="normal">E</mi><mo href="./36.2#E5" stretchy="false">)</mo></mrow></msup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" mathvariant="bold">x</mi><mo>)</mo></mrow></mrow></math>: elliptic umbilic canonical integral function</a>,
<a href="./36.1#p2.t1.r4" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./36.1#p2.t1.r4">y</mi></math>: real parameter</a>,
<a href="./36.1#p2.t1.r4" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m83.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./36.1#p2.t1.r4">z</mi></math>: real parameter</a> and
<a href="./36.1#p2.t1.r4" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m76.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./36.1#p2.t1.r4">x</mi></math>: real parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E26" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">36.2.26</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E26.png" altimg-height="41px" altimg-valign="-15px" altimg-width="445px" alttext="\Psi^{(\mathrm{E})}\left(-\tfrac{1}{2}x\mp\tfrac{\sqrt{3}}{2}y,\pm\tfrac{\sqrt%
{3}}{2}x-\tfrac{1}{2}y,z\right)=\Psi^{(\mathrm{E})}\left(x,y,z\right)," display="block"><mrow><mrow><mrow><msup><mi href="./36.2#E5" mathvariant="normal">Ψ</mi><mrow><mo href="./36.2#E5" stretchy="false">(</mo><mi href="./36.2#E5" mathvariant="normal">E</mi><mo href="./36.2#E5" stretchy="false">)</mo></mrow></msup><mo></mo><mrow><mo>(</mo><mrow><mrow><mrow><mo>-</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./36.1#p2.t1.r4">x</mi></mrow></mrow><mo>∓</mo><mrow><mstyle displaystyle="false"><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./36.1#p2.t1.r4">y</mi></mrow></mrow><mo>,</mo><mrow><mrow><mo>±</mo><mrow><mstyle displaystyle="false"><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./36.1#p2.t1.r4">x</mi></mrow></mrow><mo>-</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./36.1#p2.t1.r4">y</mi></mrow></mrow><mo>,</mo><mi href="./36.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msup><mi href="./36.2#E5" mathvariant="normal">Ψ</mi><mrow><mo href="./36.2#E5" stretchy="false">(</mo><mi href="./36.2#E5" mathvariant="normal">E</mi><mo href="./36.2#E5" stretchy="false">)</mo></mrow></msup><mo></mo><mrow><mo>(</mo><mrow><mi href="./36.1#p2.t1.r4">x</mi><mo>,</mo><mi href="./36.1#p2.t1.r4">y</mi><mo>,</mo><mi href="./36.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E26.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./36.2#E5" title="(36.2.5) ‣ Canonical Integrals ‣ §36.2(i) Definitions ‣ §36.2 Catastrophes and Canonical Integrals ‣ Properties ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m41.png" altimg-height="26px" altimg-valign="-7px" altimg-width="75px" alttext="\Psi^{(\mathrm{E})}\left(\NVar{\mathbf{x}}\right)" display="inline"><mrow><msup><mi href="./36.2#E5" mathvariant="normal">Ψ</mi><mrow><mo href="./36.2#E5" stretchy="false">(</mo><mi href="./36.2#E5" mathvariant="normal">E</mi><mo href="./36.2#E5" stretchy="false">)</mo></mrow></msup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" mathvariant="bold">x</mi><mo>)</mo></mrow></mrow></math>: elliptic umbilic canonical integral function</a>,
<a href="./36.1#p2.t1.r4" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./36.1#p2.t1.r4">y</mi></math>: real parameter</a>,
<a href="./36.1#p2.t1.r4" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m83.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./36.1#p2.t1.r4">z</mi></math>: real parameter</a> and
<a href="./36.1#p2.t1.r4" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m76.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./36.1#p2.t1.r4">x</mi></math>: real parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">(rotation by <math class="ltx_Math" altimg="m67.png" altimg-height="27px" altimg-valign="-9px" altimg-width="45px" alttext="\pm\tfrac{2}{3}\pi" display="inline"><mrow><mo>±</mo><mrow><mfrac><mn>2</mn><mn>3</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow></math> in <math class="ltx_Math" altimg="m75.png" altimg-height="16px" altimg-valign="-6px" altimg-width="35px" alttext="x,y" display="inline"><mrow><mi href="./36.1#p2.t1.r4">x</mi><mo>,</mo><mi href="./36.1#p2.t1.r4">y</mi></mrow></math> plane).</p>
<table id="E27" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">36.2.27</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E27.png" altimg-height="29px" altimg-valign="-7px" altimg-width="260px" alttext="\Psi^{(\mathrm{H})}\left(x,y,z\right)=\Psi^{(\mathrm{H})}\left(y,x,z\right)." display="block"><mrow><mrow><mrow><msup><mi href="./36.2#E5" mathvariant="normal">Ψ</mi><mrow><mo href="./36.2#E5" stretchy="false">(</mo><mi href="./36.2#E5" mathvariant="normal">H</mi><mo href="./36.2#E5" stretchy="false">)</mo></mrow></msup><mo></mo><mrow><mo>(</mo><mrow><mi href="./36.1#p2.t1.r4">x</mi><mo>,</mo><mi href="./36.1#p2.t1.r4">y</mi><mo>,</mo><mi href="./36.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msup><mi href="./36.2#E5" mathvariant="normal">Ψ</mi><mrow><mo href="./36.2#E5" stretchy="false">(</mo><mi href="./36.2#E5" mathvariant="normal">H</mi><mo href="./36.2#E5" stretchy="false">)</mo></mrow></msup><mo></mo><mrow><mo>(</mo><mrow><mi href="./36.1#p2.t1.r4">y</mi><mo>,</mo><mi href="./36.1#p2.t1.r4">x</mi><mo>,</mo><mi href="./36.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E27.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./36.2#E5" title="(36.2.5) ‣ Canonical Integrals ‣ §36.2(i) Definitions ‣ §36.2 Catastrophes and Canonical Integrals ‣ Properties ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="26px" altimg-valign="-7px" altimg-width="76px" alttext="\Psi^{(\mathrm{H})}\left(\NVar{\mathbf{x}}\right)" display="inline"><mrow><msup><mi href="./36.2#E5" mathvariant="normal">Ψ</mi><mrow><mo href="./36.2#E5" stretchy="false">(</mo><mi href="./36.2#E5" mathvariant="normal">H</mi><mo href="./36.2#E5" stretchy="false">)</mo></mrow></msup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" mathvariant="bold">x</mi><mo>)</mo></mrow></mrow></math>: hyperbolic umbilic canonical integral function</a>,
<a href="./36.1#p2.t1.r4" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./36.1#p2.t1.r4">y</mi></math>: real parameter</a>,
<a href="./36.1#p2.t1.r4" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m83.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./36.1#p2.t1.r4">z</mi></math>: real parameter</a> and
<a href="./36.1#p2.t1.r4" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m76.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./36.1#p2.t1.r4">x</mi></math>: real parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="iv" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§36.2(iv) </span>Addendum to </dd>
</dl>
</div>
</div>
<div id="SS4.p1" class="ltx_para">
<table id="E28" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">36.2.28</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_Math ltx_eqn_cell ltx_align_center"><math class="ltx_Math" alttext="\Psi^{(\mathrm{E})}\left(0,0,z\right)={\Psi^{\ast}}^{(\mathrm{E})}(0,0,-z)\\
=2\pi\sqrt{\frac{\pi z}{27}}\exp\left(\frac{2}{27}iz^{3}\right)\*\left(J_{-1/6%
}\left(\frac{2}{27}z^{3}\right)+iJ_{1/6}\left(\frac{2}{27}z^{3}\right)\right)," display="block"><mrow><mtable align="baseline 1" columnalign="left" columnspacing="0.1em" displaystyle="true"><mtr><mtd><mrow><msup><mi href="./36.2#E5" mathvariant="normal">Ψ</mi><mrow><mo href="./36.2#E5" stretchy="false">(</mo><mi href="./36.2#E5" mathvariant="normal">E</mi><mo href="./36.2#E5" stretchy="false">)</mo></mrow></msup><mo></mo><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mn>0</mn><mo>,</mo><mi href="./36.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow></mtd><mtd><mo>=</mo><mrow><mmultiscripts><mi mathvariant="normal">Ψ</mi><none></none><mo>∗</mo><none></none><mrow><mo stretchy="false">(</mo><mi mathvariant="normal">E</mi><mo stretchy="false">)</mo></mrow></mmultiscripts><mo></mo><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>,</mo><mrow><mo>-</mo><mi href="./36.1#p2.t1.r4">z</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><msqrt><mfrac><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi href="./36.1#p2.t1.r4">z</mi></mrow><mn>27</mn></mfrac></msqrt><mo></mo><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mrow><mo>(</mo><mrow><mfrac><mn>2</mn><mn>27</mn></mfrac><mo></mo><mi mathvariant="normal">i</mi><mo></mo><msup><mi href="./36.1#p2.t1.r4">z</mi><mn>3</mn></msup></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mo>(</mo><mrow><mrow><msub><mi href="./10.2#E2">J</mi><mrow><mo>-</mo><mrow><mn>1</mn><mo>/</mo><mn>6</mn></mrow></mrow></msub><mo></mo><mrow><mo>(</mo><mrow><mfrac><mn>2</mn><mn>27</mn></mfrac><mo></mo><msup><mi href="./36.1#p2.t1.r4">z</mi><mn>3</mn></msup></mrow><mo>)</mo></mrow></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mrow><mn>1</mn><mo>/</mo><mn>6</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mrow><mfrac><mn>2</mn><mn>27</mn></mfrac><mo></mo><msup><mi href="./36.1#p2.t1.r4">z</mi><mn>3</mn></msup></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>)</mo></mrow></mrow><mo>,</mo></mtd></mtr></mtable></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m84.png" altimg-height="19px" altimg-valign="-5px" altimg-width="51px" alttext="z\geq 0" display="inline"><mrow><mi href="./36.1#p2.t1.r4">z</mi><mo>≥</mo><mn>0</mn></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E28.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./36.2#E5" title="(36.2.5) ‣ Canonical Integrals ‣ §36.2(i) Definitions ‣ §36.2 Catastrophes and Canonical Integrals ‣ Properties ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m41.png" altimg-height="26px" altimg-valign="-7px" altimg-width="75px" alttext="\Psi^{(\mathrm{E})}\left(\NVar{\mathbf{x}}\right)" display="inline"><mrow><msup><mi href="./36.2#E5" mathvariant="normal">Ψ</mi><mrow><mo href="./36.2#E5" stretchy="false">(</mo><mi href="./36.2#E5" mathvariant="normal">E</mi><mo href="./36.2#E5" stretchy="false">)</mo></mrow></msup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" mathvariant="bold">x</mi><mo>)</mo></mrow></mrow></math>: elliptic umbilic canonical integral function</a>,
<a href="./4.2#E19" title="(4.2.19) ‣ §4.2(iii) The Exponential Function ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="16px" altimg-valign="-6px" altimg-width="48px" alttext="\exp\NVar{z}" display="inline"><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: exponential function</a> and
<a href="./36.1#p2.t1.r4" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m83.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./36.1#p2.t1.r4">z</mi></math>: real parameter</a>
</dd>
<dt>Addition (effective with 1.0.5):</dt>
<dd>
This equation has been added. For the proof see <cite class="ltx_cite ltx_citemacro_citet">Berry and Howls (</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS4.p2" class="ltx_para">
<table id="E29" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">36.2.29</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E29.png" altimg-height="56px" altimg-valign="-21px" altimg-width="624px" alttext="\Psi^{(\mathrm{H})}\left(0,0,z\right)={\Psi^{\ast}}^{(\mathrm{H})}(0,0,-z)=%
\frac{2^{1/3}}{\sqrt{3}}\exp\left(\frac{1}{27}iz^{3}\right)\Psi^{(\mathrm{E})}%
\left(0,0,-\frac{z}{2^{2/3}}\right)," display="block"><mrow><mrow><mrow><msup><mi href="./36.2#E5" mathvariant="normal">Ψ</mi><mrow><mo href="./36.2#E5" stretchy="false">(</mo><mi href="./36.2#E5" mathvariant="normal">H</mi><mo href="./36.2#E5" stretchy="false">)</mo></mrow></msup><mo></mo><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mn>0</mn><mo>,</mo><mi href="./36.1#p2.t1.r4">z</mi></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mmultiscripts><mi mathvariant="normal">Ψ</mi><none></none><mo>∗</mo><none></none><mrow><mo stretchy="false">(</mo><mi mathvariant="normal">H</mi><mo stretchy="false">)</mo></mrow></mmultiscripts><mo></mo><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>,</mo><mrow><mo>-</mo><mi href="./36.1#p2.t1.r4">z</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mfrac><msup><mn>2</mn><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup><msqrt><mn>3</mn></msqrt></mfrac><mo></mo><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>27</mn></mfrac><mo></mo><mi mathvariant="normal">i</mi><mo></mo><msup><mi href="./36.1#p2.t1.r4">z</mi><mn>3</mn></msup></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><msup><mi href="./36.2#E5" mathvariant="normal">Ψ</mi><mrow><mo href="./36.2#E5" stretchy="false">(</mo><mi href="./36.2#E5" mathvariant="normal">E</mi><mo href="./36.2#E5" stretchy="false">)</mo></mrow></msup><mo></mo><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mn>0</mn><mo>,</mo><mrow><mo>-</mo><mfrac><mi href="./36.1#p2.t1.r4">z</mi><msup><mn>2</mn><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup></mfrac></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m27.png" altimg-height="17px" altimg-valign="-4px" altimg-width="123px" alttext="-\infty<z<\infty" display="inline"><mrow><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mo><</mo><mi href="./36.1#p2.t1.r4">z</mi><mo><</mo><mi mathvariant="normal">∞</mi></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E29.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./36.2#E5" title="(36.2.5) ‣ Canonical Integrals ‣ §36.2(i) Definitions ‣ §36.2 Catastrophes and Canonical Integrals ‣ Properties ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m41.png" altimg-height="26px" altimg-valign="-7px" altimg-width="75px" alttext="\Psi^{(\mathrm{E})}\left(\NVar{\mathbf{x}}\right)" display="inline"><mrow><msup><mi href="./36.2#E5" mathvariant="normal">Ψ</mi><mrow><mo href="./36.2#E5" stretchy="false">(</mo><mi href="./36.2#E5" mathvariant="normal">E</mi><mo href="./36.2#E5" stretchy="false">)</mo></mrow></msup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" mathvariant="bold">x</mi><mo>)</mo></mrow></mrow></math>: elliptic umbilic canonical integral function</a>,
<a href="./4.2#E19" title="(4.2.19) ‣ §4.2(iii) The Exponential Function ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="16px" altimg-valign="-6px" altimg-width="48px" alttext="\exp\NVar{z}" display="inline"><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: exponential function</a>,
<a href="./36.2#E5" title="(36.2.5) ‣ Canonical Integrals ‣ §36.2(i) Definitions ‣ §36.2 Catastrophes and Canonical Integrals ‣ Properties ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="26px" altimg-valign="-7px" altimg-width="76px" alttext="\Psi^{(\mathrm{H})}\left(\NVar{\mathbf{x}}\right)" display="inline"><mrow><msup><mi href="./36.2#E5" mathvariant="normal">Ψ</mi><mrow><mo href="./36.2#E5" stretchy="false">(</mo><mi href="./36.2#E5" mathvariant="normal">H</mi><mo href="./36.2#E5" stretchy="false">)</mo></mrow></msup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" mathvariant="bold">x</mi><mo>)</mo></mrow></mrow></math>: hyperbolic umbilic canonical integral function</a> and
<a href="./36.1#p2.t1.r4" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m83.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./36.1#p2.t1.r4">z</mi></math>: real parameter</a>
</dd>
<dt>Addition (effective with 1.0.5):</dt>
<dd>
This equation has been added. For the proof see <cite class="ltx_cite ltx_citemacro_citet">Berry and Howls (</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS4.p3" class="ltx_para">
<p class="ltx_p">Here the functions <math class="ltx_Math" altimg="m85.png" altimg-height="23px" altimg-valign="-2px" altimg-width="53px" alttext="{\Psi^{\ast}}^{(\mathrm{E})}" display="inline"><mmultiscripts><mi mathvariant="normal">Ψ</mi><none></none><mo>∗</mo><none></none><mrow><mo stretchy="false">(</mo><mi mathvariant="normal">E</mi><mo stretchy="false">)</mo></mrow></mmultiscripts></math> and <math class="ltx_Math" altimg="m86.png" altimg-height="23px" altimg-valign="-2px" altimg-width="54px" alttext="{\Psi^{\ast}}^{(\mathrm{H})}" display="inline"><mmultiscripts><mi mathvariant="normal">Ψ</mi><none></none><mo>∗</mo><none></none><mrow><mo stretchy="false">(</mo><mi mathvariant="normal">H</mi><mo stretchy="false">)</mo></mrow></mmultiscripts></math>
are the complex conjugates of the functions <math class="ltx_Math" altimg="m40.png" altimg-height="21px" altimg-valign="-2px" altimg-width="44px" alttext="\Psi^{(\mathrm{E})}" display="inline"><msup><mi href="./36.2#E5" mathvariant="normal">Ψ</mi><mrow><mo href="./36.2#E5" stretchy="false">(</mo><mi href="./36.2#E5" mathvariant="normal">E</mi><mo href="./36.2#E5" stretchy="false">)</mo></mrow></msup></math> and <math class="ltx_Math" altimg="m43.png" altimg-height="21px" altimg-valign="-2px" altimg-width="45px" alttext="\Psi^{(\mathrm{H})}" display="inline"><msup><mi href="./36.2#E5" mathvariant="normal">Ψ</mi><mrow><mo href="./36.2#E5" stretchy="false">(</mo><mi href="./36.2#E5" mathvariant="normal">H</mi><mo href="./36.2#E5" stretchy="false">)</mo></mrow></msup></math>, respectively.</p>
</div>
</section>
</section>
</div>
<div class="ltx_page_footer">
<span></div>
</div>
</body></text>
</html>
</page>
<page>
<!DOCTYPE html><html>
<head>
<title>DLMF: 36.4 Bifurcation Sets</title>
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<div class="ltx_page_navlogo"></dd>
</dl>
</div>
</div>
<div id="Px1.p1" class="ltx_para">
<p class="ltx_p">These are real solutions <math class="ltx_Math" altimg="m46.png" altimg-height="24px" altimg-valign="-8px" altimg-width="47px" alttext="t_{j}(\mathbf{x})" display="inline"><mrow><msub><mi href="./36.4#Px1.p1">t</mi><mi>j</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi mathvariant="bold">x</mi><mo stretchy="false">)</mo></mrow></mrow></math>,
<math class="ltx_Math" altimg="m25.png" altimg-height="23px" altimg-valign="-7px" altimg-width="222px" alttext="1\leq j\leq j_{\max}(\mathbf{x})\leq K+1" display="inline"><mrow><mn>1</mn><mo>≤</mo><mi>j</mi><mo>≤</mo><mrow><msub><mi>j</mi><mi>max</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi mathvariant="bold">x</mi><mo stretchy="false">)</mo></mrow></mrow><mo>≤</mo><mrow><mi href="./36.1#p2.t1.r3">K</mi><mo>+</mo><mn>1</mn></mrow></mrow></math>, of
</p>
<table id="E1" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">36.4.1</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E1.png" altimg-height="47px" altimg-valign="-16px" altimg-width="184px" alttext="\frac{\partial}{\partial t}\Phi_{K}\left(t_{j}(\mathbf{x});\mathbf{x}\right)=0." display="block"><mrow><mrow><mrow><mfrac><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./36.1#p2.t1.r2">t</mi></mrow></mfrac><mo></mo><mrow><msub><mi href="./36.2#E1" mathvariant="normal">Φ</mi><mi href="./36.1#p2.t1.r3">K</mi></msub><mo></mo><mrow><mo>(</mo><mrow><msub><mi href="./36.4#Px1.p1">t</mi><mi>j</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi mathvariant="bold">x</mi><mo stretchy="false">)</mo></mrow></mrow><mo>;</mo><mi mathvariant="bold">x</mi><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mn>0</mn></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./36.2#E1" title="(36.2.1) ‣ Normal Forms Associated with Canonical Integrals: Cuspoid Catastrophe with Codimension K ‣ §36.2(i) Definitions ‣ §36.2 Catastrophes and Canonical Integrals ‣ Properties ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m31.png" altimg-height="23px" altimg-valign="-7px" altimg-width="81px" alttext="\Phi_{\NVar{K}}\left(\NVar{t};\NVar{\mathbf{x}}\right)" display="inline"><mrow><msub><mi href="./36.2#E1" mathvariant="normal">Φ</mi><mi class="ltx_nvar" href="./36.1#p2.t1.r3">K</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./36.1#p2.t1.r2">t</mi><mo>;</mo><mi class="ltx_nvar" mathvariant="bold">x</mi><mo>)</mo></mrow></mrow></math>: cuspoid catastrophe of codimension <math class="ltx_Math" altimg="m29.png" altimg-height="18px" altimg-valign="-2px" altimg-width="23px" alttext="K" display="inline"><mi href="./36.1#p2.t1.r3">K</mi></math></a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m34.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m41.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m47.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m47.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./36.1#p2.t1.r2" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m45.png" altimg-height="17px" altimg-valign="-2px" altimg-width="11px" alttext="t" display="inline"><mi href="./36.1#p2.t1.r2">t</mi></math>: variable</a>,
<a href="./36.1#p2.t1.r3" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="18px" altimg-valign="-2px" altimg-width="23px" alttext="K" display="inline"><mi href="./36.1#p2.t1.r3">K</mi></math>: codimension</a> and
<a href="./36.4#Px1.p1" title="Critical Points for Cuspoids ‣ §36.4(i) Formulas ‣ §36.4 Bifurcation Sets ‣ Properties ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m46.png" altimg-height="24px" altimg-valign="-8px" altimg-width="47px" alttext="t_{j}(\mathbf{x})" display="inline"><mrow><msub><mi href="./36.4#Px1.p1">t</mi><mi>j</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi mathvariant="bold">x</mi><mo stretchy="false">)</mo></mrow></mrow></math>: solutions</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px2" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Critical Points for Umbilics</h3>
<div id="Px2.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px2.p1" class="ltx_para">
<p class="ltx_p">These are real solutions <math class="ltx_Math" altimg="m40.png" altimg-height="24px" altimg-valign="-8px" altimg-width="122px" alttext="\{s_{j}(\mathbf{x}),t_{j}(\mathbf{x})\}" display="inline"><mrow><mo stretchy="false">{</mo><mrow><msub><mi href="./36.1#p2.t1.r2">s</mi><mi>j</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi mathvariant="bold">x</mi><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo><mrow><msub><mi href="./36.4#Px1.p1">t</mi><mi>j</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi mathvariant="bold">x</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">}</mo></mrow></math>,
<math class="ltx_Math" altimg="m24.png" altimg-height="23px" altimg-valign="-7px" altimg-width="179px" alttext="1\leq j\leq j_{\max}(\mathbf{x})\leq 4" display="inline"><mrow><mn>1</mn><mo>≤</mo><mi>j</mi><mo>≤</mo><mrow><msub><mi>j</mi><mi>max</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi mathvariant="bold">x</mi><mo stretchy="false">)</mo></mrow></mrow><mo>≤</mo><mn>4</mn></mrow></math>, of</p>
<table id="E2" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex1" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">36.4.2</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m13.png" altimg-height="47px" altimg-valign="-16px" altimg-width="209px" alttext="\displaystyle\frac{\partial}{\partial s}\Phi^{(\mathrm{U})}\left(s_{j}(\mathbf%
{x}),t_{j}(\mathbf{x});\mathbf{x}\right)" display="inline"><mrow><mstyle displaystyle="true"><mfrac><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./36.1#p2.t1.r2">s</mi></mrow></mfrac></mstyle><mo></mo><mrow><msup><mi href="./36.2#Px2" mathvariant="normal">Φ</mi><mrow><mo href="./36.2#Px2" stretchy="false">(</mo><mi href="./36.2#Px2" mathvariant="normal">U</mi><mo href="./36.2#Px2" stretchy="false">)</mo></mrow></msup><mo></mo><mrow><mo>(</mo><mrow><msub><mi href="./36.1#p2.t1.r2">s</mi><mi>j</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi mathvariant="bold">x</mi><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo><mrow><msub><mi href="./36.4#Px1.p1">t</mi><mi>j</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi mathvariant="bold">x</mi><mo stretchy="false">)</mo></mrow></mrow><mo>;</mo><mi mathvariant="bold">x</mi><mo>)</mo></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m7.png" altimg-height="22px" altimg-valign="-6px" altimg-width="43px" alttext="\displaystyle=0," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mn>0</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex2" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m14.png" altimg-height="47px" altimg-valign="-16px" altimg-width="206px" alttext="\displaystyle\frac{\partial}{\partial t}\Phi^{(\mathrm{U})}\left(s_{j}(\mathbf%
{x}),t_{j}(\mathbf{x});\mathbf{x}\right)" display="inline"><mrow><mstyle displaystyle="true"><mfrac><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./36.1#p2.t1.r2">t</mi></mrow></mfrac></mstyle><mo></mo><mrow><msup><mi href="./36.2#Px2" mathvariant="normal">Φ</mi><mrow><mo href="./36.2#Px2" stretchy="false">(</mo><mi href="./36.2#Px2" mathvariant="normal">U</mi><mo href="./36.2#Px2" stretchy="false">)</mo></mrow></msup><mo></mo><mrow><mo>(</mo><mrow><msub><mi href="./36.1#p2.t1.r2">s</mi><mi>j</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi mathvariant="bold">x</mi><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo><mrow><msub><mi href="./36.4#Px1.p1">t</mi><mi>j</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi mathvariant="bold">x</mi><mo stretchy="false">)</mo></mrow></mrow><mo>;</mo><mi mathvariant="bold">x</mi><mo>)</mo></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m8.png" altimg-height="18px" altimg-valign="-2px" altimg-width="43px" alttext="\displaystyle=0." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mn>0</mn></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m34.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m41.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m47.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m47.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./36.2#Px2" title="Normal Forms for Umbilic Catastrophes with Codimension = K 3 ‣ §36.2(i) Definitions ‣ §36.2 Catastrophes and Canonical Integrals ‣ Properties ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m30.png" altimg-height="26px" altimg-valign="-7px" altimg-width="109px" alttext="\Phi^{(\mathrm{U})}\left(\NVar{s},\NVar{t};\NVar{\mathbf{x}}\right)" display="inline"><mrow><msup><mi href="./36.2#Px2" mathvariant="normal">Φ</mi><mrow><mo href="./36.2#Px2" stretchy="false">(</mo><mi href="./36.2#Px2" mathvariant="normal">U</mi><mo href="./36.2#Px2" stretchy="false">)</mo></mrow></msup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./36.1#p2.t1.r2">s</mi><mo>,</mo><mi class="ltx_nvar" href="./36.1#p2.t1.r2">t</mi><mo>;</mo><mi class="ltx_nvar" mathvariant="bold">x</mi><mo>)</mo></mrow></mrow></math>: elliptic umbilic catastrophe for <math class="ltx_Math" altimg="m36.png" altimg-height="18px" altimg-valign="-2px" altimg-width="106px" alttext="\mathrm{U}=\mathrm{E}\mbox{ or }\mathrm{K}" display="inline"><mrow><mi mathvariant="normal">U</mi><mo>=</mo><mrow><mi mathvariant="normal">E</mi><mo></mo><mtext> or </mtext><mo></mo><mi mathvariant="normal">K</mi></mrow></mrow></math></a>,
<a href="./36.1#p2.t1.r2" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m45.png" altimg-height="17px" altimg-valign="-2px" altimg-width="11px" alttext="t" display="inline"><mi href="./36.1#p2.t1.r2">t</mi></math>: variable</a>,
<a href="./36.1#p2.t1.r2" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="s" display="inline"><mi href="./36.1#p2.t1.r2">s</mi></math>: variable</a> and
<a href="./36.4#Px1.p1" title="Critical Points for Cuspoids ‣ §36.4(i) Formulas ‣ §36.4 Bifurcation Sets ‣ Properties ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m46.png" altimg-height="24px" altimg-valign="-8px" altimg-width="47px" alttext="t_{j}(\mathbf{x})" display="inline"><mrow><msub><mi href="./36.4#Px1.p1">t</mi><mi>j</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi mathvariant="bold">x</mi><mo stretchy="false">)</mo></mrow></mrow></math>: solutions</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px3" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Bifurcation (Catastrophe) Set for Cuspoids</h3>
<div id="Px3.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px3.p1" class="ltx_para">
<p class="ltx_p">This is the codimension-one surface in <math class="ltx_Math" altimg="m35.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\mathbf{x}" display="inline"><mi mathvariant="bold">x</mi></math> space where critical points
coalesce, satisfying () and</p>
<table id="E3" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">36.4.3</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E3.png" altimg-height="51px" altimg-valign="-18px" altimg-width="157px" alttext="\frac{{\partial}^{2}}{{\partial t}^{2}}\Phi_{K}\left(t;\mathbf{x}\right)=0." display="block"><mrow><mrow><mrow><mfrac><mpadded width="-1.7pt"><msup><mo href="./1.5#E3">∂</mo><mn>2</mn></msup></mpadded><msup><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./36.1#p2.t1.r2">t</mi></mrow><mn>2</mn></msup></mfrac><mo></mo><mrow><msub><mi href="./36.2#E1" mathvariant="normal">Φ</mi><mi href="./36.1#p2.t1.r3">K</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./36.1#p2.t1.r2">t</mi><mo>;</mo><mi mathvariant="bold">x</mi><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mn>0</mn></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./36.2#E1" title="(36.2.1) ‣ Normal Forms Associated with Canonical Integrals: Cuspoid Catastrophe with Codimension K ‣ §36.2(i) Definitions ‣ §36.2 Catastrophes and Canonical Integrals ‣ Properties ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m31.png" altimg-height="23px" altimg-valign="-7px" altimg-width="81px" alttext="\Phi_{\NVar{K}}\left(\NVar{t};\NVar{\mathbf{x}}\right)" display="inline"><mrow><msub><mi href="./36.2#E1" mathvariant="normal">Φ</mi><mi class="ltx_nvar" href="./36.1#p2.t1.r3">K</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./36.1#p2.t1.r2">t</mi><mo>;</mo><mi class="ltx_nvar" mathvariant="bold">x</mi><mo>)</mo></mrow></mrow></math>: cuspoid catastrophe of codimension <math class="ltx_Math" altimg="m29.png" altimg-height="18px" altimg-valign="-2px" altimg-width="23px" alttext="K" display="inline"><mi href="./36.1#p2.t1.r3">K</mi></math></a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m34.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m41.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m47.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m47.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./36.1#p2.t1.r2" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m45.png" altimg-height="17px" altimg-valign="-2px" altimg-width="11px" alttext="t" display="inline"><mi href="./36.1#p2.t1.r2">t</mi></math>: variable</a> and
<a href="./36.1#p2.t1.r3" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="18px" altimg-valign="-2px" altimg-width="23px" alttext="K" display="inline"><mi href="./36.1#p2.t1.r3">K</mi></math>: codimension</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px4" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Bifurcation (Catastrophe) Set for Umbilics</h3>
<div id="Px4.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px4.p1" class="ltx_para">
<p class="ltx_p">This is the codimension-one surface in <math class="ltx_Math" altimg="m35.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\mathbf{x}" display="inline"><mi mathvariant="bold">x</mi></math> space where critical points
coalesce, satisfying () and</p>
<table id="E4" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">36.4.4</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E4.png" altimg-height="58px" altimg-valign="-21px" altimg-width="534px" alttext="\frac{{\partial}^{2}}{{\partial s}^{2}}\Phi^{(\mathrm{U})}\left(s,t;\mathbf{x}%
\right)\frac{{\partial}^{2}}{{\partial t}^{2}}\Phi^{(\mathrm{U})}\left(s,t;%
\mathbf{x}\right)-\left(\frac{{\partial}^{2}}{\partial s\partial t}\Phi^{(%
\mathrm{U})}\left(s,t;\mathbf{x}\right)\right)^{2}=0." display="block"><mrow><mrow><mrow><mrow><mfrac><mpadded width="-1.7pt"><msup><mo href="./1.5#E3">∂</mo><mn>2</mn></msup></mpadded><msup><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./36.1#p2.t1.r2">s</mi></mrow><mn>2</mn></msup></mfrac><mo></mo><mrow><mrow><msup><mi href="./36.2#Px2" mathvariant="normal">Φ</mi><mrow><mo href="./36.2#Px2" stretchy="false">(</mo><mi href="./36.2#Px2" mathvariant="normal">U</mi><mo href="./36.2#Px2" stretchy="false">)</mo></mrow></msup><mo></mo><mrow><mo>(</mo><mi href="./36.1#p2.t1.r2">s</mi><mo>,</mo><mi href="./36.1#p2.t1.r2">t</mi><mo>;</mo><mi mathvariant="bold">x</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mfrac><mpadded width="-1.7pt"><msup><mo href="./1.5#E3">∂</mo><mn>2</mn></msup></mpadded><msup><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./36.1#p2.t1.r2">t</mi></mrow><mn>2</mn></msup></mfrac><mo></mo><mrow><msup><mi href="./36.2#Px2" mathvariant="normal">Φ</mi><mrow><mo href="./36.2#Px2" stretchy="false">(</mo><mi href="./36.2#Px2" mathvariant="normal">U</mi><mo href="./36.2#Px2" stretchy="false">)</mo></mrow></msup><mo></mo><mrow><mo>(</mo><mi href="./36.1#p2.t1.r2">s</mi><mo>,</mo><mi href="./36.1#p2.t1.r2">t</mi><mo>;</mo><mi mathvariant="bold">x</mi><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>-</mo><msup><mrow><mo>(</mo><mrow><mfrac><mrow><mpadded width="-1.1pt"><msup><mo href="./1.5#E3">∂</mo><mn>2</mn></msup></mpadded><mo href="./1.5#E3"></mo></mrow><mrow><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./36.1#p2.t1.r2">s</mi></mrow><mo></mo><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./36.1#p2.t1.r2">t</mi></mrow></mrow></mfrac><mo></mo><mrow><msup><mi href="./36.2#Px2" mathvariant="normal">Φ</mi><mrow><mo href="./36.2#Px2" stretchy="false">(</mo><mi href="./36.2#Px2" mathvariant="normal">U</mi><mo href="./36.2#Px2" stretchy="false">)</mo></mrow></msup><mo></mo><mrow><mo>(</mo><mi href="./36.1#p2.t1.r2">s</mi><mo>,</mo><mi href="./36.1#p2.t1.r2">t</mi><mo>;</mo><mi mathvariant="bold">x</mi><mo>)</mo></mrow></mrow></mrow><mo>)</mo></mrow><mn>2</mn></msup></mrow><mo>=</mo><mn>0</mn></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E4.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m34.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m41.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m47.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m47.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./36.2#Px2" title="Normal Forms for Umbilic Catastrophes with Codimension = K 3 ‣ §36.2(i) Definitions ‣ §36.2 Catastrophes and Canonical Integrals ‣ Properties ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m30.png" altimg-height="26px" altimg-valign="-7px" altimg-width="109px" alttext="\Phi^{(\mathrm{U})}\left(\NVar{s},\NVar{t};\NVar{\mathbf{x}}\right)" display="inline"><mrow><msup><mi href="./36.2#Px2" mathvariant="normal">Φ</mi><mrow><mo href="./36.2#Px2" stretchy="false">(</mo><mi href="./36.2#Px2" mathvariant="normal">U</mi><mo href="./36.2#Px2" stretchy="false">)</mo></mrow></msup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./36.1#p2.t1.r2">s</mi><mo>,</mo><mi class="ltx_nvar" href="./36.1#p2.t1.r2">t</mi><mo>;</mo><mi class="ltx_nvar" mathvariant="bold">x</mi><mo>)</mo></mrow></mrow></math>: elliptic umbilic catastrophe for <math class="ltx_Math" altimg="m36.png" altimg-height="18px" altimg-valign="-2px" altimg-width="106px" alttext="\mathrm{U}=\mathrm{E}\mbox{ or }\mathrm{K}" display="inline"><mrow><mi mathvariant="normal">U</mi><mo>=</mo><mrow><mi mathvariant="normal">E</mi><mo></mo><mtext> or </mtext><mo></mo><mi mathvariant="normal">K</mi></mrow></mrow></math></a>,
<a href="./36.1#p2.t1.r2" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m45.png" altimg-height="17px" altimg-valign="-2px" altimg-width="11px" alttext="t" display="inline"><mi href="./36.1#p2.t1.r2">t</mi></math>: variable</a> and
<a href="./36.1#p2.t1.r2" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="s" display="inline"><mi href="./36.1#p2.t1.r2">s</mi></math>: variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px5" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Special Cases</h3>
<div id="Px5.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px5.p1" class="ltx_para">
<p class="ltx_p"><math class="ltx_Math" altimg="m26.png" altimg-height="18px" altimg-valign="-2px" altimg-width="59px" alttext="K=1" display="inline"><mrow><mi href="./36.1#p2.t1.r3">K</mi><mo>=</mo><mn>1</mn></mrow></math>, fold bifurcation set:
</p>
<table id="E5" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">36.4.5</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E5.png" altimg-height="18px" altimg-valign="-2px" altimg-width="59px" alttext="x=0." display="block"><mrow><mrow><mi href="./36.1#p2.t1.r4">x</mi><mo>=</mo><mn>0</mn></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E5.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./36.1#p2.t1.r4" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./36.1#p2.t1.r4">x</mi></math>: real parameter</a></dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="Px5.p2" class="ltx_para">
<p class="ltx_p"><math class="ltx_Math" altimg="m27.png" altimg-height="18px" altimg-valign="-2px" altimg-width="59px" alttext="K=2" display="inline"><mrow><mi href="./36.1#p2.t1.r3">K</mi><mo>=</mo><mn>2</mn></mrow></math>, cusp bifurcation set:
</p>
<table id="E6" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">36.4.6</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E6.png" altimg-height="27px" altimg-valign="-6px" altimg-width="123px" alttext="27x^{2}=-8y^{3}." display="block"><mrow><mrow><mrow><mn>27</mn><mo></mo><msup><mi href="./36.1#p2.t1.r4">x</mi><mn>2</mn></msup></mrow><mo>=</mo><mrow><mo>-</mo><mrow><mn>8</mn><mo></mo><msup><mi href="./36.1#p2.t1.r4">y</mi><mn>3</mn></msup></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E6.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./36.1#p2.t1.r4" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./36.1#p2.t1.r4">y</mi></math>: real parameter</a> and
<a href="./36.1#p2.t1.r4" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./36.1#p2.t1.r4">x</mi></math>: real parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="Px5.p3" class="ltx_para">
<p class="ltx_p"><math class="ltx_Math" altimg="m28.png" altimg-height="18px" altimg-valign="-2px" altimg-width="59px" alttext="K=3" display="inline"><mrow><mi href="./36.1#p2.t1.r3">K</mi><mo>=</mo><mn>3</mn></mrow></math>, swallowtail bifurcation set:
</p>
<table id="E7" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex3" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="3" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">36.4.7</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m16.png" altimg-height="14px" altimg-valign="-2px" altimg-width="17px" alttext="\displaystyle x" display="inline"><mi href="./36.1#p2.t1.r4">x</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m9.png" altimg-height="28px" altimg-valign="-7px" altimg-width="135px" alttext="\displaystyle=3t^{2}(z+5t^{2})," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mn>3</mn><mo></mo><msup><mi href="./36.1#p2.t1.r2">t</mi><mn>2</mn></msup><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./36.1#p2.t1.r4">z</mi><mo>+</mo><mrow><mn>5</mn><mo></mo><msup><mi href="./36.1#p2.t1.r2">t</mi><mn>2</mn></msup></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex4" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m17.png" altimg-height="18px" altimg-valign="-6px" altimg-width="17px" alttext="\displaystyle y" display="inline"><mi href="./36.1#p2.t1.r4">y</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell">
<math class="ltx_Math" altimg="m6.png" altimg-height="28px" altimg-valign="-7px" altimg-width="146px" alttext="\displaystyle=-t(3z+10t^{2})" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mo>-</mo><mrow><mi href="./36.1#p2.t1.r2">t</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>3</mn><mo></mo><mi href="./36.1#p2.t1.r4">z</mi></mrow><mo>+</mo><mrow><mn>10</mn><mo></mo><msup><mi href="./36.1#p2.t1.r2">t</mi><mn>2</mn></msup></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow></math>,</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m21.png" altimg-height="18px" altimg-valign="-4px" altimg-width="120px" alttext="-\infty<t<\infty" display="inline"><mrow><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mo><</mo><mi href="./36.1#p2.t1.r2">t</mi><mo><</mo><mi mathvariant="normal">∞</mi></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E7.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./36.1#p2.t1.r4" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./36.1#p2.t1.r4">y</mi></math>: real parameter</a>,
<a href="./36.1#p2.t1.r4" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./36.1#p2.t1.r4">z</mi></math>: real parameter</a>,
<a href="./36.1#p2.t1.r2" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m45.png" altimg-height="17px" altimg-valign="-2px" altimg-width="11px" alttext="t" display="inline"><mi href="./36.1#p2.t1.r2">t</mi></math>: variable</a> and
<a href="./36.1#p2.t1.r4" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./36.1#p2.t1.r4">x</mi></math>: real parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="Px5.p4" class="ltx_para">
<p class="ltx_p">Swallowtail self-intersection line:</p>
<table id="E8" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex5" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="3" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">36.4.8</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m17.png" altimg-height="18px" altimg-valign="-6px" altimg-width="17px" alttext="\displaystyle y" display="inline"><mi href="./36.1#p2.t1.r4">y</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m7.png" altimg-height="22px" altimg-valign="-6px" altimg-width="43px" alttext="\displaystyle=0," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mn>0</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex6" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m18.png" altimg-height="14px" altimg-valign="-2px" altimg-width="16px" alttext="\displaystyle z" display="inline"><mi href="./36.1#p2.t1.r4">z</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m15.png" altimg-height="22px" altimg-valign="-6px" altimg-width="43px" alttext="\displaystyle\leq 0," display="inline"><mrow><mrow><mi></mi><mo>≤</mo><mn>0</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex7" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m16.png" altimg-height="14px" altimg-valign="-2px" altimg-width="17px" alttext="\displaystyle x" display="inline"><mi href="./36.1#p2.t1.r4">x</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m12.png" altimg-height="30px" altimg-valign="-9px" altimg-width="72px" alttext="\displaystyle=\tfrac{9}{20}z^{2}." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mfrac><mn>9</mn><mn>20</mn></mfrac><mo></mo><msup><mi href="./36.1#p2.t1.r4">z</mi><mn>2</mn></msup></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E8.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./36.1#p2.t1.r4" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./36.1#p2.t1.r4">y</mi></math>: real parameter</a>,
<a href="./36.1#p2.t1.r4" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./36.1#p2.t1.r4">z</mi></math>: real parameter</a> and
<a href="./36.1#p2.t1.r4" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./36.1#p2.t1.r4">x</mi></math>: real parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="Px5.p5" class="ltx_para">
<p class="ltx_p">Swallowtail cusp lines (ribs):</p>
<table id="E9" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex8" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="3" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">36.4.9</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m18.png" altimg-height="14px" altimg-valign="-2px" altimg-width="16px" alttext="\displaystyle z" display="inline"><mi href="./36.1#p2.t1.r4">z</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m15.png" altimg-height="22px" altimg-valign="-6px" altimg-width="43px" alttext="\displaystyle\leq 0," display="inline"><mrow><mrow><mi></mi><mo>≤</mo><mn>0</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex9" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m16.png" altimg-height="14px" altimg-valign="-2px" altimg-width="17px" alttext="\displaystyle x" display="inline"><mi href="./36.1#p2.t1.r4">x</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m5.png" altimg-height="30px" altimg-valign="-9px" altimg-width="88px" alttext="\displaystyle=-\tfrac{3}{20}z^{2}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo>-</mo><mrow><mfrac><mn>3</mn><mn>20</mn></mfrac><mo></mo><msup><mi href="./36.1#p2.t1.r4">z</mi><mn>2</mn></msup></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex10" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m1.png" altimg-height="27px" altimg-valign="-6px" altimg-width="45px" alttext="\displaystyle 10y^{2}" display="inline"><mrow><mn>10</mn><mo></mo><msup><mi href="./36.1#p2.t1.r4">y</mi><mn>2</mn></msup></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m2.png" altimg-height="25px" altimg-valign="-4px" altimg-width="77px" alttext="\displaystyle=-4z^{3}." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo>-</mo><mrow><mn>4</mn><mo></mo><msup><mi href="./36.1#p2.t1.r4">z</mi><mn>3</mn></msup></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E9.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./36.1#p2.t1.r4" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./36.1#p2.t1.r4">y</mi></math>: real parameter</a>,
<a href="./36.1#p2.t1.r4" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./36.1#p2.t1.r4">z</mi></math>: real parameter</a> and
<a href="./36.1#p2.t1.r4" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./36.1#p2.t1.r4">x</mi></math>: real parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="Px5.p6" class="ltx_para">
<p class="ltx_p">Elliptic umbilic bifurcation set (codimension three):
for fixed <math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./36.1#p2.t1.r4">z</mi></math>, the section of the bifurcation set is a three-cusped astroid
</p>
<table id="E10" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex11" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="3" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">36.4.10</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m16.png" altimg-height="14px" altimg-valign="-2px" altimg-width="17px" alttext="\displaystyle x" display="inline"><mi href="./36.1#p2.t1.r4">x</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m10.png" altimg-height="30px" altimg-valign="-9px" altimg-width="246px" alttext="\displaystyle=\tfrac{1}{3}z^{2}(-\cos\left(2\phi\right)-2\cos\phi)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac><mo></mo><msup><mi href="./36.1#p2.t1.r4">z</mi><mn>2</mn></msup><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mo>-</mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><mi>ϕ</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>-</mo><mrow><mn>2</mn><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi>ϕ</mi></mrow></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex12" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m17.png" altimg-height="18px" altimg-valign="-6px" altimg-width="17px" alttext="\displaystyle y" display="inline"><mi href="./36.1#p2.t1.r4">y</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell">
<math class="ltx_Math" altimg="m11.png" altimg-height="30px" altimg-valign="-9px" altimg-width="217px" alttext="\displaystyle=\tfrac{1}{3}z^{2}(\sin\left(2\phi\right)-2\sin\phi)" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac><mo></mo><msup><mi href="./36.1#p2.t1.r4">z</mi><mn>2</mn></msup><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><mi>ϕ</mi></mrow><mo>)</mo></mrow></mrow><mo>-</mo><mrow><mn>2</mn><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi>ϕ</mi></mrow></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></math>,</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m23.png" altimg-height="21px" altimg-valign="-6px" altimg-width="101px" alttext="0\leq\phi\leq 2\pi" display="inline"><mrow><mn>0</mn><mo>≤</mo><mi>ϕ</mi><mo>≤</mo><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E10.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m32.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./36.1#p2.t1.r4" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./36.1#p2.t1.r4">y</mi></math>: real parameter</a>,
<a href="./36.1#p2.t1.r4" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./36.1#p2.t1.r4">z</mi></math>: real parameter</a> and
<a href="./36.1#p2.t1.r4" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./36.1#p2.t1.r4">x</mi></math>: real parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="Px5.p7" class="ltx_para">
<p class="ltx_p">Elliptic umbilic cusp lines (ribs):</p>
<table id="E11" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">36.4.11</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E11.png" altimg-height="30px" altimg-valign="-9px" altimg-width="234px" alttext="x+iy=-z^{2}\exp\left(\tfrac{2}{3}i\pi m\right)," display="block"><mrow><mrow><mrow><mi href="./36.1#p2.t1.r4">x</mi><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./36.1#p2.t1.r4">y</mi></mrow></mrow><mo>=</mo><mrow><mo>-</mo><mrow><msup><mi href="./36.1#p2.t1.r4">z</mi><mn>2</mn></msup><mo></mo><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mrow><mo>(</mo><mrow><mstyle displaystyle="false"><mfrac><mn>2</mn><mn>3</mn></mfrac></mstyle><mo></mo><mi mathvariant="normal">i</mi><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi href="./36.1#p2.t1.r1">m</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m42.png" altimg-height="20px" altimg-valign="-6px" altimg-width="96px" alttext="m=0,1,2" display="inline"><mrow><mi href="./36.1#p2.t1.r1">m</mi><mo>=</mo><mrow><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E11.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.2#E19" title="(4.2.19) ‣ §4.2(iii) The Exponential Function ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m33.png" altimg-height="16px" altimg-valign="-6px" altimg-width="48px" alttext="\exp\NVar{z}" display="inline"><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: exponential function</a>,
<a href="./36.1#p2.t1.r4" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./36.1#p2.t1.r4">y</mi></math>: real parameter</a>,
<a href="./36.1#p2.t1.r4" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./36.1#p2.t1.r4">z</mi></math>: real parameter</a>,
<a href="./36.1#p2.t1.r1" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./36.1#p2.t1.r1">n</mi></math>: integer</a> and
<a href="./36.1#p2.t1.r4" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./36.1#p2.t1.r4">x</mi></math>: real parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="Px5.p8" class="ltx_para">
<p class="ltx_p">Hyperbolic umbilic bifurcation set (codimension three):
</p>
<table id="E12" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex13" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="3" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">36.4.12</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m16.png" altimg-height="14px" altimg-valign="-2px" altimg-width="17px" alttext="\displaystyle x" display="inline"><mi href="./36.1#p2.t1.r4">x</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m4.png" altimg-height="30px" altimg-valign="-9px" altimg-width="287px" alttext="\displaystyle=-\tfrac{1}{12}z^{2}(\exp\left(2\tau\right)\pm 2\exp\left(-\tau%
\right))," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>12</mn></mfrac><mo></mo><msup><mi href="./36.1#p2.t1.r4">z</mi><mn>2</mn></msup><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><mi>τ</mi></mrow><mo>)</mo></mrow></mrow><mo>±</mo><mrow><mn>2</mn><mo></mo><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mi>τ</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex14" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m17.png" altimg-height="18px" altimg-valign="-6px" altimg-width="17px" alttext="\displaystyle y" display="inline"><mi href="./36.1#p2.t1.r4">y</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell">
<math class="ltx_Math" altimg="m3.png" altimg-height="30px" altimg-valign="-9px" altimg-width="281px" alttext="\displaystyle=-\tfrac{1}{12}z^{2}(\exp\left(-2\tau\right)\pm 2\exp\left(\tau%
\right))" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>12</mn></mfrac><mo></mo><msup><mi href="./36.1#p2.t1.r4">z</mi><mn>2</mn></msup><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mrow><mn>2</mn><mo></mo><mi>τ</mi></mrow></mrow><mo>)</mo></mrow></mrow><mo>±</mo><mrow><mn>2</mn><mo></mo><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mrow><mo>(</mo><mi>τ</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow></math>,</td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m22.png" altimg-height="19px" altimg-valign="-5px" altimg-width="129px" alttext="-\infty\leq\tau<\infty." display="inline"><mrow><mrow><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mo>≤</mo><mi>τ</mi><mo><</mo><mi mathvariant="normal">∞</mi></mrow><mo>.</mo></mrow></math></span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E12.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.2#E19" title="(4.2.19) ‣ §4.2(iii) The Exponential Function ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m33.png" altimg-height="16px" altimg-valign="-6px" altimg-width="48px" alttext="\exp\NVar{z}" display="inline"><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: exponential function</a>,
<a href="./36.1#p2.t1.r4" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./36.1#p2.t1.r4">y</mi></math>: real parameter</a>,
<a href="./36.1#p2.t1.r4" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./36.1#p2.t1.r4">z</mi></math>: real parameter</a> and
<a href="./36.1#p2.t1.r4" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./36.1#p2.t1.r4">x</mi></math>: real parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">The <math class="ltx_Math" altimg="m19.png" altimg-height="17px" altimg-valign="-4px" altimg-width="20px" alttext="+" display="inline"><mo>+</mo></math> sign labels the cusped sheet; the <math class="ltx_Math" altimg="m20.png" altimg-height="17px" altimg-valign="-4px" altimg-width="20px" alttext="-" display="inline"><mo>-</mo></math> sign labels the sheet that is
smooth for <math class="ltx_Math" altimg="m50.png" altimg-height="21px" altimg-valign="-6px" altimg-width="51px" alttext="z\not=0" display="inline"><mrow><mi href="./36.1#p2.t1.r4">z</mi><mo>≠</mo><mn>0</mn></mrow></math> (see Figure ).</p>
</div>
<div id="Px5.p9" class="ltx_para">
<p class="ltx_p">Hyperbolic umbilic cusp line (rib):</p>
<table id="E13" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">36.4.13</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E13.png" altimg-height="30px" altimg-valign="-9px" altimg-width="134px" alttext="x=y=-\tfrac{1}{4}z^{2}." display="block"><mrow><mrow><mi href="./36.1#p2.t1.r4">x</mi><mo>=</mo><mi href="./36.1#p2.t1.r4">y</mi><mo>=</mo><mrow><mo>-</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>4</mn></mfrac></mstyle><mo></mo><msup><mi href="./36.1#p2.t1.r4">z</mi><mn>2</mn></msup></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E13.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./36.1#p2.t1.r4" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./36.1#p2.t1.r4">y</mi></math>: real parameter</a>,
<a href="./36.1#p2.t1.r4" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./36.1#p2.t1.r4">z</mi></math>: real parameter</a> and
<a href="./36.1#p2.t1.r4" title="§36.1 Special Notation ‣ Notation ‣ Chapter 36 Integrals with Coalescing Saddles" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./36.1#p2.t1.r4">x</mi></math>: real parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="Px5.p10" class="ltx_para">
<p class="ltx_p">For derivations of the results in this subsection see
<cite class="ltx_cite ltx_citemacro_citet">Poston and Stewart (</div>
</div>
</body></text>
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<page>
<!DOCTYPE html><html>
<head>
<title>DLMF: 8.12 Uniform Asymptotic Expansions for Large Parameter</title>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
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<div class="ltx_page_navlogo"></dd>
</dl>
</div>
</div>
<div id="p1" class="ltx_para">
<p class="ltx_p">Define
</p>
<table id="E1" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex1" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">8.12.1</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m39.png" altimg-height="19px" altimg-valign="-2px" altimg-width="18px" alttext="\displaystyle\lambda" display="inline"><mi href="./8.12#p1">λ</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m25.png" altimg-height="25px" altimg-valign="-7px" altimg-width="63px" alttext="\displaystyle=z/a," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./8.1#p1.t1.r2">z</mi><mo>/</mo><mi href="./8.1#p1.t1.r3">a</mi></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex2" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m37.png" altimg-height="18px" altimg-valign="-6px" altimg-width="17px" alttext="\displaystyle\eta" display="inline"><mi href="./8.12#p1">η</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m15.png" altimg-height="31px" altimg-valign="-7px" altimg-width="202px" alttext="\displaystyle=\left(2(\lambda-1-\ln\lambda)\right)^{1/2}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><msup><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./8.12#p1">λ</mi><mo>-</mo><mn>1</mn><mo>-</mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi href="./8.12#p1">λ</mi></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a>,
<a href="./8.1#p1.t1.r2" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m121.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./8.1#p1.t1.r2">z</mi></math>: complex variable</a>,
<a href="./8.1#p1.t1.r3" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m98.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./8.1#p1.t1.r3">a</mi></math>: parameter</a>,
<a href="./8.12#p1" title="§8.12 Uniform Asymptotic Expansions for Large Parameter ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="\eta(\lambda)" display="inline"><mrow><mi href="./8.12#p1">η</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./8.12#p1">λ</mi><mo stretchy="false">)</mo></mrow></mrow></math></a> and
<a href="./8.12#p1" title="§8.12 Uniform Asymptotic Expansions for Large Parameter ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="18px" altimg-valign="-2px" altimg-width="16px" alttext="\lambda" display="inline"><mi href="./8.12#p1">λ</mi></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where the branch of the square root is continuous and satisfies
<math class="ltx_Math" altimg="m78.png" altimg-height="23px" altimg-valign="-7px" altimg-width="115px" alttext="\eta(\lambda)\sim\lambda-1" display="inline"><mrow><mrow><mi href="./8.12#p1">η</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./8.12#p1">λ</mi><mo stretchy="false">)</mo></mrow></mrow><mo href="./2.1#E1">∼</mo><mrow><mi href="./8.12#p1">λ</mi><mo>-</mo><mn>1</mn></mrow></mrow></math> as <math class="ltx_Math" altimg="m83.png" altimg-height="18px" altimg-valign="-2px" altimg-width="57px" alttext="\lambda\to 1" display="inline"><mrow><mi href="./8.12#p1">λ</mi><mo>→</mo><mn>1</mn></mrow></math>. Then</p>
<table id="E2" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex3" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">8.12.2</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m40.png" altimg-height="30px" altimg-valign="-9px" altimg-width="38px" alttext="\displaystyle\tfrac{1}{2}\eta^{2}" display="inline"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><msup><mi href="./8.12#p1">η</mi><mn>2</mn></msup></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m14.png" altimg-height="23px" altimg-valign="-6px" altimg-width="134px" alttext="\displaystyle=\lambda-1-\ln\lambda," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./8.12#p1">λ</mi><mo>-</mo><mn>1</mn><mo>-</mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi href="./8.12#p1">λ</mi></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex4" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m38.png" altimg-height="47px" altimg-valign="-16px" altimg-width="34px" alttext="\displaystyle\frac{\mathrm{d}\eta}{\mathrm{d}\lambda}" display="inline"><mstyle displaystyle="true"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi href="./8.12#p1">η</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi href="./8.12#p1">λ</mi></mrow></mfrac></mstyle></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m13.png" altimg-height="50px" altimg-valign="-20px" altimg-width="83px" alttext="\displaystyle=\frac{\lambda-1}{\lambda\eta}." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mstyle displaystyle="true"><mfrac><mrow><mi href="./8.12#p1">λ</mi><mo>-</mo><mn>1</mn></mrow><mrow><mi href="./8.12#p1">λ</mi><mo></mo><mi href="./8.12#p1">η</mi></mrow></mfrac></mstyle></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#E4" title="(1.4.4) ‣ §1.4(iii) Derivatives ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m80.png" altimg-height="29px" altimg-valign="-9px" altimg-width="27px" alttext="\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: derivative of <math class="ltx_Math" altimg="m104.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m119.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a>,
<a href="./8.12#p1" title="§8.12 Uniform Asymptotic Expansions for Large Parameter ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="\eta(\lambda)" display="inline"><mrow><mi href="./8.12#p1">η</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./8.12#p1">λ</mi><mo stretchy="false">)</mo></mrow></mrow></math></a> and
<a href="./8.12#p1" title="§8.12 Uniform Asymptotic Expansions for Large Parameter ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="18px" altimg-valign="-2px" altimg-width="16px" alttext="\lambda" display="inline"><mi href="./8.12#p1">λ</mi></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Also, denote</p>
<table id="E3" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">8.12.3</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E3.png" altimg-height="41px" altimg-valign="-15px" altimg-width="337px" alttext="P\left(a,z\right)=\tfrac{1}{2}\operatorname{erfc}\left(-\eta\sqrt{a/2}\right)-%
S(a,\eta)," display="block"><mrow><mrow><mrow><mi href="./8.2#E4">P</mi><mo></mo><mrow><mo>(</mo><mi href="./8.1#p1.t1.r3">a</mi><mo>,</mo><mi href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mrow><mi href="./7.2#E2">erfc</mi><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mrow><mi href="./8.12#p1">η</mi><mo></mo><msqrt><mrow><mi href="./8.1#p1.t1.r3">a</mi><mo>/</mo><mn>2</mn></mrow></msqrt></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>-</mo><mrow><mi href="./8.12#p1">S</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./8.1#p1.t1.r3">a</mi><mo>,</mo><mi href="./8.12#p1">η</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./7.2#E2" title="(7.2.2) ‣ §7.2(i) Error Functions ‣ §7.2 Definitions ‣ Properties ‣ Chapter 7 Error Functions, Dawson’s and Fresnel Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m90.png" altimg-height="18px" altimg-valign="-2px" altimg-width="49px" alttext="\operatorname{erfc}\NVar{z}" display="inline"><mrow><mi href="./7.2#E2">erfc</mi><mo></mo><mi class="ltx_nvar" href="./7.1#p1.t1.r2">z</mi></mrow></math>: complementary error function</a>,
<a href="./8.2#E4" title="(8.2.4) ‣ §8.2(i) Definitions ‣ §8.2 Definitions and Basic Properties ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="23px" altimg-valign="-7px" altimg-width="68px" alttext="P\left(\NVar{a},\NVar{z}\right)" display="inline"><mrow><mi href="./8.2#E4">P</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./8.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: normalized incomplete gamma function</a>,
<a href="./8.1#p1.t1.r2" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m121.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./8.1#p1.t1.r2">z</mi></math>: complex variable</a>,
<a href="./8.1#p1.t1.r3" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m98.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./8.1#p1.t1.r3">a</mi></math>: parameter</a>,
<a href="./8.12#p1" title="§8.12 Uniform Asymptotic Expansions for Large Parameter ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="\eta(\lambda)" display="inline"><mrow><mi href="./8.12#p1">η</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./8.12#p1">λ</mi><mo stretchy="false">)</mo></mrow></mrow></math></a> and
<a href="./8.12#p1" title="§8.12 Uniform Asymptotic Expansions for Large Parameter ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m64.png" altimg-height="23px" altimg-valign="-7px" altimg-width="63px" alttext="S(a,\eta)" display="inline"><mrow><mi href="./8.12#p1">S</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./8.1#p1.t1.r3">a</mi><mo>,</mo><mi href="./8.12#p1">η</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E4" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">8.12.4</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E4.png" altimg-height="41px" altimg-valign="-15px" altimg-width="321px" alttext="Q\left(a,z\right)=\tfrac{1}{2}\operatorname{erfc}\left(\eta\sqrt{a/2}\right)+S%
(a,\eta)," display="block"><mrow><mrow><mrow><mi href="./8.2#E4">Q</mi><mo></mo><mrow><mo>(</mo><mi href="./8.1#p1.t1.r3">a</mi><mo>,</mo><mi href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mrow><mi href="./7.2#E2">erfc</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./8.12#p1">η</mi><mo></mo><msqrt><mrow><mi href="./8.1#p1.t1.r3">a</mi><mo>/</mo><mn>2</mn></mrow></msqrt></mrow><mo>)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mi href="./8.12#p1">S</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./8.1#p1.t1.r3">a</mi><mo>,</mo><mi href="./8.12#p1">η</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E4.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./7.2#E2" title="(7.2.2) ‣ §7.2(i) Error Functions ‣ §7.2 Definitions ‣ Properties ‣ Chapter 7 Error Functions, Dawson’s and Fresnel Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m90.png" altimg-height="18px" altimg-valign="-2px" altimg-width="49px" alttext="\operatorname{erfc}\NVar{z}" display="inline"><mrow><mi href="./7.2#E2">erfc</mi><mo></mo><mi class="ltx_nvar" href="./7.1#p1.t1.r2">z</mi></mrow></math>: complementary error function</a>,
<a href="./8.2#E4" title="(8.2.4) ‣ §8.2(i) Definitions ‣ §8.2 Definitions and Basic Properties ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m62.png" altimg-height="23px" altimg-valign="-7px" altimg-width="68px" alttext="Q\left(\NVar{a},\NVar{z}\right)" display="inline"><mrow><mi href="./8.2#E4">Q</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./8.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: normalized incomplete gamma function</a>,
<a href="./8.1#p1.t1.r2" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m121.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./8.1#p1.t1.r2">z</mi></math>: complex variable</a>,
<a href="./8.1#p1.t1.r3" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m98.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./8.1#p1.t1.r3">a</mi></math>: parameter</a>,
<a href="./8.12#p1" title="§8.12 Uniform Asymptotic Expansions for Large Parameter ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="\eta(\lambda)" display="inline"><mrow><mi href="./8.12#p1">η</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./8.12#p1">λ</mi><mo stretchy="false">)</mo></mrow></mrow></math></a> and
<a href="./8.12#p1" title="§8.12 Uniform Asymptotic Expansions for Large Parameter ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m64.png" altimg-height="23px" altimg-valign="-7px" altimg-width="63px" alttext="S(a,\eta)" display="inline"><mrow><mi href="./8.12#p1">S</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./8.1#p1.t1.r3">a</mi><mo>,</mo><mi href="./8.12#p1">η</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E5" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">8.12.5</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E5.png" altimg-height="54px" altimg-valign="-21px" altimg-width="511px" alttext="\frac{e^{\pm\pi ia}}{2i\sin(\pi a)}Q\left(-a,ze^{\pm\pi i}\right)=\pm\tfrac{1}%
{2}\operatorname{erfc}\left(\pm i\eta\sqrt{a/2}\right)-iT(a,\eta)," display="block"><mrow><mrow><mrow><mfrac><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>±</mo><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi><mo></mo><mi href="./8.1#p1.t1.r3">a</mi></mrow></mrow></msup><mrow><mn>2</mn><mo></mo><mi mathvariant="normal">i</mi><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi href="./8.1#p1.t1.r3">a</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></mfrac><mo></mo><mrow><mi href="./8.2#E4">Q</mi><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mi href="./8.1#p1.t1.r3">a</mi></mrow><mo>,</mo><mrow><mi href="./8.1#p1.t1.r2">z</mi><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>±</mo><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi></mrow></mrow></msup></mrow><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mrow><mrow><mo>±</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mrow><mi href="./7.2#E2">erfc</mi><mo></mo><mrow><mo>(</mo><mrow><mo>±</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./8.12#p1">η</mi><mo></mo><msqrt><mrow><mi href="./8.1#p1.t1.r3">a</mi><mo>/</mo><mn>2</mn></mrow></msqrt></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>-</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mrow><mi href="./8.12#p1">T</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./8.1#p1.t1.r3">a</mi><mo>,</mo><mi href="./8.12#p1">η</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E5.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m67.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m94.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./7.2#E2" title="(7.2.2) ‣ §7.2(i) Error Functions ‣ §7.2 Definitions ‣ Properties ‣ Chapter 7 Error Functions, Dawson’s and Fresnel Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m90.png" altimg-height="18px" altimg-valign="-2px" altimg-width="49px" alttext="\operatorname{erfc}\NVar{z}" display="inline"><mrow><mi href="./7.2#E2">erfc</mi><mo></mo><mi class="ltx_nvar" href="./7.1#p1.t1.r2">z</mi></mrow></math>: complementary error function</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m86.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./8.2#E2" title="(8.2.2) ‣ §8.2(i) Definitions ‣ §8.2 Definitions and Basic Properties ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="23px" altimg-valign="-7px" altimg-width="65px" alttext="\Gamma\left(\NVar{a},\NVar{z}\right)" display="inline"><mrow><mi href="./8.2#E2" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./8.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: incomplete gamma function</a>,
<a href="./8.2#E4" title="(8.2.4) ‣ §8.2(i) Definitions ‣ §8.2 Definitions and Basic Properties ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m62.png" altimg-height="23px" altimg-valign="-7px" altimg-width="68px" alttext="Q\left(\NVar{a},\NVar{z}\right)" display="inline"><mrow><mi href="./8.2#E4">Q</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./8.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: normalized incomplete gamma function</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m97.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./8.1#p1.t1.r2" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m121.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./8.1#p1.t1.r2">z</mi></math>: complex variable</a>,
<a href="./8.1#p1.t1.r3" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m98.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./8.1#p1.t1.r3">a</mi></math>: parameter</a>,
<a href="./8.12#p1" title="§8.12 Uniform Asymptotic Expansions for Large Parameter ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="\eta(\lambda)" display="inline"><mrow><mi href="./8.12#p1">η</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./8.12#p1">λ</mi><mo stretchy="false">)</mo></mrow></mrow></math></a> and
<a href="./8.12#p1" title="§8.12 Uniform Asymptotic Expansions for Large Parameter ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m65.png" altimg-height="23px" altimg-valign="-7px" altimg-width="64px" alttext="T(a,\eta)" display="inline"><mrow><mi href="./8.12#p1">T</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./8.1#p1.t1.r3">a</mi><mo>,</mo><mi href="./8.12#p1">η</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function</a>
</dd>
<dt>Change (effective with 1.0.16):</dt>
<dd>
To be consistent with the notation used in (), the original equation,
<table id="Ex5" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="m29.png" altimg-height="49px" altimg-valign="-16px" altimg-width="551px" alttext="\Gamma\left(a+1\right)\frac{e^{\pm\pi ia}}{2\pi i}\Gamma\left(-a,ze^{\pm\pi i}%
\right)=\mp\tfrac{1}{2}\operatorname{erfc}\left(\pm i\eta\sqrt{a/2}\right)+iT(%
a,\eta)," display="block"><mrow><mrow><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./8.1#p1.t1.r3">a</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mo></mo><mfrac><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>±</mo><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi><mo></mo><mi href="./8.1#p1.t1.r3">a</mi></mrow></mrow></msup><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi></mrow></mfrac><mo></mo><mrow><mi href="./8.2#E2" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mi href="./8.1#p1.t1.r3">a</mi></mrow><mo>,</mo><mrow><mi href="./8.1#p1.t1.r2">z</mi><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>±</mo><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi></mrow></mrow></msup></mrow><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mrow><mrow><mo>∓</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mrow><mi href="./7.2#E2">erfc</mi><mo></mo><mrow><mo>(</mo><mrow><mo>±</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./8.12#p1">η</mi><mo></mo><msqrt><mrow><mi href="./8.1#p1.t1.r3">a</mi><mo>/</mo><mn>2</mn></mrow></msqrt></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mrow><mi href="./8.12#p1">T</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./8.1#p1.t1.r3">a</mi><mo>,</mo><mi href="./8.12#p1">η</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
</table>
was rewritten.
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and</p>
<table id="E6" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">8.12.6</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E6.png" altimg-height="65px" altimg-valign="-27px" altimg-width="620px" alttext="z^{-a}\gamma^{*}\left(-a,-z\right)=\cos\left(\pi a\right)-2\sin\left(\pi a%
\right)\left(\frac{e^{\frac{1}{2}a\eta^{2}}}{\sqrt{\pi}}F\left(\eta\sqrt{a/2}%
\right)+T(a,\eta)\right)," display="block"><mrow><mrow><mrow><msup><mi href="./8.1#p1.t1.r2">z</mi><mrow><mo>-</mo><mi href="./8.1#p1.t1.r3">a</mi></mrow></msup><mo></mo><mrow><msup><mi href="./8.2#E6">γ</mi><mo href="./8.2#E6">*</mo></msup><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mi href="./8.1#p1.t1.r3">a</mi></mrow><mo>,</mo><mrow><mo>-</mo><mi href="./8.1#p1.t1.r2">z</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mrow><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi href="./8.1#p1.t1.r3">a</mi></mrow><mo>)</mo></mrow></mrow><mo>-</mo><mrow><mn>2</mn><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi href="./8.1#p1.t1.r3">a</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mo>(</mo><mrow><mrow><mfrac><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./8.1#p1.t1.r3">a</mi><mo></mo><msup><mi href="./8.12#p1">η</mi><mn>2</mn></msup></mrow></msup><msqrt><mi href="./3.12#E1">π</mi></msqrt></mfrac><mo></mo><mrow><mi href="./7.2#E5">F</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./8.12#p1">η</mi><mo></mo><msqrt><mrow><mi href="./8.1#p1.t1.r3">a</mi><mo>/</mo><mn>2</mn></mrow></msqrt></mrow><mo>)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mi href="./8.12#p1">T</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./8.1#p1.t1.r3">a</mi><mo>,</mo><mi href="./8.12#p1">η</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E6.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./7.2#E5" title="(7.2.5) ‣ §7.2(ii) Dawson’s Integral ‣ §7.2 Definitions ‣ Properties ‣ Chapter 7 Error Functions, Dawson’s and Fresnel Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="23px" altimg-valign="-7px" altimg-width="49px" alttext="F\left(\NVar{z}\right)" display="inline"><mrow><mi href="./7.2#E5">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./7.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: Dawson’s integral</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m94.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m75.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m86.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./8.2#E6" title="(8.2.6) ‣ §8.2(i) Definitions ‣ §8.2 Definitions and Basic Properties ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m81.png" altimg-height="23px" altimg-valign="-7px" altimg-width="73px" alttext="\gamma^{*}\left(\NVar{a},\NVar{z}\right)" display="inline"><mrow><msup><mi href="./8.2#E6">γ</mi><mo href="./8.2#E6">*</mo></msup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./8.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: incomplete gamma function</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m97.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./8.1#p1.t1.r2" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m121.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./8.1#p1.t1.r2">z</mi></math>: complex variable</a>,
<a href="./8.1#p1.t1.r3" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m98.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./8.1#p1.t1.r3">a</mi></math>: parameter</a>,
<a href="./8.12#p1" title="§8.12 Uniform Asymptotic Expansions for Large Parameter ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="\eta(\lambda)" display="inline"><mrow><mi href="./8.12#p1">η</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./8.12#p1">λ</mi><mo stretchy="false">)</mo></mrow></mrow></math></a> and
<a href="./8.12#p1" title="§8.12 Uniform Asymptotic Expansions for Large Parameter ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m65.png" altimg-height="23px" altimg-valign="-7px" altimg-width="64px" alttext="T(a,\eta)" display="inline"><mrow><mi href="./8.12#p1">T</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./8.1#p1.t1.r3">a</mi><mo>,</mo><mi href="./8.12#p1">η</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m57.png" altimg-height="23px" altimg-valign="-7px" altimg-width="50px" alttext="F\left(x\right)" display="inline"><mrow><mi href="./7.2#E5">F</mi><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></math> is Dawson’s integral; see §. Then as
<math class="ltx_Math" altimg="m99.png" altimg-height="13px" altimg-valign="-2px" altimg-width="66px" alttext="a\to\infty" display="inline"><mrow><mi href="./8.1#p1.t1.r3">a</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></math> in the sector <math class="ltx_Math" altimg="m125.png" altimg-height="23px" altimg-valign="-7px" altimg-width="176px" alttext="|\operatorname{ph}a|\leq\pi-\delta(<\pi)" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mi href="./8.1#p1.t1.r3">a</mi></mrow><mo stretchy="false">|</mo></mrow><mo>≤</mo><mrow><mrow><mi href="./3.12#E1">π</mi><mo>-</mo><mi href="./8.1#p1.t1.r5">δ</mi></mrow><mspace width="veryverythickmathspace"></mspace><mrow><mo stretchy="false">(</mo><mrow><mi></mi><mo><</mo><mi href="./3.12#E1">π</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></math>,
</p>
<table id="E7" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">8.12.7</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E7.png" altimg-height="65px" altimg-valign="-28px" altimg-width="274px" alttext="S(a,\eta)\sim\frac{e^{-\frac{1}{2}a\eta^{2}}}{\sqrt{2\pi a}}\sum_{k=0}^{\infty%
}c_{k}(\eta)a^{-k}," display="block"><mrow><mrow><mrow><mi href="./8.12#p1">S</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./8.1#p1.t1.r3">a</mi><mo>,</mo><mi href="./8.12#p1">η</mi><mo stretchy="false">)</mo></mrow></mrow><mo href="./2.1#SS3.p1">∼</mo><mrow><mfrac><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./8.1#p1.t1.r3">a</mi><mo></mo><msup><mi href="./8.12#p1">η</mi><mn>2</mn></msup></mrow></mrow></msup><msqrt><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi href="./8.1#p1.t1.r3">a</mi></mrow></msqrt></mfrac><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./8.1#p1.t1.r4">k</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><mrow><msub><mi href="./8.12#p2">c</mi><mi href="./8.1#p1.t1.r4">k</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./8.12#p1">η</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><msup><mi href="./8.1#p1.t1.r3">a</mi><mrow><mo>-</mo><mi href="./8.1#p1.t1.r4">k</mi></mrow></msup></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E7.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./2.1#SS3.p1" title="§2.1(iii) Asymptotic Expansions ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m96.png" altimg-height="12px" altimg-valign="-2px" altimg-width="20px" alttext="\sim" display="inline"><mo href="./2.1#SS3.p1">∼</mo></math>: Poincaré asymptotic expansion</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m94.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m86.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./8.1#p1.t1.r3" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m98.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./8.1#p1.t1.r3">a</mi></math>: parameter</a>,
<a href="./8.1#p1.t1.r4" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m110.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./8.1#p1.t1.r4">k</mi></math>: nonnegative integer</a>,
<a href="./8.12#p1" title="§8.12 Uniform Asymptotic Expansions for Large Parameter ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="\eta(\lambda)" display="inline"><mrow><mi href="./8.12#p1">η</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./8.12#p1">λ</mi><mo stretchy="false">)</mo></mrow></mrow></math></a>,
<a href="./8.12#p1" title="§8.12 Uniform Asymptotic Expansions for Large Parameter ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m64.png" altimg-height="23px" altimg-valign="-7px" altimg-width="63px" alttext="S(a,\eta)" display="inline"><mrow><mi href="./8.12#p1">S</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./8.1#p1.t1.r3">a</mi><mo>,</mo><mi href="./8.12#p1">η</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function</a> and
<a href="./8.12#p2" title="§8.12 Uniform Asymptotic Expansions for Large Parameter ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m100.png" altimg-height="23px" altimg-valign="-7px" altimg-width="49px" alttext="c_{k}(\eta)" display="inline"><mrow><msub><mi href="./8.12#p2">c</mi><mi href="./8.1#p1.t1.r4">k</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./8.12#p1">η</mi><mo stretchy="false">)</mo></mrow></mrow></math>: coefficients</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E8" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">8.12.8</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E8.png" altimg-height="65px" altimg-valign="-28px" altimg-width="296px" alttext="T(a,\eta)\sim\frac{e^{\frac{1}{2}a\eta^{2}}}{\sqrt{2\pi a}}\sum_{k=0}^{\infty}%
c_{k}(\eta)(-a)^{-k}," display="block"><mrow><mrow><mrow><mi href="./8.12#p1">T</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./8.1#p1.t1.r3">a</mi><mo>,</mo><mi href="./8.12#p1">η</mi><mo stretchy="false">)</mo></mrow></mrow><mo href="./2.1#SS3.p1">∼</mo><mrow><mfrac><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./8.1#p1.t1.r3">a</mi><mo></mo><msup><mi href="./8.12#p1">η</mi><mn>2</mn></msup></mrow></msup><msqrt><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi href="./8.1#p1.t1.r3">a</mi></mrow></msqrt></mfrac><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./8.1#p1.t1.r4">k</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><mrow><msub><mi href="./8.12#p2">c</mi><mi href="./8.1#p1.t1.r4">k</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./8.12#p1">η</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mi href="./8.1#p1.t1.r3">a</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mi href="./8.1#p1.t1.r4">k</mi></mrow></msup></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E8.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./2.1#SS3.p1" title="§2.1(iii) Asymptotic Expansions ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m96.png" altimg-height="12px" altimg-valign="-2px" altimg-width="20px" alttext="\sim" display="inline"><mo href="./2.1#SS3.p1">∼</mo></math>: Poincaré asymptotic expansion</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m94.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m86.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./8.1#p1.t1.r3" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m98.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./8.1#p1.t1.r3">a</mi></math>: parameter</a>,
<a href="./8.1#p1.t1.r4" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m110.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./8.1#p1.t1.r4">k</mi></math>: nonnegative integer</a>,
<a href="./8.12#p1" title="§8.12 Uniform Asymptotic Expansions for Large Parameter ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="\eta(\lambda)" display="inline"><mrow><mi href="./8.12#p1">η</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./8.12#p1">λ</mi><mo stretchy="false">)</mo></mrow></mrow></math></a>,
<a href="./8.12#p1" title="§8.12 Uniform Asymptotic Expansions for Large Parameter ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m65.png" altimg-height="23px" altimg-valign="-7px" altimg-width="64px" alttext="T(a,\eta)" display="inline"><mrow><mi href="./8.12#p1">T</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./8.1#p1.t1.r3">a</mi><mo>,</mo><mi href="./8.12#p1">η</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function</a> and
<a href="./8.12#p2" title="§8.12 Uniform Asymptotic Expansions for Large Parameter ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m100.png" altimg-height="23px" altimg-valign="-7px" altimg-width="49px" alttext="c_{k}(\eta)" display="inline"><mrow><msub><mi href="./8.12#p2">c</mi><mi href="./8.1#p1.t1.r4">k</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./8.12#p1">η</mi><mo stretchy="false">)</mo></mrow></mrow></math>: coefficients</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">in each case uniformly with respect to <math class="ltx_Math" altimg="m82.png" altimg-height="18px" altimg-valign="-2px" altimg-width="16px" alttext="\lambda" display="inline"><mi href="./8.12#p1">λ</mi></math> in the sector
<math class="ltx_Math" altimg="m124.png" altimg-height="23px" altimg-valign="-7px" altimg-width="138px" alttext="|\operatorname{ph}\lambda|\leq 2\pi-\delta" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mi href="./8.12#p1">λ</mi></mrow><mo stretchy="false">|</mo></mrow><mo>≤</mo><mrow><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo>-</mo><mi href="./8.1#p1.t1.r5">δ</mi></mrow></mrow></math> (<math class="ltx_Math" altimg="m52.png" altimg-height="17px" altimg-valign="-3px" altimg-width="47px" alttext="<2\pi" display="inline"><mrow><mi></mi><mo><</mo><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow></math>).</p>
</div>
<div id="p2" class="ltx_para">
<p class="ltx_p">With <math class="ltx_Math" altimg="m88.png" altimg-height="21px" altimg-valign="-6px" altimg-width="89px" alttext="\mu=\lambda-1" display="inline"><mrow><mi href="./8.12#p2">μ</mi><mo>=</mo><mrow><mi href="./8.12#p1">λ</mi><mo>-</mo><mn>1</mn></mrow></mrow></math>, the coefficients <math class="ltx_Math" altimg="m100.png" altimg-height="23px" altimg-valign="-7px" altimg-width="49px" alttext="c_{k}(\eta)" display="inline"><mrow><msub><mi href="./8.12#p2">c</mi><mi href="./8.1#p1.t1.r4">k</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./8.12#p1">η</mi><mo stretchy="false">)</mo></mrow></mrow></math> are given by
</p>
<table id="E9" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex6" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">8.12.9</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m42.png" altimg-height="25px" altimg-valign="-7px" altimg-width="50px" alttext="\displaystyle c_{0}(\eta)" display="inline"><mrow><msub><mi href="./8.12#p2">c</mi><mn>0</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./8.12#p1">η</mi><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m12.png" altimg-height="49px" altimg-valign="-20px" altimg-width="89px" alttext="\displaystyle=\frac{1}{\mu}-\frac{1}{\eta}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><mfrac><mn>1</mn><mi href="./8.12#p2">μ</mi></mfrac></mstyle><mo>-</mo><mstyle displaystyle="true"><mfrac><mn>1</mn><mi href="./8.12#p1">η</mi></mfrac></mstyle></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex7" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m44.png" altimg-height="25px" altimg-valign="-7px" altimg-width="50px" alttext="\displaystyle c_{1}(\eta)" display="inline"><mrow><msub><mi href="./8.12#p2">c</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./8.12#p1">η</mi><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m11.png" altimg-height="49px" altimg-valign="-20px" altimg-width="218px" alttext="\displaystyle=\frac{1}{\eta^{3}}-\frac{1}{\mu^{3}}-\frac{1}{\mu^{2}}-\frac{1}{%
12\mu}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><mfrac><mn>1</mn><msup><mi href="./8.12#p1">η</mi><mn>3</mn></msup></mfrac></mstyle><mo>-</mo><mstyle displaystyle="true"><mfrac><mn>1</mn><msup><mi href="./8.12#p2">μ</mi><mn>3</mn></msup></mfrac></mstyle><mo>-</mo><mstyle displaystyle="true"><mfrac><mn>1</mn><msup><mi href="./8.12#p2">μ</mi><mn>2</mn></msup></mfrac></mstyle><mo>-</mo><mstyle displaystyle="true"><mfrac><mn>1</mn><mrow><mn>12</mn><mo></mo><mi href="./8.12#p2">μ</mi></mrow></mfrac></mstyle></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E9.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./8.12#p1" title="§8.12 Uniform Asymptotic Expansions for Large Parameter ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="\eta(\lambda)" display="inline"><mrow><mi href="./8.12#p1">η</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./8.12#p1">λ</mi><mo stretchy="false">)</mo></mrow></mrow></math></a>,
<a href="./8.12#p2" title="§8.12 Uniform Asymptotic Expansions for Large Parameter ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m89.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\mu" display="inline"><mi href="./8.12#p2">μ</mi></math></a> and
<a href="./8.12#p2" title="§8.12 Uniform Asymptotic Expansions for Large Parameter ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m100.png" altimg-height="23px" altimg-valign="-7px" altimg-width="49px" alttext="c_{k}(\eta)" display="inline"><mrow><msub><mi href="./8.12#p2">c</mi><mi href="./8.1#p1.t1.r4">k</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./8.12#p1">η</mi><mo stretchy="false">)</mo></mrow></mrow></math>: coefficients</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E10" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">8.12.10</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E10.png" altimg-height="50px" altimg-valign="-20px" altimg-width="289px" alttext="c_{k}(\eta)=\frac{1}{\eta}\frac{\mathrm{d}}{\mathrm{d}\eta}c_{k-1}(\eta)+(-1)^%
{k}\frac{g_{k}}{\mu}," display="block"><mrow><mrow><mrow><msub><mi href="./8.12#p2">c</mi><mi href="./8.1#p1.t1.r4">k</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./8.12#p1">η</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mfrac><mn>1</mn><mi href="./8.12#p1">η</mi></mfrac><mo></mo><mrow><mfrac><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi href="./8.12#p1">η</mi></mrow></mfrac><mo></mo><mrow><msub><mi href="./8.12#p2">c</mi><mrow><mi href="./8.1#p1.t1.r4">k</mi><mo>-</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./8.12#p1">η</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow><mo>+</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./8.1#p1.t1.r4">k</mi></msup><mo></mo><mfrac><msub><mi href="./8.12#p2">g</mi><mi href="./8.1#p1.t1.r4">k</mi></msub><mi href="./8.12#p2">μ</mi></mfrac></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m108.png" altimg-height="21px" altimg-valign="-6px" altimg-width="103px" alttext="k=1,2,\dots" display="inline"><mrow><mi href="./8.1#p1.t1.r4">k</mi><mo>=</mo><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E10.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#E4" title="(1.4.4) ‣ §1.4(iii) Derivatives ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m80.png" altimg-height="29px" altimg-valign="-9px" altimg-width="27px" alttext="\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: derivative of <math class="ltx_Math" altimg="m104.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m119.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./8.1#p1.t1.r4" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m110.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./8.1#p1.t1.r4">k</mi></math>: nonnegative integer</a>,
<a href="./8.12#p1" title="§8.12 Uniform Asymptotic Expansions for Large Parameter ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="\eta(\lambda)" display="inline"><mrow><mi href="./8.12#p1">η</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./8.12#p1">λ</mi><mo stretchy="false">)</mo></mrow></mrow></math></a>,
<a href="./8.12#p2" title="§8.12 Uniform Asymptotic Expansions for Large Parameter ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m89.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\mu" display="inline"><mi href="./8.12#p2">μ</mi></math></a>,
<a href="./8.12#p2" title="§8.12 Uniform Asymptotic Expansions for Large Parameter ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m100.png" altimg-height="23px" altimg-valign="-7px" altimg-width="49px" alttext="c_{k}(\eta)" display="inline"><mrow><msub><mi href="./8.12#p2">c</mi><mi href="./8.1#p1.t1.r4">k</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./8.12#p1">η</mi><mo stretchy="false">)</mo></mrow></mrow></math>: coefficients</a> and
<a href="./8.12#p2" title="§8.12 Uniform Asymptotic Expansions for Large Parameter ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m105.png" altimg-height="16px" altimg-valign="-6px" altimg-width="24px" alttext="g_{k}" display="inline"><msub><mi href="./8.12#p2">g</mi><mi href="./8.1#p1.t1.r4">k</mi></msub></math>: coefficients</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m105.png" altimg-height="16px" altimg-valign="-6px" altimg-width="24px" alttext="g_{k}" display="inline"><msub><mi href="./8.12#p2">g</mi><mi href="./8.1#p1.t1.r4">k</mi></msub></math>, <math class="ltx_Math" altimg="m107.png" altimg-height="21px" altimg-valign="-6px" altimg-width="122px" alttext="k=0,1,2,\dots" display="inline"><mrow><mi href="./8.1#p1.t1.r4">k</mi><mo>=</mo><mrow><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>, are the coefficients that appear in the
asymptotic expansion () of <math class="ltx_Math" altimg="m69.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(z\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>. The right-hand
sides of equations () have removable
singularities at <math class="ltx_Math" altimg="m79.png" altimg-height="20px" altimg-valign="-6px" altimg-width="51px" alttext="\eta=0" display="inline"><mrow><mi href="./8.12#p1">η</mi><mo>=</mo><mn>0</mn></mrow></math>, and the Maclaurin series expansion of <math class="ltx_Math" altimg="m100.png" altimg-height="23px" altimg-valign="-7px" altimg-width="49px" alttext="c_{k}(\eta)" display="inline"><mrow><msub><mi href="./8.12#p2">c</mi><mi href="./8.1#p1.t1.r4">k</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./8.12#p1">η</mi><mo stretchy="false">)</mo></mrow></mrow></math>
is given by
</p>
<table id="E11" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">8.12.11</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E11.png" altimg-height="64px" altimg-valign="-27px" altimg-width="172px" alttext="c_{k}(\eta)=\sum_{n=0}^{\infty}d_{k,n}\eta^{n}," display="block"><mrow><mrow><mrow><msub><mi href="./8.12#p2">c</mi><mi href="./8.1#p1.t1.r4">k</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./8.12#p1">η</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./8.1#p1.t1.r4">n</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><msub><mi href="./8.12#E19">d</mi><mrow><mi href="./8.1#p1.t1.r4">k</mi><mo>,</mo><mi href="./8.1#p1.t1.r4">n</mi></mrow></msub><mo></mo><msup><mi href="./8.12#p1">η</mi><mi href="./8.1#p1.t1.r4">n</mi></msup></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m123.png" altimg-height="24px" altimg-valign="-7px" altimg-width="91px" alttext="|\eta|<2\sqrt{\pi}" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mi href="./8.12#p1">η</mi><mo stretchy="false">|</mo></mrow><mo><</mo><mrow><mn>2</mn><mo></mo><msqrt><mi href="./3.12#E1">π</mi></msqrt></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E11.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m94.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./8.1#p1.t1.r4" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m110.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./8.1#p1.t1.r4">k</mi></math>: nonnegative integer</a>,
<a href="./8.1#p1.t1.r4" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m114.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./8.1#p1.t1.r4">n</mi></math>: nonnegative integer</a>,
<a href="./8.12#p1" title="§8.12 Uniform Asymptotic Expansions for Large Parameter ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="\eta(\lambda)" display="inline"><mrow><mi href="./8.12#p1">η</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./8.12#p1">λ</mi><mo stretchy="false">)</mo></mrow></mrow></math></a>,
<a href="./8.12#E19" title="(8.12.19) ‣ §8.12 Uniform Asymptotic Expansions for Large Parameter ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m101.png" altimg-height="23px" altimg-valign="-7px" altimg-width="58px" alttext="d(\pm\chi)" display="inline"><mrow><mi href="./8.12#E19">d</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mo>±</mo><mi href="./8.12#E19">χ</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></math></a> and
<a href="./8.12#p2" title="§8.12 Uniform Asymptotic Expansions for Large Parameter ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m100.png" altimg-height="23px" altimg-valign="-7px" altimg-width="49px" alttext="c_{k}(\eta)" display="inline"><mrow><msub><mi href="./8.12#p2">c</mi><mi href="./8.1#p1.t1.r4">k</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./8.12#p1">η</mi><mo stretchy="false">)</mo></mrow></mrow></math>: coefficients</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m102.png" altimg-height="27px" altimg-valign="-9px" altimg-width="91px" alttext="d_{0,0}=-\tfrac{1}{3}" display="inline"><mrow><msub><mi href="./8.12#E19">d</mi><mrow><mn>0</mn><mo>,</mo><mn>0</mn></mrow></msub><mo>=</mo><mrow><mo>-</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></mrow></mrow></math>,
</p>
<table id="E12" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex8" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="4" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">8.12.12</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m50.png" altimg-height="25px" altimg-valign="-8px" altimg-width="40px" alttext="\displaystyle d_{0,n}" display="inline"><msub><mi href="./8.12#E19">d</mi><mrow><mn>0</mn><mo>,</mo><mi href="./8.1#p1.t1.r4">n</mi></mrow></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m2.png" altimg-height="25px" altimg-valign="-7px" altimg-width="138px" alttext="\displaystyle=(n+2)\alpha_{n+2}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./8.1#p1.t1.r4">n</mi><mo>+</mo><mn>2</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msub><mi href="./8.12#p2">α</mi><mrow><mi href="./8.1#p1.t1.r4">n</mi><mo>+</mo><mn>2</mn></mrow></msub></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m116.png" altimg-height="19px" altimg-valign="-5px" altimg-width="53px" alttext="n\geq 1" display="inline"><mrow><mi href="./8.1#p1.t1.r4">n</mi><mo>≥</mo><mn>1</mn></mrow></math>,</span></td></tr>
<tr id="Ex9" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m51.png" altimg-height="25px" altimg-valign="-8px" altimg-width="41px" alttext="\displaystyle d_{k,n}" display="inline"><msub><mi href="./8.12#E19">d</mi><mrow><mi href="./8.1#p1.t1.r4">k</mi><mo>,</mo><mi href="./8.1#p1.t1.r4">n</mi></mrow></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m1.png" altimg-height="29px" altimg-valign="-8px" altimg-width="298px" alttext="\displaystyle=(-1)^{k}g_{k}d_{0,n}+(n+2)d_{k-1,n+2}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./8.1#p1.t1.r4">k</mi></msup><mo></mo><msub><mi href="./8.12#p2">g</mi><mi href="./8.1#p1.t1.r4">k</mi></msub><mo></mo><msub><mi href="./8.12#E19">d</mi><mrow><mn>0</mn><mo>,</mo><mi href="./8.1#p1.t1.r4">n</mi></mrow></msub></mrow><mo>+</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./8.1#p1.t1.r4">n</mi><mo>+</mo><mn>2</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msub><mi href="./8.12#E19">d</mi><mrow><mrow><mi href="./8.1#p1.t1.r4">k</mi><mo>-</mo><mn>1</mn></mrow><mo>,</mo><mrow><mi href="./8.1#p1.t1.r4">n</mi><mo>+</mo><mn>2</mn></mrow></mrow></msub></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m115.png" altimg-height="19px" altimg-valign="-5px" altimg-width="53px" alttext="n\geq 0" display="inline"><mrow><mi href="./8.1#p1.t1.r4">n</mi><mo>≥</mo><mn>0</mn></mrow></math>, <math class="ltx_Math" altimg="m111.png" altimg-height="20px" altimg-valign="-5px" altimg-width="52px" alttext="k\geq 1" display="inline"><mrow><mi href="./8.1#p1.t1.r4">k</mi><mo>≥</mo><mn>1</mn></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E12.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./8.1#p1.t1.r4" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m110.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./8.1#p1.t1.r4">k</mi></math>: nonnegative integer</a>,
<a href="./8.1#p1.t1.r4" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m114.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./8.1#p1.t1.r4">n</mi></math>: nonnegative integer</a>,
<a href="./8.12#E19" title="(8.12.19) ‣ §8.12 Uniform Asymptotic Expansions for Large Parameter ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m101.png" altimg-height="23px" altimg-valign="-7px" altimg-width="58px" alttext="d(\pm\chi)" display="inline"><mrow><mi href="./8.12#E19">d</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mo>±</mo><mi href="./8.12#E19">χ</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></math></a>,
<a href="./8.12#p2" title="§8.12 Uniform Asymptotic Expansions for Large Parameter ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m105.png" altimg-height="16px" altimg-valign="-6px" altimg-width="24px" alttext="g_{k}" display="inline"><msub><mi href="./8.12#p2">g</mi><mi href="./8.1#p1.t1.r4">k</mi></msub></math>: coefficients</a> and
<a href="./8.12#p2" title="§8.12 Uniform Asymptotic Expansions for Large Parameter ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m73.png" altimg-height="17px" altimg-valign="-7px" altimg-width="48px" alttext="\alpha_{n+2}" display="inline"><msub><mi href="./8.12#p2">α</mi><mrow><mi href="./8.1#p1.t1.r4">n</mi><mo>+</mo><mn>2</mn></mrow></msub></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and <math class="ltx_Math" altimg="m72.png" altimg-height="16px" altimg-valign="-6px" altimg-width="89px" alttext="\alpha_{3},\alpha_{4},\dots" display="inline"><mrow><msub><mi href="./8.12#p2">α</mi><mn>3</mn></msub><mo>,</mo><msub><mi href="./8.12#p2">α</mi><mn>4</mn></msub><mo>,</mo><mi mathvariant="normal">…</mi></mrow></math> are defined by
</p>
<table id="E13" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">8.12.13</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E13.png" altimg-height="64px" altimg-valign="-27px" altimg-width="254px" alttext="\lambda-1=\eta+\tfrac{1}{3}\eta^{2}+\sum_{n=3}^{\infty}\alpha_{n}\eta^{n}," display="block"><mrow><mrow><mrow><mi href="./8.12#p1">λ</mi><mo>-</mo><mn>1</mn></mrow><mo>=</mo><mrow><mi href="./8.12#p1">η</mi><mo>+</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>3</mn></mfrac></mstyle><mo></mo><msup><mi href="./8.12#p1">η</mi><mn>2</mn></msup></mrow><mo>+</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./8.1#p1.t1.r4">n</mi><mo>=</mo><mn>3</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><msub><mi href="./8.12#p2">α</mi><mi href="./8.1#p1.t1.r4">n</mi></msub><mo></mo><msup><mi href="./8.12#p1">η</mi><mi href="./8.1#p1.t1.r4">n</mi></msup></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m123.png" altimg-height="24px" altimg-valign="-7px" altimg-width="91px" alttext="|\eta|<2\sqrt{\pi}" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mi href="./8.12#p1">η</mi><mo stretchy="false">|</mo></mrow><mo><</mo><mrow><mn>2</mn><mo></mo><msqrt><mi href="./3.12#E1">π</mi></msqrt></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E13.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m94.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./8.1#p1.t1.r4" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m114.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./8.1#p1.t1.r4">n</mi></math>: nonnegative integer</a>,
<a href="./8.12#p1" title="§8.12 Uniform Asymptotic Expansions for Large Parameter ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="\eta(\lambda)" display="inline"><mrow><mi href="./8.12#p1">η</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./8.12#p1">λ</mi><mo stretchy="false">)</mo></mrow></mrow></math></a>,
<a href="./8.12#p1" title="§8.12 Uniform Asymptotic Expansions for Large Parameter ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="18px" altimg-valign="-2px" altimg-width="16px" alttext="\lambda" display="inline"><mi href="./8.12#p1">λ</mi></math></a> and
<a href="./8.12#p2" title="§8.12 Uniform Asymptotic Expansions for Large Parameter ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m73.png" altimg-height="17px" altimg-valign="-7px" altimg-width="48px" alttext="\alpha_{n+2}" display="inline"><msub><mi href="./8.12#p2">α</mi><mrow><mi href="./8.1#p1.t1.r4">n</mi><mo>+</mo><mn>2</mn></mrow></msub></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">In particular,</p>
<table id="E14" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex10" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="6" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">8.12.14</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m30.png" altimg-height="17px" altimg-valign="-5px" altimg-width="28px" alttext="\displaystyle\alpha_{3}" display="inline"><msub><mi href="./8.12#p2">α</mi><mn>3</mn></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m21.png" altimg-height="29px" altimg-valign="-9px" altimg-width="53px" alttext="\displaystyle=\tfrac{1}{36}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mfrac><mn>1</mn><mn>36</mn></mfrac></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex11" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m31.png" altimg-height="17px" altimg-valign="-5px" altimg-width="28px" alttext="\displaystyle\alpha_{4}" display="inline"><msub><mi href="./8.12#p2">α</mi><mn>4</mn></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m5.png" altimg-height="29px" altimg-valign="-9px" altimg-width="77px" alttext="\displaystyle=-\tfrac{1}{270}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo>-</mo><mfrac><mn>1</mn><mn>270</mn></mfrac></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex12" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m32.png" altimg-height="17px" altimg-valign="-5px" altimg-width="28px" alttext="\displaystyle\alpha_{5}" display="inline"><msub><mi href="./8.12#p2">α</mi><mn>5</mn></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m23.png" altimg-height="29px" altimg-valign="-9px" altimg-width="69px" alttext="\displaystyle=\tfrac{1}{4320}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mfrac><mn>1</mn><mn>4320</mn></mfrac></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex13" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m33.png" altimg-height="17px" altimg-valign="-5px" altimg-width="28px" alttext="\displaystyle\alpha_{6}" display="inline"><msub><mi href="./8.12#p2">α</mi><mn>6</mn></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m18.png" altimg-height="29px" altimg-valign="-9px" altimg-width="77px" alttext="\displaystyle=\tfrac{1}{17010}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mfrac><mn>1</mn><mn>17010</mn></mfrac></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex14" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m34.png" altimg-height="17px" altimg-valign="-5px" altimg-width="28px" alttext="\displaystyle\alpha_{7}" display="inline"><msub><mi href="./8.12#p2">α</mi><mn>7</mn></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m4.png" altimg-height="29px" altimg-valign="-9px" altimg-width="113px" alttext="\displaystyle=-\tfrac{139}{54\;43200}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo>-</mo><mfrac><mn>139</mn><mn>54 43200</mn></mfrac></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex15" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m35.png" altimg-height="17px" altimg-valign="-5px" altimg-width="28px" alttext="\displaystyle\alpha_{8}" display="inline"><msub><mi href="./8.12#p2">α</mi><mn>8</mn></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m19.png" altimg-height="29px" altimg-valign="-9px" altimg-width="90px" alttext="\displaystyle=\tfrac{1}{2\;04120}." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mfrac><mn>1</mn><mn>2 04120</mn></mfrac></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E14.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./8.12#p2" title="§8.12 Uniform Asymptotic Expansions for Large Parameter ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m73.png" altimg-height="17px" altimg-valign="-7px" altimg-width="48px" alttext="\alpha_{n+2}" display="inline"><msub><mi href="./8.12#p2">α</mi><mrow><mi href="./8.1#p1.t1.r4">n</mi><mo>+</mo><mn>2</mn></mrow></msub></math></a></dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">For numerical values of <math class="ltx_Math" altimg="m103.png" altimg-height="23px" altimg-valign="-8px" altimg-width="39px" alttext="d_{k,n}" display="inline"><msub><mi href="./8.12#E19">d</mi><mrow><mi href="./8.1#p1.t1.r4">k</mi><mo>,</mo><mi href="./8.1#p1.t1.r4">n</mi></mrow></msub></math> to 30D for <math class="ltx_Math" altimg="m106.png" altimg-height="23px" altimg-valign="-7px" altimg-width="87px" alttext="k=0(1)9" display="inline"><mrow><mi href="./8.1#p1.t1.r4">k</mi><mo>=</mo><mrow><mn>0</mn><mo></mo><mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mo></mo><mn>9</mn></mrow></mrow></math> and <math class="ltx_Math" altimg="m113.png" altimg-height="23px" altimg-valign="-7px" altimg-width="104px" alttext="n=0(1)N_{k}" display="inline"><mrow><mi href="./8.1#p1.t1.r4">n</mi><mo>=</mo><mrow><mn>0</mn><mo></mo><mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mo></mo><msub><mi>N</mi><mi href="./8.1#p1.t1.r4">k</mi></msub></mrow></mrow></math>,
where <math class="ltx_Math" altimg="m59.png" altimg-height="23px" altimg-valign="-7px" altimg-width="163px" alttext="N_{k}=28-4\left\lfloor k/2\right\rfloor" display="inline"><mrow><msub><mi>N</mi><mi href="./8.1#p1.t1.r4">k</mi></msub><mo>=</mo><mrow><mn>28</mn><mo>-</mo><mrow><mn>4</mn><mo></mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r16">⌊</mo><mrow><mi href="./8.1#p1.t1.r4">k</mi><mo>/</mo><mn>2</mn></mrow><mo href="./front/introduction#Sx4.p1.t1.r16">⌋</mo></mrow></mrow></mrow></mrow></math>, see <cite class="ltx_cite ltx_citemacro_citet">DiDonato and Morris ()</cite>.</p>
</div>
<div id="p3" class="ltx_para">
<p class="ltx_p">Special cases are given by
</p>
<table id="E15" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">8.12.15</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E15.png" altimg-height="64px" altimg-valign="-28px" altimg-width="308px" alttext="Q\left(a,a\right)\sim\frac{1}{2}+\frac{1}{\sqrt{2\pi a}}\sum_{k=0}^{\infty}c_{%
k}(0)a^{-k}," display="block"><mrow><mrow><mrow><mi href="./8.2#E4">Q</mi><mo></mo><mrow><mo>(</mo><mi href="./8.1#p1.t1.r3">a</mi><mo>,</mo><mi href="./8.1#p1.t1.r3">a</mi><mo>)</mo></mrow></mrow><mo href="./2.1#SS3.p1">∼</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>+</mo><mrow><mfrac><mn>1</mn><msqrt><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi href="./8.1#p1.t1.r3">a</mi></mrow></msqrt></mfrac><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./8.1#p1.t1.r4">k</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><mrow><msub><mi href="./8.12#p2">c</mi><mi href="./8.1#p1.t1.r4">k</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></mrow><mo></mo><msup><mi href="./8.1#p1.t1.r3">a</mi><mrow><mo>-</mo><mi href="./8.1#p1.t1.r4">k</mi></mrow></msup></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m126.png" altimg-height="23px" altimg-valign="-7px" altimg-width="127px" alttext="|\operatorname{ph}a|\leq\pi-\delta" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mi href="./8.1#p1.t1.r3">a</mi></mrow><mo stretchy="false">|</mo></mrow><mo>≤</mo><mrow><mi href="./3.12#E1">π</mi><mo>-</mo><mi href="./8.1#p1.t1.r5">δ</mi></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E15.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./2.1#SS3.p1" title="§2.1(iii) Asymptotic Expansions ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m96.png" altimg-height="12px" altimg-valign="-2px" altimg-width="20px" alttext="\sim" display="inline"><mo href="./2.1#SS3.p1">∼</mo></math>: Poincaré asymptotic expansion</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m94.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./8.2#E4" title="(8.2.4) ‣ §8.2(i) Definitions ‣ §8.2 Definitions and Basic Properties ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m62.png" altimg-height="23px" altimg-valign="-7px" altimg-width="68px" alttext="Q\left(\NVar{a},\NVar{z}\right)" display="inline"><mrow><mi href="./8.2#E4">Q</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./8.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: normalized incomplete gamma function</a>,
<a href="./1.9#E7" title="(1.9.7) ‣ Modulus and Phase ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m93.png" altimg-height="21px" altimg-valign="-6px" altimg-width="26px" alttext="\operatorname{ph}" display="inline"><mi href="./1.9#E7">ph</mi></math>: phase</a>,
<a href="./8.1#p1.t1.r3" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m98.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./8.1#p1.t1.r3">a</mi></math>: parameter</a>,
<a href="./8.1#p1.t1.r4" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m110.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./8.1#p1.t1.r4">k</mi></math>: nonnegative integer</a>,
<a href="./8.1#p1.t1.r5" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m76.png" altimg-height="18px" altimg-valign="-2px" altimg-width="14px" alttext="\delta" display="inline"><mi href="./8.1#p1.t1.r5">δ</mi></math>: small positive constant</a> and
<a href="./8.12#p2" title="§8.12 Uniform Asymptotic Expansions for Large Parameter ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m100.png" altimg-height="23px" altimg-valign="-7px" altimg-width="49px" alttext="c_{k}(\eta)" display="inline"><mrow><msub><mi href="./8.12#p2">c</mi><mi href="./8.1#p1.t1.r4">k</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./8.12#p1">η</mi><mo stretchy="false">)</mo></mrow></mrow></math>: coefficients</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E16" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">8.12.16</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E16.png" altimg-height="64px" altimg-valign="-28px" altimg-width="502px" alttext="\frac{e^{\pm\pi ia}}{2i\sin\left(\pi a\right)}Q\left(-a,ae^{\pm\pi i}\right)%
\sim\pm\frac{1}{2}-\frac{i}{\sqrt{2\pi a}}\sum_{k=0}^{\infty}c_{k}(0)(-a)^{-k}," display="block"><mrow><mrow><mrow><mfrac><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>±</mo><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi><mo></mo><mi href="./8.1#p1.t1.r3">a</mi></mrow></mrow></msup><mrow><mn>2</mn><mo></mo><mi mathvariant="normal">i</mi><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi href="./8.1#p1.t1.r3">a</mi></mrow><mo>)</mo></mrow></mrow></mrow></mfrac><mo></mo><mrow><mi href="./8.2#E4">Q</mi><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mi href="./8.1#p1.t1.r3">a</mi></mrow><mo>,</mo><mrow><mi href="./8.1#p1.t1.r3">a</mi><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>±</mo><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi></mrow></mrow></msup></mrow><mo>)</mo></mrow></mrow></mrow><mo href="./2.1#SS3.p1">∼</mo><mrow><mrow><mo>±</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>-</mo><mrow><mfrac><mi mathvariant="normal">i</mi><msqrt><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi href="./8.1#p1.t1.r3">a</mi></mrow></msqrt></mfrac><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./8.1#p1.t1.r4">k</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><mrow><msub><mi href="./8.12#p2">c</mi><mi href="./8.1#p1.t1.r4">k</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></mrow><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mi href="./8.1#p1.t1.r3">a</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mi href="./8.1#p1.t1.r4">k</mi></mrow></msup></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m126.png" altimg-height="23px" altimg-valign="-7px" altimg-width="127px" alttext="|\operatorname{ph}a|\leq\pi-\delta" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mi href="./8.1#p1.t1.r3">a</mi></mrow><mo stretchy="false">|</mo></mrow><mo>≤</mo><mrow><mi href="./3.12#E1">π</mi><mo>-</mo><mi href="./8.1#p1.t1.r5">δ</mi></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E16.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./2.1#SS3.p1" title="§2.1(iii) Asymptotic Expansions ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m96.png" altimg-height="12px" altimg-valign="-2px" altimg-width="20px" alttext="\sim" display="inline"><mo href="./2.1#SS3.p1">∼</mo></math>: Poincaré asymptotic expansion</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m94.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m86.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./8.2#E4" title="(8.2.4) ‣ §8.2(i) Definitions ‣ §8.2 Definitions and Basic Properties ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m62.png" altimg-height="23px" altimg-valign="-7px" altimg-width="68px" alttext="Q\left(\NVar{a},\NVar{z}\right)" display="inline"><mrow><mi href="./8.2#E4">Q</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./8.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: normalized incomplete gamma function</a>,
<a href="./1.9#E7" title="(1.9.7) ‣ Modulus and Phase ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m93.png" altimg-height="21px" altimg-valign="-6px" altimg-width="26px" alttext="\operatorname{ph}" display="inline"><mi href="./1.9#E7">ph</mi></math>: phase</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m97.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./8.1#p1.t1.r3" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m98.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./8.1#p1.t1.r3">a</mi></math>: parameter</a>,
<a href="./8.1#p1.t1.r4" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m110.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./8.1#p1.t1.r4">k</mi></math>: nonnegative integer</a>,
<a href="./8.1#p1.t1.r5" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m76.png" altimg-height="18px" altimg-valign="-2px" altimg-width="14px" alttext="\delta" display="inline"><mi href="./8.1#p1.t1.r5">δ</mi></math>: small positive constant</a> and
<a href="./8.12#p2" title="§8.12 Uniform Asymptotic Expansions for Large Parameter ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m100.png" altimg-height="23px" altimg-valign="-7px" altimg-width="49px" alttext="c_{k}(\eta)" display="inline"><mrow><msub><mi href="./8.12#p2">c</mi><mi href="./8.1#p1.t1.r4">k</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./8.12#p1">η</mi><mo stretchy="false">)</mo></mrow></mrow></math>: coefficients</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where</p>
<table id="E17" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex16" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="6" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">8.12.17</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m41.png" altimg-height="25px" altimg-valign="-7px" altimg-width="49px" alttext="\displaystyle c_{0}(0)" display="inline"><mrow><msub><mi href="./8.12#p2">c</mi><mn>0</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m6.png" altimg-height="29px" altimg-valign="-9px" altimg-width="61px" alttext="\displaystyle=-\tfrac{1}{3}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo>-</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex17" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m43.png" altimg-height="25px" altimg-valign="-7px" altimg-width="49px" alttext="\displaystyle c_{1}(0)" display="inline"><mrow><msub><mi href="./8.12#p2">c</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m7.png" altimg-height="29px" altimg-valign="-9px" altimg-width="77px" alttext="\displaystyle=-\tfrac{1}{540}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo>-</mo><mfrac><mn>1</mn><mn>540</mn></mfrac></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex18" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m45.png" altimg-height="25px" altimg-valign="-7px" altimg-width="49px" alttext="\displaystyle c_{2}(0)" display="inline"><mrow><msub><mi href="./8.12#p2">c</mi><mn>2</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m24.png" altimg-height="29px" altimg-valign="-9px" altimg-width="69px" alttext="\displaystyle=\tfrac{25}{6048}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mfrac><mn>25</mn><mn>6048</mn></mfrac></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex19" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m46.png" altimg-height="25px" altimg-valign="-7px" altimg-width="49px" alttext="\displaystyle c_{3}(0)" display="inline"><mrow><msub><mi href="./8.12#p2">c</mi><mn>3</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m17.png" altimg-height="29px" altimg-valign="-9px" altimg-width="90px" alttext="\displaystyle=\tfrac{101}{1\;55520}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mfrac><mn>101</mn><mn>1 55520</mn></mfrac></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex20" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m47.png" altimg-height="25px" altimg-valign="-7px" altimg-width="49px" alttext="\displaystyle c_{4}(0)" display="inline"><mrow><msub><mi href="./8.12#p2">c</mi><mn>4</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m9.png" altimg-height="29px" altimg-valign="-9px" altimg-width="137px" alttext="\displaystyle=-\tfrac{31\;84811}{36951\;55200}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo>-</mo><mfrac><mn>31 84811</mn><mn>36951 55200</mn></mfrac></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex21" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m48.png" altimg-height="25px" altimg-valign="-7px" altimg-width="49px" alttext="\displaystyle c_{5}(0)" display="inline"><mrow><msub><mi href="./8.12#p2">c</mi><mn>5</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m8.png" altimg-height="29px" altimg-valign="-9px" altimg-width="137px" alttext="\displaystyle=-\tfrac{27\;45493}{81517\;36320}." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo>-</mo><mfrac><mn>27 45493</mn><mn>81517 36320</mn></mfrac></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E17.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./8.12#p2" title="§8.12 Uniform Asymptotic Expansions for Large Parameter ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m100.png" altimg-height="23px" altimg-valign="-7px" altimg-width="49px" alttext="c_{k}(\eta)" display="inline"><mrow><msub><mi href="./8.12#p2">c</mi><mi href="./8.1#p1.t1.r4">k</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./8.12#p1">η</mi><mo stretchy="false">)</mo></mrow></mrow></math>: coefficients</a></dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="p4" class="ltx_para">
<p class="ltx_p">For error bounds for ()</cite>. For the
asymptotic behavior of <math class="ltx_Math" altimg="m100.png" altimg-height="23px" altimg-valign="-7px" altimg-width="49px" alttext="c_{k}(\eta)" display="inline"><mrow><msub><mi href="./8.12#p2">c</mi><mi href="./8.1#p1.t1.r4">k</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./8.12#p1">η</mi><mo stretchy="false">)</mo></mrow></mrow></math> as <math class="ltx_Math" altimg="m112.png" altimg-height="18px" altimg-valign="-2px" altimg-width="66px" alttext="k\to\infty" display="inline"><mrow><mi href="./8.1#p1.t1.r4">k</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></math> see
<cite class="ltx_cite ltx_citemacro_citet">Dunster<span class="ltx_text ltx_bib_etal"> et al.</span> () for
<math class="ltx_Math" altimg="m63.png" altimg-height="23px" altimg-valign="-7px" altimg-width="68px" alttext="Q\left(a,z\right)" display="inline"><mrow><mi href="./8.2#E4">Q</mi><mo></mo><mrow><mo>(</mo><mi href="./8.1#p1.t1.r3">a</mi><mo>,</mo><mi href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>.</p>
</div>
<div id="p5" class="ltx_para">
<p class="ltx_p">A different type of uniform expansion with coefficients that do not possess a
removable singularity at <math class="ltx_Math" altimg="m120.png" altimg-height="13px" altimg-valign="-2px" altimg-width="52px" alttext="z=a" display="inline"><mrow><mi href="./8.1#p1.t1.r2">z</mi><mo>=</mo><mi href="./8.1#p1.t1.r3">a</mi></mrow></math> is given by
</p>
<table id="E18" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">8.12.18</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E18.png" altimg-height="66px" altimg-valign="-28px" altimg-width="492px" alttext="\rselection{Q\left(a,z\right)\\
P\left(a,z\right)}\sim\frac{z^{a-\frac{1}{2}}e^{-z}}{\Gamma\left(a\right)}{%
\left(d(\pm\chi)\sum_{k=0}^{\infty}\frac{A_{k}(\chi)}{z^{k/2}}\mp\sum_{k=1}^{%
\infty}\frac{B_{k}(\chi)}{z^{k/2}}\right)}," display="block"><mrow><mrow><mrow><mtable displaystyle="true" rowspacing="0pt"><mtr><mtd columnalign="center"><mrow><mi href="./8.2#E4">Q</mi><mo></mo><mrow><mo>(</mo><mi href="./8.1#p1.t1.r3">a</mi><mo>,</mo><mi href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mtd></mtr><mtr><mtd columnalign="center"><mrow><mi href="./8.2#E4">P</mi><mo></mo><mrow><mo>(</mo><mi href="./8.1#p1.t1.r3">a</mi><mo>,</mo><mi href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mtd></mtr></mtable><mo>}</mo></mrow><mo href="./2.1#SS3.p1">∼</mo><mrow><mfrac><mrow><msup><mi href="./8.1#p1.t1.r2">z</mi><mrow><mi href="./8.1#p1.t1.r3">a</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mi href="./8.1#p1.t1.r2">z</mi></mrow></msup></mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi href="./8.1#p1.t1.r3">a</mi><mo>)</mo></mrow></mrow></mfrac><mo></mo><mrow><mo>(</mo><mrow><mrow><mrow><mi href="./8.12#E19">d</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mo>±</mo><mi href="./8.12#E19">χ</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./8.1#p1.t1.r4">k</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mfrac><mrow><msub><mi href="./8.12#E20">A</mi><mi href="./8.1#p1.t1.r4">k</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./8.12#E19">χ</mi><mo stretchy="false">)</mo></mrow></mrow><msup><mi href="./8.1#p1.t1.r2">z</mi><mrow><mi href="./8.1#p1.t1.r4">k</mi><mo>/</mo><mn>2</mn></mrow></msup></mfrac></mrow></mrow><mo>∓</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./8.1#p1.t1.r4">k</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mfrac><mrow><msub><mi href="./8.12#E20">B</mi><mi href="./8.1#p1.t1.r4">k</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./8.12#E19">χ</mi><mo stretchy="false">)</mo></mrow></mrow><msup><mi href="./8.1#p1.t1.r2">z</mi><mrow><mi href="./8.1#p1.t1.r4">k</mi><mo>/</mo><mn>2</mn></mrow></msup></mfrac></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E18.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m67.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./2.1#SS3.p1" title="§2.1(iii) Asymptotic Expansions ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m96.png" altimg-height="12px" altimg-valign="-2px" altimg-width="20px" alttext="\sim" display="inline"><mo href="./2.1#SS3.p1">∼</mo></math>: Poincaré asymptotic expansion</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m86.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./8.2#E4" title="(8.2.4) ‣ §8.2(i) Definitions ‣ §8.2 Definitions and Basic Properties ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="23px" altimg-valign="-7px" altimg-width="68px" alttext="P\left(\NVar{a},\NVar{z}\right)" display="inline"><mrow><mi href="./8.2#E4">P</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./8.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: normalized incomplete gamma function</a>,
<a href="./8.2#E4" title="(8.2.4) ‣ §8.2(i) Definitions ‣ §8.2 Definitions and Basic Properties ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m62.png" altimg-height="23px" altimg-valign="-7px" altimg-width="68px" alttext="Q\left(\NVar{a},\NVar{z}\right)" display="inline"><mrow><mi href="./8.2#E4">Q</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./8.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: normalized incomplete gamma function</a>,
<a href="./8.1#p1.t1.r2" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m121.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./8.1#p1.t1.r2">z</mi></math>: complex variable</a>,
<a href="./8.1#p1.t1.r3" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m98.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./8.1#p1.t1.r3">a</mi></math>: parameter</a>,
<a href="./8.1#p1.t1.r4" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m110.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./8.1#p1.t1.r4">k</mi></math>: nonnegative integer</a>,
<a href="./8.12#E19" title="(8.12.19) ‣ §8.12 Uniform Asymptotic Expansions for Large Parameter ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="16px" altimg-valign="-6px" altimg-width="17px" alttext="\chi" display="inline"><mi href="./8.12#E19">χ</mi></math></a>,
<a href="./8.12#E19" title="(8.12.19) ‣ §8.12 Uniform Asymptotic Expansions for Large Parameter ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m101.png" altimg-height="23px" altimg-valign="-7px" altimg-width="58px" alttext="d(\pm\chi)" display="inline"><mrow><mi href="./8.12#E19">d</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mo>±</mo><mi href="./8.12#E19">χ</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></math></a>,
<a href="./8.12#E20" title="(8.12.20) ‣ §8.12 Uniform Asymptotic Expansions for Large Parameter ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="23px" altimg-valign="-7px" altimg-width="57px" alttext="A_{k}(\chi)" display="inline"><mrow><msub><mi href="./8.12#E20">A</mi><mi href="./8.1#p1.t1.r4">k</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./8.12#E19">χ</mi><mo stretchy="false">)</mo></mrow></mrow></math></a> and
<a href="./8.12#E20" title="(8.12.20) ‣ §8.12 Uniform Asymptotic Expansions for Large Parameter ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="23px" altimg-valign="-7px" altimg-width="57px" alttext="B_{k}(\chi)" display="inline"><mrow><msub><mi href="./8.12#E20">B</mi><mi href="./8.1#p1.t1.r4">k</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./8.12#E19">χ</mi><mo stretchy="false">)</mo></mrow></mrow></math></a>
</dd>
<dt>Errata (effective with 1.0.16):</dt>
<dd>
The original <math class="ltx_Math" altimg="m95.png" altimg-height="17px" altimg-valign="-4px" altimg-width="20px" alttext="\pm" display="inline"><mo>±</mo></math> in front of the second summation was replaced by <math class="ltx_Math" altimg="m87.png" altimg-height="17px" altimg-valign="-4px" altimg-width="20px" alttext="\mp" display="inline"><mo>∓</mo></math> to correct an error in <cite class="ltx_cite ltx_citemacro_citet">Paris (</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">for <math class="ltx_Math" altimg="m122.png" altimg-height="13px" altimg-valign="-2px" altimg-width="65px" alttext="z\to\infty" display="inline"><mrow><mi href="./8.1#p1.t1.r2">z</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></math> in <math class="ltx_Math" altimg="m84.png" altimg-height="27px" altimg-valign="-9px" altimg-width="102px" alttext="\left|\operatorname{ph}z\right|<\frac{1}{2}\pi" display="inline"><mrow><mrow><mo>|</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mi href="./8.1#p1.t1.r2">z</mi></mrow><mo>|</mo></mrow><mo><</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow></math>, with
<math class="ltx_Math" altimg="m71.png" altimg-height="23px" altimg-valign="-7px" altimg-width="116px" alttext="\Re(z-a)\leq 0" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./8.1#p1.t1.r2">z</mi><mo>-</mo><mi href="./8.1#p1.t1.r3">a</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>≤</mo><mn>0</mn></mrow></math> for <math class="ltx_Math" altimg="m61.png" altimg-height="23px" altimg-valign="-7px" altimg-width="68px" alttext="P\left(a,z\right)" display="inline"><mrow><mi href="./8.2#E4">P</mi><mo></mo><mrow><mo>(</mo><mi href="./8.1#p1.t1.r3">a</mi><mo>,</mo><mi href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> and <math class="ltx_Math" altimg="m70.png" altimg-height="23px" altimg-valign="-7px" altimg-width="116px" alttext="\Re(z-a)\geq 0" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./8.1#p1.t1.r2">z</mi><mo>-</mo><mi href="./8.1#p1.t1.r3">a</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>≥</mo><mn>0</mn></mrow></math> for
<math class="ltx_Math" altimg="m63.png" altimg-height="23px" altimg-valign="-7px" altimg-width="68px" alttext="Q\left(a,z\right)" display="inline"><mrow><mi href="./8.2#E4">Q</mi><mo></mo><mrow><mo>(</mo><mi href="./8.1#p1.t1.r3">a</mi><mo>,</mo><mi href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>. Here</p>
<table id="E19" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex22" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">8.12.19</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m36.png" altimg-height="18px" altimg-valign="-6px" altimg-width="19px" alttext="\displaystyle\chi" display="inline"><mi href="./8.12#E19">χ</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m3.png" altimg-height="27px" altimg-valign="-7px" altimg-width="130px" alttext="\displaystyle=(z-a)/\sqrt{z}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./8.1#p1.t1.r2">z</mi><mo>-</mo><mi href="./8.1#p1.t1.r3">a</mi></mrow><mo stretchy="false">)</mo></mrow><mo>/</mo><msqrt><mi href="./8.1#p1.t1.r2">z</mi></msqrt></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex23" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m49.png" altimg-height="25px" altimg-valign="-7px" altimg-width="60px" alttext="\displaystyle d(\pm\chi)" display="inline"><mrow><mi href="./8.12#E19">d</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mo>±</mo><mi href="./8.12#E19">χ</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m16.png" altimg-height="44px" altimg-valign="-15px" altimg-width="251px" alttext="\displaystyle=\sqrt{\tfrac{1}{2}\pi}e^{\chi^{2}/2}\operatorname{erfc}\left(\pm%
\chi/\sqrt{2}\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msqrt><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow></msqrt><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><msup><mi href="./8.12#E19">χ</mi><mn>2</mn></msup><mo>/</mo><mn>2</mn></mrow></msup><mo></mo><mrow><mi href="./7.2#E2">erfc</mi><mo></mo><mrow><mo>(</mo><mrow><mo>±</mo><mrow><mi href="./8.12#E19">χ</mi><mo>/</mo><msqrt><mn>2</mn></msqrt></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E19.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd>
<span class="ltx_text"><math class="ltx_Math" altimg="m74.png" altimg-height="16px" altimg-valign="-6px" altimg-width="17px" alttext="\chi" display="inline"><mi href="./8.12#E19">χ</mi></math> (locally)</span> and
<span class="ltx_text"><math class="ltx_Math" altimg="m101.png" altimg-height="23px" altimg-valign="-7px" altimg-width="58px" alttext="d(\pm\chi)" display="inline"><mrow><mi href="./8.12#E19">d</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mo>±</mo><mi href="./8.12#E19">χ</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></math> (locally)</span>
</dd>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m94.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./7.2#E2" title="(7.2.2) ‣ §7.2(i) Error Functions ‣ §7.2 Definitions ‣ Properties ‣ Chapter 7 Error Functions, Dawson’s and Fresnel Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m90.png" altimg-height="18px" altimg-valign="-2px" altimg-width="49px" alttext="\operatorname{erfc}\NVar{z}" display="inline"><mrow><mi href="./7.2#E2">erfc</mi><mo></mo><mi class="ltx_nvar" href="./7.1#p1.t1.r2">z</mi></mrow></math>: complementary error function</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m86.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./8.1#p1.t1.r2" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m121.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./8.1#p1.t1.r2">z</mi></math>: complex variable</a> and
<a href="./8.1#p1.t1.r3" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m98.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./8.1#p1.t1.r3">a</mi></math>: parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and</p>
<table id="E20" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex24" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="3" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">8.12.20</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m26.png" altimg-height="25px" altimg-valign="-7px" altimg-width="58px" alttext="\displaystyle A_{0}(\chi)" display="inline"><mrow><msub><mi href="./8.12#E20">A</mi><mn>0</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./8.12#E19">χ</mi><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m10.png" altimg-height="22px" altimg-valign="-6px" altimg-width="43px" alttext="\displaystyle=1," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mn>1</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex25" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m27.png" altimg-height="25px" altimg-valign="-7px" altimg-width="58px" alttext="\displaystyle A_{1}(\chi)" display="inline"><mrow><msub><mi href="./8.12#E20">A</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./8.12#E19">χ</mi><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m20.png" altimg-height="30px" altimg-valign="-9px" altimg-width="116px" alttext="\displaystyle=\tfrac{1}{2}\chi+\tfrac{1}{6}\chi^{3}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./8.12#E19">χ</mi></mrow><mo>+</mo><mrow><mfrac><mn>1</mn><mn>6</mn></mfrac><mo></mo><msup><mi href="./8.12#E19">χ</mi><mn>3</mn></msup></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex26" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m28.png" altimg-height="25px" altimg-valign="-7px" altimg-width="58px" alttext="\displaystyle B_{1}(\chi)" display="inline"><mrow><msub><mi href="./8.12#E20">B</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./8.12#E19">χ</mi><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m22.png" altimg-height="30px" altimg-valign="-9px" altimg-width="104px" alttext="\displaystyle=\tfrac{1}{3}+\tfrac{1}{6}\chi^{2}." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac><mo>+</mo><mrow><mfrac><mn>1</mn><mn>6</mn></mfrac><mo></mo><msup><mi href="./8.12#E19">χ</mi><mn>2</mn></msup></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E20.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd>
<span class="ltx_text"><math class="ltx_Math" altimg="m53.png" altimg-height="23px" altimg-valign="-7px" altimg-width="57px" alttext="A_{k}(\chi)" display="inline"><mrow><msub><mi href="./8.12#E20">A</mi><mi href="./8.1#p1.t1.r4">k</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./8.12#E19">χ</mi><mo stretchy="false">)</mo></mrow></mrow></math> (locally)</span> and
<span class="ltx_text"><math class="ltx_Math" altimg="m54.png" altimg-height="23px" altimg-valign="-7px" altimg-width="57px" alttext="B_{k}(\chi)" display="inline"><mrow><msub><mi href="./8.12#E20">B</mi><mi href="./8.1#p1.t1.r4">k</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./8.12#E19">χ</mi><mo stretchy="false">)</mo></mrow></mrow></math> (locally)</span>
</dd>
<dt>Symbols:</dt>
<dd>
<a href="./8.1#p1.t1.r4" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m110.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./8.1#p1.t1.r4">k</mi></math>: nonnegative integer</a> and
<a href="./8.12#E19" title="(8.12.19) ‣ §8.12 Uniform Asymptotic Expansions for Large Parameter ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="16px" altimg-valign="-6px" altimg-width="17px" alttext="\chi" display="inline"><mi href="./8.12#E19">χ</mi></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Higher coefficients <math class="ltx_Math" altimg="m53.png" altimg-height="23px" altimg-valign="-7px" altimg-width="57px" alttext="A_{k}(\chi)" display="inline"><mrow><msub><mi href="./8.12#E20">A</mi><mi href="./8.1#p1.t1.r4">k</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./8.12#E19">χ</mi><mo stretchy="false">)</mo></mrow></mrow></math>, <math class="ltx_Math" altimg="m54.png" altimg-height="23px" altimg-valign="-7px" altimg-width="57px" alttext="B_{k}(\chi)" display="inline"><mrow><msub><mi href="./8.12#E20">B</mi><mi href="./8.1#p1.t1.r4">k</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./8.12#E19">χ</mi><mo stretchy="false">)</mo></mrow></mrow></math>, up to <math class="ltx_Math" altimg="m109.png" altimg-height="18px" altimg-valign="-2px" altimg-width="52px" alttext="k=8" display="inline"><mrow><mi href="./8.1#p1.t1.r4">k</mi><mo>=</mo><mn>8</mn></mrow></math>, are given in
<cite class="ltx_cite ltx_citemacro_citet">Paris ()</cite>.</p>
</div>
<div id="p6" class="ltx_para">
<p class="ltx_p">Lastly, a uniform approximation for <math class="ltx_Math" altimg="m68.png" altimg-height="23px" altimg-valign="-7px" altimg-width="77px" alttext="\Gamma\left(a,ax\right)" display="inline"><mrow><mi href="./8.2#E2" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi href="./8.1#p1.t1.r3">a</mi><mo>,</mo><mrow><mi href="./8.1#p1.t1.r3">a</mi><mo></mo><mi>x</mi></mrow><mo>)</mo></mrow></mrow></math> for large <math class="ltx_Math" altimg="m98.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./8.1#p1.t1.r3">a</mi></math>, with
error bounds, can be found in <cite class="ltx_cite ltx_citemacro_citet">Dunster ()</cite>.</p>
</div>
<div id="p7" class="ltx_para">
<p class="ltx_p">For other uniform asymptotic approximations of the incomplete gamma functions
in terms of the function <math class="ltx_Math" altimg="m91.png" altimg-height="18px" altimg-valign="-2px" altimg-width="36px" alttext="\operatorname{erfc}" display="inline"><mi href="./7.2#E2">erfc</mi></math> see <cite class="ltx_cite ltx_citemacro_citet">Paris (</dd>
</dl>
</div>
</div>
<div id="Px1.p1" class="ltx_para">
<p class="ltx_p">For asymptotic expansions, as <math class="ltx_Math" altimg="m99.png" altimg-height="13px" altimg-valign="-2px" altimg-width="66px" alttext="a\to\infty" display="inline"><mrow><mi href="./8.1#p1.t1.r3">a</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></math>, of the <em class="ltx_emph ltx_font_italic">inverse function</em>
<math class="ltx_Math" altimg="m118.png" altimg-height="23px" altimg-valign="-7px" altimg-width="98px" alttext="x=x(a,q)" display="inline"><mrow><mi>x</mi><mo>=</mo><mrow><mi>x</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./8.1#p1.t1.r3">a</mi><mo>,</mo><mi>q</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></math> that satisfies the equation</p>
<table id="E21" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">8.12.21</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E21.png" altimg-height="25px" altimg-valign="-7px" altimg-width="108px" alttext="Q\left(a,x\right)=q" display="block"><mrow><mrow><mi href="./8.2#E4">Q</mi><mo></mo><mrow><mo>(</mo><mi href="./8.1#p1.t1.r3">a</mi><mo>,</mo><mi>x</mi><mo>)</mo></mrow></mrow><mo>=</mo><mi>q</mi></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E21.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./8.2#E4" title="(8.2.4) ‣ §8.2(i) Definitions ‣ §8.2 Definitions and Basic Properties ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m62.png" altimg-height="23px" altimg-valign="-7px" altimg-width="68px" alttext="Q\left(\NVar{a},\NVar{z}\right)" display="inline"><mrow><mi href="./8.2#E4">Q</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./8.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: normalized incomplete gamma function</a> and
<a href="./8.1#p1.t1.r3" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m98.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./8.1#p1.t1.r3">a</mi></math>: parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">see <cite class="ltx_cite ltx_citemacro_citet">Temme ()</cite>. These expansions involve the inverse error function
<math class="ltx_Math" altimg="m92.png" altimg-height="23px" altimg-valign="-7px" altimg-width="92px" alttext="\operatorname{inverfc}\left(x\right)" display="inline"><mrow><mi href="./7.17#E1">inverfc</mi><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></math> (§), and are uniform with respect to
<math class="ltx_Math" altimg="m117.png" altimg-height="23px" altimg-valign="-7px" altimg-width="78px" alttext="q\in[0,1]" display="inline"><mrow><mi>q</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">[</mo><mn>0</mn><mo href="./front/introduction#Sx4.p1.t1.r29">,</mo><mn>1</mn><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">]</mo></mrow></mrow></math>. As a special case,</p>
<table id="E22" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">8.12.22</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E22.png" altimg-height="30px" altimg-valign="-9px" altimg-width="503px" alttext="x(a,\tfrac{1}{2})\sim a-\tfrac{1}{3}+\tfrac{8}{405}a^{-1}+\tfrac{184}{25515}a^%
{-2}+\tfrac{2248}{34\;44525}a^{-3}+\cdots," display="block"><mrow><mrow><mrow><mi>x</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./8.1#p1.t1.r3">a</mi><mo>,</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo stretchy="false">)</mo></mrow></mrow><mo href="./2.1#SS3.p1">∼</mo><mrow><mrow><mi href="./8.1#p1.t1.r3">a</mi><mo>-</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>3</mn></mfrac></mstyle></mrow><mo>+</mo><mrow><mstyle displaystyle="false"><mfrac><mn>8</mn><mn>405</mn></mfrac></mstyle><mo></mo><msup><mi href="./8.1#p1.t1.r3">a</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></mrow><mo>+</mo><mrow><mstyle displaystyle="false"><mfrac><mn>184</mn><mn>25515</mn></mfrac></mstyle><mo></mo><msup><mi href="./8.1#p1.t1.r3">a</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></mrow><mo>+</mo><mrow><mstyle displaystyle="false"><mfrac><mn>2248</mn><mn>34 44525</mn></mfrac></mstyle><mo></mo><msup><mi href="./8.1#p1.t1.r3">a</mi><mrow><mo>-</mo><mn>3</mn></mrow></msup></mrow><mo>+</mo><mi mathvariant="normal">⋯</mi></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m99.png" altimg-height="13px" altimg-valign="-2px" altimg-width="66px" alttext="a\to\infty" display="inline"><mrow><mi href="./8.1#p1.t1.r3">a</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E22.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./2.1#SS3.p1" title="§2.1(iii) Asymptotic Expansions ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m96.png" altimg-height="12px" altimg-valign="-2px" altimg-width="20px" alttext="\sim" display="inline"><mo href="./2.1#SS3.p1">∼</mo></math>: Poincaré asymptotic expansion</a> and
<a href="./8.1#p1.t1.r3" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m98.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./8.1#p1.t1.r3">a</mi></math>: parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
</section>
</div>
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<span></div>
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<title>DLMF: 8.19 Generalized Exponential Integral</title>
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<div class="ltx_page_navlogo"><a href="./8.18" title="§8.18 Asymptotic Expansions of I x ( a , b ) ‣ Related Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref" rel="prev"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">8.18 </span>Asymptotic Expansions of <math class="ltx_Math" altimg="m28.png" altimg-height="23px" altimg-valign="-7px" altimg-width="70px" alttext="I_{x}\left(a,b\right)" display="inline"><mrow><msub><mi href="./8.17#E2">I</mi><mi href="./8.17#SS1.p1">x</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./8.17#SS1.p1">a</mi><mo>,</mo><mi href="./8.17#SS1.p1">b</mi><mo>)</mo></mrow></mrow></math></span></a><a href="./8.20" title="§8.20 Asymptotic Expansions of E p ( z ) ‣ Related Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref" rel="next"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">8.20 </span>Asymptotic Expansions of <math class="ltx_Math" altimg="m24.png" altimg-height="24px" altimg-valign="-8px" altimg-width="57px" alttext="E_{p}\left(z\right)" display="inline"><mrow><msub><mi href="./8.19#E1">E</mi><mi href="./8.1#p1.t1.r3">p</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math></span></a>
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<div class="ltx_page_content">
<section class="ltx_section ltx_leqno">
<h1 class="ltx_title ltx_title_section">
<span class="ltx_tag ltx_tag_section">§8.19 </span>Generalized Exponential Integral</h1>
<div id="info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="SS1.p1" class="ltx_para">
<p class="ltx_p">For <math class="ltx_Math" altimg="m67.png" altimg-height="21px" altimg-valign="-6px" altimg-width="72px" alttext="p,z\in\mathbb{C}" display="inline"><mrow><mrow><mi href="./8.1#p1.t1.r3">p</mi><mo>,</mo><mi href="./8.1#p1.t1.r2">z</mi></mrow><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mi href="./front/introduction#Sx4.p1.t1.r1">ℂ</mi></mrow></math></p>
<table id="E1" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">8.19.1</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E1.png" altimg-height="29px" altimg-valign="-8px" altimg-width="229px" alttext="E_{p}\left(z\right)=z^{p-1}\Gamma\left(1-p,z\right)." display="block"><mrow><mrow><mrow><msub><mi href="./8.19#E1">E</mi><mi href="./8.1#p1.t1.r3">p</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msup><mi href="./8.1#p1.t1.r2">z</mi><mrow><mi href="./8.1#p1.t1.r3">p</mi><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mrow><mi href="./8.2#E2" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mi href="./8.1#p1.t1.r3">p</mi></mrow><mo>,</mo><mi href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m20.png" altimg-height="24px" altimg-valign="-8px" altimg-width="57px" alttext="E_{\NVar{p}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./8.19#E1">E</mi><mi class="ltx_nvar" href="./8.1#p1.t1.r3">p</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: generalized exponential integral</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./8.2#E2" title="(8.2.2) ‣ §8.2(i) Definitions ‣ §8.2 Definitions and Basic Properties ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m32.png" altimg-height="23px" altimg-valign="-7px" altimg-width="65px" alttext="\Gamma\left(\NVar{a},\NVar{z}\right)" display="inline"><mrow><mi href="./8.2#E2" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./8.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: incomplete gamma function</a>,
<a href="./8.1#p1.t1.r2" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./8.1#p1.t1.r2">z</mi></math>: complex variable</a> and
<a href="./8.1#p1.t1.r3" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m71.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="p" display="inline"><mi href="./8.1#p1.t1.r3">p</mi></math>: parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">5.1.45</span></span> <span class="ltx_origref"><span class="ltx_tag">6.5.9</span> (Definition extended to general values of <math class="ltx_Math" altimg="m71.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="p" display="inline"><mi href="./8.1#p1.t1.r3">p</mi></math>.)</span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Most properties of <math class="ltx_Math" altimg="m24.png" altimg-height="24px" altimg-valign="-8px" altimg-width="57px" alttext="E_{p}\left(z\right)" display="inline"><mrow><msub><mi href="./8.19#E1">E</mi><mi href="./8.1#p1.t1.r3">p</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> follow straightforwardly from those of
<math class="ltx_Math" altimg="m34.png" altimg-height="23px" altimg-valign="-7px" altimg-width="65px" alttext="\Gamma\left(a,z\right)" display="inline"><mrow><mi href="./8.2#E2" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi href="./8.1#p1.t1.r3">a</mi><mo>,</mo><mi href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>. For an extensive treatment of <math class="ltx_Math" altimg="m18.png" altimg-height="23px" altimg-valign="-7px" altimg-width="57px" alttext="E_{1}\left(z\right)" display="inline"><mrow><msub><mi href="./8.19#E1">E</mi><mn>1</mn></msub><mo></mo><mrow><mo>(</mo><mi href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> see
Chapter .</p>
<table id="E2" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">8.19.2</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E2.png" altimg-height="53px" altimg-valign="-20px" altimg-width="226px" alttext="E_{p}\left(z\right)=z^{p-1}\int_{z}^{\infty}\frac{e^{-t}}{t^{p}}\mathrm{d}t." display="block"><mrow><mrow><mrow><msub><mi href="./8.19#E1">E</mi><mi href="./8.1#p1.t1.r3">p</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msup><mi href="./8.1#p1.t1.r2">z</mi><mrow><mi href="./8.1#p1.t1.r3">p</mi><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi href="./8.1#p1.t1.r2">z</mi><mi mathvariant="normal">∞</mi></msubsup><mrow><mfrac><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mi>t</mi></mrow></msup><msup><mi>t</mi><mi href="./8.1#p1.t1.r3">p</mi></msup></mfrac><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m75.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./8.19#E1" title="(8.19.1) ‣ §8.19(i) Definition and Integral Representations ‣ §8.19 Generalized Exponential Integral ‣ Related Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="24px" altimg-valign="-8px" altimg-width="57px" alttext="E_{\NVar{p}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./8.19#E1">E</mi><mi class="ltx_nvar" href="./8.1#p1.t1.r3">p</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: generalized exponential integral</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./8.1#p1.t1.r2" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./8.1#p1.t1.r2">z</mi></math>: complex variable</a> and
<a href="./8.1#p1.t1.r3" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m71.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="p" display="inline"><mi href="./8.1#p1.t1.r3">p</mi></math>: parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">When the path of integration excludes the origin and does not cross the
negative real axis () defines the <em class="ltx_emph ltx_font_italic">principal value</em> of
<math class="ltx_Math" altimg="m24.png" altimg-height="24px" altimg-valign="-8px" altimg-width="57px" alttext="E_{p}\left(z\right)" display="inline"><mrow><msub><mi href="./8.19#E1">E</mi><mi href="./8.1#p1.t1.r3">p</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>, and <em class="ltx_emph ltx_font_italic">unless indicated otherwise</em> in
the DLMF principal
values are assumed.</p>
</div>
<section id="Px1" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Other Integral Representations</h3>
<div id="Px1.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px1.p1" class="ltx_para">
<table id="EGx1" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E3">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">8.19.3</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m5.png" altimg-height="26px" altimg-valign="-8px" altimg-width="59px" alttext="\displaystyle E_{p}\left(z\right)" display="inline"><mrow><msub><mi href="./8.19#E1">E</mi><mi href="./8.1#p1.t1.r3">p</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m3.png" altimg-height="53px" altimg-valign="-20px" altimg-width="133px" alttext="\displaystyle=\int_{1}^{\infty}\frac{e^{-zt}}{t^{p}}\mathrm{d}t," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>1</mn><mi mathvariant="normal">∞</mi></msubsup></mstyle><mrow><mstyle displaystyle="true"><mfrac><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mi href="./8.1#p1.t1.r2">z</mi><mo></mo><mi>t</mi></mrow></mrow></msup><msup><mi>t</mi><mi href="./8.1#p1.t1.r3">p</mi></msup></mfrac></mstyle><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m80.png" altimg-height="27px" altimg-valign="-9px" altimg-width="106px" alttext="|\operatorname{ph}z|<\tfrac{1}{2}\pi" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mi href="./8.1#p1.t1.r2">z</mi></mrow><mo stretchy="false">|</mo></mrow><mo><</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m75.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./8.19#E1" title="(8.19.1) ‣ §8.19(i) Definition and Integral Representations ‣ §8.19 Generalized Exponential Integral ‣ Related Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="24px" altimg-valign="-8px" altimg-width="57px" alttext="E_{\NVar{p}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./8.19#E1">E</mi><mi class="ltx_nvar" href="./8.1#p1.t1.r3">p</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: generalized exponential integral</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.9#E7" title="(1.9.7) ‣ Modulus and Phase ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="21px" altimg-valign="-6px" altimg-width="26px" alttext="\operatorname{ph}" display="inline"><mi href="./1.9#E7">ph</mi></math>: phase</a>,
<a href="./8.1#p1.t1.r2" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./8.1#p1.t1.r2">z</mi></math>: complex variable</a> and
<a href="./8.1#p1.t1.r3" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m71.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="p" display="inline"><mi href="./8.1#p1.t1.r3">p</mi></math>: parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">5.1.4</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E4">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">8.19.4</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m5.png" altimg-height="26px" altimg-valign="-8px" altimg-width="59px" alttext="\displaystyle E_{p}\left(z\right)" display="inline"><mrow><msub><mi href="./8.19#E1">E</mi><mi href="./8.1#p1.t1.r3">p</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m2.png" altimg-height="55px" altimg-valign="-21px" altimg-width="248px" alttext="\displaystyle=\frac{z^{p-1}e^{-z}}{\Gamma\left(p\right)}\int_{0}^{\infty}\frac%
{t^{p-1}e^{-zt}}{1+t}\mathrm{d}t," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><mfrac><mrow><msup><mi href="./8.1#p1.t1.r2">z</mi><mrow><mi href="./8.1#p1.t1.r3">p</mi><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mi href="./8.1#p1.t1.r2">z</mi></mrow></msup></mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi href="./8.1#p1.t1.r3">p</mi><mo>)</mo></mrow></mrow></mfrac></mstyle><mo></mo><mrow><mstyle displaystyle="true"><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup></mstyle><mrow><mstyle displaystyle="true"><mfrac><mrow><msup><mi>t</mi><mrow><mi href="./8.1#p1.t1.r3">p</mi><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mi href="./8.1#p1.t1.r2">z</mi><mo></mo><mi>t</mi></mrow></mrow></msup></mrow><mrow><mn>1</mn><mo>+</mo><mi>t</mi></mrow></mfrac></mstyle><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m80.png" altimg-height="27px" altimg-valign="-9px" altimg-width="106px" alttext="|\operatorname{ph}z|<\tfrac{1}{2}\pi" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mi href="./8.1#p1.t1.r2">z</mi></mrow><mo stretchy="false">|</mo></mrow><mo><</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow></math>, <math class="ltx_Math" altimg="m36.png" altimg-height="21px" altimg-valign="-6px" altimg-width="65px" alttext="\Re p>0" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./8.1#p1.t1.r3">p</mi></mrow><mo>></mo><mn>0</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E4.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m33.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m75.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./8.19#E1" title="(8.19.1) ‣ §8.19(i) Definition and Integral Representations ‣ §8.19 Generalized Exponential Integral ‣ Related Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="24px" altimg-valign="-8px" altimg-width="57px" alttext="E_{\NVar{p}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./8.19#E1">E</mi><mi class="ltx_nvar" href="./8.1#p1.t1.r3">p</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: generalized exponential integral</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.9#E7" title="(1.9.7) ‣ Modulus and Phase ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="21px" altimg-valign="-6px" altimg-width="26px" alttext="\operatorname{ph}" display="inline"><mi href="./1.9#E7">ph</mi></math>: phase</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m40.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./8.1#p1.t1.r2" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./8.1#p1.t1.r2">z</mi></math>: complex variable</a> and
<a href="./8.1#p1.t1.r3" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m71.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="p" display="inline"><mi href="./8.1#p1.t1.r3">p</mi></math>: parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
</div>
<div id="Px1.p2" class="ltx_para">
<p class="ltx_p">Integral representations of Mellin–Barnes type for <math class="ltx_Math" altimg="m24.png" altimg-height="24px" altimg-valign="-8px" altimg-width="57px" alttext="E_{p}\left(z\right)" display="inline"><mrow><msub><mi href="./8.19#E1">E</mi><mi href="./8.1#p1.t1.r3">p</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> follow
immediately from (</div>
<figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 8.19.1: </span><math class="ltx_Math" altimg="m23.png" altimg-height="24px" altimg-valign="-8px" altimg-width="58px" alttext="E_{p}\left(x\right)" display="inline"><mrow><msub><mi href="./8.19#E1">E</mi><mi href="./8.1#p1.t1.r3">p</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./8.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow></math>,
<math class="ltx_Math" altimg="m14.png" altimg-height="19px" altimg-valign="-5px" altimg-width="89px" alttext="0\leq x\leq 3" display="inline"><mrow><mn>0</mn><mo>≤</mo><mi href="./8.1#p1.t1.r1">x</mi><mo>≤</mo><mn>3</mn></mrow></math>, <math class="ltx_Math" altimg="m13.png" altimg-height="20px" altimg-valign="-6px" altimg-width="87px" alttext="0\leq p\leq 8" display="inline"><mrow><mn>0</mn><mo>≤</mo><mi href="./8.1#p1.t1.r3">p</mi><mo>≤</mo><mn>8</mn></mrow></math>.
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./8.19#E1" title="(8.19.1) ‣ §8.19(i) Definition and Integral Representations ‣ §8.19 Generalized Exponential Integral ‣ Related Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="24px" altimg-valign="-8px" altimg-width="57px" alttext="E_{\NVar{p}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./8.19#E1">E</mi><mi class="ltx_nvar" href="./8.1#p1.t1.r3">p</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: generalized exponential integral</a>,
<a href="./8.1#p1.t1.r1" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m75.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./8.1#p1.t1.r1">x</mi></math>: real variable</a> and
<a href="./8.1#p1.t1.r3" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m71.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="p" display="inline"><mi href="./8.1#p1.t1.r3">p</mi></math>: parameter</a>
</dd>
</dl>
</div>
</div>
</figure>
<div id="SS2.p1" class="ltx_para">
<p class="ltx_p">In Figures </div>
<figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 8.19.2: </span><math class="ltx_Math" altimg="m21.png" altimg-height="28px" altimg-valign="-12px" altimg-width="103px" alttext="E_{\frac{1}{2}}\left(x+iy\right)" display="inline"><mrow><msub><mi href="./8.19#E1">E</mi><mfrac><mn>1</mn><mn>2</mn></mfrac></msub><mo></mo><mrow><mo>(</mo><mrow><mi href="./8.1#p1.t1.r1">x</mi><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi>y</mi></mrow></mrow><mo>)</mo></mrow></mrow></math>,
<math class="ltx_Math" altimg="m11.png" altimg-height="19px" altimg-valign="-5px" altimg-width="104px" alttext="-4\leq x\leq 4" display="inline"><mrow><mrow><mo>-</mo><mn>4</mn></mrow><mo>≤</mo><mi href="./8.1#p1.t1.r1">x</mi><mo>≤</mo><mn>4</mn></mrow></math>, <math class="ltx_Math" altimg="m12.png" altimg-height="20px" altimg-valign="-6px" altimg-width="103px" alttext="-4\leq y\leq 4" display="inline"><mrow><mrow><mo>-</mo><mn>4</mn></mrow><mo>≤</mo><mi>y</mi><mo>≤</mo><mn>4</mn></mrow></math>.
Principal value.
There is a branch cut along the negative real axis.
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./8.19#E1" title="(8.19.1) ‣ §8.19(i) Definition and Integral Representations ‣ §8.19 Generalized Exponential Integral ‣ Related Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="24px" altimg-valign="-8px" altimg-width="57px" alttext="E_{\NVar{p}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./8.19#E1">E</mi><mi class="ltx_nvar" href="./8.1#p1.t1.r3">p</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: generalized exponential integral</a> and
<a href="./8.1#p1.t1.r1" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m75.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./8.1#p1.t1.r1">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</figure>
</td>
<td class="ltx_subfigure">
<figure id="F3" class="ltx_figure">
<div class="ltx_vizimg">
</div>
<figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 8.19.3: </span><math class="ltx_Math" altimg="m17.png" altimg-height="23px" altimg-valign="-7px" altimg-width="100px" alttext="E_{1}\left(x+iy\right)" display="inline"><mrow><msub><mi href="./8.19#E1">E</mi><mn>1</mn></msub><mo></mo><mrow><mo>(</mo><mrow><mi href="./8.1#p1.t1.r1">x</mi><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi>y</mi></mrow></mrow><mo>)</mo></mrow></mrow></math>,
<math class="ltx_Math" altimg="m11.png" altimg-height="19px" altimg-valign="-5px" altimg-width="104px" alttext="-4\leq x\leq 4" display="inline"><mrow><mrow><mo>-</mo><mn>4</mn></mrow><mo>≤</mo><mi href="./8.1#p1.t1.r1">x</mi><mo>≤</mo><mn>4</mn></mrow></math>, <math class="ltx_Math" altimg="m12.png" altimg-height="20px" altimg-valign="-6px" altimg-width="103px" alttext="-4\leq y\leq 4" display="inline"><mrow><mrow><mo>-</mo><mn>4</mn></mrow><mo>≤</mo><mi>y</mi><mo>≤</mo><mn>4</mn></mrow></math>.
Principal value.
There is a branch cut along the negative real axis.
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./8.19#E1" title="(8.19.1) ‣ §8.19(i) Definition and Integral Representations ‣ §8.19 Generalized Exponential Integral ‣ Related Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="24px" altimg-valign="-8px" altimg-width="57px" alttext="E_{\NVar{p}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./8.19#E1">E</mi><mi class="ltx_nvar" href="./8.1#p1.t1.r3">p</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: generalized exponential integral</a> and
<a href="./8.1#p1.t1.r1" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m75.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./8.1#p1.t1.r1">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</figure>
</td>
</tr>
</table>
</figure>
<div id="SS2.p2" class="ltx_para">
</div>
<figure id="SS2.fig2" class="ltx_figure">
<table style="width:100%;">
<tr>
<td class="ltx_subfigure">
<figure id="F4" class="ltx_figure">
<div class="ltx_vizimg">
</div>
<figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 8.19.4: </span><math class="ltx_Math" altimg="m22.png" altimg-height="28px" altimg-valign="-12px" altimg-width="103px" alttext="E_{\frac{3}{2}}\left(x+iy\right)" display="inline"><mrow><msub><mi href="./8.19#E1">E</mi><mfrac><mn>3</mn><mn>2</mn></mfrac></msub><mo></mo><mrow><mo>(</mo><mrow><mi href="./8.1#p1.t1.r1">x</mi><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi>y</mi></mrow></mrow><mo>)</mo></mrow></mrow></math>,
<math class="ltx_Math" altimg="m9.png" altimg-height="19px" altimg-valign="-5px" altimg-width="104px" alttext="-3\leq x\leq 3" display="inline"><mrow><mrow><mo>-</mo><mn>3</mn></mrow><mo>≤</mo><mi href="./8.1#p1.t1.r1">x</mi><mo>≤</mo><mn>3</mn></mrow></math>, <math class="ltx_Math" altimg="m10.png" altimg-height="20px" altimg-valign="-6px" altimg-width="103px" alttext="-3\leq y\leq 3" display="inline"><mrow><mrow><mo>-</mo><mn>3</mn></mrow><mo>≤</mo><mi>y</mi><mo>≤</mo><mn>3</mn></mrow></math>.
Principal value.
There is a branch cut along the negative real axis.
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./8.19#E1" title="(8.19.1) ‣ §8.19(i) Definition and Integral Representations ‣ §8.19 Generalized Exponential Integral ‣ Related Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="24px" altimg-valign="-8px" altimg-width="57px" alttext="E_{\NVar{p}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./8.19#E1">E</mi><mi class="ltx_nvar" href="./8.1#p1.t1.r3">p</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: generalized exponential integral</a> and
<a href="./8.1#p1.t1.r1" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m75.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./8.1#p1.t1.r1">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</figure>
</td>
<td class="ltx_subfigure">
<figure id="F5" class="ltx_figure">
<div class="ltx_vizimg">
</div>
<figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 8.19.5: </span><math class="ltx_Math" altimg="m19.png" altimg-height="23px" altimg-valign="-7px" altimg-width="100px" alttext="E_{2}\left(x+iy\right)" display="inline"><mrow><msub><mi href="./8.19#E1">E</mi><mn>2</mn></msub><mo></mo><mrow><mo>(</mo><mrow><mi href="./8.1#p1.t1.r1">x</mi><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi>y</mi></mrow></mrow><mo>)</mo></mrow></mrow></math>,
<math class="ltx_Math" altimg="m9.png" altimg-height="19px" altimg-valign="-5px" altimg-width="104px" alttext="-3\leq x\leq 3" display="inline"><mrow><mrow><mo>-</mo><mn>3</mn></mrow><mo>≤</mo><mi href="./8.1#p1.t1.r1">x</mi><mo>≤</mo><mn>3</mn></mrow></math>, <math class="ltx_Math" altimg="m10.png" altimg-height="20px" altimg-valign="-6px" altimg-width="103px" alttext="-3\leq y\leq 3" display="inline"><mrow><mrow><mo>-</mo><mn>3</mn></mrow><mo>≤</mo><mi>y</mi><mo>≤</mo><mn>3</mn></mrow></math>.
Principal value.
There is a branch cut along the negative real axis.
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./8.19#E1" title="(8.19.1) ‣ §8.19(i) Definition and Integral Representations ‣ §8.19 Generalized Exponential Integral ‣ Related Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="24px" altimg-valign="-8px" altimg-width="57px" alttext="E_{\NVar{p}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./8.19#E1">E</mi><mi class="ltx_nvar" href="./8.1#p1.t1.r3">p</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: generalized exponential integral</a> and
<a href="./8.1#p1.t1.r1" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m75.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./8.1#p1.t1.r1">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</figure>
</td>
</tr>
</table>
</figure>
</section>
<section id="iii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§8.19(iii) </span>Special Values</h2>
<div id="SS3.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="SS3.p1" class="ltx_para">
<table id="E5" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">8.19.5</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E5.png" altimg-height="28px" altimg-valign="-7px" altimg-width="153px" alttext="E_{0}\left(z\right)=z^{-1}e^{-z}," display="block"><mrow><mrow><mrow><msub><mi href="./8.19#E1">E</mi><mn>0</mn></msub><mo></mo><mrow><mo>(</mo><mi href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msup><mi href="./8.1#p1.t1.r2">z</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mi href="./8.1#p1.t1.r2">z</mi></mrow></msup></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m78.png" altimg-height="21px" altimg-valign="-6px" altimg-width="51px" alttext="z\neq 0" display="inline"><mrow><mi href="./8.1#p1.t1.r2">z</mi><mo>≠</mo><mn>0</mn></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E5.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./8.19#E1" title="(8.19.1) ‣ §8.19(i) Definition and Integral Representations ‣ §8.19 Generalized Exponential Integral ‣ Related Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="24px" altimg-valign="-8px" altimg-width="57px" alttext="E_{\NVar{p}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./8.19#E1">E</mi><mi class="ltx_nvar" href="./8.1#p1.t1.r3">p</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: generalized exponential integral</a> and
<a href="./8.1#p1.t1.r2" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./8.1#p1.t1.r2">z</mi></math>: complex variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">5.1.24</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E6" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">8.19.6</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E6.png" altimg-height="49px" altimg-valign="-20px" altimg-width="140px" alttext="E_{p}\left(0\right)=\frac{1}{p-1}," display="block"><mrow><mrow><mrow><msub><mi href="./8.19#E1">E</mi><mi href="./8.1#p1.t1.r3">p</mi></msub><mo></mo><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow></mrow><mo>=</mo><mfrac><mn>1</mn><mrow><mi href="./8.1#p1.t1.r3">p</mi><mo>-</mo><mn>1</mn></mrow></mfrac></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m37.png" altimg-height="21px" altimg-valign="-6px" altimg-width="65px" alttext="\Re p>1" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./8.1#p1.t1.r3">p</mi></mrow><mo>></mo><mn>1</mn></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E6.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./8.19#E1" title="(8.19.1) ‣ §8.19(i) Definition and Integral Representations ‣ §8.19 Generalized Exponential Integral ‣ Related Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="24px" altimg-valign="-8px" altimg-width="57px" alttext="E_{\NVar{p}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./8.19#E1">E</mi><mi class="ltx_nvar" href="./8.1#p1.t1.r3">p</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: generalized exponential integral</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m40.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a> and
<a href="./8.1#p1.t1.r3" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m71.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="p" display="inline"><mi href="./8.1#p1.t1.r3">p</mi></math>: parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">5.1.23</span> (Extended to general values of <math class="ltx_Math" altimg="m71.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="p" display="inline"><mi href="./8.1#p1.t1.r3">p</mi></math>.)</span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E7" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">8.19.7</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E7.png" altimg-height="67px" altimg-valign="-28px" altimg-width="506px" alttext="E_{n}\left(z\right)=\frac{(-z)^{n-1}}{(n-1)!}E_{1}\left(z\right)+\frac{e^{-z}}%
{(n-1)!}\sum_{k=0}^{n-2}(n-k-2)!(-z)^{k}," display="block"><mrow><mrow><mrow><msub><mi href="./8.19#E1">E</mi><mi href="./8.1#p1.t1.r4">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mfrac><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mi href="./8.1#p1.t1.r2">z</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mi href="./8.1#p1.t1.r4">n</mi><mo>-</mo><mn>1</mn></mrow></msup><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./8.1#p1.t1.r4">n</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mfrac><mo></mo><mrow><msub><mi href="./6.2#E1">E</mi><mn>1</mn></msub><mo></mo><mrow><mo>(</mo><mi href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mfrac><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mi href="./8.1#p1.t1.r2">z</mi></mrow></msup><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./8.1#p1.t1.r4">n</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mfrac><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./8.1#p1.t1.r4">k</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi href="./8.1#p1.t1.r4">n</mi><mo>-</mo><mn>2</mn></mrow></munderover><mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./8.1#p1.t1.r4">n</mi><mo>-</mo><mi href="./8.1#p1.t1.r4">k</mi><mo>-</mo><mn>2</mn></mrow><mo stretchy="false">)</mo></mrow><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mi href="./8.1#p1.t1.r2">z</mi></mrow><mo stretchy="false">)</mo></mrow><mi href="./8.1#p1.t1.r4">k</mi></msup></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m64.png" altimg-height="20px" altimg-valign="-6px" altimg-width="104px" alttext="n=2,3,\dots" display="inline"><mrow><mi href="./8.1#p1.t1.r4">n</mi><mo>=</mo><mrow><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E7.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./6.2#E1" title="(6.2.1) ‣ §6.2(i) Exponential and Logarithmic Integrals ‣ §6.2 Definitions and Interrelations ‣ Properties ‣ Chapter 6 Exponential, Logarithmic, Sine, and Cosine Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m16.png" altimg-height="23px" altimg-valign="-7px" altimg-width="57px" alttext="E_{1}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./6.2#E1">E</mi><mn>1</mn></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./6.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: exponential integral</a>,
<a href="./8.19#E1" title="(8.19.1) ‣ §8.19(i) Definition and Integral Representations ‣ §8.19 Generalized Exponential Integral ‣ Related Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="24px" altimg-valign="-8px" altimg-width="57px" alttext="E_{\NVar{p}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./8.19#E1">E</mi><mi class="ltx_nvar" href="./8.1#p1.t1.r3">p</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: generalized exponential integral</a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m8.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m61.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./8.1#p1.t1.r2" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./8.1#p1.t1.r2">z</mi></math>: complex variable</a>,
<a href="./8.1#p1.t1.r4" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./8.1#p1.t1.r4">k</mi></math>: nonnegative integer</a> and
<a href="./8.1#p1.t1.r4" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m65.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./8.1#p1.t1.r4">n</mi></math>: nonnegative integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="iv" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§8.19(iv) </span>Series Expansions</h2>
<div id="SS4.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="SS4.p1" class="ltx_para">
<p class="ltx_p">For <math class="ltx_Math" altimg="m62.png" altimg-height="20px" altimg-valign="-6px" altimg-width="123px" alttext="n=1,2,3,\dots" display="inline"><mrow><mi href="./8.1#p1.t1.r4">n</mi><mo>=</mo><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>,</p>
<table id="E8" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">8.19.8</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E8.png" altimg-height="80px" altimg-valign="-42px" altimg-width="482px" alttext="E_{n}\left(z\right)=\frac{(-z)^{n-1}}{(n-1)!}(\psi\left(n\right)-\ln z)-\sum_{%
\begin{subarray}{c}k=0\\
k\neq n-1\end{subarray}}^{\infty}\frac{(-z)^{k}}{k!(1-n+k)}," display="block"><mrow><mrow><mrow><msub><mi href="./8.19#E1">E</mi><mi href="./8.1#p1.t1.r4">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mfrac><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mi href="./8.1#p1.t1.r2">z</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mi href="./8.1#p1.t1.r4">n</mi><mo>-</mo><mn>1</mn></mrow></msup><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./8.1#p1.t1.r4">n</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mfrac><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mi href="./8.1#p1.t1.r4">n</mi><mo>)</mo></mrow></mrow><mo>-</mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi href="./8.1#p1.t1.r2">z</mi></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>-</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mtable class="ltx_align_c" rowspacing="0.0pt"><mtr><mtd><mrow><mi href="./8.1#p1.t1.r4">k</mi><mo>=</mo><mn>0</mn></mrow></mtd></mtr><mtr><mtd><mrow><mi href="./8.1#p1.t1.r4">k</mi><mo>≠</mo><mrow><mi href="./8.1#p1.t1.r4">n</mi><mo>-</mo><mn>1</mn></mrow></mrow></mtd></mtr></mtable><mi mathvariant="normal">∞</mi></munderover><mfrac><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mi href="./8.1#p1.t1.r2">z</mi></mrow><mo stretchy="false">)</mo></mrow><mi href="./8.1#p1.t1.r4">k</mi></msup><mrow><mrow><mi href="./8.1#p1.t1.r4">k</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>1</mn><mo>-</mo><mi href="./8.1#p1.t1.r4">n</mi></mrow><mo>+</mo><mi href="./8.1#p1.t1.r4">k</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mfrac></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E8.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E2" title="(5.2.2) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="\psi\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: psi (or digamma) function</a>,
<a href="./8.19#E1" title="(8.19.1) ‣ §8.19(i) Definition and Integral Representations ‣ §8.19 Generalized Exponential Integral ‣ Related Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="24px" altimg-valign="-8px" altimg-width="57px" alttext="E_{\NVar{p}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./8.19#E1">E</mi><mi class="ltx_nvar" href="./8.1#p1.t1.r3">p</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: generalized exponential integral</a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m8.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m61.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m45.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a>,
<a href="./8.1#p1.t1.r2" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./8.1#p1.t1.r2">z</mi></math>: complex variable</a>,
<a href="./8.1#p1.t1.r4" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./8.1#p1.t1.r4">k</mi></math>: nonnegative integer</a> and
<a href="./8.1#p1.t1.r4" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m65.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./8.1#p1.t1.r4">n</mi></math>: nonnegative integer</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">5.1.12</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and</p>
<table id="E9" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">8.19.9</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E9.png" altimg-height="67px" altimg-valign="-28px" altimg-width="778px" alttext="E_{n}\left(z\right)=\frac{(-1)^{n}z^{n-1}}{(n-1)!}\ln z+\frac{e^{-z}}{(n-1)!}%
\sum_{k=1}^{n-1}(-z)^{k-1}\Gamma\left(n-k\right)+\frac{e^{-z}(-z)^{n-1}}{(n-1)%
!}\sum_{k=0}^{\infty}\frac{z^{k}}{k!}\psi\left(k+1\right)," display="block"><mrow><mrow><mrow><msub><mi href="./8.19#E1">E</mi><mi href="./8.1#p1.t1.r4">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mfrac><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./8.1#p1.t1.r4">n</mi></msup><mo></mo><msup><mi href="./8.1#p1.t1.r2">z</mi><mrow><mi href="./8.1#p1.t1.r4">n</mi><mo>-</mo><mn>1</mn></mrow></msup></mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./8.1#p1.t1.r4">n</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mfrac><mo></mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi href="./8.1#p1.t1.r2">z</mi></mrow></mrow><mo>+</mo><mrow><mfrac><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mi href="./8.1#p1.t1.r2">z</mi></mrow></msup><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./8.1#p1.t1.r4">n</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mfrac><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./8.1#p1.t1.r4">k</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi href="./8.1#p1.t1.r4">n</mi><mo>-</mo><mn>1</mn></mrow></munderover><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mi href="./8.1#p1.t1.r2">z</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mi href="./8.1#p1.t1.r4">k</mi><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./8.1#p1.t1.r4">n</mi><mo>-</mo><mi href="./8.1#p1.t1.r4">k</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>+</mo><mrow><mfrac><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mi href="./8.1#p1.t1.r2">z</mi></mrow></msup><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mi href="./8.1#p1.t1.r2">z</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mi href="./8.1#p1.t1.r4">n</mi><mo>-</mo><mn>1</mn></mrow></msup></mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./8.1#p1.t1.r4">n</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mfrac><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./8.1#p1.t1.r4">k</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><mfrac><msup><mi href="./8.1#p1.t1.r2">z</mi><mi href="./8.1#p1.t1.r4">k</mi></msup><mrow><mi href="./8.1#p1.t1.r4">k</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mfrac><mo></mo><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./8.1#p1.t1.r4">k</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E9.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m33.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./5.2#E2" title="(5.2.2) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="\psi\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: psi (or digamma) function</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./8.19#E1" title="(8.19.1) ‣ §8.19(i) Definition and Integral Representations ‣ §8.19 Generalized Exponential Integral ‣ Related Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="24px" altimg-valign="-8px" altimg-width="57px" alttext="E_{\NVar{p}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./8.19#E1">E</mi><mi class="ltx_nvar" href="./8.1#p1.t1.r3">p</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: generalized exponential integral</a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m8.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m61.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m45.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a>,
<a href="./8.1#p1.t1.r2" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./8.1#p1.t1.r2">z</mi></math>: complex variable</a>,
<a href="./8.1#p1.t1.r4" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./8.1#p1.t1.r4">k</mi></math>: nonnegative integer</a> and
<a href="./8.1#p1.t1.r4" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m65.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./8.1#p1.t1.r4">n</mi></math>: nonnegative integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">with <math class="ltx_Math" altimg="m81.png" altimg-height="23px" altimg-valign="-7px" altimg-width="93px" alttext="|\operatorname{ph}z|\leq\pi" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mi href="./8.1#p1.t1.r2">z</mi></mrow><mo stretchy="false">|</mo></mrow><mo>≤</mo><mi href="./3.12#E1">π</mi></mrow></math> in both equations. For <math class="ltx_Math" altimg="m53.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="\psi\left(x\right)" display="inline"><mrow><mi href="./5.2#E2">ψ</mi><mo></mo><mrow><mo>(</mo><mi href="./8.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow></math> see
§.</p>
</div>
<div id="SS4.p2" class="ltx_para">
<p class="ltx_p">When <math class="ltx_Math" altimg="m72.png" altimg-height="21px" altimg-valign="-6px" altimg-width="53px" alttext="p\in\mathbb{C}" display="inline"><mrow><mi href="./8.1#p1.t1.r3">p</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mi href="./front/introduction#Sx4.p1.t1.r1">ℂ</mi></mrow></math></p>
<table id="E10" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">8.19.10</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E10.png" altimg-height="64px" altimg-valign="-28px" altimg-width="380px" alttext="E_{p}\left(z\right)=z^{p-1}\Gamma\left(1-p\right)-\sum_{k=0}^{\infty}\frac{(-z%
)^{k}}{k!(1-p+k)}," display="block"><mrow><mrow><mrow><msub><mi href="./8.19#E1">E</mi><mi href="./8.1#p1.t1.r3">p</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mrow><msup><mi href="./8.1#p1.t1.r2">z</mi><mrow><mi href="./8.1#p1.t1.r3">p</mi><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mi href="./8.1#p1.t1.r3">p</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>-</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./8.1#p1.t1.r4">k</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mfrac><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mi href="./8.1#p1.t1.r2">z</mi></mrow><mo stretchy="false">)</mo></mrow><mi href="./8.1#p1.t1.r4">k</mi></msup><mrow><mrow><mi href="./8.1#p1.t1.r4">k</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>1</mn><mo>-</mo><mi href="./8.1#p1.t1.r3">p</mi></mrow><mo>+</mo><mi href="./8.1#p1.t1.r4">k</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mfrac></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E10.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m33.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./8.19#E1" title="(8.19.1) ‣ §8.19(i) Definition and Integral Representations ‣ §8.19 Generalized Exponential Integral ‣ Related Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="24px" altimg-valign="-8px" altimg-width="57px" alttext="E_{\NVar{p}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./8.19#E1">E</mi><mi class="ltx_nvar" href="./8.1#p1.t1.r3">p</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: generalized exponential integral</a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m8.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m61.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./8.1#p1.t1.r2" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./8.1#p1.t1.r2">z</mi></math>: complex variable</a>,
<a href="./8.1#p1.t1.r3" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m71.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="p" display="inline"><mi href="./8.1#p1.t1.r3">p</mi></math>: parameter</a> and
<a href="./8.1#p1.t1.r4" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./8.1#p1.t1.r4">k</mi></math>: nonnegative integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E11" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">8.19.11</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E11.png" altimg-height="66px" altimg-valign="-28px" altimg-width="451px" alttext="E_{p}\left(z\right)=\Gamma\left(1-p\right)\left(z^{p-1}-e^{-z}\sum_{k=0}^{%
\infty}\frac{z^{k}}{\Gamma\left(2-p+k\right)}\right)," display="block"><mrow><mrow><mrow><msub><mi href="./8.19#E1">E</mi><mi href="./8.1#p1.t1.r3">p</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mi href="./8.1#p1.t1.r3">p</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mo>(</mo><mrow><msup><mi href="./8.1#p1.t1.r2">z</mi><mrow><mi href="./8.1#p1.t1.r3">p</mi><mo>-</mo><mn>1</mn></mrow></msup><mo>-</mo><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mi href="./8.1#p1.t1.r2">z</mi></mrow></msup><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./8.1#p1.t1.r4">k</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mfrac><msup><mi href="./8.1#p1.t1.r2">z</mi><mi href="./8.1#p1.t1.r4">k</mi></msup><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mn>2</mn><mo>-</mo><mi href="./8.1#p1.t1.r3">p</mi></mrow><mo>+</mo><mi href="./8.1#p1.t1.r4">k</mi></mrow><mo>)</mo></mrow></mrow></mfrac></mrow></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E11.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m33.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./8.19#E1" title="(8.19.1) ‣ §8.19(i) Definition and Integral Representations ‣ §8.19 Generalized Exponential Integral ‣ Related Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="24px" altimg-valign="-8px" altimg-width="57px" alttext="E_{\NVar{p}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./8.19#E1">E</mi><mi class="ltx_nvar" href="./8.1#p1.t1.r3">p</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: generalized exponential integral</a>,
<a href="./8.1#p1.t1.r2" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./8.1#p1.t1.r2">z</mi></math>: complex variable</a>,
<a href="./8.1#p1.t1.r3" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m71.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="p" display="inline"><mi href="./8.1#p1.t1.r3">p</mi></math>: parameter</a> and
<a href="./8.1#p1.t1.r4" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./8.1#p1.t1.r4">k</mi></math>: nonnegative integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">again with <math class="ltx_Math" altimg="m81.png" altimg-height="23px" altimg-valign="-7px" altimg-width="93px" alttext="|\operatorname{ph}z|\leq\pi" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mi href="./8.1#p1.t1.r2">z</mi></mrow><mo stretchy="false">|</mo></mrow><mo>≤</mo><mi href="./3.12#E1">π</mi></mrow></math> in both equations. The right-hand sides are
replaced by their limiting forms when <math class="ltx_Math" altimg="m68.png" altimg-height="20px" altimg-valign="-6px" altimg-width="121px" alttext="p=1,2,3,\dots" display="inline"><mrow><mi href="./8.1#p1.t1.r3">p</mi><mo>=</mo><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>.</p>
</div>
</section>
<section id="v" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§8.19(v) </span>Recurrence Relation and Derivatives</h2>
<div id="SS5.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="SS5.p1" class="ltx_para">
<table id="E12" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">8.19.12</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E12.png" altimg-height="28px" altimg-valign="-8px" altimg-width="240px" alttext="pE_{p+1}\left(z\right)+zE_{p}\left(z\right)=e^{-z}." display="block"><mrow><mrow><mrow><mrow><mi href="./8.1#p1.t1.r3">p</mi><mo></mo><mrow><msub><mi href="./8.19#E1">E</mi><mrow><mi href="./8.1#p1.t1.r3">p</mi><mo>+</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mi href="./8.1#p1.t1.r2">z</mi><mo></mo><mrow><msub><mi href="./8.19#E1">E</mi><mi href="./8.1#p1.t1.r3">p</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>=</mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mi href="./8.1#p1.t1.r2">z</mi></mrow></msup></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E12.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./8.19#E1" title="(8.19.1) ‣ §8.19(i) Definition and Integral Representations ‣ §8.19 Generalized Exponential Integral ‣ Related Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="24px" altimg-valign="-8px" altimg-width="57px" alttext="E_{\NVar{p}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./8.19#E1">E</mi><mi class="ltx_nvar" href="./8.1#p1.t1.r3">p</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: generalized exponential integral</a>,
<a href="./8.1#p1.t1.r2" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./8.1#p1.t1.r2">z</mi></math>: complex variable</a> and
<a href="./8.1#p1.t1.r3" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m71.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="p" display="inline"><mi href="./8.1#p1.t1.r3">p</mi></math>: parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="EGx2" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E13">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">8.19.13</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m7.png" altimg-height="47px" altimg-valign="-16px" altimg-width="85px" alttext="\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}E_{p}\left(z\right)" display="inline"><mrow><mstyle displaystyle="true"><mfrac><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi href="./8.1#p1.t1.r2">z</mi></mrow></mfrac></mstyle><mo></mo><mrow><msub><mi href="./8.19#E1">E</mi><mi href="./8.1#p1.t1.r3">p</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m1.png" altimg-height="26px" altimg-valign="-8px" altimg-width="125px" alttext="\displaystyle=-E_{p-1}\left(z\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo>-</mo><mrow><msub><mi href="./8.19#E1">E</mi><mrow><mi href="./8.1#p1.t1.r3">p</mi><mo>-</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E13.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#E4" title="(1.4.4) ‣ §1.4(iii) Derivatives ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m41.png" altimg-height="29px" altimg-valign="-9px" altimg-width="27px" alttext="\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: derivative of <math class="ltx_Math" altimg="m56.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m75.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./8.19#E1" title="(8.19.1) ‣ §8.19(i) Definition and Integral Representations ‣ §8.19 Generalized Exponential Integral ‣ Related Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="24px" altimg-valign="-8px" altimg-width="57px" alttext="E_{\NVar{p}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./8.19#E1">E</mi><mi class="ltx_nvar" href="./8.1#p1.t1.r3">p</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: generalized exponential integral</a>,
<a href="./8.1#p1.t1.r2" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./8.1#p1.t1.r2">z</mi></math>: complex variable</a> and
<a href="./8.1#p1.t1.r3" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m71.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="p" display="inline"><mi href="./8.1#p1.t1.r3">p</mi></math>: parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">5.1.36</span> (Extended to general values of <math class="ltx_Math" altimg="m71.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="p" display="inline"><mi href="./8.1#p1.t1.r3">p</mi></math>.)</span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E14">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">8.19.14</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m6.png" altimg-height="47px" altimg-valign="-16px" altimg-width="119px" alttext="\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}(e^{z}E_{p}\left(z\right))" display="inline"><mrow><mstyle displaystyle="true"><mfrac><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi href="./8.1#p1.t1.r2">z</mi></mrow></mfrac></mstyle><mo></mo><mrow><mo stretchy="false">(</mo><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mi href="./8.1#p1.t1.r2">z</mi></msup><mo></mo><mrow><msub><mi href="./8.19#E1">E</mi><mi href="./8.1#p1.t1.r3">p</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m4.png" altimg-height="53px" altimg-valign="-21px" altimg-width="259px" alttext="\displaystyle=e^{z}E_{p}\left(z\right)\left(1+\frac{p-1}{z}\right)-\frac{1}{z}." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mi href="./8.1#p1.t1.r2">z</mi></msup><mo></mo><mrow><msub><mi href="./8.19#E1">E</mi><mi href="./8.1#p1.t1.r3">p</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mstyle displaystyle="true"><mfrac><mrow><mi href="./8.1#p1.t1.r3">p</mi><mo>-</mo><mn>1</mn></mrow><mi href="./8.1#p1.t1.r2">z</mi></mfrac></mstyle></mrow><mo>)</mo></mrow></mrow><mo>-</mo><mstyle displaystyle="true"><mfrac><mn>1</mn><mi href="./8.1#p1.t1.r2">z</mi></mfrac></mstyle></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E14.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#E4" title="(1.4.4) ‣ §1.4(iii) Derivatives ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m41.png" altimg-height="29px" altimg-valign="-9px" altimg-width="27px" alttext="\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: derivative of <math class="ltx_Math" altimg="m56.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m75.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./8.19#E1" title="(8.19.1) ‣ §8.19(i) Definition and Integral Representations ‣ §8.19 Generalized Exponential Integral ‣ Related Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="24px" altimg-valign="-8px" altimg-width="57px" alttext="E_{\NVar{p}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./8.19#E1">E</mi><mi class="ltx_nvar" href="./8.1#p1.t1.r3">p</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: generalized exponential integral</a>,
<a href="./8.1#p1.t1.r2" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./8.1#p1.t1.r2">z</mi></math>: complex variable</a> and
<a href="./8.1#p1.t1.r3" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m71.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="p" display="inline"><mi href="./8.1#p1.t1.r3">p</mi></math>: parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
</div>
<section id="Px2" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">
<math class="ltx_Math" altimg="m71.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="p" display="inline"><mi href="./8.1#p1.t1.r3">p</mi></math>-Derivatives</h3>
<div id="Px2.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px2.p1" class="ltx_para">
<p class="ltx_p">For <math class="ltx_Math" altimg="m57.png" altimg-height="20px" altimg-valign="-6px" altimg-width="120px" alttext="j=1,2,3,\dots" display="inline"><mrow><mi href="./8.19#Px2.p1">j</mi><mo>=</mo><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>,
</p>
<table id="E15" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">8.19.15</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E15.png" altimg-height="55px" altimg-valign="-22px" altimg-width="340px" alttext="\frac{{\partial}^{j}E_{p}\left(z\right)}{{\partial p}^{j}}=(-1)^{j}\int_{1}^{%
\infty}(\ln t)^{j}t^{-p}e^{-zt}\mathrm{d}t," display="block"><mrow><mrow><mfrac><mrow><mpadded width="-1.7pt"><msup><mo href="./1.5#E3">∂</mo><mi href="./8.19#Px2.p1">j</mi></msup></mpadded><mo href="./1.5#E3"></mo><mrow><msub><mi href="./8.19#E1">E</mi><mi href="./8.1#p1.t1.r3">p</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow><msup><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./8.1#p1.t1.r3">p</mi></mrow><mi href="./1.5#E3">j</mi></msup></mfrac><mo>=</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./8.19#Px2.p1">j</mi></msup><mo></mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>1</mn><mi mathvariant="normal">∞</mi></msubsup><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi>t</mi></mrow><mo stretchy="false">)</mo></mrow><mi href="./8.19#Px2.p1">j</mi></msup><mo></mo><msup><mi>t</mi><mrow><mo>-</mo><mi href="./8.1#p1.t1.r3">p</mi></mrow></msup><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mi href="./8.1#p1.t1.r2">z</mi><mo></mo><mi>t</mi></mrow></mrow></msup><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m38.png" altimg-height="18px" altimg-valign="-3px" altimg-width="65px" alttext="\Re z>0" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./8.1#p1.t1.r2">z</mi></mrow><mo>></mo><mn>0</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E15.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m75.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./8.19#E1" title="(8.19.1) ‣ §8.19(i) Definition and Integral Representations ‣ §8.19 Generalized Exponential Integral ‣ Related Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="24px" altimg-valign="-8px" altimg-width="57px" alttext="E_{\NVar{p}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./8.19#E1">E</mi><mi class="ltx_nvar" href="./8.1#p1.t1.r3">p</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: generalized exponential integral</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m45.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m42.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m56.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m75.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m75.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m40.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./8.1#p1.t1.r2" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./8.1#p1.t1.r2">z</mi></math>: complex variable</a>,
<a href="./8.1#p1.t1.r3" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m71.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="p" display="inline"><mi href="./8.1#p1.t1.r3">p</mi></math>: parameter</a> and
<a href="./8.19#Px2.p1" title="p -Derivatives ‣ §8.19(v) Recurrence Relation and Derivatives ‣ §8.19 Generalized Exponential Integral ‣ Related Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="20px" altimg-valign="-6px" altimg-width="14px" alttext="j" display="inline"><mi href="./8.19#Px2.p1">j</mi></math>: numbers</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">For properties and numerical tables see <cite class="ltx_cite ltx_citemacro_citet">Milgram ()</cite>, and also
(when <math class="ltx_Math" altimg="m69.png" altimg-height="20px" altimg-valign="-6px" altimg-width="51px" alttext="p=1" display="inline"><mrow><mi href="./8.1#p1.t1.r3">p</mi><mo>=</mo><mn>1</mn></mrow></math>) <cite class="ltx_cite ltx_citemacro_citet">MacLeod (</dd>
</dl>
</div>
</div>
<div id="SS6.p1" class="ltx_para">
<table id="E16" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">8.19.16</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E16.png" altimg-height="29px" altimg-valign="-8px" altimg-width="248px" alttext="E_{p}\left(z\right)=z^{p-1}e^{-z}U\left(p,p,z\right)." display="block"><mrow><mrow><mrow><msub><mi href="./8.19#E1">E</mi><mi href="./8.1#p1.t1.r3">p</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msup><mi href="./8.1#p1.t1.r2">z</mi><mrow><mi href="./8.1#p1.t1.r3">p</mi><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mi href="./8.1#p1.t1.r2">z</mi></mrow></msup><mo></mo><mrow><mi href="./13.2#E6">U</mi><mo></mo><mrow><mo>(</mo><mi href="./8.1#p1.t1.r3">p</mi><mo>,</mo><mi href="./8.1#p1.t1.r3">p</mi><mo>,</mo><mi href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E16.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./13.2#E6" title="(13.2.6) ‣ Standard Solutions ‣ §13.2(i) Differential Equation ‣ §13.2 Definitions and Basic Properties ‣ Kummer Functions ‣ Chapter 13 Confluent Hypergeometric Functions" class="ltx_ref"><math class="ltx_Math" altimg="m30.png" altimg-height="23px" altimg-valign="-7px" altimg-width="86px" alttext="U\left(\NVar{a},\NVar{b},\NVar{z}\right)" display="inline"><mrow><mi href="./13.2#E6">U</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">a</mi><mo>,</mo><mi class="ltx_nvar">b</mi><mo>,</mo><mi class="ltx_nvar" href="./13.1#p1.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Kummer confluent hypergeometric function</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./8.19#E1" title="(8.19.1) ‣ §8.19(i) Definition and Integral Representations ‣ §8.19 Generalized Exponential Integral ‣ Related Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="24px" altimg-valign="-8px" altimg-width="57px" alttext="E_{\NVar{p}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./8.19#E1">E</mi><mi class="ltx_nvar" href="./8.1#p1.t1.r3">p</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: generalized exponential integral</a>,
<a href="./8.1#p1.t1.r2" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./8.1#p1.t1.r2">z</mi></math>: complex variable</a> and
<a href="./8.1#p1.t1.r3" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m71.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="p" display="inline"><mi href="./8.1#p1.t1.r3">p</mi></math>: parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">For <math class="ltx_Math" altimg="m31.png" altimg-height="23px" altimg-valign="-7px" altimg-width="86px" alttext="U\left(a,b,z\right)" display="inline"><mrow><mi href="./13.2#E6">U</mi><mo></mo><mrow><mo>(</mo><mi href="./8.1#p1.t1.r3">a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> see §</dd>
</dl>
</div>
</div>
<div id="SS7.p1" class="ltx_para">
<table id="E17" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">8.19.17</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E17.png" altimg-height="65px" altimg-valign="-27px" altimg-width="420px" alttext="E_{p}\left(z\right)=e^{-z}\left(\cfrac{1}{z+\cfrac{p}{1+\cfrac{1}{z+\cfrac{p+1%
}{1+\cfrac{2}{z+\cdots}}}}}\right)," display="block"><mrow><mrow><mrow><msub><mi href="./8.19#E1">E</mi><mi href="./8.1#p1.t1.r3">p</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mi href="./8.1#p1.t1.r2">z</mi></mrow></msup><mo></mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mrow><mi href="./8.1#p1.t1.r2">z</mi><mo>+</mo></mrow></mfrac><mfrac><mi href="./8.1#p1.t1.r3">p</mi><mrow><mn>1</mn><mo>+</mo></mrow></mfrac><mfrac><mn>1</mn><mrow><mi href="./8.1#p1.t1.r2">z</mi><mo>+</mo></mrow></mfrac><mfrac><mrow><mi href="./8.1#p1.t1.r3">p</mi><mo>+</mo><mn>1</mn></mrow><mrow><mn>1</mn><mo>+</mo></mrow></mfrac><mfrac><mn>2</mn><mrow><mi href="./8.1#p1.t1.r2">z</mi><mo>+</mo></mrow></mfrac><mi mathvariant="normal">⋯</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m79.png" altimg-height="23px" altimg-valign="-7px" altimg-width="93px" alttext="|\operatorname{ph}z|<\pi" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mi href="./8.1#p1.t1.r2">z</mi></mrow><mo stretchy="false">|</mo></mrow><mo><</mo><mi href="./3.12#E1">π</mi></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E17.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./8.19#E1" title="(8.19.1) ‣ §8.19(i) Definition and Integral Representations ‣ §8.19 Generalized Exponential Integral ‣ Related Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="24px" altimg-valign="-8px" altimg-width="57px" alttext="E_{\NVar{p}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./8.19#E1">E</mi><mi class="ltx_nvar" href="./8.1#p1.t1.r3">p</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: generalized exponential integral</a>,
<a href="./1.9#E7" title="(1.9.7) ‣ Modulus and Phase ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="21px" altimg-valign="-6px" altimg-width="26px" alttext="\operatorname{ph}" display="inline"><mi href="./1.9#E7">ph</mi></math>: phase</a>,
<a href="./8.1#p1.t1.r2" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./8.1#p1.t1.r2">z</mi></math>: complex variable</a> and
<a href="./8.1#p1.t1.r3" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m71.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="p" display="inline"><mi href="./8.1#p1.t1.r3">p</mi></math>: parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">5.1.22</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS7.p2" class="ltx_para">
<p class="ltx_p">See also <cite class="ltx_cite ltx_citemacro_citet">Cuyt<span class="ltx_text ltx_bib_etal"> et al.</span> (</dd>
</dl>
</div>
</div>
<div id="SS8.p1" class="ltx_para">
<p class="ltx_p">The general function <math class="ltx_Math" altimg="m24.png" altimg-height="24px" altimg-valign="-8px" altimg-width="57px" alttext="E_{p}\left(z\right)" display="inline"><mrow><msub><mi href="./8.19#E1">E</mi><mi href="./8.1#p1.t1.r3">p</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> is attained by extending the path in
() across the negative real axis. Unless <math class="ltx_Math" altimg="m71.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="p" display="inline"><mi href="./8.1#p1.t1.r3">p</mi></math> is a nonpositive
integer, <math class="ltx_Math" altimg="m24.png" altimg-height="24px" altimg-valign="-8px" altimg-width="57px" alttext="E_{p}\left(z\right)" display="inline"><mrow><msub><mi href="./8.19#E1">E</mi><mi href="./8.1#p1.t1.r3">p</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> has a branch point at <math class="ltx_Math" altimg="m76.png" altimg-height="17px" altimg-valign="-2px" altimg-width="51px" alttext="z=0" display="inline"><mrow><mi href="./8.1#p1.t1.r2">z</mi><mo>=</mo><mn>0</mn></mrow></math>. For <math class="ltx_Math" altimg="m78.png" altimg-height="21px" altimg-valign="-6px" altimg-width="51px" alttext="z\neq 0" display="inline"><mrow><mi href="./8.1#p1.t1.r2">z</mi><mo>≠</mo><mn>0</mn></mrow></math> each
branch of <math class="ltx_Math" altimg="m24.png" altimg-height="24px" altimg-valign="-8px" altimg-width="57px" alttext="E_{p}\left(z\right)" display="inline"><mrow><msub><mi href="./8.19#E1">E</mi><mi href="./8.1#p1.t1.r3">p</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> is an entire function of <math class="ltx_Math" altimg="m71.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="p" display="inline"><mi href="./8.1#p1.t1.r3">p</mi></math>.</p>
<table id="E18" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">8.19.18</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E18.png" altimg-height="54px" altimg-valign="-21px" altimg-width="431px" alttext="E_{p}\left(ze^{2m\pi i}\right)=\frac{2\pi ie^{mp\pi i}}{\Gamma\left(p\right)}%
\frac{\sin\left(mp\pi\right)}{\sin\left(p\pi\right)}z^{p-1}+E_{p}\left(z\right)," display="block"><mrow><mrow><mrow><msub><mi href="./8.19#E1">E</mi><mi href="./8.1#p1.t1.r3">p</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi href="./8.1#p1.t1.r2">z</mi><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mn>2</mn><mo></mo><mi>m</mi><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi></mrow></msup></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mfrac><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mi>m</mi><mo></mo><mi href="./8.1#p1.t1.r3">p</mi><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi></mrow></msup></mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi href="./8.1#p1.t1.r3">p</mi><mo>)</mo></mrow></mrow></mfrac><mo></mo><mfrac><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><mi>m</mi><mo></mo><mi href="./8.1#p1.t1.r3">p</mi><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo>)</mo></mrow></mrow><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./8.1#p1.t1.r3">p</mi><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo>)</mo></mrow></mrow></mfrac><mo></mo><msup><mi href="./8.1#p1.t1.r2">z</mi><mrow><mi href="./8.1#p1.t1.r3">p</mi><mo>-</mo><mn>1</mn></mrow></msup></mrow><mo>+</mo><mrow><msub><mi href="./8.19#E1">E</mi><mi href="./8.1#p1.t1.r3">p</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m60.png" altimg-height="18px" altimg-valign="-3px" altimg-width="59px" alttext="m\in\mathbb{Z}" display="inline"><mrow><mi>m</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mi href="./front/introduction#Sx4.p2.t1.r20">ℤ</mi></mrow></math>, <math class="ltx_Math" altimg="m78.png" altimg-height="21px" altimg-valign="-6px" altimg-width="51px" alttext="z\neq 0" display="inline"><mrow><mi href="./8.1#p1.t1.r2">z</mi><mo>≠</mo><mn>0</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E18.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m33.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./front/introduction#Sx4.p1.t1.r9" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="15px" altimg-valign="-3px" altimg-width="18px" alttext="\in" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo></math>: element of</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./8.19#E1" title="(8.19.1) ‣ §8.19(i) Definition and Integral Representations ‣ §8.19 Generalized Exponential Integral ‣ Related Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="24px" altimg-valign="-8px" altimg-width="57px" alttext="E_{\NVar{p}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./8.19#E1">E</mi><mi class="ltx_nvar" href="./8.1#p1.t1.r3">p</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: generalized exponential integral</a>,
<a href="./front/introduction#Sx4.p2.t1.r20" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m46.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="\mathbb{Z}" display="inline"><mi href="./front/introduction#Sx4.p2.t1.r20">ℤ</mi></math>: set of all integers</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./8.1#p1.t1.r2" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./8.1#p1.t1.r2">z</mi></math>: complex variable</a> and
<a href="./8.1#p1.t1.r3" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m71.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="p" display="inline"><mi href="./8.1#p1.t1.r3">p</mi></math>: parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="ix" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§8.19(ix) </span>Inequalities</h2>
<div id="SS9.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="SS9.p1" class="ltx_para">
<p class="ltx_p">For <math class="ltx_Math" altimg="m62.png" altimg-height="20px" altimg-valign="-6px" altimg-width="123px" alttext="n=1,2,3,\dots" display="inline"><mrow><mi href="./8.1#p1.t1.r4">n</mi><mo>=</mo><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math> and <math class="ltx_Math" altimg="m74.png" altimg-height="17px" altimg-valign="-3px" altimg-width="52px" alttext="x>0" display="inline"><mrow><mi href="./8.1#p1.t1.r1">x</mi><mo>></mo><mn>0</mn></mrow></math>,</p>
<table id="E19" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">8.19.19</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E19.png" altimg-height="46px" altimg-valign="-16px" altimg-width="307px" alttext="\frac{n-1}{n}E_{n}\left(x\right)<E_{n+1}\left(x\right)<E_{n}\left(x\right)," display="block"><mrow><mrow><mrow><mfrac><mrow><mi href="./8.1#p1.t1.r4">n</mi><mo>-</mo><mn>1</mn></mrow><mi href="./8.1#p1.t1.r4">n</mi></mfrac><mo></mo><mrow><msub><mi href="./8.19#E1">E</mi><mi href="./8.1#p1.t1.r4">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./8.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow></mrow><mo><</mo><mrow><msub><mi href="./8.19#E1">E</mi><mrow><mi href="./8.1#p1.t1.r4">n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./8.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow><mo><</mo><mrow><msub><mi href="./8.19#E1">E</mi><mi href="./8.1#p1.t1.r4">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./8.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E19.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./8.19#E1" title="(8.19.1) ‣ §8.19(i) Definition and Integral Representations ‣ §8.19 Generalized Exponential Integral ‣ Related Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="24px" altimg-valign="-8px" altimg-width="57px" alttext="E_{\NVar{p}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./8.19#E1">E</mi><mi class="ltx_nvar" href="./8.1#p1.t1.r3">p</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: generalized exponential integral</a>,
<a href="./8.1#p1.t1.r1" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m75.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./8.1#p1.t1.r1">x</mi></math>: real variable</a> and
<a href="./8.1#p1.t1.r4" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m65.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./8.1#p1.t1.r4">n</mi></math>: nonnegative integer</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">5.1.17</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E20" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">8.19.20</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E20.png" altimg-height="29px" altimg-valign="-7px" altimg-width="275px" alttext="\left(E_{n}\left(x\right)\right)^{2}<E_{n-1}\left(x\right)E_{n+1}\left(x\right)," display="block"><mrow><mrow><msup><mrow><mo>(</mo><mrow><msub><mi href="./8.19#E1">E</mi><mi href="./8.1#p1.t1.r4">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./8.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow><mo>)</mo></mrow><mn>2</mn></msup><mo><</mo><mrow><mrow><msub><mi href="./8.19#E1">E</mi><mrow><mi href="./8.1#p1.t1.r4">n</mi><mo>-</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./8.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./8.19#E1">E</mi><mrow><mi href="./8.1#p1.t1.r4">n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./8.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E20.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./8.19#E1" title="(8.19.1) ‣ §8.19(i) Definition and Integral Representations ‣ §8.19 Generalized Exponential Integral ‣ Related Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="24px" altimg-valign="-8px" altimg-width="57px" alttext="E_{\NVar{p}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./8.19#E1">E</mi><mi class="ltx_nvar" href="./8.1#p1.t1.r3">p</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: generalized exponential integral</a>,
<a href="./8.1#p1.t1.r1" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m75.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./8.1#p1.t1.r1">x</mi></math>: real variable</a> and
<a href="./8.1#p1.t1.r4" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m65.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./8.1#p1.t1.r4">n</mi></math>: nonnegative integer</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">5.1.18</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E21" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">8.19.21</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E21.png" altimg-height="47px" altimg-valign="-17px" altimg-width="279px" alttext="\frac{1}{x+n}<e^{x}E_{n}\left(x\right)\leq\frac{1}{x+n-1}," display="block"><mrow><mrow><mfrac><mn>1</mn><mrow><mi href="./8.1#p1.t1.r1">x</mi><mo>+</mo><mi href="./8.1#p1.t1.r4">n</mi></mrow></mfrac><mo><</mo><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mi href="./8.1#p1.t1.r1">x</mi></msup><mo></mo><mrow><msub><mi href="./8.19#E1">E</mi><mi href="./8.1#p1.t1.r4">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./8.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow></mrow><mo>≤</mo><mfrac><mn>1</mn><mrow><mrow><mi href="./8.1#p1.t1.r1">x</mi><mo>+</mo><mi href="./8.1#p1.t1.r4">n</mi></mrow><mo>-</mo><mn>1</mn></mrow></mfrac></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E21.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./8.19#E1" title="(8.19.1) ‣ §8.19(i) Definition and Integral Representations ‣ §8.19 Generalized Exponential Integral ‣ Related Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="24px" altimg-valign="-8px" altimg-width="57px" alttext="E_{\NVar{p}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./8.19#E1">E</mi><mi class="ltx_nvar" href="./8.1#p1.t1.r3">p</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: generalized exponential integral</a>,
<a href="./8.1#p1.t1.r1" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m75.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./8.1#p1.t1.r1">x</mi></math>: real variable</a> and
<a href="./8.1#p1.t1.r4" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m65.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./8.1#p1.t1.r4">n</mi></math>: nonnegative integer</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">5.1.19</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E22" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">8.19.22</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E22.png" altimg-height="53px" altimg-valign="-21px" altimg-width="156px" alttext="\frac{\mathrm{d}}{\mathrm{d}x}\frac{E_{n}\left(x\right)}{E_{n-1}\left(x\right)%
}>0." display="block"><mrow><mrow><mrow><mfrac><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi href="./8.1#p1.t1.r1">x</mi></mrow></mfrac><mo></mo><mfrac><mrow><msub><mi href="./8.19#E1">E</mi><mi href="./8.1#p1.t1.r4">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./8.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow><mrow><msub><mi href="./8.19#E1">E</mi><mrow><mi href="./8.1#p1.t1.r4">n</mi><mo>-</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./8.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow></mfrac></mrow><mo>></mo><mn>0</mn></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E22.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#E4" title="(1.4.4) ‣ §1.4(iii) Derivatives ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m41.png" altimg-height="29px" altimg-valign="-9px" altimg-width="27px" alttext="\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: derivative of <math class="ltx_Math" altimg="m56.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m75.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./8.19#E1" title="(8.19.1) ‣ §8.19(i) Definition and Integral Representations ‣ §8.19 Generalized Exponential Integral ‣ Related Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="24px" altimg-valign="-8px" altimg-width="57px" alttext="E_{\NVar{p}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./8.19#E1">E</mi><mi class="ltx_nvar" href="./8.1#p1.t1.r3">p</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: generalized exponential integral</a>,
<a href="./8.1#p1.t1.r1" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m75.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./8.1#p1.t1.r1">x</mi></math>: real variable</a> and
<a href="./8.1#p1.t1.r4" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m65.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./8.1#p1.t1.r4">n</mi></math>: nonnegative integer</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">5.1.21</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="x" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§8.19(x) </span>Integrals</h2>
<div id="SS10.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="SS10.p1" class="ltx_para">
<table id="E23" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">8.19.23</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E23.png" altimg-height="53px" altimg-valign="-20px" altimg-width="226px" alttext="\int_{z}^{\infty}E_{p-1}\left(t\right)\mathrm{d}t=E_{p}\left(z\right)," display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi href="./8.1#p1.t1.r2">z</mi><mi mathvariant="normal">∞</mi></msubsup><mrow><mrow><msub><mi href="./8.19#E1">E</mi><mrow><mi href="./8.1#p1.t1.r3">p</mi><mo>-</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow><mo>=</mo><mrow><msub><mi href="./8.19#E1">E</mi><mi href="./8.1#p1.t1.r3">p</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m79.png" altimg-height="23px" altimg-valign="-7px" altimg-width="93px" alttext="|\operatorname{ph}z|<\pi" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mi href="./8.1#p1.t1.r2">z</mi></mrow><mo stretchy="false">|</mo></mrow><mo><</mo><mi href="./3.12#E1">π</mi></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E23.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m75.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./8.19#E1" title="(8.19.1) ‣ §8.19(i) Definition and Integral Representations ‣ §8.19 Generalized Exponential Integral ‣ Related Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="24px" altimg-valign="-8px" altimg-width="57px" alttext="E_{\NVar{p}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./8.19#E1">E</mi><mi class="ltx_nvar" href="./8.1#p1.t1.r3">p</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: generalized exponential integral</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.9#E7" title="(1.9.7) ‣ Modulus and Phase ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="21px" altimg-valign="-6px" altimg-width="26px" alttext="\operatorname{ph}" display="inline"><mi href="./1.9#E7">ph</mi></math>: phase</a>,
<a href="./8.1#p1.t1.r2" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./8.1#p1.t1.r2">z</mi></math>: complex variable</a> and
<a href="./8.1#p1.t1.r3" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m71.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="p" display="inline"><mi href="./8.1#p1.t1.r3">p</mi></math>: parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E24" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">8.19.24</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E24.png" altimg-height="67px" altimg-valign="-28px" altimg-width="518px" alttext="\int_{0}^{\infty}e^{-at}E_{n}\left(t\right)\mathrm{d}t=\frac{(-1)^{n-1}}{a^{n}%
}\left(\ln\left(1+a\right)+\sum_{k=1}^{n-1}\frac{(-1)^{k}a^{k}}{k}\right)," display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mi href="./8.1#p1.t1.r3">a</mi><mo></mo><mi>t</mi></mrow></mrow></msup><mo></mo><mrow><msub><mi href="./8.19#E1">E</mi><mi href="./8.1#p1.t1.r4">n</mi></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow><mo>=</mo><mrow><mfrac><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mrow><mi href="./8.1#p1.t1.r4">n</mi><mo>-</mo><mn>1</mn></mrow></msup><msup><mi href="./8.1#p1.t1.r3">a</mi><mi href="./8.1#p1.t1.r4">n</mi></msup></mfrac><mo></mo><mrow><mo>(</mo><mrow><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mi href="./8.1#p1.t1.r3">a</mi></mrow><mo>)</mo></mrow></mrow><mo>+</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./8.1#p1.t1.r4">k</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi href="./8.1#p1.t1.r4">n</mi><mo>-</mo><mn>1</mn></mrow></munderover><mfrac><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./8.1#p1.t1.r4">k</mi></msup><mo></mo><msup><mi href="./8.1#p1.t1.r3">a</mi><mi href="./8.1#p1.t1.r4">k</mi></msup></mrow><mi href="./8.1#p1.t1.r4">k</mi></mfrac></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m63.png" altimg-height="20px" altimg-valign="-6px" altimg-width="104px" alttext="n=1,2,\dots" display="inline"><mrow><mi href="./8.1#p1.t1.r4">n</mi><mo>=</mo><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>, <math class="ltx_Math" altimg="m35.png" altimg-height="19px" altimg-valign="-4px" altimg-width="81px" alttext="\Re a>-1" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./8.1#p1.t1.r3">a</mi></mrow><mo>></mo><mrow><mo>-</mo><mn>1</mn></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E24.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m75.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./8.19#E1" title="(8.19.1) ‣ §8.19(i) Definition and Integral Representations ‣ §8.19 Generalized Exponential Integral ‣ Related Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="24px" altimg-valign="-8px" altimg-width="57px" alttext="E_{\NVar{p}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./8.19#E1">E</mi><mi class="ltx_nvar" href="./8.1#p1.t1.r3">p</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: generalized exponential integral</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m45.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m40.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./8.1#p1.t1.r3" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m55.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./8.1#p1.t1.r3">a</mi></math>: parameter</a>,
<a href="./8.1#p1.t1.r4" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./8.1#p1.t1.r4">k</mi></math>: nonnegative integer</a> and
<a href="./8.1#p1.t1.r4" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m65.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./8.1#p1.t1.r4">n</mi></math>: nonnegative integer</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">5.1.34</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E25" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">8.19.25</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E25.png" altimg-height="54px" altimg-valign="-20px" altimg-width="561px" alttext="\int_{0}^{\infty}e^{-at}t^{b-1}E_{p}\left(t\right)\mathrm{d}t=\frac{\Gamma%
\left(b\right)(1+a)^{-b}}{p+b-1}\*F\left(1,b;p+b;a/(1+a)\right)," display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mi href="./8.1#p1.t1.r3">a</mi><mo></mo><mi>t</mi></mrow></mrow></msup><mo></mo><msup><mi>t</mi><mrow><mi>b</mi><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mrow><msub><mi href="./8.19#E1">E</mi><mi href="./8.1#p1.t1.r3">p</mi></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow><mo>=</mo><mrow><mfrac><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi>b</mi><mo>)</mo></mrow></mrow><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>+</mo><mi href="./8.1#p1.t1.r3">a</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mi>b</mi></mrow></msup></mrow><mrow><mrow><mi href="./8.1#p1.t1.r3">p</mi><mo>+</mo><mi>b</mi></mrow><mo>-</mo><mn>1</mn></mrow></mfrac><mo></mo><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mn>1</mn><mo>,</mo><mi>b</mi><mo>;</mo><mrow><mi href="./8.1#p1.t1.r3">p</mi><mo>+</mo><mi>b</mi></mrow><mo>;</mo><mrow><mi href="./8.1#p1.t1.r3">a</mi><mo>/</mo><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>+</mo><mi href="./8.1#p1.t1.r3">a</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m35.png" altimg-height="19px" altimg-valign="-4px" altimg-width="81px" alttext="\Re a>-1" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./8.1#p1.t1.r3">a</mi></mrow><mo>></mo><mrow><mo>-</mo><mn>1</mn></mrow></mrow></math>, <math class="ltx_Math" altimg="m39.png" altimg-height="23px" altimg-valign="-7px" altimg-width="114px" alttext="\Re(p+b)>1" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./8.1#p1.t1.r3">p</mi><mo>+</mo><mi>b</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>></mo><mn>1</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E25.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m33.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./15.2#E1" title="(15.2.1) ‣ §15.2(i) Gauss Series ‣ §15.2 Definitions and Analytical Properties ‣ Properties ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m25.png" altimg-height="23px" altimg-valign="-7px" altimg-width="103px" alttext="F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m27.png" altimg-height="39px" altimg-valign="-15px" altimg-width="91px" alttext="F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi></mfrac><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: <math class="ltx_Math" altimg="m15.png" altimg-height="23px" altimg-valign="-7px" altimg-width="139px" alttext="={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow></math>
Gauss’ hypergeometric function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m75.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./8.19#E1" title="(8.19.1) ‣ §8.19(i) Definition and Integral Representations ‣ §8.19 Generalized Exponential Integral ‣ Related Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="24px" altimg-valign="-8px" altimg-width="57px" alttext="E_{\NVar{p}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./8.19#E1">E</mi><mi class="ltx_nvar" href="./8.1#p1.t1.r3">p</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: generalized exponential integral</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m40.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./8.1#p1.t1.r3" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m55.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./8.1#p1.t1.r3">a</mi></math>: parameter</a> and
<a href="./8.1#p1.t1.r3" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m71.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="p" display="inline"><mi href="./8.1#p1.t1.r3">p</mi></math>: parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E26" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">8.19.26</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E26.png" altimg-height="52px" altimg-valign="-20px" altimg-width="310px" alttext="\int_{0}^{\infty}E_{p}\left(t\right)E_{q}\left(t\right)\mathrm{d}t=\frac{L(p)+%
L(q)}{p+q-1}," display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><mrow><mrow><msub><mi href="./8.19#E1">E</mi><mi href="./8.1#p1.t1.r3">p</mi></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./8.19#E1">E</mi><mi>q</mi></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow><mo>=</mo><mfrac><mrow><mrow><mi href="./8.19#SS10.p1">L</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./8.1#p1.t1.r3">p</mi><mo stretchy="false">)</mo></mrow></mrow><mo>+</mo><mrow><mi href="./8.19#SS10.p1">L</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>q</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mrow><mrow><mi href="./8.1#p1.t1.r3">p</mi><mo>+</mo><mi>q</mi></mrow><mo>-</mo><mn>1</mn></mrow></mfrac></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m70.png" altimg-height="20px" altimg-valign="-6px" altimg-width="51px" alttext="p>0" display="inline"><mrow><mi href="./8.1#p1.t1.r3">p</mi><mo>></mo><mn>0</mn></mrow></math>, <math class="ltx_Math" altimg="m73.png" altimg-height="20px" altimg-valign="-6px" altimg-width="50px" alttext="q>0" display="inline"><mrow><mi>q</mi><mo>></mo><mn>0</mn></mrow></math>, <math class="ltx_Math" altimg="m66.png" altimg-height="20px" altimg-valign="-6px" altimg-width="85px" alttext="p+q>1" display="inline"><mrow><mrow><mi href="./8.1#p1.t1.r3">p</mi><mo>+</mo><mi>q</mi></mrow><mo>></mo><mn>1</mn></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E26.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m75.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./8.19#E1" title="(8.19.1) ‣ §8.19(i) Definition and Integral Representations ‣ §8.19 Generalized Exponential Integral ‣ Related Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="24px" altimg-valign="-8px" altimg-width="57px" alttext="E_{\NVar{p}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./8.19#E1">E</mi><mi class="ltx_nvar" href="./8.1#p1.t1.r3">p</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: generalized exponential integral</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./8.1#p1.t1.r3" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m71.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="p" display="inline"><mi href="./8.1#p1.t1.r3">p</mi></math>: parameter</a> and
<a href="./8.19#SS10.p1" title="§8.19(x) Integrals ‣ §8.19 Generalized Exponential Integral ‣ Related Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="L(p)" display="inline"><mrow><mi href="./8.19#SS10.p1">L</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./8.1#p1.t1.r3">p</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where
</p>
<table id="E27" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">8.19.27</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E27.png" altimg-height="53px" altimg-valign="-20px" altimg-width="413px" alttext="L(p)=\int_{0}^{\infty}e^{-t}E_{p}\left(t\right)\mathrm{d}t=\frac{1}{2p}F\left(%
1,1;1+p;\tfrac{1}{2}\right)," display="block"><mrow><mrow><mrow><mi href="./8.19#SS10.p1">L</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./8.1#p1.t1.r3">p</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mi>t</mi></mrow></msup><mo></mo><mrow><msub><mi href="./8.19#E1">E</mi><mi href="./8.1#p1.t1.r3">p</mi></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow><mo>=</mo><mrow><mfrac><mn>1</mn><mrow><mn>2</mn><mo></mo><mi href="./8.1#p1.t1.r3">p</mi></mrow></mfrac><mo></mo><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>;</mo><mrow><mn>1</mn><mo>+</mo><mi href="./8.1#p1.t1.r3">p</mi></mrow><mo>;</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m70.png" altimg-height="20px" altimg-valign="-6px" altimg-width="51px" alttext="p>0" display="inline"><mrow><mi href="./8.1#p1.t1.r3">p</mi><mo>></mo><mn>0</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E27.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./15.2#E1" title="(15.2.1) ‣ §15.2(i) Gauss Series ‣ §15.2 Definitions and Analytical Properties ‣ Properties ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m25.png" altimg-height="23px" altimg-valign="-7px" altimg-width="103px" alttext="F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m27.png" altimg-height="39px" altimg-valign="-15px" altimg-width="91px" alttext="F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi></mfrac><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: <math class="ltx_Math" altimg="m15.png" altimg-height="23px" altimg-valign="-7px" altimg-width="139px" alttext="={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow></math>
Gauss’ hypergeometric function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m75.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./8.19#E1" title="(8.19.1) ‣ §8.19(i) Definition and Integral Representations ‣ §8.19 Generalized Exponential Integral ‣ Related Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="24px" altimg-valign="-8px" altimg-width="57px" alttext="E_{\NVar{p}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./8.19#E1">E</mi><mi class="ltx_nvar" href="./8.1#p1.t1.r3">p</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: generalized exponential integral</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./8.1#p1.t1.r3" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m71.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="p" display="inline"><mi href="./8.1#p1.t1.r3">p</mi></math>: parameter</a> and
<a href="./8.19#SS10.p1" title="§8.19(x) Integrals ‣ §8.19 Generalized Exponential Integral ‣ Related Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="L(p)" display="inline"><mrow><mi href="./8.19#SS10.p1">L</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./8.1#p1.t1.r3">p</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">For the hypergeometric function <math class="ltx_Math" altimg="m26.png" altimg-height="23px" altimg-valign="-7px" altimg-width="103px" alttext="F\left(a,b;c;z\right)" display="inline"><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi href="./8.1#p1.t1.r3">a</mi><mo>,</mo><mi>b</mi><mo>;</mo><mi>c</mi><mo>;</mo><mi href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> see
§. When <math class="ltx_Math" altimg="m68.png" altimg-height="20px" altimg-valign="-6px" altimg-width="121px" alttext="p=1,2,3,\dots" display="inline"><mrow><mi href="./8.1#p1.t1.r3">p</mi><mo>=</mo><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>, <math class="ltx_Math" altimg="m29.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="L(p)" display="inline"><mrow><mi href="./8.19#SS10.p1">L</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./8.1#p1.t1.r3">p</mi><mo stretchy="false">)</mo></mrow></mrow></math> can also be evaluated via
().</p>
</div>
<div id="SS10.p2" class="ltx_para">
<p class="ltx_p">For collections of integrals involving <math class="ltx_Math" altimg="m24.png" altimg-height="24px" altimg-valign="-8px" altimg-width="57px" alttext="E_{p}\left(z\right)" display="inline"><mrow><msub><mi href="./8.19#E1">E</mi><mi href="./8.1#p1.t1.r3">p</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>, especially for
integer <math class="ltx_Math" altimg="m71.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="p" display="inline"><mi href="./8.1#p1.t1.r3">p</mi></math>, see <cite class="ltx_cite ltx_citemacro_citet">Apelblat ( is available.<span id="pagesettings"></span></span>
<a href="./8.18" title="§8.18 Asymptotic Expansions of I x ( a , b ) ‣ Related Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref" rel="prev"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">8.18 </span>Asymptotic Expansions of <math class="ltx_Math" altimg="m28.png" altimg-height="23px" altimg-valign="-7px" altimg-width="70px" alttext="I_{x}\left(a,b\right)" display="inline"><mrow><msub><mi href="./8.17#E2">I</mi><mi href="./8.17#SS1.p1">x</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./8.17#SS1.p1">a</mi><mo>,</mo><mi href="./8.17#SS1.p1">b</mi><mo>)</mo></mrow></mrow></math></span></a><a href="./8.20" title="§8.20 Asymptotic Expansions of E p ( z ) ‣ Related Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref" rel="next"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">8.20 </span>Asymptotic Expansions of <math class="ltx_Math" altimg="m24.png" altimg-height="24px" altimg-valign="-8px" altimg-width="57px" alttext="E_{p}\left(z\right)" display="inline"><mrow><msub><mi href="./8.19#E1">E</mi><mi href="./8.1#p1.t1.r3">p</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./8.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math></span></a>
</div>
</div>
</body></text>
</html>
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<head>
<title>DLMF: 22.16 Related Functions</title>
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<h6>Contents</h6>
<ul class="ltx_toclist ltx_toclist_section">
<li class="ltx_tocentry"><a href="#i"><span class="ltx_tag ltx_tag_subsection">§22.16(i) </span>Jacobi’s Amplitude (<math class="ltx_Math" altimg="m47.png" altimg-height="13px" altimg-valign="-2px" altimg-width="31px" alttext="\operatorname{am}" display="inline"><mi href="./22.16#E1">am</mi></math>) Function</a></li>
<li class="ltx_tocentry"></li>
</ul>
<section id="i" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§22.16(i) </span>Jacobi’s Amplitude (<math class="ltx_Math" altimg="m47.png" altimg-height="13px" altimg-valign="-2px" altimg-width="31px" alttext="\operatorname{am}" display="inline"><mi href="./22.16#E1">am</mi></math>) Function</h2>
<div id="SS1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Keywords:</dt>
<dd><a class="ltx_keyword" href="./idx/#amplitudeamfunction">amplitude (<math class="ltx_Math" altimg="m47.png" altimg-height="13px" altimg-valign="-2px" altimg-width="31px" alttext="\operatorname{am}" display="inline"><mi href="./22.16#E1">am</mi></math>) function</a></dd>
</dl>
</div>
</div>
<section id="Px1" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Definition</h3>
<div id="Px1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Keywords:</dt>
<dd><a class="ltx_keyword" href="./idx/#amplitudeamfunction">amplitude (<math class="ltx_Math" altimg="m47.png" altimg-height="13px" altimg-valign="-2px" altimg-width="31px" alttext="\operatorname{am}" display="inline"><mi href="./22.16#E1">am</mi></math>) function</a></dd>
</dl>
</div>
</div>
<div id="Px1.p1" class="ltx_para">
<table id="E1" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">22.16.1</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E1.png" altimg-height="25px" altimg-valign="-7px" altimg-width="262px" alttext="\operatorname{am}\left(x,k\right)=\operatorname{Arcsin}\left(\operatorname{sn}%
\left(x,k\right)\right)," display="block"><mrow><mrow><mrow><mi href="./22.16#E1">am</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mi href="./4.23#E1">Arcsin</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./22.2#E4">sn</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m82.png" altimg-height="18px" altimg-valign="-3px" altimg-width="54px" alttext="x\in\mathbb{R}" display="inline"><mrow><mi href="./22.1#p2.t1.r1">x</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mi href="./front/introduction#Sx4.p2.t1.r14">ℝ</mi></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m48.png" altimg-height="23px" altimg-valign="-7px" altimg-width="81px" alttext="\operatorname{am}\left(\NVar{x},\NVar{k}\right)" display="inline"><mrow><mi href="./22.16#E1">am</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobi’s amplitude function</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./22.2#E4" title="(22.2.4) ‣ §22.2 Definitions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m65.png" altimg-height="23px" altimg-valign="-7px" altimg-width="72px" alttext="\operatorname{sn}\left(\NVar{z},\NVar{k}\right)" display="inline"><mrow><mi href="./22.2#E4">sn</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r2">z</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobian elliptic function</a>,
<a href="./front/introduction#Sx4.p1.t1.r9" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="15px" altimg-valign="-3px" altimg-width="18px" alttext="\in" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo></math>: element of</a>,
<a href="./4.23#E1" title="(4.23.1) ‣ §4.23(i) General Definitions ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m46.png" altimg-height="18px" altimg-valign="-2px" altimg-width="74px" alttext="\operatorname{Arcsin}\NVar{z}" display="inline"><mrow><mi href="./4.23#E1">Arcsin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: general arcsine function</a>,
<a href="./front/introduction#Sx4.p2.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\mathbb{R}" display="inline"><mi href="./front/introduction#Sx4.p2.t1.r14">ℝ</mi></math>: real line</a>,
<a href="./22.1#p2.t1.r1" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./22.1#p2.t1.r1">x</mi></math>: real</a> and
<a href="./22.1#p2.t1.r3" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./22.1#p2.t1.r3">k</mi></math>: modulus</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where the inverse sine has its principal value when <math class="ltx_Math" altimg="m23.png" altimg-height="20px" altimg-valign="-5px" altimg-width="121px" alttext="-K\leq x\leq K" display="inline"><mrow><mrow><mo>-</mo><mrow><mi href="./19.2#E8">K</mi><mo></mo></mrow></mrow><mo>≤</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>≤</mo><mrow><mi href="./19.2#E8">K</mi><mo></mo></mrow></mrow></math> and is
defined by continuity elsewhere. See Figure .
<math class="ltx_Math" altimg="m49.png" altimg-height="23px" altimg-valign="-7px" altimg-width="81px" alttext="\operatorname{am}\left(x,k\right)" display="inline"><mrow><mi href="./22.16#E1">am</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math> is an infinitely differentiable function of <math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./22.1#p2.t1.r1">x</mi></math>.</p>
</div>
</section>
<section id="Px2" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Quasi-Periodicity</h3>
<div id="Px2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Keywords:</dt>
<dd><a class="ltx_keyword" href="./idx/#amplitudeamfunction">amplitude (<math class="ltx_Math" altimg="m47.png" altimg-height="13px" altimg-valign="-2px" altimg-width="31px" alttext="\operatorname{am}" display="inline"><mi href="./22.16#E1">am</mi></math>) function</a></dd>
</dl>
</div>
</div>
<div id="Px2.p1" class="ltx_para">
<table id="E2" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">22.16.2</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E2.png" altimg-height="25px" altimg-valign="-7px" altimg-width="279px" alttext="\operatorname{am}\left(x+2K,k\right)=\operatorname{am}\left(x,k\right)+\pi." display="block"><mrow><mrow><mrow><mi href="./22.16#E1">am</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./22.1#p2.t1.r1">x</mi><mo>+</mo><mrow><mn>2</mn><mo></mo><mrow><mi href="./19.2#E8">K</mi><mo></mo></mrow></mrow></mrow><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mi href="./22.16#E1">am</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo>+</mo><mi href="./3.12#E1">π</mi></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./22.16#E1" title="(22.16.1) ‣ Definition ‣ §22.16(i) Jacobi’s Amplitude ( am ) Function ‣ §22.16 Related Functions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="23px" altimg-valign="-7px" altimg-width="81px" alttext="\operatorname{am}\left(\NVar{x},\NVar{k}\right)" display="inline"><mrow><mi href="./22.16#E1">am</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobi’s amplitude function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m67.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./19.2#E8" title="(19.2.8) ‣ §19.2(ii) Legendre’s Integrals ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m31.png" altimg-height="23px" altimg-valign="-7px" altimg-width="52px" alttext="K\left(\NVar{k}\right)" display="inline"><mrow><mi href="./19.2#E8">K</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Legendre’s complete elliptic integral of the first kind</a>,
<a href="./22.1#p2.t1.r1" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./22.1#p2.t1.r1">x</mi></math>: real</a> and
<a href="./22.1#p2.t1.r3" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./22.1#p2.t1.r3">k</mi></math>: modulus</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px3" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Integral Representation</h3>
<div id="Px3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Keywords:</dt>
<dd><a class="ltx_keyword" href="./idx/#amplitudeamfunction">amplitude (<math class="ltx_Math" altimg="m47.png" altimg-height="13px" altimg-valign="-2px" altimg-width="31px" alttext="\operatorname{am}" display="inline"><mi href="./22.16#E1">am</mi></math>) function</a></dd>
</dl>
</div>
</div>
<div id="Px3.p1" class="ltx_para">
<table id="E3" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">22.16.3</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E3.png" altimg-height="53px" altimg-valign="-20px" altimg-width="238px" alttext="\operatorname{am}\left(x,k\right)=\int_{0}^{x}\operatorname{dn}\left(t,k\right%
)\mathrm{d}t." display="block"><mrow><mrow><mrow><mi href="./22.16#E1">am</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi href="./22.1#p2.t1.r1">x</mi></msubsup><mrow><mrow><mi href="./22.2#E6">dn</mi><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./22.16#E1" title="(22.16.1) ‣ Definition ‣ §22.16(i) Jacobi’s Amplitude ( am ) Function ‣ §22.16 Related Functions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="23px" altimg-valign="-7px" altimg-width="81px" alttext="\operatorname{am}\left(\NVar{x},\NVar{k}\right)" display="inline"><mrow><mi href="./22.16#E1">am</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobi’s amplitude function</a>,
<a href="./22.2#E6" title="(22.2.6) ‣ §22.2 Definitions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="23px" altimg-valign="-7px" altimg-width="75px" alttext="\operatorname{dn}\left(\NVar{z},\NVar{k}\right)" display="inline"><mrow><mi href="./22.2#E6">dn</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r2">z</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobian elliptic function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m45.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./22.1#p2.t1.r1" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./22.1#p2.t1.r1">x</mi></math>: real</a> and
<a href="./22.1#p2.t1.r3" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./22.1#p2.t1.r3">k</mi></math>: modulus</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px4" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Special Values</h3>
<div id="Px4.info" class="ltx_metadata ltx_info">
, <a class="ltx_keyword" href="./idx/#amplitudeamfunction">amplitude (<math class="ltx_Math" altimg="m47.png" altimg-height="13px" altimg-valign="-2px" altimg-width="31px" alttext="\operatorname{am}" display="inline"><mi href="./22.16#E1">am</mi></math>) function</a>
</dd>
</dl>
</div>
</div>
<div id="Px4.p1" class="ltx_para">
<table id="EGx1" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E4">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">22.16.4</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m17.png" altimg-height="25px" altimg-valign="-7px" altimg-width="82px" alttext="\displaystyle\operatorname{am}\left(x,0\right)" display="inline"><mrow><mi href="./22.16#E1">am</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mn>0</mn><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m13.png" altimg-height="18px" altimg-valign="-6px" altimg-width="44px" alttext="\displaystyle=x," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mi href="./22.1#p2.t1.r1">x</mi></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E4.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./22.16#E1" title="(22.16.1) ‣ Definition ‣ §22.16(i) Jacobi’s Amplitude ( am ) Function ‣ §22.16 Related Functions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="23px" altimg-valign="-7px" altimg-width="81px" alttext="\operatorname{am}\left(\NVar{x},\NVar{k}\right)" display="inline"><mrow><mi href="./22.16#E1">am</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobi’s amplitude function</a> and
<a href="./22.1#p2.t1.r1" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./22.1#p2.t1.r1">x</mi></math>: real</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E5">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">22.16.5</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m18.png" altimg-height="25px" altimg-valign="-7px" altimg-width="82px" alttext="\displaystyle\operatorname{am}\left(x,1\right)" display="inline"><mrow><mi href="./22.16#E1">am</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mn>1</mn><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m10.png" altimg-height="25px" altimg-valign="-7px" altimg-width="87px" alttext="\displaystyle=\operatorname{gd}\left(x\right)." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./4.23#E39">gd</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>)</mo></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E5.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.23#E39" title="(4.23.39) ‣ §4.23(viii) Gudermannian Function ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m57.png" altimg-height="21px" altimg-valign="-6px" altimg-width="40px" alttext="\operatorname{gd}\NVar{x}" display="inline"><mrow><mi href="./4.23#E39">gd</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r3">x</mi></mrow></math>: Gudermannian function</a>,
<a href="./22.16#E1" title="(22.16.1) ‣ Definition ‣ §22.16(i) Jacobi’s Amplitude ( am ) Function ‣ §22.16 Related Functions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="23px" altimg-valign="-7px" altimg-width="81px" alttext="\operatorname{am}\left(\NVar{x},\NVar{k}\right)" display="inline"><mrow><mi href="./22.16#E1">am</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobi’s amplitude function</a> and
<a href="./22.1#p2.t1.r1" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./22.1#p2.t1.r1">x</mi></math>: real</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
<p class="ltx_p">For the Gudermannian function <math class="ltx_Math" altimg="m58.png" altimg-height="23px" altimg-valign="-7px" altimg-width="56px" alttext="\operatorname{gd}\left(x\right)" display="inline"><mrow><mi href="./4.23#E39">gd</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>)</mo></mrow></mrow></math> see §.</p>
</div>
</section>
<section id="Px5" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Approximation for Small <math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./22.1#p2.t1.r1">x</mi></math>
</h3>
<div id="Px5.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Keywords:</dt>
<dd><a class="ltx_keyword" href="./idx/#amplitudeamfunction">amplitude (<math class="ltx_Math" altimg="m47.png" altimg-height="13px" altimg-valign="-2px" altimg-width="31px" alttext="\operatorname{am}" display="inline"><mi href="./22.16#E1">am</mi></math>) function</a></dd>
</dl>
</div>
</div>
<div id="Px5.p1" class="ltx_para">
<table id="E6" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">22.16.6</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E6.png" altimg-height="50px" altimg-valign="-16px" altimg-width="429px" alttext="\operatorname{am}\left(x,k\right)=x-k^{2}\frac{x^{3}}{3!}+k^{2}\left(4+k^{2}%
\right)\frac{x^{5}}{5!}+O\left(x^{7}\right)." display="block"><mrow><mrow><mrow><mi href="./22.16#E1">am</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mi href="./22.1#p2.t1.r1">x</mi><mo>-</mo><mrow><msup><mi href="./22.1#p2.t1.r3">k</mi><mn>2</mn></msup><mo></mo><mfrac><msup><mi href="./22.1#p2.t1.r1">x</mi><mn>3</mn></msup><mrow><mn>3</mn><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mfrac></mrow></mrow><mo>+</mo><mrow><msup><mi href="./22.1#p2.t1.r3">k</mi><mn>2</mn></msup><mo></mo><mrow><mo>(</mo><mrow><mn>4</mn><mo>+</mo><msup><mi href="./22.1#p2.t1.r3">k</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow><mo></mo><mfrac><msup><mi href="./22.1#p2.t1.r1">x</mi><mn>5</mn></msup><mrow><mn>5</mn><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mfrac></mrow><mo>+</mo><mrow><mi href="./2.1#E3">O</mi><mo></mo><mrow><mo>(</mo><msup><mi href="./22.1#p2.t1.r1">x</mi><mn>7</mn></msup><mo>)</mo></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E6.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./2.1#E3" title="(2.1.3) ‣ §2.1(i) Asymptotic and Order Symbols ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m33.png" altimg-height="23px" altimg-valign="-7px" altimg-width="50px" alttext="O\left(\NVar{x}\right)" display="inline"><mrow><mi href="./2.1#E3">O</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>)</mo></mrow></mrow></math>: order not exceeding</a>,
<a href="./22.16#E1" title="(22.16.1) ‣ Definition ‣ §22.16(i) Jacobi’s Amplitude ( am ) Function ‣ §22.16 Related Functions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="23px" altimg-valign="-7px" altimg-width="81px" alttext="\operatorname{am}\left(\NVar{x},\NVar{k}\right)" display="inline"><mrow><mi href="./22.16#E1">am</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobi’s amplitude function</a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m79.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./22.1#p2.t1.r1" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./22.1#p2.t1.r1">x</mi></math>: real</a> and
<a href="./22.1#p2.t1.r3" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./22.1#p2.t1.r3">k</mi></math>: modulus</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px6" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Approximations for Small <math class="ltx_Math" altimg="m77.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./22.1#p2.t1.r3">k</mi></math>, <math class="ltx_Math" altimg="m78.png" altimg-height="19px" altimg-valign="-2px" altimg-width="21px" alttext="k^{\prime}" display="inline"><msup><mi href="./22.1#p2.t1.r4">k</mi><mo>′</mo></msup></math>
</h3>
<div id="Px6.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Keywords:</dt>
<dd><a class="ltx_keyword" href="./idx/#amplitudeamfunction">amplitude (<math class="ltx_Math" altimg="m47.png" altimg-height="13px" altimg-valign="-2px" altimg-width="31px" alttext="\operatorname{am}" display="inline"><mi href="./22.16#E1">am</mi></math>) function</a></dd>
</dl>
</div>
</div>
<div id="Px6.p1" class="ltx_para">
<table id="E7" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">22.16.7</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E7.png" altimg-height="30px" altimg-valign="-9px" altimg-width="403px" alttext="\operatorname{am}\left(x,k\right)=x-\tfrac{1}{4}k^{2}(x-\sin x\cos x)+O\left(k%
^{4}\right)," display="block"><mrow><mrow><mrow><mi href="./22.16#E1">am</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mi href="./22.1#p2.t1.r1">x</mi><mo>-</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>4</mn></mfrac></mstyle><mo></mo><msup><mi href="./22.1#p2.t1.r3">k</mi><mn>2</mn></msup><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./22.1#p2.t1.r1">x</mi><mo>-</mo><mrow><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi href="./22.1#p2.t1.r1">x</mi></mrow><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi href="./22.1#p2.t1.r1">x</mi></mrow></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mi href="./2.1#E3">O</mi><mo></mo><mrow><mo>(</mo><msup><mi href="./22.1#p2.t1.r3">k</mi><mn>4</mn></msup><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E7.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./2.1#E3" title="(2.1.3) ‣ §2.1(i) Asymptotic and Order Symbols ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m33.png" altimg-height="23px" altimg-valign="-7px" altimg-width="50px" alttext="O\left(\NVar{x}\right)" display="inline"><mrow><mi href="./2.1#E3">O</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>)</mo></mrow></mrow></math>: order not exceeding</a>,
<a href="./22.16#E1" title="(22.16.1) ‣ Definition ‣ §22.16(i) Jacobi’s Amplitude ( am ) Function ‣ §22.16 Related Functions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="23px" altimg-valign="-7px" altimg-width="81px" alttext="\operatorname{am}\left(\NVar{x},\NVar{k}\right)" display="inline"><mrow><mi href="./22.16#E1">am</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobi’s amplitude function</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m34.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m68.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./22.1#p2.t1.r1" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./22.1#p2.t1.r1">x</mi></math>: real</a> and
<a href="./22.1#p2.t1.r3" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./22.1#p2.t1.r3">k</mi></math>: modulus</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">16.13.4</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E8" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">22.16.8</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E8.png" altimg-height="41px" altimg-valign="-15px" altimg-width="521px" alttext="\operatorname{am}\left(x,k\right)=\operatorname{gd}x-\tfrac{1}{4}{k^{\prime}}^%
{2}(x-\sinh x\cosh x)\operatorname{sech}x+O\left({k^{\prime}}^{4}\right)." display="block"><mrow><mrow><mrow><mi href="./22.16#E1">am</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mrow><mi href="./4.23#E39">gd</mi><mo></mo><mi href="./22.1#p2.t1.r1">x</mi></mrow><mo>-</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>4</mn></mfrac></mstyle><mo></mo><mmultiscripts><mi href="./22.1#p2.t1.r4">k</mi><none></none><mo>′</mo><none></none><mn>2</mn></mmultiscripts><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./22.1#p2.t1.r1">x</mi><mo>-</mo><mrow><mrow><mi href="./4.28#E1">sinh</mi><mo></mo><mi href="./22.1#p2.t1.r1">x</mi></mrow><mo></mo><mrow><mi href="./4.28#E2">cosh</mi><mo></mo><mi href="./22.1#p2.t1.r1">x</mi></mrow></mrow></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mi href="./4.28#E6">sech</mi><mo></mo><mi href="./22.1#p2.t1.r1">x</mi></mrow></mrow></mrow><mo>+</mo><mrow><mi href="./2.1#E3">O</mi><mo></mo><mrow><mo>(</mo><mmultiscripts><mi href="./22.1#p2.t1.r4">k</mi><none></none><mo>′</mo><none></none><mn>4</mn></mmultiscripts><mo>)</mo></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E8.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./2.1#E3" title="(2.1.3) ‣ §2.1(i) Asymptotic and Order Symbols ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m33.png" altimg-height="23px" altimg-valign="-7px" altimg-width="50px" alttext="O\left(\NVar{x}\right)" display="inline"><mrow><mi href="./2.1#E3">O</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>)</mo></mrow></mrow></math>: order not exceeding</a>,
<a href="./4.23#E39" title="(4.23.39) ‣ §4.23(viii) Gudermannian Function ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m57.png" altimg-height="21px" altimg-valign="-6px" altimg-width="40px" alttext="\operatorname{gd}\NVar{x}" display="inline"><mrow><mi href="./4.23#E39">gd</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r3">x</mi></mrow></math>: Gudermannian function</a>,
<a href="./22.16#E1" title="(22.16.1) ‣ Definition ‣ §22.16(i) Jacobi’s Amplitude ( am ) Function ‣ §22.16 Related Functions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="23px" altimg-valign="-7px" altimg-width="81px" alttext="\operatorname{am}\left(\NVar{x},\NVar{k}\right)" display="inline"><mrow><mi href="./22.16#E1">am</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobi’s amplitude function</a>,
<a href="./4.28#E2" title="(4.28.2) ‣ §4.28 Definitions and Periodicity ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m35.png" altimg-height="18px" altimg-valign="-2px" altimg-width="56px" alttext="\cosh\NVar{z}" display="inline"><mrow><mi href="./4.28#E2">cosh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: hyperbolic cosine function</a>,
<a href="./4.28#E6" title="(4.28.6) ‣ §4.28 Definitions and Periodicity ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m64.png" altimg-height="18px" altimg-valign="-2px" altimg-width="54px" alttext="\operatorname{sech}\NVar{z}" display="inline"><mrow><mi href="./4.28#E6">sech</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: hyperbolic secant function</a>,
<a href="./4.28#E1" title="(4.28.1) ‣ §4.28 Definitions and Periodicity ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="18px" altimg-valign="-2px" altimg-width="53px" alttext="\sinh\NVar{z}" display="inline"><mrow><mi href="./4.28#E1">sinh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: hyperbolic sine function</a>,
<a href="./22.1#p2.t1.r1" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./22.1#p2.t1.r1">x</mi></math>: real</a>,
<a href="./22.1#p2.t1.r3" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./22.1#p2.t1.r3">k</mi></math>: modulus</a> and
<a href="./22.1#p2.t1.r4" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="19px" altimg-valign="-2px" altimg-width="21px" alttext="k^{\prime}" display="inline"><msup><mi href="./22.1#p2.t1.r4">k</mi><mo>′</mo></msup></math>: complementary modulus</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">16.15.4</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px7" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Fourier Series</h3>
<div id="Px7.info" class="ltx_metadata ltx_info">
, <a class="ltx_keyword" href="./idx/#amplitudeamfunction">amplitude (<math class="ltx_Math" altimg="m47.png" altimg-height="13px" altimg-valign="-2px" altimg-width="31px" alttext="\operatorname{am}" display="inline"><mi href="./22.16#E1">am</mi></math>) function</a>
</dd>
</dl>
</div>
</div>
<div id="Px7.p1" class="ltx_para">
<p class="ltx_p">With <math class="ltx_Math" altimg="m80.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="q" display="inline"><mi href="./22.2#E1">q</mi></math> as in () and <math class="ltx_Math" altimg="m73.png" altimg-height="23px" altimg-valign="-7px" altimg-width="118px" alttext="\zeta=\pi x/(2K)" display="inline"><mrow><mi>ζ</mi><mo>=</mo><mrow><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi href="./22.1#p2.t1.r1">x</mi></mrow><mo>/</mo><mrow><mo stretchy="false">(</mo><mrow><mn>2</mn><mo></mo><mrow><mi href="./19.2#E8">K</mi><mo></mo></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></math>,</p>
<table id="E9" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">22.16.9</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E9.png" altimg-height="64px" altimg-valign="-27px" altimg-width="334px" alttext="\operatorname{am}\left(x,k\right)=\frac{\pi}{2K}x+2\sum_{n=1}^{\infty}\frac{q^%
{n}\sin\left(2n\zeta\right)}{n(1+q^{2n})}." display="block"><mrow><mrow><mrow><mi href="./22.16#E1">am</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mfrac><mi href="./3.12#E1">π</mi><mrow><mn>2</mn><mo></mo><mrow><mi href="./19.2#E8">K</mi><mo></mo></mrow></mrow></mfrac><mo></mo><mi href="./22.1#p2.t1.r1">x</mi></mrow><mo>+</mo><mrow><mn>2</mn><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mfrac><mrow><msup><mi href="./22.2#E1">q</mi><mi>n</mi></msup><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><mi>n</mi><mo></mo><mi>ζ</mi></mrow><mo>)</mo></mrow></mrow></mrow><mrow><mi>n</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>+</mo><msup><mi href="./22.2#E1">q</mi><mrow><mn>2</mn><mo></mo><mi>n</mi></mrow></msup></mrow><mo stretchy="false">)</mo></mrow></mrow></mfrac></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E9.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./22.16#E1" title="(22.16.1) ‣ Definition ‣ §22.16(i) Jacobi’s Amplitude ( am ) Function ‣ §22.16 Related Functions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="23px" altimg-valign="-7px" altimg-width="81px" alttext="\operatorname{am}\left(\NVar{x},\NVar{k}\right)" display="inline"><mrow><mi href="./22.16#E1">am</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobi’s amplitude function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m67.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./19.2#E8" title="(19.2.8) ‣ §19.2(ii) Legendre’s Integrals ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m31.png" altimg-height="23px" altimg-valign="-7px" altimg-width="52px" alttext="K\left(\NVar{k}\right)" display="inline"><mrow><mi href="./19.2#E8">K</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Legendre’s complete elliptic integral of the first kind</a>,
<a href="./22.2#E1" title="(22.2.1) ‣ §22.2 Definitions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m80.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="q" display="inline"><mi href="./22.2#E1">q</mi></math>: nome</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m68.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./22.1#p2.t1.r1" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./22.1#p2.t1.r1">x</mi></math>: real</a> and
<a href="./22.1#p2.t1.r3" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./22.1#p2.t1.r3">k</mi></math>: modulus</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px8" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Relation to Elliptic Integrals</h3>
<div id="Px8.info" class="ltx_metadata ltx_info">
, <a class="ltx_keyword" href="./idx/#amplitudeamfunction">amplitude (<math class="ltx_Math" altimg="m47.png" altimg-height="13px" altimg-valign="-2px" altimg-width="31px" alttext="\operatorname{am}" display="inline"><mi href="./22.16#E1">am</mi></math>) function</a>
</dd>
</dl>
</div>
</div>
<div id="Px8.p1" class="ltx_para">
<p class="ltx_p">If <math class="ltx_Math" altimg="m23.png" altimg-height="20px" altimg-valign="-5px" altimg-width="121px" alttext="-K\leq x\leq K" display="inline"><mrow><mrow><mo>-</mo><mrow><mi href="./19.2#E8">K</mi><mo></mo></mrow></mrow><mo>≤</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>≤</mo><mrow><mi href="./19.2#E8">K</mi><mo></mo></mrow></mrow></math>, then the following four equations
are equivalent:</p>
<table id="E10" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">22.16.10</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E10.png" altimg-height="25px" altimg-valign="-7px" altimg-width="119px" alttext="x=F\left(\phi,k\right)," display="block"><mrow><mrow><mi href="./22.1#p2.t1.r1">x</mi><mo>=</mo><mrow><mi href="./19.2#E4">F</mi><mo></mo><mrow><mo>(</mo><mi>ϕ</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E10.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.2#E4" title="(19.2.4) ‣ §19.2(ii) Legendre’s Integrals ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="23px" altimg-valign="-7px" altimg-width="70px" alttext="F\left(\NVar{\phi},\NVar{k}\right)" display="inline"><mrow><mi href="./19.2#E4">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r2">ϕ</mi><mo>,</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Legendre’s incomplete elliptic integral of the first kind</a>,
<a href="./22.1#p2.t1.r1" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./22.1#p2.t1.r1">x</mi></math>: real</a> and
<a href="./22.1#p2.t1.r3" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./22.1#p2.t1.r3">k</mi></math>: modulus</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E11" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">22.16.11</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E11.png" altimg-height="25px" altimg-valign="-7px" altimg-width="127px" alttext="\operatorname{am}\left(x,k\right)=\phi," display="block"><mrow><mrow><mrow><mi href="./22.16#E1">am</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo>=</mo><mi>ϕ</mi></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E11.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./22.16#E1" title="(22.16.1) ‣ Definition ‣ §22.16(i) Jacobi’s Amplitude ( am ) Function ‣ §22.16 Related Functions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="23px" altimg-valign="-7px" altimg-width="81px" alttext="\operatorname{am}\left(\NVar{x},\NVar{k}\right)" display="inline"><mrow><mi href="./22.16#E1">am</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobi’s amplitude function</a>,
<a href="./22.1#p2.t1.r1" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./22.1#p2.t1.r1">x</mi></math>: real</a> and
<a href="./22.1#p2.t1.r3" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./22.1#p2.t1.r3">k</mi></math>: modulus</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E12" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">22.16.12</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E12.png" altimg-height="25px" altimg-valign="-7px" altimg-width="297px" alttext="\operatorname{sn}\left(x,k\right)=\sin\phi=\sin\left(\operatorname{am}\left(x,%
k\right)\right)," display="block"><mrow><mrow><mrow><mi href="./22.2#E4">sn</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi>ϕ</mi></mrow><mo>=</mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./22.16#E1">am</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E12.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./22.16#E1" title="(22.16.1) ‣ Definition ‣ §22.16(i) Jacobi’s Amplitude ( am ) Function ‣ §22.16 Related Functions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="23px" altimg-valign="-7px" altimg-width="81px" alttext="\operatorname{am}\left(\NVar{x},\NVar{k}\right)" display="inline"><mrow><mi href="./22.16#E1">am</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobi’s amplitude function</a>,
<a href="./22.2#E4" title="(22.2.4) ‣ §22.2 Definitions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m65.png" altimg-height="23px" altimg-valign="-7px" altimg-width="72px" alttext="\operatorname{sn}\left(\NVar{z},\NVar{k}\right)" display="inline"><mrow><mi href="./22.2#E4">sn</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r2">z</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobian elliptic function</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m68.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./22.1#p2.t1.r1" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./22.1#p2.t1.r1">x</mi></math>: real</a> and
<a href="./22.1#p2.t1.r3" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./22.1#p2.t1.r3">k</mi></math>: modulus</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E13" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">22.16.13</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E13.png" altimg-height="25px" altimg-valign="-7px" altimg-width="302px" alttext="\operatorname{cn}\left(x,k\right)=\cos\phi=\cos\left(\operatorname{am}\left(x,%
k\right)\right)." display="block"><mrow><mrow><mrow><mi href="./22.2#E5">cn</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi>ϕ</mi></mrow><mo>=</mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./22.16#E1">am</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E13.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./22.16#E1" title="(22.16.1) ‣ Definition ‣ §22.16(i) Jacobi’s Amplitude ( am ) Function ‣ §22.16 Related Functions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="23px" altimg-valign="-7px" altimg-width="81px" alttext="\operatorname{am}\left(\NVar{x},\NVar{k}\right)" display="inline"><mrow><mi href="./22.16#E1">am</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobi’s amplitude function</a>,
<a href="./22.2#E5" title="(22.2.5) ‣ §22.2 Definitions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="23px" altimg-valign="-7px" altimg-width="73px" alttext="\operatorname{cn}\left(\NVar{z},\NVar{k}\right)" display="inline"><mrow><mi href="./22.2#E5">cn</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r2">z</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobian elliptic function</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m34.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./22.1#p2.t1.r1" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./22.1#p2.t1.r1">x</mi></math>: real</a> and
<a href="./22.1#p2.t1.r3" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./22.1#p2.t1.r3">k</mi></math>: modulus</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">For <math class="ltx_Math" altimg="m30.png" altimg-height="23px" altimg-valign="-7px" altimg-width="70px" alttext="F\left(\phi,k\right)" display="inline"><mrow><mi href="./19.2#E4">F</mi><mo></mo><mrow><mo>(</mo><mi>ϕ</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math> see §</dd>
</dl>
</div>
</div>
<div id="Px9.p1" class="ltx_para">
<p class="ltx_p">For <math class="ltx_Math" altimg="m22.png" altimg-height="19px" altimg-valign="-4px" altimg-width="121px" alttext="-K<x<K" display="inline"><mrow><mrow><mo>-</mo><mrow><mi href="./19.2#E8">K</mi><mo></mo></mrow></mrow><mo><</mo><mi href="./22.1#p2.t1.r1">x</mi><mo><</mo><mrow><mi href="./19.2#E8">K</mi><mo></mo></mrow></mrow></math>,
</p>
<table id="E14" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">22.16.14</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E14.png" altimg-height="57px" altimg-valign="-20px" altimg-width="288px" alttext="\mathcal{E}\left(x,k\right)=\int_{0}^{\operatorname{sn}\left(x,k\right)}\sqrt{%
\frac{1-k^{2}t^{2}}{1-t^{2}}}\mathrm{d}t;" display="block"><mrow><mrow><mrow><mi class="ltx_font_mathcaligraphic" href="./22.16#E14">ℰ</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mrow><mi href="./22.2#E4">sn</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></msubsup><mrow><msqrt><mfrac><mrow><mn>1</mn><mo>-</mo><mrow><msup><mi href="./22.1#p2.t1.r3">k</mi><mn>2</mn></msup><mo></mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mrow><mrow><mn>1</mn><mo>-</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac></msqrt><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow><mo>;</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E14.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m40.png" altimg-height="23px" altimg-valign="-7px" altimg-width="67px" alttext="\mathcal{E}\left(\NVar{x},\NVar{k}\right)" display="inline"><mrow><mi class="ltx_font_mathcaligraphic" href="./22.16#E14">ℰ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobi’s epsilon function</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./22.2#E4" title="(22.2.4) ‣ §22.2 Definitions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m65.png" altimg-height="23px" altimg-valign="-7px" altimg-width="72px" alttext="\operatorname{sn}\left(\NVar{z},\NVar{k}\right)" display="inline"><mrow><mi href="./22.2#E4">sn</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r2">z</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobian elliptic function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m45.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./22.1#p2.t1.r1" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./22.1#p2.t1.r1">x</mi></math>: real</a> and
<a href="./22.1#p2.t1.r3" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./22.1#p2.t1.r3">k</mi></math>: modulus</a>
</dd>
<dt>Errata (effective with 1.0.1):</dt>
<dd>
Originally appeared with the upper limit as <math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./22.1#p2.t1.r1">x</mi></math>, rather than <math class="ltx_Math" altimg="m66.png" altimg-height="23px" altimg-valign="-7px" altimg-width="73px" alttext="\operatorname{sn}\left(x,k\right)" display="inline"><mrow><mi href="./22.2#E4">sn</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>:
<table id="Ex1" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="m15.png" altimg-height="57px" altimg-valign="-20px" altimg-width="242px" alttext="\mathcal{E}\left(x,k\right)=\int_{0}^{x}\sqrt{\frac{1-k^{2}t^{2}}{1-t^{2}}}%
\mathrm{d}t" display="block"><mrow><mrow><mi class="ltx_font_mathcaligraphic" href="./22.16#E14">ℰ</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi href="./22.1#p2.t1.r1">x</mi></msubsup><mrow><msqrt><mfrac><mrow><mn>1</mn><mo>-</mo><mrow><msup><mi href="./22.1#p2.t1.r3">k</mi><mn>2</mn></msup><mo></mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mrow><mrow><mn>1</mn><mo>-</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac></msqrt><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
</table>
<p><span class="ltx_font_italic">Reported 2010-07-08 by Charles Karney</span></p>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">compare (.</p>
</div>
<div id="Px9.p2" class="ltx_para">
<table id="EGx2" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E15">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">22.16.15</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m16.png" altimg-height="25px" altimg-valign="-7px" altimg-width="68px" alttext="\displaystyle\mathcal{E}\left(x,k\right)" display="inline"><mrow><mi class="ltx_font_mathcaligraphic" href="./22.16#E14">ℰ</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m6.png" altimg-height="53px" altimg-valign="-20px" altimg-width="236px" alttext="\displaystyle=-k^{2}\int_{0}^{x}{\operatorname{sn}^{2}}\left(t,k\right)\mathrm%
{d}t+x," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mo>-</mo><mrow><msup><mi href="./22.1#p2.t1.r3">k</mi><mn>2</mn></msup><mo></mo><mrow><mstyle displaystyle="true"><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi href="./22.1#p2.t1.r1">x</mi></msubsup></mstyle><mrow><mrow><msup><mi href="./22.2#E4">sn</mi><mn>2</mn></msup><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow></mrow><mo>+</mo><mi href="./22.1#p2.t1.r1">x</mi></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E15.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./22.16#E14" title="(22.16.14) ‣ Integral Representations ‣ §22.16(ii) Jacobi’s Epsilon Function ‣ §22.16 Related Functions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m40.png" altimg-height="23px" altimg-valign="-7px" altimg-width="67px" alttext="\mathcal{E}\left(\NVar{x},\NVar{k}\right)" display="inline"><mrow><mi class="ltx_font_mathcaligraphic" href="./22.16#E14">ℰ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobi’s epsilon function</a>,
<a href="./22.2#E4" title="(22.2.4) ‣ §22.2 Definitions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m65.png" altimg-height="23px" altimg-valign="-7px" altimg-width="72px" alttext="\operatorname{sn}\left(\NVar{z},\NVar{k}\right)" display="inline"><mrow><mi href="./22.2#E4">sn</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r2">z</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobian elliptic function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m45.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./22.1#p2.t1.r1" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./22.1#p2.t1.r1">x</mi></math>: real</a> and
<a href="./22.1#p2.t1.r3" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./22.1#p2.t1.r3">k</mi></math>: modulus</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">16.26.1</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E16">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">22.16.16</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m16.png" altimg-height="25px" altimg-valign="-7px" altimg-width="68px" alttext="\displaystyle\mathcal{E}\left(x,k\right)" display="inline"><mrow><mi class="ltx_font_mathcaligraphic" href="./22.16#E14">ℰ</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m11.png" altimg-height="53px" altimg-valign="-20px" altimg-width="247px" alttext="\displaystyle=k^{2}\int_{0}^{x}{\operatorname{cn}^{2}}\left(t,k\right)\mathrm{%
d}t+{k^{\prime}}^{2}x," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><msup><mi href="./22.1#p2.t1.r3">k</mi><mn>2</mn></msup><mo></mo><mrow><mstyle displaystyle="true"><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi href="./22.1#p2.t1.r1">x</mi></msubsup></mstyle><mrow><mrow><msup><mi href="./22.2#E5">cn</mi><mn>2</mn></msup><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow><mo>+</mo><mrow><mmultiscripts><mi href="./22.1#p2.t1.r4">k</mi><none></none><mo>′</mo><none></none><mn>2</mn></mmultiscripts><mo></mo><mi href="./22.1#p2.t1.r1">x</mi></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E16.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./22.16#E14" title="(22.16.14) ‣ Integral Representations ‣ §22.16(ii) Jacobi’s Epsilon Function ‣ §22.16 Related Functions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m40.png" altimg-height="23px" altimg-valign="-7px" altimg-width="67px" alttext="\mathcal{E}\left(\NVar{x},\NVar{k}\right)" display="inline"><mrow><mi class="ltx_font_mathcaligraphic" href="./22.16#E14">ℰ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobi’s epsilon function</a>,
<a href="./22.2#E5" title="(22.2.5) ‣ §22.2 Definitions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="23px" altimg-valign="-7px" altimg-width="73px" alttext="\operatorname{cn}\left(\NVar{z},\NVar{k}\right)" display="inline"><mrow><mi href="./22.2#E5">cn</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r2">z</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobian elliptic function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m45.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./22.1#p2.t1.r1" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./22.1#p2.t1.r1">x</mi></math>: real</a>,
<a href="./22.1#p2.t1.r3" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./22.1#p2.t1.r3">k</mi></math>: modulus</a> and
<a href="./22.1#p2.t1.r4" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="19px" altimg-valign="-2px" altimg-width="21px" alttext="k^{\prime}" display="inline"><msup><mi href="./22.1#p2.t1.r4">k</mi><mo>′</mo></msup></math>: complementary modulus</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">16.26.2</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E17">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">22.16.17</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m16.png" altimg-height="25px" altimg-valign="-7px" altimg-width="68px" alttext="\displaystyle\mathcal{E}\left(x,k\right)" display="inline"><mrow><mi class="ltx_font_mathcaligraphic" href="./22.16#E14">ℰ</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m9.png" altimg-height="53px" altimg-valign="-20px" altimg-width="164px" alttext="\displaystyle=\int_{0}^{x}{\operatorname{dn}^{2}}\left(t,k\right)\mathrm{d}t." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi href="./22.1#p2.t1.r1">x</mi></msubsup></mstyle><mrow><mrow><msup><mi href="./22.2#E6">dn</mi><mn>2</mn></msup><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E17.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./22.16#E14" title="(22.16.14) ‣ Integral Representations ‣ §22.16(ii) Jacobi’s Epsilon Function ‣ §22.16 Related Functions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m40.png" altimg-height="23px" altimg-valign="-7px" altimg-width="67px" alttext="\mathcal{E}\left(\NVar{x},\NVar{k}\right)" display="inline"><mrow><mi class="ltx_font_mathcaligraphic" href="./22.16#E14">ℰ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobi’s epsilon function</a>,
<a href="./22.2#E6" title="(22.2.6) ‣ §22.2 Definitions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="23px" altimg-valign="-7px" altimg-width="75px" alttext="\operatorname{dn}\left(\NVar{z},\NVar{k}\right)" display="inline"><mrow><mi href="./22.2#E6">dn</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r2">z</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobian elliptic function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m45.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./22.1#p2.t1.r1" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./22.1#p2.t1.r1">x</mi></math>: real</a> and
<a href="./22.1#p2.t1.r3" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./22.1#p2.t1.r3">k</mi></math>: modulus</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">16.26.3</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
</div>
<div id="Px9.p3" class="ltx_para">
</div>
<div id="Px9.p4" class="ltx_para">
<table id="EGx3" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E18">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">22.16.18</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m16.png" altimg-height="25px" altimg-valign="-7px" altimg-width="68px" alttext="\displaystyle\mathcal{E}\left(x,k\right)" display="inline"><mrow><mi class="ltx_font_mathcaligraphic" href="./22.16#E14">ℰ</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m5.png" altimg-height="53px" altimg-valign="-20px" altimg-width="430px" alttext="\displaystyle=-k^{2}\int_{0}^{x}{\operatorname{cd}^{2}}\left(t,k\right)\mathrm%
{d}t+x+k^{2}\operatorname{sn}\left(x,k\right)\operatorname{cd}\left(x,k\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mo>-</mo><mrow><msup><mi href="./22.1#p2.t1.r3">k</mi><mn>2</mn></msup><mo></mo><mrow><mstyle displaystyle="true"><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi href="./22.1#p2.t1.r1">x</mi></msubsup></mstyle><mrow><mrow><msup><mi href="./22.2#E8">cd</mi><mn>2</mn></msup><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow></mrow><mo>+</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>+</mo><mrow><msup><mi href="./22.1#p2.t1.r3">k</mi><mn>2</mn></msup><mo></mo><mrow><mi href="./22.2#E4">sn</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./22.2#E8">cd</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E18.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./22.16#E14" title="(22.16.14) ‣ Integral Representations ‣ §22.16(ii) Jacobi’s Epsilon Function ‣ §22.16 Related Functions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m40.png" altimg-height="23px" altimg-valign="-7px" altimg-width="67px" alttext="\mathcal{E}\left(\NVar{x},\NVar{k}\right)" display="inline"><mrow><mi class="ltx_font_mathcaligraphic" href="./22.16#E14">ℰ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobi’s epsilon function</a>,
<a href="./22.2#E8" title="(22.2.8) ‣ §22.2 Definitions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="23px" altimg-valign="-7px" altimg-width="73px" alttext="\operatorname{cd}\left(\NVar{z},\NVar{k}\right)" display="inline"><mrow><mi href="./22.2#E8">cd</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r2">z</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobian elliptic function</a>,
<a href="./22.2#E4" title="(22.2.4) ‣ §22.2 Definitions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m65.png" altimg-height="23px" altimg-valign="-7px" altimg-width="72px" alttext="\operatorname{sn}\left(\NVar{z},\NVar{k}\right)" display="inline"><mrow><mi href="./22.2#E4">sn</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r2">z</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobian elliptic function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m45.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./22.1#p2.t1.r1" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./22.1#p2.t1.r1">x</mi></math>: real</a> and
<a href="./22.1#p2.t1.r3" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./22.1#p2.t1.r3">k</mi></math>: modulus</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">16.26.4</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E19">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">22.16.19</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m16.png" altimg-height="25px" altimg-valign="-7px" altimg-width="68px" alttext="\displaystyle\mathcal{E}\left(x,k\right)" display="inline"><mrow><mi class="ltx_font_mathcaligraphic" href="./22.16#E14">ℰ</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m12.png" altimg-height="53px" altimg-valign="-20px" altimg-width="464px" alttext="\displaystyle=k^{2}{k^{\prime}}^{2}\int_{0}^{x}{\operatorname{sd}^{2}}\left(t,%
k\right)\mathrm{d}t+{k^{\prime}}^{2}x+k^{2}\operatorname{sn}\left(x,k\right)%
\operatorname{cd}\left(x,k\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><msup><mi href="./22.1#p2.t1.r3">k</mi><mn>2</mn></msup><mo></mo><mmultiscripts><mi href="./22.1#p2.t1.r4">k</mi><none></none><mo>′</mo><none></none><mn>2</mn></mmultiscripts><mo></mo><mrow><mstyle displaystyle="true"><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi href="./22.1#p2.t1.r1">x</mi></msubsup></mstyle><mrow><mrow><msup><mi href="./22.2#E7">sd</mi><mn>2</mn></msup><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow><mo>+</mo><mrow><mmultiscripts><mi href="./22.1#p2.t1.r4">k</mi><none></none><mo>′</mo><none></none><mn>2</mn></mmultiscripts><mo></mo><mi href="./22.1#p2.t1.r1">x</mi></mrow><mo>+</mo><mrow><msup><mi href="./22.1#p2.t1.r3">k</mi><mn>2</mn></msup><mo></mo><mrow><mi href="./22.2#E4">sn</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./22.2#E8">cd</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E19.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./22.16#E14" title="(22.16.14) ‣ Integral Representations ‣ §22.16(ii) Jacobi’s Epsilon Function ‣ §22.16 Related Functions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m40.png" altimg-height="23px" altimg-valign="-7px" altimg-width="67px" alttext="\mathcal{E}\left(\NVar{x},\NVar{k}\right)" display="inline"><mrow><mi class="ltx_font_mathcaligraphic" href="./22.16#E14">ℰ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobi’s epsilon function</a>,
<a href="./22.2#E8" title="(22.2.8) ‣ §22.2 Definitions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="23px" altimg-valign="-7px" altimg-width="73px" alttext="\operatorname{cd}\left(\NVar{z},\NVar{k}\right)" display="inline"><mrow><mi href="./22.2#E8">cd</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r2">z</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobian elliptic function</a>,
<a href="./22.2#E7" title="(22.2.7) ‣ §22.2 Definitions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="23px" altimg-valign="-7px" altimg-width="72px" alttext="\operatorname{sd}\left(\NVar{z},\NVar{k}\right)" display="inline"><mrow><mi href="./22.2#E7">sd</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r2">z</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobian elliptic function</a>,
<a href="./22.2#E4" title="(22.2.4) ‣ §22.2 Definitions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m65.png" altimg-height="23px" altimg-valign="-7px" altimg-width="72px" alttext="\operatorname{sn}\left(\NVar{z},\NVar{k}\right)" display="inline"><mrow><mi href="./22.2#E4">sn</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r2">z</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobian elliptic function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m45.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./22.1#p2.t1.r1" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./22.1#p2.t1.r1">x</mi></math>: real</a>,
<a href="./22.1#p2.t1.r3" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./22.1#p2.t1.r3">k</mi></math>: modulus</a> and
<a href="./22.1#p2.t1.r4" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="19px" altimg-valign="-2px" altimg-width="21px" alttext="k^{\prime}" display="inline"><msup><mi href="./22.1#p2.t1.r4">k</mi><mo>′</mo></msup></math>: complementary modulus</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">16.26.5</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E20">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">22.16.20</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m16.png" altimg-height="25px" altimg-valign="-7px" altimg-width="68px" alttext="\displaystyle\mathcal{E}\left(x,k\right)" display="inline"><mrow><mi class="ltx_font_mathcaligraphic" href="./22.16#E14">ℰ</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m14.png" altimg-height="53px" altimg-valign="-20px" altimg-width="386px" alttext="\displaystyle={k^{\prime}}^{2}\int_{0}^{x}{\operatorname{nd}^{2}}\left(t,k%
\right)\mathrm{d}t+k^{2}\operatorname{sn}\left(x,k\right)\operatorname{cd}%
\left(x,k\right)." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mmultiscripts><mi href="./22.1#p2.t1.r4">k</mi><none></none><mo>′</mo><none></none><mn>2</mn></mmultiscripts><mo></mo><mrow><mstyle displaystyle="true"><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi href="./22.1#p2.t1.r1">x</mi></msubsup></mstyle><mrow><mrow><msup><mi href="./22.2#E6">nd</mi><mn>2</mn></msup><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow><mo>+</mo><mrow><msup><mi href="./22.1#p2.t1.r3">k</mi><mn>2</mn></msup><mo></mo><mrow><mi href="./22.2#E4">sn</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./22.2#E8">cd</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E20.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./22.16#E14" title="(22.16.14) ‣ Integral Representations ‣ §22.16(ii) Jacobi’s Epsilon Function ‣ §22.16 Related Functions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m40.png" altimg-height="23px" altimg-valign="-7px" altimg-width="67px" alttext="\mathcal{E}\left(\NVar{x},\NVar{k}\right)" display="inline"><mrow><mi class="ltx_font_mathcaligraphic" href="./22.16#E14">ℰ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobi’s epsilon function</a>,
<a href="./22.2#E8" title="(22.2.8) ‣ §22.2 Definitions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="23px" altimg-valign="-7px" altimg-width="73px" alttext="\operatorname{cd}\left(\NVar{z},\NVar{k}\right)" display="inline"><mrow><mi href="./22.2#E8">cd</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r2">z</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobian elliptic function</a>,
<a href="./22.2#E6" title="(22.2.6) ‣ §22.2 Definitions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="23px" altimg-valign="-7px" altimg-width="75px" alttext="\operatorname{nd}\left(\NVar{z},\NVar{k}\right)" display="inline"><mrow><mi href="./22.2#E6">nd</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r2">z</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobian elliptic function</a>,
<a href="./22.2#E4" title="(22.2.4) ‣ §22.2 Definitions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m65.png" altimg-height="23px" altimg-valign="-7px" altimg-width="72px" alttext="\operatorname{sn}\left(\NVar{z},\NVar{k}\right)" display="inline"><mrow><mi href="./22.2#E4">sn</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r2">z</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobian elliptic function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m45.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./22.1#p2.t1.r1" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./22.1#p2.t1.r1">x</mi></math>: real</a>,
<a href="./22.1#p2.t1.r3" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./22.1#p2.t1.r3">k</mi></math>: modulus</a> and
<a href="./22.1#p2.t1.r4" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="19px" altimg-valign="-2px" altimg-width="21px" alttext="k^{\prime}" display="inline"><msup><mi href="./22.1#p2.t1.r4">k</mi><mo>′</mo></msup></math>: complementary modulus</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">16.26.6</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
</div>
<div id="Px9.p5" class="ltx_para">
<p class="ltx_p">In Equations (),
<math class="ltx_Math" altimg="m21.png" altimg-height="19px" altimg-valign="-4px" altimg-width="125px" alttext="-K<x<K." display="inline"><mrow><mrow><mrow><mo>-</mo><mrow><mi href="./19.2#E8">K</mi><mo></mo></mrow></mrow><mo><</mo><mi href="./22.1#p2.t1.r1">x</mi><mo><</mo><mrow><mi href="./19.2#E8">K</mi><mo></mo></mrow></mrow><mo>.</mo></mrow></math></p>
<table id="EGx4" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E21">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">22.16.21</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m16.png" altimg-height="25px" altimg-valign="-7px" altimg-width="68px" alttext="\displaystyle\mathcal{E}\left(x,k\right)" display="inline"><mrow><mi class="ltx_font_mathcaligraphic" href="./22.16#E14">ℰ</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m4.png" altimg-height="53px" altimg-valign="-20px" altimg-width="387px" alttext="\displaystyle=-\int_{0}^{x}{\operatorname{dc}^{2}}\left(t,k\right)\mathrm{d}t+%
x+\operatorname{sn}\left(x,k\right)\operatorname{dc}\left(x,k\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mo>-</mo><mrow><mstyle displaystyle="true"><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi href="./22.1#p2.t1.r1">x</mi></msubsup></mstyle><mrow><mrow><msup><mi href="./22.2#E8">dc</mi><mn>2</mn></msup><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow><mo>+</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>+</mo><mrow><mrow><mi href="./22.2#E4">sn</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./22.2#E8">dc</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E21.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./22.16#E14" title="(22.16.14) ‣ Integral Representations ‣ §22.16(ii) Jacobi’s Epsilon Function ‣ §22.16 Related Functions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m40.png" altimg-height="23px" altimg-valign="-7px" altimg-width="67px" alttext="\mathcal{E}\left(\NVar{x},\NVar{k}\right)" display="inline"><mrow><mi class="ltx_font_mathcaligraphic" href="./22.16#E14">ℰ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobi’s epsilon function</a>,
<a href="./22.2#E8" title="(22.2.8) ‣ §22.2 Definitions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="23px" altimg-valign="-7px" altimg-width="73px" alttext="\operatorname{dc}\left(\NVar{z},\NVar{k}\right)" display="inline"><mrow><mi href="./22.2#E8">dc</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r2">z</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobian elliptic function</a>,
<a href="./22.2#E4" title="(22.2.4) ‣ §22.2 Definitions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m65.png" altimg-height="23px" altimg-valign="-7px" altimg-width="72px" alttext="\operatorname{sn}\left(\NVar{z},\NVar{k}\right)" display="inline"><mrow><mi href="./22.2#E4">sn</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r2">z</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobian elliptic function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m45.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./22.1#p2.t1.r1" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./22.1#p2.t1.r1">x</mi></math>: real</a> and
<a href="./22.1#p2.t1.r3" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./22.1#p2.t1.r3">k</mi></math>: modulus</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">16.26.7</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E22">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">22.16.22</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m16.png" altimg-height="25px" altimg-valign="-7px" altimg-width="68px" alttext="\displaystyle\mathcal{E}\left(x,k\right)" display="inline"><mrow><mi class="ltx_font_mathcaligraphic" href="./22.16#E14">ℰ</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m7.png" altimg-height="53px" altimg-valign="-20px" altimg-width="438px" alttext="\displaystyle=-{k^{\prime}}^{2}\int_{0}^{x}{\operatorname{nc}^{2}}\left(t,k%
\right)\mathrm{d}t+{k^{\prime}}^{2}x+\operatorname{sn}\left(x,k\right)%
\operatorname{dc}\left(x,k\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mo>-</mo><mrow><mmultiscripts><mi href="./22.1#p2.t1.r4">k</mi><none></none><mo>′</mo><none></none><mn>2</mn></mmultiscripts><mo></mo><mrow><mstyle displaystyle="true"><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi href="./22.1#p2.t1.r1">x</mi></msubsup></mstyle><mrow><mrow><msup><mi href="./22.2#E5">nc</mi><mn>2</mn></msup><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow></mrow><mo>+</mo><mrow><mmultiscripts><mi href="./22.1#p2.t1.r4">k</mi><none></none><mo>′</mo><none></none><mn>2</mn></mmultiscripts><mo></mo><mi href="./22.1#p2.t1.r1">x</mi></mrow><mo>+</mo><mrow><mrow><mi href="./22.2#E4">sn</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./22.2#E8">dc</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E22.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./22.16#E14" title="(22.16.14) ‣ Integral Representations ‣ §22.16(ii) Jacobi’s Epsilon Function ‣ §22.16 Related Functions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m40.png" altimg-height="23px" altimg-valign="-7px" altimg-width="67px" alttext="\mathcal{E}\left(\NVar{x},\NVar{k}\right)" display="inline"><mrow><mi class="ltx_font_mathcaligraphic" href="./22.16#E14">ℰ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobi’s epsilon function</a>,
<a href="./22.2#E8" title="(22.2.8) ‣ §22.2 Definitions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="23px" altimg-valign="-7px" altimg-width="73px" alttext="\operatorname{dc}\left(\NVar{z},\NVar{k}\right)" display="inline"><mrow><mi href="./22.2#E8">dc</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r2">z</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobian elliptic function</a>,
<a href="./22.2#E5" title="(22.2.5) ‣ §22.2 Definitions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="23px" altimg-valign="-7px" altimg-width="73px" alttext="\operatorname{nc}\left(\NVar{z},\NVar{k}\right)" display="inline"><mrow><mi href="./22.2#E5">nc</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r2">z</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobian elliptic function</a>,
<a href="./22.2#E4" title="(22.2.4) ‣ §22.2 Definitions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m65.png" altimg-height="23px" altimg-valign="-7px" altimg-width="72px" alttext="\operatorname{sn}\left(\NVar{z},\NVar{k}\right)" display="inline"><mrow><mi href="./22.2#E4">sn</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r2">z</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobian elliptic function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m45.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./22.1#p2.t1.r1" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./22.1#p2.t1.r1">x</mi></math>: real</a>,
<a href="./22.1#p2.t1.r3" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./22.1#p2.t1.r3">k</mi></math>: modulus</a> and
<a href="./22.1#p2.t1.r4" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="19px" altimg-valign="-2px" altimg-width="21px" alttext="k^{\prime}" display="inline"><msup><mi href="./22.1#p2.t1.r4">k</mi><mo>′</mo></msup></math>: complementary modulus</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">16.26.8</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E23">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">22.16.23</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m16.png" altimg-height="25px" altimg-valign="-7px" altimg-width="68px" alttext="\displaystyle\mathcal{E}\left(x,k\right)" display="inline"><mrow><mi class="ltx_font_mathcaligraphic" href="./22.16#E14">ℰ</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m8.png" altimg-height="53px" altimg-valign="-20px" altimg-width="373px" alttext="\displaystyle=-{k^{\prime}}^{2}\int_{0}^{x}{\operatorname{sc}^{2}}\left(t,k%
\right)\mathrm{d}t+\operatorname{sn}\left(x,k\right)\operatorname{dc}\left(x,k%
\right)." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mo>-</mo><mrow><mmultiscripts><mi href="./22.1#p2.t1.r4">k</mi><none></none><mo>′</mo><none></none><mn>2</mn></mmultiscripts><mo></mo><mrow><mstyle displaystyle="true"><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi href="./22.1#p2.t1.r1">x</mi></msubsup></mstyle><mrow><mrow><msup><mi href="./22.2#E9">sc</mi><mn>2</mn></msup><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow></mrow><mo>+</mo><mrow><mrow><mi href="./22.2#E4">sn</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./22.2#E8">dc</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E23.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./22.16#E14" title="(22.16.14) ‣ Integral Representations ‣ §22.16(ii) Jacobi’s Epsilon Function ‣ §22.16 Related Functions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m40.png" altimg-height="23px" altimg-valign="-7px" altimg-width="67px" alttext="\mathcal{E}\left(\NVar{x},\NVar{k}\right)" display="inline"><mrow><mi class="ltx_font_mathcaligraphic" href="./22.16#E14">ℰ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobi’s epsilon function</a>,
<a href="./22.2#E8" title="(22.2.8) ‣ §22.2 Definitions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="23px" altimg-valign="-7px" altimg-width="73px" alttext="\operatorname{dc}\left(\NVar{z},\NVar{k}\right)" display="inline"><mrow><mi href="./22.2#E8">dc</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r2">z</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobian elliptic function</a>,
<a href="./22.2#E9" title="(22.2.9) ‣ §22.2 Definitions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m62.png" altimg-height="23px" altimg-valign="-7px" altimg-width="70px" alttext="\operatorname{sc}\left(\NVar{z},\NVar{k}\right)" display="inline"><mrow><mi href="./22.2#E9">sc</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r2">z</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobian elliptic function</a>,
<a href="./22.2#E4" title="(22.2.4) ‣ §22.2 Definitions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m65.png" altimg-height="23px" altimg-valign="-7px" altimg-width="72px" alttext="\operatorname{sn}\left(\NVar{z},\NVar{k}\right)" display="inline"><mrow><mi href="./22.2#E4">sn</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r2">z</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobian elliptic function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m45.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./22.1#p2.t1.r1" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./22.1#p2.t1.r1">x</mi></math>: real</a>,
<a href="./22.1#p2.t1.r3" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./22.1#p2.t1.r3">k</mi></math>: modulus</a> and
<a href="./22.1#p2.t1.r4" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="19px" altimg-valign="-2px" altimg-width="21px" alttext="k^{\prime}" display="inline"><msup><mi href="./22.1#p2.t1.r4">k</mi><mo>′</mo></msup></math>: complementary modulus</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">16.26.9</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
</div>
<div id="Px9.p6" class="ltx_para">
<p class="ltx_p">In Equations (),
<math class="ltx_Math" altimg="m20.png" altimg-height="19px" altimg-valign="-4px" altimg-width="141px" alttext="-2K<x<2K" display="inline"><mrow><mrow><mo>-</mo><mrow><mn>2</mn><mo></mo><mrow><mi href="./19.2#E8">K</mi><mo></mo></mrow></mrow></mrow><mo><</mo><mi href="./22.1#p2.t1.r1">x</mi><mo><</mo><mrow><mn>2</mn><mo></mo><mrow><mi href="./19.2#E8">K</mi><mo></mo></mrow></mrow></mrow></math>.</p>
<table id="EGx5" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E24">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">22.16.24</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m16.png" altimg-height="25px" altimg-valign="-7px" altimg-width="68px" alttext="\displaystyle\mathcal{E}\left(x,k\right)" display="inline"><mrow><mi class="ltx_font_mathcaligraphic" href="./22.16#E14">ℰ</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m3.png" altimg-height="53px" altimg-valign="-20px" altimg-width="514px" alttext="\displaystyle=-\int_{0}^{x}\left({\operatorname{ns}^{2}}\left(t,k\right)-t^{-2%
}\right)\mathrm{d}t+x^{-1}+x-\operatorname{cn}\left(x,k\right)\operatorname{ds%
}\left(x,k\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mrow><mo>-</mo><mrow><mstyle displaystyle="true"><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi href="./22.1#p2.t1.r1">x</mi></msubsup></mstyle><mrow><mrow><mo>(</mo><mrow><mrow><msup><mi href="./22.2#E4">ns</mi><mn>2</mn></msup><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo>-</mo><msup><mi>t</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></mrow><mo>)</mo></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow><mo>+</mo><msup><mi href="./22.1#p2.t1.r1">x</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo>+</mo><mi href="./22.1#p2.t1.r1">x</mi></mrow><mo>-</mo><mrow><mrow><mi href="./22.2#E5">cn</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./22.2#E7">ds</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E24.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./22.16#E14" title="(22.16.14) ‣ Integral Representations ‣ §22.16(ii) Jacobi’s Epsilon Function ‣ §22.16 Related Functions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m40.png" altimg-height="23px" altimg-valign="-7px" altimg-width="67px" alttext="\mathcal{E}\left(\NVar{x},\NVar{k}\right)" display="inline"><mrow><mi class="ltx_font_mathcaligraphic" href="./22.16#E14">ℰ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobi’s epsilon function</a>,
<a href="./22.2#E5" title="(22.2.5) ‣ §22.2 Definitions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="23px" altimg-valign="-7px" altimg-width="73px" alttext="\operatorname{cn}\left(\NVar{z},\NVar{k}\right)" display="inline"><mrow><mi href="./22.2#E5">cn</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r2">z</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobian elliptic function</a>,
<a href="./22.2#E7" title="(22.2.7) ‣ §22.2 Definitions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="23px" altimg-valign="-7px" altimg-width="72px" alttext="\operatorname{ds}\left(\NVar{z},\NVar{k}\right)" display="inline"><mrow><mi href="./22.2#E7">ds</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r2">z</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobian elliptic function</a>,
<a href="./22.2#E4" title="(22.2.4) ‣ §22.2 Definitions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m61.png" altimg-height="23px" altimg-valign="-7px" altimg-width="72px" alttext="\operatorname{ns}\left(\NVar{z},\NVar{k}\right)" display="inline"><mrow><mi href="./22.2#E4">ns</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r2">z</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobian elliptic function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m45.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./22.1#p2.t1.r1" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./22.1#p2.t1.r1">x</mi></math>: real</a> and
<a href="./22.1#p2.t1.r3" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./22.1#p2.t1.r3">k</mi></math>: modulus</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">16.26.10</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E25">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">22.16.25</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m16.png" altimg-height="25px" altimg-valign="-7px" altimg-width="68px" alttext="\displaystyle\mathcal{E}\left(x,k\right)" display="inline"><mrow><mi class="ltx_font_mathcaligraphic" href="./22.16#E14">ℰ</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m2.png" altimg-height="53px" altimg-valign="-20px" altimg-width="540px" alttext="\displaystyle=-\int_{0}^{x}\left({\operatorname{ds}^{2}}\left(t,k\right)-t^{-2%
}\right)\mathrm{d}t+x^{-1}+{k^{\prime}}^{2}x-\operatorname{cn}\left(x,k\right)%
\operatorname{ds}\left(x,k\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mrow><mo>-</mo><mrow><mstyle displaystyle="true"><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi href="./22.1#p2.t1.r1">x</mi></msubsup></mstyle><mrow><mrow><mo>(</mo><mrow><mrow><msup><mi href="./22.2#E7">ds</mi><mn>2</mn></msup><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo>-</mo><msup><mi>t</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></mrow><mo>)</mo></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow><mo>+</mo><msup><mi href="./22.1#p2.t1.r1">x</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo>+</mo><mrow><mmultiscripts><mi href="./22.1#p2.t1.r4">k</mi><none></none><mo>′</mo><none></none><mn>2</mn></mmultiscripts><mo></mo><mi href="./22.1#p2.t1.r1">x</mi></mrow></mrow><mo>-</mo><mrow><mrow><mi href="./22.2#E5">cn</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./22.2#E7">ds</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E25.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./22.16#E14" title="(22.16.14) ‣ Integral Representations ‣ §22.16(ii) Jacobi’s Epsilon Function ‣ §22.16 Related Functions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m40.png" altimg-height="23px" altimg-valign="-7px" altimg-width="67px" alttext="\mathcal{E}\left(\NVar{x},\NVar{k}\right)" display="inline"><mrow><mi class="ltx_font_mathcaligraphic" href="./22.16#E14">ℰ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobi’s epsilon function</a>,
<a href="./22.2#E5" title="(22.2.5) ‣ §22.2 Definitions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="23px" altimg-valign="-7px" altimg-width="73px" alttext="\operatorname{cn}\left(\NVar{z},\NVar{k}\right)" display="inline"><mrow><mi href="./22.2#E5">cn</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r2">z</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobian elliptic function</a>,
<a href="./22.2#E7" title="(22.2.7) ‣ §22.2 Definitions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="23px" altimg-valign="-7px" altimg-width="72px" alttext="\operatorname{ds}\left(\NVar{z},\NVar{k}\right)" display="inline"><mrow><mi href="./22.2#E7">ds</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r2">z</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobian elliptic function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m45.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./22.1#p2.t1.r1" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./22.1#p2.t1.r1">x</mi></math>: real</a>,
<a href="./22.1#p2.t1.r3" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./22.1#p2.t1.r3">k</mi></math>: modulus</a> and
<a href="./22.1#p2.t1.r4" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="19px" altimg-valign="-2px" altimg-width="21px" alttext="k^{\prime}" display="inline"><msup><mi href="./22.1#p2.t1.r4">k</mi><mo>′</mo></msup></math>: complementary modulus</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">16.26.11</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E26">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">22.16.26</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m16.png" altimg-height="25px" altimg-valign="-7px" altimg-width="68px" alttext="\displaystyle\mathcal{E}\left(x,k\right)" display="inline"><mrow><mi class="ltx_font_mathcaligraphic" href="./22.16#E14">ℰ</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m1.png" altimg-height="53px" altimg-valign="-20px" altimg-width="476px" alttext="\displaystyle=-\int_{0}^{x}\left({\operatorname{cs}^{2}}\left(t,k\right)-t^{-2%
}\right)\mathrm{d}t+x^{-1}-\operatorname{cn}\left(x,k\right)\operatorname{ds}%
\left(x,k\right)." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mrow><mo>-</mo><mrow><mstyle displaystyle="true"><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi href="./22.1#p2.t1.r1">x</mi></msubsup></mstyle><mrow><mrow><mo>(</mo><mrow><mrow><msup><mi href="./22.2#E9">cs</mi><mn>2</mn></msup><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo>-</mo><msup><mi>t</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></mrow><mo>)</mo></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow><mo>+</mo><msup><mi href="./22.1#p2.t1.r1">x</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></mrow><mo>-</mo><mrow><mrow><mi href="./22.2#E5">cn</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./22.2#E7">ds</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E26.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./22.16#E14" title="(22.16.14) ‣ Integral Representations ‣ §22.16(ii) Jacobi’s Epsilon Function ‣ §22.16 Related Functions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m40.png" altimg-height="23px" altimg-valign="-7px" altimg-width="67px" alttext="\mathcal{E}\left(\NVar{x},\NVar{k}\right)" display="inline"><mrow><mi class="ltx_font_mathcaligraphic" href="./22.16#E14">ℰ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobi’s epsilon function</a>,
<a href="./22.2#E5" title="(22.2.5) ‣ §22.2 Definitions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="23px" altimg-valign="-7px" altimg-width="73px" alttext="\operatorname{cn}\left(\NVar{z},\NVar{k}\right)" display="inline"><mrow><mi href="./22.2#E5">cn</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r2">z</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobian elliptic function</a>,
<a href="./22.2#E9" title="(22.2.9) ‣ §22.2 Definitions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="23px" altimg-valign="-7px" altimg-width="70px" alttext="\operatorname{cs}\left(\NVar{z},\NVar{k}\right)" display="inline"><mrow><mi href="./22.2#E9">cs</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r2">z</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobian elliptic function</a>,
<a href="./22.2#E7" title="(22.2.7) ‣ §22.2 Definitions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="23px" altimg-valign="-7px" altimg-width="72px" alttext="\operatorname{ds}\left(\NVar{z},\NVar{k}\right)" display="inline"><mrow><mi href="./22.2#E7">ds</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r2">z</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobian elliptic function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m45.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./22.1#p2.t1.r1" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./22.1#p2.t1.r1">x</mi></math>: real</a> and
<a href="./22.1#p2.t1.r3" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./22.1#p2.t1.r3">k</mi></math>: modulus</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">16.26.12</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
</div>
</section>
<section id="Px10" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Quasi-Addition and Quasi-Periodic Formulas</h3>
<div id="Px10.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px10.p1" class="ltx_para">
<table id="E27" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">22.16.27</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E27.png" altimg-height="28px" altimg-valign="-7px" altimg-width="657px" alttext="\mathcal{E}\left(x_{1}+x_{2},k\right)=\mathcal{E}\left(x_{1},k\right)+\mathcal%
{E}\left(x_{2},k\right)-k^{2}\operatorname{sn}\left(x_{1},k\right)%
\operatorname{sn}\left(x_{2},k\right)\operatorname{sn}\left(x_{1}+x_{2},k%
\right)," display="block"><mrow><mrow><mrow><mi class="ltx_font_mathcaligraphic" href="./22.16#E14">ℰ</mi><mo></mo><mrow><mo>(</mo><mrow><msub><mi href="./22.1#p2.t1.r1">x</mi><mn>1</mn></msub><mo>+</mo><msub><mi href="./22.1#p2.t1.r1">x</mi><mn>2</mn></msub></mrow><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mrow><mi class="ltx_font_mathcaligraphic" href="./22.16#E14">ℰ</mi><mo></mo><mrow><mo>(</mo><msub><mi href="./22.1#p2.t1.r1">x</mi><mn>1</mn></msub><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo>+</mo><mrow><mi class="ltx_font_mathcaligraphic" href="./22.16#E14">ℰ</mi><mo></mo><mrow><mo>(</mo><msub><mi href="./22.1#p2.t1.r1">x</mi><mn>2</mn></msub><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></mrow><mo>-</mo><mrow><msup><mi href="./22.1#p2.t1.r3">k</mi><mn>2</mn></msup><mo></mo><mrow><mi href="./22.2#E4">sn</mi><mo></mo><mrow><mo>(</mo><msub><mi href="./22.1#p2.t1.r1">x</mi><mn>1</mn></msub><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./22.2#E4">sn</mi><mo></mo><mrow><mo>(</mo><msub><mi href="./22.1#p2.t1.r1">x</mi><mn>2</mn></msub><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./22.2#E4">sn</mi><mo></mo><mrow><mo>(</mo><mrow><msub><mi href="./22.1#p2.t1.r1">x</mi><mn>1</mn></msub><mo>+</mo><msub><mi href="./22.1#p2.t1.r1">x</mi><mn>2</mn></msub></mrow><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E27.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./22.16#E14" title="(22.16.14) ‣ Integral Representations ‣ §22.16(ii) Jacobi’s Epsilon Function ‣ §22.16 Related Functions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m40.png" altimg-height="23px" altimg-valign="-7px" altimg-width="67px" alttext="\mathcal{E}\left(\NVar{x},\NVar{k}\right)" display="inline"><mrow><mi class="ltx_font_mathcaligraphic" href="./22.16#E14">ℰ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobi’s epsilon function</a>,
<a href="./22.2#E4" title="(22.2.4) ‣ §22.2 Definitions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m65.png" altimg-height="23px" altimg-valign="-7px" altimg-width="72px" alttext="\operatorname{sn}\left(\NVar{z},\NVar{k}\right)" display="inline"><mrow><mi href="./22.2#E4">sn</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r2">z</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobian elliptic function</a>,
<a href="./22.1#p2.t1.r1" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./22.1#p2.t1.r1">x</mi></math>: real</a> and
<a href="./22.1#p2.t1.r3" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./22.1#p2.t1.r3">k</mi></math>: modulus</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E28" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">22.16.28</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E28.png" altimg-height="28px" altimg-valign="-7px" altimg-width="468px" alttext="\mathcal{E}\left(x+K,k\right)=\mathcal{E}\left(x,k\right)+E\left(k\right)-k^{2%
}\operatorname{sn}\left(x,k\right)\operatorname{cd}\left(x,k\right)," display="block"><mrow><mrow><mrow><mi class="ltx_font_mathcaligraphic" href="./22.16#E14">ℰ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./22.1#p2.t1.r1">x</mi><mo>+</mo><mrow><mi href="./19.2#E8">K</mi><mo></mo></mrow></mrow><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mrow><mi class="ltx_font_mathcaligraphic" href="./22.16#E14">ℰ</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo>+</mo><mrow><mi href="./19.2#E8">E</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></mrow><mo>-</mo><mrow><msup><mi href="./22.1#p2.t1.r3">k</mi><mn>2</mn></msup><mo></mo><mrow><mi href="./22.2#E4">sn</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./22.2#E8">cd</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E28.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./22.16#E14" title="(22.16.14) ‣ Integral Representations ‣ §22.16(ii) Jacobi’s Epsilon Function ‣ §22.16 Related Functions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m40.png" altimg-height="23px" altimg-valign="-7px" altimg-width="67px" alttext="\mathcal{E}\left(\NVar{x},\NVar{k}\right)" display="inline"><mrow><mi class="ltx_font_mathcaligraphic" href="./22.16#E14">ℰ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobi’s epsilon function</a>,
<a href="./22.2#E8" title="(22.2.8) ‣ §22.2 Definitions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="23px" altimg-valign="-7px" altimg-width="73px" alttext="\operatorname{cd}\left(\NVar{z},\NVar{k}\right)" display="inline"><mrow><mi href="./22.2#E8">cd</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r2">z</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobian elliptic function</a>,
<a href="./22.2#E4" title="(22.2.4) ‣ §22.2 Definitions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m65.png" altimg-height="23px" altimg-valign="-7px" altimg-width="72px" alttext="\operatorname{sn}\left(\NVar{z},\NVar{k}\right)" display="inline"><mrow><mi href="./22.2#E4">sn</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r2">z</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobian elliptic function</a>,
<a href="./19.2#E8" title="(19.2.8) ‣ §19.2(ii) Legendre’s Integrals ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m31.png" altimg-height="23px" altimg-valign="-7px" altimg-width="52px" alttext="K\left(\NVar{k}\right)" display="inline"><mrow><mi href="./19.2#E8">K</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Legendre’s complete elliptic integral of the first kind</a>,
<a href="./19.2#E8" title="(19.2.8) ‣ §19.2(ii) Legendre’s Integrals ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m26.png" altimg-height="23px" altimg-valign="-7px" altimg-width="50px" alttext="E\left(\NVar{k}\right)" display="inline"><mrow><mi href="./19.2#E8">E</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Legendre’s complete elliptic integral of the second kind</a>,
<a href="./22.1#p2.t1.r1" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./22.1#p2.t1.r1">x</mi></math>: real</a> and
<a href="./22.1#p2.t1.r3" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./22.1#p2.t1.r3">k</mi></math>: modulus</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E29" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">22.16.29</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E29.png" altimg-height="25px" altimg-valign="-7px" altimg-width="294px" alttext="\mathcal{E}\left(x+2K,k\right)=\mathcal{E}\left(x,k\right)+2\!E\left(k\right)." display="block"><mrow><mrow><mrow><mi class="ltx_font_mathcaligraphic" href="./22.16#E14">ℰ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./22.1#p2.t1.r1">x</mi><mo>+</mo><mrow><mn>2</mn><mo></mo><mrow><mi href="./19.2#E8">K</mi><mo></mo></mrow></mrow></mrow><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mi class="ltx_font_mathcaligraphic" href="./22.16#E14">ℰ</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo>+</mo><mrow><mpadded width="-1.7pt"><mn>2</mn></mpadded><mo></mo><mrow><mi href="./19.2#E8">E</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E29.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./22.16#E14" title="(22.16.14) ‣ Integral Representations ‣ §22.16(ii) Jacobi’s Epsilon Function ‣ §22.16 Related Functions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m40.png" altimg-height="23px" altimg-valign="-7px" altimg-width="67px" alttext="\mathcal{E}\left(\NVar{x},\NVar{k}\right)" display="inline"><mrow><mi class="ltx_font_mathcaligraphic" href="./22.16#E14">ℰ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobi’s epsilon function</a>,
<a href="./19.2#E8" title="(19.2.8) ‣ §19.2(ii) Legendre’s Integrals ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m31.png" altimg-height="23px" altimg-valign="-7px" altimg-width="52px" alttext="K\left(\NVar{k}\right)" display="inline"><mrow><mi href="./19.2#E8">K</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Legendre’s complete elliptic integral of the first kind</a>,
<a href="./19.2#E8" title="(19.2.8) ‣ §19.2(ii) Legendre’s Integrals ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m26.png" altimg-height="23px" altimg-valign="-7px" altimg-width="50px" alttext="E\left(\NVar{k}\right)" display="inline"><mrow><mi href="./19.2#E8">E</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Legendre’s complete elliptic integral of the second kind</a>,
<a href="./22.1#p2.t1.r1" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./22.1#p2.t1.r1">x</mi></math>: real</a> and
<a href="./22.1#p2.t1.r3" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./22.1#p2.t1.r3">k</mi></math>: modulus</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">For <math class="ltx_Math" altimg="m28.png" altimg-height="23px" altimg-valign="-7px" altimg-width="50px" alttext="E\left(k\right)" display="inline"><mrow><mi href="./19.2#E8">E</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math> see §</dd>
</dl>
</div>
</div>
<div id="Px11.p1" class="ltx_para">
<table id="E30" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">22.16.30</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E30.png" altimg-height="53px" altimg-valign="-21px" altimg-width="419px" alttext="\mathcal{E}\left(x,k\right)=\frac{1}{{\theta_{3}^{2}}\left(0,q\right)\theta_{4%
}\left(\xi,q\right)}\frac{\mathrm{d}}{\mathrm{d}\xi}\theta_{4}\left(\xi,q%
\right)+\frac{E\left(k\right)}{K\left(k\right)}x," display="block"><mrow><mrow><mrow><mi class="ltx_font_mathcaligraphic" href="./22.16#E14">ℰ</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mfrac><mn>1</mn><mrow><mrow><msubsup><mi href="./20.2#i">θ</mi><mn>3</mn><mn>2</mn></msubsup><mo></mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi href="./22.2#E1">q</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./20.2#i">θ</mi><mn>4</mn></msub><mo></mo><mrow><mo>(</mo><mi>ξ</mi><mo>,</mo><mi href="./22.2#E1">q</mi><mo>)</mo></mrow></mrow></mrow></mfrac><mo></mo><mrow><mfrac><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi>ξ</mi></mrow></mfrac><mo></mo><mrow><msub><mi href="./20.2#i">θ</mi><mn>4</mn></msub><mo></mo><mrow><mo>(</mo><mi>ξ</mi><mo>,</mo><mi href="./22.2#E1">q</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>+</mo><mrow><mfrac><mrow><mi href="./19.2#E8">E</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow><mrow><mi href="./19.2#E8">K</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></mfrac><mo></mo><mi href="./22.1#p2.t1.r1">x</mi></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E30.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./22.16#E14" title="(22.16.14) ‣ Integral Representations ‣ §22.16(ii) Jacobi’s Epsilon Function ‣ §22.16 Related Functions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m40.png" altimg-height="23px" altimg-valign="-7px" altimg-width="67px" alttext="\mathcal{E}\left(\NVar{x},\NVar{k}\right)" display="inline"><mrow><mi class="ltx_font_mathcaligraphic" href="./22.16#E14">ℰ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobi’s epsilon function</a>,
<a href="./20.2#i" title="§20.2(i) Fourier Series ‣ §20.2 Definitions and Periodic Properties ‣ Properties ‣ Chapter 20 Theta Functions" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="24px" altimg-valign="-8px" altimg-width="69px" alttext="\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)" display="inline"><mrow><msub><mi href="./20.2#i">θ</mi><mi class="ltx_nvar">j</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./20.1#p2.t1.r3">z</mi><mo>,</mo><mi class="ltx_nvar" href="./20.1#p2.t1.r4">q</mi><mo>)</mo></mrow></mrow></math>: theta function</a>,
<a href="./19.2#E8" title="(19.2.8) ‣ §19.2(ii) Legendre’s Integrals ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m31.png" altimg-height="23px" altimg-valign="-7px" altimg-width="52px" alttext="K\left(\NVar{k}\right)" display="inline"><mrow><mi href="./19.2#E8">K</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Legendre’s complete elliptic integral of the first kind</a>,
<a href="./19.2#E8" title="(19.2.8) ‣ §19.2(ii) Legendre’s Integrals ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m26.png" altimg-height="23px" altimg-valign="-7px" altimg-width="50px" alttext="E\left(\NVar{k}\right)" display="inline"><mrow><mi href="./19.2#E8">E</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Legendre’s complete elliptic integral of the second kind</a>,
<a href="./1.4#E4" title="(1.4.4) ‣ §1.4(iii) Derivatives ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m36.png" altimg-height="29px" altimg-valign="-9px" altimg-width="27px" alttext="\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: derivative of <math class="ltx_Math" altimg="m74.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./22.2#E1" title="(22.2.1) ‣ §22.2 Definitions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m80.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="q" display="inline"><mi href="./22.2#E1">q</mi></math>: nome</a>,
<a href="./22.1#p2.t1.r1" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./22.1#p2.t1.r1">x</mi></math>: real</a> and
<a href="./22.1#p2.t1.r3" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./22.1#p2.t1.r3">k</mi></math>: modulus</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m72.png" altimg-height="25px" altimg-valign="-7px" altimg-width="128px" alttext="\xi=x/{\theta_{3}^{2}}\left(0,q\right)" display="inline"><mrow><mi>ξ</mi><mo>=</mo><mrow><mi href="./22.1#p2.t1.r1">x</mi><mo>/</mo><mrow><msubsup><mi href="./20.2#i">θ</mi><mn>3</mn><mn>2</mn></msubsup><mo></mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi href="./22.2#E1">q</mi><mo>)</mo></mrow></mrow></mrow></mrow></math>. For <math class="ltx_Math" altimg="m71.png" altimg-height="23px" altimg-valign="-8px" altimg-width="22px" alttext="\theta_{j}" display="inline"><msub><mi href="./20.2#i">θ</mi><mi>j</mi></msub></math> see §. For <math class="ltx_Math" altimg="m28.png" altimg-height="23px" altimg-valign="-7px" altimg-width="50px" alttext="E\left(k\right)" display="inline"><mrow><mi href="./19.2#E8">E</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math> see §.</p>
</div>
</section>
<section id="Px12" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Relation to the Elliptic Integral <math class="ltx_Math" altimg="m27.png" altimg-height="23px" altimg-valign="-7px" altimg-width="71px" alttext="E\left(\phi,k\right)" display="inline"><mrow><mi href="./19.2#E5">E</mi><mo></mo><mrow><mo>(</mo><mi>ϕ</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>
</h3>
<div id="Px12.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px12.p1" class="ltx_para">
<table id="E31" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">22.16.31</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E31.png" altimg-height="25px" altimg-valign="-7px" altimg-width="238px" alttext="E\left(\operatorname{am}\left(x,k\right),k\right)=\mathcal{E}\left(x,k\right)," display="block"><mrow><mrow><mrow><mi href="./19.2#E5">E</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./22.16#E1">am</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mi class="ltx_font_mathcaligraphic" href="./22.16#E14">ℰ</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m23.png" altimg-height="20px" altimg-valign="-5px" altimg-width="121px" alttext="-K\leq x\leq K" display="inline"><mrow><mrow><mo>-</mo><mrow><mi href="./19.2#E8">K</mi><mo></mo></mrow></mrow><mo>≤</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>≤</mo><mrow><mi href="./19.2#E8">K</mi><mo></mo></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E31.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./22.16#E14" title="(22.16.14) ‣ Integral Representations ‣ §22.16(ii) Jacobi’s Epsilon Function ‣ §22.16 Related Functions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m40.png" altimg-height="23px" altimg-valign="-7px" altimg-width="67px" alttext="\mathcal{E}\left(\NVar{x},\NVar{k}\right)" display="inline"><mrow><mi class="ltx_font_mathcaligraphic" href="./22.16#E14">ℰ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobi’s epsilon function</a>,
<a href="./22.16#E1" title="(22.16.1) ‣ Definition ‣ §22.16(i) Jacobi’s Amplitude ( am ) Function ‣ §22.16 Related Functions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="23px" altimg-valign="-7px" altimg-width="81px" alttext="\operatorname{am}\left(\NVar{x},\NVar{k}\right)" display="inline"><mrow><mi href="./22.16#E1">am</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobi’s amplitude function</a>,
<a href="./19.2#E8" title="(19.2.8) ‣ §19.2(ii) Legendre’s Integrals ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m31.png" altimg-height="23px" altimg-valign="-7px" altimg-width="52px" alttext="K\left(\NVar{k}\right)" display="inline"><mrow><mi href="./19.2#E8">K</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Legendre’s complete elliptic integral of the first kind</a>,
<a href="./19.2#E5" title="(19.2.5) ‣ §19.2(ii) Legendre’s Integrals ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m25.png" altimg-height="23px" altimg-valign="-7px" altimg-width="71px" alttext="E\left(\NVar{\phi},\NVar{k}\right)" display="inline"><mrow><mi href="./19.2#E5">E</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r2">ϕ</mi><mo>,</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Legendre’s incomplete elliptic integral of the second kind</a>,
<a href="./22.1#p2.t1.r1" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./22.1#p2.t1.r1">x</mi></math>: real</a> and
<a href="./22.1#p2.t1.r3" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./22.1#p2.t1.r3">k</mi></math>: modulus</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">For <math class="ltx_Math" altimg="m27.png" altimg-height="23px" altimg-valign="-7px" altimg-width="71px" alttext="E\left(\phi,k\right)" display="inline"><mrow><mi href="./19.2#E5">E</mi><mo></mo><mrow><mo>(</mo><mi>ϕ</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math> see §</dd>
</dl>
</div>
</div>
<div id="Px13.p1" class="ltx_para">
<p class="ltx_p">With <math class="ltx_Math" altimg="m28.png" altimg-height="23px" altimg-valign="-7px" altimg-width="50px" alttext="E\left(k\right)" display="inline"><mrow><mi href="./19.2#E8">E</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math> and <math class="ltx_Math" altimg="m32.png" altimg-height="23px" altimg-valign="-7px" altimg-width="52px" alttext="K\left(k\right)" display="inline"><mrow><mi href="./19.2#E8">K</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math> as in § and
<math class="ltx_Math" altimg="m82.png" altimg-height="18px" altimg-valign="-3px" altimg-width="54px" alttext="x\in\mathbb{R}" display="inline"><mrow><mi href="./22.1#p2.t1.r1">x</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mi href="./front/introduction#Sx4.p2.t1.r14">ℝ</mi></mrow></math>,</p>
<table id="E32" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">22.16.32</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E32.png" altimg-height="25px" altimg-valign="-7px" altimg-width="318px" alttext="\mathrm{Z}\left(x|k\right)=\mathcal{E}\left(x,k\right)-(E\left(k\right)/K\left%
(k\right))x." display="block"><mrow><mrow><mrow><mi href="./22.16#E32" mathvariant="normal">Z</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo stretchy="false">|</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mi class="ltx_font_mathcaligraphic" href="./22.16#E14">ℰ</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo>-</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mi href="./19.2#E8">E</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo>/</mo><mrow><mi href="./19.2#E8">K</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi href="./22.1#p2.t1.r1">x</mi></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E32.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m43.png" altimg-height="23px" altimg-valign="-7px" altimg-width="63px" alttext="\mathrm{Z}\left(\NVar{x}|\NVar{k}\right)" display="inline"><mrow><mi href="./22.16#E32" mathvariant="normal">Z</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r1">x</mi><mo stretchy="false">|</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobi’s zeta function</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./22.16#E14" title="(22.16.14) ‣ Integral Representations ‣ §22.16(ii) Jacobi’s Epsilon Function ‣ §22.16 Related Functions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m40.png" altimg-height="23px" altimg-valign="-7px" altimg-width="67px" alttext="\mathcal{E}\left(\NVar{x},\NVar{k}\right)" display="inline"><mrow><mi class="ltx_font_mathcaligraphic" href="./22.16#E14">ℰ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobi’s epsilon function</a>,
<a href="./19.2#E8" title="(19.2.8) ‣ §19.2(ii) Legendre’s Integrals ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m31.png" altimg-height="23px" altimg-valign="-7px" altimg-width="52px" alttext="K\left(\NVar{k}\right)" display="inline"><mrow><mi href="./19.2#E8">K</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Legendre’s complete elliptic integral of the first kind</a>,
<a href="./19.2#E8" title="(19.2.8) ‣ §19.2(ii) Legendre’s Integrals ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m26.png" altimg-height="23px" altimg-valign="-7px" altimg-width="50px" alttext="E\left(\NVar{k}\right)" display="inline"><mrow><mi href="./19.2#E8">E</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Legendre’s complete elliptic integral of the second kind</a>,
<a href="./22.1#p2.t1.r1" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./22.1#p2.t1.r1">x</mi></math>: real</a> and
<a href="./22.1#p2.t1.r3" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./22.1#p2.t1.r3">k</mi></math>: modulus</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">See Figure . (Sometimes in the literature
<math class="ltx_Math" altimg="m44.png" altimg-height="23px" altimg-valign="-7px" altimg-width="63px" alttext="\mathrm{Z}\left(x|k\right)" display="inline"><mrow><mi href="./22.16#E32" mathvariant="normal">Z</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo stretchy="false">|</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math> is denoted by <math class="ltx_Math" altimg="m42.png" altimg-height="25px" altimg-valign="-7px" altimg-width="141px" alttext="\mathrm{Z}(\operatorname{am}\left(x,k\right),k^{2})" display="inline"><mrow><mi href="./22.16#E32" mathvariant="normal">Z</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./22.16#E1">am</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo>,</mo><msup><mi href="./22.1#p2.t1.r3">k</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow></mrow></math>.)</p>
</div>
</section>
<section id="Px14" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Properties</h3>
<div id="Px14.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px14.p1" class="ltx_para">
<p class="ltx_p"><math class="ltx_Math" altimg="m44.png" altimg-height="23px" altimg-valign="-7px" altimg-width="63px" alttext="\mathrm{Z}\left(x|k\right)" display="inline"><mrow><mi href="./22.16#E32" mathvariant="normal">Z</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo stretchy="false">|</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math> satisfies the same quasi-addition formula as the function
<math class="ltx_Math" altimg="m41.png" altimg-height="23px" altimg-valign="-7px" altimg-width="67px" alttext="\mathcal{E}\left(x,k\right)" display="inline"><mrow><mi class="ltx_font_mathcaligraphic" href="./22.16#E14">ℰ</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>, given by (). Also,</p>
<table id="E33" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">22.16.33</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E33.png" altimg-height="28px" altimg-valign="-7px" altimg-width="392px" alttext="\mathrm{Z}\left(x+K|k\right)=\mathrm{Z}\left(x|k\right)-k^{2}\operatorname{sn}%
\left(x,k\right)\operatorname{cd}\left(x,k\right)," display="block"><mrow><mrow><mrow><mi href="./22.16#E32" mathvariant="normal">Z</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./22.1#p2.t1.r1">x</mi><mo>+</mo><mrow><mi href="./19.2#E8">K</mi><mo></mo></mrow></mrow><mo stretchy="false">|</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mi href="./22.16#E32" mathvariant="normal">Z</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo stretchy="false">|</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo>-</mo><mrow><msup><mi href="./22.1#p2.t1.r3">k</mi><mn>2</mn></msup><mo></mo><mrow><mi href="./22.2#E4">sn</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./22.2#E8">cd</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E33.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./22.16#E32" title="(22.16.32) ‣ Definition ‣ §22.16(iii) Jacobi’s Zeta Function ‣ §22.16 Related Functions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="23px" altimg-valign="-7px" altimg-width="63px" alttext="\mathrm{Z}\left(\NVar{x}|\NVar{k}\right)" display="inline"><mrow><mi href="./22.16#E32" mathvariant="normal">Z</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r1">x</mi><mo stretchy="false">|</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobi’s zeta function</a>,
<a href="./22.2#E8" title="(22.2.8) ‣ §22.2 Definitions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="23px" altimg-valign="-7px" altimg-width="73px" alttext="\operatorname{cd}\left(\NVar{z},\NVar{k}\right)" display="inline"><mrow><mi href="./22.2#E8">cd</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r2">z</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobian elliptic function</a>,
<a href="./22.2#E4" title="(22.2.4) ‣ §22.2 Definitions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m65.png" altimg-height="23px" altimg-valign="-7px" altimg-width="72px" alttext="\operatorname{sn}\left(\NVar{z},\NVar{k}\right)" display="inline"><mrow><mi href="./22.2#E4">sn</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r2">z</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobian elliptic function</a>,
<a href="./19.2#E8" title="(19.2.8) ‣ §19.2(ii) Legendre’s Integrals ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m31.png" altimg-height="23px" altimg-valign="-7px" altimg-width="52px" alttext="K\left(\NVar{k}\right)" display="inline"><mrow><mi href="./19.2#E8">K</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Legendre’s complete elliptic integral of the first kind</a>,
<a href="./22.1#p2.t1.r1" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./22.1#p2.t1.r1">x</mi></math>: real</a> and
<a href="./22.1#p2.t1.r3" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./22.1#p2.t1.r3">k</mi></math>: modulus</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E34" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">22.16.34</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E34.png" altimg-height="25px" altimg-valign="-7px" altimg-width="212px" alttext="\mathrm{Z}\left(x+2K|k\right)=\mathrm{Z}\left(x|k\right)." display="block"><mrow><mrow><mrow><mi href="./22.16#E32" mathvariant="normal">Z</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./22.1#p2.t1.r1">x</mi><mo>+</mo><mrow><mn>2</mn><mo></mo><mrow><mi href="./19.2#E8">K</mi><mo></mo></mrow></mrow></mrow><mo stretchy="false">|</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mi href="./22.16#E32" mathvariant="normal">Z</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo stretchy="false">|</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E34.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./22.16#E32" title="(22.16.32) ‣ Definition ‣ §22.16(iii) Jacobi’s Zeta Function ‣ §22.16 Related Functions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="23px" altimg-valign="-7px" altimg-width="63px" alttext="\mathrm{Z}\left(\NVar{x}|\NVar{k}\right)" display="inline"><mrow><mi href="./22.16#E32" mathvariant="normal">Z</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r1">x</mi><mo stretchy="false">|</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobi’s zeta function</a>,
<a href="./19.2#E8" title="(19.2.8) ‣ §19.2(ii) Legendre’s Integrals ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m31.png" altimg-height="23px" altimg-valign="-7px" altimg-width="52px" alttext="K\left(\NVar{k}\right)" display="inline"><mrow><mi href="./19.2#E8">K</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Legendre’s complete elliptic integral of the first kind</a>,
<a href="./22.1#p2.t1.r1" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./22.1#p2.t1.r1">x</mi></math>: real</a> and
<a href="./22.1#p2.t1.r3" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./22.1#p2.t1.r3">k</mi></math>: modulus</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
</section>
<section id="iv" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§22.16(iv) </span>Graphs</h2>
<div id="SS4.info" class="ltx_metadata ltx_info">
<figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 22.16.1: </span>Jacobi’s amplitude function <math class="ltx_Math" altimg="m49.png" altimg-height="23px" altimg-valign="-7px" altimg-width="81px" alttext="\operatorname{am}\left(x,k\right)" display="inline"><mrow><mi href="./22.16#E1">am</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>
for <math class="ltx_Math" altimg="m24.png" altimg-height="19px" altimg-valign="-5px" altimg-width="111px" alttext="0\leq x\leq 10\pi" display="inline"><mrow><mn>0</mn><mo>≤</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>≤</mo><mrow><mn>10</mn><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow></math>
and <math class="ltx_Math" altimg="m76.png" altimg-height="21px" altimg-valign="-6px" altimg-width="230px" alttext="k=0.4,0.7,0.99,0.999999" display="inline"><mrow><mi href="./22.1#p2.t1.r3">k</mi><mo>=</mo><mrow><mn>0.4</mn><mo>,</mo><mn>0.7</mn><mo>,</mo><mn>0.99</mn><mo>,</mo><mn>0.999999</mn></mrow></mrow></math>.
Values of <math class="ltx_Math" altimg="m77.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./22.1#p2.t1.r3">k</mi></math> greater than 1 are illustrated in
Figure <div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./22.16#E1" title="(22.16.1) ‣ Definition ‣ §22.16(i) Jacobi’s Amplitude ( am ) Function ‣ §22.16 Related Functions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="23px" altimg-valign="-7px" altimg-width="81px" alttext="\operatorname{am}\left(\NVar{x},\NVar{k}\right)" display="inline"><mrow><mi href="./22.16#E1">am</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobi’s amplitude function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m67.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./22.1#p2.t1.r1" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./22.1#p2.t1.r1">x</mi></math>: real</a> and
<a href="./22.1#p2.t1.r3" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./22.1#p2.t1.r3">k</mi></math>: modulus</a>
</dd>
</dl>
</div>
</div>
</figure>
<figure id="F2" class="ltx_figure"><figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 22.16.2: </span>Jacobi’s epsilon function <math class="ltx_Math" altimg="m41.png" altimg-height="23px" altimg-valign="-7px" altimg-width="67px" alttext="\mathcal{E}\left(x,k\right)" display="inline"><mrow><mi class="ltx_font_mathcaligraphic" href="./22.16#E14">ℰ</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>
for <math class="ltx_Math" altimg="m24.png" altimg-height="19px" altimg-valign="-5px" altimg-width="111px" alttext="0\leq x\leq 10\pi" display="inline"><mrow><mn>0</mn><mo>≤</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>≤</mo><mrow><mn>10</mn><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow></math>
and <math class="ltx_Math" altimg="m76.png" altimg-height="21px" altimg-valign="-6px" altimg-width="230px" alttext="k=0.4,0.7,0.99,0.999999" display="inline"><mrow><mi href="./22.1#p2.t1.r3">k</mi><mo>=</mo><mrow><mn>0.4</mn><mo>,</mo><mn>0.7</mn><mo>,</mo><mn>0.99</mn><mo>,</mo><mn>0.999999</mn></mrow></mrow></math>.
(These graphs are similar to those in Figure ), and the graphs of
<math class="ltx_Math" altimg="m55.png" altimg-height="23px" altimg-valign="-7px" altimg-width="76px" alttext="\operatorname{dn}\left(x,k\right)" display="inline"><mrow><mi href="./22.2#E6">dn</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math> in §<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./22.16#E14" title="(22.16.14) ‣ Integral Representations ‣ §22.16(ii) Jacobi’s Epsilon Function ‣ §22.16 Related Functions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m40.png" altimg-height="23px" altimg-valign="-7px" altimg-width="67px" alttext="\mathcal{E}\left(\NVar{x},\NVar{k}\right)" display="inline"><mrow><mi class="ltx_font_mathcaligraphic" href="./22.16#E14">ℰ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobi’s epsilon function</a>,
<a href="./22.2#E6" title="(22.2.6) ‣ §22.2 Definitions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="23px" altimg-valign="-7px" altimg-width="75px" alttext="\operatorname{dn}\left(\NVar{z},\NVar{k}\right)" display="inline"><mrow><mi href="./22.2#E6">dn</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r2">z</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobian elliptic function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m67.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./22.1#p2.t1.r1" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./22.1#p2.t1.r1">x</mi></math>: real</a> and
<a href="./22.1#p2.t1.r3" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./22.1#p2.t1.r3">k</mi></math>: modulus</a>
</dd>
</dl>
</div>
</div>
</figure>
<figure id="F3" class="ltx_figure"><figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 22.16.3: </span>Jacobi’s zeta function <math class="ltx_Math" altimg="m44.png" altimg-height="23px" altimg-valign="-7px" altimg-width="63px" alttext="\mathrm{Z}\left(x|k\right)" display="inline"><mrow><mi href="./22.16#E32" mathvariant="normal">Z</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo stretchy="false">|</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>
for <math class="ltx_Math" altimg="m24.png" altimg-height="19px" altimg-valign="-5px" altimg-width="111px" alttext="0\leq x\leq 10\pi" display="inline"><mrow><mn>0</mn><mo>≤</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>≤</mo><mrow><mn>10</mn><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow></math>
and <math class="ltx_Math" altimg="m76.png" altimg-height="21px" altimg-valign="-6px" altimg-width="230px" alttext="k=0.4,0.7,0.99,0.999999" display="inline"><mrow><mi href="./22.1#p2.t1.r3">k</mi><mo>=</mo><mrow><mn>0.4</mn><mo>,</mo><mn>0.7</mn><mo>,</mo><mn>0.99</mn><mo>,</mo><mn>0.999999</mn></mrow></mrow></math>.
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./22.16#E32" title="(22.16.32) ‣ Definition ‣ §22.16(iii) Jacobi’s Zeta Function ‣ §22.16 Related Functions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="23px" altimg-valign="-7px" altimg-width="63px" alttext="\mathrm{Z}\left(\NVar{x}|\NVar{k}\right)" display="inline"><mrow><mi href="./22.16#E32" mathvariant="normal">Z</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r1">x</mi><mo stretchy="false">|</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobi’s zeta function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m67.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./22.1#p2.t1.r1" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./22.1#p2.t1.r1">x</mi></math>: real</a> and
<a href="./22.1#p2.t1.r3" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./22.1#p2.t1.r3">k</mi></math>: modulus</a>
</dd>
</dl>
</div>
</div>
</figure>
</section>
</section>
</div>
<div class="ltx_page_footer">
<span></div>
</div>
</body></text>
</html>
</page>
<page>
<!DOCTYPE html><html>
<head>
<title>DLMF: 22.19 Physical Applications</title>
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<div class="ltx_page_navlogo">, <a class="ltx_keyword" href="./idx/#amplitudeamfunction">amplitude (<math class="ltx_Math" altimg="m35.png" altimg-height="13px" altimg-valign="-2px" altimg-width="31px" alttext="\operatorname{am}" display="inline"><mi href="./22.16#E1">am</mi></math>) function</a>, </dd>
</dl>
</div>
</div>
<div id="SS1.p1" class="ltx_para">
<p class="ltx_p">With appropriate scalings, Newton’s equation of motion for a pendulum with a
mass in a gravitational field constrained to move in a vertical plane at a
fixed distance from a fulcrum is</p>
<table id="E1" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">22.19.1</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E1.png" altimg-height="51px" altimg-valign="-18px" altimg-width="175px" alttext="\frac{{\mathrm{d}}^{2}\theta(t)}{{\mathrm{d}t}^{2}}=-\sin\theta(t)," display="block"><mrow><mrow><mfrac><mrow><mpadded lspace="-1.7pt" width="-4.5pt"><msup><mo href="./1.4#E4">d</mo><mn>2</mn></msup></mpadded><mrow><mi href="./22.19#SS1.p1">θ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><msup><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi>t</mi></mrow><mn>2</mn></msup></mfrac><mo>=</mo><mrow><mo>-</mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mi href="./22.19#SS1.p1">θ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#E4" title="(1.4.4) ‣ §1.4(iii) Derivatives ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m31.png" altimg-height="29px" altimg-valign="-9px" altimg-width="27px" alttext="\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: derivative of <math class="ltx_Math" altimg="m64.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m45.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a> and
<a href="./22.19#SS1.p1" title="§22.19(i) Classical Dynamics: The Pendulum ‣ §22.19 Physical Applications ‣ Applications ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="23px" altimg-valign="-7px" altimg-width="37px" alttext="\theta(t)" display="inline"><mrow><mi href="./22.19#SS1.p1">θ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math>: angular displacement</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p"><math class="ltx_Math" altimg="m55.png" altimg-height="18px" altimg-valign="-2px" altimg-width="14px" alttext="\theta" display="inline"><mi href="./22.19#SS1.p1">θ</mi></math> being the angular displacement from the point of stable equilibrium,
<math class="ltx_Math" altimg="m53.png" altimg-height="18px" altimg-valign="-2px" altimg-width="51px" alttext="\theta=0" display="inline"><mrow><mi href="./22.19#SS1.p1">θ</mi><mo>=</mo><mn>0</mn></mrow></math>. The bounded <math class="ltx_Math" altimg="m1.png" altimg-height="23px" altimg-valign="-7px" altimg-width="122px" alttext="(-\pi\leq\theta\leq\pi)" display="inline"><mrow><mo stretchy="false">(</mo><mrow><mrow><mo>-</mo><mi href="./3.12#E1">π</mi></mrow><mo>≤</mo><mi href="./22.19#SS1.p1">θ</mi><mo>≤</mo><mi href="./3.12#E1">π</mi></mrow><mo stretchy="false">)</mo></mrow></math> oscillatory solution of
() is traditionally written</p>
<table id="E2" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">22.19.2</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E2.png" altimg-height="53px" altimg-valign="-21px" altimg-width="391px" alttext="\sin\left(\tfrac{1}{2}\theta(t)\right)=\sin\left(\frac{1}{2}\alpha\right)%
\operatorname{sn}\left(t+K,\sin\left(\tfrac{1}{2}\alpha\right)\right)," display="block"><mrow><mrow><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mrow><mi href="./22.19#SS1.p1">θ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./22.19#SS1.p1">α</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./22.2#E4">sn</mi><mo></mo><mrow><mo>(</mo><mrow><mi>t</mi><mo>+</mo><mrow><mi href="./19.2#E8">K</mi><mo></mo></mrow></mrow><mo>,</mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./22.19#SS1.p1">α</mi></mrow><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./22.2#E4" title="(22.2.4) ‣ §22.2 Definitions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="23px" altimg-valign="-7px" altimg-width="72px" alttext="\operatorname{sn}\left(\NVar{z},\NVar{k}\right)" display="inline"><mrow><mi href="./22.2#E4">sn</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r2">z</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobian elliptic function</a>,
<a href="./19.2#E8" title="(19.2.8) ‣ §19.2(ii) Legendre’s Integrals ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m21.png" altimg-height="23px" altimg-valign="-7px" altimg-width="52px" alttext="K\left(\NVar{k}\right)" display="inline"><mrow><mi href="./19.2#E8">K</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Legendre’s complete elliptic integral of the first kind</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m45.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./22.19#SS1.p1" title="§22.19(i) Classical Dynamics: The Pendulum ‣ §22.19 Physical Applications ‣ Applications ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="23px" altimg-valign="-7px" altimg-width="37px" alttext="\theta(t)" display="inline"><mrow><mi href="./22.19#SS1.p1">θ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math>: angular displacement</a> and
<a href="./22.19#SS1.p1" title="§22.19(i) Classical Dynamics: The Pendulum ‣ §22.19 Physical Applications ‣ Applications ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m27.png" altimg-height="13px" altimg-valign="-2px" altimg-width="17px" alttext="\alpha" display="inline"><mi href="./22.19#SS1.p1">α</mi></math>: initial displacement</a>
</dd>
<dt>Errata (effective with 1.0.8):</dt>
<dd>
Originally the first argument to the function <math class="ltx_Math" altimg="m42.png" altimg-height="13px" altimg-valign="-2px" altimg-width="23px" alttext="\operatorname{sn}" display="inline"><mi href="./22.2#E4">sn</mi></math> was given incorrectly
as <math class="ltx_Math" altimg="m79.png" altimg-height="17px" altimg-valign="-2px" altimg-width="11px" alttext="t" display="inline"><mi>t</mi></math>. The correct argument is <math class="ltx_Math" altimg="m77.png" altimg-height="19px" altimg-valign="-4px" altimg-width="54px" alttext="t+K" display="inline"><mrow><mi>t</mi><mo>+</mo><mrow><mi href="./19.2#E8">K</mi><mo></mo></mrow></mrow></math>.
<p><span class="ltx_font_italic">Reported 2014-03-05 by Svante Janson</span></p>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">for an initial angular displacement <math class="ltx_Math" altimg="m27.png" altimg-height="13px" altimg-valign="-2px" altimg-width="17px" alttext="\alpha" display="inline"><mi href="./22.19#SS1.p1">α</mi></math>, with <math class="ltx_Math" altimg="m33.png" altimg-height="23px" altimg-valign="-7px" altimg-width="95px" alttext="\ifrac{\mathrm{d}\theta}{\mathrm{d}t}=0" display="inline"><mrow><mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi href="./22.19#SS1.p1">θ</mi></mrow><mo href="./1.4#E4">/</mo><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi>t</mi></mrow></mrow><mo>=</mo><mn>0</mn></mrow></math> at
time <math class="ltx_Math" altimg="m4.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math>; see <cite class="ltx_cite ltx_citemacro_citet">Lawden (, pp. 114–117)</cite>. The period is
<math class="ltx_Math" altimg="m11.png" altimg-height="27px" altimg-valign="-9px" altimg-width="122px" alttext="4\!K\left(\sin\left(\frac{1}{2}\alpha\right)\right)" display="inline"><mrow><mpadded width="-1.7pt"><mn>4</mn></mpadded><mo></mo><mrow><mi href="./19.2#E8">K</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./22.19#SS1.p1">α</mi></mrow><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow></mrow></math>. The periodicity and symmetry of the
pendulum imply that the motion in each four
intervals <math class="ltx_Math" altimg="m56.png" altimg-height="23px" altimg-valign="-7px" altimg-width="101px" alttext="\theta\in(0,\pm\alpha)" display="inline"><mrow><mi href="./22.19#SS1.p1">θ</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mn>0</mn><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mrow><mo>±</mo><mi href="./22.19#SS1.p1">α</mi></mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></mrow></math> and <math class="ltx_Math" altimg="m57.png" altimg-height="23px" altimg-valign="-7px" altimg-width="101px" alttext="\theta\in(\pm\alpha,0)" display="inline"><mrow><mi href="./22.19#SS1.p1">θ</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mrow><mo>±</mo><mi href="./22.19#SS1.p1">α</mi></mrow><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mn>0</mn><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></mrow></math> have the same “quarter periods”
<math class="ltx_Math" altimg="m20.png" altimg-height="27px" altimg-valign="-9px" altimg-width="161px" alttext="K=K\left(\sin\left(\frac{1}{2}\alpha\right)\right)" display="inline"><mrow><mrow><mi href="./19.2#E8">K</mi><mo></mo></mrow><mo>=</mo><mrow><mi href="./19.2#E8">K</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./22.19#SS1.p1">α</mi></mrow><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow></mrow></math>. Thus the offset <math class="ltx_Math" altimg="m77.png" altimg-height="19px" altimg-valign="-4px" altimg-width="54px" alttext="t+K" display="inline"><mrow><mi>t</mi><mo>+</mo><mrow><mi href="./19.2#E8">K</mi><mo></mo></mrow></mrow></math> in as the motion starts <math class="ltx_Math" altimg="m51.png" altimg-height="23px" altimg-valign="-7px" altimg-width="79px" alttext="\theta(0)=\alpha" display="inline"><mrow><mrow><mi href="./22.19#SS1.p1">θ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mi href="./22.19#SS1.p1">α</mi></mrow></math>, rather than <math class="ltx_Math" altimg="m50.png" altimg-height="23px" altimg-valign="-7px" altimg-width="76px" alttext="\theta(0)=0" display="inline"><mrow><mrow><mi href="./22.19#SS1.p1">θ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mn>0</mn></mrow></math> as in ,
which follows. The angle <math class="ltx_Math" altimg="m26.png" altimg-height="13px" altimg-valign="-2px" altimg-width="56px" alttext="\alpha=\pi" display="inline"><mrow><mi href="./22.19#SS1.p1">α</mi><mo>=</mo><mi href="./3.12#E1">π</mi></mrow></math> is a
<em class="ltx_emph ltx_font_italic">separatrix</em>,
separating oscillatory and unbounded motion. With the same
initial conditions, if the sign of gravity is reversed then the new period is
<math class="ltx_Math" altimg="m16.png" altimg-height="27px" altimg-valign="-9px" altimg-width="128px" alttext="4\!{K^{\prime}}\left(\sin\left(\frac{1}{2}\alpha\right)\right)" display="inline"><mrow><mpadded width="-1.7pt"><mn>4</mn></mpadded><mo></mo><mrow><msup><mi href="./19.2#E9">K</mi><mo href="./19.2#E9">′</mo></msup><mo></mo><mrow><mo>(</mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./22.19#SS1.p1">α</mi></mrow><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow></mrow></math>; see <cite class="ltx_cite ltx_citemacro_citet">Whittaker ()</cite> writes:</p>
<table id="E3" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">22.19.3</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E3.png" altimg-height="41px" altimg-valign="-15px" altimg-width="268px" alttext="\theta(t)=2\operatorname{am}\left(t\sqrt{E/2},\sqrt{2/E}\right)," display="block"><mrow><mrow><mrow><mi href="./22.19#SS1.p1">θ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mn>2</mn><mo></mo><mrow><mi href="./22.16#E1">am</mi><mo></mo><mrow><mo>(</mo><mrow><mi>t</mi><mo></mo><msqrt><mrow><mi href="./22.19#SS1.p2">E</mi><mo>/</mo><mn>2</mn></mrow></msqrt></mrow><mo>,</mo><msqrt><mrow><mn>2</mn><mo>/</mo><mi href="./22.19#SS1.p2">E</mi></mrow></msqrt><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./22.16#E1" title="(22.16.1) ‣ Definition ‣ §22.16(i) Jacobi’s Amplitude ( am ) Function ‣ §22.16 Related Functions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m36.png" altimg-height="23px" altimg-valign="-7px" altimg-width="81px" alttext="\operatorname{am}\left(\NVar{x},\NVar{k}\right)" display="inline"><mrow><mi href="./22.16#E1">am</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobi’s amplitude function</a>,
<a href="./22.19#SS1.p1" title="§22.19(i) Classical Dynamics: The Pendulum ‣ §22.19 Physical Applications ‣ Applications ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="23px" altimg-valign="-7px" altimg-width="37px" alttext="\theta(t)" display="inline"><mrow><mi href="./22.19#SS1.p1">θ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math>: angular displacement</a> and
<a href="./22.19#SS1.p2" title="§22.19(i) Classical Dynamics: The Pendulum ‣ §22.19 Physical Applications ‣ Applications ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="E" display="inline"><mi href="./22.19#SS1.p2">E</mi></math>: energy</a>
</dd>
<dt>Errata (effective with 1.0.8):</dt>
<dd>
Originally the first argument to the function <math class="ltx_Math" altimg="m35.png" altimg-height="13px" altimg-valign="-2px" altimg-width="31px" alttext="\operatorname{am}" display="inline"><mi href="./22.16#E1">am</mi></math> was given incorrectly
as <math class="ltx_Math" altimg="m79.png" altimg-height="17px" altimg-valign="-2px" altimg-width="11px" alttext="t" display="inline"><mi>t</mi></math>. The correct argument is <math class="ltx_Math" altimg="m80.png" altimg-height="28px" altimg-valign="-8px" altimg-width="67px" alttext="t\sqrt{E/2}" display="inline"><mrow><mi>t</mi><mo></mo><msqrt><mrow><mi href="./22.19#SS1.p2">E</mi><mo>/</mo><mn>2</mn></mrow></msqrt></mrow></math>.
<p><span class="ltx_font_italic">Reported 2014-03-05 by Svante Janson</span></p>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">for the initial conditions <math class="ltx_Math" altimg="m50.png" altimg-height="23px" altimg-valign="-7px" altimg-width="76px" alttext="\theta(0)=0" display="inline"><mrow><mrow><mi href="./22.19#SS1.p1">θ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mn>0</mn></mrow></math>, the point of stable equilibrium for
<math class="ltx_Math" altimg="m17.png" altimg-height="18px" altimg-valign="-2px" altimg-width="57px" alttext="E=0" display="inline"><mrow><mi href="./22.19#SS1.p2">E</mi><mo>=</mo><mn>0</mn></mrow></math>, and <math class="ltx_Math" altimg="m32.png" altimg-height="27px" altimg-valign="-7px" altimg-width="150px" alttext="\ifrac{\mathrm{d}\theta(t)}{\mathrm{d}t}=\sqrt{2E}" display="inline"><mrow><mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mrow><mi href="./22.19#SS1.p1">θ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo href="./1.4#E4">/</mo><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi>t</mi></mrow></mrow><mo>=</mo><msqrt><mrow><mn>2</mn><mo></mo><mi href="./22.19#SS1.p2">E</mi></mrow></msqrt></mrow></math>. Here
<math class="ltx_Math" altimg="m18.png" altimg-height="27px" altimg-valign="-9px" altimg-width="282px" alttext="E=\frac{1}{2}(\ifrac{\mathrm{d}\theta(t)}{\mathrm{d}t})^{2}+1-\cos\theta(t)" display="inline"><mrow><mi href="./22.19#SS1.p2">E</mi><mo>=</mo><mrow><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mrow><mi href="./22.19#SS1.p1">θ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo href="./1.4#E4">/</mo><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi>t</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow><mo>+</mo><mn>1</mn></mrow><mo>-</mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mi href="./22.19#SS1.p1">θ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow></mrow></math> is the
<em class="ltx_emph ltx_font_italic">energy</em>, which is a first integral of the motion.
This formulation gives the bounded and unbounded solutions from the same
formula (), for <math class="ltx_Math" altimg="m73.png" altimg-height="20px" altimg-valign="-5px" altimg-width="52px" alttext="k\geq 1" display="inline"><mrow><mi href="./22.1#p2.t1.r3">k</mi><mo>≥</mo><mn>1</mn></mrow></math> and <math class="ltx_Math" altimg="m74.png" altimg-height="20px" altimg-valign="-5px" altimg-width="52px" alttext="k\leq 1" display="inline"><mrow><mi href="./22.1#p2.t1.r3">k</mi><mo>≤</mo><mn>1</mn></mrow></math>, respectively.
Also, <math class="ltx_Math" altimg="m52.png" altimg-height="23px" altimg-valign="-7px" altimg-width="37px" alttext="\theta(t)" display="inline"><mrow><mi href="./22.19#SS1.p1">θ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math> is not restricted to the principal range <math class="ltx_Math" altimg="m2.png" altimg-height="20px" altimg-valign="-5px" altimg-width="107px" alttext="-\pi\leq\theta\leq\pi" display="inline"><mrow><mrow><mo>-</mo><mi href="./3.12#E1">π</mi></mrow><mo>≤</mo><mi href="./22.19#SS1.p1">θ</mi><mo>≤</mo><mi href="./3.12#E1">π</mi></mrow></math>. Figure shows the nature of the solutions
<math class="ltx_Math" altimg="m52.png" altimg-height="23px" altimg-valign="-7px" altimg-width="37px" alttext="\theta(t)" display="inline"><mrow><mi href="./22.19#SS1.p1">θ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math> of () by graphing <math class="ltx_Math" altimg="m37.png" altimg-height="23px" altimg-valign="-7px" altimg-width="81px" alttext="\operatorname{am}\left(x,k\right)" display="inline"><mrow><mi href="./22.16#E1">am</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math> for both
<math class="ltx_Math" altimg="m5.png" altimg-height="20px" altimg-valign="-5px" altimg-width="88px" alttext="0\leq k\leq 1" display="inline"><mrow><mn>0</mn><mo>≤</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>≤</mo><mn>1</mn></mrow></math>, as in Figure , and <math class="ltx_Math" altimg="m73.png" altimg-height="20px" altimg-valign="-5px" altimg-width="52px" alttext="k\geq 1" display="inline"><mrow><mi href="./22.1#p2.t1.r3">k</mi><mo>≥</mo><mn>1</mn></mrow></math>, where it
is periodic.</p>
</div>
<figure id="F1" class="ltx_figure"><figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 22.19.1: </span>Jacobi’s amplitude function <math class="ltx_Math" altimg="m37.png" altimg-height="23px" altimg-valign="-7px" altimg-width="81px" alttext="\operatorname{am}\left(x,k\right)" display="inline"><mrow><mi href="./22.16#E1">am</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>
for <math class="ltx_Math" altimg="m6.png" altimg-height="19px" altimg-valign="-5px" altimg-width="111px" alttext="0\leq x\leq 10\pi" display="inline"><mrow><mn>0</mn><mo>≤</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>≤</mo><mrow><mn>10</mn><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow></math>
and <math class="ltx_Math" altimg="m66.png" altimg-height="21px" altimg-valign="-6px" altimg-width="215px" alttext="k=0.5,0.9999,1.0001,2" display="inline"><mrow><mi href="./22.1#p2.t1.r3">k</mi><mo>=</mo><mrow><mn>0.5</mn><mo>,</mo><mn>0.9999</mn><mo>,</mo><mn>1.0001</mn><mo>,</mo><mn>2</mn></mrow></mrow></math>.
When <math class="ltx_Math" altimg="m65.png" altimg-height="18px" altimg-valign="-3px" altimg-width="52px" alttext="k<1" display="inline"><mrow><mi href="./22.1#p2.t1.r3">k</mi><mo><</mo><mn>1</mn></mrow></math>, <math class="ltx_Math" altimg="m37.png" altimg-height="23px" altimg-valign="-7px" altimg-width="81px" alttext="\operatorname{am}\left(x,k\right)" display="inline"><mrow><mi href="./22.16#E1">am</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math> increases monotonically indicating
that the motion of the pendulum is unbounded in <math class="ltx_Math" altimg="m55.png" altimg-height="18px" altimg-valign="-2px" altimg-width="14px" alttext="\theta" display="inline"><mi href="./22.19#SS1.p1">θ</mi></math>,
corresponding to free rotation about the fulcrum; compare Figure
. As <math class="ltx_Math" altimg="m75.png" altimg-height="19px" altimg-valign="-4px" altimg-width="72px" alttext="k\to 1-" display="inline"><mrow><mi href="./22.1#p2.t1.r3">k</mi><mo>→</mo><mrow><mn>1</mn><mo>-</mo></mrow></mrow></math>, plateaus are seen as the motion
approaches the separatrix where <math class="ltx_Math" altimg="m54.png" altimg-height="18px" altimg-valign="-2px" altimg-width="65px" alttext="\theta=n\pi" display="inline"><mrow><mi href="./22.19#SS1.p1">θ</mi><mo>=</mo><mrow><mi>n</mi><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow></math>,
<math class="ltx_Math" altimg="m76.png" altimg-height="20px" altimg-valign="-6px" altimg-width="135px" alttext="n=\pm 1,\pm 2,\ldots" display="inline"><mrow><mi>n</mi><mo>=</mo><mrow><mrow><mo>±</mo><mn>1</mn></mrow><mo>,</mo><mrow><mo>±</mo><mn>2</mn></mrow><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>, at which points the motion is time
independent for <math class="ltx_Math" altimg="m70.png" altimg-height="18px" altimg-valign="-2px" altimg-width="52px" alttext="k=1" display="inline"><mrow><mi href="./22.1#p2.t1.r3">k</mi><mo>=</mo><mn>1</mn></mrow></math>. This corresponds to the pendulum being
“upside down” at a point of unstable equilibrium.
For <math class="ltx_Math" altimg="m71.png" altimg-height="18px" altimg-valign="-3px" altimg-width="52px" alttext="k>1" display="inline"><mrow><mi href="./22.1#p2.t1.r3">k</mi><mo>></mo><mn>1</mn></mrow></math>, the motion is periodic in <math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./22.1#p2.t1.r1">x</mi></math>, corresponding to bounded
oscillatory motion.
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./22.16#E1" title="(22.16.1) ‣ Definition ‣ §22.16(i) Jacobi’s Amplitude ( am ) Function ‣ §22.16 Related Functions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m36.png" altimg-height="23px" altimg-valign="-7px" altimg-width="81px" alttext="\operatorname{am}\left(\NVar{x},\NVar{k}\right)" display="inline"><mrow><mi href="./22.16#E1">am</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobi’s amplitude function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./22.1#p2.t1.r1" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./22.1#p2.t1.r1">x</mi></math>: real</a>,
<a href="./22.1#p2.t1.r3" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./22.1#p2.t1.r3">k</mi></math>: modulus</a> and
<a href="./22.19#SS1.p1" title="§22.19(i) Classical Dynamics: The Pendulum ‣ §22.19 Physical Applications ‣ Applications ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="23px" altimg-valign="-7px" altimg-width="37px" alttext="\theta(t)" display="inline"><mrow><mi href="./22.19#SS1.p1">θ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math>: angular displacement</a>
</dd>
</dl>
</div>
</div>
</figure>
</section>
<section id="ii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§22.19(ii) </span>Classical Dynamics: The Quartic Oscillator</h2>
<div id="SS2.info" class="ltx_metadata ltx_info">
,
<a href="./errata/#Sx9.I15.ix3" title="Paragraph Case III: = V ( x ) + - 1 2 x 2 1 4 β x 4 in §22.19(ii) ‣ Version 1.0.9 (August 29, 2014) ‣ Errata" class="ltx_ref"><span class="ltx_text ltx_ref_tag">Paragraph <span class="ltx_text ltx_ref_title">Case III: <math class="ltx_Math" altimg="m22.png" altimg-height="27px" altimg-valign="-9px" altimg-width="192px" alttext="V(x)=-\frac{1}{2}x^{2}+\frac{1}{4}\beta x^{4}" display="inline"><mrow><mrow><mi href="./22.19#SS2.p1">V</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./22.19#SS2.p1">x</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><msup><mi href="./22.19#SS2.p1">x</mi><mn>2</mn></msup></mrow></mrow><mo>+</mo><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac><mo></mo><mi href="./22.19#SS2.p2">β</mi><mo></mo><msup><mi href="./22.19#SS2.p1">x</mi><mn>4</mn></msup></mrow></mrow></mrow></math></span> in §<span class="ltx_text ltx_ref_tag">22.19(ii)</span></span></a>
</dd>
</dl>
</div>
</div>
<div id="SS2.p1" class="ltx_para">
<p class="ltx_p">Classical motion in one dimension is described by Newton’s equation</p>
<table id="E4" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">22.19.4</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E4.png" altimg-height="51px" altimg-valign="-18px" altimg-width="171px" alttext="\frac{{\mathrm{d}}^{2}x(t)}{{\mathrm{d}t}^{2}}=-\frac{\mathrm{d}V(x)}{\mathrm{%
d}x}," display="block"><mrow><mrow><mfrac><mrow><mpadded lspace="-1.7pt" width="-4.5pt"><msup><mo href="./1.4#E4">d</mo><mn>2</mn></msup></mpadded><mrow><mi href="./22.19#SS2.p1">x</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./22.19#SS2.p1">t</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><msup><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi href="./22.19#SS2.p1">t</mi></mrow><mn>2</mn></msup></mfrac><mo>=</mo><mrow><mo>-</mo><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mrow><mi href="./22.19#SS2.p1">V</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./22.19#SS2.p1">x</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi href="./22.19#SS2.p1">x</mi></mrow></mfrac></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E4.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#E4" title="(1.4.4) ‣ §1.4(iii) Derivatives ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m31.png" altimg-height="29px" altimg-valign="-9px" altimg-width="27px" alttext="\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: derivative of <math class="ltx_Math" altimg="m64.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./22.19#SS2.p1" title="§22.19(ii) Classical Dynamics: The Quartic Oscillator ‣ §22.19 Physical Applications ‣ Applications ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m25.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="V(x)" display="inline"><mrow><mi href="./22.19#SS2.p1">V</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./22.19#SS2.p1">x</mi><mo stretchy="false">)</mo></mrow></mrow></math>: potential energy</a>,
<a href="./22.19#SS2.p1" title="§22.19(ii) Classical Dynamics: The Quartic Oscillator ‣ §22.19 Physical Applications ‣ Applications ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="23px" altimg-valign="-7px" altimg-width="38px" alttext="x(t)" display="inline"><mrow><mi href="./22.19#SS2.p1">x</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./22.19#SS2.p1">t</mi><mo stretchy="false">)</mo></mrow></mrow></math>: coordinate</a> and
<a href="./22.19#SS2.p1" title="§22.19(ii) Classical Dynamics: The Quartic Oscillator ‣ §22.19 Physical Applications ‣ Applications ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m79.png" altimg-height="17px" altimg-valign="-2px" altimg-width="11px" alttext="t" display="inline"><mi href="./22.19#SS2.p1">t</mi></math>: time</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m25.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="V(x)" display="inline"><mrow><mi href="./22.19#SS2.p1">V</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./22.19#SS2.p1">x</mi><mo stretchy="false">)</mo></mrow></mrow></math> is the potential energy, and <math class="ltx_Math" altimg="m82.png" altimg-height="23px" altimg-valign="-7px" altimg-width="38px" alttext="x(t)" display="inline"><mrow><mi href="./22.19#SS2.p1">x</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./22.19#SS2.p1">t</mi><mo stretchy="false">)</mo></mrow></mrow></math> is the coordinate as a
function of time <math class="ltx_Math" altimg="m79.png" altimg-height="17px" altimg-valign="-2px" altimg-width="11px" alttext="t" display="inline"><mi href="./22.19#SS2.p1">t</mi></math>. The potential</p>
<table id="E5" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">22.19.5</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E5.png" altimg-height="30px" altimg-valign="-9px" altimg-width="194px" alttext="V(x)=\pm\tfrac{1}{2}x^{2}\pm\tfrac{1}{4}\beta x^{4}" display="block"><mrow><mrow><mi href="./22.19#SS2.p1">V</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./22.19#SS2.p1">x</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mo>±</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><msup><mi href="./22.19#SS2.p1">x</mi><mn>2</mn></msup></mrow></mrow><mo>±</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>4</mn></mfrac></mstyle><mo></mo><mi href="./22.19#SS2.p2">β</mi><mo></mo><msup><mi href="./22.19#SS2.p1">x</mi><mn>4</mn></msup></mrow></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E5.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./22.19#SS2.p1" title="§22.19(ii) Classical Dynamics: The Quartic Oscillator ‣ §22.19 Physical Applications ‣ Applications ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m25.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="V(x)" display="inline"><mrow><mi href="./22.19#SS2.p1">V</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./22.19#SS2.p1">x</mi><mo stretchy="false">)</mo></mrow></mrow></math>: potential energy</a>,
<a href="./22.19#SS2.p1" title="§22.19(ii) Classical Dynamics: The Quartic Oscillator ‣ §22.19 Physical Applications ‣ Applications ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="23px" altimg-valign="-7px" altimg-width="38px" alttext="x(t)" display="inline"><mrow><mi href="./22.19#SS2.p1">x</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./22.19#SS2.p1">t</mi><mo stretchy="false">)</mo></mrow></mrow></math>: coordinate</a> and
<a href="./22.19#SS2.p2" title="§22.19(ii) Classical Dynamics: The Quartic Oscillator ‣ §22.19 Physical Applications ‣ Applications ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m28.png" altimg-height="21px" altimg-valign="-6px" altimg-width="17px" alttext="\beta" display="inline"><mi href="./22.19#SS2.p2">β</mi></math>: real positive</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">plays a prototypal role in classical mechanics
(<cite class="ltx_cite ltx_citemacro_citet">Lawden ()</cite>).</p>
</div>
<div id="SS2.p2" class="ltx_para">
<p class="ltx_p">For <math class="ltx_Math" altimg="m28.png" altimg-height="21px" altimg-valign="-6px" altimg-width="17px" alttext="\beta" display="inline"><mi href="./22.19#SS2.p2">β</mi></math> real and positive, three of the four possible combinations of signs
give rise to bounded oscillatory motions. We consider the case of a particle of
mass 1, initially held at rest at displacement <math class="ltx_Math" altimg="m59.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./22.19#SS2.p2">a</mi></math> from the origin and then
released at time <math class="ltx_Math" altimg="m78.png" altimg-height="17px" altimg-valign="-2px" altimg-width="48px" alttext="t=0" display="inline"><mrow><mi href="./22.19#SS2.p1">t</mi><mo>=</mo><mn>0</mn></mrow></math>. The subsequent position as a function of time,
<math class="ltx_Math" altimg="m82.png" altimg-height="23px" altimg-valign="-7px" altimg-width="38px" alttext="x(t)" display="inline"><mrow><mi href="./22.19#SS2.p1">x</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./22.19#SS2.p1">t</mi><mo stretchy="false">)</mo></mrow></mrow></math>, for the three cases is given with results expressed in terms of <math class="ltx_Math" altimg="m59.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./22.19#SS2.p2">a</mi></math> and
the dimensionless parameter <math class="ltx_Math" altimg="m29.png" altimg-height="27px" altimg-valign="-9px" altimg-width="86px" alttext="\eta=\frac{1}{2}\beta a^{2}" display="inline"><mrow><mi href="./22.19#SS2.p2">η</mi><mo>=</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./22.19#SS2.p2">β</mi><mo></mo><msup><mi href="./22.19#SS2.p2">a</mi><mn>2</mn></msup></mrow></mrow></math>.</p>
</div>
<section id="Px1" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Case I: <math class="ltx_Math" altimg="m23.png" altimg-height="27px" altimg-valign="-9px" altimg-width="177px" alttext="V(x)=\frac{1}{2}x^{2}+\frac{1}{4}\beta x^{4}" display="inline"><mrow><mrow><mi href="./22.19#SS2.p1">V</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./22.19#SS2.p1">x</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><msup><mi href="./22.19#SS2.p1">x</mi><mn>2</mn></msup></mrow><mo>+</mo><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac><mo></mo><mi href="./22.19#SS2.p2">β</mi><mo></mo><msup><mi href="./22.19#SS2.p1">x</mi><mn>4</mn></msup></mrow></mrow></mrow></math>
</h3>
<div id="Px1.info" class="ltx_metadata ltx_info">
, pp. 117–119)</cite>. The subsequent time evolution is always
oscillatory with period <math class="ltx_Math" altimg="m12.png" altimg-height="25px" altimg-valign="-7px" altimg-width="144px" alttext="4\!K\left(k\right)/\sqrt{1+2\eta}" display="inline"><mrow><mrow><mpadded width="-1.7pt"><mn>4</mn></mpadded><mo></mo><mrow><mi href="./19.2#E8">K</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></mrow><mo>/</mo><msqrt><mrow><mn>1</mn><mo>+</mo><mrow><mn>2</mn><mo></mo><mi href="./22.19#SS2.p2">η</mi></mrow></mrow></msqrt></mrow></math> and modulus <math class="ltx_Math" altimg="m67.png" altimg-height="28px" altimg-valign="-8px" altimg-width="148px" alttext="k=1/\sqrt{2+\eta^{-1}}" display="inline"><mrow><mi href="./22.1#p2.t1.r3">k</mi><mo>=</mo><mrow><mn>1</mn><mo>/</mo><msqrt><mrow><mn>2</mn><mo>+</mo><msup><mi href="./22.19#SS2.p2">η</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></mrow></msqrt></mrow></mrow></math>:</p>
<table id="E6" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">22.19.6</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E6.png" altimg-height="41px" altimg-valign="-15px" altimg-width="238px" alttext="x(t)=a\operatorname{cn}\left(t\sqrt{1+2\eta},k\right)." display="block"><mrow><mrow><mrow><mi href="./22.19#SS2.p1">x</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./22.19#SS2.p1">t</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mi href="./22.19#SS2.p2">a</mi><mo></mo><mrow><mi href="./22.2#E5">cn</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./22.19#SS2.p1">t</mi><mo></mo><msqrt><mrow><mn>1</mn><mo>+</mo><mrow><mn>2</mn><mo></mo><mi href="./22.19#SS2.p2">η</mi></mrow></mrow></msqrt></mrow><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E6.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./22.2#E5" title="(22.2.5) ‣ §22.2 Definitions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="23px" altimg-valign="-7px" altimg-width="73px" alttext="\operatorname{cn}\left(\NVar{z},\NVar{k}\right)" display="inline"><mrow><mi href="./22.2#E5">cn</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r2">z</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobian elliptic function</a>,
<a href="./22.1#p2.t1.r3" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./22.1#p2.t1.r3">k</mi></math>: modulus</a>,
<a href="./22.19#SS2.p1" title="§22.19(ii) Classical Dynamics: The Quartic Oscillator ‣ §22.19 Physical Applications ‣ Applications ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="23px" altimg-valign="-7px" altimg-width="38px" alttext="x(t)" display="inline"><mrow><mi href="./22.19#SS2.p1">x</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./22.19#SS2.p1">t</mi><mo stretchy="false">)</mo></mrow></mrow></math>: coordinate</a>,
<a href="./22.19#SS2.p1" title="§22.19(ii) Classical Dynamics: The Quartic Oscillator ‣ §22.19 Physical Applications ‣ Applications ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m79.png" altimg-height="17px" altimg-valign="-2px" altimg-width="11px" alttext="t" display="inline"><mi href="./22.19#SS2.p1">t</mi></math>: time</a>,
<a href="./22.19#SS2.p2" title="§22.19(ii) Classical Dynamics: The Quartic Oscillator ‣ §22.19 Physical Applications ‣ Applications ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./22.19#SS2.p2">a</mi></math>: displacement</a> and
<a href="./22.19#SS2.p2" title="§22.19(ii) Classical Dynamics: The Quartic Oscillator ‣ §22.19 Physical Applications ‣ Applications ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m30.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="\eta" display="inline"><mi href="./22.19#SS2.p2">η</mi></math>: parameter</a>
</dd>
<dt>Errata (effective with 1.0.9):</dt>
<dd>
Originally the term <math class="ltx_Math" altimg="m46.png" altimg-height="24px" altimg-valign="-7px" altimg-width="76px" alttext="\sqrt{1+2\eta}" display="inline"><msqrt><mrow><mn>1</mn><mo>+</mo><mrow><mn>2</mn><mo></mo><mi href="./22.19#SS2.p2">η</mi></mrow></mrow></msqrt></math> was given incorrectly as <math class="ltx_Math" altimg="m47.png" altimg-height="24px" altimg-valign="-7px" altimg-width="66px" alttext="\sqrt{1+\eta}" display="inline"><msqrt><mrow><mn>1</mn><mo>+</mo><mi href="./22.19#SS2.p2">η</mi></mrow></msqrt></math>
in this equation and in the line above.
Additionally, for improved clarity, the modulus <math class="ltx_Math" altimg="m72.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./22.1#p2.t1.r3">k</mi></math> has been defined in the line above.
Previously <math class="ltx_Math" altimg="m7.png" altimg-height="28px" altimg-valign="-8px" altimg-width="110px" alttext="1/\sqrt{2+\eta^{-1}}" display="inline"><mrow><mn>1</mn><mo>/</mo><msqrt><mrow><mn>2</mn><mo>+</mo><msup><mi href="./22.19#SS2.p2">η</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></mrow></msqrt></mrow></math> was given explicitly in the equation.
<p><span class="ltx_font_italic">Reported 2014-04-28 by Svante Janson</span></p>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px2" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Case II: <math class="ltx_Math" altimg="m24.png" altimg-height="27px" altimg-valign="-9px" altimg-width="177px" alttext="V(x)=\frac{1}{2}x^{2}-\frac{1}{4}\beta x^{4}" display="inline"><mrow><mrow><mi href="./22.19#SS2.p1">V</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./22.19#SS2.p1">x</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><msup><mi href="./22.19#SS2.p1">x</mi><mn>2</mn></msup></mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac><mo></mo><mi href="./22.19#SS2.p2">β</mi><mo></mo><msup><mi href="./22.19#SS2.p1">x</mi><mn>4</mn></msup></mrow></mrow></mrow></math>
</h3>
<div id="Px2.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px2.p1" class="ltx_para">
<p class="ltx_p">There is bounded oscillatory motion near <math class="ltx_Math" altimg="m83.png" altimg-height="17px" altimg-valign="-2px" altimg-width="52px" alttext="x=0" display="inline"><mrow><mi href="./22.19#SS2.p1">x</mi><mo>=</mo><mn>0</mn></mrow></math>, with period
<math class="ltx_Math" altimg="m13.png" altimg-height="25px" altimg-valign="-7px" altimg-width="134px" alttext="4\!K\left(k\right)/\sqrt{1-\eta}" display="inline"><mrow><mrow><mpadded width="-1.7pt"><mn>4</mn></mpadded><mo></mo><mrow><mi href="./19.2#E8">K</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></mrow><mo>/</mo><msqrt><mrow><mn>1</mn><mo>-</mo><mi href="./22.19#SS2.p2">η</mi></mrow></msqrt></mrow></math>, and modulus <math class="ltx_Math" altimg="m69.png" altimg-height="28px" altimg-valign="-8px" altimg-width="148px" alttext="k=1/\sqrt{\eta^{-1}-1}" display="inline"><mrow><mi href="./22.1#p2.t1.r3">k</mi><mo>=</mo><mrow><mn>1</mn><mo>/</mo><msqrt><mrow><msup><mi href="./22.19#SS2.p2">η</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo>-</mo><mn>1</mn></mrow></msqrt></mrow></mrow></math>, for initial displacements with
<math class="ltx_Math" altimg="m88.png" altimg-height="28px" altimg-valign="-8px" altimg-width="105px" alttext="|a|\leq\sqrt{1/\beta}" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mi href="./22.19#SS2.p2">a</mi><mo stretchy="false">|</mo></mrow><mo>≤</mo><msqrt><mrow><mn>1</mn><mo>/</mo><mi href="./22.19#SS2.p2">β</mi></mrow></msqrt></mrow></math>.</p>
<table id="E7" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">22.19.7</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E7.png" altimg-height="41px" altimg-valign="-15px" altimg-width="227px" alttext="x(t)=a\operatorname{sn}\left(t\sqrt{1-\eta},k\right)." display="block"><mrow><mrow><mrow><mi href="./22.19#SS2.p1">x</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./22.19#SS2.p1">t</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mi href="./22.19#SS2.p2">a</mi><mo></mo><mrow><mi href="./22.2#E4">sn</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./22.19#SS2.p1">t</mi><mo></mo><msqrt><mrow><mn>1</mn><mo>-</mo><mi href="./22.19#SS2.p2">η</mi></mrow></msqrt></mrow><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E7.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./22.2#E4" title="(22.2.4) ‣ §22.2 Definitions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="23px" altimg-valign="-7px" altimg-width="72px" alttext="\operatorname{sn}\left(\NVar{z},\NVar{k}\right)" display="inline"><mrow><mi href="./22.2#E4">sn</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r2">z</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobian elliptic function</a>,
<a href="./22.1#p2.t1.r3" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./22.1#p2.t1.r3">k</mi></math>: modulus</a>,
<a href="./22.19#SS2.p1" title="§22.19(ii) Classical Dynamics: The Quartic Oscillator ‣ §22.19 Physical Applications ‣ Applications ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="23px" altimg-valign="-7px" altimg-width="38px" alttext="x(t)" display="inline"><mrow><mi href="./22.19#SS2.p1">x</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./22.19#SS2.p1">t</mi><mo stretchy="false">)</mo></mrow></mrow></math>: coordinate</a>,
<a href="./22.19#SS2.p1" title="§22.19(ii) Classical Dynamics: The Quartic Oscillator ‣ §22.19 Physical Applications ‣ Applications ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m79.png" altimg-height="17px" altimg-valign="-2px" altimg-width="11px" alttext="t" display="inline"><mi href="./22.19#SS2.p1">t</mi></math>: time</a>,
<a href="./22.19#SS2.p2" title="§22.19(ii) Classical Dynamics: The Quartic Oscillator ‣ §22.19 Physical Applications ‣ Applications ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./22.19#SS2.p2">a</mi></math>: displacement</a> and
<a href="./22.19#SS2.p2" title="§22.19(ii) Classical Dynamics: The Quartic Oscillator ‣ §22.19 Physical Applications ‣ Applications ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m30.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="\eta" display="inline"><mi href="./22.19#SS2.p2">η</mi></math>: parameter</a>
</dd>
<dt>Clarification (effective with 1.0.9):</dt>
<dd>
For added clarity, the modulus <math class="ltx_Math" altimg="m72.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./22.1#p2.t1.r3">k</mi></math> has been defined in the line above.
Previously <math class="ltx_Math" altimg="m9.png" altimg-height="28px" altimg-valign="-8px" altimg-width="110px" alttext="1/\sqrt{\eta^{-1}-1}" display="inline"><mrow><mn>1</mn><mo>/</mo><msqrt><mrow><msup><mi href="./22.19#SS2.p2">η</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo>-</mo><mn>1</mn></mrow></msqrt></mrow></math> was given explicitly in the equation.
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">As <math class="ltx_Math" altimg="m61.png" altimg-height="28px" altimg-valign="-8px" altimg-width="98px" alttext="a\to\sqrt{1/\beta}" display="inline"><mrow><mi href="./22.19#SS2.p2">a</mi><mo>→</mo><msqrt><mrow><mn>1</mn><mo>/</mo><mi href="./22.19#SS2.p2">β</mi></mrow></msqrt></mrow></math> from below the period diverges since
<math class="ltx_Math" altimg="m58.png" altimg-height="28px" altimg-valign="-8px" altimg-width="109px" alttext="a=\pm\sqrt{1/\beta}" display="inline"><mrow><mi href="./22.19#SS2.p2">a</mi><mo>=</mo><mrow><mo>±</mo><msqrt><mrow><mn>1</mn><mo>/</mo><mi href="./22.19#SS2.p2">β</mi></mrow></msqrt></mrow></mrow></math> are points of unstable equilibrium.</p>
</div>
</section>
<section id="Px3" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Case III: <math class="ltx_Math" altimg="m22.png" altimg-height="27px" altimg-valign="-9px" altimg-width="192px" alttext="V(x)=-\frac{1}{2}x^{2}+\frac{1}{4}\beta x^{4}" display="inline"><mrow><mrow><mi href="./22.19#SS2.p1">V</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./22.19#SS2.p1">x</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><msup><mi href="./22.19#SS2.p1">x</mi><mn>2</mn></msup></mrow></mrow><mo>+</mo><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac><mo></mo><mi href="./22.19#SS2.p2">β</mi><mo></mo><msup><mi href="./22.19#SS2.p1">x</mi><mn>4</mn></msup></mrow></mrow></mrow></math>
</h3>
<div id="Px3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Errata (effective with 1.0.9):</dt>
<dd>
Two corrections have been made in this paragraph.
First, the correct range of the initial displacement <math class="ltx_Math" altimg="m59.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./22.19#SS2.p2">a</mi></math> is <math class="ltx_Math" altimg="m48.png" altimg-height="28px" altimg-valign="-8px" altimg-width="183px" alttext="\sqrt{1/\beta}\leq|a|<\sqrt{2/\beta}" display="inline"><mrow><msqrt><mrow><mn>1</mn><mo>/</mo><mi href="./22.19#SS2.p2">β</mi></mrow></msqrt><mo>≤</mo><mrow><mo stretchy="false">|</mo><mi href="./22.19#SS2.p2">a</mi><mo stretchy="false">|</mo></mrow><mo><</mo><msqrt><mrow><mn>2</mn><mo>/</mo><mi href="./22.19#SS2.p2">β</mi></mrow></msqrt></mrow></math>.
Previously it was <math class="ltx_Math" altimg="m49.png" altimg-height="28px" altimg-valign="-8px" altimg-width="183px" alttext="\sqrt{1/\beta}\leq|a|\leq\sqrt{2/\beta}" display="inline"><mrow><msqrt><mrow><mn>1</mn><mo>/</mo><mi href="./22.19#SS2.p2">β</mi></mrow></msqrt><mo>≤</mo><mrow><mo stretchy="false">|</mo><mi href="./22.19#SS2.p2">a</mi><mo stretchy="false">|</mo></mrow><mo>≤</mo><msqrt><mrow><mn>2</mn><mo>/</mo><mi href="./22.19#SS2.p2">β</mi></mrow></msqrt></mrow></math>.
Second, the correct period of the oscillations is <math class="ltx_Math" altimg="m10.png" altimg-height="25px" altimg-valign="-9px" altimg-width="100px" alttext="2\!K\left(k\right)/\sqrt{\eta}" display="inline"><mrow><mrow><mpadded width="-1.7pt"><mn>2</mn></mpadded><mo></mo><mrow><mi href="./19.2#E8">K</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></mrow><mo>/</mo><msqrt><mi href="./22.19#SS2.p2">η</mi></msqrt></mrow></math>.
Previously it was given incorrectly as <math class="ltx_Math" altimg="m15.png" altimg-height="25px" altimg-valign="-9px" altimg-width="100px" alttext="4\!K\left(k\right)/\sqrt{\eta}" display="inline"><mrow><mrow><mpadded width="-1.7pt"><mn>4</mn></mpadded><mo></mo><mrow><mi href="./19.2#E8">K</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></mrow><mo>/</mo><msqrt><mi href="./22.19#SS2.p2">η</mi></msqrt></mrow></math>.
<p><span class="ltx_font_italic">Reported 2014-04-28 by Svante Janson</span></p>
</dd>
<dt>Errata (effective with 1.0.9):</dt>
<dd>
For improved clarity, the modulus <math class="ltx_Math" altimg="m72.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./22.1#p2.t1.r3">k</mi></math> has been defined in the line above.
Previously <math class="ltx_Math" altimg="m8.png" altimg-height="28px" altimg-valign="-8px" altimg-width="110px" alttext="1/\sqrt{2-\eta^{-1}}" display="inline"><mrow><mn>1</mn><mo>/</mo><msqrt><mrow><mn>2</mn><mo>-</mo><msup><mi href="./22.19#SS2.p2">η</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></mrow></msqrt></mrow></math> was given explicitly in the equation.
<p><span class="ltx_font_italic">Reported 2014-04-28 by Svante Janson</span></p>
</dd>
</dl>
</div>
</div>
<div id="Px3.p1" class="ltx_para">
<p class="ltx_p">Two types of oscillatory motion are possible. For an initial displacement with
<math class="ltx_Math" altimg="m48.png" altimg-height="28px" altimg-valign="-8px" altimg-width="183px" alttext="\sqrt{1/\beta}\leq|a|<\sqrt{2/\beta}" display="inline"><mrow><msqrt><mrow><mn>1</mn><mo>/</mo><mi href="./22.19#SS2.p2">β</mi></mrow></msqrt><mo>≤</mo><mrow><mo stretchy="false">|</mo><mi href="./22.19#SS2.p2">a</mi><mo stretchy="false">|</mo></mrow><mo><</mo><msqrt><mrow><mn>2</mn><mo>/</mo><mi href="./22.19#SS2.p2">β</mi></mrow></msqrt></mrow></math>, bounded oscillations take place
near one of the two points of stable equilibrium <math class="ltx_Math" altimg="m84.png" altimg-height="28px" altimg-valign="-8px" altimg-width="110px" alttext="x=\pm\sqrt{1/\beta}" display="inline"><mrow><mi href="./22.19#SS2.p1">x</mi><mo>=</mo><mrow><mo>±</mo><msqrt><mrow><mn>1</mn><mo>/</mo><mi href="./22.19#SS2.p2">β</mi></mrow></msqrt></mrow></mrow></math>.
Such oscillations, of period <math class="ltx_Math" altimg="m10.png" altimg-height="25px" altimg-valign="-9px" altimg-width="100px" alttext="2\!K\left(k\right)/\sqrt{\eta}" display="inline"><mrow><mrow><mpadded width="-1.7pt"><mn>2</mn></mpadded><mo></mo><mrow><mi href="./19.2#E8">K</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></mrow><mo>/</mo><msqrt><mi href="./22.19#SS2.p2">η</mi></msqrt></mrow></math>, with modulus <math class="ltx_Math" altimg="m68.png" altimg-height="28px" altimg-valign="-8px" altimg-width="148px" alttext="k=1/\sqrt{2-\eta^{-1}}" display="inline"><mrow><mi href="./22.1#p2.t1.r3">k</mi><mo>=</mo><mrow><mn>1</mn><mo>/</mo><msqrt><mrow><mn>2</mn><mo>-</mo><msup><mi href="./22.19#SS2.p2">η</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></mrow></msqrt></mrow></mrow></math>
are given by:</p>
<table id="E8" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">22.19.8</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E8.png" altimg-height="26px" altimg-valign="-8px" altimg-width="185px" alttext="x(t)=a\operatorname{dn}\left(t\sqrt{\eta},k\right)." display="block"><mrow><mrow><mrow><mi href="./22.19#SS2.p1">x</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./22.19#SS2.p1">t</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mi href="./22.19#SS2.p2">a</mi><mo></mo><mrow><mi href="./22.2#E6">dn</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./22.19#SS2.p1">t</mi><mo></mo><msqrt><mi href="./22.19#SS2.p2">η</mi></msqrt></mrow><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E8.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./22.2#E6" title="(22.2.6) ‣ §22.2 Definitions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m41.png" altimg-height="23px" altimg-valign="-7px" altimg-width="75px" alttext="\operatorname{dn}\left(\NVar{z},\NVar{k}\right)" display="inline"><mrow><mi href="./22.2#E6">dn</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r2">z</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobian elliptic function</a>,
<a href="./22.1#p2.t1.r3" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./22.1#p2.t1.r3">k</mi></math>: modulus</a>,
<a href="./22.19#SS2.p1" title="§22.19(ii) Classical Dynamics: The Quartic Oscillator ‣ §22.19 Physical Applications ‣ Applications ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="23px" altimg-valign="-7px" altimg-width="38px" alttext="x(t)" display="inline"><mrow><mi href="./22.19#SS2.p1">x</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./22.19#SS2.p1">t</mi><mo stretchy="false">)</mo></mrow></mrow></math>: coordinate</a>,
<a href="./22.19#SS2.p1" title="§22.19(ii) Classical Dynamics: The Quartic Oscillator ‣ §22.19 Physical Applications ‣ Applications ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m79.png" altimg-height="17px" altimg-valign="-2px" altimg-width="11px" alttext="t" display="inline"><mi href="./22.19#SS2.p1">t</mi></math>: time</a>,
<a href="./22.19#SS2.p2" title="§22.19(ii) Classical Dynamics: The Quartic Oscillator ‣ §22.19 Physical Applications ‣ Applications ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./22.19#SS2.p2">a</mi></math>: displacement</a> and
<a href="./22.19#SS2.p2" title="§22.19(ii) Classical Dynamics: The Quartic Oscillator ‣ §22.19 Physical Applications ‣ Applications ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m30.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="\eta" display="inline"><mi href="./22.19#SS2.p2">η</mi></math>: parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">As <math class="ltx_Math" altimg="m62.png" altimg-height="28px" altimg-valign="-8px" altimg-width="98px" alttext="a\to\sqrt{2/\beta}" display="inline"><mrow><mi href="./22.19#SS2.p2">a</mi><mo>→</mo><msqrt><mrow><mn>2</mn><mo>/</mo><mi href="./22.19#SS2.p2">β</mi></mrow></msqrt></mrow></math> from below the period diverges since <math class="ltx_Math" altimg="m83.png" altimg-height="17px" altimg-valign="-2px" altimg-width="52px" alttext="x=0" display="inline"><mrow><mi href="./22.19#SS2.p1">x</mi><mo>=</mo><mn>0</mn></mrow></math> is a
point of unstable equlilibrium. For initial displacement with
<math class="ltx_Math" altimg="m87.png" altimg-height="28px" altimg-valign="-8px" altimg-width="105px" alttext="|a|\geq\sqrt{2/\beta}" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mi href="./22.19#SS2.p2">a</mi><mo stretchy="false">|</mo></mrow><mo>≥</mo><msqrt><mrow><mn>2</mn><mo>/</mo><mi href="./22.19#SS2.p2">β</mi></mrow></msqrt></mrow></math> the motion extends over the full range
<math class="ltx_Math" altimg="m3.png" altimg-height="19px" altimg-valign="-5px" altimg-width="105px" alttext="-a\leq x\leq a" display="inline"><mrow><mrow><mo>-</mo><mi href="./22.19#SS2.p2">a</mi></mrow><mo>≤</mo><mi href="./22.19#SS2.p1">x</mi><mo>≤</mo><mi href="./22.19#SS2.p2">a</mi></mrow></math>:</p>
<table id="E9" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">22.19.9</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E9.png" altimg-height="41px" altimg-valign="-15px" altimg-width="238px" alttext="x(t)=a\operatorname{cn}\left(t\sqrt{2\eta-1},k\right)," display="block"><mrow><mrow><mrow><mi href="./22.19#SS2.p1">x</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./22.19#SS2.p1">t</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mi href="./22.19#SS2.p2">a</mi><mo></mo><mrow><mi href="./22.2#E5">cn</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./22.19#SS2.p1">t</mi><mo></mo><msqrt><mrow><mrow><mn>2</mn><mo></mo><mi href="./22.19#SS2.p2">η</mi></mrow><mo>-</mo><mn>1</mn></mrow></msqrt></mrow><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E9.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./22.2#E5" title="(22.2.5) ‣ §22.2 Definitions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="23px" altimg-valign="-7px" altimg-width="73px" alttext="\operatorname{cn}\left(\NVar{z},\NVar{k}\right)" display="inline"><mrow><mi href="./22.2#E5">cn</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r2">z</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobian elliptic function</a>,
<a href="./22.1#p2.t1.r3" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./22.1#p2.t1.r3">k</mi></math>: modulus</a>,
<a href="./22.19#SS2.p1" title="§22.19(ii) Classical Dynamics: The Quartic Oscillator ‣ §22.19 Physical Applications ‣ Applications ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="23px" altimg-valign="-7px" altimg-width="38px" alttext="x(t)" display="inline"><mrow><mi href="./22.19#SS2.p1">x</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./22.19#SS2.p1">t</mi><mo stretchy="false">)</mo></mrow></mrow></math>: coordinate</a>,
<a href="./22.19#SS2.p1" title="§22.19(ii) Classical Dynamics: The Quartic Oscillator ‣ §22.19 Physical Applications ‣ Applications ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m79.png" altimg-height="17px" altimg-valign="-2px" altimg-width="11px" alttext="t" display="inline"><mi href="./22.19#SS2.p1">t</mi></math>: time</a>,
<a href="./22.19#SS2.p2" title="§22.19(ii) Classical Dynamics: The Quartic Oscillator ‣ §22.19 Physical Applications ‣ Applications ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./22.19#SS2.p2">a</mi></math>: displacement</a> and
<a href="./22.19#SS2.p2" title="§22.19(ii) Classical Dynamics: The Quartic Oscillator ‣ §22.19 Physical Applications ‣ Applications ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m30.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="\eta" display="inline"><mi href="./22.19#SS2.p2">η</mi></math>: parameter</a>
</dd>
<dt>Clarification (effective with 1.0.9):</dt>
<dd>
For added clarity, the modulus <math class="ltx_Math" altimg="m72.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./22.1#p2.t1.r3">k</mi></math> has been defined in the line below.
Previously <math class="ltx_Math" altimg="m8.png" altimg-height="28px" altimg-valign="-8px" altimg-width="110px" alttext="1/\sqrt{2-\eta^{-1}}" display="inline"><mrow><mn>1</mn><mo>/</mo><msqrt><mrow><mn>2</mn><mo>-</mo><msup><mi href="./22.19#SS2.p2">η</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></mrow></msqrt></mrow></math> was given explicitly in the equation.
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">with period <math class="ltx_Math" altimg="m14.png" altimg-height="25px" altimg-valign="-7px" altimg-width="144px" alttext="4\!K\left(k\right)/\sqrt{2\eta-1}" display="inline"><mrow><mrow><mpadded width="-1.7pt"><mn>4</mn></mpadded><mo></mo><mrow><mi href="./19.2#E8">K</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></mrow><mo>/</mo><msqrt><mrow><mrow><mn>2</mn><mo></mo><mi href="./22.19#SS2.p2">η</mi></mrow><mo>-</mo><mn>1</mn></mrow></msqrt></mrow></math> and modulus <math class="ltx_Math" altimg="m68.png" altimg-height="28px" altimg-valign="-8px" altimg-width="148px" alttext="k=1/\sqrt{2-\eta^{-1}}" display="inline"><mrow><mi href="./22.1#p2.t1.r3">k</mi><mo>=</mo><mrow><mn>1</mn><mo>/</mo><msqrt><mrow><mn>2</mn><mo>-</mo><msup><mi href="./22.19#SS2.p2">η</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></mrow></msqrt></mrow></mrow></math>.
As <math class="ltx_Math" altimg="m34.png" altimg-height="28px" altimg-valign="-8px" altimg-width="109px" alttext="\left|a\right|\to\sqrt{1/\beta}" display="inline"><mrow><mrow><mo>|</mo><mi href="./22.19#SS2.p2">a</mi><mo>|</mo></mrow><mo>→</mo><msqrt><mrow><mn>1</mn><mo>/</mo><mi href="./22.19#SS2.p2">β</mi></mrow></msqrt></mrow></math> from above the period again diverges. Both the <math class="ltx_Math" altimg="m40.png" altimg-height="18px" altimg-valign="-2px" altimg-width="26px" alttext="\operatorname{dn}" display="inline"><mi href="./22.2#E6">dn</mi></math> and <math class="ltx_Math" altimg="m38.png" altimg-height="13px" altimg-valign="-2px" altimg-width="24px" alttext="\operatorname{cn}" display="inline"><mi href="./22.2#E5">cn</mi></math>
solutions approach <math class="ltx_Math" altimg="m60.png" altimg-height="18px" altimg-valign="-2px" altimg-width="65px" alttext="a\operatorname{sech}t" display="inline"><mrow><mi href="./22.19#SS2.p2">a</mi><mo></mo><mrow><mi href="./4.28#E6">sech</mi><mo></mo><mi href="./22.19#SS2.p1">t</mi></mrow></mrow></math> as <math class="ltx_Math" altimg="m62.png" altimg-height="28px" altimg-valign="-8px" altimg-width="98px" alttext="a\to\sqrt{2/\beta}" display="inline"><mrow><mi href="./22.19#SS2.p2">a</mi><mo>→</mo><msqrt><mrow><mn>2</mn><mo>/</mo><mi href="./22.19#SS2.p2">β</mi></mrow></msqrt></mrow></math> from the
appropriate directions.</p>
</div>
</section>
</section>
<section id="iii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§22.19(iii) </span>Nonlinear ODEs and PDEs</h2>
<div id="SS3.info" class="ltx_metadata ltx_info">
, Chapter 1)</cite>). Hyperelliptic
functions
<math class="ltx_Math" altimg="m81.png" altimg-height="23px" altimg-valign="-7px" altimg-width="41px" alttext="u(z)" display="inline"><mrow><mi href="./22.19#SS4.p1">u</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./22.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> are solutions of the equation <math class="ltx_Math" altimg="m86.png" altimg-height="28px" altimg-valign="-9px" altimg-width="179px" alttext="z=\int_{0}^{u}(f(x))^{-1/2}\mathrm{d}x" display="inline"><mrow><mi href="./22.1#p2.t1.r2">z</mi><mo>=</mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi href="./22.19#SS4.p1">u</mi></msubsup><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./22.19#SS4.p1">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></mrow></msup><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./22.1#p2.t1.r1">x</mi></mrow></mrow></mrow></mrow></math>,
where <math class="ltx_Math" altimg="m63.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="f(x)" display="inline"><mrow><mi href="./22.19#SS4.p1">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo stretchy="false">)</mo></mrow></mrow></math> is a polynomial of degree higher than 4. Elementary discussions of
this topic appear in <cite class="ltx_cite ltx_citemacro_citet">Lawden (</div>
</div>
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<title>DLMF: 29.10 Lamé Functions with Imaginary Periods</title>
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</dl>
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<div id="p1" class="ltx_para">
<p class="ltx_p">The substitutions
</p>
<table id="E1" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">29.10.1</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E1.png" altimg-height="26px" altimg-valign="-7px" altimg-width="163px" alttext="h=\nu(\nu+1)-h^{\prime}," display="block"><mrow><mrow><mi href="./29.1#p2.t1.r4">h</mi><mo>=</mo><mrow><mrow><mi href="./29.1#p2.t1.r4">ν</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./29.1#p2.t1.r4">ν</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>-</mo><msup><mi href="./29.1#p2.t1.r4">h</mi><mo>′</mo></msup></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./29.1#p2.t1.r4" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m10.png" altimg-height="18px" altimg-valign="-2px" altimg-width="16px" alttext="h" display="inline"><mi href="./29.1#p2.t1.r4">h</mi></math>: real parameter</a> and
<a href="./29.1#p2.t1.r4" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m7.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./29.1#p2.t1.r4">ν</mi></math>: real parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E2" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">29.10.2</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E2.png" altimg-height="26px" altimg-valign="-7px" altimg-width="168px" alttext="z^{\prime}=\mathrm{i}(z-\!K\!-\mathrm{i}\!{K^{\prime}}\!)," display="block"><mrow><mrow><msup><mi href="./29.1#p2.t1.r3">z</mi><mo>′</mo></msup><mo>=</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./29.1#p2.t1.r3">z</mi><mo rspace="0.8pt">-</mo><mpadded width="-1.7pt"><mrow><mi href="./19.2#E8">K</mi><mo></mo></mrow></mpadded><mo rspace="0.8pt">-</mo><mrow><mpadded width="-1.7pt"><mi mathvariant="normal">i</mi></mpadded><mo></mo><mpadded width="-1.7pt"><mrow><msup><mi href="./19.2#E9">K</mi><mo href="./19.2#E9">′</mo></msup><mo></mo></mrow></mpadded></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./19.2#E9" title="(19.2.9) ‣ §19.2(ii) Legendre’s Integrals ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m16.png" altimg-height="24px" altimg-valign="-7px" altimg-width="58px" alttext="{K^{\prime}}\left(\NVar{k}\right)" display="inline"><mrow><msup><mi href="./19.2#E9">K</mi><mo href="./19.2#E9">′</mo></msup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Legendre’s complementary complete elliptic integral of the first kind</a>,
<a href="./19.2#E8" title="(19.2.8) ‣ §19.2(ii) Legendre’s Integrals ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m3.png" altimg-height="23px" altimg-valign="-7px" altimg-width="52px" alttext="K\left(\NVar{k}\right)" display="inline"><mrow><mi href="./19.2#E8">K</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Legendre’s complete elliptic integral of the first kind</a>,
<a href="./29.1#p2.t1.r3" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m15.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./29.1#p2.t1.r3">z</mi></math>: complex variable</a> and
<a href="./29.1#p2.t1.r4" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m11.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./29.1#p2.t1.r4">k</mi></math>: real parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">transform () into</p>
<table id="E3" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">29.10.3</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E3.png" altimg-height="51px" altimg-valign="-18px" altimg-width="370px" alttext="\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z^{\prime}}^{2}}+(h^{\prime}-\nu(\nu+1){k^%
{\prime}}^{2}{\operatorname{sn}^{2}}\left(z^{\prime},k^{\prime}\right))w=0." display="block"><mrow><mrow><mrow><mfrac><mrow><mpadded lspace="-1.7pt" width="-4.5pt"><msup><mo href="./1.4#E4">d</mo><mn>2</mn></msup></mpadded><mi href="./29.2#i">w</mi></mrow><msup><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><msup><mi href="./29.1#p2.t1.r3">z</mi><mo>′</mo></msup></mrow><mn>2</mn></msup></mfrac><mo>+</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><msup><mi href="./29.1#p2.t1.r4">h</mi><mo>′</mo></msup><mo>-</mo><mrow><mi href="./29.1#p2.t1.r4">ν</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./29.1#p2.t1.r4">ν</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mmultiscripts><mi href="./29.1#p2.t1.r4">k</mi><none></none><mo>′</mo><none></none><mn>2</mn></mmultiscripts><mo></mo><mrow><msup><mi href="./22.2#E4">sn</mi><mn>2</mn></msup><mo></mo><mrow><mo>(</mo><msup><mi href="./29.1#p2.t1.r3">z</mi><mo>′</mo></msup><mo>,</mo><msup><mi href="./29.1#p2.t1.r4">k</mi><mo>′</mo></msup><mo>)</mo></mrow></mrow></mrow></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi href="./29.2#i">w</mi></mrow></mrow><mo>=</mo><mn>0</mn></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./22.2#E4" title="(22.2.4) ‣ §22.2 Definitions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m8.png" altimg-height="23px" altimg-valign="-7px" altimg-width="72px" alttext="\operatorname{sn}\left(\NVar{z},\NVar{k}\right)" display="inline"><mrow><mi href="./22.2#E4">sn</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r2">z</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobian elliptic function</a>,
<a href="./1.4#E4" title="(1.4.4) ‣ §1.4(iii) Derivatives ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m4.png" altimg-height="29px" altimg-valign="-9px" altimg-width="27px" alttext="\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: derivative of <math class="ltx_Math" altimg="m9.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m14.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./29.1#p2.t1.r3" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m15.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./29.1#p2.t1.r3">z</mi></math>: complex variable</a>,
<a href="./29.1#p2.t1.r4" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m10.png" altimg-height="18px" altimg-valign="-2px" altimg-width="16px" alttext="h" display="inline"><mi href="./29.1#p2.t1.r4">h</mi></math>: real parameter</a>,
<a href="./29.1#p2.t1.r4" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m11.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./29.1#p2.t1.r4">k</mi></math>: real parameter</a>,
<a href="./29.1#p2.t1.r4" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m7.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./29.1#p2.t1.r4">ν</mi></math>: real parameter</a> and
<a href="./29.2#i" title="§29.2(i) Lamé’s Equation ‣ §29.2 Differential Equations ‣ Lamé Functions ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m13.png" altimg-height="23px" altimg-valign="-7px" altimg-width="45px" alttext="w(z)" display="inline"><mrow><mi href="./29.2#i">w</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./29.1#p2.t1.r3">z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: solution</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">In consequence, the functions</p>
<table id="E4" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex1" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="4" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">29.10.4</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E4a.png" altimg-height="39px" altimg-valign="-15px" altimg-width="236px" alttext="\mathit{Ec}^{2m}_{\nu}\left(\mathrm{i}(z-\!K\!-\mathrm{i}\!{K^{\prime}}\!),{k^%
{\prime}}^{2}\right)," display="inline"><mrow><mrow><msubsup><mi href="./29.3#SS4.p1" mathvariant="italic">Ec</mi><mi href="./29.1#p2.t1.r4">ν</mi><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">m</mi></mrow></msubsup><mo></mo><mrow><mo>(</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./29.1#p2.t1.r3">z</mi><mo rspace="0.8pt">-</mo><mpadded width="-1.7pt"><mrow><mi href="./19.2#E8">K</mi><mo></mo></mrow></mpadded><mo rspace="0.8pt">-</mo><mrow><mpadded width="-1.7pt"><mi mathvariant="normal">i</mi></mpadded><mo></mo><mpadded width="-1.7pt"><mrow><msup><mi href="./19.2#E9">K</mi><mo href="./19.2#E9">′</mo></msup><mo></mo></mrow></mpadded></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo><mmultiscripts><mi href="./29.1#p2.t1.r4">k</mi><none></none><mo>′</mo><none></none><mn>2</mn></mmultiscripts><mo>)</mo></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex2" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E4b.png" altimg-height="39px" altimg-valign="-15px" altimg-width="256px" alttext="\mathit{Ec}^{2m+1}_{\nu}\left(\mathrm{i}(z-\!K\!-\mathrm{i}\!{K^{\prime}}\!),{%
k^{\prime}}^{2}\right)," display="inline"><mrow><mrow><msubsup><mi href="./29.3#SS4.p1" mathvariant="italic">Ec</mi><mi href="./29.1#p2.t1.r4">ν</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">m</mi></mrow><mo>+</mo><mn>1</mn></mrow></msubsup><mo></mo><mrow><mo>(</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./29.1#p2.t1.r3">z</mi><mo rspace="0.8pt">-</mo><mpadded width="-1.7pt"><mrow><mi href="./19.2#E8">K</mi><mo></mo></mrow></mpadded><mo rspace="0.8pt">-</mo><mrow><mpadded width="-1.7pt"><mi mathvariant="normal">i</mi></mpadded><mo></mo><mpadded width="-1.7pt"><mrow><msup><mi href="./19.2#E9">K</mi><mo href="./19.2#E9">′</mo></msup><mo></mo></mrow></mpadded></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo><mmultiscripts><mi href="./29.1#p2.t1.r4">k</mi><none></none><mo>′</mo><none></none><mn>2</mn></mmultiscripts><mo>)</mo></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex3" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E4c.png" altimg-height="39px" altimg-valign="-15px" altimg-width="256px" alttext="\mathit{Es}^{2m+1}_{\nu}\left(\mathrm{i}(z-\!K\!-\mathrm{i}\!{K^{\prime}}\!),{%
k^{\prime}}^{2}\right)," display="inline"><mrow><mrow><msubsup><mi href="./29.3#SS4.p1" mathvariant="italic">Es</mi><mi href="./29.1#p2.t1.r4">ν</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">m</mi></mrow><mo>+</mo><mn>1</mn></mrow></msubsup><mo></mo><mrow><mo>(</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./29.1#p2.t1.r3">z</mi><mo rspace="0.8pt">-</mo><mpadded width="-1.7pt"><mrow><mi href="./19.2#E8">K</mi><mo></mo></mrow></mpadded><mo rspace="0.8pt">-</mo><mrow><mpadded width="-1.7pt"><mi mathvariant="normal">i</mi></mpadded><mo></mo><mpadded width="-1.7pt"><mrow><msup><mi href="./19.2#E9">K</mi><mo href="./19.2#E9">′</mo></msup><mo></mo></mrow></mpadded></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo><mmultiscripts><mi href="./29.1#p2.t1.r4">k</mi><none></none><mo>′</mo><none></none><mn>2</mn></mmultiscripts><mo>)</mo></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex4" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E4d.png" altimg-height="39px" altimg-valign="-15px" altimg-width="256px" alttext="\mathit{Es}^{2m+2}_{\nu}\left(\mathrm{i}(z-\!K\!-\mathrm{i}\!{K^{\prime}}\!),{%
k^{\prime}}^{2}\right)," display="inline"><mrow><mrow><msubsup><mi href="./29.3#SS4.p1" mathvariant="italic">Es</mi><mi href="./29.1#p2.t1.r4">ν</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">m</mi></mrow><mo>+</mo><mn>2</mn></mrow></msubsup><mo></mo><mrow><mo>(</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./29.1#p2.t1.r3">z</mi><mo rspace="0.8pt">-</mo><mpadded width="-1.7pt"><mrow><mi href="./19.2#E8">K</mi><mo></mo></mrow></mpadded><mo rspace="0.8pt">-</mo><mrow><mpadded width="-1.7pt"><mi mathvariant="normal">i</mi></mpadded><mo></mo><mpadded width="-1.7pt"><mrow><msup><mi href="./19.2#E9">K</mi><mo href="./19.2#E9">′</mo></msup><mo></mo></mrow></mpadded></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo><mmultiscripts><mi href="./29.1#p2.t1.r4">k</mi><none></none><mo>′</mo><none></none><mn>2</mn></mmultiscripts><mo>)</mo></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E4.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./29.3#SS4.p1" title="§29.3(iv) Lamé Functions ‣ §29.3 Definitions and Basic Properties ‣ Lamé Functions ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m5.png" altimg-height="27px" altimg-valign="-9px" altimg-width="104px" alttext="\mathit{Ec}^{\NVar{m}}_{\NVar{\nu}}\left(\NVar{z},\NVar{k^{2}}\right)" display="inline"><mrow><msubsup><mi href="./29.3#SS4.p1" mathvariant="italic">Ec</mi><mi class="ltx_nvar" href="./29.1#p2.t1.r4">ν</mi><mi class="ltx_nvar" href="./29.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./29.1#p2.t1.r3">z</mi><mo>,</mo><msup><mi class="ltx_nvar" href="./29.1#p2.t1.r4">k</mi><mn class="ltx_nvar">2</mn></msup><mo>)</mo></mrow></mrow></math>: Lamé function</a>,
<a href="./29.3#SS4.p1" title="§29.3(iv) Lamé Functions ‣ §29.3 Definitions and Basic Properties ‣ Lamé Functions ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m6.png" altimg-height="27px" altimg-valign="-9px" altimg-width="103px" alttext="\mathit{Es}^{\NVar{m}}_{\NVar{\nu}}\left(\NVar{z},\NVar{k^{2}}\right)" display="inline"><mrow><msubsup><mi href="./29.3#SS4.p1" mathvariant="italic">Es</mi><mi class="ltx_nvar" href="./29.1#p2.t1.r4">ν</mi><mi class="ltx_nvar" href="./29.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./29.1#p2.t1.r3">z</mi><mo>,</mo><msup><mi class="ltx_nvar" href="./29.1#p2.t1.r4">k</mi><mn class="ltx_nvar">2</mn></msup><mo>)</mo></mrow></mrow></math>: Lamé function</a>,
<a href="./19.2#E9" title="(19.2.9) ‣ §19.2(ii) Legendre’s Integrals ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m16.png" altimg-height="24px" altimg-valign="-7px" altimg-width="58px" alttext="{K^{\prime}}\left(\NVar{k}\right)" display="inline"><mrow><msup><mi href="./19.2#E9">K</mi><mo href="./19.2#E9">′</mo></msup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Legendre’s complementary complete elliptic integral of the first kind</a>,
<a href="./19.2#E8" title="(19.2.8) ‣ §19.2(ii) Legendre’s Integrals ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m3.png" altimg-height="23px" altimg-valign="-7px" altimg-width="52px" alttext="K\left(\NVar{k}\right)" display="inline"><mrow><mi href="./19.2#E8">K</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Legendre’s complete elliptic integral of the first kind</a>,
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m12.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./29.1#p2.t1.r1">m</mi></math>: nonnegative integer</a>,
<a href="./29.1#p2.t1.r3" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m15.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./29.1#p2.t1.r3">z</mi></math>: complex variable</a>,
<a href="./29.1#p2.t1.r4" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m11.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./29.1#p2.t1.r4">k</mi></math>: real parameter</a> and
<a href="./29.1#p2.t1.r4" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m7.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./29.1#p2.t1.r4">ν</mi></math>: real parameter</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">are solutions of (). The first and the fourth functions have
period <math class="ltx_Math" altimg="m1.png" altimg-height="19px" altimg-valign="-2px" altimg-width="40px" alttext="2\mathrm{i}\!{K^{\prime}}\!" display="inline"><mrow><mn>2</mn><mo></mo><mpadded width="-1.7pt"><mi mathvariant="normal">i</mi></mpadded><mo></mo><mpadded width="-1.7pt"><mrow><msup><mi href="./19.2#E9">K</mi><mo href="./19.2#E9">′</mo></msup><mo></mo></mrow></mpadded></mrow></math>; the second and the third have period
<math class="ltx_Math" altimg="m2.png" altimg-height="19px" altimg-valign="-2px" altimg-width="40px" alttext="4\mathrm{i}\!{K^{\prime}}\!" display="inline"><mrow><mn>4</mn><mo></mo><mpadded width="-1.7pt"><mi mathvariant="normal">i</mi></mpadded><mo></mo><mpadded width="-1.7pt"><mrow><msup><mi href="./19.2#E9">K</mi><mo href="./19.2#E9">′</mo></msup><mo></mo></mrow></mpadded></mrow></math>.</p>
</div>
<div id="p2" class="ltx_para">
<p class="ltx_p">For these results and further information see <cite class="ltx_cite ltx_citemacro_citet">Erdélyi<span class="ltx_text ltx_bib_etal"> et al.</span> (</div>
</div>
</body></text>
</html>
</page>
<page>
<!DOCTYPE html><html>
<head>
<title>DLMF: 22.20 Methods of Computation</title>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
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<text><body class="color_default textfont_default titlefont_default fontsize_default navbar_default">
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<div class="ltx_page_navlogo"></dd>
</dl>
</div>
</div>
<div id="SS1.p1" class="ltx_para">
<p class="ltx_p">A powerful way of computing the twelve Jacobian elliptic functions for real or
complex values of both the argument <math class="ltx_Math" altimg="m89.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./22.1#p2.t1.r2">z</mi></math> and the modulus <math class="ltx_Math" altimg="m71.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./22.1#p2.t1.r3">k</mi></math> is to use the
definitions in terms of theta functions given in §</dd>
</dl>
</div>
</div>
<div id="SS2.p1" class="ltx_para">
<p class="ltx_p">Given real or complex numbers <math class="ltx_Math" altimg="m57.png" altimg-height="21px" altimg-valign="-6px" altimg-width="50px" alttext="a_{0},b_{0}" display="inline"><mrow><msub><mi href="./22.20#SS2.p1">a</mi><mn>0</mn></msub><mo>,</mo><msub><mi href="./22.20#SS2.p1">b</mi><mn>0</mn></msub></mrow></math>, with <math class="ltx_Math" altimg="m60.png" altimg-height="23px" altimg-valign="-7px" altimg-width="51px" alttext="b_{0}/a_{0}" display="inline"><mrow><msub><mi href="./22.20#SS2.p1">b</mi><mn>0</mn></msub><mo>/</mo><msub><mi href="./22.20#SS2.p1">a</mi><mn>0</mn></msub></mrow></math> not real and negative,
define</p>
<table id="E1" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex1" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="3" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">22.20.1</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m14.png" altimg-height="17px" altimg-valign="-5px" altimg-width="27px" alttext="\displaystyle a_{n}" display="inline"><msub><mi href="./22.20#SS2.p1">a</mi><mi href="./22.20#SS2.p1">n</mi></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m7.png" altimg-height="29px" altimg-valign="-9px" altimg-width="173px" alttext="\displaystyle=\tfrac{1}{2}\left(a_{n-1}+b_{n-1}\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mrow><mo>(</mo><mrow><msub><mi href="./22.20#SS2.p1">a</mi><mrow><mi href="./22.20#SS2.p1">n</mi><mo>-</mo><mn>1</mn></mrow></msub><mo>+</mo><msub><mi href="./22.20#SS2.p1">b</mi><mrow><mi href="./22.20#SS2.p1">n</mi><mo>-</mo><mn>1</mn></mrow></msub></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex2" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m15.png" altimg-height="22px" altimg-valign="-5px" altimg-width="25px" alttext="\displaystyle b_{n}" display="inline"><msub><mi href="./22.20#SS2.p1">b</mi><mi href="./22.20#SS2.p1">n</mi></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m5.png" altimg-height="31px" altimg-valign="-7px" altimg-width="156px" alttext="\displaystyle=\left(a_{n-1}b_{n-1}\right)^{1/2}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><msup><mrow><mo>(</mo><mrow><msub><mi href="./22.20#SS2.p1">a</mi><mrow><mi href="./22.20#SS2.p1">n</mi><mo>-</mo><mn>1</mn></mrow></msub><mo></mo><msub><mi href="./22.20#SS2.p1">b</mi><mrow><mi href="./22.20#SS2.p1">n</mi><mo>-</mo><mn>1</mn></mrow></msub></mrow><mo>)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex3" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m16.png" altimg-height="17px" altimg-valign="-5px" altimg-width="26px" alttext="\displaystyle c_{n}" display="inline"><msub><mi href="./22.20#SS2.p1">c</mi><mi href="./22.20#SS2.p1">n</mi></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m8.png" altimg-height="29px" altimg-valign="-9px" altimg-width="173px" alttext="\displaystyle=\tfrac{1}{2}\left(a_{n-1}-b_{n-1}\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mrow><mo>(</mo><mrow><msub><mi href="./22.20#SS2.p1">a</mi><mrow><mi href="./22.20#SS2.p1">n</mi><mo>-</mo><mn>1</mn></mrow></msub><mo>-</mo><msub><mi href="./22.20#SS2.p1">b</mi><mrow><mi href="./22.20#SS2.p1">n</mi><mo>-</mo><mn>1</mn></mrow></msub></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./22.20#SS2.p1" title="§22.20(ii) Arithmetic-Geometric Mean ‣ §22.20 Methods of Computation ‣ Computation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="16px" altimg-valign="-5px" altimg-width="26px" alttext="a_{n}" display="inline"><msub><mi href="./22.20#SS2.p1">a</mi><mi href="./22.20#SS2.p1">n</mi></msub></math>: numbers</a>,
<a href="./22.20#SS2.p1" title="§22.20(ii) Arithmetic-Geometric Mean ‣ §22.20 Methods of Computation ‣ Computation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m62.png" altimg-height="21px" altimg-valign="-5px" altimg-width="24px" alttext="b_{n}" display="inline"><msub><mi href="./22.20#SS2.p1">b</mi><mi href="./22.20#SS2.p1">n</mi></msub></math>: numbers</a>,
<a href="./22.20#SS2.p1" title="§22.20(ii) Arithmetic-Geometric Mean ‣ §22.20 Methods of Computation ‣ Computation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="16px" altimg-valign="-5px" altimg-width="24px" alttext="c_{n}" display="inline"><msub><mi href="./22.20#SS2.p1">c</mi><mi href="./22.20#SS2.p1">n</mi></msub></math>: numbers</a> and
<a href="./22.20#SS2.p1" title="§22.20(ii) Arithmetic-Geometric Mean ‣ §22.20 Methods of Computation ‣ Computation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m80.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./22.20#SS2.p1">n</mi></math>: positive</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">for <math class="ltx_Math" altimg="m81.png" altimg-height="19px" altimg-valign="-5px" altimg-width="53px" alttext="n\geq 1" display="inline"><mrow><mi href="./22.20#SS2.p1">n</mi><mo>≥</mo><mn>1</mn></mrow></math>, where the square root is chosen
so that <math class="ltx_Math" altimg="m44.png" altimg-height="27px" altimg-valign="-9px" altimg-width="261px" alttext="\operatorname{ph}b_{n}=\tfrac{1}{2}(\operatorname{ph}a_{n-1}+\operatorname{ph}%
b_{n-1})" display="inline"><mrow><mrow><mi href="./1.9#E7">ph</mi><mo></mo><msub><mi href="./22.20#SS2.p1">b</mi><mi href="./22.20#SS2.p1">n</mi></msub></mrow><mo>=</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mi href="./1.9#E7">ph</mi><mo></mo><msub><mi href="./22.20#SS2.p1">a</mi><mrow><mi href="./22.20#SS2.p1">n</mi><mo>-</mo><mn>1</mn></mrow></msub></mrow><mo>+</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><msub><mi href="./22.20#SS2.p1">b</mi><mrow><mi href="./22.20#SS2.p1">n</mi><mo>-</mo><mn>1</mn></mrow></msub></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></math>,
where <math class="ltx_Math" altimg="m42.png" altimg-height="22px" altimg-valign="-7px" altimg-width="72px" alttext="\operatorname{ph}a_{n-1}" display="inline"><mrow><mi href="./1.9#E7">ph</mi><mo></mo><msub><mi href="./22.20#SS2.p1">a</mi><mrow><mi href="./22.20#SS2.p1">n</mi><mo>-</mo><mn>1</mn></mrow></msub></mrow></math> and <math class="ltx_Math" altimg="m43.png" altimg-height="22px" altimg-valign="-7px" altimg-width="70px" alttext="\operatorname{ph}b_{n-1}" display="inline"><mrow><mi href="./1.9#E7">ph</mi><mo></mo><msub><mi href="./22.20#SS2.p1">b</mi><mrow><mi href="./22.20#SS2.p1">n</mi><mo>-</mo><mn>1</mn></mrow></msub></mrow></math> are chosen so that
their difference is numerically less than <math class="ltx_Math" altimg="m52.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>.
Then as
<math class="ltx_Math" altimg="m82.png" altimg-height="13px" altimg-valign="-2px" altimg-width="67px" alttext="n\to\infty" display="inline"><mrow><mi href="./22.20#SS2.p1">n</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></math> sequences <math class="ltx_Math" altimg="m55.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\{a_{n}\}" display="inline"><mrow><mo stretchy="false">{</mo><msub><mi href="./22.20#SS2.p1">a</mi><mi href="./22.20#SS2.p1">n</mi></msub><mo stretchy="false">}</mo></mrow></math>, <math class="ltx_Math" altimg="m56.png" altimg-height="23px" altimg-valign="-7px" altimg-width="44px" alttext="\{b_{n}\}" display="inline"><mrow><mo stretchy="false">{</mo><msub><mi href="./22.20#SS2.p1">b</mi><mi href="./22.20#SS2.p1">n</mi></msub><mo stretchy="false">}</mo></mrow></math> converge to a common limit
<math class="ltx_Math" altimg="m25.png" altimg-height="23px" altimg-valign="-7px" altimg-width="135px" alttext="M=M(a_{0},b_{0})" display="inline"><mrow><mi href="./22.20#SS2.p1">M</mi><mo>=</mo><mrow><mi href="./22.20#SS2.p1">M</mi><mo></mo><mrow><mo stretchy="false">(</mo><msub><mi href="./22.20#SS2.p1">a</mi><mn>0</mn></msub><mo>,</mo><msub><mi href="./22.20#SS2.p1">b</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mrow></mrow></math>, the <em class="ltx_emph ltx_font_italic">arithmetic-geometric mean</em> of <math class="ltx_Math" altimg="m57.png" altimg-height="21px" altimg-valign="-6px" altimg-width="50px" alttext="a_{0},b_{0}" display="inline"><mrow><msub><mi href="./22.20#SS2.p1">a</mi><mn>0</mn></msub><mo>,</mo><msub><mi href="./22.20#SS2.p1">b</mi><mn>0</mn></msub></mrow></math>. And since</p>
<table id="E2" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">22.20.2</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E2.png" altimg-height="29px" altimg-valign="-7px" altimg-width="436px" alttext="\max\left(\left|a_{n}-M\right|,\left|b_{n}-M\right|,\left|c_{n}\right|\right)%
\leq\text{(const.)}\times 2^{-2^{n}}," display="block"><mrow><mrow><mrow><mi>max</mi><mo></mo><mrow><mo>(</mo><mrow><mo>|</mo><mrow><msub><mi href="./22.20#SS2.p1">a</mi><mi href="./22.20#SS2.p1">n</mi></msub><mo>-</mo><mi href="./22.20#SS2.p1">M</mi></mrow><mo>|</mo></mrow><mo>,</mo><mrow><mo>|</mo><mrow><msub><mi href="./22.20#SS2.p1">b</mi><mi href="./22.20#SS2.p1">n</mi></msub><mo>-</mo><mi href="./22.20#SS2.p1">M</mi></mrow><mo>|</mo></mrow><mo>,</mo><mrow><mo>|</mo><msub><mi href="./22.20#SS2.p1">c</mi><mi href="./22.20#SS2.p1">n</mi></msub><mo>|</mo></mrow><mo>)</mo></mrow></mrow><mo>≤</mo><mrow><mtext>(const.)</mtext><mo>×</mo><msup><mn>2</mn><mrow><mo>-</mo><msup><mn>2</mn><mi href="./22.20#SS2.p1">n</mi></msup></mrow></msup></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./22.20#SS2.p1" title="§22.20(ii) Arithmetic-Geometric Mean ‣ §22.20 Methods of Computation ‣ Computation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="16px" altimg-valign="-5px" altimg-width="26px" alttext="a_{n}" display="inline"><msub><mi href="./22.20#SS2.p1">a</mi><mi href="./22.20#SS2.p1">n</mi></msub></math>: numbers</a>,
<a href="./22.20#SS2.p1" title="§22.20(ii) Arithmetic-Geometric Mean ‣ §22.20 Methods of Computation ‣ Computation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m62.png" altimg-height="21px" altimg-valign="-5px" altimg-width="24px" alttext="b_{n}" display="inline"><msub><mi href="./22.20#SS2.p1">b</mi><mi href="./22.20#SS2.p1">n</mi></msub></math>: numbers</a>,
<a href="./22.20#SS2.p1" title="§22.20(ii) Arithmetic-Geometric Mean ‣ §22.20 Methods of Computation ‣ Computation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="16px" altimg-valign="-5px" altimg-width="24px" alttext="c_{n}" display="inline"><msub><mi href="./22.20#SS2.p1">c</mi><mi href="./22.20#SS2.p1">n</mi></msub></math>: numbers</a>,
<a href="./22.20#SS2.p1" title="§22.20(ii) Arithmetic-Geometric Mean ‣ §22.20 Methods of Computation ‣ Computation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m80.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./22.20#SS2.p1">n</mi></math>: positive</a> and
<a href="./22.20#SS2.p1" title="§22.20(ii) Arithmetic-Geometric Mean ‣ §22.20 Methods of Computation ‣ Computation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m23.png" altimg-height="23px" altimg-valign="-7px" altimg-width="87px" alttext="M(a_{0},b_{0})" display="inline"><mrow><mi href="./22.20#SS2.p1">M</mi><mo></mo><mrow><mo stretchy="false">(</mo><msub><mi href="./22.20#SS2.p1">a</mi><mn>0</mn></msub><mo>,</mo><msub><mi href="./22.20#SS2.p1">b</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mrow></math>: limit</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">convergence is very rapid.</p>
</div>
<div id="SS2.p2" class="ltx_para">
<p class="ltx_p">For <math class="ltx_Math" altimg="m88.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./22.1#p2.t1.r1">x</mi></math> real and <math class="ltx_Math" altimg="m72.png" altimg-height="23px" altimg-valign="-7px" altimg-width="84px" alttext="k\in(0,1)" display="inline"><mrow><mi href="./22.1#p2.t1.r3">k</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mn>0</mn><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mn>1</mn><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></mrow></math>, use () with <math class="ltx_Math" altimg="m58.png" altimg-height="20px" altimg-valign="-5px" altimg-width="60px" alttext="a_{0}=1" display="inline"><mrow><msub><mi href="./22.20#SS2.p1">a</mi><mn>0</mn></msub><mo>=</mo><mn>1</mn></mrow></math>,
<math class="ltx_Math" altimg="m61.png" altimg-height="24px" altimg-valign="-7px" altimg-width="134px" alttext="b_{0}=k^{\prime}\in(0,1)" display="inline"><mrow><msub><mi href="./22.20#SS2.p1">b</mi><mn>0</mn></msub><mo>=</mo><msup><mi href="./22.1#p2.t1.r4">k</mi><mo>′</mo></msup><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mn>0</mn><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mn>1</mn><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></mrow></math>, <math class="ltx_Math" altimg="m63.png" altimg-height="21px" altimg-valign="-5px" altimg-width="59px" alttext="c_{0}=k" display="inline"><mrow><msub><mi href="./22.20#SS2.p1">c</mi><mn>0</mn></msub><mo>=</mo><mi href="./22.1#p2.t1.r3">k</mi></mrow></math>, and continue until <math class="ltx_Math" altimg="m65.png" altimg-height="16px" altimg-valign="-5px" altimg-width="28px" alttext="c_{N}" display="inline"><msub><mi href="./22.20#SS2.p1">c</mi><mi>N</mi></msub></math> is zero to the
required accuracy. Next, compute <math class="ltx_Math" altimg="m50.png" altimg-height="22px" altimg-valign="-7px" altimg-width="153px" alttext="\phi_{N},\phi_{N-1},\dots,\phi_{0}" display="inline"><mrow><msub><mi href="./22.20#SS2.p2">ϕ</mi><mi>N</mi></msub><mo>,</mo><msub><mi href="./22.20#SS2.p2">ϕ</mi><mrow><mi>N</mi><mo>-</mo><mn>1</mn></mrow></msub><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><msub><mi href="./22.20#SS2.p2">ϕ</mi><mn>0</mn></msub></mrow></math>, where
</p>
<table id="E3" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">22.20.3</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E3.png" altimg-height="27px" altimg-valign="-6px" altimg-width="127px" alttext="\phi_{N}=2^{N}a_{N}x," display="block"><mrow><mrow><msub><mi href="./22.20#SS2.p2">ϕ</mi><mi>N</mi></msub><mo>=</mo><mrow><msup><mn>2</mn><mi>N</mi></msup><mo></mo><msub><mi href="./22.20#SS2.p1">a</mi><mi>N</mi></msub><mo></mo><mi href="./22.1#p2.t1.r1">x</mi></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./22.1#p2.t1.r1" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m88.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./22.1#p2.t1.r1">x</mi></math>: real</a>,
<a href="./22.20#SS2.p1" title="§22.20(ii) Arithmetic-Geometric Mean ‣ §22.20 Methods of Computation ‣ Computation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="16px" altimg-valign="-5px" altimg-width="26px" alttext="a_{n}" display="inline"><msub><mi href="./22.20#SS2.p1">a</mi><mi href="./22.20#SS2.p1">n</mi></msub></math>: numbers</a> and
<a href="./22.20#SS2.p2" title="§22.20(ii) Arithmetic-Geometric Mean ‣ §22.20 Methods of Computation ‣ Computation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="21px" altimg-valign="-6px" altimg-width="31px" alttext="\phi_{N}" display="inline"><msub><mi href="./22.20#SS2.p2">ϕ</mi><mi>N</mi></msub></math>: approximations</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E4" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">22.20.4</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E4.png" altimg-height="53px" altimg-valign="-21px" altimg-width="343px" alttext="\phi_{n-1}=\frac{1}{2}\left(\phi_{n}+\operatorname{arcsin}\left(\frac{c_{n}}{a%
_{n}}\sin\phi_{n}\right)\right)," display="block"><mrow><mrow><msub><mi href="./22.20#SS2.p2">ϕ</mi><mrow><mi href="./22.20#SS2.p1">n</mi><mo>-</mo><mn>1</mn></mrow></msub><mo>=</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mrow><mo>(</mo><mrow><msub><mi href="./22.20#SS2.p2">ϕ</mi><mi href="./22.20#SS2.p1">n</mi></msub><mo>+</mo><mrow><mi href="./4.23#SS2.p1">arcsin</mi><mo></mo><mrow><mo>(</mo><mrow><mfrac><msub><mi href="./22.20#SS2.p1">c</mi><mi href="./22.20#SS2.p1">n</mi></msub><msub><mi href="./22.20#SS2.p1">a</mi><mi href="./22.20#SS2.p1">n</mi></msub></mfrac><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><msub><mi href="./22.20#SS2.p2">ϕ</mi><mi href="./22.20#SS2.p1">n</mi></msub></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E4.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.23#SS2.p1" title="§4.23(ii) Principal Values ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m31.png" altimg-height="18px" altimg-valign="-2px" altimg-width="69px" alttext="\operatorname{arcsin}\NVar{z}" display="inline"><mrow><mi href="./4.23#SS2.p1">arcsin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: arcsine function</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./22.20#SS2.p1" title="§22.20(ii) Arithmetic-Geometric Mean ‣ §22.20 Methods of Computation ‣ Computation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="16px" altimg-valign="-5px" altimg-width="26px" alttext="a_{n}" display="inline"><msub><mi href="./22.20#SS2.p1">a</mi><mi href="./22.20#SS2.p1">n</mi></msub></math>: numbers</a>,
<a href="./22.20#SS2.p1" title="§22.20(ii) Arithmetic-Geometric Mean ‣ §22.20 Methods of Computation ‣ Computation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="16px" altimg-valign="-5px" altimg-width="24px" alttext="c_{n}" display="inline"><msub><mi href="./22.20#SS2.p1">c</mi><mi href="./22.20#SS2.p1">n</mi></msub></math>: numbers</a>,
<a href="./22.20#SS2.p1" title="§22.20(ii) Arithmetic-Geometric Mean ‣ §22.20 Methods of Computation ‣ Computation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m80.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./22.20#SS2.p1">n</mi></math>: positive</a> and
<a href="./22.20#SS2.p2" title="§22.20(ii) Arithmetic-Geometric Mean ‣ §22.20 Methods of Computation ‣ Computation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="21px" altimg-valign="-6px" altimg-width="31px" alttext="\phi_{N}" display="inline"><msub><mi href="./22.20#SS2.p2">ϕ</mi><mi>N</mi></msub></math>: approximations</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and the inverse sine has its principal value (§). Then</p>
<table id="E5" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex4" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="3" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">22.20.5</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m13.png" altimg-height="25px" altimg-valign="-7px" altimg-width="75px" alttext="\displaystyle\operatorname{sn}\left(x,k\right)" display="inline"><mrow><mi href="./22.2#E4">sn</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m6.png" altimg-height="23px" altimg-valign="-6px" altimg-width="81px" alttext="\displaystyle=\sin\phi_{0}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><msub><mi href="./22.20#SS2.p2">ϕ</mi><mn>0</mn></msub></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex5" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m11.png" altimg-height="25px" altimg-valign="-7px" altimg-width="76px" alttext="\displaystyle\operatorname{cn}\left(x,k\right)" display="inline"><mrow><mi href="./22.2#E5">cn</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m1.png" altimg-height="23px" altimg-valign="-6px" altimg-width="83px" alttext="\displaystyle=\cos\phi_{0}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><msub><mi href="./22.20#SS2.p2">ϕ</mi><mn>0</mn></msub></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex6" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m12.png" altimg-height="25px" altimg-valign="-7px" altimg-width="78px" alttext="\displaystyle\operatorname{dn}\left(x,k\right)" display="inline"><mrow><mi href="./22.2#E6">dn</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m2.png" altimg-height="51px" altimg-valign="-21px" altimg-width="149px" alttext="\displaystyle=\frac{\cos\phi_{0}}{\cos\left(\phi_{1}-\phi_{0}\right)}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mstyle displaystyle="true"><mfrac><mrow><mi href="./4.14#E2">cos</mi><mo></mo><msub><mi href="./22.20#SS2.p2">ϕ</mi><mn>0</mn></msub></mrow><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><msub><mi href="./22.20#SS2.p2">ϕ</mi><mn>1</mn></msub><mo>-</mo><msub><mi href="./22.20#SS2.p2">ϕ</mi><mn>0</mn></msub></mrow><mo>)</mo></mrow></mrow></mfrac></mstyle></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E5.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./22.2#E5" title="(22.2.5) ‣ §22.2 Definitions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m34.png" altimg-height="23px" altimg-valign="-7px" altimg-width="73px" alttext="\operatorname{cn}\left(\NVar{z},\NVar{k}\right)" display="inline"><mrow><mi href="./22.2#E5">cn</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r2">z</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobian elliptic function</a>,
<a href="./22.2#E6" title="(22.2.6) ‣ §22.2 Definitions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="23px" altimg-valign="-7px" altimg-width="75px" alttext="\operatorname{dn}\left(\NVar{z},\NVar{k}\right)" display="inline"><mrow><mi href="./22.2#E6">dn</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r2">z</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobian elliptic function</a>,
<a href="./22.2#E4" title="(22.2.4) ‣ §22.2 Definitions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="23px" altimg-valign="-7px" altimg-width="72px" alttext="\operatorname{sn}\left(\NVar{z},\NVar{k}\right)" display="inline"><mrow><mi href="./22.2#E4">sn</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r2">z</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobian elliptic function</a>,
<a href="./19.2#E8" title="(19.2.8) ‣ §19.2(ii) Legendre’s Integrals ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m21.png" altimg-height="23px" altimg-valign="-7px" altimg-width="52px" alttext="K\left(\NVar{k}\right)" display="inline"><mrow><mi href="./19.2#E8">K</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Legendre’s complete elliptic integral of the first kind</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m27.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./22.1#p2.t1.r1" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m88.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./22.1#p2.t1.r1">x</mi></math>: real</a>,
<a href="./22.1#p2.t1.r3" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m71.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./22.1#p2.t1.r3">k</mi></math>: modulus</a> and
<a href="./22.20#SS2.p2" title="§22.20(ii) Arithmetic-Geometric Mean ‣ §22.20 Methods of Computation ‣ Computation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="21px" altimg-valign="-6px" altimg-width="31px" alttext="\phi_{N}" display="inline"><msub><mi href="./22.20#SS2.p2">ϕ</mi><mi>N</mi></msub></math>: approximations</a>
</dd>
<dt>Note (effective with 1.0.10):</dt>
<dd>
A note was added after () to deal with cases when computation
of <math class="ltx_Math" altimg="m41.png" altimg-height="23px" altimg-valign="-7px" altimg-width="76px" alttext="\operatorname{dn}\left(x,k\right)" display="inline"><mrow><mi href="./22.2#E6">dn</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math> with () becomes numerically unstable
near <math class="ltx_Math" altimg="m87.png" altimg-height="18px" altimg-valign="-2px" altimg-width="61px" alttext="x=K" display="inline"><mrow><mi href="./22.1#p2.t1.r1">x</mi><mo>=</mo><mrow><mi href="./19.2#E8">K</mi><mo></mo></mrow></mrow></math>.
<p><span class="ltx_font_italic">Suggested 2014-10-20 by Hartmut Henkel</span></p>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and the subsidiary functions can be found using ().
This formula for <math class="ltx_Math" altimg="m35.png" altimg-height="18px" altimg-valign="-2px" altimg-width="26px" alttext="\operatorname{dn}" display="inline"><mi href="./22.2#E6">dn</mi></math> becomes unstable near <math class="ltx_Math" altimg="m87.png" altimg-height="18px" altimg-valign="-2px" altimg-width="61px" alttext="x=K" display="inline"><mrow><mi href="./22.1#p2.t1.r1">x</mi><mo>=</mo><mi href="./19.2#E8">K</mi></mrow></math>.
If only the value of <math class="ltx_Math" altimg="m41.png" altimg-height="23px" altimg-valign="-7px" altimg-width="76px" alttext="\operatorname{dn}\left(x,k\right)" display="inline"><mrow><mi href="./22.2#E6">dn</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math> at <math class="ltx_Math" altimg="m87.png" altimg-height="18px" altimg-valign="-2px" altimg-width="61px" alttext="x=K" display="inline"><mrow><mi href="./22.1#p2.t1.r1">x</mi><mo>=</mo><mi href="./19.2#E8">K</mi></mrow></math> is required then the
exact value is in the table .
If both <math class="ltx_Math" altimg="m71.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./22.1#p2.t1.r3">k</mi></math> and <math class="ltx_Math" altimg="m88.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./22.1#p2.t1.r1">x</mi></math> are real then <math class="ltx_Math" altimg="m35.png" altimg-height="18px" altimg-valign="-2px" altimg-width="26px" alttext="\operatorname{dn}" display="inline"><mi href="./22.2#E6">dn</mi></math> is strictly positive and
<math class="ltx_Math" altimg="m40.png" altimg-height="28px" altimg-valign="-8px" altimg-width="255px" alttext="\operatorname{dn}\left(x,k\right)=\sqrt{1-k^{2}{\operatorname{sn}^{2}}\left(x,%
k\right)}" display="inline"><mrow><mrow><mi href="./22.2#E6">dn</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow><mo>=</mo><msqrt><mrow><mn>1</mn><mo>-</mo><mrow><msup><mi href="./22.1#p2.t1.r3">k</mi><mn>2</mn></msup><mo></mo><mrow><msup><mi href="./22.2#E4">sn</mi><mn>2</mn></msup><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></mrow></mrow></msqrt></mrow></math> which follows from
(). If either <math class="ltx_Math" altimg="m71.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./22.1#p2.t1.r3">k</mi></math> or <math class="ltx_Math" altimg="m88.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./22.1#p2.t1.r1">x</mi></math> is complex then ()
gives the definition of <math class="ltx_Math" altimg="m41.png" altimg-height="23px" altimg-valign="-7px" altimg-width="76px" alttext="\operatorname{dn}\left(x,k\right)" display="inline"><mrow><mi href="./22.2#E6">dn</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math> as a quotient of theta functions.</p>
</div>
<div id="SS2.p3" class="ltx_para">
<p class="ltx_p">See also <cite class="ltx_cite ltx_citemacro_citet">Wachspress (</dd>
</dl>
</div>
</div>
<div id="Px1.p1" class="ltx_para">
<p class="ltx_p">To compute <math class="ltx_Math" altimg="m45.png" altimg-height="13px" altimg-valign="-2px" altimg-width="23px" alttext="\operatorname{sn}" display="inline"><mi href="./22.2#E4">sn</mi></math>, <math class="ltx_Math" altimg="m32.png" altimg-height="13px" altimg-valign="-2px" altimg-width="24px" alttext="\operatorname{cn}" display="inline"><mi href="./22.2#E5">cn</mi></math>, <math class="ltx_Math" altimg="m35.png" altimg-height="18px" altimg-valign="-2px" altimg-width="26px" alttext="\operatorname{dn}" display="inline"><mi href="./22.2#E6">dn</mi></math> to 10D when <math class="ltx_Math" altimg="m86.png" altimg-height="17px" altimg-valign="-2px" altimg-width="68px" alttext="x=0.8" display="inline"><mrow><mi href="./22.1#p2.t1.r1">x</mi><mo>=</mo><mn>0.8</mn></mrow></math>,
<math class="ltx_Math" altimg="m68.png" altimg-height="18px" altimg-valign="-2px" altimg-width="77px" alttext="k=0.65" display="inline"><mrow><mi href="./22.1#p2.t1.r3">k</mi><mo>=</mo><mn>0.65</mn></mrow></math>.</p>
</div>
<div id="Px1.p2" class="ltx_para">
<p class="ltx_p">Four iterations of () lead to <math class="ltx_Math" altimg="m64.png" altimg-height="23px" altimg-valign="-5px" altimg-width="147px" alttext="c_{4}=\Sci{6.5}{-12}" display="inline"><mrow><msub><mi href="./22.20#SS2.p1">c</mi><mn>4</mn></msub><mo>=</mo><mn mathvariant="italic">6.5×10⁻¹²</mn></mrow></math>. From
() we obtain <math class="ltx_Math" altimg="m49.png" altimg-height="21px" altimg-valign="-6px" altimg-width="172px" alttext="\phi_{1}=1.40213\;91827" display="inline"><mrow><msub><mi href="./22.20#SS2.p2">ϕ</mi><mn>1</mn></msub><mo>=</mo><mn>1.40213 91827</mn></mrow></math>
and <math class="ltx_Math" altimg="m48.png" altimg-height="21px" altimg-valign="-6px" altimg-width="172px" alttext="\phi_{0}=0.76850\;92170" display="inline"><mrow><msub><mi href="./22.20#SS2.p2">ϕ</mi><mn>0</mn></msub><mo>=</mo><mn>0.76850 92170</mn></mrow></math>. Then from (),
<math class="ltx_Math" altimg="m46.png" altimg-height="23px" altimg-valign="-7px" altimg-width="259px" alttext="\operatorname{sn}\left(0.8,0.65\right)=0.69506\;42165" display="inline"><mrow><mrow><mi href="./22.2#E4">sn</mi><mo></mo><mrow><mo>(</mo><mn>0.8</mn><mo>,</mo><mn>0.65</mn><mo>)</mo></mrow></mrow><mo>=</mo><mn>0.69506 42165</mn></mrow></math>,
<math class="ltx_Math" altimg="m33.png" altimg-height="23px" altimg-valign="-7px" altimg-width="260px" alttext="\operatorname{cn}\left(0.8,0.65\right)=0.71894\;76580" display="inline"><mrow><mrow><mi href="./22.2#E5">cn</mi><mo></mo><mrow><mo>(</mo><mn>0.8</mn><mo>,</mo><mn>0.65</mn><mo>)</mo></mrow></mrow><mo>=</mo><mn>0.71894 76580</mn></mrow></math>,
<math class="ltx_Math" altimg="m38.png" altimg-height="23px" altimg-valign="-7px" altimg-width="262px" alttext="\operatorname{dn}\left(0.8,0.65\right)=0.89212\;34349" display="inline"><mrow><mrow><mi href="./22.2#E6">dn</mi><mo></mo><mrow><mo>(</mo><mn>0.8</mn><mo>,</mo><mn>0.65</mn><mo>)</mo></mrow></mrow><mo>=</mo><mn>0.89212 34349</mn></mrow></math>.</p>
</div>
</section>
</section>
<section id="iii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§22.20(iii) </span>Landen Transformations</h2>
<div id="SS3.info" class="ltx_metadata ltx_info">
, <math class="ltx_Math" altimg="m71.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./22.1#p2.t1.r3">k</mi></math> or <math class="ltx_Math" altimg="m78.png" altimg-height="19px" altimg-valign="-2px" altimg-width="21px" alttext="k^{\prime}" display="inline"><msup><mi href="./22.1#p2.t1.r4">k</mi><mo>′</mo></msup></math> can always be made sufficently small to enable
the approximations given in §</dd>
</dl>
</div>
</div>
<div id="Px2.p1" class="ltx_para">
<p class="ltx_p">To compute <math class="ltx_Math" altimg="m41.png" altimg-height="23px" altimg-valign="-7px" altimg-width="76px" alttext="\operatorname{dn}\left(x,k\right)" display="inline"><mrow><mi href="./22.2#E6">dn</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math> to 6D for <math class="ltx_Math" altimg="m85.png" altimg-height="17px" altimg-valign="-2px" altimg-width="68px" alttext="x=0.2" display="inline"><mrow><mi href="./22.1#p2.t1.r1">x</mi><mo>=</mo><mn>0.2</mn></mrow></math>, <math class="ltx_Math" altimg="m74.png" altimg-height="20px" altimg-valign="-2px" altimg-width="86px" alttext="k^{2}=0.19" display="inline"><mrow><msup><mi href="./22.1#p2.t1.r3">k</mi><mn>2</mn></msup><mo>=</mo><mn>0.19</mn></mrow></math>, <math class="ltx_Math" altimg="m75.png" altimg-height="19px" altimg-valign="-2px" altimg-width="73px" alttext="k^{\prime}=0.9" display="inline"><mrow><msup><mi href="./22.1#p2.t1.r4">k</mi><mo>′</mo></msup><mo>=</mo><mn>0.9</mn></mrow></math>.</p>
</div>
<div id="Px2.p2" class="ltx_para">
<p class="ltx_p">From (), <math class="ltx_Math" altimg="m79.png" altimg-height="27px" altimg-valign="-9px" altimg-width="71px" alttext="k_{1}=\tfrac{1}{19}" display="inline"><mrow><msub><mi href="./22.1#p2.t1.r3">k</mi><mn>1</mn></msub><mo>=</mo><mfrac><mn>1</mn><mn>19</mn></mfrac></mrow></math> and <math class="ltx_Math" altimg="m84.png" altimg-height="23px" altimg-valign="-7px" altimg-width="157px" alttext="x/(1+k_{1})=0.19" display="inline"><mrow><mrow><mi href="./22.1#p2.t1.r1">x</mi><mo>/</mo><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>+</mo><msub><mi href="./22.1#p2.t1.r3">k</mi><mn>1</mn></msub></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mn>0.19</mn></mrow></math>. From
the first two terms in () we find
<math class="ltx_Math" altimg="m36.png" altimg-height="27px" altimg-valign="-9px" altimg-width="215px" alttext="\operatorname{dn}\left(0.19,\tfrac{1}{19}\right)=0.999951" display="inline"><mrow><mrow><mi href="./22.2#E6">dn</mi><mo></mo><mrow><mo>(</mo><mn>0.19</mn><mo>,</mo><mfrac><mn>1</mn><mn>19</mn></mfrac><mo>)</mo></mrow></mrow><mo>=</mo><mn>0.999951</mn></mrow></math>. Then by using
() we have <math class="ltx_Math" altimg="m37.png" altimg-height="29px" altimg-valign="-9px" altimg-width="236px" alttext="\operatorname{dn}\left(0.2,\sqrt{0.19}\right)=0.996253" display="inline"><mrow><mrow><mi href="./22.2#E6">dn</mi><mo></mo><mrow><mo>(</mo><mn>0.2</mn><mo>,</mo><msqrt><mn>0.19</mn></msqrt><mo>)</mo></mrow></mrow><mo>=</mo><mn>0.996253</mn></mrow></math>.</p>
</div>
<div id="Px2.p3" class="ltx_para">
<p class="ltx_p">If needed, the corresponding values of <math class="ltx_Math" altimg="m45.png" altimg-height="13px" altimg-valign="-2px" altimg-width="23px" alttext="\operatorname{sn}" display="inline"><mi href="./22.2#E4">sn</mi></math> and <math class="ltx_Math" altimg="m32.png" altimg-height="13px" altimg-valign="-2px" altimg-width="24px" alttext="\operatorname{cn}" display="inline"><mi href="./22.2#E5">cn</mi></math> can be found
subsequently by applying (</dd>
</dl>
</div>
</div>
<div id="SS4.p1" class="ltx_para">
<p class="ltx_p">If either <math class="ltx_Math" altimg="m54.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\tau" display="inline"><mi href="./22.1#p2.t1.r7">τ</mi></math> or <math class="ltx_Math" altimg="m83.png" altimg-height="24px" altimg-valign="-6px" altimg-width="75px" alttext="q=e^{i\pi\tau}" display="inline"><mrow><mi href="./22.2#E1">q</mi><mo>=</mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi href="./22.1#p2.t1.r7">τ</mi></mrow></msup></mrow></math> is given, then we use
<math class="ltx_Math" altimg="m70.png" altimg-height="25px" altimg-valign="-7px" altimg-width="187px" alttext="k={\theta_{2}^{2}}\left(0,q\right)/{\theta_{3}^{2}}\left(0,q\right)" display="inline"><mrow><mi href="./22.1#p2.t1.r3">k</mi><mo>=</mo><mrow><mrow><msubsup><mi href="./20.2#i">θ</mi><mn>2</mn><mn>2</mn></msubsup><mo></mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi href="./22.2#E1">q</mi><mo>)</mo></mrow></mrow><mo>/</mo><mrow><msubsup><mi href="./20.2#i">θ</mi><mn>3</mn><mn>2</mn></msubsup><mo></mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi href="./22.2#E1">q</mi><mo>)</mo></mrow></mrow></mrow></mrow></math>,
<math class="ltx_Math" altimg="m77.png" altimg-height="25px" altimg-valign="-7px" altimg-width="193px" alttext="k^{\prime}={\theta_{4}^{2}}\left(0,q\right)/{\theta_{3}^{2}}\left(0,q\right)" display="inline"><mrow><msup><mi href="./22.1#p2.t1.r4">k</mi><mo>′</mo></msup><mo>=</mo><mrow><mrow><msubsup><mi href="./20.2#i">θ</mi><mn>4</mn><mn>2</mn></msubsup><mo></mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi href="./22.2#E1">q</mi><mo>)</mo></mrow></mrow><mo>/</mo><mrow><msubsup><mi href="./20.2#i">θ</mi><mn>3</mn><mn>2</mn></msubsup><mo></mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi href="./22.2#E1">q</mi><mo>)</mo></mrow></mrow></mrow></mrow></math>,
<math class="ltx_Math" altimg="m19.png" altimg-height="27px" altimg-valign="-9px" altimg-width="140px" alttext="K=\frac{1}{2}\pi{\theta_{3}^{2}}\left(0,q\right)" display="inline"><mrow><mrow><mi href="./19.2#E8">K</mi><mo></mo></mrow><mo>=</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mrow><msubsup><mi href="./20.2#i">θ</mi><mn>3</mn><mn>2</mn></msubsup><mo></mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi href="./22.2#E1">q</mi><mo>)</mo></mrow></mrow></mrow></mrow></math>, and <math class="ltx_Math" altimg="m22.png" altimg-height="20px" altimg-valign="-4px" altimg-width="106px" alttext="K^{\prime}=-i\tau K" display="inline"><mrow><mrow><msup><mi href="./19.2#E9">K</mi><mo href="./19.2#E9">′</mo></msup><mo></mo></mrow><mo>=</mo><mrow><mo>-</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./22.1#p2.t1.r7">τ</mi><mo></mo><mrow><mi href="./19.2#E8">K</mi><mo></mo></mrow></mrow></mrow></mrow></math>, obtaining
the values of the theta functions as in §.</p>
</div>
<div id="SS4.p2" class="ltx_para">
<p class="ltx_p">If <math class="ltx_Math" altimg="m67.png" altimg-height="22px" altimg-valign="-6px" altimg-width="41px" alttext="k,k^{\prime}" display="inline"><mrow><mi href="./22.1#p2.t1.r3">k</mi><mo>,</mo><msup><mi href="./22.1#p2.t1.r4">k</mi><mo>′</mo></msup></mrow></math> are given with <math class="ltx_Math" altimg="m73.png" altimg-height="24px" altimg-valign="-4px" altimg-width="111px" alttext="k^{2}+{k^{\prime}}^{2}=1" display="inline"><mrow><mrow><msup><mi href="./22.1#p2.t1.r3">k</mi><mn>2</mn></msup><mo>+</mo><mmultiscripts><mi href="./22.1#p2.t1.r4">k</mi><none></none><mo>′</mo><none></none><mn>2</mn></mmultiscripts></mrow><mo>=</mo><mn>1</mn></mrow></math> and
<math class="ltx_Math" altimg="m26.png" altimg-height="24px" altimg-valign="-7px" altimg-width="107px" alttext="\Im k^{\prime}/\Im k<0" display="inline"><mrow><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℑ</mi><mo></mo><msup><mi href="./22.1#p2.t1.r4">k</mi><mo>′</mo></msup></mrow><mo>/</mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℑ</mi><mo></mo><mi href="./22.1#p2.t1.r3">k</mi></mrow></mrow><mo><</mo><mn>0</mn></mrow></math>, then <math class="ltx_Math" altimg="m17.png" altimg-height="22px" altimg-valign="-6px" altimg-width="54px" alttext="K,K^{\prime}" display="inline"><mrow><mrow><mi href="./19.2#E8">K</mi><mo></mo></mrow><mo>,</mo><mrow><msup><mi href="./19.2#E9">K</mi><mo href="./19.2#E9">′</mo></msup><mo></mo></mrow></mrow></math> can be
found from</p>
<table id="E6" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex7" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">22.20.6</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m9.png" altimg-height="19px" altimg-valign="-2px" altimg-width="24px" alttext="\displaystyle K" display="inline"><mrow><mi href="./19.2#E8">K</mi><mo></mo></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m4.png" altimg-height="46px" altimg-valign="-21px" altimg-width="120px" alttext="\displaystyle=\frac{\pi}{2M(1,k^{\prime})}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mstyle displaystyle="true"><mfrac><mi href="./3.12#E1">π</mi><mrow><mn>2</mn><mo></mo><mrow><mi href="./22.20#SS2.p1">M</mi><mo></mo><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><msup><mi href="./22.1#p2.t1.r4">k</mi><mo>′</mo></msup><mo stretchy="false">)</mo></mrow></mrow></mrow></mfrac></mstyle></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex8" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m10.png" altimg-height="21px" altimg-valign="-2px" altimg-width="30px" alttext="\displaystyle K^{\prime}" display="inline"><mrow><msup><mi href="./19.2#E9">K</mi><mo href="./19.2#E9">′</mo></msup><mo></mo></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m3.png" altimg-height="46px" altimg-valign="-21px" altimg-width="114px" alttext="\displaystyle=\frac{\pi}{2M(1,k)}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mstyle displaystyle="true"><mfrac><mi href="./3.12#E1">π</mi><mrow><mn>2</mn><mo></mo><mrow><mi href="./22.20#SS2.p1">M</mi><mo></mo><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></mfrac></mstyle></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E6.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./19.2#E9" title="(19.2.9) ‣ §19.2(ii) Legendre’s Integrals ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m90.png" altimg-height="24px" altimg-valign="-7px" altimg-width="58px" alttext="{K^{\prime}}\left(\NVar{k}\right)" display="inline"><mrow><msup><mi href="./19.2#E9">K</mi><mo href="./19.2#E9">′</mo></msup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Legendre’s complementary complete elliptic integral of the first kind</a>,
<a href="./19.2#E8" title="(19.2.8) ‣ §19.2(ii) Legendre’s Integrals ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals" class="ltx_ref"><math class="ltx_Math" altimg="m21.png" altimg-height="23px" altimg-valign="-7px" altimg-width="52px" alttext="K\left(\NVar{k}\right)" display="inline"><mrow><mi href="./19.2#E8">K</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./19.1#p1.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Legendre’s complete elliptic integral of the first kind</a>,
<a href="./22.1#p2.t1.r3" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m71.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./22.1#p2.t1.r3">k</mi></math>: modulus</a>,
<a href="./22.1#p2.t1.r4" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="19px" altimg-valign="-2px" altimg-width="21px" alttext="k^{\prime}" display="inline"><msup><mi href="./22.1#p2.t1.r4">k</mi><mo>′</mo></msup></math>: complementary modulus</a> and
<a href="./22.20#SS2.p1" title="§22.20(ii) Arithmetic-Geometric Mean ‣ §22.20 Methods of Computation ‣ Computation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m23.png" altimg-height="23px" altimg-valign="-7px" altimg-width="87px" alttext="M(a_{0},b_{0})" display="inline"><mrow><mi href="./22.20#SS2.p1">M</mi><mo></mo><mrow><mo stretchy="false">(</mo><msub><mi href="./22.20#SS2.p1">a</mi><mn>0</mn></msub><mo>,</mo><msub><mi href="./22.20#SS2.p1">b</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mrow></math>: limit</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">using the arithmetic-geometric mean.</p>
</div>
<section id="Px3" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Example 1</h3>
<div id="Px3.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px3.p1" class="ltx_para">
<p class="ltx_p">If <math class="ltx_Math" altimg="m69.png" altimg-height="27px" altimg-valign="-7px" altimg-width="132px" alttext="k=k^{\prime}=1/\sqrt{2}" display="inline"><mrow><mi href="./22.1#p2.t1.r3">k</mi><mo>=</mo><msup><mi href="./22.1#p2.t1.r4">k</mi><mo>′</mo></msup><mo>=</mo><mrow><mn>1</mn><mo>/</mo><msqrt><mn>2</mn></msqrt></mrow></mrow></math>, then three iterations of () give
<math class="ltx_Math" altimg="m24.png" altimg-height="18px" altimg-valign="-2px" altimg-width="173px" alttext="M=0.84721\;30848" display="inline"><mrow><mi href="./22.20#SS2.p1">M</mi><mo>=</mo><mn>0.84721 30848</mn></mrow></math>, and from () <math class="ltx_Math" altimg="m20.png" altimg-height="23px" altimg-valign="-7px" altimg-width="265px" alttext="K=\pi/(2M)=1.85407\;46773" display="inline"><mrow><mrow><mi href="./19.2#E8">K</mi><mo></mo></mrow><mo>=</mo><mrow><mi href="./3.12#E1">π</mi><mo>/</mo><mrow><mo stretchy="false">(</mo><mrow><mn>2</mn><mo></mo><mi href="./22.20#SS2.p1">M</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mn>1.85407 46773</mn></mrow></math>
— in agreement with the value of
<math class="ltx_Math" altimg="m28.png" altimg-height="32px" altimg-valign="-9px" altimg-width="149px" alttext="\left(\Gamma\left(\tfrac{1}{4}\right)\right)^{2}/\left(4\sqrt{\pi}\right)" display="inline"><mrow><msup><mrow><mo>(</mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mo>)</mo></mrow></mrow><mo>)</mo></mrow><mn>2</mn></msup><mo>/</mo><mrow><mo>(</mo><mrow><mn>4</mn><mo></mo><msqrt><mi href="./3.12#E1">π</mi></msqrt></mrow><mo>)</mo></mrow></mrow></math>;
compare (</dd>
</dl>
</div>
</div>
<div id="Px4.p1" class="ltx_para">
<p class="ltx_p">If <math class="ltx_Math" altimg="m76.png" altimg-height="20px" altimg-valign="-4px" altimg-width="89px" alttext="k^{\prime}=1-i" display="inline"><mrow><msup><mi href="./22.1#p2.t1.r4">k</mi><mo>′</mo></msup><mo>=</mo><mrow><mn>1</mn><mo>-</mo><mi mathvariant="normal">i</mi></mrow></mrow></math>, then four iterations of () give
<math class="ltx_Math" altimg="m18.png" altimg-height="19px" altimg-valign="-4px" altimg-width="322px" alttext="K=1.23969\;74481+i0.56499\;30988" display="inline"><mrow><mrow><mi href="./19.2#E8">K</mi><mo></mo></mrow><mo>=</mo><mrow><mn>1.23969 74481</mn><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mn>0.56499 30988</mn></mrow></mrow></mrow></math>.</p>
</div>
</section>
</section>
<section id="v" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§22.20(v) </span>Inverse Functions</h2>
<div id="SS5.info" class="ltx_metadata ltx_info">
, <a class="ltx_keyword" href="./idx/#amplitudeamfunction">amplitude (<math class="ltx_Math" altimg="m29.png" altimg-height="13px" altimg-valign="-2px" altimg-width="31px" alttext="\operatorname{am}" display="inline"><mi href="./22.16#E1">am</mi></math>) function</a>
</dd>
</dl>
</div>
</div>
<div id="SS6.p1" class="ltx_para">
<p class="ltx_p"><math class="ltx_Math" altimg="m30.png" altimg-height="23px" altimg-valign="-7px" altimg-width="81px" alttext="\operatorname{am}\left(x,k\right)" display="inline"><mrow><mi href="./22.16#E1">am</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math> can be computed from its definition ()</cite> shows how to apply the arithmetic-geometric mean to
compute <math class="ltx_Math" altimg="m30.png" altimg-height="23px" altimg-valign="-7px" altimg-width="81px" alttext="\operatorname{am}\left(x,k\right)" display="inline"><mrow><mi href="./22.16#E1">am</mi><mo></mo><mrow><mo>(</mo><mi href="./22.1#p2.t1.r1">x</mi><mo>,</mo><mi href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>.</p>
</div>
<div id="SS6.p2" class="ltx_para">
<p class="ltx_p">Jacobi’s epsilon function can be computed from its representation
(</div>
</div>
</body></text>
</html>
</page>
<page>
<!DOCTYPE html><html>
<head>
<title>DLMF: 29.15 Fourier Series and Chebyshev Series</title>
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<div class="ltx_page_navlogo"></dd>
</dl>
</div>
</div>
<section id="Px1" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Polynomial <math class="ltx_Math" altimg="m53.png" altimg-height="27px" altimg-valign="-9px" altimg-width="110px" alttext="\mathit{uE}^{m}_{2n}\left(z,k^{2}\right)" display="inline"><mrow><msubsup><mi href="./29.12#E1" mathvariant="italic">uE</mi><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">n</mi></mrow><mi href="./29.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo>(</mo><mi href="./29.1#p2.t1.r3">z</mi><mo>,</mo><msup><mi href="./29.1#p2.t1.r4">k</mi><mn>2</mn></msup><mo>)</mo></mrow></mrow></math>
</h3>
<div id="Px1.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px1.p1" class="ltx_para">
<p class="ltx_p">When <math class="ltx_Math" altimg="m57.png" altimg-height="17px" altimg-valign="-2px" altimg-width="64px" alttext="\nu=2n" display="inline"><mrow><mi href="./29.1#p2.t1.r4">ν</mi><mo>=</mo><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">n</mi></mrow></mrow></math>, <math class="ltx_Math" altimg="m68.png" altimg-height="20px" altimg-valign="-6px" altimg-width="133px" alttext="m=0,1,\dots,n" display="inline"><mrow><mi href="./29.1#p2.t1.r1">m</mi><mo>=</mo><mrow><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><mi href="./29.1#p2.t1.r1">n</mi></mrow></mrow></math>, the Fourier series ()
terminates:</p>
<table id="E1" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">29.15.1</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E1.png" altimg-height="66px" altimg-valign="-30px" altimg-width="353px" alttext="\mathit{uE}^{m}_{2n}\left(z,k^{2}\right)=\tfrac{1}{2}A_{0}+\sum_{p=1}^{n}A_{2p%
}\cos\left(2p\phi\right)." display="block"><mrow><mrow><mrow><msubsup><mi href="./29.12#E1" mathvariant="italic">uE</mi><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">n</mi></mrow><mi href="./29.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo>(</mo><mi href="./29.1#p2.t1.r3">z</mi><mo>,</mo><msup><mi href="./29.1#p2.t1.r4">k</mi><mn>2</mn></msup><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><msub><mi href="./29.6#SS2.p1">A</mi><mn>0</mn></msub></mrow><mo>+</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./29.1#p2.t1.r1">p</mi><mo>=</mo><mn>1</mn></mrow><mi href="./29.1#p2.t1.r1">n</mi></munderover><mrow><msub><mi href="./29.6#SS2.p1">A</mi><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow></msub><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi><mo></mo><mi href="./29.2#E5">ϕ</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./29.12#E1" title="(29.12.1) ‣ §29.12(i) Elliptic-Function Form ‣ §29.12 Definitions ‣ Lamé Polynomials ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="27px" altimg-valign="-9px" altimg-width="110px" alttext="\mathit{uE}^{\NVar{m}}_{2\NVar{n}}\left(\NVar{z},\NVar{k^{2}}\right)" display="inline"><mrow><msubsup><mi href="./29.12#E1" mathvariant="italic">uE</mi><mrow><mn>2</mn><mo></mo><mi class="ltx_nvar" href="./29.1#p2.t1.r1">n</mi></mrow><mi class="ltx_nvar" href="./29.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./29.1#p2.t1.r3">z</mi><mo>,</mo><msup><mi class="ltx_nvar" href="./29.1#p2.t1.r4">k</mi><mn class="ltx_nvar">2</mn></msup><mo>)</mo></mrow></mrow></math>: Lamé polynomial</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m32.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./29.1#p2.t1.r1">m</mi></math>: nonnegative integer</a>,
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./29.1#p2.t1.r1">n</mi></math>: nonnegative integer</a>,
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="p" display="inline"><mi href="./29.1#p2.t1.r1">p</mi></math>: nonnegative integer</a>,
<a href="./29.1#p2.t1.r3" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m73.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./29.1#p2.t1.r3">z</mi></math>: complex variable</a>,
<a href="./29.1#p2.t1.r4" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m67.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./29.1#p2.t1.r4">k</mi></math>: real parameter</a>,
<a href="./29.2#E5" title="(29.2.5) ‣ §29.2(ii) Other Forms ‣ §29.2 Differential Equations ‣ Lamé Functions ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m62.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="\phi" display="inline"><mi href="./29.2#E5">ϕ</mi></math>: change of variable</a> and
<a href="./29.6#SS2.p1" title="§29.6(ii) Function Ec + 2 m 1 ν ( z , k 2 ) ‣ §29.6 Fourier Series ‣ Lamé Functions ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m17.png" altimg-height="23px" altimg-valign="-8px" altimg-width="36px" alttext="A_{2p}" display="inline"><msub><mi href="./29.6#SS2.p1">A</mi><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow></msub></math>: coefficients</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">A convenient way of constructing the coefficients, together with the
eigenvalues, is as follows. Equations (), with
<math class="ltx_Math" altimg="m71.png" altimg-height="20px" altimg-valign="-6px" altimg-width="126px" alttext="p=1,2,\dots,n" display="inline"><mrow><mi href="./29.1#p2.t1.r1">p</mi><mo>=</mo><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><mi href="./29.1#p2.t1.r1">n</mi></mrow></mrow></math>, (), and <math class="ltx_Math" altimg="m16.png" altimg-height="22px" altimg-valign="-7px" altimg-width="95px" alttext="A_{2n+2}=0" display="inline"><mrow><msub><mi href="./29.6#SS2.p1">A</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mn>2</mn></mrow></msub><mo>=</mo><mn>0</mn></mrow></math> can be cast as an
algebraic eigenvalue problem in the following way. Let</p>
<table id="E2" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">29.15.2</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E2.png" altimg-height="166px" altimg-valign="-77px" altimg-width="332px" alttext="\mathbf{M}=\begin{bmatrix}\beta_{0}&\alpha_{0}&0&\cdots&0\\
\gamma_{1}&\beta_{1}&\alpha_{1}&\ddots&\vdots\\
0&\ddots&\ddots&\ddots&0\\
\vdots&\ddots&\gamma_{n-1}&\beta_{n-1}&\alpha_{n-1}\\
0&\cdots&0&\gamma_{n}&\beta_{n}\end{bmatrix}" display="block"><mrow><mi mathvariant="bold">M</mi><mo>=</mo><mrow><mo>[</mo><mtable columnspacing="5pt" displaystyle="true" rowspacing="0pt"><mtr><mtd columnalign="center"><msub><mi href="./29.3#E12">β</mi><mn>0</mn></msub></mtd><mtd columnalign="center"><msub><mi href="./29.3#E11">α</mi><mn>0</mn></msub></mtd><mtd columnalign="center"><mn>0</mn></mtd><mtd columnalign="center"><mi mathvariant="normal">⋯</mi></mtd><mtd columnalign="center"><mn>0</mn></mtd></mtr><mtr><mtd columnalign="center"><msub><mi href="./29.3#E12">γ</mi><mn>1</mn></msub></mtd><mtd columnalign="center"><msub><mi href="./29.3#E12">β</mi><mn>1</mn></msub></mtd><mtd columnalign="center"><msub><mi href="./29.3#E11">α</mi><mn>1</mn></msub></mtd><mtd columnalign="center"><mi mathvariant="normal">⋱</mi></mtd><mtd columnalign="center"><mi mathvariant="normal">⋮</mi></mtd></mtr><mtr><mtd columnalign="center"><mn>0</mn></mtd><mtd columnalign="center"><mi mathvariant="normal">⋱</mi></mtd><mtd columnalign="center"><mi mathvariant="normal">⋱</mi></mtd><mtd columnalign="center"><mi mathvariant="normal">⋱</mi></mtd><mtd columnalign="center"><mn>0</mn></mtd></mtr><mtr><mtd columnalign="center"><mi mathvariant="normal">⋮</mi></mtd><mtd columnalign="center"><mi mathvariant="normal">⋱</mi></mtd><mtd columnalign="center"><msub><mi href="./29.3#E12">γ</mi><mrow><mi href="./29.1#p2.t1.r1">n</mi><mo>-</mo><mn>1</mn></mrow></msub></mtd><mtd columnalign="center"><msub><mi href="./29.3#E12">β</mi><mrow><mi href="./29.1#p2.t1.r1">n</mi><mo>-</mo><mn>1</mn></mrow></msub></mtd><mtd columnalign="center"><msub><mi href="./29.3#E11">α</mi><mrow><mi href="./29.1#p2.t1.r1">n</mi><mo>-</mo><mn>1</mn></mrow></msub></mtd></mtr><mtr><mtd columnalign="center"><mn>0</mn></mtd><mtd columnalign="center"><mi mathvariant="normal">⋯</mi></mtd><mtd columnalign="center"><mn>0</mn></mtd><mtd columnalign="center"><msub><mi href="./29.3#E12">γ</mi><mi href="./29.1#p2.t1.r1">n</mi></msub></mtd><mtd columnalign="center"><msub><mi href="./29.3#E12">β</mi><mi href="./29.1#p2.t1.r1">n</mi></msub></mtd></mtr></mtable><mo>]</mo></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./29.1#p2.t1.r1">n</mi></math>: nonnegative integer</a>,
<a href="./29.3#E11" title="(29.3.11) ‣ §29.3(iii) Continued Fractions ‣ §29.3 Definitions and Basic Properties ‣ Lamé Functions ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m30.png" altimg-height="18px" altimg-valign="-8px" altimg-width="26px" alttext="\alpha_{p}" display="inline"><msub><mi href="./29.3#E11">α</mi><mi href="./29.1#p2.t1.r1">p</mi></msub></math></a>,
<a href="./29.3#E12" title="(29.3.12) ‣ §29.3(iii) Continued Fractions ‣ §29.3 Definitions and Basic Properties ‣ Lamé Functions ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m31.png" altimg-height="23px" altimg-valign="-8px" altimg-width="25px" alttext="\beta_{p}" display="inline"><msub><mi href="./29.3#E12">β</mi><mi href="./29.1#p2.t1.r1">p</mi></msub></math></a> and
<a href="./29.3#E12" title="(29.3.12) ‣ §29.3(iii) Continued Fractions ‣ §29.3 Definitions and Basic Properties ‣ Lamé Functions ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m35.png" altimg-height="18px" altimg-valign="-8px" altimg-width="24px" alttext="\gamma_{p}" display="inline"><msub><mi href="./29.3#E12">γ</mi><mi href="./29.1#p2.t1.r1">p</mi></msub></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">be the tridiagonal matrix with <math class="ltx_Math" altimg="m30.png" altimg-height="18px" altimg-valign="-8px" altimg-width="26px" alttext="\alpha_{p}" display="inline"><msub><mi href="./29.3#E11">α</mi><mi href="./29.1#p2.t1.r1">p</mi></msub></math>, <math class="ltx_Math" altimg="m31.png" altimg-height="23px" altimg-valign="-8px" altimg-width="25px" alttext="\beta_{p}" display="inline"><msub><mi href="./29.3#E12">β</mi><mi href="./29.1#p2.t1.r1">p</mi></msub></math>, <math class="ltx_Math" altimg="m35.png" altimg-height="18px" altimg-valign="-8px" altimg-width="24px" alttext="\gamma_{p}" display="inline"><msub><mi href="./29.3#E12">γ</mi><mi href="./29.1#p2.t1.r1">p</mi></msub></math> as in
().
Let the eigenvalues of <math class="ltx_Math" altimg="m36.png" altimg-height="18px" altimg-valign="-2px" altimg-width="26px" alttext="\mathbf{M}" display="inline"><mi mathvariant="bold">M</mi></math> be <math class="ltx_Math" altimg="m24.png" altimg-height="23px" altimg-valign="-8px" altimg-width="30px" alttext="H_{p}" display="inline"><msub><mi href="./29.15#Px1.p1">H</mi><mi href="./29.1#p2.t1.r1">p</mi></msub></math> with
</p>
<table id="E3" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">29.15.3</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E3.png" altimg-height="23px" altimg-valign="-6px" altimg-width="193px" alttext="H_{0}<H_{1}<\cdots<H_{n}," display="block"><mrow><mrow><msub><mi href="./29.15#Px1.p1">H</mi><mn>0</mn></msub><mo><</mo><msub><mi href="./29.15#Px1.p1">H</mi><mn>1</mn></msub><mo><</mo><mi mathvariant="normal">⋯</mi><mo><</mo><msub><mi href="./29.15#Px1.p1">H</mi><mi href="./29.1#p2.t1.r1">n</mi></msub></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./29.1#p2.t1.r1">n</mi></math>: nonnegative integer</a> and
<a href="./29.15#Px1.p1" title="Polynomial uE m 2 n ( z , k 2 ) ‣ §29.15(i) Fourier Coefficients ‣ §29.15 Fourier Series and Chebyshev Series ‣ Lamé Polynomials ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m24.png" altimg-height="23px" altimg-valign="-8px" altimg-width="30px" alttext="H_{p}" display="inline"><msub><mi href="./29.15#Px1.p1">H</mi><mi href="./29.1#p2.t1.r1">p</mi></msub></math>: eigenvalues</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and also let</p>
<table id="E4" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">29.15.4</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E4.png" altimg-height="28px" altimg-valign="-7px" altimg-width="164px" alttext="[A_{0},A_{2},\dots,A_{2n}]^{\mathrm{T}}" display="block"><msup><mrow><mo stretchy="false">[</mo><msub><mi href="./29.6#SS2.p1">A</mi><mn>0</mn></msub><mo>,</mo><msub><mi href="./29.6#SS2.p1">A</mi><mn>2</mn></msub><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><msub><mi href="./29.6#SS2.p1">A</mi><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">n</mi></mrow></msub><mo stretchy="false">]</mo></mrow><mi mathvariant="normal">T</mi></msup></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E4.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./29.1#p2.t1.r1">n</mi></math>: nonnegative integer</a> and
<a href="./29.6#SS2.p1" title="§29.6(ii) Function Ec + 2 m 1 ν ( z , k 2 ) ‣ §29.6 Fourier Series ‣ Lamé Functions ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m17.png" altimg-height="23px" altimg-valign="-8px" altimg-width="36px" alttext="A_{2p}" display="inline"><msub><mi href="./29.6#SS2.p1">A</mi><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow></msub></math>: coefficients</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">be the eigenvector corresponding to <math class="ltx_Math" altimg="m23.png" altimg-height="21px" altimg-valign="-5px" altimg-width="36px" alttext="H_{m}" display="inline"><msub><mi href="./29.15#Px1.p1">H</mi><mi href="./29.1#p2.t1.r1">m</mi></msub></math> and normalized so that
</p>
<table id="E5" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">29.15.5</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E5.png" altimg-height="66px" altimg-valign="-30px" altimg-width="172px" alttext="\tfrac{1}{2}A_{0}^{2}+\sum_{p=1}^{n}A_{2p}^{2}=1" display="block"><mrow><mrow><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><msubsup><mi href="./29.6#SS2.p1">A</mi><mn>0</mn><mn>2</mn></msubsup></mrow><mo>+</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./29.1#p2.t1.r1">p</mi><mo>=</mo><mn>1</mn></mrow><mi href="./29.1#p2.t1.r1">n</mi></munderover><msubsup><mi href="./29.6#SS2.p1">A</mi><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow><mn>2</mn></msubsup></mrow></mrow><mo>=</mo><mn>1</mn></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E5.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./29.1#p2.t1.r1">n</mi></math>: nonnegative integer</a>,
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="p" display="inline"><mi href="./29.1#p2.t1.r1">p</mi></math>: nonnegative integer</a> and
<a href="./29.6#SS2.p1" title="§29.6(ii) Function Ec + 2 m 1 ν ( z , k 2 ) ‣ §29.6 Fourier Series ‣ Lamé Functions ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m17.png" altimg-height="23px" altimg-valign="-8px" altimg-width="36px" alttext="A_{2p}" display="inline"><msub><mi href="./29.6#SS2.p1">A</mi><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow></msub></math>: coefficients</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and</p>
<table id="E6" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">29.15.6</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E6.png" altimg-height="66px" altimg-valign="-30px" altimg-width="177px" alttext="\tfrac{1}{2}A_{0}+\sum_{p=1}^{n}A_{2p}>0." display="block"><mrow><mrow><mrow><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><msub><mi href="./29.6#SS2.p1">A</mi><mn>0</mn></msub></mrow><mo>+</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./29.1#p2.t1.r1">p</mi><mo>=</mo><mn>1</mn></mrow><mi href="./29.1#p2.t1.r1">n</mi></munderover><msub><mi href="./29.6#SS2.p1">A</mi><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow></msub></mrow></mrow><mo>></mo><mn>0</mn></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E6.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./29.1#p2.t1.r1">n</mi></math>: nonnegative integer</a>,
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="p" display="inline"><mi href="./29.1#p2.t1.r1">p</mi></math>: nonnegative integer</a> and
<a href="./29.6#SS2.p1" title="§29.6(ii) Function Ec + 2 m 1 ν ( z , k 2 ) ‣ §29.6 Fourier Series ‣ Lamé Functions ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m17.png" altimg-height="23px" altimg-valign="-8px" altimg-width="36px" alttext="A_{2p}" display="inline"><msub><mi href="./29.6#SS2.p1">A</mi><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow></msub></math>: coefficients</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Then</p>
<table id="E7" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">29.15.7</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E22.png" altimg-height="30px" altimg-valign="-9px" altimg-width="290px" alttext="a^{2m}_{\nu}\left(k^{2}\right)=\tfrac{1}{2}(H_{m}+\nu(\nu+1)k^{2})," display="block"><mrow><mrow><mrow><msubsup><mi href="./29.3#SS1.p1">a</mi><mi href="./29.1#p2.t1.r4">ν</mi><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">m</mi></mrow></msubsup><mo></mo><mrow><mo>(</mo><msup><mi href="./29.1#p2.t1.r4">k</mi><mn>2</mn></msup><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mrow><mo stretchy="false">(</mo><mrow><msub><mi href="./29.15#Px1.p1">H</mi><mi href="./29.1#p2.t1.r1">m</mi></msub><mo>+</mo><mrow><mi href="./29.1#p2.t1.r4">ν</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./29.1#p2.t1.r4">ν</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msup><mi href="./29.1#p2.t1.r4">k</mi><mn>2</mn></msup></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E7.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./29.3#SS1.p1" title="§29.3(i) Eigenvalues ‣ §29.3 Definitions and Basic Properties ‣ Lamé Functions ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m65.png" altimg-height="27px" altimg-valign="-9px" altimg-width="67px" alttext="a^{\NVar{n}}_{\NVar{\nu}}\left(\NVar{k^{2}}\right)" display="inline"><mrow><msubsup><mi href="./29.3#SS1.p1">a</mi><mi class="ltx_nvar" href="./29.1#p2.t1.r4">ν</mi><mi class="ltx_nvar" href="./29.1#p2.t1.r1">n</mi></msubsup><mo></mo><mrow><mo>(</mo><msup><mi class="ltx_nvar" href="./29.1#p2.t1.r4">k</mi><mn class="ltx_nvar">2</mn></msup><mo>)</mo></mrow></mrow></math>: eigenvalues of Lamé’s equation</a>,
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./29.1#p2.t1.r1">m</mi></math>: nonnegative integer</a>,
<a href="./29.1#p2.t1.r4" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m67.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./29.1#p2.t1.r4">k</mi></math>: real parameter</a>,
<a href="./29.1#p2.t1.r4" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./29.1#p2.t1.r4">ν</mi></math>: real parameter</a> and
<a href="./29.15#Px1.p1" title="Polynomial uE m 2 n ( z , k 2 ) ‣ §29.15(i) Fourier Coefficients ‣ §29.15 Fourier Series and Chebyshev Series ‣ Lamé Polynomials ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m24.png" altimg-height="23px" altimg-valign="-8px" altimg-width="30px" alttext="H_{p}" display="inline"><msub><mi href="./29.15#Px1.p1">H</mi><mi href="./29.1#p2.t1.r1">p</mi></msub></math>: eigenvalues</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and () applies, with <math class="ltx_Math" altimg="m62.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="\phi" display="inline"><mi href="./29.2#E5">ϕ</mi></math> again defined as in
().</p>
</div>
<div id="Px1.p2" class="ltx_para">
</div>
</section>
<section id="Px2" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Polynomial <math class="ltx_Math" altimg="m44.png" altimg-height="27px" altimg-valign="-9px" altimg-width="128px" alttext="\mathit{sE}^{m}_{2n+1}\left(z,k^{2}\right)" display="inline"><mrow><msubsup><mi href="./29.12#E2" mathvariant="italic">sE</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mn>1</mn></mrow><mi href="./29.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo>(</mo><mi href="./29.1#p2.t1.r3">z</mi><mo>,</mo><msup><mi href="./29.1#p2.t1.r4">k</mi><mn>2</mn></msup><mo>)</mo></mrow></mrow></math>
</h3>
<div id="Px2.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px2.p1" class="ltx_para">
<p class="ltx_p">When <math class="ltx_Math" altimg="m54.png" altimg-height="18px" altimg-valign="-4px" altimg-width="98px" alttext="\nu=2n+1" display="inline"><mrow><mi href="./29.1#p2.t1.r4">ν</mi><mo>=</mo><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mn>1</mn></mrow></mrow></math>, <math class="ltx_Math" altimg="m68.png" altimg-height="20px" altimg-valign="-6px" altimg-width="133px" alttext="m=0,1,\dots,n" display="inline"><mrow><mi href="./29.1#p2.t1.r1">m</mi><mo>=</mo><mrow><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><mi href="./29.1#p2.t1.r1">n</mi></mrow></mrow></math>, the Fourier series ()
terminates:</p>
<table id="E8" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">29.15.8</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E8.png" altimg-height="66px" altimg-valign="-30px" altimg-width="380px" alttext="\mathit{sE}^{m}_{2n+1}\left(z,k^{2}\right)=\sum_{p=0}^{n}A_{2p+1}\cos\left((2p%
+1)\phi\right)." display="block"><mrow><mrow><mrow><msubsup><mi href="./29.12#E2" mathvariant="italic">sE</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mn>1</mn></mrow><mi href="./29.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo>(</mo><mi href="./29.1#p2.t1.r3">z</mi><mo>,</mo><msup><mi href="./29.1#p2.t1.r4">k</mi><mn>2</mn></msup><mo>)</mo></mrow></mrow><mo>=</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./29.1#p2.t1.r1">p</mi><mo>=</mo><mn>0</mn></mrow><mi href="./29.1#p2.t1.r1">n</mi></munderover><mrow><msub><mi href="./29.6#SS2.p1">A</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow><mo>+</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi href="./29.2#E5">ϕ</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E8.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./29.12#E2" title="(29.12.2) ‣ §29.12(i) Elliptic-Function Form ‣ §29.12 Definitions ‣ Lamé Polynomials ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="27px" altimg-valign="-9px" altimg-width="128px" alttext="\mathit{sE}^{\NVar{m}}_{2\NVar{n}+1}\left(\NVar{z},\NVar{k^{2}}\right)" display="inline"><mrow><msubsup><mi href="./29.12#E2" mathvariant="italic">sE</mi><mrow><mrow><mn>2</mn><mo></mo><mi class="ltx_nvar" href="./29.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mn>1</mn></mrow><mi class="ltx_nvar" href="./29.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./29.1#p2.t1.r3">z</mi><mo>,</mo><msup><mi class="ltx_nvar" href="./29.1#p2.t1.r4">k</mi><mn class="ltx_nvar">2</mn></msup><mo>)</mo></mrow></mrow></math>: Lamé polynomial</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m32.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./29.1#p2.t1.r1">m</mi></math>: nonnegative integer</a>,
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./29.1#p2.t1.r1">n</mi></math>: nonnegative integer</a>,
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="p" display="inline"><mi href="./29.1#p2.t1.r1">p</mi></math>: nonnegative integer</a>,
<a href="./29.1#p2.t1.r3" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m73.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./29.1#p2.t1.r3">z</mi></math>: complex variable</a>,
<a href="./29.1#p2.t1.r4" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m67.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./29.1#p2.t1.r4">k</mi></math>: real parameter</a>,
<a href="./29.2#E5" title="(29.2.5) ‣ §29.2(ii) Other Forms ‣ §29.2 Differential Equations ‣ Lamé Functions ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m62.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="\phi" display="inline"><mi href="./29.2#E5">ϕ</mi></math>: change of variable</a> and
<a href="./29.6#SS2.p1" title="§29.6(ii) Function Ec + 2 m 1 ν ( z , k 2 ) ‣ §29.6 Fourier Series ‣ Lamé Functions ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m17.png" altimg-height="23px" altimg-valign="-8px" altimg-width="36px" alttext="A_{2p}" display="inline"><msub><mi href="./29.6#SS2.p1">A</mi><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow></msub></math>: coefficients</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">In () replace <math class="ltx_Math" altimg="m30.png" altimg-height="18px" altimg-valign="-8px" altimg-width="26px" alttext="\alpha_{p}" display="inline"><msub><mi href="./29.3#E11">α</mi><mi href="./29.1#p2.t1.r1">p</mi></msub></math>, <math class="ltx_Math" altimg="m31.png" altimg-height="23px" altimg-valign="-8px" altimg-width="25px" alttext="\beta_{p}" display="inline"><msub><mi href="./29.3#E12">β</mi><mi href="./29.1#p2.t1.r1">p</mi></msub></math>, and <math class="ltx_Math" altimg="m35.png" altimg-height="18px" altimg-valign="-8px" altimg-width="24px" alttext="\gamma_{p}" display="inline"><msub><mi href="./29.3#E12">γ</mi><mi href="./29.1#p2.t1.r1">p</mi></msub></math> as in
() by
</p>
<table id="E9" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">29.15.9</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E9.png" altimg-height="28px" altimg-valign="-7px" altimg-width="190px" alttext="[A_{1},A_{3},\dots,A_{2n+1}]^{\mathrm{T}}," display="block"><mrow><msup><mrow><mo stretchy="false">[</mo><msub><mi href="./29.6#SS2.p1">A</mi><mn>1</mn></msub><mo>,</mo><msub><mi href="./29.6#SS2.p1">A</mi><mn>3</mn></msub><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><msub><mi href="./29.6#SS2.p1">A</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">]</mo></mrow><mi mathvariant="normal">T</mi></msup><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E9.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./29.1#p2.t1.r1">n</mi></math>: nonnegative integer</a> and
<a href="./29.6#SS2.p1" title="§29.6(ii) Function Ec + 2 m 1 ν ( z , k 2 ) ‣ §29.6 Fourier Series ‣ Lamé Functions ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m17.png" altimg-height="23px" altimg-valign="-8px" altimg-width="36px" alttext="A_{2p}" display="inline"><msub><mi href="./29.6#SS2.p1">A</mi><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow></msub></math>: coefficients</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E10" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">29.15.10</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E10.png" altimg-height="66px" altimg-valign="-30px" altimg-width="137px" alttext="\sum_{p=0}^{n}A_{2p+1}^{2}=1," display="block"><mrow><mrow><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./29.1#p2.t1.r1">p</mi><mo>=</mo><mn>0</mn></mrow><mi href="./29.1#p2.t1.r1">n</mi></munderover><msubsup><mi href="./29.6#SS2.p1">A</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow><mo>+</mo><mn>1</mn></mrow><mn>2</mn></msubsup></mrow><mo>=</mo><mn>1</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E10.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./29.1#p2.t1.r1">n</mi></math>: nonnegative integer</a>,
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="p" display="inline"><mi href="./29.1#p2.t1.r1">p</mi></math>: nonnegative integer</a> and
<a href="./29.6#SS2.p1" title="§29.6(ii) Function Ec + 2 m 1 ν ( z , k 2 ) ‣ §29.6 Fourier Series ‣ Lamé Functions ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m17.png" altimg-height="23px" altimg-valign="-8px" altimg-width="36px" alttext="A_{2p}" display="inline"><msub><mi href="./29.6#SS2.p1">A</mi><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow></msub></math>: coefficients</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E11" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">29.15.11</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E11.png" altimg-height="66px" altimg-valign="-30px" altimg-width="137px" alttext="\sum_{p=0}^{n}A_{2p+1}>0." display="block"><mrow><mrow><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./29.1#p2.t1.r1">p</mi><mo>=</mo><mn>0</mn></mrow><mi href="./29.1#p2.t1.r1">n</mi></munderover><msub><mi href="./29.6#SS2.p1">A</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow><mo>+</mo><mn>1</mn></mrow></msub></mrow><mo>></mo><mn>0</mn></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E11.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./29.1#p2.t1.r1">n</mi></math>: nonnegative integer</a>,
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="p" display="inline"><mi href="./29.1#p2.t1.r1">p</mi></math>: nonnegative integer</a> and
<a href="./29.6#SS2.p1" title="§29.6(ii) Function Ec + 2 m 1 ν ( z , k 2 ) ‣ §29.6 Fourier Series ‣ Lamé Functions ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m17.png" altimg-height="23px" altimg-valign="-8px" altimg-width="36px" alttext="A_{2p}" display="inline"><msub><mi href="./29.6#SS2.p1">A</mi><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow></msub></math>: coefficients</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Then</p>
<table id="E12" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">29.15.12</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E32.png" altimg-height="30px" altimg-valign="-9px" altimg-width="310px" alttext="a^{2m+1}_{\nu}\left(k^{2}\right)=\tfrac{1}{2}(H_{m}+\nu(\nu+1)k^{2})," display="block"><mrow><mrow><mrow><msubsup><mi href="./29.3#SS1.p1">a</mi><mi href="./29.1#p2.t1.r4">ν</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">m</mi></mrow><mo>+</mo><mn>1</mn></mrow></msubsup><mo></mo><mrow><mo>(</mo><msup><mi href="./29.1#p2.t1.r4">k</mi><mn>2</mn></msup><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mrow><mo stretchy="false">(</mo><mrow><msub><mi href="./29.15#Px1.p1">H</mi><mi href="./29.1#p2.t1.r1">m</mi></msub><mo>+</mo><mrow><mi href="./29.1#p2.t1.r4">ν</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./29.1#p2.t1.r4">ν</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msup><mi href="./29.1#p2.t1.r4">k</mi><mn>2</mn></msup></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E12.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./29.3#SS1.p1" title="§29.3(i) Eigenvalues ‣ §29.3 Definitions and Basic Properties ‣ Lamé Functions ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m65.png" altimg-height="27px" altimg-valign="-9px" altimg-width="67px" alttext="a^{\NVar{n}}_{\NVar{\nu}}\left(\NVar{k^{2}}\right)" display="inline"><mrow><msubsup><mi href="./29.3#SS1.p1">a</mi><mi class="ltx_nvar" href="./29.1#p2.t1.r4">ν</mi><mi class="ltx_nvar" href="./29.1#p2.t1.r1">n</mi></msubsup><mo></mo><mrow><mo>(</mo><msup><mi class="ltx_nvar" href="./29.1#p2.t1.r4">k</mi><mn class="ltx_nvar">2</mn></msup><mo>)</mo></mrow></mrow></math>: eigenvalues of Lamé’s equation</a>,
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./29.1#p2.t1.r1">m</mi></math>: nonnegative integer</a>,
<a href="./29.1#p2.t1.r4" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m67.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./29.1#p2.t1.r4">k</mi></math>: real parameter</a>,
<a href="./29.1#p2.t1.r4" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./29.1#p2.t1.r4">ν</mi></math>: real parameter</a> and
<a href="./29.15#Px1.p1" title="Polynomial uE m 2 n ( z , k 2 ) ‣ §29.15(i) Fourier Coefficients ‣ §29.15 Fourier Series and Chebyshev Series ‣ Lamé Polynomials ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m24.png" altimg-height="23px" altimg-valign="-8px" altimg-width="30px" alttext="H_{p}" display="inline"><msub><mi href="./29.15#Px1.p1">H</mi><mi href="./29.1#p2.t1.r1">p</mi></msub></math>: eigenvalues</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and () applies.</p>
</div>
</section>
<section id="Px3" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Polynomial <math class="ltx_Math" altimg="m38.png" altimg-height="27px" altimg-valign="-9px" altimg-width="129px" alttext="\mathit{cE}^{m}_{2n+1}\left(z,k^{2}\right)" display="inline"><mrow><msubsup><mi href="./29.12#E3" mathvariant="italic">cE</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mn>1</mn></mrow><mi href="./29.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo>(</mo><mi href="./29.1#p2.t1.r3">z</mi><mo>,</mo><msup><mi href="./29.1#p2.t1.r4">k</mi><mn>2</mn></msup><mo>)</mo></mrow></mrow></math>
</h3>
<div id="Px3.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px3.p1" class="ltx_para">
<p class="ltx_p">When <math class="ltx_Math" altimg="m54.png" altimg-height="18px" altimg-valign="-4px" altimg-width="98px" alttext="\nu=2n+1" display="inline"><mrow><mi href="./29.1#p2.t1.r4">ν</mi><mo>=</mo><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mn>1</mn></mrow></mrow></math>, <math class="ltx_Math" altimg="m68.png" altimg-height="20px" altimg-valign="-6px" altimg-width="133px" alttext="m=0,1,\dots,n" display="inline"><mrow><mi href="./29.1#p2.t1.r1">m</mi><mo>=</mo><mrow><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><mi href="./29.1#p2.t1.r1">n</mi></mrow></mrow></math>, the Fourier series ()
terminates:</p>
<table id="E13" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">29.15.13</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E13.png" altimg-height="66px" altimg-valign="-30px" altimg-width="379px" alttext="\mathit{cE}^{m}_{2n+1}\left(z,k^{2}\right)=\sum_{p=0}^{n}B_{2p+1}\sin\left((2p%
+1)\phi\right)." display="block"><mrow><mrow><mrow><msubsup><mi href="./29.12#E3" mathvariant="italic">cE</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mn>1</mn></mrow><mi href="./29.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo>(</mo><mi href="./29.1#p2.t1.r3">z</mi><mo>,</mo><msup><mi href="./29.1#p2.t1.r4">k</mi><mn>2</mn></msup><mo>)</mo></mrow></mrow><mo>=</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./29.1#p2.t1.r1">p</mi><mo>=</mo><mn>0</mn></mrow><mi href="./29.1#p2.t1.r1">n</mi></munderover><mrow><msub><mi href="./29.6#SS4.p1">B</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow><mo>+</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi href="./29.2#E5">ϕ</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E13.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./29.12#E3" title="(29.12.3) ‣ §29.12(i) Elliptic-Function Form ‣ §29.12 Definitions ‣ Lamé Polynomials ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="27px" altimg-valign="-9px" altimg-width="129px" alttext="\mathit{cE}^{\NVar{m}}_{2\NVar{n}+1}\left(\NVar{z},\NVar{k^{2}}\right)" display="inline"><mrow><msubsup><mi href="./29.12#E3" mathvariant="italic">cE</mi><mrow><mrow><mn>2</mn><mo></mo><mi class="ltx_nvar" href="./29.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mn>1</mn></mrow><mi class="ltx_nvar" href="./29.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./29.1#p2.t1.r3">z</mi><mo>,</mo><msup><mi class="ltx_nvar" href="./29.1#p2.t1.r4">k</mi><mn class="ltx_nvar">2</mn></msup><mo>)</mo></mrow></mrow></math>: Lamé polynomial</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./29.1#p2.t1.r1">m</mi></math>: nonnegative integer</a>,
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./29.1#p2.t1.r1">n</mi></math>: nonnegative integer</a>,
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="p" display="inline"><mi href="./29.1#p2.t1.r1">p</mi></math>: nonnegative integer</a>,
<a href="./29.1#p2.t1.r3" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m73.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./29.1#p2.t1.r3">z</mi></math>: complex variable</a>,
<a href="./29.1#p2.t1.r4" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m67.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./29.1#p2.t1.r4">k</mi></math>: real parameter</a>,
<a href="./29.2#E5" title="(29.2.5) ‣ §29.2(ii) Other Forms ‣ §29.2 Differential Equations ‣ Lamé Functions ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m62.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="\phi" display="inline"><mi href="./29.2#E5">ϕ</mi></math>: change of variable</a> and
<a href="./29.6#SS4.p1" title="§29.6(iv) Function Es + 2 m 2 ν ( z , k 2 ) ‣ §29.6 Fourier Series ‣ Lamé Functions ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m18.png" altimg-height="23px" altimg-valign="-8px" altimg-width="57px" alttext="B_{2p+1}" display="inline"><msub><mi href="./29.6#SS4.p1">B</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow><mo>+</mo><mn>1</mn></mrow></msub></math>: coefficents</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">In () replace <math class="ltx_Math" altimg="m30.png" altimg-height="18px" altimg-valign="-8px" altimg-width="26px" alttext="\alpha_{p}" display="inline"><msub><mi href="./29.3#E11">α</mi><mi href="./29.1#p2.t1.r1">p</mi></msub></math>, <math class="ltx_Math" altimg="m31.png" altimg-height="23px" altimg-valign="-8px" altimg-width="25px" alttext="\beta_{p}" display="inline"><msub><mi href="./29.3#E12">β</mi><mi href="./29.1#p2.t1.r1">p</mi></msub></math>, and <math class="ltx_Math" altimg="m35.png" altimg-height="18px" altimg-valign="-8px" altimg-width="24px" alttext="\gamma_{p}" display="inline"><msub><mi href="./29.3#E12">γ</mi><mi href="./29.1#p2.t1.r1">p</mi></msub></math> as in
() by
</p>
<table id="E14" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">29.15.14</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E14.png" altimg-height="28px" altimg-valign="-7px" altimg-width="190px" alttext="[B_{1},B_{3},\dots,B_{2n+1}]^{\mathrm{T}}," display="block"><mrow><msup><mrow><mo stretchy="false">[</mo><msub><mi href="./29.6#SS4.p1">B</mi><mn>1</mn></msub><mo>,</mo><msub><mi href="./29.6#SS4.p1">B</mi><mn>3</mn></msub><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><msub><mi href="./29.6#SS4.p1">B</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">]</mo></mrow><mi mathvariant="normal">T</mi></msup><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E14.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./29.1#p2.t1.r1">n</mi></math>: nonnegative integer</a> and
<a href="./29.6#SS4.p1" title="§29.6(iv) Function Es + 2 m 2 ν ( z , k 2 ) ‣ §29.6 Fourier Series ‣ Lamé Functions ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m18.png" altimg-height="23px" altimg-valign="-8px" altimg-width="57px" alttext="B_{2p+1}" display="inline"><msub><mi href="./29.6#SS4.p1">B</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow><mo>+</mo><mn>1</mn></mrow></msub></math>: coefficents</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E15" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">29.15.15</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E15.png" altimg-height="66px" altimg-valign="-30px" altimg-width="137px" alttext="\sum_{p=0}^{n}B_{2p+1}^{2}=1," display="block"><mrow><mrow><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./29.1#p2.t1.r1">p</mi><mo>=</mo><mn>0</mn></mrow><mi href="./29.1#p2.t1.r1">n</mi></munderover><msubsup><mi href="./29.6#SS4.p1">B</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow><mo>+</mo><mn>1</mn></mrow><mn>2</mn></msubsup></mrow><mo>=</mo><mn>1</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E15.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./29.1#p2.t1.r1">n</mi></math>: nonnegative integer</a>,
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="p" display="inline"><mi href="./29.1#p2.t1.r1">p</mi></math>: nonnegative integer</a> and
<a href="./29.6#SS4.p1" title="§29.6(iv) Function Es + 2 m 2 ν ( z , k 2 ) ‣ §29.6 Fourier Series ‣ Lamé Functions ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m18.png" altimg-height="23px" altimg-valign="-8px" altimg-width="57px" alttext="B_{2p+1}" display="inline"><msub><mi href="./29.6#SS4.p1">B</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow><mo>+</mo><mn>1</mn></mrow></msub></math>: coefficents</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E16" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">29.15.16</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E16.png" altimg-height="66px" altimg-valign="-30px" altimg-width="203px" alttext="\sum_{p=0}^{n}(2p+1)B_{2p+1}>0." display="block"><mrow><mrow><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./29.1#p2.t1.r1">p</mi><mo>=</mo><mn>0</mn></mrow><mi href="./29.1#p2.t1.r1">n</mi></munderover><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msub><mi href="./29.6#SS4.p1">B</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow><mo>+</mo><mn>1</mn></mrow></msub></mrow></mrow><mo>></mo><mn>0</mn></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E16.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./29.1#p2.t1.r1">n</mi></math>: nonnegative integer</a>,
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="p" display="inline"><mi href="./29.1#p2.t1.r1">p</mi></math>: nonnegative integer</a> and
<a href="./29.6#SS4.p1" title="§29.6(iv) Function Es + 2 m 2 ν ( z , k 2 ) ‣ §29.6 Fourier Series ‣ Lamé Functions ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m18.png" altimg-height="23px" altimg-valign="-8px" altimg-width="57px" alttext="B_{2p+1}" display="inline"><msub><mi href="./29.6#SS4.p1">B</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow><mo>+</mo><mn>1</mn></mrow></msub></math>: coefficents</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Then</p>
<table id="E17" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">29.15.17</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E37.png" altimg-height="30px" altimg-valign="-9px" altimg-width="308px" alttext="b^{2m+1}_{\nu}\left(k^{2}\right)=\tfrac{1}{2}(H_{m}+\nu(\nu+1)k^{2})," display="block"><mrow><mrow><mrow><msubsup><mi href="./29.3#SS1.p1">b</mi><mi href="./29.1#p2.t1.r4">ν</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">m</mi></mrow><mo>+</mo><mn>1</mn></mrow></msubsup><mo></mo><mrow><mo>(</mo><msup><mi href="./29.1#p2.t1.r4">k</mi><mn>2</mn></msup><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mrow><mo stretchy="false">(</mo><mrow><msub><mi href="./29.15#Px1.p1">H</mi><mi href="./29.1#p2.t1.r1">m</mi></msub><mo>+</mo><mrow><mi href="./29.1#p2.t1.r4">ν</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./29.1#p2.t1.r4">ν</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msup><mi href="./29.1#p2.t1.r4">k</mi><mn>2</mn></msup></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E17.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./29.3#SS1.p1" title="§29.3(i) Eigenvalues ‣ §29.3 Definitions and Basic Properties ‣ Lamé Functions ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="27px" altimg-valign="-9px" altimg-width="65px" alttext="b^{\NVar{n}}_{\NVar{\nu}}\left(\NVar{k^{2}}\right)" display="inline"><mrow><msubsup><mi href="./29.3#SS1.p1">b</mi><mi class="ltx_nvar" href="./29.1#p2.t1.r4">ν</mi><mi class="ltx_nvar" href="./29.1#p2.t1.r1">n</mi></msubsup><mo></mo><mrow><mo>(</mo><msup><mi class="ltx_nvar" href="./29.1#p2.t1.r4">k</mi><mn class="ltx_nvar">2</mn></msup><mo>)</mo></mrow></mrow></math>: eigenvalues of Lamé’s equation</a>,
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./29.1#p2.t1.r1">m</mi></math>: nonnegative integer</a>,
<a href="./29.1#p2.t1.r4" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m67.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./29.1#p2.t1.r4">k</mi></math>: real parameter</a>,
<a href="./29.1#p2.t1.r4" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./29.1#p2.t1.r4">ν</mi></math>: real parameter</a> and
<a href="./29.15#Px1.p1" title="Polynomial uE m 2 n ( z , k 2 ) ‣ §29.15(i) Fourier Coefficients ‣ §29.15 Fourier Series and Chebyshev Series ‣ Lamé Polynomials ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m24.png" altimg-height="23px" altimg-valign="-8px" altimg-width="30px" alttext="H_{p}" display="inline"><msub><mi href="./29.15#Px1.p1">H</mi><mi href="./29.1#p2.t1.r1">p</mi></msub></math>: eigenvalues</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and () applies.</p>
</div>
</section>
<section id="Px4" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Polynomial <math class="ltx_Math" altimg="m42.png" altimg-height="27px" altimg-valign="-9px" altimg-width="130px" alttext="\mathit{dE}^{m}_{2n+1}\left(z,k^{2}\right)" display="inline"><mrow><msubsup><mi href="./29.12#E4" mathvariant="italic">dE</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mn>1</mn></mrow><mi href="./29.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo>(</mo><mi href="./29.1#p2.t1.r3">z</mi><mo>,</mo><msup><mi href="./29.1#p2.t1.r4">k</mi><mn>2</mn></msup><mo>)</mo></mrow></mrow></math>
</h3>
<div id="Px4.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px4.p1" class="ltx_para">
<p class="ltx_p">When <math class="ltx_Math" altimg="m54.png" altimg-height="18px" altimg-valign="-4px" altimg-width="98px" alttext="\nu=2n+1" display="inline"><mrow><mi href="./29.1#p2.t1.r4">ν</mi><mo>=</mo><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mn>1</mn></mrow></mrow></math>, <math class="ltx_Math" altimg="m68.png" altimg-height="20px" altimg-valign="-6px" altimg-width="133px" alttext="m=0,1,\dots,n" display="inline"><mrow><mi href="./29.1#p2.t1.r1">m</mi><mo>=</mo><mrow><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><mi href="./29.1#p2.t1.r1">n</mi></mrow></mrow></math>, the Fourier series ()
terminates:
</p>
<table id="E18" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">29.15.18</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E18.png" altimg-height="68px" altimg-valign="-30px" altimg-width="477px" alttext="\mathit{dE}^{m}_{2n+1}\left(z,k^{2}\right)=\operatorname{dn}\left(z,k\right)%
\left(\tfrac{1}{2}C_{0}+\sum_{p=1}^{n}C_{2p}\cos\left(2p\phi\right)\right)." display="block"><mrow><mrow><mrow><msubsup><mi href="./29.12#E4" mathvariant="italic">dE</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mn>1</mn></mrow><mi href="./29.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo>(</mo><mi href="./29.1#p2.t1.r3">z</mi><mo>,</mo><msup><mi href="./29.1#p2.t1.r4">k</mi><mn>2</mn></msup><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mi href="./22.2#E6">dn</mi><mo></mo><mrow><mo>(</mo><mi href="./29.1#p2.t1.r3">z</mi><mo>,</mo><mi href="./29.1#p2.t1.r4">k</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mo>(</mo><mrow><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><msub><mi href="./29.6#SS2.p2">C</mi><mn>0</mn></msub></mrow><mo>+</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./29.1#p2.t1.r1">p</mi><mo>=</mo><mn>1</mn></mrow><mi href="./29.1#p2.t1.r1">n</mi></munderover><mrow><msub><mi href="./29.6#SS2.p2">C</mi><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow></msub><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi><mo></mo><mi href="./29.2#E5">ϕ</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E18.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./22.2#E6" title="(22.2.6) ‣ §22.2 Definitions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="23px" altimg-valign="-7px" altimg-width="75px" alttext="\operatorname{dn}\left(\NVar{z},\NVar{k}\right)" display="inline"><mrow><mi href="./22.2#E6">dn</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r2">z</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobian elliptic function</a>,
<a href="./29.12#E4" title="(29.12.4) ‣ §29.12(i) Elliptic-Function Form ‣ §29.12 Definitions ‣ Lamé Polynomials ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m41.png" altimg-height="27px" altimg-valign="-9px" altimg-width="130px" alttext="\mathit{dE}^{\NVar{m}}_{2\NVar{n}+1}\left(\NVar{z},\NVar{k^{2}}\right)" display="inline"><mrow><msubsup><mi href="./29.12#E4" mathvariant="italic">dE</mi><mrow><mrow><mn>2</mn><mo></mo><mi class="ltx_nvar" href="./29.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mn>1</mn></mrow><mi class="ltx_nvar" href="./29.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./29.1#p2.t1.r3">z</mi><mo>,</mo><msup><mi class="ltx_nvar" href="./29.1#p2.t1.r4">k</mi><mn class="ltx_nvar">2</mn></msup><mo>)</mo></mrow></mrow></math>: Lamé polynomial</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m32.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./29.1#p2.t1.r1">m</mi></math>: nonnegative integer</a>,
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./29.1#p2.t1.r1">n</mi></math>: nonnegative integer</a>,
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="p" display="inline"><mi href="./29.1#p2.t1.r1">p</mi></math>: nonnegative integer</a>,
<a href="./29.1#p2.t1.r3" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m73.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./29.1#p2.t1.r3">z</mi></math>: complex variable</a>,
<a href="./29.1#p2.t1.r4" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m67.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./29.1#p2.t1.r4">k</mi></math>: real parameter</a>,
<a href="./29.2#E5" title="(29.2.5) ‣ §29.2(ii) Other Forms ‣ §29.2 Differential Equations ‣ Lamé Functions ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m62.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="\phi" display="inline"><mi href="./29.2#E5">ϕ</mi></math>: change of variable</a> and
<a href="./29.6#SS2.p2" title="§29.6(ii) Function Ec + 2 m 1 ν ( z , k 2 ) ‣ §29.6 Fourier Series ‣ Lamé Functions ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="23px" altimg-valign="-8px" altimg-width="56px" alttext="C_{2p+1}" display="inline"><msub><mi href="./29.6#SS2.p2">C</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow><mo>+</mo><mn>1</mn></mrow></msub></math>: coefficents</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">In () replace <math class="ltx_Math" altimg="m30.png" altimg-height="18px" altimg-valign="-8px" altimg-width="26px" alttext="\alpha_{p}" display="inline"><msub><mi href="./29.3#E11">α</mi><mi href="./29.1#p2.t1.r1">p</mi></msub></math>, <math class="ltx_Math" altimg="m31.png" altimg-height="23px" altimg-valign="-8px" altimg-width="25px" alttext="\beta_{p}" display="inline"><msub><mi href="./29.3#E12">β</mi><mi href="./29.1#p2.t1.r1">p</mi></msub></math>, and <math class="ltx_Math" altimg="m35.png" altimg-height="18px" altimg-valign="-8px" altimg-width="24px" alttext="\gamma_{p}" display="inline"><msub><mi href="./29.3#E12">γ</mi><mi href="./29.1#p2.t1.r1">p</mi></msub></math> as in
() by
</p>
<table id="E19" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">29.15.19</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E19.png" altimg-height="28px" altimg-valign="-7px" altimg-width="168px" alttext="[C_{0},C_{2},\dots,C_{2n}]^{\mathrm{T}}," display="block"><mrow><msup><mrow><mo stretchy="false">[</mo><msub><mi href="./29.6#SS2.p2">C</mi><mn>0</mn></msub><mo>,</mo><msub><mi href="./29.6#SS2.p2">C</mi><mn>2</mn></msub><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><msub><mi href="./29.6#SS2.p2">C</mi><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">n</mi></mrow></msub><mo stretchy="false">]</mo></mrow><mi mathvariant="normal">T</mi></msup><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E19.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./29.1#p2.t1.r1">n</mi></math>: nonnegative integer</a> and
<a href="./29.6#SS2.p2" title="§29.6(ii) Function Ec + 2 m 1 ν ( z , k 2 ) ‣ §29.6 Fourier Series ‣ Lamé Functions ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="23px" altimg-valign="-8px" altimg-width="56px" alttext="C_{2p+1}" display="inline"><msub><mi href="./29.6#SS2.p2">C</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow><mo>+</mo><mn>1</mn></mrow></msub></math>: coefficents</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E20" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">29.15.20</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E20.png" altimg-height="69px" altimg-valign="-30px" altimg-width="473px" alttext="\left(1-\tfrac{1}{2}k^{2}\right)\left(\tfrac{1}{2}C_{0}^{2}+\sum_{p=1}^{n}C_{2%
p}^{2}\right)-\tfrac{1}{2}k^{2}\sum_{p=0}^{n-1}C_{2p}C_{2p+2}=1," display="block"><mrow><mrow><mrow><mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><msup><mi href="./29.1#p2.t1.r4">k</mi><mn>2</mn></msup></mrow></mrow><mo>)</mo></mrow><mo></mo><mrow><mo>(</mo><mrow><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><msubsup><mi href="./29.6#SS2.p2">C</mi><mn>0</mn><mn>2</mn></msubsup></mrow><mo>+</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./29.1#p2.t1.r1">p</mi><mo>=</mo><mn>1</mn></mrow><mi href="./29.1#p2.t1.r1">n</mi></munderover><msubsup><mi href="./29.6#SS2.p2">C</mi><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow><mn>2</mn></msubsup></mrow></mrow><mo>)</mo></mrow></mrow><mo>-</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><msup><mi href="./29.1#p2.t1.r4">k</mi><mn>2</mn></msup><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./29.1#p2.t1.r1">p</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi href="./29.1#p2.t1.r1">n</mi><mo>-</mo><mn>1</mn></mrow></munderover><mrow><msub><mi href="./29.6#SS2.p2">C</mi><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow></msub><mo></mo><msub><mi href="./29.6#SS2.p2">C</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow><mo>+</mo><mn>2</mn></mrow></msub></mrow></mrow></mrow></mrow><mo>=</mo><mn>1</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E20.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./29.1#p2.t1.r1">n</mi></math>: nonnegative integer</a>,
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="p" display="inline"><mi href="./29.1#p2.t1.r1">p</mi></math>: nonnegative integer</a>,
<a href="./29.1#p2.t1.r4" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m67.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./29.1#p2.t1.r4">k</mi></math>: real parameter</a> and
<a href="./29.6#SS2.p2" title="§29.6(ii) Function Ec + 2 m 1 ν ( z , k 2 ) ‣ §29.6 Fourier Series ‣ Lamé Functions ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="23px" altimg-valign="-8px" altimg-width="56px" alttext="C_{2p+1}" display="inline"><msub><mi href="./29.6#SS2.p2">C</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow><mo>+</mo><mn>1</mn></mrow></msub></math>: coefficents</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E21" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">29.15.21</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E21.png" altimg-height="66px" altimg-valign="-30px" altimg-width="176px" alttext="\tfrac{1}{2}C_{0}+\sum_{p=1}^{n}C_{2p}>0." display="block"><mrow><mrow><mrow><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><msub><mi href="./29.6#SS2.p2">C</mi><mn>0</mn></msub></mrow><mo>+</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./29.1#p2.t1.r1">p</mi><mo>=</mo><mn>1</mn></mrow><mi href="./29.1#p2.t1.r1">n</mi></munderover><msub><mi href="./29.6#SS2.p2">C</mi><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow></msub></mrow></mrow><mo>></mo><mn>0</mn></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E21.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./29.1#p2.t1.r1">n</mi></math>: nonnegative integer</a>,
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="p" display="inline"><mi href="./29.1#p2.t1.r1">p</mi></math>: nonnegative integer</a> and
<a href="./29.6#SS2.p2" title="§29.6(ii) Function Ec + 2 m 1 ν ( z , k 2 ) ‣ §29.6 Fourier Series ‣ Lamé Functions ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="23px" altimg-valign="-8px" altimg-width="56px" alttext="C_{2p+1}" display="inline"><msub><mi href="./29.6#SS2.p2">C</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow><mo>+</mo><mn>1</mn></mrow></msub></math>: coefficents</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Then</p>
<table id="E22" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">29.15.22</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E22.png" altimg-height="30px" altimg-valign="-9px" altimg-width="290px" alttext="a^{2m}_{\nu}\left(k^{2}\right)=\tfrac{1}{2}(H_{m}+\nu(\nu+1)k^{2})," display="block"><mrow><mrow><mrow><msubsup><mi href="./29.3#SS1.p1">a</mi><mi href="./29.1#p2.t1.r4">ν</mi><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">m</mi></mrow></msubsup><mo></mo><mrow><mo>(</mo><msup><mi href="./29.1#p2.t1.r4">k</mi><mn>2</mn></msup><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mrow><mo stretchy="false">(</mo><mrow><msub><mi href="./29.15#Px1.p1">H</mi><mi href="./29.1#p2.t1.r1">m</mi></msub><mo>+</mo><mrow><mi href="./29.1#p2.t1.r4">ν</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./29.1#p2.t1.r4">ν</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msup><mi href="./29.1#p2.t1.r4">k</mi><mn>2</mn></msup></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E22.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./29.3#SS1.p1" title="§29.3(i) Eigenvalues ‣ §29.3 Definitions and Basic Properties ‣ Lamé Functions ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m65.png" altimg-height="27px" altimg-valign="-9px" altimg-width="67px" alttext="a^{\NVar{n}}_{\NVar{\nu}}\left(\NVar{k^{2}}\right)" display="inline"><mrow><msubsup><mi href="./29.3#SS1.p1">a</mi><mi class="ltx_nvar" href="./29.1#p2.t1.r4">ν</mi><mi class="ltx_nvar" href="./29.1#p2.t1.r1">n</mi></msubsup><mo></mo><mrow><mo>(</mo><msup><mi class="ltx_nvar" href="./29.1#p2.t1.r4">k</mi><mn class="ltx_nvar">2</mn></msup><mo>)</mo></mrow></mrow></math>: eigenvalues of Lamé’s equation</a>,
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./29.1#p2.t1.r1">m</mi></math>: nonnegative integer</a>,
<a href="./29.1#p2.t1.r4" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m67.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./29.1#p2.t1.r4">k</mi></math>: real parameter</a>,
<a href="./29.1#p2.t1.r4" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./29.1#p2.t1.r4">ν</mi></math>: real parameter</a> and
<a href="./29.15#Px1.p1" title="Polynomial uE m 2 n ( z , k 2 ) ‣ §29.15(i) Fourier Coefficients ‣ §29.15 Fourier Series and Chebyshev Series ‣ Lamé Polynomials ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m24.png" altimg-height="23px" altimg-valign="-8px" altimg-width="30px" alttext="H_{p}" display="inline"><msub><mi href="./29.15#Px1.p1">H</mi><mi href="./29.1#p2.t1.r1">p</mi></msub></math>: eigenvalues</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and () applies.</p>
</div>
</section>
<section id="Px5" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Polynomial <math class="ltx_Math" altimg="m46.png" altimg-height="27px" altimg-valign="-9px" altimg-width="137px" alttext="\mathit{scE}^{m}_{2n+2}\left(z,k^{2}\right)" display="inline"><mrow><msubsup><mi href="./29.12#E5" mathvariant="italic">scE</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mn>2</mn></mrow><mi href="./29.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo>(</mo><mi href="./29.1#p2.t1.r3">z</mi><mo>,</mo><msup><mi href="./29.1#p2.t1.r4">k</mi><mn>2</mn></msup><mo>)</mo></mrow></mrow></math>
</h3>
<div id="Px5.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px5.p1" class="ltx_para">
<p class="ltx_p">When <math class="ltx_Math" altimg="m55.png" altimg-height="18px" altimg-valign="-4px" altimg-width="98px" alttext="\nu=2n+2" display="inline"><mrow><mi href="./29.1#p2.t1.r4">ν</mi><mo>=</mo><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mn>2</mn></mrow></mrow></math>, <math class="ltx_Math" altimg="m68.png" altimg-height="20px" altimg-valign="-6px" altimg-width="133px" alttext="m=0,1,\dots,n" display="inline"><mrow><mi href="./29.1#p2.t1.r1">m</mi><mo>=</mo><mrow><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><mi href="./29.1#p2.t1.r1">n</mi></mrow></mrow></math>, the Fourier series ()
terminates:
</p>
<table id="E23" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">29.15.23</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E23.png" altimg-height="66px" altimg-valign="-30px" altimg-width="387px" alttext="\mathit{scE}^{m}_{2n+2}\left(z,k^{2}\right)=\sum_{p=0}^{n}B_{2p+2}\sin\left((2%
p+2)\phi\right)." display="block"><mrow><mrow><mrow><msubsup><mi href="./29.12#E5" mathvariant="italic">scE</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mn>2</mn></mrow><mi href="./29.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo>(</mo><mi href="./29.1#p2.t1.r3">z</mi><mo>,</mo><msup><mi href="./29.1#p2.t1.r4">k</mi><mn>2</mn></msup><mo>)</mo></mrow></mrow><mo>=</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./29.1#p2.t1.r1">p</mi><mo>=</mo><mn>0</mn></mrow><mi href="./29.1#p2.t1.r1">n</mi></munderover><mrow><msub><mi href="./29.6#SS4.p1">B</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow><mo>+</mo><mn>2</mn></mrow></msub><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow><mo>+</mo><mn>2</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi href="./29.2#E5">ϕ</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E23.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./29.12#E5" title="(29.12.5) ‣ §29.12(i) Elliptic-Function Form ‣ §29.12 Definitions ‣ Lamé Polynomials ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m45.png" altimg-height="27px" altimg-valign="-9px" altimg-width="137px" alttext="\mathit{scE}^{\NVar{m}}_{2\NVar{n}+2}\left(\NVar{z},\NVar{k^{2}}\right)" display="inline"><mrow><msubsup><mi href="./29.12#E5" mathvariant="italic">scE</mi><mrow><mrow><mn>2</mn><mo></mo><mi class="ltx_nvar" href="./29.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mn>2</mn></mrow><mi class="ltx_nvar" href="./29.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./29.1#p2.t1.r3">z</mi><mo>,</mo><msup><mi class="ltx_nvar" href="./29.1#p2.t1.r4">k</mi><mn class="ltx_nvar">2</mn></msup><mo>)</mo></mrow></mrow></math>: Lamé polynomial</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./29.1#p2.t1.r1">m</mi></math>: nonnegative integer</a>,
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./29.1#p2.t1.r1">n</mi></math>: nonnegative integer</a>,
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="p" display="inline"><mi href="./29.1#p2.t1.r1">p</mi></math>: nonnegative integer</a>,
<a href="./29.1#p2.t1.r3" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m73.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./29.1#p2.t1.r3">z</mi></math>: complex variable</a>,
<a href="./29.1#p2.t1.r4" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m67.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./29.1#p2.t1.r4">k</mi></math>: real parameter</a>,
<a href="./29.2#E5" title="(29.2.5) ‣ §29.2(ii) Other Forms ‣ §29.2 Differential Equations ‣ Lamé Functions ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m62.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="\phi" display="inline"><mi href="./29.2#E5">ϕ</mi></math>: change of variable</a> and
<a href="./29.6#SS4.p1" title="§29.6(iv) Function Es + 2 m 2 ν ( z , k 2 ) ‣ §29.6 Fourier Series ‣ Lamé Functions ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m18.png" altimg-height="23px" altimg-valign="-8px" altimg-width="57px" alttext="B_{2p+1}" display="inline"><msub><mi href="./29.6#SS4.p1">B</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow><mo>+</mo><mn>1</mn></mrow></msub></math>: coefficents</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">In () replace <math class="ltx_Math" altimg="m30.png" altimg-height="18px" altimg-valign="-8px" altimg-width="26px" alttext="\alpha_{p}" display="inline"><msub><mi href="./29.3#E11">α</mi><mi href="./29.1#p2.t1.r1">p</mi></msub></math>, <math class="ltx_Math" altimg="m31.png" altimg-height="23px" altimg-valign="-8px" altimg-width="25px" alttext="\beta_{p}" display="inline"><msub><mi href="./29.3#E12">β</mi><mi href="./29.1#p2.t1.r1">p</mi></msub></math>, and <math class="ltx_Math" altimg="m35.png" altimg-height="18px" altimg-valign="-8px" altimg-width="24px" alttext="\gamma_{p}" display="inline"><msub><mi href="./29.3#E12">γ</mi><mi href="./29.1#p2.t1.r1">p</mi></msub></math> as in
() by
</p>
<table id="E24" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">29.15.24</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E24.png" altimg-height="28px" altimg-valign="-7px" altimg-width="190px" alttext="[B_{2},B_{4},\dots,B_{2n+2}]^{\mathrm{T}}," display="block"><mrow><msup><mrow><mo stretchy="false">[</mo><msub><mi href="./29.6#SS4.p1">B</mi><mn>2</mn></msub><mo>,</mo><msub><mi href="./29.6#SS4.p1">B</mi><mn>4</mn></msub><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><msub><mi href="./29.6#SS4.p1">B</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mn>2</mn></mrow></msub><mo stretchy="false">]</mo></mrow><mi mathvariant="normal">T</mi></msup><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E24.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./29.1#p2.t1.r1">n</mi></math>: nonnegative integer</a> and
<a href="./29.6#SS4.p1" title="§29.6(iv) Function Es + 2 m 2 ν ( z , k 2 ) ‣ §29.6 Fourier Series ‣ Lamé Functions ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m18.png" altimg-height="23px" altimg-valign="-8px" altimg-width="57px" alttext="B_{2p+1}" display="inline"><msub><mi href="./29.6#SS4.p1">B</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow><mo>+</mo><mn>1</mn></mrow></msub></math>: coefficents</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E25" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">29.15.25</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E25.png" altimg-height="66px" altimg-valign="-30px" altimg-width="137px" alttext="\sum_{p=0}^{n}B_{2p+2}^{2}=1," display="block"><mrow><mrow><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./29.1#p2.t1.r1">p</mi><mo>=</mo><mn>0</mn></mrow><mi href="./29.1#p2.t1.r1">n</mi></munderover><msubsup><mi href="./29.6#SS4.p1">B</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow><mo>+</mo><mn>2</mn></mrow><mn>2</mn></msubsup></mrow><mo>=</mo><mn>1</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E25.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./29.1#p2.t1.r1">n</mi></math>: nonnegative integer</a>,
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="p" display="inline"><mi href="./29.1#p2.t1.r1">p</mi></math>: nonnegative integer</a> and
<a href="./29.6#SS4.p1" title="§29.6(iv) Function Es + 2 m 2 ν ( z , k 2 ) ‣ §29.6 Fourier Series ‣ Lamé Functions ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m18.png" altimg-height="23px" altimg-valign="-8px" altimg-width="57px" alttext="B_{2p+1}" display="inline"><msub><mi href="./29.6#SS4.p1">B</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow><mo>+</mo><mn>1</mn></mrow></msub></math>: coefficents</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E26" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">29.15.26</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E26.png" altimg-height="66px" altimg-valign="-30px" altimg-width="203px" alttext="\sum_{p=0}^{n}(2p+2)B_{2p+2}>0." display="block"><mrow><mrow><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./29.1#p2.t1.r1">p</mi><mo>=</mo><mn>0</mn></mrow><mi href="./29.1#p2.t1.r1">n</mi></munderover><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow><mo>+</mo><mn>2</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msub><mi href="./29.6#SS4.p1">B</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow><mo>+</mo><mn>2</mn></mrow></msub></mrow></mrow><mo>></mo><mn>0</mn></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E26.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./29.1#p2.t1.r1">n</mi></math>: nonnegative integer</a>,
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="p" display="inline"><mi href="./29.1#p2.t1.r1">p</mi></math>: nonnegative integer</a> and
<a href="./29.6#SS4.p1" title="§29.6(iv) Function Es + 2 m 2 ν ( z , k 2 ) ‣ §29.6 Fourier Series ‣ Lamé Functions ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m18.png" altimg-height="23px" altimg-valign="-8px" altimg-width="57px" alttext="B_{2p+1}" display="inline"><msub><mi href="./29.6#SS4.p1">B</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow><mo>+</mo><mn>1</mn></mrow></msub></math>: coefficents</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Then</p>
<table id="E27" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">29.15.27</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E42.png" altimg-height="30px" altimg-valign="-9px" altimg-width="308px" alttext="b^{2m+2}_{\nu}\left(k^{2}\right)=\tfrac{1}{2}(H_{m}+\nu(\nu+1)k^{2})," display="block"><mrow><mrow><mrow><msubsup><mi href="./29.3#SS1.p1">b</mi><mi href="./29.1#p2.t1.r4">ν</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">m</mi></mrow><mo>+</mo><mn>2</mn></mrow></msubsup><mo></mo><mrow><mo>(</mo><msup><mi href="./29.1#p2.t1.r4">k</mi><mn>2</mn></msup><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mrow><mo stretchy="false">(</mo><mrow><msub><mi href="./29.15#Px1.p1">H</mi><mi href="./29.1#p2.t1.r1">m</mi></msub><mo>+</mo><mrow><mi href="./29.1#p2.t1.r4">ν</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./29.1#p2.t1.r4">ν</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msup><mi href="./29.1#p2.t1.r4">k</mi><mn>2</mn></msup></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E27.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./29.3#SS1.p1" title="§29.3(i) Eigenvalues ‣ §29.3 Definitions and Basic Properties ‣ Lamé Functions ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="27px" altimg-valign="-9px" altimg-width="65px" alttext="b^{\NVar{n}}_{\NVar{\nu}}\left(\NVar{k^{2}}\right)" display="inline"><mrow><msubsup><mi href="./29.3#SS1.p1">b</mi><mi class="ltx_nvar" href="./29.1#p2.t1.r4">ν</mi><mi class="ltx_nvar" href="./29.1#p2.t1.r1">n</mi></msubsup><mo></mo><mrow><mo>(</mo><msup><mi class="ltx_nvar" href="./29.1#p2.t1.r4">k</mi><mn class="ltx_nvar">2</mn></msup><mo>)</mo></mrow></mrow></math>: eigenvalues of Lamé’s equation</a>,
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./29.1#p2.t1.r1">m</mi></math>: nonnegative integer</a>,
<a href="./29.1#p2.t1.r4" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m67.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./29.1#p2.t1.r4">k</mi></math>: real parameter</a>,
<a href="./29.1#p2.t1.r4" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./29.1#p2.t1.r4">ν</mi></math>: real parameter</a> and
<a href="./29.15#Px1.p1" title="Polynomial uE m 2 n ( z , k 2 ) ‣ §29.15(i) Fourier Coefficients ‣ §29.15 Fourier Series and Chebyshev Series ‣ Lamé Polynomials ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m24.png" altimg-height="23px" altimg-valign="-8px" altimg-width="30px" alttext="H_{p}" display="inline"><msub><mi href="./29.15#Px1.p1">H</mi><mi href="./29.1#p2.t1.r1">p</mi></msub></math>: eigenvalues</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and () applies.</p>
</div>
</section>
<section id="Px6" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Polynomial <math class="ltx_Math" altimg="m50.png" altimg-height="27px" altimg-valign="-9px" altimg-width="138px" alttext="\mathit{sdE}^{m}_{2n+2}\left(z,k^{2}\right)" display="inline"><mrow><msubsup><mi href="./29.12#E6" mathvariant="italic">sdE</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mn>2</mn></mrow><mi href="./29.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo>(</mo><mi href="./29.1#p2.t1.r3">z</mi><mo>,</mo><msup><mi href="./29.1#p2.t1.r4">k</mi><mn>2</mn></msup><mo>)</mo></mrow></mrow></math>
</h3>
<div id="Px6.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px6.p1" class="ltx_para">
<p class="ltx_p">When <math class="ltx_Math" altimg="m55.png" altimg-height="18px" altimg-valign="-4px" altimg-width="98px" alttext="\nu=2n+2" display="inline"><mrow><mi href="./29.1#p2.t1.r4">ν</mi><mo>=</mo><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mn>2</mn></mrow></mrow></math>, <math class="ltx_Math" altimg="m68.png" altimg-height="20px" altimg-valign="-6px" altimg-width="133px" alttext="m=0,1,\dots,n" display="inline"><mrow><mi href="./29.1#p2.t1.r1">m</mi><mo>=</mo><mrow><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><mi href="./29.1#p2.t1.r1">n</mi></mrow></mrow></math>, the Fourier series ()
terminates:
</p>
<table id="E28" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">29.15.28</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E28.png" altimg-height="66px" altimg-valign="-30px" altimg-width="464px" alttext="\mathit{sdE}^{m}_{2n+2}\left(z,k^{2}\right)=\operatorname{dn}\left(z,k\right)%
\sum_{p=0}^{n}C_{2p+1}\cos\left((2p+1)\phi\right)." display="block"><mrow><mrow><mrow><msubsup><mi href="./29.12#E6" mathvariant="italic">sdE</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mn>2</mn></mrow><mi href="./29.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo>(</mo><mi href="./29.1#p2.t1.r3">z</mi><mo>,</mo><msup><mi href="./29.1#p2.t1.r4">k</mi><mn>2</mn></msup><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mi href="./22.2#E6">dn</mi><mo></mo><mrow><mo>(</mo><mi href="./29.1#p2.t1.r3">z</mi><mo>,</mo><mi href="./29.1#p2.t1.r4">k</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./29.1#p2.t1.r1">p</mi><mo>=</mo><mn>0</mn></mrow><mi href="./29.1#p2.t1.r1">n</mi></munderover><mrow><msub><mi href="./29.6#SS2.p2">C</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow><mo>+</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi href="./29.2#E5">ϕ</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E28.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./22.2#E6" title="(22.2.6) ‣ §22.2 Definitions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="23px" altimg-valign="-7px" altimg-width="75px" alttext="\operatorname{dn}\left(\NVar{z},\NVar{k}\right)" display="inline"><mrow><mi href="./22.2#E6">dn</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r2">z</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobian elliptic function</a>,
<a href="./29.12#E6" title="(29.12.6) ‣ §29.12(i) Elliptic-Function Form ‣ §29.12 Definitions ‣ Lamé Polynomials ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="27px" altimg-valign="-9px" altimg-width="138px" alttext="\mathit{sdE}^{\NVar{m}}_{2\NVar{n}+2}\left(\NVar{z},\NVar{k^{2}}\right)" display="inline"><mrow><msubsup><mi href="./29.12#E6" mathvariant="italic">sdE</mi><mrow><mrow><mn>2</mn><mo></mo><mi class="ltx_nvar" href="./29.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mn>2</mn></mrow><mi class="ltx_nvar" href="./29.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./29.1#p2.t1.r3">z</mi><mo>,</mo><msup><mi class="ltx_nvar" href="./29.1#p2.t1.r4">k</mi><mn class="ltx_nvar">2</mn></msup><mo>)</mo></mrow></mrow></math>: Lamé polynomial</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m32.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./29.1#p2.t1.r1">m</mi></math>: nonnegative integer</a>,
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./29.1#p2.t1.r1">n</mi></math>: nonnegative integer</a>,
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="p" display="inline"><mi href="./29.1#p2.t1.r1">p</mi></math>: nonnegative integer</a>,
<a href="./29.1#p2.t1.r3" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m73.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./29.1#p2.t1.r3">z</mi></math>: complex variable</a>,
<a href="./29.1#p2.t1.r4" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m67.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./29.1#p2.t1.r4">k</mi></math>: real parameter</a>,
<a href="./29.2#E5" title="(29.2.5) ‣ §29.2(ii) Other Forms ‣ §29.2 Differential Equations ‣ Lamé Functions ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m62.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="\phi" display="inline"><mi href="./29.2#E5">ϕ</mi></math>: change of variable</a> and
<a href="./29.6#SS2.p2" title="§29.6(ii) Function Ec + 2 m 1 ν ( z , k 2 ) ‣ §29.6 Fourier Series ‣ Lamé Functions ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="23px" altimg-valign="-8px" altimg-width="56px" alttext="C_{2p+1}" display="inline"><msub><mi href="./29.6#SS2.p2">C</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow><mo>+</mo><mn>1</mn></mrow></msub></math>: coefficents</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">In () replace <math class="ltx_Math" altimg="m30.png" altimg-height="18px" altimg-valign="-8px" altimg-width="26px" alttext="\alpha_{p}" display="inline"><msub><mi href="./29.3#E11">α</mi><mi href="./29.1#p2.t1.r1">p</mi></msub></math>, <math class="ltx_Math" altimg="m31.png" altimg-height="23px" altimg-valign="-8px" altimg-width="25px" alttext="\beta_{p}" display="inline"><msub><mi href="./29.3#E12">β</mi><mi href="./29.1#p2.t1.r1">p</mi></msub></math>, and <math class="ltx_Math" altimg="m35.png" altimg-height="18px" altimg-valign="-8px" altimg-width="24px" alttext="\gamma_{p}" display="inline"><msub><mi href="./29.3#E12">γ</mi><mi href="./29.1#p2.t1.r1">p</mi></msub></math> as in
() by
</p>
<table id="E29" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">29.15.29</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E29.png" altimg-height="28px" altimg-valign="-7px" altimg-width="188px" alttext="[C_{1},C_{3},\dots,C_{2n+1}]^{\mathrm{T}}," display="block"><mrow><msup><mrow><mo stretchy="false">[</mo><msub><mi href="./29.6#SS2.p2">C</mi><mn>1</mn></msub><mo>,</mo><msub><mi href="./29.6#SS2.p2">C</mi><mn>3</mn></msub><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><msub><mi href="./29.6#SS2.p2">C</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">]</mo></mrow><mi mathvariant="normal">T</mi></msup><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E29.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./29.1#p2.t1.r1">n</mi></math>: nonnegative integer</a> and
<a href="./29.6#SS2.p2" title="§29.6(ii) Function Ec + 2 m 1 ν ( z , k 2 ) ‣ §29.6 Fourier Series ‣ Lamé Functions ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="23px" altimg-valign="-8px" altimg-width="56px" alttext="C_{2p+1}" display="inline"><msub><mi href="./29.6#SS2.p2">C</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow><mo>+</mo><mn>1</mn></mrow></msub></math>: coefficents</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E30" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">29.15.30</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E30.png" altimg-height="69px" altimg-valign="-30px" altimg-width="520px" alttext="\left(1-\tfrac{1}{2}k^{2}\right)\sum_{p=0}^{n}C_{2p+1}^{2}-{\tfrac{1}{2}k^{2}%
\left(\tfrac{1}{2}C_{1}^{2}+\sum_{p=0}^{n-1}C_{2p+1}C_{2p+3}\right)=1}," display="block"><mrow><mrow><mrow><mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><msup><mi href="./29.1#p2.t1.r4">k</mi><mn>2</mn></msup></mrow></mrow><mo>)</mo></mrow><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./29.1#p2.t1.r1">p</mi><mo>=</mo><mn>0</mn></mrow><mi href="./29.1#p2.t1.r1">n</mi></munderover><msubsup><mi href="./29.6#SS2.p2">C</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow><mo>+</mo><mn>1</mn></mrow><mn>2</mn></msubsup></mrow></mrow><mo>-</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><msup><mi href="./29.1#p2.t1.r4">k</mi><mn>2</mn></msup><mo></mo><mrow><mo>(</mo><mrow><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><msubsup><mi href="./29.6#SS2.p2">C</mi><mn>1</mn><mn>2</mn></msubsup></mrow><mo>+</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./29.1#p2.t1.r1">p</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi href="./29.1#p2.t1.r1">n</mi><mo>-</mo><mn>1</mn></mrow></munderover><mrow><msub><mi href="./29.6#SS2.p2">C</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow><mo>+</mo><mn>1</mn></mrow></msub><mo></mo><msub><mi href="./29.6#SS2.p2">C</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow><mo>+</mo><mn>3</mn></mrow></msub></mrow></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mn>1</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E30.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./29.1#p2.t1.r1">n</mi></math>: nonnegative integer</a>,
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="p" display="inline"><mi href="./29.1#p2.t1.r1">p</mi></math>: nonnegative integer</a>,
<a href="./29.1#p2.t1.r4" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m67.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./29.1#p2.t1.r4">k</mi></math>: real parameter</a> and
<a href="./29.6#SS2.p2" title="§29.6(ii) Function Ec + 2 m 1 ν ( z , k 2 ) ‣ §29.6 Fourier Series ‣ Lamé Functions ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="23px" altimg-valign="-8px" altimg-width="56px" alttext="C_{2p+1}" display="inline"><msub><mi href="./29.6#SS2.p2">C</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow><mo>+</mo><mn>1</mn></mrow></msub></math>: coefficents</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E31" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">29.15.31</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E31.png" altimg-height="66px" altimg-valign="-30px" altimg-width="136px" alttext="\sum_{p=0}^{n}C_{2p+1}>0." display="block"><mrow><mrow><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./29.1#p2.t1.r1">p</mi><mo>=</mo><mn>0</mn></mrow><mi href="./29.1#p2.t1.r1">n</mi></munderover><msub><mi href="./29.6#SS2.p2">C</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow><mo>+</mo><mn>1</mn></mrow></msub></mrow><mo>></mo><mn>0</mn></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E31.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./29.1#p2.t1.r1">n</mi></math>: nonnegative integer</a>,
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="p" display="inline"><mi href="./29.1#p2.t1.r1">p</mi></math>: nonnegative integer</a> and
<a href="./29.6#SS2.p2" title="§29.6(ii) Function Ec + 2 m 1 ν ( z , k 2 ) ‣ §29.6 Fourier Series ‣ Lamé Functions ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="23px" altimg-valign="-8px" altimg-width="56px" alttext="C_{2p+1}" display="inline"><msub><mi href="./29.6#SS2.p2">C</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow><mo>+</mo><mn>1</mn></mrow></msub></math>: coefficents</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Then</p>
<table id="E32" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">29.15.32</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E32.png" altimg-height="30px" altimg-valign="-9px" altimg-width="310px" alttext="a^{2m+1}_{\nu}\left(k^{2}\right)=\tfrac{1}{2}(H_{m}+\nu(\nu+1)k^{2})," display="block"><mrow><mrow><mrow><msubsup><mi href="./29.3#SS1.p1">a</mi><mi href="./29.1#p2.t1.r4">ν</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">m</mi></mrow><mo>+</mo><mn>1</mn></mrow></msubsup><mo></mo><mrow><mo>(</mo><msup><mi href="./29.1#p2.t1.r4">k</mi><mn>2</mn></msup><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mrow><mo stretchy="false">(</mo><mrow><msub><mi href="./29.15#Px1.p1">H</mi><mi href="./29.1#p2.t1.r1">m</mi></msub><mo>+</mo><mrow><mi href="./29.1#p2.t1.r4">ν</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./29.1#p2.t1.r4">ν</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msup><mi href="./29.1#p2.t1.r4">k</mi><mn>2</mn></msup></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E32.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./29.3#SS1.p1" title="§29.3(i) Eigenvalues ‣ §29.3 Definitions and Basic Properties ‣ Lamé Functions ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m65.png" altimg-height="27px" altimg-valign="-9px" altimg-width="67px" alttext="a^{\NVar{n}}_{\NVar{\nu}}\left(\NVar{k^{2}}\right)" display="inline"><mrow><msubsup><mi href="./29.3#SS1.p1">a</mi><mi class="ltx_nvar" href="./29.1#p2.t1.r4">ν</mi><mi class="ltx_nvar" href="./29.1#p2.t1.r1">n</mi></msubsup><mo></mo><mrow><mo>(</mo><msup><mi class="ltx_nvar" href="./29.1#p2.t1.r4">k</mi><mn class="ltx_nvar">2</mn></msup><mo>)</mo></mrow></mrow></math>: eigenvalues of Lamé’s equation</a>,
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./29.1#p2.t1.r1">m</mi></math>: nonnegative integer</a>,
<a href="./29.1#p2.t1.r4" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m67.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./29.1#p2.t1.r4">k</mi></math>: real parameter</a>,
<a href="./29.1#p2.t1.r4" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./29.1#p2.t1.r4">ν</mi></math>: real parameter</a> and
<a href="./29.15#Px1.p1" title="Polynomial uE m 2 n ( z , k 2 ) ‣ §29.15(i) Fourier Coefficients ‣ §29.15 Fourier Series and Chebyshev Series ‣ Lamé Polynomials ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m24.png" altimg-height="23px" altimg-valign="-8px" altimg-width="30px" alttext="H_{p}" display="inline"><msub><mi href="./29.15#Px1.p1">H</mi><mi href="./29.1#p2.t1.r1">p</mi></msub></math>: eigenvalues</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and () applies.</p>
</div>
</section>
<section id="Px7" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Polynomial <math class="ltx_Math" altimg="m40.png" altimg-height="27px" altimg-valign="-9px" altimg-width="138px" alttext="\mathit{cdE}^{m}_{2n+2}\left(z,k^{2}\right)" display="inline"><mrow><msubsup><mi href="./29.12#E7" mathvariant="italic">cdE</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mn>2</mn></mrow><mi href="./29.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo>(</mo><mi href="./29.1#p2.t1.r3">z</mi><mo>,</mo><msup><mi href="./29.1#p2.t1.r4">k</mi><mn>2</mn></msup><mo>)</mo></mrow></mrow></math>
</h3>
<div id="Px7.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px7.p1" class="ltx_para">
<p class="ltx_p">When <math class="ltx_Math" altimg="m55.png" altimg-height="18px" altimg-valign="-4px" altimg-width="98px" alttext="\nu=2n+2" display="inline"><mrow><mi href="./29.1#p2.t1.r4">ν</mi><mo>=</mo><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mn>2</mn></mrow></mrow></math>, <math class="ltx_Math" altimg="m68.png" altimg-height="20px" altimg-valign="-6px" altimg-width="133px" alttext="m=0,1,\dots,n" display="inline"><mrow><mi href="./29.1#p2.t1.r1">m</mi><mo>=</mo><mrow><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><mi href="./29.1#p2.t1.r1">n</mi></mrow></mrow></math>, the Fourier series ()
terminates:</p>
<table id="E33" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">29.15.33</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E33.png" altimg-height="66px" altimg-valign="-30px" altimg-width="464px" alttext="\mathit{cdE}^{m}_{2n+2}\left(z,k^{2}\right)=\operatorname{dn}\left(z,k\right)%
\sum_{p=0}^{n}D_{2p+1}\sin\left((2p+1)\phi\right)." display="block"><mrow><mrow><mrow><msubsup><mi href="./29.12#E7" mathvariant="italic">cdE</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mn>2</mn></mrow><mi href="./29.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo>(</mo><mi href="./29.1#p2.t1.r3">z</mi><mo>,</mo><msup><mi href="./29.1#p2.t1.r4">k</mi><mn>2</mn></msup><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mi href="./22.2#E6">dn</mi><mo></mo><mrow><mo>(</mo><mi href="./29.1#p2.t1.r3">z</mi><mo>,</mo><mi href="./29.1#p2.t1.r4">k</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./29.1#p2.t1.r1">p</mi><mo>=</mo><mn>0</mn></mrow><mi href="./29.1#p2.t1.r1">n</mi></munderover><mrow><msub><mi href="./29.6#SS4.p2">D</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow><mo>+</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi href="./29.2#E5">ϕ</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E33.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./22.2#E6" title="(22.2.6) ‣ §22.2 Definitions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="23px" altimg-valign="-7px" altimg-width="75px" alttext="\operatorname{dn}\left(\NVar{z},\NVar{k}\right)" display="inline"><mrow><mi href="./22.2#E6">dn</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r2">z</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobian elliptic function</a>,
<a href="./29.12#E7" title="(29.12.7) ‣ §29.12(i) Elliptic-Function Form ‣ §29.12 Definitions ‣ Lamé Polynomials ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="27px" altimg-valign="-9px" altimg-width="138px" alttext="\mathit{cdE}^{\NVar{m}}_{2\NVar{n}+2}\left(\NVar{z},\NVar{k^{2}}\right)" display="inline"><mrow><msubsup><mi href="./29.12#E7" mathvariant="italic">cdE</mi><mrow><mrow><mn>2</mn><mo></mo><mi class="ltx_nvar" href="./29.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mn>2</mn></mrow><mi class="ltx_nvar" href="./29.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./29.1#p2.t1.r3">z</mi><mo>,</mo><msup><mi class="ltx_nvar" href="./29.1#p2.t1.r4">k</mi><mn class="ltx_nvar">2</mn></msup><mo>)</mo></mrow></mrow></math>: Lamé polynomial</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./29.1#p2.t1.r1">m</mi></math>: nonnegative integer</a>,
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./29.1#p2.t1.r1">n</mi></math>: nonnegative integer</a>,
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="p" display="inline"><mi href="./29.1#p2.t1.r1">p</mi></math>: nonnegative integer</a>,
<a href="./29.1#p2.t1.r3" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m73.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./29.1#p2.t1.r3">z</mi></math>: complex variable</a>,
<a href="./29.1#p2.t1.r4" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m67.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./29.1#p2.t1.r4">k</mi></math>: real parameter</a>,
<a href="./29.2#E5" title="(29.2.5) ‣ §29.2(ii) Other Forms ‣ §29.2 Differential Equations ‣ Lamé Functions ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m62.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="\phi" display="inline"><mi href="./29.2#E5">ϕ</mi></math>: change of variable</a> and
<a href="./29.6#SS4.p2" title="§29.6(iv) Function Es + 2 m 2 ν ( z , k 2 ) ‣ §29.6 Fourier Series ‣ Lamé Functions ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="23px" altimg-valign="-8px" altimg-width="38px" alttext="D_{2p}" display="inline"><msub><mi href="./29.6#SS4.p2">D</mi><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow></msub></math>: coefficents</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">In () replace <math class="ltx_Math" altimg="m30.png" altimg-height="18px" altimg-valign="-8px" altimg-width="26px" alttext="\alpha_{p}" display="inline"><msub><mi href="./29.3#E11">α</mi><mi href="./29.1#p2.t1.r1">p</mi></msub></math>, <math class="ltx_Math" altimg="m31.png" altimg-height="23px" altimg-valign="-8px" altimg-width="25px" alttext="\beta_{p}" display="inline"><msub><mi href="./29.3#E12">β</mi><mi href="./29.1#p2.t1.r1">p</mi></msub></math>, and <math class="ltx_Math" altimg="m35.png" altimg-height="18px" altimg-valign="-8px" altimg-width="24px" alttext="\gamma_{p}" display="inline"><msub><mi href="./29.3#E12">γ</mi><mi href="./29.1#p2.t1.r1">p</mi></msub></math> as in
() by
</p>
<table id="E34" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">29.15.34</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E34.png" altimg-height="28px" altimg-valign="-7px" altimg-width="194px" alttext="[D_{1},D_{3},\dots,D_{2n+1}]^{\mathrm{T}}," display="block"><mrow><msup><mrow><mo stretchy="false">[</mo><msub><mi href="./29.6#SS4.p2">D</mi><mn>1</mn></msub><mo>,</mo><msub><mi href="./29.6#SS4.p2">D</mi><mn>3</mn></msub><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><msub><mi href="./29.6#SS4.p2">D</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">]</mo></mrow><mi mathvariant="normal">T</mi></msup><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E34.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./29.1#p2.t1.r1">n</mi></math>: nonnegative integer</a> and
<a href="./29.6#SS4.p2" title="§29.6(iv) Function Es + 2 m 2 ν ( z , k 2 ) ‣ §29.6 Fourier Series ‣ Lamé Functions ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="23px" altimg-valign="-8px" altimg-width="38px" alttext="D_{2p}" display="inline"><msub><mi href="./29.6#SS4.p2">D</mi><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow></msub></math>: coefficents</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E35" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">29.15.35</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E35.png" altimg-height="69px" altimg-valign="-30px" altimg-width="528px" alttext="\left(1-\tfrac{1}{2}k^{2}\right)\sum_{p=0}^{n}D_{2p+1}^{2}+{\tfrac{1}{2}k^{2}%
\left(\tfrac{1}{2}D_{1}^{2}-\sum_{p=0}^{n-1}D_{2p+1}D_{2p+3}\right)=1}," display="block"><mrow><mrow><mrow><mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><msup><mi href="./29.1#p2.t1.r4">k</mi><mn>2</mn></msup></mrow></mrow><mo>)</mo></mrow><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./29.1#p2.t1.r1">p</mi><mo>=</mo><mn>0</mn></mrow><mi href="./29.1#p2.t1.r1">n</mi></munderover><msubsup><mi href="./29.6#SS4.p2">D</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow><mo>+</mo><mn>1</mn></mrow><mn>2</mn></msubsup></mrow></mrow><mo>+</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><msup><mi href="./29.1#p2.t1.r4">k</mi><mn>2</mn></msup><mo></mo><mrow><mo>(</mo><mrow><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><msubsup><mi href="./29.6#SS4.p2">D</mi><mn>1</mn><mn>2</mn></msubsup></mrow><mo>-</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./29.1#p2.t1.r1">p</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi href="./29.1#p2.t1.r1">n</mi><mo>-</mo><mn>1</mn></mrow></munderover><mrow><msub><mi href="./29.6#SS4.p2">D</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow><mo>+</mo><mn>1</mn></mrow></msub><mo></mo><msub><mi href="./29.6#SS4.p2">D</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow><mo>+</mo><mn>3</mn></mrow></msub></mrow></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mn>1</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E35.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./29.1#p2.t1.r1">n</mi></math>: nonnegative integer</a>,
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="p" display="inline"><mi href="./29.1#p2.t1.r1">p</mi></math>: nonnegative integer</a>,
<a href="./29.1#p2.t1.r4" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m67.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./29.1#p2.t1.r4">k</mi></math>: real parameter</a> and
<a href="./29.6#SS4.p2" title="§29.6(iv) Function Es + 2 m 2 ν ( z , k 2 ) ‣ §29.6 Fourier Series ‣ Lamé Functions ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="23px" altimg-valign="-8px" altimg-width="38px" alttext="D_{2p}" display="inline"><msub><mi href="./29.6#SS4.p2">D</mi><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow></msub></math>: coefficents</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E36" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">29.15.36</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E36.png" altimg-height="66px" altimg-valign="-30px" altimg-width="205px" alttext="\sum_{p=0}^{n}(2p+1)D_{2p+1}>0." display="block"><mrow><mrow><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./29.1#p2.t1.r1">p</mi><mo>=</mo><mn>0</mn></mrow><mi href="./29.1#p2.t1.r1">n</mi></munderover><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msub><mi href="./29.6#SS4.p2">D</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow><mo>+</mo><mn>1</mn></mrow></msub></mrow></mrow><mo>></mo><mn>0</mn></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E36.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./29.1#p2.t1.r1">n</mi></math>: nonnegative integer</a>,
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="p" display="inline"><mi href="./29.1#p2.t1.r1">p</mi></math>: nonnegative integer</a> and
<a href="./29.6#SS4.p2" title="§29.6(iv) Function Es + 2 m 2 ν ( z , k 2 ) ‣ §29.6 Fourier Series ‣ Lamé Functions ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="23px" altimg-valign="-8px" altimg-width="38px" alttext="D_{2p}" display="inline"><msub><mi href="./29.6#SS4.p2">D</mi><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow></msub></math>: coefficents</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Then</p>
<table id="E37" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">29.15.37</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E37.png" altimg-height="30px" altimg-valign="-9px" altimg-width="308px" alttext="b^{2m+1}_{\nu}\left(k^{2}\right)=\tfrac{1}{2}(H_{m}+\nu(\nu+1)k^{2})," display="block"><mrow><mrow><mrow><msubsup><mi href="./29.3#SS1.p1">b</mi><mi href="./29.1#p2.t1.r4">ν</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">m</mi></mrow><mo>+</mo><mn>1</mn></mrow></msubsup><mo></mo><mrow><mo>(</mo><msup><mi href="./29.1#p2.t1.r4">k</mi><mn>2</mn></msup><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mrow><mo stretchy="false">(</mo><mrow><msub><mi href="./29.15#Px1.p1">H</mi><mi href="./29.1#p2.t1.r1">m</mi></msub><mo>+</mo><mrow><mi href="./29.1#p2.t1.r4">ν</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./29.1#p2.t1.r4">ν</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msup><mi href="./29.1#p2.t1.r4">k</mi><mn>2</mn></msup></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E37.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./29.3#SS1.p1" title="§29.3(i) Eigenvalues ‣ §29.3 Definitions and Basic Properties ‣ Lamé Functions ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="27px" altimg-valign="-9px" altimg-width="65px" alttext="b^{\NVar{n}}_{\NVar{\nu}}\left(\NVar{k^{2}}\right)" display="inline"><mrow><msubsup><mi href="./29.3#SS1.p1">b</mi><mi class="ltx_nvar" href="./29.1#p2.t1.r4">ν</mi><mi class="ltx_nvar" href="./29.1#p2.t1.r1">n</mi></msubsup><mo></mo><mrow><mo>(</mo><msup><mi class="ltx_nvar" href="./29.1#p2.t1.r4">k</mi><mn class="ltx_nvar">2</mn></msup><mo>)</mo></mrow></mrow></math>: eigenvalues of Lamé’s equation</a>,
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./29.1#p2.t1.r1">m</mi></math>: nonnegative integer</a>,
<a href="./29.1#p2.t1.r4" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m67.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./29.1#p2.t1.r4">k</mi></math>: real parameter</a>,
<a href="./29.1#p2.t1.r4" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./29.1#p2.t1.r4">ν</mi></math>: real parameter</a> and
<a href="./29.15#Px1.p1" title="Polynomial uE m 2 n ( z , k 2 ) ‣ §29.15(i) Fourier Coefficients ‣ §29.15 Fourier Series and Chebyshev Series ‣ Lamé Polynomials ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m24.png" altimg-height="23px" altimg-valign="-8px" altimg-width="30px" alttext="H_{p}" display="inline"><msub><mi href="./29.15#Px1.p1">H</mi><mi href="./29.1#p2.t1.r1">p</mi></msub></math>: eigenvalues</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and () applies.</p>
</div>
</section>
<section id="Px8" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Polynomial <math class="ltx_Math" altimg="m48.png" altimg-height="27px" altimg-valign="-9px" altimg-width="146px" alttext="\mathit{scdE}^{m}_{2n+3}\left(z,k^{2}\right)" display="inline"><mrow><msubsup><mi href="./29.12#E8" mathvariant="italic">scdE</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mn>3</mn></mrow><mi href="./29.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo>(</mo><mi href="./29.1#p2.t1.r3">z</mi><mo>,</mo><msup><mi href="./29.1#p2.t1.r4">k</mi><mn>2</mn></msup><mo>)</mo></mrow></mrow></math>
</h3>
<div id="Px8.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px8.p1" class="ltx_para">
<p class="ltx_p">When <math class="ltx_Math" altimg="m56.png" altimg-height="18px" altimg-valign="-4px" altimg-width="98px" alttext="\nu=2n+3" display="inline"><mrow><mi href="./29.1#p2.t1.r4">ν</mi><mo>=</mo><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mn>3</mn></mrow></mrow></math>, <math class="ltx_Math" altimg="m68.png" altimg-height="20px" altimg-valign="-6px" altimg-width="133px" alttext="m=0,1,\dots,n" display="inline"><mrow><mi href="./29.1#p2.t1.r1">m</mi><mo>=</mo><mrow><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><mi href="./29.1#p2.t1.r1">n</mi></mrow></mrow></math>, the Fourier series ()
terminates:
</p>
<table id="E38" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">29.15.38</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E38.png" altimg-height="66px" altimg-valign="-30px" altimg-width="472px" alttext="\mathit{scdE}^{m}_{2n+3}\left(z,k^{2}\right)=\operatorname{dn}\left(z,k\right)%
\sum_{p=0}^{n}D_{2p+2}\sin\left((2p+2)\phi\right)." display="block"><mrow><mrow><mrow><msubsup><mi href="./29.12#E8" mathvariant="italic">scdE</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mn>3</mn></mrow><mi href="./29.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo>(</mo><mi href="./29.1#p2.t1.r3">z</mi><mo>,</mo><msup><mi href="./29.1#p2.t1.r4">k</mi><mn>2</mn></msup><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mi href="./22.2#E6">dn</mi><mo></mo><mrow><mo>(</mo><mi href="./29.1#p2.t1.r3">z</mi><mo>,</mo><mi href="./29.1#p2.t1.r4">k</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./29.1#p2.t1.r1">p</mi><mo>=</mo><mn>0</mn></mrow><mi href="./29.1#p2.t1.r1">n</mi></munderover><mrow><msub><mi href="./29.6#SS4.p2">D</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow><mo>+</mo><mn>2</mn></mrow></msub><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow><mo>+</mo><mn>2</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi href="./29.2#E5">ϕ</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E38.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./22.2#E6" title="(22.2.6) ‣ §22.2 Definitions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="23px" altimg-valign="-7px" altimg-width="75px" alttext="\operatorname{dn}\left(\NVar{z},\NVar{k}\right)" display="inline"><mrow><mi href="./22.2#E6">dn</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r2">z</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobian elliptic function</a>,
<a href="./29.12#E8" title="(29.12.8) ‣ §29.12(i) Elliptic-Function Form ‣ §29.12 Definitions ‣ Lamé Polynomials ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="27px" altimg-valign="-9px" altimg-width="146px" alttext="\mathit{scdE}^{\NVar{m}}_{2\NVar{n}+3}\left(\NVar{z},\NVar{k^{2}}\right)" display="inline"><mrow><msubsup><mi href="./29.12#E8" mathvariant="italic">scdE</mi><mrow><mrow><mn>2</mn><mo></mo><mi class="ltx_nvar" href="./29.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mn>3</mn></mrow><mi class="ltx_nvar" href="./29.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./29.1#p2.t1.r3">z</mi><mo>,</mo><msup><mi class="ltx_nvar" href="./29.1#p2.t1.r4">k</mi><mn class="ltx_nvar">2</mn></msup><mo>)</mo></mrow></mrow></math>: Lamé polynomial</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./29.1#p2.t1.r1">m</mi></math>: nonnegative integer</a>,
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./29.1#p2.t1.r1">n</mi></math>: nonnegative integer</a>,
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="p" display="inline"><mi href="./29.1#p2.t1.r1">p</mi></math>: nonnegative integer</a>,
<a href="./29.1#p2.t1.r3" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m73.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./29.1#p2.t1.r3">z</mi></math>: complex variable</a>,
<a href="./29.1#p2.t1.r4" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m67.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./29.1#p2.t1.r4">k</mi></math>: real parameter</a>,
<a href="./29.2#E5" title="(29.2.5) ‣ §29.2(ii) Other Forms ‣ §29.2 Differential Equations ‣ Lamé Functions ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m62.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="\phi" display="inline"><mi href="./29.2#E5">ϕ</mi></math>: change of variable</a> and
<a href="./29.6#SS4.p2" title="§29.6(iv) Function Es + 2 m 2 ν ( z , k 2 ) ‣ §29.6 Fourier Series ‣ Lamé Functions ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="23px" altimg-valign="-8px" altimg-width="38px" alttext="D_{2p}" display="inline"><msub><mi href="./29.6#SS4.p2">D</mi><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow></msub></math>: coefficents</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">In () replace <math class="ltx_Math" altimg="m30.png" altimg-height="18px" altimg-valign="-8px" altimg-width="26px" alttext="\alpha_{p}" display="inline"><msub><mi href="./29.3#E11">α</mi><mi href="./29.1#p2.t1.r1">p</mi></msub></math>, <math class="ltx_Math" altimg="m31.png" altimg-height="23px" altimg-valign="-8px" altimg-width="25px" alttext="\beta_{p}" display="inline"><msub><mi href="./29.3#E12">β</mi><mi href="./29.1#p2.t1.r1">p</mi></msub></math>, and <math class="ltx_Math" altimg="m35.png" altimg-height="18px" altimg-valign="-8px" altimg-width="24px" alttext="\gamma_{p}" display="inline"><msub><mi href="./29.3#E12">γ</mi><mi href="./29.1#p2.t1.r1">p</mi></msub></math> as in
() by
</p>
<table id="E39" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">29.15.39</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E39.png" altimg-height="28px" altimg-valign="-7px" altimg-width="194px" alttext="[D_{2},D_{4},\dots,D_{2n+2}]^{\mathrm{T}}," display="block"><mrow><msup><mrow><mo stretchy="false">[</mo><msub><mi href="./29.6#SS4.p2">D</mi><mn>2</mn></msub><mo>,</mo><msub><mi href="./29.6#SS4.p2">D</mi><mn>4</mn></msub><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><msub><mi href="./29.6#SS4.p2">D</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mn>2</mn></mrow></msub><mo stretchy="false">]</mo></mrow><mi mathvariant="normal">T</mi></msup><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E39.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./29.1#p2.t1.r1">n</mi></math>: nonnegative integer</a> and
<a href="./29.6#SS4.p2" title="§29.6(iv) Function Es + 2 m 2 ν ( z , k 2 ) ‣ §29.6 Fourier Series ‣ Lamé Functions ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="23px" altimg-valign="-8px" altimg-width="38px" alttext="D_{2p}" display="inline"><msub><mi href="./29.6#SS4.p2">D</mi><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow></msub></math>: coefficents</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E40" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">29.15.40</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E40.png" altimg-height="66px" altimg-valign="-30px" altimg-width="411px" alttext="\left(1-\tfrac{1}{2}k^{2}\right)\sum_{p=0}^{n}D_{2p+2}^{2}-\tfrac{1}{2}k^{2}%
\sum_{p=1}^{n}D_{2p}D_{2p+2}=1," display="block"><mrow><mrow><mrow><mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><msup><mi href="./29.1#p2.t1.r4">k</mi><mn>2</mn></msup></mrow></mrow><mo>)</mo></mrow><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./29.1#p2.t1.r1">p</mi><mo>=</mo><mn>0</mn></mrow><mi href="./29.1#p2.t1.r1">n</mi></munderover><msubsup><mi href="./29.6#SS4.p2">D</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow><mo>+</mo><mn>2</mn></mrow><mn>2</mn></msubsup></mrow></mrow><mo>-</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><msup><mi href="./29.1#p2.t1.r4">k</mi><mn>2</mn></msup><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./29.1#p2.t1.r1">p</mi><mo>=</mo><mn>1</mn></mrow><mi href="./29.1#p2.t1.r1">n</mi></munderover><mrow><msub><mi href="./29.6#SS4.p2">D</mi><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow></msub><mo></mo><msub><mi href="./29.6#SS4.p2">D</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow><mo>+</mo><mn>2</mn></mrow></msub></mrow></mrow></mrow></mrow><mo>=</mo><mn>1</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E40.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./29.1#p2.t1.r1">n</mi></math>: nonnegative integer</a>,
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="p" display="inline"><mi href="./29.1#p2.t1.r1">p</mi></math>: nonnegative integer</a>,
<a href="./29.1#p2.t1.r4" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m67.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./29.1#p2.t1.r4">k</mi></math>: real parameter</a> and
<a href="./29.6#SS4.p2" title="§29.6(iv) Function Es + 2 m 2 ν ( z , k 2 ) ‣ §29.6 Fourier Series ‣ Lamé Functions ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="23px" altimg-valign="-8px" altimg-width="38px" alttext="D_{2p}" display="inline"><msub><mi href="./29.6#SS4.p2">D</mi><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow></msub></math>: coefficents</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E41" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">29.15.41</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E41.png" altimg-height="66px" altimg-valign="-30px" altimg-width="205px" alttext="\sum_{p=0}^{n}(2p+2)D_{2p+2}>0." display="block"><mrow><mrow><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./29.1#p2.t1.r1">p</mi><mo>=</mo><mn>0</mn></mrow><mi href="./29.1#p2.t1.r1">n</mi></munderover><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow><mo>+</mo><mn>2</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msub><mi href="./29.6#SS4.p2">D</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow><mo>+</mo><mn>2</mn></mrow></msub></mrow></mrow><mo>></mo><mn>0</mn></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E41.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./29.1#p2.t1.r1">n</mi></math>: nonnegative integer</a>,
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="p" display="inline"><mi href="./29.1#p2.t1.r1">p</mi></math>: nonnegative integer</a> and
<a href="./29.6#SS4.p2" title="§29.6(iv) Function Es + 2 m 2 ν ( z , k 2 ) ‣ §29.6 Fourier Series ‣ Lamé Functions ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="23px" altimg-valign="-8px" altimg-width="38px" alttext="D_{2p}" display="inline"><msub><mi href="./29.6#SS4.p2">D</mi><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow></msub></math>: coefficents</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Then</p>
<table id="E42" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">29.15.42</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E42.png" altimg-height="30px" altimg-valign="-9px" altimg-width="308px" alttext="b^{2m+2}_{\nu}\left(k^{2}\right)=\tfrac{1}{2}(H_{m}+\nu(\nu+1)k^{2})," display="block"><mrow><mrow><mrow><msubsup><mi href="./29.3#SS1.p1">b</mi><mi href="./29.1#p2.t1.r4">ν</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">m</mi></mrow><mo>+</mo><mn>2</mn></mrow></msubsup><mo></mo><mrow><mo>(</mo><msup><mi href="./29.1#p2.t1.r4">k</mi><mn>2</mn></msup><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mrow><mo stretchy="false">(</mo><mrow><msub><mi href="./29.15#Px1.p1">H</mi><mi href="./29.1#p2.t1.r1">m</mi></msub><mo>+</mo><mrow><mi href="./29.1#p2.t1.r4">ν</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./29.1#p2.t1.r4">ν</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msup><mi href="./29.1#p2.t1.r4">k</mi><mn>2</mn></msup></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E42.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./29.3#SS1.p1" title="§29.3(i) Eigenvalues ‣ §29.3 Definitions and Basic Properties ‣ Lamé Functions ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="27px" altimg-valign="-9px" altimg-width="65px" alttext="b^{\NVar{n}}_{\NVar{\nu}}\left(\NVar{k^{2}}\right)" display="inline"><mrow><msubsup><mi href="./29.3#SS1.p1">b</mi><mi class="ltx_nvar" href="./29.1#p2.t1.r4">ν</mi><mi class="ltx_nvar" href="./29.1#p2.t1.r1">n</mi></msubsup><mo></mo><mrow><mo>(</mo><msup><mi class="ltx_nvar" href="./29.1#p2.t1.r4">k</mi><mn class="ltx_nvar">2</mn></msup><mo>)</mo></mrow></mrow></math>: eigenvalues of Lamé’s equation</a>,
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./29.1#p2.t1.r1">m</mi></math>: nonnegative integer</a>,
<a href="./29.1#p2.t1.r4" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m67.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./29.1#p2.t1.r4">k</mi></math>: real parameter</a>,
<a href="./29.1#p2.t1.r4" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./29.1#p2.t1.r4">ν</mi></math>: real parameter</a> and
<a href="./29.15#Px1.p1" title="Polynomial uE m 2 n ( z , k 2 ) ‣ §29.15(i) Fourier Coefficients ‣ §29.15 Fourier Series and Chebyshev Series ‣ Lamé Polynomials ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m24.png" altimg-height="23px" altimg-valign="-8px" altimg-width="30px" alttext="H_{p}" display="inline"><msub><mi href="./29.15#Px1.p1">H</mi><mi href="./29.1#p2.t1.r1">p</mi></msub></math>: eigenvalues</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and (</dd>
</dl>
</div>
</div>
<div id="SS2.p1" class="ltx_para">
<p class="ltx_p">The Chebyshev polynomial <math class="ltx_Math" altimg="m26.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="T" display="inline"><mi href="./18.3#T1.t1.r4">T</mi></math> of the first kind (§)
satisfies <math class="ltx_Math" altimg="m33.png" altimg-height="24px" altimg-valign="-8px" altimg-width="180px" alttext="\cos\left(p\phi\right)=T_{p}\left(\cos\phi\right)" display="inline"><mrow><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./29.1#p2.t1.r1">p</mi><mo></mo><mi>ϕ</mi></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msub><mi href="./18.3#T1.t1.r4">T</mi><mi href="./29.1#p2.t1.r1">p</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi>ϕ</mi></mrow><mo>)</mo></mrow></mrow></mrow></math>. Since
() implies that <math class="ltx_Math" altimg="m34.png" altimg-height="23px" altimg-valign="-7px" altimg-width="140px" alttext="\cos\phi=\operatorname{sn}\left(z,k\right)" display="inline"><mrow><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi>ϕ</mi></mrow><mo>=</mo><mrow><mi href="./22.2#E4">sn</mi><mo></mo><mrow><mo>(</mo><mi href="./29.1#p2.t1.r3">z</mi><mo>,</mo><mi href="./29.1#p2.t1.r4">k</mi><mo>)</mo></mrow></mrow></mrow></math>,
() can be rewritten in the form
</p>
<table id="E43" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">29.15.43</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E43.png" altimg-height="66px" altimg-valign="-30px" altimg-width="388px" alttext="\mathit{uE}^{m}_{2n}\left(z,k^{2}\right)=\tfrac{1}{2}A_{0}+\sum_{p=1}^{n}A_{2p%
}T_{2p}\left(\operatorname{sn}\left(z,k\right)\right)." display="block"><mrow><mrow><mrow><msubsup><mi href="./29.12#E1" mathvariant="italic">uE</mi><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">n</mi></mrow><mi href="./29.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo>(</mo><mi href="./29.1#p2.t1.r3">z</mi><mo>,</mo><msup><mi href="./29.1#p2.t1.r4">k</mi><mn>2</mn></msup><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><msub><mi href="./29.6#SS1.p1">A</mi><mn>0</mn></msub></mrow><mo>+</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./29.1#p2.t1.r1">p</mi><mo>=</mo><mn>1</mn></mrow><mi href="./29.1#p2.t1.r1">n</mi></munderover><mrow><msub><mi href="./29.6#SS1.p1">A</mi><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow></msub><mo></mo><mrow><msub><mi href="./18.3#T1.t1.r4">T</mi><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mrow><mi href="./22.2#E4">sn</mi><mo></mo><mrow><mo>(</mo><mi href="./29.1#p2.t1.r3">z</mi><mo>,</mo><mi href="./29.1#p2.t1.r4">k</mi><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E43.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./18.3#T1.t1.r4" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m27.png" altimg-height="23px" altimg-valign="-7px" altimg-width="57px" alttext="T_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r4">T</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Chebyshev polynomial of the first kind</a>,
<a href="./22.2#E4" title="(22.2.4) ‣ §22.2 Definitions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m61.png" altimg-height="23px" altimg-valign="-7px" altimg-width="72px" alttext="\operatorname{sn}\left(\NVar{z},\NVar{k}\right)" display="inline"><mrow><mi href="./22.2#E4">sn</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r2">z</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobian elliptic function</a>,
<a href="./29.12#E1" title="(29.12.1) ‣ §29.12(i) Elliptic-Function Form ‣ §29.12 Definitions ‣ Lamé Polynomials ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="27px" altimg-valign="-9px" altimg-width="110px" alttext="\mathit{uE}^{\NVar{m}}_{2\NVar{n}}\left(\NVar{z},\NVar{k^{2}}\right)" display="inline"><mrow><msubsup><mi href="./29.12#E1" mathvariant="italic">uE</mi><mrow><mn>2</mn><mo></mo><mi class="ltx_nvar" href="./29.1#p2.t1.r1">n</mi></mrow><mi class="ltx_nvar" href="./29.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./29.1#p2.t1.r3">z</mi><mo>,</mo><msup><mi class="ltx_nvar" href="./29.1#p2.t1.r4">k</mi><mn class="ltx_nvar">2</mn></msup><mo>)</mo></mrow></mrow></math>: Lamé polynomial</a>,
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./29.1#p2.t1.r1">m</mi></math>: nonnegative integer</a>,
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./29.1#p2.t1.r1">n</mi></math>: nonnegative integer</a>,
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="p" display="inline"><mi href="./29.1#p2.t1.r1">p</mi></math>: nonnegative integer</a>,
<a href="./29.1#p2.t1.r3" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m73.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./29.1#p2.t1.r3">z</mi></math>: complex variable</a>,
<a href="./29.1#p2.t1.r4" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m67.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./29.1#p2.t1.r4">k</mi></math>: real parameter</a> and
<a href="./29.6#SS1.p1" title="§29.6(i) Function Ec 2 m ν ( z , k 2 ) ‣ §29.6 Fourier Series ‣ Lamé Functions ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m17.png" altimg-height="23px" altimg-valign="-8px" altimg-width="36px" alttext="A_{2p}" display="inline"><msub><mi href="./29.6#SS1.p1">A</mi><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow></msub></math>: coefficients</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">This determines the polynomial <math class="ltx_Math" altimg="m25.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="P" display="inline"><mi href="./29.12#SS1.p4">P</mi></math> of degree <math class="ltx_Math" altimg="m70.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./29.1#p2.t1.r1">n</mi></math> for which
<math class="ltx_Math" altimg="m52.png" altimg-height="27px" altimg-valign="-9px" altimg-width="245px" alttext="\mathit{uE}^{m}_{2n}\left(z,k^{2}\right)=P({\operatorname{sn}^{2}}\left(z,k%
\right))" display="inline"><mrow><mrow><msubsup><mi href="./29.12#E1" mathvariant="italic">uE</mi><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">n</mi></mrow><mi href="./29.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo>(</mo><mi href="./29.1#p2.t1.r3">z</mi><mo>,</mo><msup><mi href="./29.1#p2.t1.r4">k</mi><mn>2</mn></msup><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mi href="./29.12#SS1.p4">P</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><msup><mi href="./22.2#E4">sn</mi><mn>2</mn></msup><mo></mo><mrow><mo>(</mo><mi href="./29.1#p2.t1.r3">z</mi><mo>,</mo><mi href="./29.1#p2.t1.r4">k</mi><mo>)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></math>; compare Table
. The set of coefficients of this polynomial (without
normalization) can also be found directly as an eigenvector of an
<math class="ltx_Math" altimg="m15.png" altimg-height="23px" altimg-valign="-7px" altimg-width="152px" alttext="(n+1)\times(n+1)" display="inline"><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./29.1#p2.t1.r1">n</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo>×</mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./29.1#p2.t1.r1">n</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow></math> tridiagonal matrix; see <cite class="ltx_cite ltx_citemacro_citet">Arscott and Khabaza ()</cite>.</p>
</div>
<div id="SS2.p2" class="ltx_para">
</div>
<div id="SS2.p3" class="ltx_para">
<p class="ltx_p">Using also <math class="ltx_Math" altimg="m64.png" altimg-height="24px" altimg-valign="-8px" altimg-width="285px" alttext="\sin\left((p+1)\phi\right)=(\sin\phi)U_{p}\left(\cos\phi\right)" display="inline"><mrow><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./29.1#p2.t1.r1">p</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi>ϕ</mi></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi>ϕ</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><msub><mi href="./18.3#T1.t1.r5">U</mi><mi href="./29.1#p2.t1.r1">p</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi>ϕ</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow></math>, with
<math class="ltx_Math" altimg="m28.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="U" display="inline"><mi href="./18.3#T1.t1.r5">U</mi></math> denoting the Chebyshev polynomial of the second kind
(§), we obtain
</p>
<table id="EGx1" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E44">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">29.15.44</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m11.png" altimg-height="30px" altimg-valign="-9px" altimg-width="130px" alttext="\displaystyle\mathit{sE}^{m}_{2n+1}\left(z,k^{2}\right)" display="inline"><mrow><msubsup><mi href="./29.12#E2" mathvariant="italic">sE</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mn>1</mn></mrow><mi href="./29.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo>(</mo><mi href="./29.1#p2.t1.r3">z</mi><mo>,</mo><msup><mi href="./29.1#p2.t1.r4">k</mi><mn>2</mn></msup><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m7.png" altimg-height="66px" altimg-valign="-30px" altimg-width="256px" alttext="\displaystyle=\sum_{p=0}^{n}A_{2p+1}T_{2p+1}\left(\operatorname{sn}\left(z,k%
\right)\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./29.1#p2.t1.r1">p</mi><mo>=</mo><mn>0</mn></mrow><mi href="./29.1#p2.t1.r1">n</mi></munderover></mstyle><mrow><msub><mi href="./29.6#SS2.p1">A</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow><mo>+</mo><mn>1</mn></mrow></msub><mo></mo><mrow><msub><mi href="./18.3#T1.t1.r4">T</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow><mo>+</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mrow><mi href="./22.2#E4">sn</mi><mo></mo><mrow><mo>(</mo><mi href="./29.1#p2.t1.r3">z</mi><mo>,</mo><mi href="./29.1#p2.t1.r4">k</mi><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E44.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./18.3#T1.t1.r4" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m27.png" altimg-height="23px" altimg-valign="-7px" altimg-width="57px" alttext="T_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r4">T</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Chebyshev polynomial of the first kind</a>,
<a href="./22.2#E4" title="(22.2.4) ‣ §22.2 Definitions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m61.png" altimg-height="23px" altimg-valign="-7px" altimg-width="72px" alttext="\operatorname{sn}\left(\NVar{z},\NVar{k}\right)" display="inline"><mrow><mi href="./22.2#E4">sn</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r2">z</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobian elliptic function</a>,
<a href="./29.12#E2" title="(29.12.2) ‣ §29.12(i) Elliptic-Function Form ‣ §29.12 Definitions ‣ Lamé Polynomials ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="27px" altimg-valign="-9px" altimg-width="128px" alttext="\mathit{sE}^{\NVar{m}}_{2\NVar{n}+1}\left(\NVar{z},\NVar{k^{2}}\right)" display="inline"><mrow><msubsup><mi href="./29.12#E2" mathvariant="italic">sE</mi><mrow><mrow><mn>2</mn><mo></mo><mi class="ltx_nvar" href="./29.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mn>1</mn></mrow><mi class="ltx_nvar" href="./29.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./29.1#p2.t1.r3">z</mi><mo>,</mo><msup><mi class="ltx_nvar" href="./29.1#p2.t1.r4">k</mi><mn class="ltx_nvar">2</mn></msup><mo>)</mo></mrow></mrow></math>: Lamé polynomial</a>,
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./29.1#p2.t1.r1">m</mi></math>: nonnegative integer</a>,
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./29.1#p2.t1.r1">n</mi></math>: nonnegative integer</a>,
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="p" display="inline"><mi href="./29.1#p2.t1.r1">p</mi></math>: nonnegative integer</a>,
<a href="./29.1#p2.t1.r3" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m73.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./29.1#p2.t1.r3">z</mi></math>: complex variable</a>,
<a href="./29.1#p2.t1.r4" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m67.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./29.1#p2.t1.r4">k</mi></math>: real parameter</a> and
<a href="./29.6#SS2.p1" title="§29.6(ii) Function Ec + 2 m 1 ν ( z , k 2 ) ‣ §29.6 Fourier Series ‣ Lamé Functions ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m17.png" altimg-height="23px" altimg-valign="-8px" altimg-width="36px" alttext="A_{2p}" display="inline"><msub><mi href="./29.6#SS2.p1">A</mi><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow></msub></math>: coefficients</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E45">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">29.15.45</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m8.png" altimg-height="30px" altimg-valign="-9px" altimg-width="131px" alttext="\displaystyle\mathit{cE}^{m}_{2n+1}\left(z,k^{2}\right)" display="inline"><mrow><msubsup><mi href="./29.12#E3" mathvariant="italic">cE</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mn>1</mn></mrow><mi href="./29.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo>(</mo><mi href="./29.1#p2.t1.r3">z</mi><mo>,</mo><msup><mi href="./29.1#p2.t1.r4">k</mi><mn>2</mn></msup><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m3.png" altimg-height="66px" altimg-valign="-30px" altimg-width="310px" alttext="\displaystyle=\operatorname{cn}\left(z,k\right)\sum_{p=0}^{n}B_{2p+1}U_{2p}%
\left(\operatorname{sn}\left(z,k\right)\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mi href="./22.2#E5">cn</mi><mo></mo><mrow><mo>(</mo><mi href="./29.1#p2.t1.r3">z</mi><mo>,</mo><mi href="./29.1#p2.t1.r4">k</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mstyle displaystyle="true"><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./29.1#p2.t1.r1">p</mi><mo>=</mo><mn>0</mn></mrow><mi href="./29.1#p2.t1.r1">n</mi></munderover></mstyle><mrow><msub><mi href="./29.6#SS3.p1">B</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow><mo>+</mo><mn>1</mn></mrow></msub><mo></mo><mrow><msub><mi href="./18.3#T1.t1.r5">U</mi><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mrow><mi href="./22.2#E4">sn</mi><mo></mo><mrow><mo>(</mo><mi href="./29.1#p2.t1.r3">z</mi><mo>,</mo><mi href="./29.1#p2.t1.r4">k</mi><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E45.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./18.3#T1.t1.r5" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="23px" altimg-valign="-7px" altimg-width="59px" alttext="U_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r5">U</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Chebyshev polynomial of the second kind</a>,
<a href="./22.2#E5" title="(22.2.5) ‣ §22.2 Definitions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="23px" altimg-valign="-7px" altimg-width="73px" alttext="\operatorname{cn}\left(\NVar{z},\NVar{k}\right)" display="inline"><mrow><mi href="./22.2#E5">cn</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r2">z</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobian elliptic function</a>,
<a href="./22.2#E4" title="(22.2.4) ‣ §22.2 Definitions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m61.png" altimg-height="23px" altimg-valign="-7px" altimg-width="72px" alttext="\operatorname{sn}\left(\NVar{z},\NVar{k}\right)" display="inline"><mrow><mi href="./22.2#E4">sn</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r2">z</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobian elliptic function</a>,
<a href="./29.12#E3" title="(29.12.3) ‣ §29.12(i) Elliptic-Function Form ‣ §29.12 Definitions ‣ Lamé Polynomials ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="27px" altimg-valign="-9px" altimg-width="129px" alttext="\mathit{cE}^{\NVar{m}}_{2\NVar{n}+1}\left(\NVar{z},\NVar{k^{2}}\right)" display="inline"><mrow><msubsup><mi href="./29.12#E3" mathvariant="italic">cE</mi><mrow><mrow><mn>2</mn><mo></mo><mi class="ltx_nvar" href="./29.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mn>1</mn></mrow><mi class="ltx_nvar" href="./29.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./29.1#p2.t1.r3">z</mi><mo>,</mo><msup><mi class="ltx_nvar" href="./29.1#p2.t1.r4">k</mi><mn class="ltx_nvar">2</mn></msup><mo>)</mo></mrow></mrow></math>: Lamé polynomial</a>,
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./29.1#p2.t1.r1">m</mi></math>: nonnegative integer</a>,
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./29.1#p2.t1.r1">n</mi></math>: nonnegative integer</a>,
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="p" display="inline"><mi href="./29.1#p2.t1.r1">p</mi></math>: nonnegative integer</a>,
<a href="./29.1#p2.t1.r3" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m73.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./29.1#p2.t1.r3">z</mi></math>: complex variable</a>,
<a href="./29.1#p2.t1.r4" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m67.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./29.1#p2.t1.r4">k</mi></math>: real parameter</a> and
<a href="./29.6#SS3.p1" title="§29.6(iii) Function Es + 2 m 1 ν ( z , k 2 ) ‣ §29.6 Fourier Series ‣ Lamé Functions ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m18.png" altimg-height="23px" altimg-valign="-8px" altimg-width="57px" alttext="B_{2p+1}" display="inline"><msub><mi href="./29.6#SS3.p1">B</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow><mo>+</mo><mn>1</mn></mrow></msub></math>: coefficents</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E46">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">29.15.46</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m10.png" altimg-height="30px" altimg-valign="-9px" altimg-width="132px" alttext="\displaystyle\mathit{dE}^{m}_{2n+1}\left(z,k^{2}\right)" display="inline"><mrow><msubsup><mi href="./29.12#E4" mathvariant="italic">dE</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mn>1</mn></mrow><mi href="./29.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo>(</mo><mi href="./29.1#p2.t1.r3">z</mi><mo>,</mo><msup><mi href="./29.1#p2.t1.r4">k</mi><mn>2</mn></msup><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m5.png" altimg-height="68px" altimg-valign="-30px" altimg-width="381px" alttext="\displaystyle=\operatorname{dn}\left(z,k\right)\left(\tfrac{1}{2}C_{0}+\sum_{p%
=1}^{n}C_{2p}T_{2p}\left(\operatorname{sn}\left(z,k\right)\right)\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mi href="./22.2#E6">dn</mi><mo></mo><mrow><mo>(</mo><mi href="./29.1#p2.t1.r3">z</mi><mo>,</mo><mi href="./29.1#p2.t1.r4">k</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mo>(</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><msub><mi href="./29.6#SS1.p4">C</mi><mn>0</mn></msub></mrow><mo>+</mo><mrow><mstyle displaystyle="true"><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./29.1#p2.t1.r1">p</mi><mo>=</mo><mn>1</mn></mrow><mi href="./29.1#p2.t1.r1">n</mi></munderover></mstyle><mrow><msub><mi href="./29.6#SS1.p4">C</mi><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow></msub><mo></mo><mrow><msub><mi href="./18.3#T1.t1.r4">T</mi><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mrow><mi href="./22.2#E4">sn</mi><mo></mo><mrow><mo>(</mo><mi href="./29.1#p2.t1.r3">z</mi><mo>,</mo><mi href="./29.1#p2.t1.r4">k</mi><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E46.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./18.3#T1.t1.r4" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m27.png" altimg-height="23px" altimg-valign="-7px" altimg-width="57px" alttext="T_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r4">T</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Chebyshev polynomial of the first kind</a>,
<a href="./22.2#E6" title="(22.2.6) ‣ §22.2 Definitions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="23px" altimg-valign="-7px" altimg-width="75px" alttext="\operatorname{dn}\left(\NVar{z},\NVar{k}\right)" display="inline"><mrow><mi href="./22.2#E6">dn</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r2">z</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobian elliptic function</a>,
<a href="./22.2#E4" title="(22.2.4) ‣ §22.2 Definitions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m61.png" altimg-height="23px" altimg-valign="-7px" altimg-width="72px" alttext="\operatorname{sn}\left(\NVar{z},\NVar{k}\right)" display="inline"><mrow><mi href="./22.2#E4">sn</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r2">z</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobian elliptic function</a>,
<a href="./29.12#E4" title="(29.12.4) ‣ §29.12(i) Elliptic-Function Form ‣ §29.12 Definitions ‣ Lamé Polynomials ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m41.png" altimg-height="27px" altimg-valign="-9px" altimg-width="130px" alttext="\mathit{dE}^{\NVar{m}}_{2\NVar{n}+1}\left(\NVar{z},\NVar{k^{2}}\right)" display="inline"><mrow><msubsup><mi href="./29.12#E4" mathvariant="italic">dE</mi><mrow><mrow><mn>2</mn><mo></mo><mi class="ltx_nvar" href="./29.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mn>1</mn></mrow><mi class="ltx_nvar" href="./29.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./29.1#p2.t1.r3">z</mi><mo>,</mo><msup><mi class="ltx_nvar" href="./29.1#p2.t1.r4">k</mi><mn class="ltx_nvar">2</mn></msup><mo>)</mo></mrow></mrow></math>: Lamé polynomial</a>,
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./29.1#p2.t1.r1">m</mi></math>: nonnegative integer</a>,
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./29.1#p2.t1.r1">n</mi></math>: nonnegative integer</a>,
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="p" display="inline"><mi href="./29.1#p2.t1.r1">p</mi></math>: nonnegative integer</a>,
<a href="./29.1#p2.t1.r3" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m73.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./29.1#p2.t1.r3">z</mi></math>: complex variable</a>,
<a href="./29.1#p2.t1.r4" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m67.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./29.1#p2.t1.r4">k</mi></math>: real parameter</a> and
<a href="./29.6#SS1.p4" title="§29.6(i) Function Ec 2 m ν ( z , k 2 ) ‣ §29.6 Fourier Series ‣ Lamé Functions ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="23px" altimg-valign="-8px" altimg-width="36px" alttext="C_{2p}" display="inline"><msub><mi href="./29.6#SS1.p4">C</mi><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow></msub></math>: coefficents</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E47">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">29.15.47</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m12.png" altimg-height="30px" altimg-valign="-9px" altimg-width="139px" alttext="\displaystyle\mathit{scE}^{m}_{2n+2}\left(z,k^{2}\right)" display="inline"><mrow><msubsup><mi href="./29.12#E5" mathvariant="italic">scE</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mn>2</mn></mrow><mi href="./29.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo>(</mo><mi href="./29.1#p2.t1.r3">z</mi><mo>,</mo><msup><mi href="./29.1#p2.t1.r4">k</mi><mn>2</mn></msup><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m4.png" altimg-height="66px" altimg-valign="-30px" altimg-width="330px" alttext="\displaystyle=\operatorname{cn}\left(z,k\right)\sum_{p=0}^{n}B_{2p+2}U_{2p+1}%
\left(\operatorname{sn}\left(z,k\right)\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mi href="./22.2#E5">cn</mi><mo></mo><mrow><mo>(</mo><mi href="./29.1#p2.t1.r3">z</mi><mo>,</mo><mi href="./29.1#p2.t1.r4">k</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mstyle displaystyle="true"><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./29.1#p2.t1.r1">p</mi><mo>=</mo><mn>0</mn></mrow><mi href="./29.1#p2.t1.r1">n</mi></munderover></mstyle><mrow><msub><mi href="./29.6#SS4.p1">B</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow><mo>+</mo><mn>2</mn></mrow></msub><mo></mo><mrow><msub><mi href="./18.3#T1.t1.r5">U</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow><mo>+</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mrow><mi href="./22.2#E4">sn</mi><mo></mo><mrow><mo>(</mo><mi href="./29.1#p2.t1.r3">z</mi><mo>,</mo><mi href="./29.1#p2.t1.r4">k</mi><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E47.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./18.3#T1.t1.r5" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="23px" altimg-valign="-7px" altimg-width="59px" alttext="U_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r5">U</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Chebyshev polynomial of the second kind</a>,
<a href="./22.2#E5" title="(22.2.5) ‣ §22.2 Definitions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="23px" altimg-valign="-7px" altimg-width="73px" alttext="\operatorname{cn}\left(\NVar{z},\NVar{k}\right)" display="inline"><mrow><mi href="./22.2#E5">cn</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r2">z</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobian elliptic function</a>,
<a href="./22.2#E4" title="(22.2.4) ‣ §22.2 Definitions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m61.png" altimg-height="23px" altimg-valign="-7px" altimg-width="72px" alttext="\operatorname{sn}\left(\NVar{z},\NVar{k}\right)" display="inline"><mrow><mi href="./22.2#E4">sn</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r2">z</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobian elliptic function</a>,
<a href="./29.12#E5" title="(29.12.5) ‣ §29.12(i) Elliptic-Function Form ‣ §29.12 Definitions ‣ Lamé Polynomials ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m45.png" altimg-height="27px" altimg-valign="-9px" altimg-width="137px" alttext="\mathit{scE}^{\NVar{m}}_{2\NVar{n}+2}\left(\NVar{z},\NVar{k^{2}}\right)" display="inline"><mrow><msubsup><mi href="./29.12#E5" mathvariant="italic">scE</mi><mrow><mrow><mn>2</mn><mo></mo><mi class="ltx_nvar" href="./29.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mn>2</mn></mrow><mi class="ltx_nvar" href="./29.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./29.1#p2.t1.r3">z</mi><mo>,</mo><msup><mi class="ltx_nvar" href="./29.1#p2.t1.r4">k</mi><mn class="ltx_nvar">2</mn></msup><mo>)</mo></mrow></mrow></math>: Lamé polynomial</a>,
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./29.1#p2.t1.r1">m</mi></math>: nonnegative integer</a>,
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./29.1#p2.t1.r1">n</mi></math>: nonnegative integer</a>,
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="p" display="inline"><mi href="./29.1#p2.t1.r1">p</mi></math>: nonnegative integer</a>,
<a href="./29.1#p2.t1.r3" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m73.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./29.1#p2.t1.r3">z</mi></math>: complex variable</a>,
<a href="./29.1#p2.t1.r4" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m67.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./29.1#p2.t1.r4">k</mi></math>: real parameter</a> and
<a href="./29.6#SS4.p1" title="§29.6(iv) Function Es + 2 m 2 ν ( z , k 2 ) ‣ §29.6 Fourier Series ‣ Lamé Functions ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m18.png" altimg-height="23px" altimg-valign="-8px" altimg-width="57px" alttext="B_{2p+1}" display="inline"><msub><mi href="./29.6#SS4.p1">B</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow><mo>+</mo><mn>1</mn></mrow></msub></math>: coefficents</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E48">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">29.15.48</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m14.png" altimg-height="30px" altimg-valign="-9px" altimg-width="140px" alttext="\displaystyle\mathit{sdE}^{m}_{2n+2}\left(z,k^{2}\right)" display="inline"><mrow><msubsup><mi href="./29.12#E6" mathvariant="italic">sdE</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mn>2</mn></mrow><mi href="./29.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo>(</mo><mi href="./29.1#p2.t1.r3">z</mi><mo>,</mo><msup><mi href="./29.1#p2.t1.r4">k</mi><mn>2</mn></msup><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m6.png" altimg-height="66px" altimg-valign="-30px" altimg-width="329px" alttext="\displaystyle=\operatorname{dn}\left(z,k\right)\sum_{p=0}^{n}C_{2p+1}T_{2p+1}%
\left(\operatorname{sn}\left(z,k\right)\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mi href="./22.2#E6">dn</mi><mo></mo><mrow><mo>(</mo><mi href="./29.1#p2.t1.r3">z</mi><mo>,</mo><mi href="./29.1#p2.t1.r4">k</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mstyle displaystyle="true"><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./29.1#p2.t1.r1">p</mi><mo>=</mo><mn>0</mn></mrow><mi href="./29.1#p2.t1.r1">n</mi></munderover></mstyle><mrow><msub><mi href="./29.6#SS2.p2">C</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow><mo>+</mo><mn>1</mn></mrow></msub><mo></mo><mrow><msub><mi href="./18.3#T1.t1.r4">T</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow><mo>+</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mrow><mi href="./22.2#E4">sn</mi><mo></mo><mrow><mo>(</mo><mi href="./29.1#p2.t1.r3">z</mi><mo>,</mo><mi href="./29.1#p2.t1.r4">k</mi><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E48.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./18.3#T1.t1.r4" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m27.png" altimg-height="23px" altimg-valign="-7px" altimg-width="57px" alttext="T_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r4">T</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Chebyshev polynomial of the first kind</a>,
<a href="./22.2#E6" title="(22.2.6) ‣ §22.2 Definitions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="23px" altimg-valign="-7px" altimg-width="75px" alttext="\operatorname{dn}\left(\NVar{z},\NVar{k}\right)" display="inline"><mrow><mi href="./22.2#E6">dn</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r2">z</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobian elliptic function</a>,
<a href="./22.2#E4" title="(22.2.4) ‣ §22.2 Definitions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m61.png" altimg-height="23px" altimg-valign="-7px" altimg-width="72px" alttext="\operatorname{sn}\left(\NVar{z},\NVar{k}\right)" display="inline"><mrow><mi href="./22.2#E4">sn</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r2">z</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobian elliptic function</a>,
<a href="./29.12#E6" title="(29.12.6) ‣ §29.12(i) Elliptic-Function Form ‣ §29.12 Definitions ‣ Lamé Polynomials ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="27px" altimg-valign="-9px" altimg-width="138px" alttext="\mathit{sdE}^{\NVar{m}}_{2\NVar{n}+2}\left(\NVar{z},\NVar{k^{2}}\right)" display="inline"><mrow><msubsup><mi href="./29.12#E6" mathvariant="italic">sdE</mi><mrow><mrow><mn>2</mn><mo></mo><mi class="ltx_nvar" href="./29.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mn>2</mn></mrow><mi class="ltx_nvar" href="./29.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./29.1#p2.t1.r3">z</mi><mo>,</mo><msup><mi class="ltx_nvar" href="./29.1#p2.t1.r4">k</mi><mn class="ltx_nvar">2</mn></msup><mo>)</mo></mrow></mrow></math>: Lamé polynomial</a>,
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./29.1#p2.t1.r1">m</mi></math>: nonnegative integer</a>,
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./29.1#p2.t1.r1">n</mi></math>: nonnegative integer</a>,
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="p" display="inline"><mi href="./29.1#p2.t1.r1">p</mi></math>: nonnegative integer</a>,
<a href="./29.1#p2.t1.r3" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m73.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./29.1#p2.t1.r3">z</mi></math>: complex variable</a>,
<a href="./29.1#p2.t1.r4" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m67.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./29.1#p2.t1.r4">k</mi></math>: real parameter</a> and
<a href="./29.6#SS2.p2" title="§29.6(ii) Function Ec + 2 m 1 ν ( z , k 2 ) ‣ §29.6 Fourier Series ‣ Lamé Functions ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="23px" altimg-valign="-8px" altimg-width="56px" alttext="C_{2p+1}" display="inline"><msub><mi href="./29.6#SS2.p2">C</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow><mo>+</mo><mn>1</mn></mrow></msub></math>: coefficents</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E49">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">29.15.49</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m9.png" altimg-height="30px" altimg-valign="-9px" altimg-width="140px" alttext="\displaystyle\mathit{cdE}^{m}_{2n+2}\left(z,k^{2}\right)" display="inline"><mrow><msubsup><mi href="./29.12#E7" mathvariant="italic">cdE</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mn>2</mn></mrow><mi href="./29.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo>(</mo><mi href="./29.1#p2.t1.r3">z</mi><mo>,</mo><msup><mi href="./29.1#p2.t1.r4">k</mi><mn>2</mn></msup><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m1.png" altimg-height="66px" altimg-valign="-30px" altimg-width="386px" alttext="\displaystyle=\operatorname{cn}\left(z,k\right)\operatorname{dn}\left(z,k%
\right)\sum_{p=0}^{n}D_{2p+1}U_{2p}\left(\operatorname{sn}\left(z,k\right)%
\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mi href="./22.2#E5">cn</mi><mo></mo><mrow><mo>(</mo><mi href="./29.1#p2.t1.r3">z</mi><mo>,</mo><mi href="./29.1#p2.t1.r4">k</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./22.2#E6">dn</mi><mo></mo><mrow><mo>(</mo><mi href="./29.1#p2.t1.r3">z</mi><mo>,</mo><mi href="./29.1#p2.t1.r4">k</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mstyle displaystyle="true"><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./29.1#p2.t1.r1">p</mi><mo>=</mo><mn>0</mn></mrow><mi href="./29.1#p2.t1.r1">n</mi></munderover></mstyle><mrow><msub><mi href="./29.6#SS4.p2">D</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow><mo>+</mo><mn>1</mn></mrow></msub><mo></mo><mrow><msub><mi href="./18.3#T1.t1.r5">U</mi><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mrow><mi href="./22.2#E4">sn</mi><mo></mo><mrow><mo>(</mo><mi href="./29.1#p2.t1.r3">z</mi><mo>,</mo><mi href="./29.1#p2.t1.r4">k</mi><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E49.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./18.3#T1.t1.r5" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="23px" altimg-valign="-7px" altimg-width="59px" alttext="U_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r5">U</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Chebyshev polynomial of the second kind</a>,
<a href="./22.2#E5" title="(22.2.5) ‣ §22.2 Definitions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="23px" altimg-valign="-7px" altimg-width="73px" alttext="\operatorname{cn}\left(\NVar{z},\NVar{k}\right)" display="inline"><mrow><mi href="./22.2#E5">cn</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r2">z</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobian elliptic function</a>,
<a href="./22.2#E6" title="(22.2.6) ‣ §22.2 Definitions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="23px" altimg-valign="-7px" altimg-width="75px" alttext="\operatorname{dn}\left(\NVar{z},\NVar{k}\right)" display="inline"><mrow><mi href="./22.2#E6">dn</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r2">z</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobian elliptic function</a>,
<a href="./22.2#E4" title="(22.2.4) ‣ §22.2 Definitions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m61.png" altimg-height="23px" altimg-valign="-7px" altimg-width="72px" alttext="\operatorname{sn}\left(\NVar{z},\NVar{k}\right)" display="inline"><mrow><mi href="./22.2#E4">sn</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r2">z</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobian elliptic function</a>,
<a href="./29.12#E7" title="(29.12.7) ‣ §29.12(i) Elliptic-Function Form ‣ §29.12 Definitions ‣ Lamé Polynomials ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="27px" altimg-valign="-9px" altimg-width="138px" alttext="\mathit{cdE}^{\NVar{m}}_{2\NVar{n}+2}\left(\NVar{z},\NVar{k^{2}}\right)" display="inline"><mrow><msubsup><mi href="./29.12#E7" mathvariant="italic">cdE</mi><mrow><mrow><mn>2</mn><mo></mo><mi class="ltx_nvar" href="./29.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mn>2</mn></mrow><mi class="ltx_nvar" href="./29.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./29.1#p2.t1.r3">z</mi><mo>,</mo><msup><mi class="ltx_nvar" href="./29.1#p2.t1.r4">k</mi><mn class="ltx_nvar">2</mn></msup><mo>)</mo></mrow></mrow></math>: Lamé polynomial</a>,
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./29.1#p2.t1.r1">m</mi></math>: nonnegative integer</a>,
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./29.1#p2.t1.r1">n</mi></math>: nonnegative integer</a>,
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="p" display="inline"><mi href="./29.1#p2.t1.r1">p</mi></math>: nonnegative integer</a>,
<a href="./29.1#p2.t1.r3" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m73.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./29.1#p2.t1.r3">z</mi></math>: complex variable</a>,
<a href="./29.1#p2.t1.r4" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m67.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./29.1#p2.t1.r4">k</mi></math>: real parameter</a> and
<a href="./29.6#SS4.p2" title="§29.6(iv) Function Es + 2 m 2 ν ( z , k 2 ) ‣ §29.6 Fourier Series ‣ Lamé Functions ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="23px" altimg-valign="-8px" altimg-width="38px" alttext="D_{2p}" display="inline"><msub><mi href="./29.6#SS4.p2">D</mi><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow></msub></math>: coefficents</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E50">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">29.15.50</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m13.png" altimg-height="30px" altimg-valign="-9px" altimg-width="148px" alttext="\displaystyle\mathit{scdE}^{m}_{2n+3}\left(z,k^{2}\right)" display="inline"><mrow><msubsup><mi href="./29.12#E8" mathvariant="italic">scdE</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mn>3</mn></mrow><mi href="./29.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo>(</mo><mi href="./29.1#p2.t1.r3">z</mi><mo>,</mo><msup><mi href="./29.1#p2.t1.r4">k</mi><mn>2</mn></msup><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m2.png" altimg-height="66px" altimg-valign="-30px" altimg-width="406px" alttext="\displaystyle=\operatorname{cn}\left(z,k\right)\operatorname{dn}\left(z,k%
\right)\sum_{p=0}^{n}D_{2p+2}U_{2p+1}\left(\operatorname{sn}\left(z,k\right)%
\right)." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mi href="./22.2#E5">cn</mi><mo></mo><mrow><mo>(</mo><mi href="./29.1#p2.t1.r3">z</mi><mo>,</mo><mi href="./29.1#p2.t1.r4">k</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./22.2#E6">dn</mi><mo></mo><mrow><mo>(</mo><mi href="./29.1#p2.t1.r3">z</mi><mo>,</mo><mi href="./29.1#p2.t1.r4">k</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mstyle displaystyle="true"><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./29.1#p2.t1.r1">p</mi><mo>=</mo><mn>0</mn></mrow><mi href="./29.1#p2.t1.r1">n</mi></munderover></mstyle><mrow><msub><mi href="./29.6#SS3.p2">D</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow><mo>+</mo><mn>2</mn></mrow></msub><mo></mo><mrow><msub><mi href="./18.3#T1.t1.r5">U</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow><mo>+</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mrow><mi href="./22.2#E4">sn</mi><mo></mo><mrow><mo>(</mo><mi href="./29.1#p2.t1.r3">z</mi><mo>,</mo><mi href="./29.1#p2.t1.r4">k</mi><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E50.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./18.3#T1.t1.r5" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="23px" altimg-valign="-7px" altimg-width="59px" alttext="U_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r5">U</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Chebyshev polynomial of the second kind</a>,
<a href="./22.2#E5" title="(22.2.5) ‣ §22.2 Definitions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="23px" altimg-valign="-7px" altimg-width="73px" alttext="\operatorname{cn}\left(\NVar{z},\NVar{k}\right)" display="inline"><mrow><mi href="./22.2#E5">cn</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r2">z</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobian elliptic function</a>,
<a href="./22.2#E6" title="(22.2.6) ‣ §22.2 Definitions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="23px" altimg-valign="-7px" altimg-width="75px" alttext="\operatorname{dn}\left(\NVar{z},\NVar{k}\right)" display="inline"><mrow><mi href="./22.2#E6">dn</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r2">z</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobian elliptic function</a>,
<a href="./22.2#E4" title="(22.2.4) ‣ §22.2 Definitions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref"><math class="ltx_Math" altimg="m61.png" altimg-height="23px" altimg-valign="-7px" altimg-width="72px" alttext="\operatorname{sn}\left(\NVar{z},\NVar{k}\right)" display="inline"><mrow><mi href="./22.2#E4">sn</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r2">z</mi><mo>,</mo><mi class="ltx_nvar" href="./22.1#p2.t1.r3">k</mi><mo>)</mo></mrow></mrow></math>: Jacobian elliptic function</a>,
<a href="./29.12#E8" title="(29.12.8) ‣ §29.12(i) Elliptic-Function Form ‣ §29.12 Definitions ‣ Lamé Polynomials ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="27px" altimg-valign="-9px" altimg-width="146px" alttext="\mathit{scdE}^{\NVar{m}}_{2\NVar{n}+3}\left(\NVar{z},\NVar{k^{2}}\right)" display="inline"><mrow><msubsup><mi href="./29.12#E8" mathvariant="italic">scdE</mi><mrow><mrow><mn>2</mn><mo></mo><mi class="ltx_nvar" href="./29.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mn>3</mn></mrow><mi class="ltx_nvar" href="./29.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./29.1#p2.t1.r3">z</mi><mo>,</mo><msup><mi class="ltx_nvar" href="./29.1#p2.t1.r4">k</mi><mn class="ltx_nvar">2</mn></msup><mo>)</mo></mrow></mrow></math>: Lamé polynomial</a>,
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./29.1#p2.t1.r1">m</mi></math>: nonnegative integer</a>,
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./29.1#p2.t1.r1">n</mi></math>: nonnegative integer</a>,
<a href="./29.1#p2.t1.r1" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="p" display="inline"><mi href="./29.1#p2.t1.r1">p</mi></math>: nonnegative integer</a>,
<a href="./29.1#p2.t1.r3" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m73.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./29.1#p2.t1.r3">z</mi></math>: complex variable</a>,
<a href="./29.1#p2.t1.r4" title="§29.1 Special Notation ‣ Notation ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m67.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./29.1#p2.t1.r4">k</mi></math>: real parameter</a> and
<a href="./29.6#SS3.p2" title="§29.6(iii) Function Es + 2 m 1 ν ( z , k 2 ) ‣ §29.6 Fourier Series ‣ Lamé Functions ‣ Chapter 29 Lamé Functions" class="ltx_ref"><math class="ltx_Math" altimg="m21.png" altimg-height="23px" altimg-valign="-8px" altimg-width="58px" alttext="D_{2p+1}" display="inline"><msub><mi href="./29.6#SS3.p2">D</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./29.1#p2.t1.r1">p</mi></mrow><mo>+</mo><mn>1</mn></mrow></msub></math>: coefficents</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
</div>
<div id="SS2.p4" class="ltx_para">
<p class="ltx_p">For explicit formulas for Lamé polynomials of low degree, see
<cite class="ltx_cite ltx_citemacro_citet">Arscott (</div>
</div>
</body></text>
</html>
</page>
<page>
<!DOCTYPE html><html>
<head>
<title>DLMF: 14.13 Trigonometric Expansions</title>
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<text><body class="color_default textfont_default titlefont_default fontsize_default navbar_default">
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<div class="ltx_page_navlogo">)</cite>, but without any statement of conditions on <math class="ltx_Math" altimg="m22.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./14.1#p1.t1.r4">ν</mi></math> and <math class="ltx_Math" altimg="m20.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\mu" display="inline"><mi href="./14.1#p1.t1.r4">μ</mi></math>. To
obtain these conditions we see that by combination of (),
<table id="Ex1" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_Math ltx_eqn_cell ltx_align_center"><math class="ltx_Math" alttext="\pm\frac{1}{2}\pi i\mathsf{P}^{\mu}_{\nu}\left(\cos\theta\right)+\mathsf{Q}^{%
\mu}_{\nu}\left(\cos\theta\right)=\pi^{\frac{1}{2}}\Gamma\left(\nu+\mu+1\right%
)(2\sin\theta)^{\mu}e^{\pm(\nu+\mu+1)i\theta}\*\mathbf{F}\left(\nu+\mu+1,\mu+%
\frac{1}{2};\nu+\frac{3}{2};e^{\pm 2i\theta}\right)." display="block"><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mrow><mrow><mo>±</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi><mo></mo><mrow><msubsup><mi href="./14.3#E1" mathvariant="sans-serif">P</mi><mi href="./14.1#p1.t1.r4">ν</mi><mi href="./14.1#p1.t1.r4">μ</mi></msubsup><mo></mo><mrow><mo>(</mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi href="./14.13#p1">θ</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>+</mo><mrow><msubsup><mi href="./14.3#E2" mathvariant="sans-serif">Q</mi><mi href="./14.1#p1.t1.r4">ν</mi><mi href="./14.1#p1.t1.r4">μ</mi></msubsup><mo></mo><mrow><mo>(</mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi href="./14.13#p1">θ</mi></mrow><mo>)</mo></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>=</mo><mrow><msup><mi href="./3.12#E1">π</mi><mfrac><mn>1</mn><mn>2</mn></mfrac></msup><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./14.1#p1.t1.r4">ν</mi><mo>+</mo><mi href="./14.1#p1.t1.r4">μ</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mn>2</mn><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi href="./14.13#p1">θ</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mi href="./14.1#p1.t1.r4">μ</mi></msup><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>±</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./14.1#p1.t1.r4">ν</mi><mo>+</mo><mi href="./14.1#p1.t1.r4">μ</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi mathvariant="normal">i</mi><mo></mo><mi href="./14.13#p1">θ</mi></mrow></mrow></msup><mo></mo><mrow><mi href="./15.2#E2" mathvariant="bold">F</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./14.1#p1.t1.r4">ν</mi><mo>+</mo><mi href="./14.1#p1.t1.r4">μ</mi><mo>+</mo><mn>1</mn></mrow><mo>,</mo><mrow><mi href="./14.1#p1.t1.r4">μ</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>;</mo><mrow><mi href="./14.1#p1.t1.r4">ν</mi><mo>+</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow><mo>;</mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>±</mo><mrow><mn>2</mn><mo></mo><mi mathvariant="normal">i</mi><mo></mo><mi href="./14.13#p1">θ</mi></mrow></mrow></msup><mo>)</mo></mrow></mrow></mrow><mo>.</mo></mtd></mtr></mtable></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
</table>
We then apply the results given in the last paragraph of §</dd>
</dl>
</div>
</div>
<div id="p1" class="ltx_para">
<p class="ltx_p">When <math class="ltx_Math" altimg="m11.png" altimg-height="18px" altimg-valign="-3px" altimg-width="89px" alttext="0<\theta<\pi" display="inline"><mrow><mn>0</mn><mo><</mo><mi href="./14.13#p1">θ</mi><mo><</mo><mi href="./3.12#E1">π</mi></mrow></math>, and <math class="ltx_Math" altimg="m21.png" altimg-height="19px" altimg-valign="-6px" altimg-width="52px" alttext="\nu+\mu" display="inline"><mrow><mi href="./14.1#p1.t1.r4">ν</mi><mo>+</mo><mi href="./14.1#p1.t1.r4">μ</mi></mrow></math> is not a negative integer,
</p>
<table id="EGx1" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E1">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">14.13.1</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m5.png" altimg-height="25px" altimg-valign="-7px" altimg-width="88px" alttext="\displaystyle\mathsf{P}^{\mu}_{\nu}\left(\cos\theta\right)" display="inline"><mrow><msubsup><mi href="./14.3#E1" mathvariant="sans-serif">P</mi><mi href="./14.1#p1.t1.r4">ν</mi><mi href="./14.1#p1.t1.r4">μ</mi></msubsup><mo></mo><mrow><mo>(</mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi href="./14.13#p1">θ</mi></mrow><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m3.png" altimg-height="64px" altimg-valign="-28px" altimg-width="622px" alttext="\displaystyle=\frac{2^{\mu+1}(\sin\theta)^{\mu}}{\pi^{1/2}}\*\sum_{k=0}^{%
\infty}\frac{\Gamma\left(\nu+\mu+k+1\right)}{\Gamma\left(\nu+k+\frac{3}{2}%
\right)}\frac{{\left(\mu+\frac{1}{2}\right)_{k}}}{k!}\*\sin\left((\nu+\mu+2k+1%
)\theta\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><mfrac><mrow><msup><mn>2</mn><mrow><mi href="./14.1#p1.t1.r4">μ</mi><mo>+</mo><mn>1</mn></mrow></msup><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi href="./14.13#p1">θ</mi></mrow><mo stretchy="false">)</mo></mrow><mi href="./14.1#p1.t1.r4">μ</mi></msup></mrow><msup><mi href="./3.12#E1">π</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mfrac></mstyle><mo></mo><mrow><mstyle displaystyle="true"><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover></mstyle><mrow><mstyle displaystyle="true"><mfrac><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./14.1#p1.t1.r4">ν</mi><mo>+</mo><mi href="./14.1#p1.t1.r4">μ</mi><mo>+</mo><mi>k</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./14.1#p1.t1.r4">ν</mi><mo>+</mo><mi>k</mi><mo>+</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow></mfrac></mstyle><mo></mo><mstyle displaystyle="true"><mfrac><msub><mrow><mo href="./5.2#iii">(</mo><mrow><mi href="./14.1#p1.t1.r4">μ</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo href="./5.2#iii">)</mo></mrow><mi>k</mi></msub><mrow><mi>k</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mfrac></mstyle><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./14.1#p1.t1.r4">ν</mi><mo>+</mo><mi href="./14.1#p1.t1.r4">μ</mi><mo>+</mo><mrow><mn>2</mn><mo></mo><mi>k</mi></mrow><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi href="./14.13#p1">θ</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m13.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./14.3#E1" title="(14.3.1) ‣ Ferrers Function of the First Kind ‣ §14.3(i) Interval - 1 < x < 1 ‣ §14.3 Definitions and Hypergeometric Representations ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m16.png" altimg-height="23px" altimg-valign="-7px" altimg-width="58px" alttext="\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msubsup><mi href="./14.3#E1" mathvariant="sans-serif">P</mi><mi class="ltx_nvar" href="./14.1#p1.t1.r4">ν</mi><mi class="ltx_nvar" href="./14.1#p1.t1.r4">μ</mi></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./14.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow></math>: Ferrers function of the first kind</a>,
<a href="./5.2#iii" title="§5.2(iii) Pochhammer’s Symbol ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m27.png" altimg-height="24px" altimg-valign="-8px" altimg-width="41px" alttext="{\left(\NVar{a}\right)_{\NVar{n}}}" display="inline"><msub><mrow><mo href="./5.2#iii">(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r5">a</mi><mo href="./5.2#iii">)</mo></mrow><mi class="ltx_nvar" href="./5.1#p2.t1.r1">n</mi></msub></math>: Pochhammer’s symbol (or shifted factorial)</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m23.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m15.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m10.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m25.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m24.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./14.1#p1.t1.r4" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\mu" display="inline"><mi href="./14.1#p1.t1.r4">μ</mi></math>: general order</a>,
<a href="./14.1#p1.t1.r4" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./14.1#p1.t1.r4">ν</mi></math>: general degree</a> and
<a href="./14.13#p1" title="§14.13 Trigonometric Expansions ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m11.png" altimg-height="18px" altimg-valign="-3px" altimg-width="89px" alttext="0<\theta<\pi" display="inline"><mrow><mn>0</mn><mo><</mo><mi href="./14.13#p1">θ</mi><mo><</mo><mi href="./3.12#E1">π</mi></mrow></math>: variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">8.7.1</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E2">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">14.13.2</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m7.png" altimg-height="25px" altimg-valign="-7px" altimg-width="90px" alttext="\displaystyle\mathsf{Q}^{\mu}_{\nu}\left(\cos\theta\right)" display="inline"><mrow><msubsup><mi href="./14.3#E2" mathvariant="sans-serif">Q</mi><mi href="./14.1#p1.t1.r4">ν</mi><mi href="./14.1#p1.t1.r4">μ</mi></msubsup><mo></mo><mrow><mo>(</mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi href="./14.13#p1">θ</mi></mrow><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m4.png" altimg-height="64px" altimg-valign="-28px" altimg-width="636px" alttext="\displaystyle=\pi^{1/2}2^{\mu}(\sin\theta)^{\mu}\*\sum_{k=0}^{\infty}\frac{%
\Gamma\left(\nu+\mu+k+1\right)}{\Gamma\left(\nu+k+\frac{3}{2}\right)}\frac{{%
\left(\mu+\frac{1}{2}\right)_{k}}}{k!}\*\cos\left((\nu+\mu+2k+1)\theta\right)." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msup><mi href="./3.12#E1">π</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo></mo><msup><mn>2</mn><mi href="./14.1#p1.t1.r4">μ</mi></msup><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi href="./14.13#p1">θ</mi></mrow><mo stretchy="false">)</mo></mrow><mi href="./14.1#p1.t1.r4">μ</mi></msup><mo></mo><mrow><mstyle displaystyle="true"><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover></mstyle><mrow><mstyle displaystyle="true"><mfrac><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./14.1#p1.t1.r4">ν</mi><mo>+</mo><mi href="./14.1#p1.t1.r4">μ</mi><mo>+</mo><mi>k</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./14.1#p1.t1.r4">ν</mi><mo>+</mo><mi>k</mi><mo>+</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow></mfrac></mstyle><mo></mo><mstyle displaystyle="true"><mfrac><msub><mrow><mo href="./5.2#iii">(</mo><mrow><mi href="./14.1#p1.t1.r4">μ</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo href="./5.2#iii">)</mo></mrow><mi>k</mi></msub><mrow><mi>k</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mfrac></mstyle><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./14.1#p1.t1.r4">ν</mi><mo>+</mo><mi href="./14.1#p1.t1.r4">μ</mi><mo>+</mo><mrow><mn>2</mn><mo></mo><mi>k</mi></mrow><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi href="./14.13#p1">θ</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m13.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./14.3#E2" title="(14.3.2) ‣ Ferrers Function of the Second Kind ‣ §14.3(i) Interval - 1 < x < 1 ‣ §14.3 Definitions and Hypergeometric Representations ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m18.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="\mathsf{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msubsup><mi href="./14.3#E2" mathvariant="sans-serif">Q</mi><mi class="ltx_nvar" href="./14.1#p1.t1.r4">ν</mi><mi class="ltx_nvar" href="./14.1#p1.t1.r4">μ</mi></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./14.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow></math>: Ferrers function of the second kind</a>,
<a href="./5.2#iii" title="§5.2(iii) Pochhammer’s Symbol ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m27.png" altimg-height="24px" altimg-valign="-8px" altimg-width="41px" alttext="{\left(\NVar{a}\right)_{\NVar{n}}}" display="inline"><msub><mrow><mo href="./5.2#iii">(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r5">a</mi><mo href="./5.2#iii">)</mo></mrow><mi class="ltx_nvar" href="./5.1#p2.t1.r1">n</mi></msub></math>: Pochhammer’s symbol (or shifted factorial)</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m23.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m15.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m10.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m25.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m24.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./14.1#p1.t1.r4" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\mu" display="inline"><mi href="./14.1#p1.t1.r4">μ</mi></math>: general order</a>,
<a href="./14.1#p1.t1.r4" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./14.1#p1.t1.r4">ν</mi></math>: general degree</a> and
<a href="./14.13#p1" title="§14.13 Trigonometric Expansions ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m11.png" altimg-height="18px" altimg-valign="-3px" altimg-width="89px" alttext="0<\theta<\pi" display="inline"><mrow><mn>0</mn><mo><</mo><mi href="./14.13#p1">θ</mi><mo><</mo><mi href="./3.12#E1">π</mi></mrow></math>: variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">8.7.2</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
<p class="ltx_p">These Fourier series converge absolutely when <math class="ltx_Math" altimg="m14.png" altimg-height="21px" altimg-valign="-6px" altimg-width="67px" alttext="\Re\mu<0" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./14.1#p1.t1.r4">μ</mi></mrow><mo><</mo><mn>0</mn></mrow></math>, and conditionally when <math class="ltx_Math" altimg="m22.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./14.1#p1.t1.r4">ν</mi></math>
is real and <math class="ltx_Math" altimg="m12.png" altimg-height="27px" altimg-valign="-9px" altimg-width="92px" alttext="0\leq\mu<\frac{1}{2}" display="inline"><mrow><mn>0</mn><mo>≤</mo><mi href="./14.1#p1.t1.r4">μ</mi><mo><</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math>.</p>
</div>
<div id="p2" class="ltx_para">
<p class="ltx_p">In particular,</p>
<table id="EGx2" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E3">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">14.13.3</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m6.png" altimg-height="25px" altimg-valign="-7px" altimg-width="88px" alttext="\displaystyle\mathsf{P}_{n}\left(\cos\theta\right)" display="inline"><mrow><msub><mi href="./14.2#SS2.p2" mathvariant="sans-serif">P</mi><mi href="./14.1#p1.t1.r3">n</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi href="./14.13#p1">θ</mi></mrow><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m2.png" altimg-height="64px" altimg-valign="-28px" altimg-width="776px" alttext="\displaystyle=\frac{2^{2n+2}(n!)^{2}}{\pi(2n+1)!}\*\sum_{k=0}^{\infty}\frac{1%
\cdot 3\cdots(2k-1)}{k!}\*\frac{(n+1)(n+2)\cdots(n+k)}{(2n+3)(2n+5)\cdots(2n+2%
k+1)}\*\sin\left((n+2k+1)\theta\right)," display="inline"><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><mfrac><mrow><msup><mn>2</mn><mrow><mrow><mn>2</mn><mo></mo><mi href="./14.1#p1.t1.r3">n</mi></mrow><mo>+</mo><mn>2</mn></mrow></msup><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./14.1#p1.t1.r3">n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow><mrow><mi href="./3.12#E1">π</mi><mo></mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>2</mn><mo></mo><mi href="./14.1#p1.t1.r3">n</mi></mrow><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mrow></mfrac></mstyle><mo></mo><mrow><mstyle displaystyle="true"><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover></mstyle><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mstyle displaystyle="true"><mfrac><mrow><mrow><mn>1</mn><mo>⋅</mo><mn>3</mn></mrow><mo></mo><mi mathvariant="normal">⋯</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>2</mn><mo></mo><mi>k</mi></mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mrow><mi>k</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mfrac></mstyle><mo></mo><mstyle displaystyle="true"><mfrac><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./14.1#p1.t1.r3">n</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./14.1#p1.t1.r3">n</mi><mo>+</mo><mn>2</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi mathvariant="normal">⋯</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./14.1#p1.t1.r3">n</mi><mo>+</mo><mi>k</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>2</mn><mo></mo><mi href="./14.1#p1.t1.r3">n</mi></mrow><mo>+</mo><mn>3</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>2</mn><mo></mo><mi href="./14.1#p1.t1.r3">n</mi></mrow><mo>+</mo><mn>5</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi mathvariant="normal">⋯</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>2</mn><mo></mo><mi href="./14.1#p1.t1.r3">n</mi></mrow><mo>+</mo><mrow><mn>2</mn><mo></mo><mi>k</mi></mrow><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow></mfrac></mstyle></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>×</mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./14.1#p1.t1.r3">n</mi><mo>+</mo><mrow><mn>2</mn><mo></mo><mi>k</mi></mrow><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi href="./14.13#p1">θ</mi></mrow><mo>)</mo></mrow></mrow><mo>,</mo></mtd></mtr></mtable></mrow></mrow></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m23.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m15.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m10.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m25.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./14.2#SS2.p2" title="§14.2(ii) Associated Legendre Equation ‣ §14.2 Differential Equations ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m17.png" altimg-height="25px" altimg-valign="-7px" altimg-width="137px" alttext="\mathsf{P}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{P}^{0}_{\nu}\left(x\right)" display="inline"><mrow><mrow><msub><mi href="./14.2#SS2.p2" mathvariant="sans-serif">P</mi><mi class="ltx_nvar" href="./14.1#p1.t1.r4">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./14.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msubsup><mi href="./14.3#E1" mathvariant="sans-serif">P</mi><mi href="./14.1#p1.t1.r4">ν</mi><mn>0</mn></msubsup><mo></mo><mrow><mo>(</mo><mi href="./14.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow></mrow></math>: Ferrers function of the first kind</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m24.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./14.1#p1.t1.r3" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m26.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./14.1#p1.t1.r3">n</mi></math>: nonnegative integer</a> and
<a href="./14.13#p1" title="§14.13 Trigonometric Expansions ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m11.png" altimg-height="18px" altimg-valign="-3px" altimg-width="89px" alttext="0<\theta<\pi" display="inline"><mrow><mn>0</mn><mo><</mo><mi href="./14.13#p1">θ</mi><mo><</mo><mi href="./3.12#E1">π</mi></mrow></math>: variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">8.7.3</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E4">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">14.13.4</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m8.png" altimg-height="25px" altimg-valign="-7px" altimg-width="90px" alttext="\displaystyle\mathsf{Q}_{n}\left(\cos\theta\right)" display="inline"><mrow><msub><mi href="./14.2#SS2.p2" mathvariant="sans-serif">Q</mi><mi href="./14.1#p1.t1.r3">n</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi href="./14.13#p1">θ</mi></mrow><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m1.png" altimg-height="64px" altimg-valign="-28px" altimg-width="778px" alttext="\displaystyle=\frac{2^{2n+1}(n!)^{2}}{(2n+1)!}\*\sum_{k=0}^{\infty}\frac{1%
\cdot 3\cdots(2k-1)}{k!}\*\frac{(n+1)(n+2)\cdots(n+k)}{(2n+3)(2n+5)\cdots(2n+2%
k+1)}\*\cos\left((n+2k+1)\theta\right)," display="inline"><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><mfrac><mrow><msup><mn>2</mn><mrow><mrow><mn>2</mn><mo></mo><mi href="./14.1#p1.t1.r3">n</mi></mrow><mo>+</mo><mn>1</mn></mrow></msup><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./14.1#p1.t1.r3">n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>2</mn><mo></mo><mi href="./14.1#p1.t1.r3">n</mi></mrow><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mfrac></mstyle><mo></mo><mrow><mstyle displaystyle="true"><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover></mstyle><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mstyle displaystyle="true"><mfrac><mrow><mrow><mn>1</mn><mo>⋅</mo><mn>3</mn></mrow><mo></mo><mi mathvariant="normal">⋯</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>2</mn><mo></mo><mi>k</mi></mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mrow><mi>k</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mfrac></mstyle><mo></mo><mstyle displaystyle="true"><mfrac><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./14.1#p1.t1.r3">n</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./14.1#p1.t1.r3">n</mi><mo>+</mo><mn>2</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi mathvariant="normal">⋯</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./14.1#p1.t1.r3">n</mi><mo>+</mo><mi>k</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>2</mn><mo></mo><mi href="./14.1#p1.t1.r3">n</mi></mrow><mo>+</mo><mn>3</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>2</mn><mo></mo><mi href="./14.1#p1.t1.r3">n</mi></mrow><mo>+</mo><mn>5</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi mathvariant="normal">⋯</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>2</mn><mo></mo><mi href="./14.1#p1.t1.r3">n</mi></mrow><mo>+</mo><mrow><mn>2</mn><mo></mo><mi>k</mi></mrow><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow></mfrac></mstyle></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>×</mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./14.1#p1.t1.r3">n</mi><mo>+</mo><mrow><mn>2</mn><mo></mo><mi>k</mi></mrow><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi href="./14.13#p1">θ</mi></mrow><mo>)</mo></mrow></mrow><mo>,</mo></mtd></mtr></mtable></mrow></mrow></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E4.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m15.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m10.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m25.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./14.2#SS2.p2" title="§14.2(ii) Associated Legendre Equation ‣ §14.2 Differential Equations ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="25px" altimg-valign="-7px" altimg-width="140px" alttext="\mathsf{Q}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{Q}^{0}_{\nu}\left(x\right)" display="inline"><mrow><mrow><msub><mi href="./14.2#SS2.p2" mathvariant="sans-serif">Q</mi><mi class="ltx_nvar" href="./14.1#p1.t1.r4">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./14.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msubsup><mi href="./14.3#E2" mathvariant="sans-serif">Q</mi><mi href="./14.1#p1.t1.r4">ν</mi><mn>0</mn></msubsup><mo></mo><mrow><mo>(</mo><mi href="./14.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow></mrow></math>: Ferrers function of the second kind</a>,
<a href="./14.1#p1.t1.r3" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m26.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./14.1#p1.t1.r3">n</mi></math>: nonnegative integer</a> and
<a href="./14.13#p1" title="§14.13 Trigonometric Expansions ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m11.png" altimg-height="18px" altimg-valign="-3px" altimg-width="89px" alttext="0<\theta<\pi" display="inline"><mrow><mn>0</mn><mo><</mo><mi href="./14.13#p1">θ</mi><mo><</mo><mi href="./3.12#E1">π</mi></mrow></math>: variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">8.7.4</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
<p class="ltx_p">with conditional convergence for each.</p>
</div>
<div id="p3" class="ltx_para">
<p class="ltx_p">For other trigonometric expansions see <cite class="ltx_cite ltx_citemacro_citet">Erdélyi<span class="ltx_text ltx_bib_etal"> et al.</span> (</div>
</div>
</body></text>
</html>
</page>
<page>
<!DOCTYPE html><html>
<head>
<title>DLMF: 14.19 Toroidal (or Ring) Functions</title>
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<div class="ltx_page_navlogo"></dd>
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<div id="SS1.p1" class="ltx_para">
<p class="ltx_p">When <math class="ltx_Math" altimg="m37.png" altimg-height="27px" altimg-valign="-9px" altimg-width="91px" alttext="\nu=n-\frac{1}{2}" display="inline"><mrow><mi href="./14.1#p1.t1.r4">ν</mi><mo>=</mo><mrow><mi href="./14.1#p1.t1.r3">n</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow></math>, <math class="ltx_Math" altimg="m48.png" altimg-height="20px" altimg-valign="-6px" altimg-width="123px" alttext="n=0,1,2,\dots" display="inline"><mrow><mi href="./14.1#p1.t1.r3">n</mi><mo>=</mo><mrow><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>, <math class="ltx_Math" altimg="m33.png" altimg-height="21px" altimg-valign="-6px" altimg-width="55px" alttext="\mu\in\mathbb{R}" display="inline"><mrow><mi href="./14.1#p1.t1.r4">μ</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mi href="./front/introduction#Sx4.p2.t1.r14">ℝ</mi></mrow></math>, and
<math class="ltx_Math" altimg="m51.png" altimg-height="23px" altimg-valign="-7px" altimg-width="94px" alttext="x\in(1,\infty)" display="inline"><mrow><mi href="./14.19#SS1.p1">x</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mn>1</mn><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi mathvariant="normal">∞</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></mrow></math> solutions of () are known as
<em class="ltx_emph ltx_font_italic">toroidal</em> or <em class="ltx_emph ltx_font_italic">ring functions</em>. This form of the differential equation
arises when Laplace’s equation is transformed into <em class="ltx_emph ltx_font_italic">toroidal coordinates</em>
<math class="ltx_Math" altimg="m12.png" altimg-height="23px" altimg-valign="-7px" altimg-width="70px" alttext="(\eta,\theta,\phi)" display="inline"><mrow><mo stretchy="false">(</mo><mi href="./14.19#SS1.p1">η</mi><mo>,</mo><mi href="./14.19#SS1.p1">θ</mi><mo>,</mo><mi href="./14.19#SS1.p1">ϕ</mi><mo stretchy="false">)</mo></mrow></math>, which are related to Cartesian coordinates <math class="ltx_Math" altimg="m13.png" altimg-height="23px" altimg-valign="-7px" altimg-width="70px" alttext="(x,y,z)" display="inline"><mrow><mo stretchy="false">(</mo><mi href="./14.19#SS1.p1">x</mi><mo>,</mo><mi href="./14.19#SS1.p1">y</mi><mo>,</mo><mi href="./14.19#SS1.p1">z</mi><mo stretchy="false">)</mo></mrow></math> by
</p>
<table id="E1" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex1" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="3" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">14.19.1</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m9.png" altimg-height="14px" altimg-valign="-2px" altimg-width="17px" alttext="\displaystyle x" display="inline"><mi href="./14.19#SS1.p1">x</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m4.png" altimg-height="50px" altimg-valign="-20px" altimg-width="153px" alttext="\displaystyle=\frac{c\sinh\eta\cos\phi}{\cosh\eta-\cos\theta}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mstyle displaystyle="true"><mfrac><mrow><mi>c</mi><mo></mo><mrow><mi href="./4.28#E1">sinh</mi><mo></mo><mi href="./14.19#SS1.p1">η</mi></mrow><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi href="./14.19#SS1.p1">ϕ</mi></mrow></mrow><mrow><mrow><mi href="./4.28#E2">cosh</mi><mo></mo><mi href="./14.19#SS1.p1">η</mi></mrow><mo>-</mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi href="./14.19#SS1.p1">θ</mi></mrow></mrow></mfrac></mstyle></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex2" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m10.png" altimg-height="18px" altimg-valign="-6px" altimg-width="17px" alttext="\displaystyle y" display="inline"><mi href="./14.19#SS1.p1">y</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m5.png" altimg-height="50px" altimg-valign="-20px" altimg-width="153px" alttext="\displaystyle=\frac{c\sinh\eta\sin\phi}{\cosh\eta-\cos\theta}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mstyle displaystyle="true"><mfrac><mrow><mi>c</mi><mo></mo><mrow><mi href="./4.28#E1">sinh</mi><mo></mo><mi href="./14.19#SS1.p1">η</mi></mrow><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi href="./14.19#SS1.p1">ϕ</mi></mrow></mrow><mrow><mrow><mi href="./4.28#E2">cosh</mi><mo></mo><mi href="./14.19#SS1.p1">η</mi></mrow><mo>-</mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi href="./14.19#SS1.p1">θ</mi></mrow></mrow></mfrac></mstyle></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex3" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m11.png" altimg-height="14px" altimg-valign="-2px" altimg-width="16px" alttext="\displaystyle z" display="inline"><mi href="./14.19#SS1.p1">z</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m3.png" altimg-height="50px" altimg-valign="-20px" altimg-width="153px" alttext="\displaystyle=\frac{c\sin\theta}{\cosh\eta-\cos\theta}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mstyle displaystyle="true"><mfrac><mrow><mi>c</mi><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi href="./14.19#SS1.p1">θ</mi></mrow></mrow><mrow><mrow><mi href="./4.28#E2">cosh</mi><mo></mo><mi href="./14.19#SS1.p1">η</mi></mrow><mo>-</mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi href="./14.19#SS1.p1">θ</mi></mrow></mrow></mfrac></mstyle></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m21.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./4.28#E2" title="(4.28.2) ‣ §4.28 Definitions and Periodicity ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="18px" altimg-valign="-2px" altimg-width="56px" alttext="\cosh\NVar{z}" display="inline"><mrow><mi href="./4.28#E2">cosh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: hyperbolic cosine function</a>,
<a href="./4.28#E1" title="(4.28.1) ‣ §4.28 Definitions and Periodicity ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m42.png" altimg-height="18px" altimg-valign="-2px" altimg-width="53px" alttext="\sinh\NVar{z}" display="inline"><mrow><mi href="./4.28#E1">sinh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: hyperbolic sine function</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m41.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./14.19#SS1.p1" title="§14.19(i) Introduction ‣ §14.19 Toroidal (or Ring) Functions ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./14.19#SS1.p1">x</mi></math>: Cartesian coordinate</a>,
<a href="./14.19#SS1.p1" title="§14.19(i) Introduction ‣ §14.19 Toroidal (or Ring) Functions ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./14.19#SS1.p1">y</mi></math>: Cartesian coordinate</a>,
<a href="./14.19#SS1.p1" title="§14.19(i) Introduction ‣ §14.19 Toroidal (or Ring) Functions ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./14.19#SS1.p1">z</mi></math>: Cartesian coordinate</a>,
<a href="./14.19#SS1.p1" title="§14.19(i) Introduction ‣ §14.19 Toroidal (or Ring) Functions ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m24.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="\eta" display="inline"><mi href="./14.19#SS1.p1">η</mi></math>: toroidal coordinate</a>,
<a href="./14.19#SS1.p1" title="§14.19(i) Introduction ‣ §14.19 Toroidal (or Ring) Functions ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="18px" altimg-valign="-2px" altimg-width="14px" alttext="\theta" display="inline"><mi href="./14.19#SS1.p1">θ</mi></math>: toroidal coordinate</a> and
<a href="./14.19#SS1.p1" title="§14.19(i) Introduction ‣ §14.19 Toroidal (or Ring) Functions ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="\phi" display="inline"><mi href="./14.19#SS1.p1">ϕ</mi></math>: toroidal coordinate</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where the constant <math class="ltx_Math" altimg="m45.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="c" display="inline"><mi>c</mi></math> is a scaling factor. Most required properties of
toroidal functions come directly from the results for
<math class="ltx_Math" altimg="m15.png" altimg-height="23px" altimg-valign="-7px" altimg-width="61px" alttext="P^{\mu}_{\nu}\left(x\right)" display="inline"><mrow><msubsup><mi href="./14.21#SS1.p1">P</mi><mi href="./14.1#p1.t1.r4">ν</mi><mi href="./14.1#p1.t1.r4">μ</mi></msubsup><mo></mo><mrow><mo>(</mo><mi href="./14.19#SS1.p1">x</mi><mo>)</mo></mrow></mrow></math> and <math class="ltx_Math" altimg="m20.png" altimg-height="23px" altimg-valign="-7px" altimg-width="62px" alttext="\boldsymbol{Q}^{\mu}_{\nu}\left(x\right)" display="inline"><mrow><msubsup><mi href="./14.21#SS1.p1" mathvariant="bold-italic">Q</mi><mi href="./14.1#p1.t1.r4">ν</mi><mi href="./14.1#p1.t1.r4">μ</mi></msubsup><mo></mo><mrow><mo>(</mo><mi href="./14.19#SS1.p1">x</mi><mo>)</mo></mrow></mrow></math>. In particular,
for <math class="ltx_Math" altimg="m31.png" altimg-height="20px" altimg-valign="-6px" altimg-width="53px" alttext="\mu=0" display="inline"><mrow><mi href="./14.1#p1.t1.r4">μ</mi><mo>=</mo><mn>0</mn></mrow></math> and <math class="ltx_Math" altimg="m36.png" altimg-height="27px" altimg-valign="-9px" altimg-width="70px" alttext="\nu=\pm\frac{1}{2}" display="inline"><mrow><mi href="./14.1#p1.t1.r4">ν</mi><mo>=</mo><mrow><mo>±</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow></math> see §) with <math class="ltx_Math" altimg="m53.png" altimg-height="27px" altimg-valign="-9px" altimg-width="90px" alttext="z=\frac{1}{2}-\mu" display="inline"><mrow><mi href="./14.1#p1.t1.r2">z</mi><mo>=</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>-</mo><mi href="./14.1#p1.t1.r4">μ</mi></mrow></mrow></math>.
</dd>
</dl>
</div>
</div>
<div id="SS2.p1" class="ltx_para">
<p class="ltx_p">With <math class="ltx_Math" altimg="m26.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\mathbf{F}" display="inline"><mi href="./15.2#E2" mathvariant="bold">F</mi></math> as in § and <math class="ltx_Math" altimg="m44.png" altimg-height="21px" altimg-valign="-6px" altimg-width="50px" alttext="\xi>0" display="inline"><mrow><mi href="./14.19#SS2.p1">ξ</mi><mo>></mo><mn>0</mn></mrow></math>,
</p>
<table id="E2" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">14.19.2</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E2.png" altimg-height="57px" altimg-valign="-21px" altimg-width="722px" alttext="P^{\mu}_{\nu-\frac{1}{2}}\left(\cosh\xi\right)=\frac{\Gamma\left(\frac{1}{2}-%
\mu\right)}{\pi^{1/2}\left(1-e^{-2\xi}\right)^{\mu}e^{(\nu+(1/2))\xi}}\*%
\mathbf{F}\left(\tfrac{1}{2}-\mu,\tfrac{1}{2}+\nu-\mu;1-2\mu;1-e^{-2\xi}\right)," display="block"><mrow><mrow><mrow><msubsup><mi href="./14.21#SS1.p1">P</mi><mrow><mi href="./14.1#p1.t1.r4">ν</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mi href="./14.1#p1.t1.r4">μ</mi></msubsup><mo></mo><mrow><mo>(</mo><mrow><mi href="./4.28#E2">cosh</mi><mo></mo><mi href="./14.19#SS2.p1">ξ</mi></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mfrac><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>-</mo><mi href="./14.1#p1.t1.r4">μ</mi></mrow><mo>)</mo></mrow></mrow><mrow><msup><mi href="./3.12#E1">π</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo></mo><msup><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mn>2</mn><mo></mo><mi href="./14.19#SS2.p1">ξ</mi></mrow></mrow></msup></mrow><mo>)</mo></mrow><mi href="./14.1#p1.t1.r4">μ</mi></msup><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./14.1#p1.t1.r4">ν</mi><mo>+</mo><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi href="./14.19#SS2.p1">ξ</mi></mrow></msup></mrow></mfrac><mo></mo><mrow><mi href="./15.2#E2" mathvariant="bold">F</mi><mo></mo><mrow><mo>(</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo>-</mo><mi href="./14.1#p1.t1.r4">μ</mi></mrow><mo>,</mo><mrow><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo>+</mo><mi href="./14.1#p1.t1.r4">ν</mi></mrow><mo>-</mo><mi href="./14.1#p1.t1.r4">μ</mi></mrow><mo>;</mo><mrow><mn>1</mn><mo>-</mo><mrow><mn>2</mn><mo></mo><mi href="./14.1#p1.t1.r4">μ</mi></mrow></mrow><mo>;</mo><mrow><mn>1</mn><mo>-</mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mn>2</mn><mo></mo><mi href="./14.19#SS2.p1">ξ</mi></mrow></mrow></msup></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m34.png" altimg-height="27px" altimg-valign="-9px" altimg-width="131px" alttext="\mu\neq\frac{1}{2},\frac{3}{2},\frac{5}{2},\ldots" display="inline"><mrow><mi href="./14.1#p1.t1.r4">μ</mi><mo>≠</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>,</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mo>,</mo><mfrac><mn>5</mn><mn>2</mn></mfrac><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m16.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./14.21#SS1.p1" title="§14.21(i) Associated Legendre Equation ‣ §14.21 Definitions and Basic Properties ‣ Complex Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m14.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msubsup><mi href="./14.21#SS1.p1">P</mi><mi class="ltx_nvar" href="./14.1#p1.t1.r4">ν</mi><mi class="ltx_nvar" href="./14.1#p1.t1.r4">μ</mi></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./14.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: associated Legendre function of the first kind</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m40.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m30.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./4.28#E2" title="(4.28.2) ‣ §4.28 Definitions and Periodicity ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="18px" altimg-valign="-2px" altimg-width="56px" alttext="\cosh\NVar{z}" display="inline"><mrow><mi href="./4.28#E2">cosh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: hyperbolic cosine function</a>,
<a href="./15.2#E2" title="(15.2.2) ‣ §15.2(i) Gauss Series ‣ §15.2 Definitions and Analytical Properties ‣ Properties ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m27.png" altimg-height="23px" altimg-valign="-7px" altimg-width="102px" alttext="\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E2" mathvariant="bold">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m28.png" altimg-height="39px" altimg-valign="-15px" altimg-width="90px" alttext="\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E2" mathvariant="bold">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi></mfrac><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: Olver’s hypergeometric function</a>,
<a href="./14.1#p1.t1.r4" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m32.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\mu" display="inline"><mi href="./14.1#p1.t1.r4">μ</mi></math>: general order</a>,
<a href="./14.1#p1.t1.r4" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./14.1#p1.t1.r4">ν</mi></math>: general degree</a> and
<a href="./14.19#SS2.p1" title="§14.19(ii) Hypergeometric Representations ‣ §14.19 Toroidal (or Ring) Functions ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="21px" altimg-valign="-6px" altimg-width="50px" alttext="\xi>0" display="inline"><mrow><mi href="./14.19#SS2.p1">ξ</mi><mo>></mo><mn>0</mn></mrow></math>: variable</a>
</dd>
<dt>Errata (effective with 1.0.1):</dt>
<dd>
Originally the argument to <math class="ltx_Math" altimg="m26.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\mathbf{F}" display="inline"><mi href="./15.2#E2" mathvariant="bold">F</mi></math> was incorrect and the condition on <math class="ltx_Math" altimg="m32.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\mu" display="inline"><mi href="./14.1#p1.t1.r4">μ</mi></math> too weak:
<table id="Ex4" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="m6.png" altimg-height="55px" altimg-valign="-21px" altimg-width="728px" alttext="P^{\mu}_{\nu-\frac{1}{2}}\left(\cosh\xi\right)=\frac{\Gamma\left(1-2\mu\right)%
2^{2\mu}}{\Gamma\left(1-\mu\right)\left(1-e^{-2\xi}\right)^{\mu}e^{(\nu+(1/2))%
\xi}}\*\mathbf{F}\left(\tfrac{1}{2}-\mu,\tfrac{1}{2}+\nu-\mu;1-2\mu;e^{-2\xi}%
\right)," display="block"><mrow><mrow><mrow><msubsup><mi href="./14.21#SS1.p1">P</mi><mrow><mi href="./14.1#p1.t1.r4">ν</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mi href="./14.1#p1.t1.r4">μ</mi></msubsup><mo></mo><mrow><mo>(</mo><mrow><mi href="./4.28#E2">cosh</mi><mo></mo><mi href="./14.19#SS2.p1">ξ</mi></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mfrac><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mrow><mn>2</mn><mo></mo><mi href="./14.1#p1.t1.r4">μ</mi></mrow></mrow><mo>)</mo></mrow></mrow><mo></mo><msup><mn>2</mn><mrow><mn>2</mn><mo></mo><mi href="./14.1#p1.t1.r4">μ</mi></mrow></msup></mrow><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mi href="./14.1#p1.t1.r4">μ</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><msup><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mn>2</mn><mo></mo><mi href="./14.19#SS2.p1">ξ</mi></mrow></mrow></msup></mrow><mo>)</mo></mrow><mi href="./14.1#p1.t1.r4">μ</mi></msup><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./14.1#p1.t1.r4">ν</mi><mo>+</mo><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi href="./14.19#SS2.p1">ξ</mi></mrow></msup></mrow></mfrac><mo></mo><mrow><mi href="./15.2#E2" mathvariant="bold">F</mi><mo></mo><mrow><mo>(</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo>-</mo><mi href="./14.1#p1.t1.r4">μ</mi></mrow><mo>,</mo><mrow><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo>+</mo><mi href="./14.1#p1.t1.r4">ν</mi></mrow><mo>-</mo><mi href="./14.1#p1.t1.r4">μ</mi></mrow><mo>;</mo><mrow><mn>1</mn><mo>-</mo><mrow><mn>2</mn><mo></mo><mi href="./14.1#p1.t1.r4">μ</mi></mrow></mrow><mo>;</mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mn>2</mn><mo></mo><mi href="./14.19#SS2.p1">ξ</mi></mrow></mrow></msup><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m35.png" altimg-height="27px" altimg-valign="-9px" altimg-width="56px" alttext="\mu\neq\frac{1}{2}" display="inline"><mrow><mi href="./14.1#p1.t1.r4">μ</mi><mo>≠</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math>.</span></td></tr>
</table>
Also, the factor multiplying <math class="ltx_Math" altimg="m26.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\mathbf{F}" display="inline"><mi href="./15.2#E2" mathvariant="bold">F</mi></math> was rewritten to clarify the poles, which
determine the correct condition on <math class="ltx_Math" altimg="m32.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\mu" display="inline"><mi href="./14.1#p1.t1.r4">μ</mi></math>.
<p><span class="ltx_font_italic">Reported 2010-11-02 by Alvaro Valenzuela</span></p>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E3" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">14.19.3</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E3.png" altimg-height="52px" altimg-valign="-16px" altimg-width="592px" alttext="\boldsymbol{Q}^{\mu}_{\nu-\frac{1}{2}}\left(\cosh\xi\right)=\frac{\pi^{1/2}%
\left(1-e^{-2\xi}\right)^{\mu}}{e^{(\nu+(1/2))\xi}}\*\mathbf{F}\left(\mu+%
\tfrac{1}{2},\nu+\mu+\tfrac{1}{2};\nu+1;e^{-2\xi}\right)." display="block"><mrow><mrow><mrow><msubsup><mi href="./14.21#SS1.p1" mathvariant="bold-italic">Q</mi><mrow><mi href="./14.1#p1.t1.r4">ν</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mi href="./14.1#p1.t1.r4">μ</mi></msubsup><mo></mo><mrow><mo>(</mo><mrow><mi href="./4.28#E2">cosh</mi><mo></mo><mi href="./14.19#SS2.p1">ξ</mi></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mfrac><mrow><msup><mi href="./3.12#E1">π</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo></mo><msup><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mn>2</mn><mo></mo><mi href="./14.19#SS2.p1">ξ</mi></mrow></mrow></msup></mrow><mo>)</mo></mrow><mi href="./14.1#p1.t1.r4">μ</mi></msup></mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./14.1#p1.t1.r4">ν</mi><mo>+</mo><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi href="./14.19#SS2.p1">ξ</mi></mrow></msup></mfrac><mo></mo><mrow><mi href="./15.2#E2" mathvariant="bold">F</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./14.1#p1.t1.r4">μ</mi><mo>+</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle></mrow><mo>,</mo><mrow><mi href="./14.1#p1.t1.r4">ν</mi><mo>+</mo><mi href="./14.1#p1.t1.r4">μ</mi><mo>+</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle></mrow><mo>;</mo><mrow><mi href="./14.1#p1.t1.r4">ν</mi><mo>+</mo><mn>1</mn></mrow><mo>;</mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mn>2</mn><mo></mo><mi href="./14.19#SS2.p1">ξ</mi></mrow></mrow></msup><mo>)</mo></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./14.21#SS1.p1" title="§14.21(i) Associated Legendre Equation ‣ §14.21 Definitions and Basic Properties ‣ Complex Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="23px" altimg-valign="-7px" altimg-width="61px" alttext="\boldsymbol{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msubsup><mi href="./14.21#SS1.p1" mathvariant="bold-italic">Q</mi><mi class="ltx_nvar" href="./14.1#p1.t1.r4">ν</mi><mi class="ltx_nvar" href="./14.1#p1.t1.r4">μ</mi></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./14.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: Olver’s associated Legendre function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m40.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m30.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./4.28#E2" title="(4.28.2) ‣ §4.28 Definitions and Periodicity ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="18px" altimg-valign="-2px" altimg-width="56px" alttext="\cosh\NVar{z}" display="inline"><mrow><mi href="./4.28#E2">cosh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: hyperbolic cosine function</a>,
<a href="./15.2#E2" title="(15.2.2) ‣ §15.2(i) Gauss Series ‣ §15.2 Definitions and Analytical Properties ‣ Properties ‣ Chapter 15 Hypergeometric Function" class="ltx_ref"><math class="ltx_Math" altimg="m27.png" altimg-height="23px" altimg-valign="-7px" altimg-width="102px" alttext="\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E2" mathvariant="bold">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> or <math class="ltx_Math" altimg="m28.png" altimg-height="39px" altimg-valign="-15px" altimg-width="90px" alttext="\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)" display="inline"><mrow><mi href="./15.2#E2" mathvariant="bold">F</mi><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">a</mi><mo>,</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r3">b</mi></mrow><mi class="ltx_nvar" href="./15.1#p1.t1.r3">c</mi></mfrac><mo>;</mo><mi class="ltx_nvar" href="./15.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: Olver’s hypergeometric function</a>,
<a href="./14.1#p1.t1.r4" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m32.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\mu" display="inline"><mi href="./14.1#p1.t1.r4">μ</mi></math>: general order</a>,
<a href="./14.1#p1.t1.r4" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./14.1#p1.t1.r4">ν</mi></math>: general degree</a> and
<a href="./14.19#SS2.p1" title="§14.19(ii) Hypergeometric Representations ‣ §14.19 Toroidal (or Ring) Functions ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="21px" altimg-valign="-6px" altimg-width="50px" alttext="\xi>0" display="inline"><mrow><mi href="./14.19#SS2.p1">ξ</mi><mo>></mo><mn>0</mn></mrow></math>: variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="iii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§14.19(iii) </span>Integral Representations</h2>
<div id="SS3.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="SS3.p1" class="ltx_para">
<p class="ltx_p">With <math class="ltx_Math" altimg="m44.png" altimg-height="21px" altimg-valign="-6px" altimg-width="50px" alttext="\xi>0" display="inline"><mrow><mi href="./14.19#SS3.p1">ξ</mi><mo>></mo><mn>0</mn></mrow></math>,
</p>
<table id="EGx1" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E4">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">14.19.4</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m7.png" altimg-height="32px" altimg-valign="-14px" altimg-width="123px" alttext="\displaystyle P^{m}_{n-\frac{1}{2}}\left(\cosh\xi\right)" display="inline"><mrow><msubsup><mi href="./14.21#SS1.p1">P</mi><mrow><mi href="./14.1#p1.t1.r3">n</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mi href="./14.1#p1.t1.r3">m</mi></msubsup><mo></mo><mrow><mo>(</mo><mrow><mi href="./4.28#E2">cosh</mi><mo></mo><mi href="./14.19#SS3.p1">ξ</mi></mrow><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m2.png" altimg-height="59px" altimg-valign="-24px" altimg-width="655px" alttext="\displaystyle=\frac{\Gamma\left(n+m+\frac{1}{2}\right)(\sinh\xi)^{m}}{2^{m}\pi%
^{1/2}\Gamma\left(n-m+\frac{1}{2}\right)\Gamma\left(m+\frac{1}{2}\right)}\*%
\int_{0}^{\pi}\frac{(\sin\phi)^{2m}}{(\cosh\xi+\cos\phi\sinh\xi)^{n+m+(1/2)}}%
\mathrm{d}\phi," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><mfrac><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./14.1#p1.t1.r3">n</mi><mo>+</mo><mi href="./14.1#p1.t1.r3">m</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./4.28#E1">sinh</mi><mo></mo><mi href="./14.19#SS3.p1">ξ</mi></mrow><mo stretchy="false">)</mo></mrow><mi href="./14.1#p1.t1.r3">m</mi></msup></mrow><mrow><msup><mn>2</mn><mi href="./14.1#p1.t1.r3">m</mi></msup><mo></mo><msup><mi href="./3.12#E1">π</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mi href="./14.1#p1.t1.r3">n</mi><mo>-</mo><mi href="./14.1#p1.t1.r3">m</mi></mrow><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./14.1#p1.t1.r3">m</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow></mrow></mfrac></mstyle><mo></mo><mrow><mstyle displaystyle="true"><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi href="./3.12#E1">π</mi></msubsup></mstyle><mrow><mstyle displaystyle="true"><mfrac><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi>ϕ</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mn>2</mn><mo></mo><mi href="./14.1#p1.t1.r3">m</mi></mrow></msup><msup><mrow><mo stretchy="false">(</mo><mrow><mrow><mi href="./4.28#E2">cosh</mi><mo></mo><mi href="./14.19#SS3.p1">ξ</mi></mrow><mo>+</mo><mrow><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi>ϕ</mi></mrow><mo></mo><mrow><mi href="./4.28#E1">sinh</mi><mo></mo><mi href="./14.19#SS3.p1">ξ</mi></mrow></mrow></mrow><mo stretchy="false">)</mo></mrow><mrow><mi href="./14.1#p1.t1.r3">n</mi><mo>+</mo><mi href="./14.1#p1.t1.r3">m</mi><mo>+</mo><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow><mo stretchy="false">)</mo></mrow></mrow></msup></mfrac></mstyle><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>ϕ</mi></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E4.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m16.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./14.21#SS1.p1" title="§14.21(i) Associated Legendre Equation ‣ §14.21 Definitions and Basic Properties ‣ Complex Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m14.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msubsup><mi href="./14.21#SS1.p1">P</mi><mi class="ltx_nvar" href="./14.1#p1.t1.r4">ν</mi><mi class="ltx_nvar" href="./14.1#p1.t1.r4">μ</mi></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./14.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: associated Legendre function of the first kind</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m40.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m21.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m50.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.28#E2" title="(4.28.2) ‣ §4.28 Definitions and Periodicity ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="18px" altimg-valign="-2px" altimg-width="56px" alttext="\cosh\NVar{z}" display="inline"><mrow><mi href="./4.28#E2">cosh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: hyperbolic cosine function</a>,
<a href="./4.28#E1" title="(4.28.1) ‣ §4.28 Definitions and Periodicity ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m42.png" altimg-height="18px" altimg-valign="-2px" altimg-width="53px" alttext="\sinh\NVar{z}" display="inline"><mrow><mi href="./4.28#E1">sinh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: hyperbolic sine function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m25.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m41.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./14.1#p1.t1.r3" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./14.1#p1.t1.r3">m</mi></math>: nonnegative integer</a>,
<a href="./14.1#p1.t1.r3" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./14.1#p1.t1.r3">n</mi></math>: nonnegative integer</a> and
<a href="./14.19#SS3.p1" title="§14.19(iii) Integral Representations ‣ §14.19 Toroidal (or Ring) Functions ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="21px" altimg-valign="-6px" altimg-width="50px" alttext="\xi>0" display="inline"><mrow><mi href="./14.19#SS3.p1">ξ</mi><mo>></mo><mn>0</mn></mrow></math>: variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">8.11.2</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E5">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">14.19.5</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m8.png" altimg-height="32px" altimg-valign="-14px" altimg-width="128px" alttext="\displaystyle\boldsymbol{Q}^{m}_{n-\frac{1}{2}}\left(\cosh\xi\right)" display="inline"><mrow><msubsup><mi href="./14.21#SS1.p1" mathvariant="bold-italic">Q</mi><mrow><mi href="./14.1#p1.t1.r3">n</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mi href="./14.1#p1.t1.r3">m</mi></msubsup><mo></mo><mrow><mo>(</mo><mrow><mi href="./4.28#E2">cosh</mi><mo></mo><mi href="./14.19#SS3.p1">ξ</mi></mrow><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m1.png" altimg-height="59px" altimg-valign="-24px" altimg-width="611px" alttext="\displaystyle=\frac{\Gamma\left(n+\frac{1}{2}\right)}{\Gamma\left(n+m+\tfrac{1%
}{2}\right)\Gamma\left(n-m+\frac{1}{2}\right)}\*\int_{0}^{\infty}\frac{\cosh%
\left(mt\right)}{(\cosh\xi+\cosh t\sinh\xi)^{n+(1/2)}}\mathrm{d}t," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><mfrac><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./14.1#p1.t1.r3">n</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./14.1#p1.t1.r3">n</mi><mo>+</mo><mi href="./14.1#p1.t1.r3">m</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mi href="./14.1#p1.t1.r3">n</mi><mo>-</mo><mi href="./14.1#p1.t1.r3">m</mi></mrow><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow></mrow></mfrac></mstyle><mo></mo><mrow><mstyle displaystyle="true"><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup></mstyle><mrow><mstyle displaystyle="true"><mfrac><mrow><mi href="./4.28#E2">cosh</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./14.1#p1.t1.r3">m</mi><mo></mo><mi>t</mi></mrow><mo>)</mo></mrow></mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mrow><mi href="./4.28#E2">cosh</mi><mo></mo><mi href="./14.19#SS3.p1">ξ</mi></mrow><mo>+</mo><mrow><mrow><mi href="./4.28#E2">cosh</mi><mo></mo><mi>t</mi></mrow><mo></mo><mrow><mi href="./4.28#E1">sinh</mi><mo></mo><mi href="./14.19#SS3.p1">ξ</mi></mrow></mrow></mrow><mo stretchy="false">)</mo></mrow><mrow><mi href="./14.1#p1.t1.r3">n</mi><mo>+</mo><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow><mo stretchy="false">)</mo></mrow></mrow></msup></mfrac></mstyle><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m46.png" altimg-height="27px" altimg-valign="-9px" altimg-width="97px" alttext="m<n+\tfrac{1}{2}" display="inline"><mrow><mi href="./14.1#p1.t1.r3">m</mi><mo><</mo><mrow><mi href="./14.1#p1.t1.r3">n</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E5.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m16.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./14.21#SS1.p1" title="§14.21(i) Associated Legendre Equation ‣ §14.21 Definitions and Basic Properties ‣ Complex Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="23px" altimg-valign="-7px" altimg-width="61px" alttext="\boldsymbol{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msubsup><mi href="./14.21#SS1.p1" mathvariant="bold-italic">Q</mi><mi class="ltx_nvar" href="./14.1#p1.t1.r4">ν</mi><mi class="ltx_nvar" href="./14.1#p1.t1.r4">μ</mi></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./14.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: Olver’s associated Legendre function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m50.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.28#E2" title="(4.28.2) ‣ §4.28 Definitions and Periodicity ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="18px" altimg-valign="-2px" altimg-width="56px" alttext="\cosh\NVar{z}" display="inline"><mrow><mi href="./4.28#E2">cosh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: hyperbolic cosine function</a>,
<a href="./4.28#E1" title="(4.28.1) ‣ §4.28 Definitions and Periodicity ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m42.png" altimg-height="18px" altimg-valign="-2px" altimg-width="53px" alttext="\sinh\NVar{z}" display="inline"><mrow><mi href="./4.28#E1">sinh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: hyperbolic sine function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m25.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./14.1#p1.t1.r3" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./14.1#p1.t1.r3">m</mi></math>: nonnegative integer</a>,
<a href="./14.1#p1.t1.r3" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./14.1#p1.t1.r3">n</mi></math>: nonnegative integer</a> and
<a href="./14.19#SS3.p1" title="§14.19(iii) Integral Representations ‣ §14.19 Toroidal (or Ring) Functions ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="21px" altimg-valign="-6px" altimg-width="50px" alttext="\xi>0" display="inline"><mrow><mi href="./14.19#SS3.p1">ξ</mi><mo>></mo><mn>0</mn></mrow></math>: variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">8.11.4</span> (modified and corrected)</span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
</div>
</section>
<section id="iv" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§14.19(iv) </span>Sums</h2>
<div id="SS4.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="SS4.p1" class="ltx_para">
<p class="ltx_p">With <math class="ltx_Math" altimg="m44.png" altimg-height="21px" altimg-valign="-6px" altimg-width="50px" alttext="\xi>0" display="inline"><mrow><mi href="./14.19#SS4.p1">ξ</mi><mo>></mo><mn>0</mn></mrow></math>,
</p>
<table id="E6" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">14.19.6</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E6.png" altimg-height="68px" altimg-valign="-27px" altimg-width="741px" alttext="\boldsymbol{Q}^{\mu}_{-\frac{1}{2}}\left(\cosh\xi\right)+2\sum_{n=1}^{\infty}%
\frac{\Gamma\left(\mu+n+\tfrac{1}{2}\right)}{\Gamma\left(\mu+\tfrac{1}{2}%
\right)}\boldsymbol{Q}^{\mu}_{n-\frac{1}{2}}\left(\cosh\xi\right)\cos\left(n%
\phi\right)=\dfrac{\left(\frac{1}{2}\pi\right)^{1/2}\left(\sinh\xi\right)^{\mu%
}}{\left(\cosh\xi-\cos\phi\right)^{\mu+(1/2)}}," display="block"><mrow><mrow><mrow><mrow><msubsup><mi href="./14.21#SS1.p1" mathvariant="bold-italic">Q</mi><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mi href="./14.1#p1.t1.r4">μ</mi></msubsup><mo></mo><mrow><mo>(</mo><mrow><mi href="./4.28#E2">cosh</mi><mo></mo><mi href="./14.19#SS4.p1">ξ</mi></mrow><mo>)</mo></mrow></mrow><mo>+</mo><mrow><mn>2</mn><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./14.1#p1.t1.r3">n</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><mfrac><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./14.1#p1.t1.r4">μ</mi><mo>+</mo><mi href="./14.1#p1.t1.r3">n</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./14.1#p1.t1.r4">μ</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow></mfrac><mo></mo><mrow><msubsup><mi href="./14.21#SS1.p1" mathvariant="bold-italic">Q</mi><mrow><mi href="./14.1#p1.t1.r3">n</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mi href="./14.1#p1.t1.r4">μ</mi></msubsup><mo></mo><mrow><mo>(</mo><mrow><mi href="./4.28#E2">cosh</mi><mo></mo><mi href="./14.19#SS4.p1">ξ</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./14.1#p1.t1.r3">n</mi><mo></mo><mi>ϕ</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow></mrow></mrow><mo>=</mo><mfrac><mrow><msup><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo>)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo></mo><msup><mrow><mo>(</mo><mrow><mi href="./4.28#E1">sinh</mi><mo></mo><mi href="./14.19#SS4.p1">ξ</mi></mrow><mo>)</mo></mrow><mi href="./14.1#p1.t1.r4">μ</mi></msup></mrow><msup><mrow><mo>(</mo><mrow><mrow><mi href="./4.28#E2">cosh</mi><mo></mo><mi href="./14.19#SS4.p1">ξ</mi></mrow><mo>-</mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi>ϕ</mi></mrow></mrow><mo>)</mo></mrow><mrow><mi href="./14.1#p1.t1.r4">μ</mi><mo>+</mo><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow><mo stretchy="false">)</mo></mrow></mrow></msup></mfrac></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m18.png" altimg-height="27px" altimg-valign="-9px" altimg-width="85px" alttext="\Re\mu>-\tfrac{1}{2}" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./14.1#p1.t1.r4">μ</mi></mrow><mo>></mo><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E6.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m16.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./14.21#SS1.p1" title="§14.21(i) Associated Legendre Equation ‣ §14.21 Definitions and Basic Properties ‣ Complex Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="23px" altimg-valign="-7px" altimg-width="61px" alttext="\boldsymbol{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msubsup><mi href="./14.21#SS1.p1" mathvariant="bold-italic">Q</mi><mi class="ltx_nvar" href="./14.1#p1.t1.r4">ν</mi><mi class="ltx_nvar" href="./14.1#p1.t1.r4">μ</mi></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./14.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: Olver’s associated Legendre function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m40.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m21.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./4.28#E2" title="(4.28.2) ‣ §4.28 Definitions and Periodicity ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="18px" altimg-valign="-2px" altimg-width="56px" alttext="\cosh\NVar{z}" display="inline"><mrow><mi href="./4.28#E2">cosh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: hyperbolic cosine function</a>,
<a href="./4.28#E1" title="(4.28.1) ‣ §4.28 Definitions and Periodicity ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m42.png" altimg-height="18px" altimg-valign="-2px" altimg-width="53px" alttext="\sinh\NVar{z}" display="inline"><mrow><mi href="./4.28#E1">sinh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: hyperbolic sine function</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m17.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./14.1#p1.t1.r3" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./14.1#p1.t1.r3">n</mi></math>: nonnegative integer</a>,
<a href="./14.1#p1.t1.r4" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m32.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\mu" display="inline"><mi href="./14.1#p1.t1.r4">μ</mi></math>: general order</a> and
<a href="./14.19#SS4.p1" title="§14.19(iv) Sums ‣ §14.19 Toroidal (or Ring) Functions ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="21px" altimg-valign="-6px" altimg-width="50px" alttext="\xi>0" display="inline"><mrow><mi href="./14.19#SS4.p1">ξ</mi><mo>></mo><mn>0</mn></mrow></math>: variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="v" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§14.19(v) </span>Whipple’s Formula for Toroidal Functions</h2>
<div id="SS5.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="SS5.p1" class="ltx_para">
<p class="ltx_p">With <math class="ltx_Math" altimg="m44.png" altimg-height="21px" altimg-valign="-6px" altimg-width="50px" alttext="\xi>0" display="inline"><mrow><mi href="./14.19#SS5.p1">ξ</mi><mo>></mo><mn>0</mn></mrow></math>,
</p>
<table id="E7" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">14.19.7</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E7.png" altimg-height="62px" altimg-valign="-24px" altimg-width="541px" alttext="P^{m}_{n-\frac{1}{2}}\left(\cosh\xi\right)=\frac{\Gamma\left(n+m+\tfrac{1}{2}%
\right)}{\Gamma\left(n-m+\tfrac{1}{2}\right)}\*\left(\frac{2}{\pi\sinh\xi}%
\right)^{1/2}\boldsymbol{Q}^{n}_{m-\frac{1}{2}}\left(\coth\xi\right)," display="block"><mrow><mrow><mrow><msubsup><mi href="./14.21#SS1.p1">P</mi><mrow><mi href="./14.1#p1.t1.r3">n</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mi href="./14.1#p1.t1.r3">m</mi></msubsup><mo></mo><mrow><mo>(</mo><mrow><mi href="./4.28#E2">cosh</mi><mo></mo><mi href="./14.19#SS5.p1">ξ</mi></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mfrac><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./14.1#p1.t1.r3">n</mi><mo>+</mo><mi href="./14.1#p1.t1.r3">m</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mi href="./14.1#p1.t1.r3">n</mi><mo>-</mo><mi href="./14.1#p1.t1.r3">m</mi></mrow><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow></mfrac><mo></mo><msup><mrow><mo>(</mo><mfrac><mn>2</mn><mrow><mi href="./3.12#E1">π</mi><mo></mo><mrow><mi href="./4.28#E1">sinh</mi><mo></mo><mi href="./14.19#SS5.p1">ξ</mi></mrow></mrow></mfrac><mo>)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo></mo><mrow><msubsup><mi href="./14.21#SS1.p1" mathvariant="bold-italic">Q</mi><mrow><mi href="./14.1#p1.t1.r3">m</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mi href="./14.1#p1.t1.r3">n</mi></msubsup><mo></mo><mrow><mo>(</mo><mrow><mi href="./4.28#E7">coth</mi><mo></mo><mi href="./14.19#SS5.p1">ξ</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E7.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m16.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./14.21#SS1.p1" title="§14.21(i) Associated Legendre Equation ‣ §14.21 Definitions and Basic Properties ‣ Complex Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m14.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msubsup><mi href="./14.21#SS1.p1">P</mi><mi class="ltx_nvar" href="./14.1#p1.t1.r4">ν</mi><mi class="ltx_nvar" href="./14.1#p1.t1.r4">μ</mi></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./14.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: associated Legendre function of the first kind</a>,
<a href="./14.21#SS1.p1" title="§14.21(i) Associated Legendre Equation ‣ §14.21 Definitions and Basic Properties ‣ Complex Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="23px" altimg-valign="-7px" altimg-width="61px" alttext="\boldsymbol{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msubsup><mi href="./14.21#SS1.p1" mathvariant="bold-italic">Q</mi><mi class="ltx_nvar" href="./14.1#p1.t1.r4">ν</mi><mi class="ltx_nvar" href="./14.1#p1.t1.r4">μ</mi></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./14.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: Olver’s associated Legendre function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m40.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.28#E2" title="(4.28.2) ‣ §4.28 Definitions and Periodicity ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="18px" altimg-valign="-2px" altimg-width="56px" alttext="\cosh\NVar{z}" display="inline"><mrow><mi href="./4.28#E2">cosh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: hyperbolic cosine function</a>,
<a href="./4.28#E7" title="(4.28.7) ‣ §4.28 Definitions and Periodicity ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m23.png" altimg-height="18px" altimg-valign="-2px" altimg-width="55px" alttext="\coth\NVar{z}" display="inline"><mrow><mi href="./4.28#E7">coth</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: hyperbolic cotangent function</a>,
<a href="./4.28#E1" title="(4.28.1) ‣ §4.28 Definitions and Periodicity ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m42.png" altimg-height="18px" altimg-valign="-2px" altimg-width="53px" alttext="\sinh\NVar{z}" display="inline"><mrow><mi href="./4.28#E1">sinh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: hyperbolic sine function</a>,
<a href="./14.1#p1.t1.r3" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./14.1#p1.t1.r3">m</mi></math>: nonnegative integer</a>,
<a href="./14.1#p1.t1.r3" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./14.1#p1.t1.r3">n</mi></math>: nonnegative integer</a> and
<a href="./14.19#SS5.p1" title="§14.19(v) Whipple’s Formula for Toroidal Functions ‣ §14.19 Toroidal (or Ring) Functions ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="21px" altimg-valign="-6px" altimg-width="50px" alttext="\xi>0" display="inline"><mrow><mi href="./14.19#SS5.p1">ξ</mi><mo>></mo><mn>0</mn></mrow></math>: variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E8" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">14.19.8</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E8.png" altimg-height="62px" altimg-valign="-24px" altimg-width="539px" alttext="\boldsymbol{Q}^{m}_{n-\frac{1}{2}}\left(\cosh\xi\right)=\frac{\Gamma\left(m-n+%
\tfrac{1}{2}\right)}{\Gamma\left(m+n+\tfrac{1}{2}\right)}\*\left(\frac{\pi}{2%
\sinh\xi}\right)^{1/2}P^{n}_{m-\frac{1}{2}}\left(\coth\xi\right)." display="block"><mrow><mrow><mrow><msubsup><mi href="./14.21#SS1.p1" mathvariant="bold-italic">Q</mi><mrow><mi href="./14.1#p1.t1.r3">n</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mi href="./14.1#p1.t1.r3">m</mi></msubsup><mo></mo><mrow><mo>(</mo><mrow><mi href="./4.28#E2">cosh</mi><mo></mo><mi href="./14.19#SS5.p1">ξ</mi></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mfrac><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mi href="./14.1#p1.t1.r3">m</mi><mo>-</mo><mi href="./14.1#p1.t1.r3">n</mi></mrow><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./14.1#p1.t1.r3">m</mi><mo>+</mo><mi href="./14.1#p1.t1.r3">n</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow></mfrac><mo></mo><msup><mrow><mo>(</mo><mfrac><mi href="./3.12#E1">π</mi><mrow><mn>2</mn><mo></mo><mrow><mi href="./4.28#E1">sinh</mi><mo></mo><mi href="./14.19#SS5.p1">ξ</mi></mrow></mrow></mfrac><mo>)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo></mo><mrow><msubsup><mi href="./14.21#SS1.p1">P</mi><mrow><mi href="./14.1#p1.t1.r3">m</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mi href="./14.1#p1.t1.r3">n</mi></msubsup><mo></mo><mrow><mo>(</mo><mrow><mi href="./4.28#E7">coth</mi><mo></mo><mi href="./14.19#SS5.p1">ξ</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
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<div id="E8.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m16.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./14.21#SS1.p1" title="§14.21(i) Associated Legendre Equation ‣ §14.21 Definitions and Basic Properties ‣ Complex Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m14.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msubsup><mi href="./14.21#SS1.p1">P</mi><mi class="ltx_nvar" href="./14.1#p1.t1.r4">ν</mi><mi class="ltx_nvar" href="./14.1#p1.t1.r4">μ</mi></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./14.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: associated Legendre function of the first kind</a>,
<a href="./14.21#SS1.p1" title="§14.21(i) Associated Legendre Equation ‣ §14.21 Definitions and Basic Properties ‣ Complex Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m19.png" altimg-height="23px" altimg-valign="-7px" altimg-width="61px" alttext="\boldsymbol{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msubsup><mi href="./14.21#SS1.p1" mathvariant="bold-italic">Q</mi><mi class="ltx_nvar" href="./14.1#p1.t1.r4">ν</mi><mi class="ltx_nvar" href="./14.1#p1.t1.r4">μ</mi></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./14.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: Olver’s associated Legendre function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m40.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.28#E2" title="(4.28.2) ‣ §4.28 Definitions and Periodicity ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="18px" altimg-valign="-2px" altimg-width="56px" alttext="\cosh\NVar{z}" display="inline"><mrow><mi href="./4.28#E2">cosh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: hyperbolic cosine function</a>,
<a href="./4.28#E7" title="(4.28.7) ‣ §4.28 Definitions and Periodicity ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m23.png" altimg-height="18px" altimg-valign="-2px" altimg-width="55px" alttext="\coth\NVar{z}" display="inline"><mrow><mi href="./4.28#E7">coth</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: hyperbolic cotangent function</a>,
<a href="./4.28#E1" title="(4.28.1) ‣ §4.28 Definitions and Periodicity ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m42.png" altimg-height="18px" altimg-valign="-2px" altimg-width="53px" alttext="\sinh\NVar{z}" display="inline"><mrow><mi href="./4.28#E1">sinh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: hyperbolic sine function</a>,
<a href="./14.1#p1.t1.r3" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./14.1#p1.t1.r3">m</mi></math>: nonnegative integer</a>,
<a href="./14.1#p1.t1.r3" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./14.1#p1.t1.r3">n</mi></math>: nonnegative integer</a> and
<a href="./14.19#SS5.p1" title="§14.19(v) Whipple’s Formula for Toroidal Functions ‣ §14.19 Toroidal (or Ring) Functions ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="21px" altimg-valign="-6px" altimg-width="50px" alttext="\xi>0" display="inline"><mrow><mi href="./14.19#SS5.p1">ξ</mi><mo>></mo><mn>0</mn></mrow></math>: variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
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<title>DLMF: 28.8 Asymptotic Expansions for Large q</title>
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<span class="ltx_tag ltx_tag_section">§28.8 </span>Asymptotic Expansions for Large <math class="ltx_Math" altimg="m76.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="q" display="inline"><mi href="./28.1#p2.t1.r6">q</mi></math>
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<div id="SS1.p1" class="ltx_para">
<p class="ltx_p">Denote <math class="ltx_Math" altimg="m70.png" altimg-height="24px" altimg-valign="-9px" altimg-width="69px" alttext="h=\sqrt{q}" display="inline"><mrow><mi href="./28.1#p2.t1.r6">h</mi><mo>=</mo><msqrt><mi href="./28.1#p2.t1.r6">q</mi></msqrt></mrow></math> and <math class="ltx_Math" altimg="m78.png" altimg-height="18px" altimg-valign="-4px" altimg-width="102px" alttext="s=2m+1" display="inline"><mrow><mi>s</mi><mo>=</mo><mrow><mrow><mn>2</mn><mo></mo><mi href="./28.1#p2.t1.r1">m</mi></mrow><mo>+</mo><mn>1</mn></mrow></mrow></math>. Then as <math class="ltx_Math" altimg="m72.png" altimg-height="19px" altimg-valign="-4px" altimg-width="82px" alttext="h\to+\infty" display="inline"><mrow><mi href="./28.1#p2.t1.r6">h</mi><mo>→</mo><mrow><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></mrow></math> with <math class="ltx_Math" altimg="m73.png" altimg-height="20px" altimg-valign="-6px" altimg-width="128px" alttext="m=0,1,2,\dots" display="inline"><mrow><mi href="./28.1#p2.t1.r1">m</mi><mo>=</mo><mrow><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>,</p>
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<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">28.8.1</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_Math ltx_eqn_cell ltx_align_center"><math class="ltx_Math" alttext="\rselection{a_{m}\left(h^{2}\right)\\
b_{m+1}\left(h^{2}\right)}\sim-2h^{2}+2sh-\frac{1}{8}(s^{2}+1)-\frac{1}{2^{7}h%
}(s^{3}+3s)-\frac{1}{2^{12}h^{2}}(5s^{4}+34s^{2}+9)-\frac{1}{2^{17}h^{3}}(33s^%
{5}+410s^{3}+405s)-\frac{1}{2^{20}h^{4}}(63s^{6}+1260s^{4}+2943s^{2}+486)-%
\frac{1}{2^{25}h^{5}}(527s^{7}+15617s^{5}+69001s^{3}+41607s)+\cdots." display="block"><mrow><mrow><mtable displaystyle="true" rowspacing="0pt"><mtr><mtd columnalign="center"><mrow><msub><mi href="./28.2#SS5.p1">a</mi><mi href="./28.1#p2.t1.r1">m</mi></msub><mo></mo><mrow><mo>(</mo><msup><mi href="./28.1#p2.t1.r6">h</mi><mn>2</mn></msup><mo>)</mo></mrow></mrow></mtd></mtr><mtr><mtd columnalign="center"><mrow><msub><mi href="./28.2#SS5.p1">b</mi><mrow><mi href="./28.1#p2.t1.r1">m</mi><mo>+</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><msup><mi href="./28.1#p2.t1.r6">h</mi><mn>2</mn></msup><mo>)</mo></mrow></mrow></mtd></mtr></mtable><mo>}</mo></mrow><mo href="./2.1#SS3.p1">∼</mo><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mrow><mrow><mo>-</mo><mrow><mn>2</mn><mo></mo><msup><mi href="./28.1#p2.t1.r6">h</mi><mn>2</mn></msup></mrow></mrow><mo>+</mo><mrow><mn>2</mn><mo></mo><mi>s</mi><mo></mo><mi href="./28.1#p2.t1.r6">h</mi></mrow></mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>8</mn></mfrac><mo></mo><mrow><mo stretchy="false">(</mo><mrow><msup><mi>s</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mrow><msup><mn>2</mn><mn>7</mn></msup><mo></mo><mi href="./28.1#p2.t1.r6">h</mi></mrow></mfrac><mo></mo><mrow><mo stretchy="false">(</mo><mrow><msup><mi>s</mi><mn>3</mn></msup><mo>+</mo><mrow><mn>3</mn><mo></mo><mi>s</mi></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mrow><msup><mn>2</mn><mn>12</mn></msup><mo></mo><msup><mi href="./28.1#p2.t1.r6">h</mi><mn>2</mn></msup></mrow></mfrac><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>5</mn><mo></mo><msup><mi>s</mi><mn>4</mn></msup></mrow><mo>+</mo><mrow><mn>34</mn><mo></mo><msup><mi>s</mi><mn>2</mn></msup></mrow><mo>+</mo><mn>9</mn></mrow><mo stretchy="false">)</mo></mrow></mrow></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>-</mo><mrow><mfrac><mn>1</mn><mrow><msup><mn>2</mn><mn>17</mn></msup><mo></mo><msup><mi href="./28.1#p2.t1.r6">h</mi><mn>3</mn></msup></mrow></mfrac><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>33</mn><mo></mo><msup><mi>s</mi><mn>5</mn></msup></mrow><mo>+</mo><mrow><mn>410</mn><mo></mo><msup><mi>s</mi><mn>3</mn></msup></mrow><mo>+</mo><mrow><mn>405</mn><mo></mo><mi>s</mi></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mrow><msup><mn>2</mn><mn>20</mn></msup><mo></mo><msup><mi href="./28.1#p2.t1.r6">h</mi><mn>4</mn></msup></mrow></mfrac><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>63</mn><mo></mo><msup><mi>s</mi><mn>6</mn></msup></mrow><mo>+</mo><mrow><mn>1260</mn><mo></mo><msup><mi>s</mi><mn>4</mn></msup></mrow><mo>+</mo><mrow><mn>2943</mn><mo></mo><msup><mi>s</mi><mn>2</mn></msup></mrow><mo>+</mo><mn>486</mn></mrow><mo stretchy="false">)</mo></mrow></mrow></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>-</mo><mrow><mfrac><mn>1</mn><mrow><msup><mn>2</mn><mn>25</mn></msup><mo></mo><msup><mi href="./28.1#p2.t1.r6">h</mi><mn>5</mn></msup></mrow></mfrac><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>527</mn><mo></mo><msup><mi>s</mi><mn>7</mn></msup></mrow><mo>+</mo><mrow><mn>15617</mn><mo></mo><msup><mi>s</mi><mn>5</mn></msup></mrow><mo>+</mo><mrow><mn>69001</mn><mo></mo><msup><mi>s</mi><mn>3</mn></msup></mrow><mo>+</mo><mrow><mn>41607</mn><mo></mo><mi>s</mi></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></mtd></mtr></mtable></mrow></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>+</mo><mi mathvariant="normal">⋯</mi><mo>.</mo></mtd></mtr></mtable></mrow></mrow></math></td>
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<div id="E1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./28.2#SS5.p1" title="§28.2(v) Eigenvalues a n , b n ‣ §28.2 Definitions and Basic Properties ‣ Mathieu Functions of Integer Order ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m68.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="a_{\NVar{n}}\left(\NVar{q}\right)" display="inline"><mrow><msub><mi href="./28.2#SS5.p1">a</mi><mi class="ltx_nvar" href="./28.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./28.1#p2.t1.r6">q</mi><mo>)</mo></mrow></mrow></math>: eigenvalues of Mathieu equation</a>,
<a href="./28.2#SS5.p1" title="§28.2(v) Eigenvalues a n , b n ‣ §28.2 Definitions and Basic Properties ‣ Mathieu Functions of Integer Order ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="23px" altimg-valign="-7px" altimg-width="52px" alttext="b_{\NVar{n}}\left(\NVar{q}\right)" display="inline"><mrow><msub><mi href="./28.2#SS5.p1">b</mi><mi class="ltx_nvar" href="./28.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./28.1#p2.t1.r6">q</mi><mo>)</mo></mrow></mrow></math>: eigenvalues of Mathieu equation</a>,
<a href="./2.1#SS3.p1" title="§2.1(iii) Asymptotic Expansions ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="12px" altimg-valign="-2px" altimg-width="20px" alttext="\sim" display="inline"><mo href="./2.1#SS3.p1">∼</mo></math>: Poincaré asymptotic expansion</a>,
<a href="./28.1#p2.t1.r1" title="§28.1 Special Notation ‣ Notation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./28.1#p2.t1.r1">m</mi></math>: integer</a> and
<a href="./28.1#p2.t1.r6" title="§28.1 Special Notation ‣ Notation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m71.png" altimg-height="18px" altimg-valign="-2px" altimg-width="16px" alttext="h" display="inline"><mi href="./28.1#p2.t1.r6">h</mi></math>: parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">20.2.30</span> (in different form)</span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">For error estimates see <cite class="ltx_cite ltx_citemacro_citet">Kurz (. Also,</p>
<table id="E2" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">28.8.2</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E2.png" altimg-height="60px" altimg-valign="-21px" altimg-width="772px" alttext="b_{m+1}\left(h^{2}\right)-a_{m}\left(h^{2}\right)=\frac{2^{4m+5}}{m!}\left(%
\frac{2}{\pi}\right)^{\ifrac{1}{2}}h^{m+(\ifrac{3}{2})}e^{-4h}\*{\left(1-\frac%
{6m^{2}+14m+7}{32h}+O\left(\frac{1}{h^{2}}\right)\right)}." display="block"><mrow><mrow><mrow><mrow><msub><mi href="./28.2#SS5.p1">b</mi><mrow><mi href="./28.1#p2.t1.r1">m</mi><mo>+</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><msup><mi href="./28.1#p2.t1.r6">h</mi><mn>2</mn></msup><mo>)</mo></mrow></mrow><mo>-</mo><mrow><msub><mi href="./28.2#SS5.p1">a</mi><mi href="./28.1#p2.t1.r1">m</mi></msub><mo></mo><mrow><mo>(</mo><msup><mi href="./28.1#p2.t1.r6">h</mi><mn>2</mn></msup><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mrow><mfrac><msup><mn>2</mn><mrow><mrow><mn>4</mn><mo></mo><mi href="./28.1#p2.t1.r1">m</mi></mrow><mo>+</mo><mn>5</mn></mrow></msup><mrow><mi href="./28.1#p2.t1.r1">m</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mfrac><mo></mo><msup><mrow><mo>(</mo><mfrac><mn>2</mn><mi href="./3.12#E1">π</mi></mfrac><mo>)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo></mo><msup><mi href="./28.1#p2.t1.r6">h</mi><mrow><mi href="./28.1#p2.t1.r1">m</mi><mo>+</mo><mrow><mo stretchy="false">(</mo><mrow><mn>3</mn><mo>/</mo><mn>2</mn></mrow><mo stretchy="false">)</mo></mrow></mrow></msup><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mn>4</mn><mo></mo><mi href="./28.1#p2.t1.r6">h</mi></mrow></mrow></msup><mo></mo><mrow><mo>(</mo><mrow><mrow><mn>1</mn><mo>-</mo><mfrac><mrow><mrow><mn>6</mn><mo></mo><msup><mi href="./28.1#p2.t1.r1">m</mi><mn>2</mn></msup></mrow><mo>+</mo><mrow><mn>14</mn><mo></mo><mi href="./28.1#p2.t1.r1">m</mi></mrow><mo>+</mo><mn>7</mn></mrow><mrow><mn>32</mn><mo></mo><mi href="./28.1#p2.t1.r6">h</mi></mrow></mfrac></mrow><mo>+</mo><mrow><mi href="./2.1#E3">O</mi><mo></mo><mrow><mo>(</mo><mfrac><mn>1</mn><msup><mi href="./28.1#p2.t1.r6">h</mi><mn>2</mn></msup></mfrac><mo>)</mo></mrow></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./2.1#E3" title="(2.1.3) ‣ §2.1(i) Asymptotic and Order Symbols ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m35.png" altimg-height="23px" altimg-valign="-7px" altimg-width="50px" alttext="O\left(\NVar{x}\right)" display="inline"><mrow><mi href="./2.1#E3">O</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>)</mo></mrow></mrow></math>: order not exceeding</a>,
<a href="./28.2#SS5.p1" title="§28.2(v) Eigenvalues a n , b n ‣ §28.2 Definitions and Basic Properties ‣ Mathieu Functions of Integer Order ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m68.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="a_{\NVar{n}}\left(\NVar{q}\right)" display="inline"><mrow><msub><mi href="./28.2#SS5.p1">a</mi><mi class="ltx_nvar" href="./28.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./28.1#p2.t1.r6">q</mi><mo>)</mo></mrow></mrow></math>: eigenvalues of Mathieu equation</a>,
<a href="./28.2#SS5.p1" title="§28.2(v) Eigenvalues a n , b n ‣ §28.2 Definitions and Basic Properties ‣ Mathieu Functions of Integer Order ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="23px" altimg-valign="-7px" altimg-width="52px" alttext="b_{\NVar{n}}\left(\NVar{q}\right)" display="inline"><mrow><msub><mi href="./28.2#SS5.p1">b</mi><mi class="ltx_nvar" href="./28.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./28.1#p2.t1.r6">q</mi><mo>)</mo></mrow></mrow></math>: eigenvalues of Mathieu equation</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m25.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m75.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./28.1#p2.t1.r1" title="§28.1 Special Notation ‣ Notation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./28.1#p2.t1.r1">m</mi></math>: integer</a> and
<a href="./28.1#p2.t1.r6" title="§28.1 Special Notation ‣ Notation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m71.png" altimg-height="18px" altimg-valign="-2px" altimg-width="16px" alttext="h" display="inline"><mi href="./28.1#p2.t1.r6">h</mi></math>: parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">20.2.31</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="ii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§28.8(ii) </span>Sips’ Expansions</h2>
<div id="SS2.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="SS2.p1" class="ltx_para">
<p class="ltx_p">Let <math class="ltx_Math" altimg="m79.png" altimg-height="28px" altimg-valign="-9px" altimg-width="157px" alttext="x=\tfrac{1}{2}\pi+\lambda h^{-\ifrac{1}{4}}" display="inline"><mrow><mi href="./28.1#p2.t1.r2">x</mi><mo>=</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo>+</mo><mrow><mi>λ</mi><mo></mo><msup><mi href="./28.1#p2.t1.r6">h</mi><mrow><mo>-</mo><mrow><mn>1</mn><mo>/</mo><mn>4</mn></mrow></mrow></msup></mrow></mrow></mrow></math>, where <math class="ltx_Math" altimg="m47.png" altimg-height="18px" altimg-valign="-2px" altimg-width="16px" alttext="\lambda" display="inline"><mi>λ</mi></math> is a real
constant such that <math class="ltx_Math" altimg="m85.png" altimg-height="26px" altimg-valign="-7px" altimg-width="93px" alttext="|\lambda|<2^{\ifrac{1}{4}}" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mi>λ</mi><mo stretchy="false">|</mo></mrow><mo><</mo><msup><mn>2</mn><mrow><mn>1</mn><mo>/</mo><mn>4</mn></mrow></msup></mrow></math>. Also let
<math class="ltx_Math" altimg="m65.png" altimg-height="26px" altimg-valign="-6px" altimg-width="123px" alttext="\xi=2\sqrt{h}\cos x" display="inline"><mrow><mi href="./28.8#SS2.p1">ξ</mi><mo>=</mo><mrow><mn>2</mn><mo></mo><msqrt><mi href="./28.1#p2.t1.r6">h</mi></msqrt><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi href="./28.1#p2.t1.r2">x</mi></mrow></mrow></mrow></math> and
<math class="ltx_Math" altimg="m33.png" altimg-height="31px" altimg-valign="-7px" altimg-width="221px" alttext="D_{m}\left(\xi\right)=e^{-\ifrac{\xi^{2}}{4}}\mathit{He}_{m}\left(\xi\right)" display="inline"><mrow><mrow><msub><mi href="./12.1#p3">D</mi><mi href="./28.1#p2.t1.r1">m</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./28.8#SS2.p1">ξ</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><msup><mi href="./28.8#SS2.p1">ξ</mi><mn>2</mn></msup><mo>/</mo><mn>4</mn></mrow></mrow></msup><mo></mo><mrow><msub><mi href="./18.3#T1.t1.r14" mathvariant="italic">He</mi><mi href="./28.1#p2.t1.r1">m</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./28.8#SS2.p1">ξ</mi><mo>)</mo></mrow></mrow></mrow></mrow></math>
(§). Then as <math class="ltx_Math" altimg="m72.png" altimg-height="19px" altimg-valign="-4px" altimg-width="82px" alttext="h\to+\infty" display="inline"><mrow><mi href="./28.1#p2.t1.r6">h</mi><mo>→</mo><mrow><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></mrow></math>
</p>
<table id="E3" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex1" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">28.8.3</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m12.png" altimg-height="30px" altimg-valign="-9px" altimg-width="101px" alttext="\displaystyle\mathrm{ce}_{m}\left(x,h^{2}\right)" display="inline"><mrow><msub><mi href="./28.2#SS6.p1">ce</mi><mi href="./28.1#p2.t1.r1">m</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./28.1#p2.t1.r2">x</mi><mo>,</mo><msup><mi href="./28.1#p2.t1.r6">h</mi><mn>2</mn></msup><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m3.png" altimg-height="30px" altimg-valign="-7px" altimg-width="214px" alttext="\displaystyle=\widehat{C}_{m}\left(U_{m}(\xi)+V_{m}(\xi)\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msub><mover accent="true"><mi>C</mi><mo>^</mo></mover><mi href="./28.1#p2.t1.r1">m</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mrow><msub><mi href="./28.8#E4">U</mi><mi href="./28.1#p2.t1.r1">m</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.8#SS2.p1">ξ</mi><mo stretchy="false">)</mo></mrow></mrow><mo>+</mo><mrow><msub><mi href="./28.8#E5">V</mi><mi href="./28.1#p2.t1.r1">m</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.8#SS2.p1">ξ</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex2" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m11.png" altimg-height="52px" altimg-valign="-16px" altimg-width="125px" alttext="\displaystyle\frac{\mathrm{se}_{m+1}\left(x,h^{2}\right)}{\sin x}" display="inline"><mstyle displaystyle="true"><mfrac><mrow><msub><mi href="./28.2#SS6.p1">se</mi><mrow><mi href="./28.1#p2.t1.r1">m</mi><mo>+</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./28.1#p2.t1.r2">x</mi><mo>,</mo><msup><mi href="./28.1#p2.t1.r6">h</mi><mn>2</mn></msup><mo>)</mo></mrow></mrow><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi href="./28.1#p2.t1.r2">x</mi></mrow></mfrac></mstyle></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m4.png" altimg-height="30px" altimg-valign="-7px" altimg-width="212px" alttext="\displaystyle=\widehat{S}_{m}\left(U_{m}(\xi)-V_{m}(\xi)\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msub><mover accent="true"><mi>S</mi><mo>^</mo></mover><mi href="./28.1#p2.t1.r1">m</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mrow><msub><mi href="./28.8#E4">U</mi><mi href="./28.1#p2.t1.r1">m</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.8#SS2.p1">ξ</mi><mo stretchy="false">)</mo></mrow></mrow><mo>-</mo><mrow><msub><mi href="./28.8#E5">V</mi><mi href="./28.1#p2.t1.r1">m</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.8#SS2.p1">ξ</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./28.2#SS6.p1" title="§28.2(vi) Eigenfunctions ‣ §28.2 Definitions and Basic Properties ‣ Mathieu Functions of Integer Order ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="23px" altimg-valign="-7px" altimg-width="80px" alttext="\mathrm{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)" display="inline"><mrow><msub><mi href="./28.2#SS6.p1">ce</mi><mi class="ltx_nvar" href="./28.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./28.1#p2.t1.r3">z</mi><mo>,</mo><mi class="ltx_nvar" href="./28.1#p2.t1.r6">q</mi><mo>)</mo></mrow></mrow></math>: Mathieu function</a>,
<a href="./28.2#SS6.p1" title="§28.2(vi) Eigenfunctions ‣ §28.2 Definitions and Basic Properties ‣ Mathieu Functions of Integer Order ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="23px" altimg-valign="-7px" altimg-width="79px" alttext="\mathrm{se}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)" display="inline"><mrow><msub><mi href="./28.2#SS6.p1">se</mi><mi class="ltx_nvar" href="./28.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./28.1#p2.t1.r3">z</mi><mo>,</mo><mi class="ltx_nvar" href="./28.1#p2.t1.r6">q</mi><mo>)</mo></mrow></mrow></math>: Mathieu function</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m61.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./28.1#p2.t1.r1" title="§28.1 Special Notation ‣ Notation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./28.1#p2.t1.r1">m</mi></math>: integer</a>,
<a href="./28.1#p2.t1.r6" title="§28.1 Special Notation ‣ Notation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m71.png" altimg-height="18px" altimg-valign="-2px" altimg-width="16px" alttext="h" display="inline"><mi href="./28.1#p2.t1.r6">h</mi></math>: parameter</a>,
<a href="./28.1#p2.t1.r2" title="§28.1 Special Notation ‣ Notation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./28.1#p2.t1.r2">x</mi></math>: real variable</a>,
<a href="./28.8#SS2.p1" title="§28.8(ii) Sips’ Expansions ‣ §28.8 Asymptotic Expansions for Large q ‣ Mathieu Functions of Integer Order ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="21px" altimg-valign="-6px" altimg-width="14px" alttext="\xi" display="inline"><mi href="./28.8#SS2.p1">ξ</mi></math>: variable</a>,
<a href="./28.8#E4" title="(28.8.4) ‣ §28.8(ii) Sips’ Expansions ‣ §28.8 Asymptotic Expansions for Large q ‣ Mathieu Functions of Integer Order ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="23px" altimg-valign="-7px" altimg-width="58px" alttext="U_{m}(\xi)" display="inline"><mrow><msub><mi href="./28.8#E4">U</mi><mi href="./28.1#p2.t1.r1">m</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.8#SS2.p1">ξ</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function</a> and
<a href="./28.8#E5" title="(28.8.5) ‣ §28.8(ii) Sips’ Expansions ‣ §28.8 Asymptotic Expansions for Large q ‣ Mathieu Functions of Integer Order ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="23px" altimg-valign="-7px" altimg-width="56px" alttext="V_{m}(\xi)" display="inline"><mrow><msub><mi href="./28.8#E5">V</mi><mi href="./28.1#p2.t1.r1">m</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.8#SS2.p1">ξ</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">20.9.15</span> (in slightly different form)</span> <span class="ltx_origref"><span class="ltx_tag">20.9.16</span> (in slightly different form)</span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where</p>
</div>
<div id="SS2.p2" class="ltx_para">
<table id="EGx1" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E4">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">28.8.4</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m7.png" altimg-height="25px" altimg-valign="-7px" altimg-width="60px" alttext="\displaystyle U_{m}(\xi)" display="inline"><mrow><msub><mi href="./28.8#E4">U</mi><mi href="./28.1#p2.t1.r1">m</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.8#SS2.p1">ξ</mi><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m17.png" altimg-height="107px" altimg-valign="-21px" altimg-width="889px" alttext="\displaystyle\sim D_{m}\left(\xi\right)-\frac{1}{2^{6}h}\left(D_{m+4}\left(\xi%
\right)-4!\dbinom{m}{4}D_{m-4}\left(\xi\right)\right)+\frac{1}{2^{13}h^{2}}%
\left(D_{m+8}\left(\xi\right)-2^{5}(m+2)D_{m+4}\left(\xi\right)+4!\,2^{5}(m-1)%
\dbinom{m}{4}D_{m-4}\left(\xi\right)+8!\genfrac{(}{)}{0.0pt}{}{m}{8}D_{m-8}%
\left(\xi\right)\right)+\cdots," display="inline"><mrow><mi></mi><mo href="./2.1#SS3.p1">∼</mo><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mrow><mrow><msub><mi href="./12.1#p3">D</mi><mi href="./28.1#p2.t1.r1">m</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./28.8#SS2.p1">ξ</mi><mo>)</mo></mrow></mrow><mo>-</mo><mrow><mstyle displaystyle="true"><mfrac><mn>1</mn><mrow><msup><mn>2</mn><mn>6</mn></msup><mo></mo><mi href="./28.1#p2.t1.r6">h</mi></mrow></mfrac></mstyle><mo></mo><mrow><mo>(</mo><mrow><mrow><msub><mi href="./12.1#p3">D</mi><mrow><mi href="./28.1#p2.t1.r1">m</mi><mo>+</mo><mn>4</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./28.8#SS2.p1">ξ</mi><mo>)</mo></mrow></mrow><mo>-</mo><mrow><mrow><mn>4</mn><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow><mo></mo><mrow><mo href="./1.2#i">(</mo><mstyle displaystyle="true"><mfrac linethickness="0pt"><mi href="./28.1#p2.t1.r1">m</mi><mn>4</mn></mfrac></mstyle><mo href="./1.2#i">)</mo></mrow><mo></mo><mrow><msub><mi href="./12.1#p3">D</mi><mrow><mi href="./28.1#p2.t1.r1">m</mi><mo>-</mo><mn>4</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./28.8#SS2.p1">ξ</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>+</mo><mrow><mstyle displaystyle="true"><mfrac><mn>1</mn><mrow><msup><mn>2</mn><mn>13</mn></msup><mo></mo><msup><mi href="./28.1#p2.t1.r6">h</mi><mn>2</mn></msup></mrow></mfrac></mstyle><mo></mo><mrow><mo maxsize="2.94em" minsize="2.94em">(</mo><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mrow><mrow><msub><mi href="./12.1#p3">D</mi><mrow><mi href="./28.1#p2.t1.r1">m</mi><mo>+</mo><mn>8</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./28.8#SS2.p1">ξ</mi><mo>)</mo></mrow></mrow><mo>-</mo><mrow><msup><mn>2</mn><mn>5</mn></msup><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./28.1#p2.t1.r1">m</mi><mo>+</mo><mn>2</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><msub><mi href="./12.1#p3">D</mi><mrow><mi href="./28.1#p2.t1.r1">m</mi><mo>+</mo><mn>4</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./28.8#SS2.p1">ξ</mi><mo>)</mo></mrow></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>+</mo><mrow><mrow><mn>4</mn><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow><mo></mo><msup><mn> 2</mn><mn>5</mn></msup><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./28.1#p2.t1.r1">m</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo href="./1.2#i">(</mo><mstyle displaystyle="true"><mfrac linethickness="0pt"><mi href="./28.1#p2.t1.r1">m</mi><mn>4</mn></mfrac></mstyle><mo href="./1.2#i">)</mo></mrow><mo></mo><mrow><msub><mi href="./12.1#p3">D</mi><mrow><mi href="./28.1#p2.t1.r1">m</mi><mo>-</mo><mn>4</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./28.8#SS2.p1">ξ</mi><mo>)</mo></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>+</mo><mrow><mrow><mn>8</mn><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow><mo></mo><mrow><mo href="./1.2#i">(</mo><mstyle displaystyle="true"><mfrac linethickness="0.0pt"><mi href="./28.1#p2.t1.r1">m</mi><mn>8</mn></mfrac></mstyle><mo href="./1.2#i">)</mo></mrow><mo></mo><mrow><msub><mi href="./12.1#p3">D</mi><mrow><mi href="./28.1#p2.t1.r1">m</mi><mo>-</mo><mn>8</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./28.8#SS2.p1">ξ</mi><mo>)</mo></mrow></mrow></mrow><mo maxsize="2.94em" minsize="2.94em">)</mo></mtd></mtr></mtable></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>+</mo><mi mathvariant="normal">⋯</mi><mo>,</mo></mtd></mtr></mtable></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E4.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text"><math class="ltx_Math" altimg="m38.png" altimg-height="23px" altimg-valign="-7px" altimg-width="58px" alttext="U_{m}(\xi)" display="inline"><mrow><msub><mi href="./28.8#E4">U</mi><mi href="./28.1#p2.t1.r1">m</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.8#SS2.p1">ξ</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function (locally)</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./12.1#p3" title="§12.1 Special Notation ‣ Notation ‣ Chapter 12 Parabolic Cylinder Functions" class="ltx_ref"><math class="ltx_Math" altimg="m32.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="D_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./12.1#p3">D</mi><mi class="ltx_nvar" href="./12.1#p1.t1.r4">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./12.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: parabolic cylinder function</a>,
<a href="./2.1#SS3.p1" title="§2.1(iii) Asymptotic Expansions ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="12px" altimg-valign="-2px" altimg-width="20px" alttext="\sim" display="inline"><mo href="./2.1#SS3.p1">∼</mo></math>: Poincaré asymptotic expansion</a>,
<a href="./1.2#i" title="§1.2(i) Binomial Coefficients ‣ §1.2 Elementary Algebra ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m46.png" altimg-height="27px" altimg-valign="-9px" altimg-width="37px" alttext="\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}" display="inline"><mrow><mo href="./1.2#i">(</mo><mfrac linethickness="0.0pt"><mi class="ltx_nvar" href="./1.1#p2.t1.r5">m</mi><mi class="ltx_nvar" href="./1.1#p2.t1.r5">n</mi></mfrac><mo href="./1.2#i">)</mo></mrow></math>: binomial coefficient</a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m25.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m75.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./28.1#p2.t1.r1" title="§28.1 Special Notation ‣ Notation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./28.1#p2.t1.r1">m</mi></math>: integer</a>,
<a href="./28.1#p2.t1.r6" title="§28.1 Special Notation ‣ Notation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m71.png" altimg-height="18px" altimg-valign="-2px" altimg-width="16px" alttext="h" display="inline"><mi href="./28.1#p2.t1.r6">h</mi></math>: parameter</a> and
<a href="./28.8#SS2.p1" title="§28.8(ii) Sips’ Expansions ‣ §28.8 Asymptotic Expansions for Large q ‣ Mathieu Functions of Integer Order ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="21px" altimg-valign="-6px" altimg-width="14px" alttext="\xi" display="inline"><mi href="./28.8#SS2.p1">ξ</mi></math>: variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">20.9.17</span> (in different form)</span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E5">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">28.8.5</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m8.png" altimg-height="25px" altimg-valign="-7px" altimg-width="58px" alttext="\displaystyle V_{m}(\xi)" display="inline"><mrow><msub><mi href="./28.8#E5">V</mi><mi href="./28.1#p2.t1.r1">m</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.8#SS2.p1">ξ</mi><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m19.png" altimg-height="107px" altimg-valign="-21px" altimg-width="816px" alttext="\displaystyle\sim\frac{1}{2^{4}h}\bigg{(}-D_{m+2}\left(\xi\right)-m(m-1)D_{m-2%
}\left(\xi\right)\bigg{)}+\frac{1}{2^{10}h^{2}}\left(D_{m+6}\left(\xi\right)+(%
m^{2}-25m-36)D_{m+2}\left(\xi\right)-m(m-1)(m^{2}+27m-10)D_{m-2}\left(\xi%
\right)-6!\genfrac{(}{)}{0.0pt}{}{m}{6}D_{m-6}\left(\xi\right)\right)+\cdots," display="inline"><mrow><mi></mi><mo href="./2.1#SS3.p1">∼</mo><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mrow><mstyle displaystyle="true"><mfrac><mn>1</mn><mrow><msup><mn>2</mn><mn>4</mn></msup><mo></mo><mi href="./28.1#p2.t1.r6">h</mi></mrow></mfrac></mstyle><mo></mo><mrow><mo maxsize="210%" minsize="210%">(</mo><mrow><mrow><mo>-</mo><mrow><msub><mi href="./12.1#p3">D</mi><mrow><mi href="./28.1#p2.t1.r1">m</mi><mo>+</mo><mn>2</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./28.8#SS2.p1">ξ</mi><mo>)</mo></mrow></mrow></mrow><mo>-</mo><mrow><mi href="./28.1#p2.t1.r1">m</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./28.1#p2.t1.r1">m</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><msub><mi href="./12.1#p3">D</mi><mrow><mi href="./28.1#p2.t1.r1">m</mi><mo>-</mo><mn>2</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./28.8#SS2.p1">ξ</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo maxsize="210%" minsize="210%">)</mo></mrow></mrow></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>+</mo><mrow><mstyle displaystyle="true"><mfrac><mn>1</mn><mrow><msup><mn>2</mn><mn>10</mn></msup><mo></mo><msup><mi href="./28.1#p2.t1.r6">h</mi><mn>2</mn></msup></mrow></mfrac></mstyle><mo></mo><mrow><mo maxsize="2.94em" minsize="2.94em">(</mo><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mrow><mrow><msub><mi href="./12.1#p3">D</mi><mrow><mi href="./28.1#p2.t1.r1">m</mi><mo>+</mo><mn>6</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./28.8#SS2.p1">ξ</mi><mo>)</mo></mrow></mrow><mo>+</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><msup><mi href="./28.1#p2.t1.r1">m</mi><mn>2</mn></msup><mo>-</mo><mrow><mn>25</mn><mo></mo><mi href="./28.1#p2.t1.r1">m</mi></mrow><mo>-</mo><mn>36</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><msub><mi href="./12.1#p3">D</mi><mrow><mi href="./28.1#p2.t1.r1">m</mi><mo>+</mo><mn>2</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./28.8#SS2.p1">ξ</mi><mo>)</mo></mrow></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>-</mo><mrow><mi href="./28.1#p2.t1.r1">m</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./28.1#p2.t1.r1">m</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><msup><mi href="./28.1#p2.t1.r1">m</mi><mn>2</mn></msup><mo>+</mo><mrow><mn>27</mn><mo></mo><mi href="./28.1#p2.t1.r1">m</mi></mrow></mrow><mo>-</mo><mn>10</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><msub><mi href="./12.1#p3">D</mi><mrow><mi href="./28.1#p2.t1.r1">m</mi><mo>-</mo><mn>2</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./28.8#SS2.p1">ξ</mi><mo>)</mo></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>-</mo><mrow><mrow><mn>6</mn><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow><mo></mo><mrow><mo href="./1.2#i">(</mo><mstyle displaystyle="true"><mfrac linethickness="0.0pt"><mi href="./28.1#p2.t1.r1">m</mi><mn>6</mn></mfrac></mstyle><mo href="./1.2#i">)</mo></mrow><mo></mo><mrow><msub><mi href="./12.1#p3">D</mi><mrow><mi href="./28.1#p2.t1.r1">m</mi><mo>-</mo><mn>6</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./28.8#SS2.p1">ξ</mi><mo>)</mo></mrow></mrow></mrow><mo maxsize="2.94em" minsize="2.94em">)</mo></mtd></mtr></mtable></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>+</mo><mi mathvariant="normal">⋯</mi><mo>,</mo></mtd></mtr></mtable></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E5.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text"><math class="ltx_Math" altimg="m39.png" altimg-height="23px" altimg-valign="-7px" altimg-width="56px" alttext="V_{m}(\xi)" display="inline"><mrow><msub><mi href="./28.8#E5">V</mi><mi href="./28.1#p2.t1.r1">m</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.8#SS2.p1">ξ</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function (locally)</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./12.1#p3" title="§12.1 Special Notation ‣ Notation ‣ Chapter 12 Parabolic Cylinder Functions" class="ltx_ref"><math class="ltx_Math" altimg="m32.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="D_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./12.1#p3">D</mi><mi class="ltx_nvar" href="./12.1#p1.t1.r4">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./12.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: parabolic cylinder function</a>,
<a href="./2.1#SS3.p1" title="§2.1(iii) Asymptotic Expansions ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="12px" altimg-valign="-2px" altimg-width="20px" alttext="\sim" display="inline"><mo href="./2.1#SS3.p1">∼</mo></math>: Poincaré asymptotic expansion</a>,
<a href="./1.2#i" title="§1.2(i) Binomial Coefficients ‣ §1.2 Elementary Algebra ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m46.png" altimg-height="27px" altimg-valign="-9px" altimg-width="37px" alttext="\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}" display="inline"><mrow><mo href="./1.2#i">(</mo><mfrac linethickness="0.0pt"><mi class="ltx_nvar" href="./1.1#p2.t1.r5">m</mi><mi class="ltx_nvar" href="./1.1#p2.t1.r5">n</mi></mfrac><mo href="./1.2#i">)</mo></mrow></math>: binomial coefficient</a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m25.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m75.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./28.1#p2.t1.r1" title="§28.1 Special Notation ‣ Notation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./28.1#p2.t1.r1">m</mi></math>: integer</a>,
<a href="./28.1#p2.t1.r6" title="§28.1 Special Notation ‣ Notation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m71.png" altimg-height="18px" altimg-valign="-2px" altimg-width="16px" alttext="h" display="inline"><mi href="./28.1#p2.t1.r6">h</mi></math>: parameter</a> and
<a href="./28.8#SS2.p1" title="§28.8(ii) Sips’ Expansions ‣ §28.8 Asymptotic Expansions for Large q ‣ Mathieu Functions of Integer Order ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="21px" altimg-valign="-6px" altimg-width="14px" alttext="\xi" display="inline"><mi href="./28.8#SS2.p1">ξ</mi></math>: variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">20.9.18</span> (in slightly different form)</span>
</dd>
<dt>Error (effective with 1.0.16):</dt>
<dd>
Originally the <math class="ltx_Math" altimg="m28.png" altimg-height="17px" altimg-valign="-4px" altimg-width="20px" alttext="-" display="inline"><mo>-</mo></math> in front of the <math class="ltx_Math" altimg="m31.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="6!" display="inline"><mrow><mn>6</mn><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math> was given incorrectly as <math class="ltx_Math" altimg="m26.png" altimg-height="17px" altimg-valign="-4px" altimg-width="20px" alttext="+" display="inline"><mo>+</mo></math>.
<p><span class="ltx_font_italic">Suggested 2017-02-02 by Daniel Karlsson</span></p>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
<p class="ltx_p">and</p>
<table id="EGx2" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E6">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">28.8.6</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m23.png" altimg-height="28px" altimg-valign="-5px" altimg-width="35px" alttext="\displaystyle\widehat{C}_{m}" display="inline"><msub><mover accent="true"><mi>C</mi><mo>^</mo></mover><mi href="./28.1#p2.t1.r1">m</mi></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m20.png" altimg-height="60px" altimg-valign="-21px" altimg-width="690px" alttext="\displaystyle\sim\left(\frac{\pi h}{2(m!)^{2}}\right)^{\ifrac{1}{4}}\left(1+%
\frac{2m+1}{8h}+\dfrac{m^{4}+2m^{3}+263m^{2}+262m+108}{2048h^{2}}+\cdots\right%
)^{-\ifrac{1}{2}}," display="inline"><mrow><mrow><mi></mi><mo href="./2.1#SS3.p1">∼</mo><mrow><msup><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi href="./28.1#p2.t1.r6">h</mi></mrow><mrow><mn>2</mn><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./28.1#p2.t1.r1">m</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow></mfrac></mstyle><mo>)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>4</mn></mrow></msup><mo></mo><msup><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mstyle displaystyle="true"><mfrac><mrow><mrow><mn>2</mn><mo></mo><mi href="./28.1#p2.t1.r1">m</mi></mrow><mo>+</mo><mn>1</mn></mrow><mrow><mn>8</mn><mo></mo><mi href="./28.1#p2.t1.r6">h</mi></mrow></mfrac></mstyle><mo>+</mo><mstyle displaystyle="true"><mfrac><mrow><msup><mi href="./28.1#p2.t1.r1">m</mi><mn>4</mn></msup><mo>+</mo><mrow><mn>2</mn><mo></mo><msup><mi href="./28.1#p2.t1.r1">m</mi><mn>3</mn></msup></mrow><mo>+</mo><mrow><mn>263</mn><mo></mo><msup><mi href="./28.1#p2.t1.r1">m</mi><mn>2</mn></msup></mrow><mo>+</mo><mrow><mn>262</mn><mo></mo><mi href="./28.1#p2.t1.r1">m</mi></mrow><mo>+</mo><mn>108</mn></mrow><mrow><mn>2048</mn><mo></mo><msup><mi href="./28.1#p2.t1.r6">h</mi><mn>2</mn></msup></mrow></mfrac></mstyle><mo>+</mo><mi mathvariant="normal">⋯</mi></mrow><mo>)</mo></mrow><mrow><mo>-</mo><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></mrow></msup></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E6.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./2.1#SS3.p1" title="§2.1(iii) Asymptotic Expansions ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="12px" altimg-valign="-2px" altimg-width="20px" alttext="\sim" display="inline"><mo href="./2.1#SS3.p1">∼</mo></math>: Poincaré asymptotic expansion</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m25.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m75.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./28.1#p2.t1.r1" title="§28.1 Special Notation ‣ Notation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./28.1#p2.t1.r1">m</mi></math>: integer</a> and
<a href="./28.1#p2.t1.r6" title="§28.1 Special Notation ‣ Notation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m71.png" altimg-height="18px" altimg-valign="-2px" altimg-width="16px" alttext="h" display="inline"><mi href="./28.1#p2.t1.r6">h</mi></math>: parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">20.9.19</span> (in slightly different form)</span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E7">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">28.8.7</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m24.png" altimg-height="28px" altimg-valign="-5px" altimg-width="33px" alttext="\displaystyle\widehat{S}_{m}" display="inline"><msub><mover accent="true"><mi>S</mi><mo>^</mo></mover><mi href="./28.1#p2.t1.r1">m</mi></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m21.png" altimg-height="60px" altimg-valign="-21px" altimg-width="680px" alttext="\displaystyle\sim\left(\frac{\pi h}{2(m!)^{2}}\right)^{\ifrac{1}{4}}\left(1-%
\frac{2m+1}{8h}+\dfrac{m^{4}+2m^{3}-121m^{2}-122m-84}{2048h^{2}}+\cdots\right)%
^{-\ifrac{1}{2}}." display="inline"><mrow><mrow><mi></mi><mo href="./2.1#SS3.p1">∼</mo><mrow><msup><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac><mrow><mi href="./3.12#E1">π</mi><mo></mo><mi href="./28.1#p2.t1.r6">h</mi></mrow><mrow><mn>2</mn><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./28.1#p2.t1.r1">m</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow></mfrac></mstyle><mo>)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>4</mn></mrow></msup><mo></mo><msup><mrow><mo>(</mo><mrow><mrow><mn>1</mn><mo>-</mo><mstyle displaystyle="true"><mfrac><mrow><mrow><mn>2</mn><mo></mo><mi href="./28.1#p2.t1.r1">m</mi></mrow><mo>+</mo><mn>1</mn></mrow><mrow><mn>8</mn><mo></mo><mi href="./28.1#p2.t1.r6">h</mi></mrow></mfrac></mstyle></mrow><mo>+</mo><mstyle displaystyle="true"><mfrac><mrow><mrow><msup><mi href="./28.1#p2.t1.r1">m</mi><mn>4</mn></msup><mo>+</mo><mrow><mn>2</mn><mo></mo><msup><mi href="./28.1#p2.t1.r1">m</mi><mn>3</mn></msup></mrow></mrow><mo>-</mo><mrow><mn>121</mn><mo></mo><msup><mi href="./28.1#p2.t1.r1">m</mi><mn>2</mn></msup></mrow><mo>-</mo><mrow><mn>122</mn><mo></mo><mi href="./28.1#p2.t1.r1">m</mi></mrow><mo>-</mo><mn>84</mn></mrow><mrow><mn>2048</mn><mo></mo><msup><mi href="./28.1#p2.t1.r6">h</mi><mn>2</mn></msup></mrow></mfrac></mstyle><mo>+</mo><mi mathvariant="normal">⋯</mi></mrow><mo>)</mo></mrow><mrow><mo>-</mo><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></mrow></msup></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E7.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./2.1#SS3.p1" title="§2.1(iii) Asymptotic Expansions ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="12px" altimg-valign="-2px" altimg-width="20px" alttext="\sim" display="inline"><mo href="./2.1#SS3.p1">∼</mo></math>: Poincaré asymptotic expansion</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m25.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m75.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./28.1#p2.t1.r1" title="§28.1 Special Notation ‣ Notation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./28.1#p2.t1.r1">m</mi></math>: integer</a> and
<a href="./28.1#p2.t1.r6" title="§28.1 Special Notation ‣ Notation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m71.png" altimg-height="18px" altimg-valign="-2px" altimg-width="16px" alttext="h" display="inline"><mi href="./28.1#p2.t1.r6">h</mi></math>: parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">20.9.20</span> (in slightly different form)</span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
</div>
<div id="SS2.p3" class="ltx_para">
<p class="ltx_p">These results are derived formally in
<cite class="ltx_cite ltx_citemacro_citet">Sips (</dd>
</dl>
</div>
</div>
<div id="SS3.p1" class="ltx_para">
<p class="ltx_p">Let <math class="ltx_Math" altimg="m80.png" altimg-height="28px" altimg-valign="-9px" altimg-width="157px" alttext="x=\tfrac{1}{2}\pi-\mu h^{-\ifrac{1}{4}}" display="inline"><mrow><mi href="./28.1#p2.t1.r2">x</mi><mo>=</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo>-</mo><mrow><mi>μ</mi><mo></mo><msup><mi href="./28.1#p2.t1.r6">h</mi><mrow><mo>-</mo><mrow><mn>1</mn><mo>/</mo><mn>4</mn></mrow></mrow></msup></mrow></mrow></mrow></math>, where <math class="ltx_Math" altimg="m56.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\mu" display="inline"><mi>μ</mi></math> is a constant
such that <math class="ltx_Math" altimg="m57.png" altimg-height="20px" altimg-valign="-6px" altimg-width="53px" alttext="\mu\geq 1" display="inline"><mrow><mi>μ</mi><mo>≥</mo><mn>1</mn></mrow></math>, and <math class="ltx_Math" altimg="m78.png" altimg-height="18px" altimg-valign="-4px" altimg-width="102px" alttext="s=2m+1" display="inline"><mrow><mi>s</mi><mo>=</mo><mrow><mrow><mn>2</mn><mo></mo><mi href="./28.1#p2.t1.r1">m</mi></mrow><mo>+</mo><mn>1</mn></mrow></mrow></math>. Then as <math class="ltx_Math" altimg="m72.png" altimg-height="19px" altimg-valign="-4px" altimg-width="82px" alttext="h\to+\infty" display="inline"><mrow><mi href="./28.1#p2.t1.r6">h</mi><mo>→</mo><mrow><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></mrow></math>
</p>
<table id="E8" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex3" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">28.8.8</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m9.png" altimg-height="57px" altimg-valign="-21px" altimg-width="106px" alttext="\displaystyle\dfrac{\mathrm{ce}_{m}\left(x,h^{2}\right)}{\mathrm{ce}_{m}\left(%
0,h^{2}\right)}" display="inline"><mstyle displaystyle="true"><mfrac><mrow><msub><mi href="./28.2#SS6.p1">ce</mi><mi href="./28.1#p2.t1.r1">m</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./28.1#p2.t1.r2">x</mi><mo>,</mo><msup><mi href="./28.1#p2.t1.r6">h</mi><mn>2</mn></msup><mo>)</mo></mrow></mrow><mrow><msub><mi href="./28.2#SS6.p1">ce</mi><mi href="./28.1#p2.t1.r1">m</mi></msub><mo></mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><msup><mi href="./28.1#p2.t1.r6">h</mi><mn>2</mn></msup><mo>)</mo></mrow></mrow></mfrac></mstyle></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m1.png" altimg-height="53px" altimg-valign="-19px" altimg-width="594px" alttext="\displaystyle=\dfrac{2^{m-(\ifrac{1}{2})}}{\sigma_{m}}\left(W_{m}^{+}(x)(P_{m}%
(x)-Q_{m}(x))+W_{m}^{-}(x)(P_{m}(x)+Q_{m}(x))\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><mfrac><msup><mn>2</mn><mrow><mi href="./28.1#p2.t1.r1">m</mi><mo>-</mo><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow><mo stretchy="false">)</mo></mrow></mrow></msup><msub><mi href="./28.8#E10">σ</mi><mi href="./28.1#p2.t1.r1">m</mi></msub></mfrac></mstyle><mo></mo><mrow><mo>(</mo><mrow><mrow><mrow><msubsup><mi href="./28.8#E9">W</mi><mi href="./28.1#p2.t1.r1">m</mi><mo>+</mo></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.1#p2.t1.r2">x</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><msub><mi href="./28.8#E11">P</mi><mi href="./28.1#p2.t1.r1">m</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.1#p2.t1.r2">x</mi><mo stretchy="false">)</mo></mrow></mrow><mo>-</mo><mrow><msub><mi href="./28.8#E12">Q</mi><mi href="./28.1#p2.t1.r1">m</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.1#p2.t1.r2">x</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>+</mo><mrow><mrow><msubsup><mi href="./28.8#E9">W</mi><mi href="./28.1#p2.t1.r1">m</mi><mo>-</mo></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.1#p2.t1.r2">x</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><msub><mi href="./28.8#E11">P</mi><mi href="./28.1#p2.t1.r1">m</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.1#p2.t1.r2">x</mi><mo stretchy="false">)</mo></mrow></mrow><mo>+</mo><mrow><msub><mi href="./28.8#E12">Q</mi><mi href="./28.1#p2.t1.r1">m</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.1#p2.t1.r2">x</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex4" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m10.png" altimg-height="59px" altimg-valign="-23px" altimg-width="125px" alttext="\displaystyle\dfrac{\mathrm{se}_{m+1}\left(x,h^{2}\right)}{\mathrm{se}_{m+1}'%
\left(0,h^{2}\right)}" display="inline"><mstyle displaystyle="true"><mfrac><mrow><msub><mi href="./28.2#SS6.p1">se</mi><mrow><mi href="./28.1#p2.t1.r1">m</mi><mo>+</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./28.1#p2.t1.r2">x</mi><mo>,</mo><msup><mi href="./28.1#p2.t1.r6">h</mi><mn>2</mn></msup><mo>)</mo></mrow></mrow><mrow><msubsup><mi href="./28.2#SS6.p1">se</mi><mrow><mi href="./28.1#p2.t1.r1">m</mi><mo>+</mo><mn>1</mn></mrow><mo>′</mo></msubsup><mo></mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><msup><mi href="./28.1#p2.t1.r6">h</mi><mn>2</mn></msup><mo>)</mo></mrow></mrow></mfrac></mstyle></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m2.png" altimg-height="55px" altimg-valign="-20px" altimg-width="594px" alttext="\displaystyle=\dfrac{2^{m-(\ifrac{1}{2})}}{\tau_{m+1}}\left(W_{m}^{+}(x)(P_{m}%
(x)-Q_{m}(x))-W_{m}^{-}(x)(P_{m}(x)+Q_{m}(x))\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><mfrac><msup><mn>2</mn><mrow><mi href="./28.1#p2.t1.r1">m</mi><mo>-</mo><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow><mo stretchy="false">)</mo></mrow></mrow></msup><msub><mi href="./28.8#E10">τ</mi><mrow><mi href="./28.1#p2.t1.r1">m</mi><mo>+</mo><mn>1</mn></mrow></msub></mfrac></mstyle><mo></mo><mrow><mo>(</mo><mrow><mrow><mrow><msubsup><mi href="./28.8#E9">W</mi><mi href="./28.1#p2.t1.r1">m</mi><mo>+</mo></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.1#p2.t1.r2">x</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><msub><mi href="./28.8#E11">P</mi><mi href="./28.1#p2.t1.r1">m</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.1#p2.t1.r2">x</mi><mo stretchy="false">)</mo></mrow></mrow><mo>-</mo><mrow><msub><mi href="./28.8#E12">Q</mi><mi href="./28.1#p2.t1.r1">m</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.1#p2.t1.r2">x</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>-</mo><mrow><mrow><msubsup><mi href="./28.8#E9">W</mi><mi href="./28.1#p2.t1.r1">m</mi><mo>-</mo></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.1#p2.t1.r2">x</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><msub><mi href="./28.8#E11">P</mi><mi href="./28.1#p2.t1.r1">m</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.1#p2.t1.r2">x</mi><mo stretchy="false">)</mo></mrow></mrow><mo>+</mo><mrow><msub><mi href="./28.8#E12">Q</mi><mi href="./28.1#p2.t1.r1">m</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.1#p2.t1.r2">x</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E8.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./28.2#SS6.p1" title="§28.2(vi) Eigenfunctions ‣ §28.2 Definitions and Basic Properties ‣ Mathieu Functions of Integer Order ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="23px" altimg-valign="-7px" altimg-width="80px" alttext="\mathrm{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)" display="inline"><mrow><msub><mi href="./28.2#SS6.p1">ce</mi><mi class="ltx_nvar" href="./28.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./28.1#p2.t1.r3">z</mi><mo>,</mo><mi class="ltx_nvar" href="./28.1#p2.t1.r6">q</mi><mo>)</mo></mrow></mrow></math>: Mathieu function</a>,
<a href="./28.2#SS6.p1" title="§28.2(vi) Eigenfunctions ‣ §28.2 Definitions and Basic Properties ‣ Mathieu Functions of Integer Order ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="23px" altimg-valign="-7px" altimg-width="79px" alttext="\mathrm{se}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)" display="inline"><mrow><msub><mi href="./28.2#SS6.p1">se</mi><mi class="ltx_nvar" href="./28.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./28.1#p2.t1.r3">z</mi><mo>,</mo><mi class="ltx_nvar" href="./28.1#p2.t1.r6">q</mi><mo>)</mo></mrow></mrow></math>: Mathieu function</a>,
<a href="./28.1#p2.t1.r1" title="§28.1 Special Notation ‣ Notation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./28.1#p2.t1.r1">m</mi></math>: integer</a>,
<a href="./28.1#p2.t1.r6" title="§28.1 Special Notation ‣ Notation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m71.png" altimg-height="18px" altimg-valign="-2px" altimg-width="16px" alttext="h" display="inline"><mi href="./28.1#p2.t1.r6">h</mi></math>: parameter</a>,
<a href="./28.1#p2.t1.r2" title="§28.1 Special Notation ‣ Notation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./28.1#p2.t1.r2">x</mi></math>: real variable</a>,
<a href="./28.8#E12" title="(28.8.12) ‣ §28.8(iii) Goldstein’s Expansions ‣ §28.8 Asymptotic Expansions for Large q ‣ Mathieu Functions of Integer Order ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="23px" altimg-valign="-7px" altimg-width="62px" alttext="Q_{m}(x)" display="inline"><mrow><msub><mi href="./28.8#E12">Q</mi><mi href="./28.1#p2.t1.r1">m</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.1#p2.t1.r2">x</mi><mo stretchy="false">)</mo></mrow></mrow></math></a>,
<a href="./28.8#E9" title="(28.8.9) ‣ §28.8(iii) Goldstein’s Expansions ‣ §28.8 Asymptotic Expansions for Large q ‣ Mathieu Functions of Integer Order ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m41.png" altimg-height="24px" altimg-valign="-7px" altimg-width="39px" alttext="W_{m}^{\pm}" display="inline"><msubsup><mi href="./28.8#E9">W</mi><mi href="./28.1#p2.t1.r1">m</mi><mo>±</mo></msubsup></math>: function</a>,
<a href="./28.8#E10" title="(28.8.10) ‣ §28.8(iii) Goldstein’s Expansions ‣ §28.8 Asymptotic Expansions for Large q ‣ Mathieu Functions of Integer Order ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="16px" altimg-valign="-5px" altimg-width="31px" alttext="\sigma_{m}" display="inline"><msub><mi href="./28.8#E10">σ</mi><mi href="./28.1#p2.t1.r1">m</mi></msub></math></a>,
<a href="./28.8#E10" title="(28.8.10) ‣ §28.8(iii) Goldstein’s Expansions ‣ §28.8 Asymptotic Expansions for Large q ‣ Mathieu Functions of Integer Order ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m62.png" altimg-height="17px" altimg-valign="-7px" altimg-width="48px" alttext="\tau_{m+1}" display="inline"><msub><mi href="./28.8#E10">τ</mi><mrow><mi href="./28.1#p2.t1.r1">m</mi><mo>+</mo><mn>1</mn></mrow></msub></math></a> and
<a href="./28.8#E11" title="(28.8.11) ‣ §28.8(iii) Goldstein’s Expansions ‣ §28.8 Asymptotic Expansions for Large q ‣ Mathieu Functions of Integer Order ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m36.png" altimg-height="23px" altimg-valign="-7px" altimg-width="59px" alttext="P_{m}(x)" display="inline"><mrow><msub><mi href="./28.8#E11">P</mi><mi href="./28.1#p2.t1.r1">m</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.1#p2.t1.r2">x</mi><mo stretchy="false">)</mo></mrow></mrow></math></a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">20.9.11</span> (in slightly different form)</span> <span class="ltx_origref"><span class="ltx_tag">20.9.12</span> (in slightly different form)</span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where</p>
<table id="E9" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">28.8.9</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E9.png" altimg-height="65px" altimg-valign="-27px" altimg-width="425px" alttext="W_{m}^{\pm}(x)=\frac{e^{\pm 2h\sin x}}{(\cos x)^{m+1}}\begin{cases}\left(\cos%
\left(\frac{1}{2}x+\frac{1}{4}\pi\right)\right)^{2m+1},\\
\left(\sin\left(\frac{1}{2}x+\frac{1}{4}\pi\right)\right)^{2m+1},\end{cases}" display="block"><mrow><mrow><msubsup><mi href="./28.8#E9">W</mi><mi href="./28.1#p2.t1.r1">m</mi><mo>±</mo></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.1#p2.t1.r2">x</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mfrac><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>±</mo><mrow><mn>2</mn><mo></mo><mi href="./28.1#p2.t1.r6">h</mi><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi href="./28.1#p2.t1.r2">x</mi></mrow></mrow></mrow></msup><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi href="./28.1#p2.t1.r2">x</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mi href="./28.1#p2.t1.r1">m</mi><mo>+</mo><mn>1</mn></mrow></msup></mfrac><mo></mo><mrow><mo>{</mo><mtable columnspacing="5pt" displaystyle="true" rowspacing="0pt"><mtr><mtd columnalign="left"><mrow><msup><mrow><mo>(</mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./28.1#p2.t1.r2">x</mi></mrow><mo>+</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>4</mn></mfrac></mstyle><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow><mo>)</mo></mrow></mrow><mo>)</mo></mrow><mrow><mrow><mn>2</mn><mo></mo><mi href="./28.1#p2.t1.r1">m</mi></mrow><mo>+</mo><mn>1</mn></mrow></msup><mo>,</mo></mrow></mtd><mtd></mtd></mtr><mtr><mtd columnalign="left"><mrow><msup><mrow><mo>(</mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./28.1#p2.t1.r2">x</mi></mrow><mo>+</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>4</mn></mfrac></mstyle><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow><mo>)</mo></mrow></mrow><mo>)</mo></mrow><mrow><mrow><mn>2</mn><mo></mo><mi href="./28.1#p2.t1.r1">m</mi></mrow><mo>+</mo><mn>1</mn></mrow></msup><mo>,</mo></mrow></mtd><mtd></mtd></mtr></mtable></mrow></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E9.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text"><math class="ltx_Math" altimg="m41.png" altimg-height="24px" altimg-valign="-7px" altimg-width="39px" alttext="W_{m}^{\pm}" display="inline"><msubsup><mi href="./28.8#E9">W</mi><mi href="./28.1#p2.t1.r1">m</mi><mo>±</mo></msubsup></math>: function (locally)</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m61.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./28.1#p2.t1.r1" title="§28.1 Special Notation ‣ Notation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./28.1#p2.t1.r1">m</mi></math>: integer</a>,
<a href="./28.1#p2.t1.r6" title="§28.1 Special Notation ‣ Notation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m71.png" altimg-height="18px" altimg-valign="-2px" altimg-width="16px" alttext="h" display="inline"><mi href="./28.1#p2.t1.r6">h</mi></math>: parameter</a> and
<a href="./28.1#p2.t1.r2" title="§28.1 Special Notation ‣ Notation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./28.1#p2.t1.r2">x</mi></math>: real variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">20.9.13</span> (in slightly different form)</span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and</p>
<table id="E10" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex5" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">28.8.10</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m13.png" altimg-height="17px" altimg-valign="-5px" altimg-width="33px" alttext="\displaystyle\sigma_{m}" display="inline"><msub><mi href="./28.8#E10">σ</mi><mi href="./28.1#p2.t1.r1">m</mi></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m15.png" altimg-height="50px" altimg-valign="-16px" altimg-width="366px" alttext="\displaystyle\sim 1+\dfrac{s}{2^{3}h}+\dfrac{4s^{2}+3}{2^{7}h^{2}}+\dfrac{19s^%
{3}+59s}{2^{11}h^{3}}+\cdots," display="inline"><mrow><mrow><mi></mi><mo href="./2.1#SS3.p1">∼</mo><mrow><mn>1</mn><mo>+</mo><mstyle displaystyle="true"><mfrac><mi>s</mi><mrow><msup><mn>2</mn><mn>3</mn></msup><mo></mo><mi href="./28.1#p2.t1.r6">h</mi></mrow></mfrac></mstyle><mo>+</mo><mstyle displaystyle="true"><mfrac><mrow><mrow><mn>4</mn><mo></mo><msup><mi>s</mi><mn>2</mn></msup></mrow><mo>+</mo><mn>3</mn></mrow><mrow><msup><mn>2</mn><mn>7</mn></msup><mo></mo><msup><mi href="./28.1#p2.t1.r6">h</mi><mn>2</mn></msup></mrow></mfrac></mstyle><mo>+</mo><mstyle displaystyle="true"><mfrac><mrow><mrow><mn>19</mn><mo></mo><msup><mi>s</mi><mn>3</mn></msup></mrow><mo>+</mo><mrow><mn>59</mn><mo></mo><mi>s</mi></mrow></mrow><mrow><msup><mn>2</mn><mn>11</mn></msup><mo></mo><msup><mi href="./28.1#p2.t1.r6">h</mi><mn>3</mn></msup></mrow></mfrac></mstyle><mo>+</mo><mi mathvariant="normal">⋯</mi></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex6" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m22.png" altimg-height="19px" altimg-valign="-7px" altimg-width="50px" alttext="\displaystyle\tau_{m+1}" display="inline"><msub><mi href="./28.8#E10">τ</mi><mrow><mi href="./28.1#p2.t1.r1">m</mi><mo>+</mo><mn>1</mn></mrow></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m16.png" altimg-height="50px" altimg-valign="-16px" altimg-width="356px" alttext="\displaystyle\sim 2h-\dfrac{1}{4}s-\dfrac{2s^{2}+3}{2^{6}h}-\frac{7s^{3}+47s}{%
2^{10}h^{2}}-\cdots," display="inline"><mrow><mrow><mi></mi><mo href="./2.1#SS3.p1">∼</mo><mrow><mrow><mn>2</mn><mo></mo><mi href="./28.1#p2.t1.r6">h</mi></mrow><mo>-</mo><mrow><mstyle displaystyle="true"><mfrac><mn>1</mn><mn>4</mn></mfrac></mstyle><mo></mo><mi>s</mi></mrow><mo>-</mo><mstyle displaystyle="true"><mfrac><mrow><mrow><mn>2</mn><mo></mo><msup><mi>s</mi><mn>2</mn></msup></mrow><mo>+</mo><mn>3</mn></mrow><mrow><msup><mn>2</mn><mn>6</mn></msup><mo></mo><mi href="./28.1#p2.t1.r6">h</mi></mrow></mfrac></mstyle><mo>-</mo><mstyle displaystyle="true"><mfrac><mrow><mrow><mn>7</mn><mo></mo><msup><mi>s</mi><mn>3</mn></msup></mrow><mo>+</mo><mrow><mn>47</mn><mo></mo><mi>s</mi></mrow></mrow><mrow><msup><mn>2</mn><mn>10</mn></msup><mo></mo><msup><mi href="./28.1#p2.t1.r6">h</mi><mn>2</mn></msup></mrow></mfrac></mstyle><mo>-</mo><mi mathvariant="normal">⋯</mi></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E10.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd>
<span class="ltx_text"><math class="ltx_Math" altimg="m59.png" altimg-height="16px" altimg-valign="-5px" altimg-width="31px" alttext="\sigma_{m}" display="inline"><msub><mi href="./28.8#E10">σ</mi><mi href="./28.1#p2.t1.r1">m</mi></msub></math> (locally)</span> and
<span class="ltx_text"><math class="ltx_Math" altimg="m62.png" altimg-height="17px" altimg-valign="-7px" altimg-width="48px" alttext="\tau_{m+1}" display="inline"><msub><mi href="./28.8#E10">τ</mi><mrow><mi href="./28.1#p2.t1.r1">m</mi><mo>+</mo><mn>1</mn></mrow></msub></math> (locally)</span>
</dd>
<dt>Symbols:</dt>
<dd>
<a href="./2.1#SS3.p1" title="§2.1(iii) Asymptotic Expansions ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="12px" altimg-valign="-2px" altimg-width="20px" alttext="\sim" display="inline"><mo href="./2.1#SS3.p1">∼</mo></math>: Poincaré asymptotic expansion</a>,
<a href="./28.1#p2.t1.r1" title="§28.1 Special Notation ‣ Notation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./28.1#p2.t1.r1">m</mi></math>: integer</a> and
<a href="./28.1#p2.t1.r6" title="§28.1 Special Notation ‣ Notation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m71.png" altimg-height="18px" altimg-valign="-2px" altimg-width="16px" alttext="h" display="inline"><mi href="./28.1#p2.t1.r6">h</mi></math>: parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">20.9.14</span> (in different form)</span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="EGx3" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E11">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">28.8.11</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m5.png" altimg-height="25px" altimg-valign="-7px" altimg-width="61px" alttext="\displaystyle P_{m}(x)" display="inline"><mrow><msub><mi href="./28.8#E11">P</mi><mi href="./28.1#p2.t1.r1">m</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.1#p2.t1.r2">x</mi><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m14.png" altimg-height="54px" altimg-valign="-21px" altimg-width="576px" alttext="\displaystyle\sim 1+\dfrac{s}{2^{3}h{\cos^{2}}x}+\dfrac{1}{h^{2}}\left(\dfrac{%
s^{4}+86s^{2}+105}{2^{11}{\cos^{4}}x}-\dfrac{s^{4}+22s^{2}+57}{2^{11}{\cos^{2}%
}x}\right)+\cdots," display="inline"><mrow><mrow><mi></mi><mo href="./2.1#SS3.p1">∼</mo><mrow><mn>1</mn><mo>+</mo><mstyle displaystyle="true"><mfrac><mi>s</mi><mrow><msup><mn>2</mn><mn>3</mn></msup><mo></mo><mi href="./28.1#p2.t1.r6">h</mi><mo></mo><mrow><msup><mi href="./4.14#E2">cos</mi><mn>2</mn></msup><mo></mo><mi href="./28.1#p2.t1.r2">x</mi></mrow></mrow></mfrac></mstyle><mo>+</mo><mrow><mstyle displaystyle="true"><mfrac><mn>1</mn><msup><mi href="./28.1#p2.t1.r6">h</mi><mn>2</mn></msup></mfrac></mstyle><mo></mo><mrow><mo>(</mo><mrow><mstyle displaystyle="true"><mfrac><mrow><msup><mi>s</mi><mn>4</mn></msup><mo>+</mo><mrow><mn>86</mn><mo></mo><msup><mi>s</mi><mn>2</mn></msup></mrow><mo>+</mo><mn>105</mn></mrow><mrow><msup><mn>2</mn><mn>11</mn></msup><mo></mo><mrow><msup><mi href="./4.14#E2">cos</mi><mn>4</mn></msup><mo></mo><mi href="./28.1#p2.t1.r2">x</mi></mrow></mrow></mfrac></mstyle><mo>-</mo><mstyle displaystyle="true"><mfrac><mrow><msup><mi>s</mi><mn>4</mn></msup><mo>+</mo><mrow><mn>22</mn><mo></mo><msup><mi>s</mi><mn>2</mn></msup></mrow><mo>+</mo><mn>57</mn></mrow><mrow><msup><mn>2</mn><mn>11</mn></msup><mo></mo><mrow><msup><mi href="./4.14#E2">cos</mi><mn>2</mn></msup><mo></mo><mi href="./28.1#p2.t1.r2">x</mi></mrow></mrow></mfrac></mstyle></mrow><mo>)</mo></mrow></mrow><mo>+</mo><mi mathvariant="normal">⋯</mi></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E11.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text"><math class="ltx_Math" altimg="m36.png" altimg-height="23px" altimg-valign="-7px" altimg-width="59px" alttext="P_{m}(x)" display="inline"><mrow><msub><mi href="./28.8#E11">P</mi><mi href="./28.1#p2.t1.r1">m</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.1#p2.t1.r2">x</mi><mo stretchy="false">)</mo></mrow></mrow></math> (locally)</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./2.1#SS3.p1" title="§2.1(iii) Asymptotic Expansions ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="12px" altimg-valign="-2px" altimg-width="20px" alttext="\sim" display="inline"><mo href="./2.1#SS3.p1">∼</mo></math>: Poincaré asymptotic expansion</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./28.1#p2.t1.r1" title="§28.1 Special Notation ‣ Notation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./28.1#p2.t1.r1">m</mi></math>: integer</a>,
<a href="./28.1#p2.t1.r6" title="§28.1 Special Notation ‣ Notation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m71.png" altimg-height="18px" altimg-valign="-2px" altimg-width="16px" alttext="h" display="inline"><mi href="./28.1#p2.t1.r6">h</mi></math>: parameter</a> and
<a href="./28.1#p2.t1.r2" title="§28.1 Special Notation ‣ Notation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./28.1#p2.t1.r2">x</mi></math>: real variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">20.9.14</span> (in different form)</span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E12">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">28.8.12</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m6.png" altimg-height="25px" altimg-valign="-7px" altimg-width="64px" alttext="\displaystyle Q_{m}(x)" display="inline"><mrow><msub><mi href="./28.8#E12">Q</mi><mi href="./28.1#p2.t1.r1">m</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.1#p2.t1.r2">x</mi><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m18.png" altimg-height="54px" altimg-valign="-21px" altimg-width="545px" alttext="\displaystyle\sim\dfrac{\sin x}{{\cos^{2}}x}\left(\dfrac{1}{2^{5}h}(s^{2}+3)+%
\dfrac{1}{2^{9}h^{2}}\left(s^{3}+3s+\dfrac{4s^{3}+44s}{{\cos^{2}}x}\right)%
\right)+\cdots." display="inline"><mrow><mrow><mi></mi><mo href="./2.1#SS3.p1">∼</mo><mrow><mrow><mstyle displaystyle="true"><mfrac><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi href="./28.1#p2.t1.r2">x</mi></mrow><mrow><msup><mi href="./4.14#E2">cos</mi><mn>2</mn></msup><mo></mo><mi href="./28.1#p2.t1.r2">x</mi></mrow></mfrac></mstyle><mo></mo><mrow><mo>(</mo><mrow><mrow><mstyle displaystyle="true"><mfrac><mn>1</mn><mrow><msup><mn>2</mn><mn>5</mn></msup><mo></mo><mi href="./28.1#p2.t1.r6">h</mi></mrow></mfrac></mstyle><mo></mo><mrow><mo stretchy="false">(</mo><mrow><msup><mi>s</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>+</mo><mrow><mstyle displaystyle="true"><mfrac><mn>1</mn><mrow><msup><mn>2</mn><mn>9</mn></msup><mo></mo><msup><mi href="./28.1#p2.t1.r6">h</mi><mn>2</mn></msup></mrow></mfrac></mstyle><mo></mo><mrow><mo>(</mo><mrow><msup><mi>s</mi><mn>3</mn></msup><mo>+</mo><mrow><mn>3</mn><mo></mo><mi>s</mi></mrow><mo>+</mo><mstyle displaystyle="true"><mfrac><mrow><mrow><mn>4</mn><mo></mo><msup><mi>s</mi><mn>3</mn></msup></mrow><mo>+</mo><mrow><mn>44</mn><mo></mo><mi>s</mi></mrow></mrow><mrow><msup><mi href="./4.14#E2">cos</mi><mn>2</mn></msup><mo></mo><mi href="./28.1#p2.t1.r2">x</mi></mrow></mfrac></mstyle></mrow><mo>)</mo></mrow></mrow></mrow><mo>)</mo></mrow></mrow><mo>+</mo><mi mathvariant="normal">⋯</mi></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E12.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text"><math class="ltx_Math" altimg="m37.png" altimg-height="23px" altimg-valign="-7px" altimg-width="62px" alttext="Q_{m}(x)" display="inline"><mrow><msub><mi href="./28.8#E12">Q</mi><mi href="./28.1#p2.t1.r1">m</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.1#p2.t1.r2">x</mi><mo stretchy="false">)</mo></mrow></mrow></math> (locally)</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./2.1#SS3.p1" title="§2.1(iii) Asymptotic Expansions ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="12px" altimg-valign="-2px" altimg-width="20px" alttext="\sim" display="inline"><mo href="./2.1#SS3.p1">∼</mo></math>: Poincaré asymptotic expansion</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m61.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./28.1#p2.t1.r1" title="§28.1 Special Notation ‣ Notation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./28.1#p2.t1.r1">m</mi></math>: integer</a>,
<a href="./28.1#p2.t1.r6" title="§28.1 Special Notation ‣ Notation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m71.png" altimg-height="18px" altimg-valign="-2px" altimg-width="16px" alttext="h" display="inline"><mi href="./28.1#p2.t1.r6">h</mi></math>: parameter</a> and
<a href="./28.1#p2.t1.r2" title="§28.1 Special Notation ‣ Notation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./28.1#p2.t1.r2">x</mi></math>: real variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">20.9.14</span> (in different form)</span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
</div>
</section>
<section id="iv" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§28.8(iv) </span>Uniform Approximations</h2>
<div id="SS4.info" class="ltx_metadata ltx_info">
). The approximations apply
when the parameters <math class="ltx_Math" altimg="m67.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./28.1#p2.t1.r6">a</mi></math> and <math class="ltx_Math" altimg="m76.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="q" display="inline"><mi href="./28.1#p2.t1.r6">q</mi></math> are real and large, and are uniform with
respect to various regions in the <math class="ltx_Math" altimg="m83.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./28.1#p2.t1.r3">z</mi></math>-plane. The approximants are elementary
functions, Airy functions, Bessel functions, and parabolic cylinder functions;
compare §). These approximations apply when <math class="ltx_Math" altimg="m76.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="q" display="inline"><mi href="./28.1#p2.t1.r6">q</mi></math> and <math class="ltx_Math" altimg="m67.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./28.1#p2.t1.r6">a</mi></math> are real and
<math class="ltx_Math" altimg="m77.png" altimg-height="16px" altimg-valign="-6px" altimg-width="65px" alttext="q\to\infty" display="inline"><mrow><mi href="./28.1#p2.t1.r6">q</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></math>. They are uniform with respect to <math class="ltx_Math" altimg="m67.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./28.1#p2.t1.r6">a</mi></math> when
<math class="ltx_Math" altimg="m27.png" altimg-height="23px" altimg-valign="-7px" altimg-width="172px" alttext="-2q\leq a\leq(2-\delta)q" display="inline"><mrow><mrow><mo>-</mo><mrow><mn>2</mn><mo></mo><mi href="./28.1#p2.t1.r6">q</mi></mrow></mrow><mo>≤</mo><mi href="./28.1#p2.t1.r6">a</mi><mo>≤</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mn>2</mn><mo>-</mo><mi href="./28.1#p2.t1.r5">δ</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi href="./28.1#p2.t1.r6">q</mi></mrow></mrow></math>, where <math class="ltx_Math" altimg="m45.png" altimg-height="18px" altimg-valign="-2px" altimg-width="14px" alttext="\delta" display="inline"><mi href="./28.1#p2.t1.r5">δ</mi></math> is an arbitrary constant such that
<math class="ltx_Math" altimg="m29.png" altimg-height="18px" altimg-valign="-3px" altimg-width="87px" alttext="0<\delta<4" display="inline"><mrow><mn>0</mn><mo><</mo><mi href="./28.1#p2.t1.r5">δ</mi><mo><</mo><mn>4</mn></mrow></math>, and also with respect to <math class="ltx_Math" altimg="m83.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./28.1#p2.t1.r3">z</mi></math> in the semi-infinite strip given
by <math class="ltx_Math" altimg="m30.png" altimg-height="20px" altimg-valign="-5px" altimg-width="104px" alttext="0\leq\Re z\leq\pi" display="inline"><mrow><mn>0</mn><mo>≤</mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./28.1#p2.t1.r3">z</mi></mrow><mo>≤</mo><mi href="./3.12#E1">π</mi></mrow></math> and <math class="ltx_Math" altimg="m42.png" altimg-height="20px" altimg-valign="-5px" altimg-width="65px" alttext="\Im z\geq 0" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℑ</mi><mo></mo><mi href="./28.1#p2.t1.r3">z</mi></mrow><mo>≥</mo><mn>0</mn></mrow></math>.</p>
</div>
<div id="Px2.p2" class="ltx_para">
<p class="ltx_p">The approximations are expressed in terms of Whittaker functions
<math class="ltx_Math" altimg="m40.png" altimg-height="24px" altimg-valign="-8px" altimg-width="77px" alttext="W_{\kappa,\mu}\left(z\right)" display="inline"><mrow><msub><mi href="./13.14#E3">W</mi><mrow><mi>κ</mi><mo href="./13.14#E3">,</mo><mi>μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./28.1#p2.t1.r3">z</mi><mo>)</mo></mrow></mrow></math> and <math class="ltx_Math" altimg="m34.png" altimg-height="24px" altimg-valign="-8px" altimg-width="77px" alttext="M_{\kappa,\mu}\left(z\right)" display="inline"><mrow><msub><mi href="./13.14#E2">M</mi><mrow><mi>κ</mi><mo href="./13.14#E2">,</mo><mi>μ</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./28.1#p2.t1.r3">z</mi><mo>)</mo></mrow></mrow></math> with
<math class="ltx_Math" altimg="m55.png" altimg-height="27px" altimg-valign="-9px" altimg-width="56px" alttext="\mu=\tfrac{1}{4}" display="inline"><mrow><mi>μ</mi><mo>=</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></math>; compare §. They are derived by
rigorous analysis and accompanied by strict and realistic error bounds.
With additional restrictions on <math class="ltx_Math" altimg="m83.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./28.1#p2.t1.r3">z</mi></math>, uniform asymptotic approximations for
solutions of (.</p>
</div>
<div id="Px2.p3" class="ltx_para">
<p class="ltx_p">Subsequently the asymptotic solutions involving either elementary or Whittaker
functions are identified in terms of the Floquet solutions
<math class="ltx_Math" altimg="m53.png" altimg-height="23px" altimg-valign="-7px" altimg-width="87px" alttext="\mathrm{me}_{\nu}\left(z,q\right)" display="inline"><mrow><msub><mi href="./28.12#SS2.p2">me</mi><mi href="./28.1#p2.t1.r4">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./28.1#p2.t1.r3">z</mi><mo>,</mo><mi href="./28.1#p2.t1.r6">q</mi><mo>)</mo></mrow></mrow></math> (§) and modified
Mathieu functions <math class="ltx_Math" altimg="m84.png" altimg-height="29px" altimg-valign="-7px" altimg-width="93px" alttext="{\mathrm{M}^{(j)}_{\nu}}\left(z,h\right)" display="inline"><mrow><msubsup><mi href="./28.20#SS3.p1" mathvariant="normal">M</mi><mi href="./28.1#p2.t1.r4">ν</mi><mrow><mo href="./28.20#SS3.p1" stretchy="false">(</mo><mi href="./28.1#p2.t1.r1">j</mi><mo href="./28.20#SS3.p1" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi href="./28.1#p2.t1.r3">z</mi><mo>,</mo><mi href="./28.1#p2.t1.r6">h</mi><mo>)</mo></mrow></mrow></math> (§</div>
</div>
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<div class="ltx_page_navbar">
<div class="ltx_page_navlogo"></li>
<li class="ltx_tocentry"><a href="#iii"><span class="ltx_tag ltx_tag_subsection">§14.20(iii) </span>Behavior as <math class="ltx_Math" altimg="m107.png" altimg-height="17px" altimg-valign="-2px" altimg-width="57px" alttext="x\to 1" display="inline"><mrow><mi href="./14.1#p1.t1.r1">x</mi><mo>→</mo><mn>1</mn></mrow></math></a></li>
<li class="ltx_tocentry"></li>
<li class="ltx_tocentry"><a href="#vii"><span class="ltx_tag ltx_tag_subsection">§14.20(vii) </span>Asymptotic Approximations: Large <math class="ltx_Math" altimg="m83.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\tau" display="inline"><mi href="./14.1#p1.t1.r1">τ</mi></math>, Fixed
<math class="ltx_Math" altimg="m61.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\mu" display="inline"><mi href="./14.1#p1.t1.r4">μ</mi></math></a></li>
<li class="ltx_tocentry"><a href="#viii"><span class="ltx_tag ltx_tag_subsection">§14.20(viii) </span>Asymptotic Approximations: Large <math class="ltx_Math" altimg="m83.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\tau" display="inline"><mi href="./14.1#p1.t1.r1">τ</mi></math>,
<math class="ltx_Math" altimg="m11.png" altimg-height="21px" altimg-valign="-6px" altimg-width="105px" alttext="0\leq\mu\leq A\tau" display="inline"><mrow><mn>0</mn><mo>≤</mo><mi href="./14.1#p1.t1.r4">μ</mi><mo>≤</mo><mrow><mi href="./14.20#SS8.p1">A</mi><mo></mo><mi href="./14.1#p1.t1.r1">τ</mi></mrow></mrow></math></a></li>
<li class="ltx_tocentry"><a href="#ix"><span class="ltx_tag ltx_tag_subsection">§14.20(ix) </span>Asymptotic Approximations: Large <math class="ltx_Math" altimg="m61.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\mu" display="inline"><mi href="./14.1#p1.t1.r4">μ</mi></math>,
<math class="ltx_Math" altimg="m12.png" altimg-height="21px" altimg-valign="-6px" altimg-width="105px" alttext="0\leq\tau\leq A\mu" display="inline"><mrow><mn>0</mn><mo>≤</mo><mi href="./14.1#p1.t1.r1">τ</mi><mo>≤</mo><mrow><mi href="./14.20#SS8.p1">A</mi><mo></mo><mi href="./14.1#p1.t1.r4">μ</mi></mrow></mrow></math></a></li>
<li class="ltx_tocentry"> <em class="ltx_emph ltx_font_italic">we assume that</em> <math class="ltx_Math" altimg="m65.png" altimg-height="27px" altimg-valign="-9px" altimg-width="112px" alttext="\nu=-\frac{1}{2}+i\tau" display="inline"><mrow><mi href="./14.1#p1.t1.r4">ν</mi><mo>=</mo><mrow><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./14.1#p1.t1.r1">τ</mi></mrow></mrow></mrow></math>,
<em class="ltx_emph ltx_font_italic">with</em> <math class="ltx_Math" altimg="m62.png" altimg-height="20px" altimg-valign="-6px" altimg-width="53px" alttext="\mu\geq 0" display="inline"><mrow><mi href="./14.1#p1.t1.r4">μ</mi><mo>≥</mo><mn>0</mn></mrow></math> <em class="ltx_emph ltx_font_italic">and</em> <math class="ltx_Math" altimg="m84.png" altimg-height="19px" altimg-valign="-5px" altimg-width="52px" alttext="\tau\geq 0" display="inline"><mrow><mi href="./14.1#p1.t1.r1">τ</mi><mo>≥</mo><mn>0</mn></mrow></math>. () takes the form</p>
<table id="E1" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">14.20.1</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E1.png" altimg-height="54px" altimg-valign="-21px" altimg-width="455px" alttext="\left(1-x^{2}\right)\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}x}^{2}}-2x\frac{%
\mathrm{d}w}{\mathrm{d}x}-\left(\tau^{2}+\frac{1}{4}+\frac{\mu^{2}}{1-x^{2}}%
\right)w=0." display="block"><mrow><mrow><mrow><mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><msup><mi href="./14.1#p1.t1.r1">x</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow><mo></mo><mfrac><mrow><mpadded lspace="-1.7pt" width="-4.5pt"><msup><mo href="./1.4#E4">d</mo><mn>2</mn></msup></mpadded><mi>w</mi></mrow><msup><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi href="./14.1#p1.t1.r1">x</mi></mrow><mn>2</mn></msup></mfrac></mrow><mo>-</mo><mrow><mn>2</mn><mo></mo><mi href="./14.1#p1.t1.r1">x</mi><mo></mo><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi>w</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi href="./14.1#p1.t1.r1">x</mi></mrow></mfrac></mrow><mo>-</mo><mrow><mrow><mo>(</mo><mrow><msup><mi href="./14.1#p1.t1.r1">τ</mi><mn>2</mn></msup><mo>+</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mo>+</mo><mfrac><msup><mi href="./14.1#p1.t1.r4">μ</mi><mn>2</mn></msup><mrow><mn>1</mn><mo>-</mo><msup><mi href="./14.1#p1.t1.r1">x</mi><mn>2</mn></msup></mrow></mfrac></mrow><mo>)</mo></mrow><mo></mo><mi>w</mi></mrow></mrow><mo>=</mo><mn>0</mn></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#E4" title="(1.4.4) ‣ §1.4(iii) Derivatives ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="29px" altimg-valign="-9px" altimg-width="27px" alttext="\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: derivative of <math class="ltx_Math" altimg="m99.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m103.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./14.1#p1.t1.r1" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./14.1#p1.t1.r1">x</mi></math>: real variable</a>,
<a href="./14.1#p1.t1.r1" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m83.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\tau" display="inline"><mi href="./14.1#p1.t1.r1">τ</mi></math>: real variable</a> and
<a href="./14.1#p1.t1.r4" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m61.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\mu" display="inline"><mi href="./14.1#p1.t1.r4">μ</mi></math>: general order</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Solutions are known as <em class="ltx_emph ltx_font_italic">conical</em> or <em class="ltx_emph ltx_font_italic">Mehler functions</em>. For
<math class="ltx_Math" altimg="m5.png" altimg-height="18px" altimg-valign="-4px" altimg-width="104px" alttext="-1<x<1" display="inline"><mrow><mrow><mo>-</mo><mn>1</mn></mrow><mo><</mo><mi href="./14.1#p1.t1.r1">x</mi><mo><</mo><mn>1</mn></mrow></math> and <math class="ltx_Math" altimg="m82.png" altimg-height="17px" altimg-valign="-3px" altimg-width="52px" alttext="\tau>0" display="inline"><mrow><mi href="./14.1#p1.t1.r1">τ</mi><mo>></mo><mn>0</mn></mrow></math>, a numerically satisfactory pair of
real conical functions is <math class="ltx_Math" altimg="m51.png" altimg-height="35px" altimg-valign="-15px" altimg-width="99px" alttext="\mathsf{P}^{-\mu}_{-\frac{1}{2}+i\tau}\left(x\right)" display="inline"><mrow><msubsup><mi href="./14.3#E1" mathvariant="sans-serif">P</mi><mrow><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./14.1#p1.t1.r1">τ</mi></mrow></mrow><mrow><mo>-</mo><mi href="./14.1#p1.t1.r4">μ</mi></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi href="./14.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow></math> and
<math class="ltx_Math" altimg="m49.png" altimg-height="35px" altimg-valign="-15px" altimg-width="115px" alttext="\mathsf{P}^{-\mu}_{-\frac{1}{2}+i\tau}\left(-x\right)" display="inline"><mrow><msubsup><mi href="./14.3#E1" mathvariant="sans-serif">P</mi><mrow><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./14.1#p1.t1.r1">τ</mi></mrow></mrow><mrow><mo>-</mo><mi href="./14.1#p1.t1.r4">μ</mi></mrow></msubsup><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mi href="./14.1#p1.t1.r1">x</mi></mrow><mo>)</mo></mrow></mrow></math>.
</p>
</div>
<div id="SS1.p2" class="ltx_para">
<p class="ltx_p">Another real-valued solution <math class="ltx_Math" altimg="m94.png" altimg-height="36px" altimg-valign="-15px" altimg-width="100px" alttext="\widehat{\mathsf{Q}}^{-\mu}_{-\frac{1}{2}+\mathrm{i}\tau}\left(x\right)" display="inline"><mrow><msubsup><mover accent="true"><mi href="./14.20#E2" mathvariant="sans-serif">Q</mi><mo href="./14.20#E2">^</mo></mover><mrow><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./14.1#p1.t1.r1">τ</mi></mrow></mrow><mrow><mo>-</mo><mi href="./14.1#p1.t1.r4">μ</mi></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi href="./14.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow></math>
of ()</cite>.
This is defined by
</p>
<table id="E2" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">14.20.2</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E2.png" altimg-height="41px" altimg-valign="-15px" altimg-width="530px" alttext="\widehat{\mathsf{Q}}^{-\mu}_{-\frac{1}{2}+i\tau}\left(x\right)=\Re\left(e^{\mu%
\pi i}\mathsf{Q}^{-\mu}_{-\frac{1}{2}+i\tau}\left(x\right)\right)-\tfrac{1}{2}%
\pi\sin\left(\mu\pi\right)\mathsf{P}^{-\mu}_{-\frac{1}{2}+i\tau}\left(x\right)." display="block"><mrow><mrow><mrow><msubsup><mover accent="true"><mi href="./14.20#E2" mathvariant="sans-serif">Q</mi><mo href="./14.20#E2">^</mo></mover><mrow><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./14.1#p1.t1.r1">τ</mi></mrow></mrow><mrow><mo>-</mo><mi href="./14.1#p1.t1.r4">μ</mi></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi href="./14.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mrow><mo>(</mo><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mi href="./14.1#p1.t1.r4">μ</mi><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi></mrow></msup><mo></mo><mrow><msubsup><mi href="./14.3#E2" mathvariant="sans-serif">Q</mi><mrow><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./14.1#p1.t1.r1">τ</mi></mrow></mrow><mrow><mo>-</mo><mi href="./14.1#p1.t1.r4">μ</mi></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi href="./14.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow></mrow><mo>)</mo></mrow></mrow><mo>-</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./14.1#p1.t1.r4">μ</mi><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><msubsup><mi href="./14.3#E1" mathvariant="sans-serif">P</mi><mrow><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./14.1#p1.t1.r1">τ</mi></mrow></mrow><mrow><mo>-</mo><mi href="./14.1#p1.t1.r4">μ</mi></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi href="./14.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m93.png" altimg-height="36px" altimg-valign="-15px" altimg-width="101px" alttext="\widehat{\mathsf{Q}}^{-\NVar{\mu}}_{\NVar{-\frac{1}{2}+i\tau}}\left(\NVar{x}\right)" display="inline"><mrow><msubsup><mover accent="true"><mi href="./14.20#E2" mathvariant="sans-serif">Q</mi><mo href="./14.20#E2">^</mo></mover><mrow><mrow><mo class="ltx_nvar">-</mo><mfrac class="ltx_nvar"><mn class="ltx_nvar">1</mn><mn class="ltx_nvar">2</mn></mfrac></mrow><mo class="ltx_nvar">+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi class="ltx_nvar" href="./14.1#p1.t1.r1">τ</mi></mrow></mrow><mrow><mo>-</mo><mi class="ltx_nvar" href="./14.1#p1.t1.r4">μ</mi></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./14.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow></math>: conical function</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./14.3#E1" title="(14.3.1) ‣ Ferrers Function of the First Kind ‣ §14.3(i) Interval - 1 < x < 1 ‣ §14.3 Definitions and Hypergeometric Representations ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="23px" altimg-valign="-7px" altimg-width="58px" alttext="\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msubsup><mi href="./14.3#E1" mathvariant="sans-serif">P</mi><mi class="ltx_nvar" href="./14.1#p1.t1.r4">ν</mi><mi class="ltx_nvar" href="./14.1#p1.t1.r4">μ</mi></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./14.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow></math>: Ferrers function of the first kind</a>,
<a href="./14.3#E2" title="(14.3.2) ‣ Ferrers Function of the Second Kind ‣ §14.3(i) Interval - 1 < x < 1 ‣ §14.3 Definitions and Hypergeometric Representations ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="\mathsf{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msubsup><mi href="./14.3#E2" mathvariant="sans-serif">Q</mi><mi class="ltx_nvar" href="./14.1#p1.t1.r4">ν</mi><mi class="ltx_nvar" href="./14.1#p1.t1.r4">μ</mi></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./14.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow></math>: Ferrers function of the second kind</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m68.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m42.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m28.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m75.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./14.1#p1.t1.r1" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./14.1#p1.t1.r1">x</mi></math>: real variable</a>,
<a href="./14.1#p1.t1.r1" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m83.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\tau" display="inline"><mi href="./14.1#p1.t1.r1">τ</mi></math>: real variable</a> and
<a href="./14.1#p1.t1.r4" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m61.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\mu" display="inline"><mi href="./14.1#p1.t1.r4">μ</mi></math>: general order</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Equivalently,</p>
<table id="E3" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">14.20.3</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_Math ltx_eqn_cell ltx_align_center"><math class="ltx_Math" alttext="\widehat{\mathsf{Q}}^{-\mu}_{-\frac{1}{2}+\mathrm{i}\tau}\left(x\right)=\frac{%
\pi e^{-\tau\pi}\sin\left(\mu\pi\right)\sinh\left(\tau\pi\right)}{2({\cosh^{2}%
}\left(\tau\pi\right)-{\sin^{2}}\left(\mu\pi\right))}\mathsf{P}^{-\mu}_{-\frac%
{1}{2}+\mathrm{i}\tau}\left(x\right)+\frac{\pi(e^{-\tau\pi}{\cos^{2}}\left(\mu%
\pi\right)+\sinh\left(\tau\pi\right))}{2({\cosh^{2}}\left(\tau\pi\right)-{\sin%
^{2}}\left(\mu\pi\right))}\mathsf{P}^{-\mu}_{-\frac{1}{2}+\mathrm{i}\tau}\left%
(-x\right)." display="block"><mrow><mrow><msubsup><mover accent="true"><mi href="./14.20#E2" mathvariant="sans-serif">Q</mi><mo href="./14.20#E2">^</mo></mover><mrow><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./14.1#p1.t1.r1">τ</mi></mrow></mrow><mrow><mo>-</mo><mi href="./14.1#p1.t1.r4">μ</mi></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi href="./14.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mrow><mfrac><mrow><mi href="./3.12#E1">π</mi><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mi href="./14.1#p1.t1.r1">τ</mi><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow></msup><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./14.1#p1.t1.r4">μ</mi><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./4.28#E1">sinh</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./14.1#p1.t1.r1">τ</mi><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo>)</mo></mrow></mrow></mrow><mrow><mn>2</mn><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><msup><mi href="./4.28#E2">cosh</mi><mn>2</mn></msup><mo></mo><mrow><mo>(</mo><mrow><mi href="./14.1#p1.t1.r1">τ</mi><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo>)</mo></mrow></mrow><mo>-</mo><mrow><msup><mi href="./4.14#E1">sin</mi><mn>2</mn></msup><mo></mo><mrow><mo>(</mo><mrow><mi href="./14.1#p1.t1.r4">μ</mi><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></mfrac><mo></mo><mrow><msubsup><mi href="./14.3#E1" mathvariant="sans-serif">P</mi><mrow><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./14.1#p1.t1.r1">τ</mi></mrow></mrow><mrow><mo>-</mo><mi href="./14.1#p1.t1.r4">μ</mi></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi href="./14.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>+</mo><mrow><mfrac><mrow><mi href="./3.12#E1">π</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mi href="./14.1#p1.t1.r1">τ</mi><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow></msup><mo></mo><mrow><msup><mi href="./4.14#E2">cos</mi><mn>2</mn></msup><mo></mo><mrow><mo>(</mo><mrow><mi href="./14.1#p1.t1.r4">μ</mi><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mi href="./4.28#E1">sinh</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./14.1#p1.t1.r1">τ</mi><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow><mrow><mn>2</mn><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><msup><mi href="./4.28#E2">cosh</mi><mn>2</mn></msup><mo></mo><mrow><mo>(</mo><mrow><mi href="./14.1#p1.t1.r1">τ</mi><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo>)</mo></mrow></mrow><mo>-</mo><mrow><msup><mi href="./4.14#E1">sin</mi><mn>2</mn></msup><mo></mo><mrow><mo>(</mo><mrow><mi href="./14.1#p1.t1.r4">μ</mi><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></mfrac><mo></mo><mrow><msubsup><mi href="./14.3#E1" mathvariant="sans-serif">P</mi><mrow><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./14.1#p1.t1.r1">τ</mi></mrow></mrow><mrow><mo>-</mo><mi href="./14.1#p1.t1.r4">μ</mi></mrow></msubsup><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mi href="./14.1#p1.t1.r1">x</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>.</mo></mtd></mtr></mtable></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./14.3#E1" title="(14.3.1) ‣ Ferrers Function of the First Kind ‣ §14.3(i) Interval - 1 < x < 1 ‣ §14.3 Definitions and Hypergeometric Representations ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="23px" altimg-valign="-7px" altimg-width="58px" alttext="\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msubsup><mi href="./14.3#E1" mathvariant="sans-serif">P</mi><mi class="ltx_nvar" href="./14.1#p1.t1.r4">ν</mi><mi class="ltx_nvar" href="./14.1#p1.t1.r4">μ</mi></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./14.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow></math>: Ferrers function of the first kind</a>,
<a href="./14.20#E2" title="(14.20.2) ‣ §14.20(i) Definitions and Wronskians ‣ §14.20 Conical (or Mehler) Functions ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m93.png" altimg-height="36px" altimg-valign="-15px" altimg-width="101px" alttext="\widehat{\mathsf{Q}}^{-\NVar{\mu}}_{\NVar{-\frac{1}{2}+i\tau}}\left(\NVar{x}\right)" display="inline"><mrow><msubsup><mover accent="true"><mi href="./14.20#E2" mathvariant="sans-serif">Q</mi><mo href="./14.20#E2">^</mo></mover><mrow><mrow><mo class="ltx_nvar">-</mo><mfrac class="ltx_nvar"><mn class="ltx_nvar">1</mn><mn class="ltx_nvar">2</mn></mfrac></mrow><mo class="ltx_nvar">+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi class="ltx_nvar" href="./14.1#p1.t1.r1">τ</mi></mrow></mrow><mrow><mo>-</mo><mi class="ltx_nvar" href="./14.1#p1.t1.r4">μ</mi></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./14.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow></math>: conical function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m68.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m31.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m42.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./4.28#E2" title="(4.28.2) ‣ §4.28 Definitions and Periodicity ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m32.png" altimg-height="18px" altimg-valign="-2px" altimg-width="56px" alttext="\cosh\NVar{z}" display="inline"><mrow><mi href="./4.28#E2">cosh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: hyperbolic cosine function</a>,
<a href="./4.28#E1" title="(4.28.1) ‣ §4.28 Definitions and Periodicity ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m76.png" altimg-height="18px" altimg-valign="-2px" altimg-width="53px" alttext="\sinh\NVar{z}" display="inline"><mrow><mi href="./4.28#E1">sinh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: hyperbolic sine function</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m75.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./14.1#p1.t1.r1" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./14.1#p1.t1.r1">x</mi></math>: real variable</a>,
<a href="./14.1#p1.t1.r1" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m83.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\tau" display="inline"><mi href="./14.1#p1.t1.r1">τ</mi></math>: real variable</a> and
<a href="./14.1#p1.t1.r4" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m61.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\mu" display="inline"><mi href="./14.1#p1.t1.r4">μ</mi></math>: general order</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p"><math class="ltx_Math" altimg="m94.png" altimg-height="36px" altimg-valign="-15px" altimg-width="100px" alttext="\widehat{\mathsf{Q}}^{-\mu}_{-\frac{1}{2}+\mathrm{i}\tau}\left(x\right)" display="inline"><mrow><msubsup><mover accent="true"><mi href="./14.20#E2" mathvariant="sans-serif">Q</mi><mo href="./14.20#E2">^</mo></mover><mrow><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./14.1#p1.t1.r1">τ</mi></mrow></mrow><mrow><mo>-</mo><mi href="./14.1#p1.t1.r4">μ</mi></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi href="./14.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow></math> exists
except when <math class="ltx_Math" altimg="m59.png" altimg-height="27px" altimg-valign="-9px" altimg-width="109px" alttext="\mu=\frac{1}{2},\frac{3}{2},\dots" display="inline"><mrow><mi href="./14.1#p1.t1.r4">μ</mi><mo>=</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>,</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math> and
<math class="ltx_Math" altimg="m80.png" altimg-height="17px" altimg-valign="-2px" altimg-width="52px" alttext="\tau=0" display="inline"><mrow><mi href="./14.1#p1.t1.r1">τ</mi><mo>=</mo><mn>0</mn></mrow></math>; compare §. It is an important companion
solution to <math class="ltx_Math" altimg="m48.png" altimg-height="35px" altimg-valign="-15px" altimg-width="98px" alttext="\mathsf{P}^{-\mu}_{-\frac{1}{2}+\mathrm{i}\tau}\left(x\right)" display="inline"><mrow><msubsup><mi href="./14.3#E1" mathvariant="sans-serif">P</mi><mrow><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./14.1#p1.t1.r1">τ</mi></mrow></mrow><mrow><mo>-</mo><mi href="./14.1#p1.t1.r4">μ</mi></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi href="./14.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow></math> when <math class="ltx_Math" altimg="m83.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\tau" display="inline"><mi href="./14.1#p1.t1.r1">τ</mi></math> is
large; compare §§.</p>
<table id="E4" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">14.20.4</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E4.png" altimg-height="54px" altimg-valign="-24px" altimg-width="527px" alttext="\mathscr{W}\left\{\mathsf{P}^{-\mu}_{-\frac{1}{2}+\mathrm{i}\tau}\left(x\right%
),\mathsf{P}^{-\mu}_{-\frac{1}{2}+\mathrm{i}\tau}\left(-x\right)\right\}=\frac%
{2}{|\Gamma\left(\mu+\frac{1}{2}+\mathrm{i}\tau\right)|^{2}(1-x^{2})}." display="block"><mrow><mrow><mrow><mi class="ltx_font_mathscript" href="./1.13#E4">𝒲</mi><mo></mo><mrow><mo>{</mo><mrow><msubsup><mi href="./14.3#E1" mathvariant="sans-serif">P</mi><mrow><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./14.1#p1.t1.r1">τ</mi></mrow></mrow><mrow><mo>-</mo><mi href="./14.1#p1.t1.r4">μ</mi></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi href="./14.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow><mo>,</mo><mrow><msubsup><mi href="./14.3#E1" mathvariant="sans-serif">P</mi><mrow><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./14.1#p1.t1.r1">τ</mi></mrow></mrow><mrow><mo>-</mo><mi href="./14.1#p1.t1.r4">μ</mi></mrow></msubsup><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mi href="./14.1#p1.t1.r1">x</mi></mrow><mo>)</mo></mrow></mrow><mo>}</mo></mrow></mrow><mo>=</mo><mfrac><mn>2</mn><mrow><msup><mrow><mo stretchy="false">|</mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./14.1#p1.t1.r4">μ</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./14.1#p1.t1.r1">τ</mi></mrow></mrow><mo>)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow><mn>2</mn></msup><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><msup><mi href="./14.1#p1.t1.r1">x</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow></mrow></mfrac></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E4.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m27.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./14.3#E1" title="(14.3.1) ‣ Ferrers Function of the First Kind ‣ §14.3(i) Interval - 1 < x < 1 ‣ §14.3 Definitions and Hypergeometric Representations ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="23px" altimg-valign="-7px" altimg-width="58px" alttext="\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msubsup><mi href="./14.3#E1" mathvariant="sans-serif">P</mi><mi class="ltx_nvar" href="./14.1#p1.t1.r4">ν</mi><mi class="ltx_nvar" href="./14.1#p1.t1.r4">μ</mi></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./14.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow></math>: Ferrers function of the first kind</a>,
<a href="./1.13#E4" title="(1.13.4) ‣ Wronskian ‣ §1.13(i) Existence of Solutions ‣ §1.13 Differential Equations ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="18px" altimg-valign="-2px" altimg-width="26px" alttext="\mathscr{W}" display="inline"><mi class="ltx_font_mathscript" href="./1.13#E4">𝒲</mi></math>: Wronskian</a>,
<a href="./14.1#p1.t1.r1" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./14.1#p1.t1.r1">x</mi></math>: real variable</a>,
<a href="./14.1#p1.t1.r1" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m83.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\tau" display="inline"><mi href="./14.1#p1.t1.r1">τ</mi></math>: real variable</a> and
<a href="./14.1#p1.t1.r4" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m61.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\mu" display="inline"><mi href="./14.1#p1.t1.r4">μ</mi></math>: general order</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E5" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">14.20.5</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E5.png" altimg-height="58px" altimg-valign="-25px" altimg-width="718px" alttext="\mathscr{W}\left\{\mathsf{P}^{-\mu}_{-\frac{1}{2}+\mathrm{i}\tau}\left(x\right%
),\widehat{\mathsf{Q}}^{-\mu}_{-\frac{1}{2}+\mathrm{i}\tau}\left(x\right)%
\right\}=\frac{\pi(e^{-\tau\pi}{\cos^{2}}\left(\mu\pi\right)+\sinh\left(\tau%
\pi\right))}{|\Gamma\left(\mu+\frac{1}{2}+\mathrm{i}\tau\right)|^{2}({\cosh^{2%
}}\left(\tau\pi\right)-{\sin^{2}}\left(\mu\pi\right))(1-x^{2})}," display="block"><mrow><mrow><mrow><mi class="ltx_font_mathscript" href="./1.13#E4">𝒲</mi><mo></mo><mrow><mo>{</mo><mrow><msubsup><mi href="./14.3#E1" mathvariant="sans-serif">P</mi><mrow><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./14.1#p1.t1.r1">τ</mi></mrow></mrow><mrow><mo>-</mo><mi href="./14.1#p1.t1.r4">μ</mi></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi href="./14.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow><mo>,</mo><mrow><msubsup><mover accent="true"><mi href="./14.20#E2" mathvariant="sans-serif">Q</mi><mo href="./14.20#E2">^</mo></mover><mrow><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./14.1#p1.t1.r1">τ</mi></mrow></mrow><mrow><mo>-</mo><mi href="./14.1#p1.t1.r4">μ</mi></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi href="./14.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow><mo>}</mo></mrow></mrow><mo>=</mo><mfrac><mrow><mi href="./3.12#E1">π</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mi href="./14.1#p1.t1.r1">τ</mi><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow></msup><mo></mo><mrow><msup><mi href="./4.14#E2">cos</mi><mn>2</mn></msup><mo></mo><mrow><mo>(</mo><mrow><mi href="./14.1#p1.t1.r4">μ</mi><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mi href="./4.28#E1">sinh</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./14.1#p1.t1.r1">τ</mi><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow><mrow><msup><mrow><mo stretchy="false">|</mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./14.1#p1.t1.r4">μ</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./14.1#p1.t1.r1">τ</mi></mrow></mrow><mo>)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow><mn>2</mn></msup><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><msup><mi href="./4.28#E2">cosh</mi><mn>2</mn></msup><mo></mo><mrow><mo>(</mo><mrow><mi href="./14.1#p1.t1.r1">τ</mi><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo>)</mo></mrow></mrow><mo>-</mo><mrow><msup><mi href="./4.14#E1">sin</mi><mn>2</mn></msup><mo></mo><mrow><mo>(</mo><mrow><mi href="./14.1#p1.t1.r4">μ</mi><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><msup><mi href="./14.1#p1.t1.r1">x</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow></mrow></mfrac></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E5.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m27.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./14.3#E1" title="(14.3.1) ‣ Ferrers Function of the First Kind ‣ §14.3(i) Interval - 1 < x < 1 ‣ §14.3 Definitions and Hypergeometric Representations ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="23px" altimg-valign="-7px" altimg-width="58px" alttext="\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msubsup><mi href="./14.3#E1" mathvariant="sans-serif">P</mi><mi class="ltx_nvar" href="./14.1#p1.t1.r4">ν</mi><mi class="ltx_nvar" href="./14.1#p1.t1.r4">μ</mi></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./14.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow></math>: Ferrers function of the first kind</a>,
<a href="./14.20#E2" title="(14.20.2) ‣ §14.20(i) Definitions and Wronskians ‣ §14.20 Conical (or Mehler) Functions ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m93.png" altimg-height="36px" altimg-valign="-15px" altimg-width="101px" alttext="\widehat{\mathsf{Q}}^{-\NVar{\mu}}_{\NVar{-\frac{1}{2}+i\tau}}\left(\NVar{x}\right)" display="inline"><mrow><msubsup><mover accent="true"><mi href="./14.20#E2" mathvariant="sans-serif">Q</mi><mo href="./14.20#E2">^</mo></mover><mrow><mrow><mo class="ltx_nvar">-</mo><mfrac class="ltx_nvar"><mn class="ltx_nvar">1</mn><mn class="ltx_nvar">2</mn></mfrac></mrow><mo class="ltx_nvar">+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi class="ltx_nvar" href="./14.1#p1.t1.r1">τ</mi></mrow></mrow><mrow><mo>-</mo><mi class="ltx_nvar" href="./14.1#p1.t1.r4">μ</mi></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./14.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow></math>: conical function</a>,
<a href="./1.13#E4" title="(1.13.4) ‣ Wronskian ‣ §1.13(i) Existence of Solutions ‣ §1.13 Differential Equations ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="18px" altimg-valign="-2px" altimg-width="26px" alttext="\mathscr{W}" display="inline"><mi class="ltx_font_mathscript" href="./1.13#E4">𝒲</mi></math>: Wronskian</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m68.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m31.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m42.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./4.28#E2" title="(4.28.2) ‣ §4.28 Definitions and Periodicity ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m32.png" altimg-height="18px" altimg-valign="-2px" altimg-width="56px" alttext="\cosh\NVar{z}" display="inline"><mrow><mi href="./4.28#E2">cosh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: hyperbolic cosine function</a>,
<a href="./4.28#E1" title="(4.28.1) ‣ §4.28 Definitions and Periodicity ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m76.png" altimg-height="18px" altimg-valign="-2px" altimg-width="53px" alttext="\sinh\NVar{z}" display="inline"><mrow><mi href="./4.28#E1">sinh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: hyperbolic sine function</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m75.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./14.1#p1.t1.r1" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./14.1#p1.t1.r1">x</mi></math>: real variable</a>,
<a href="./14.1#p1.t1.r1" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m83.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\tau" display="inline"><mi href="./14.1#p1.t1.r1">τ</mi></math>: real variable</a> and
<a href="./14.1#p1.t1.r4" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m61.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\mu" display="inline"><mi href="./14.1#p1.t1.r4">μ</mi></math>: general order</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">provided that <math class="ltx_Math" altimg="m94.png" altimg-height="36px" altimg-valign="-15px" altimg-width="100px" alttext="\widehat{\mathsf{Q}}^{-\mu}_{-\frac{1}{2}+\mathrm{i}\tau}\left(x\right)" display="inline"><mrow><msubsup><mover accent="true"><mi href="./14.20#E2" mathvariant="sans-serif">Q</mi><mo href="./14.20#E2">^</mo></mover><mrow><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./14.1#p1.t1.r1">τ</mi></mrow></mrow><mrow><mo>-</mo><mi href="./14.1#p1.t1.r4">μ</mi></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi href="./14.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow></math> exists.</p>
</div>
<div id="SS1.p3" class="ltx_para">
<p class="ltx_p">Lastly, for the range <math class="ltx_Math" altimg="m14.png" altimg-height="17px" altimg-valign="-3px" altimg-width="99px" alttext="1<x<\infty" display="inline"><mrow><mn>1</mn><mo><</mo><mi href="./14.1#p1.t1.r1">x</mi><mo><</mo><mi mathvariant="normal">∞</mi></mrow></math>,
<math class="ltx_Math" altimg="m22.png" altimg-height="35px" altimg-valign="-15px" altimg-width="99px" alttext="P^{-\mu}_{-\frac{1}{2}+i\tau}\left(x\right)" display="inline"><mrow><msubsup><mi href="./14.21#SS1.p1">P</mi><mrow><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./14.1#p1.t1.r1">τ</mi></mrow></mrow><mrow><mo>-</mo><mi href="./14.1#p1.t1.r4">μ</mi></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi href="./14.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow></math> is a real-valued solution of
(); in terms of
<math class="ltx_Math" altimg="m26.png" altimg-height="32px" altimg-valign="-15px" altimg-width="101px" alttext="Q^{\mu}_{-\frac{1}{2}\pm\mathrm{i}\tau}\left(x\right)" display="inline"><mrow><msubsup><mi href="./14.21#SS1.p1">Q</mi><mrow><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>±</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./14.1#p1.t1.r1">τ</mi></mrow></mrow><mi href="./14.1#p1.t1.r4">μ</mi></msubsup><mo></mo><mrow><mo>(</mo><mi href="./14.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow></math> (which are complex-valued in
general):</p>
<table id="E6" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">14.20.6</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E6.png" altimg-height="61px" altimg-valign="-28px" altimg-width="620px" alttext="P^{-\mu}_{-\frac{1}{2}+i\tau}\left(x\right)=\frac{ie^{-\mu\pi i}}{\sinh\left(%
\tau\pi\right)\left|\Gamma\left(\mu+\frac{1}{2}+i\tau\right)\right|^{2}}\*%
\left(Q^{\mu}_{-\frac{1}{2}+i\tau}\left(x\right)-Q^{\mu}_{-\frac{1}{2}-i\tau}%
\left(x\right)\right)," display="block"><mrow><mrow><mrow><msubsup><mi href="./14.21#SS1.p1">P</mi><mrow><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./14.1#p1.t1.r1">τ</mi></mrow></mrow><mrow><mo>-</mo><mi href="./14.1#p1.t1.r4">μ</mi></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi href="./14.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mfrac><mrow><mi mathvariant="normal">i</mi><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mi href="./14.1#p1.t1.r4">μ</mi><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi></mrow></mrow></msup></mrow><mrow><mrow><mi href="./4.28#E1">sinh</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./14.1#p1.t1.r1">τ</mi><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><msup><mrow><mo>|</mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./14.1#p1.t1.r4">μ</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./14.1#p1.t1.r1">τ</mi></mrow></mrow><mo>)</mo></mrow></mrow><mo>|</mo></mrow><mn>2</mn></msup></mrow></mfrac><mo></mo><mrow><mo>(</mo><mrow><mrow><msubsup><mi href="./14.21#SS1.p1">Q</mi><mrow><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./14.1#p1.t1.r1">τ</mi></mrow></mrow><mi href="./14.1#p1.t1.r4">μ</mi></msubsup><mo></mo><mrow><mo>(</mo><mi href="./14.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow><mo>-</mo><mrow><msubsup><mi href="./14.21#SS1.p1">Q</mi><mrow><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>-</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./14.1#p1.t1.r1">τ</mi></mrow></mrow><mi href="./14.1#p1.t1.r4">μ</mi></msubsup><mo></mo><mrow><mo>(</mo><mi href="./14.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m86.png" altimg-height="21px" altimg-valign="-6px" altimg-width="52px" alttext="\tau\neq 0" display="inline"><mrow><mi href="./14.1#p1.t1.r1">τ</mi><mo>≠</mo><mn>0</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E6.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m27.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./14.21#SS1.p1" title="§14.21(i) Associated Legendre Equation ‣ §14.21 Definitions and Basic Properties ‣ Complex Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m23.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msubsup><mi href="./14.21#SS1.p1">P</mi><mi class="ltx_nvar" href="./14.1#p1.t1.r4">ν</mi><mi class="ltx_nvar" href="./14.1#p1.t1.r4">μ</mi></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./14.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: associated Legendre function of the first kind</a>,
<a href="./14.21#SS1.p1" title="§14.21(i) Associated Legendre Equation ‣ §14.21 Definitions and Basic Properties ‣ Complex Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m25.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="Q^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msubsup><mi href="./14.21#SS1.p1">Q</mi><mi class="ltx_nvar" href="./14.1#p1.t1.r4">ν</mi><mi class="ltx_nvar" href="./14.1#p1.t1.r4">μ</mi></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./14.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: associated Legendre function of the second kind</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m68.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m42.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./4.28#E1" title="(4.28.1) ‣ §4.28 Definitions and Periodicity ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m76.png" altimg-height="18px" altimg-valign="-2px" altimg-width="53px" alttext="\sinh\NVar{z}" display="inline"><mrow><mi href="./4.28#E1">sinh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: hyperbolic sine function</a>,
<a href="./14.1#p1.t1.r1" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./14.1#p1.t1.r1">x</mi></math>: real variable</a>,
<a href="./14.1#p1.t1.r1" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m83.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\tau" display="inline"><mi href="./14.1#p1.t1.r1">τ</mi></math>: real variable</a> and
<a href="./14.1#p1.t1.r4" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m61.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\mu" display="inline"><mi href="./14.1#p1.t1.r4">μ</mi></math>: general order</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="ii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§14.20(ii) </span>Graphics</h2>
<div id="SS2.info" class="ltx_metadata ltx_info">
<figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 14.20.1: </span><math class="ltx_Math" altimg="m52.png" altimg-height="32px" altimg-valign="-15px" altimg-width="99px" alttext="\mathsf{P}^{0}_{-\frac{1}{2}+i\tau}\left(x\right)" display="inline"><mrow><msubsup><mi href="./14.3#E1" mathvariant="sans-serif">P</mi><mrow><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./14.1#p1.t1.r1">τ</mi></mrow></mrow><mn>0</mn></msubsup><mo></mo><mrow><mo>(</mo><mi href="./14.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow></math>, <math class="ltx_Math" altimg="m78.png" altimg-height="20px" altimg-valign="-6px" altimg-width="127px" alttext="\tau=0,1,2,4,8" display="inline"><mrow><mi href="./14.1#p1.t1.r1">τ</mi><mo>=</mo><mrow><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>8</mn></mrow></mrow></math>.
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./14.3#E1" title="(14.3.1) ‣ Ferrers Function of the First Kind ‣ §14.3(i) Interval - 1 < x < 1 ‣ §14.3 Definitions and Hypergeometric Representations ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="23px" altimg-valign="-7px" altimg-width="58px" alttext="\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msubsup><mi href="./14.3#E1" mathvariant="sans-serif">P</mi><mi class="ltx_nvar" href="./14.1#p1.t1.r4">ν</mi><mi class="ltx_nvar" href="./14.1#p1.t1.r4">μ</mi></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./14.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow></math>: Ferrers function of the first kind</a>,
<a href="./14.1#p1.t1.r1" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./14.1#p1.t1.r1">x</mi></math>: real variable</a> and
<a href="./14.1#p1.t1.r1" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m83.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\tau" display="inline"><mi href="./14.1#p1.t1.r1">τ</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</figure>
</td>
<td class="ltx_subfigure">
<figure id="F2" class="ltx_figure"><figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 14.20.2: </span><math class="ltx_Math" altimg="m96.png" altimg-height="36px" altimg-valign="-15px" altimg-width="101px" alttext="\widehat{\mathsf{Q}}^{0}_{-\frac{1}{2}+i\tau}\left(x\right)" display="inline"><mrow><msubsup><mover accent="true"><mi href="./14.20#E2" mathvariant="sans-serif">Q</mi><mo href="./14.20#E2">^</mo></mover><mrow><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./14.1#p1.t1.r1">τ</mi></mrow></mrow><mn>0</mn></msubsup><mo></mo><mrow><mo>(</mo><mi href="./14.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow></math>, <math class="ltx_Math" altimg="m79.png" altimg-height="27px" altimg-valign="-9px" altimg-width="130px" alttext="\tau=0,\tfrac{1}{2},1,2,4" display="inline"><mrow><mi href="./14.1#p1.t1.r1">τ</mi><mo>=</mo><mrow><mn>0</mn><mo>,</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>4</mn></mrow></mrow></math>.
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./14.20#E2" title="(14.20.2) ‣ §14.20(i) Definitions and Wronskians ‣ §14.20 Conical (or Mehler) Functions ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m93.png" altimg-height="36px" altimg-valign="-15px" altimg-width="101px" alttext="\widehat{\mathsf{Q}}^{-\NVar{\mu}}_{\NVar{-\frac{1}{2}+i\tau}}\left(\NVar{x}\right)" display="inline"><mrow><msubsup><mover accent="true"><mi href="./14.20#E2" mathvariant="sans-serif">Q</mi><mo href="./14.20#E2">^</mo></mover><mrow><mrow><mo class="ltx_nvar">-</mo><mfrac class="ltx_nvar"><mn class="ltx_nvar">1</mn><mn class="ltx_nvar">2</mn></mfrac></mrow><mo class="ltx_nvar">+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi class="ltx_nvar" href="./14.1#p1.t1.r1">τ</mi></mrow></mrow><mrow><mo>-</mo><mi class="ltx_nvar" href="./14.1#p1.t1.r4">μ</mi></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./14.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow></math>: conical function</a>,
<a href="./14.1#p1.t1.r1" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./14.1#p1.t1.r1">x</mi></math>: real variable</a> and
<a href="./14.1#p1.t1.r1" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m83.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\tau" display="inline"><mi href="./14.1#p1.t1.r1">τ</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</figure>
</td>
</tr>
</table>
</figure>
<figure id="SS2.fig2" class="ltx_figure">
<table style="width:100%;">
<tr>
<td class="ltx_subfigure">
<figure id="F3" class="ltx_figure"><figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 14.20.3: </span><math class="ltx_Math" altimg="m44.png" altimg-height="37px" altimg-valign="-15px" altimg-width="99px" alttext="\mathsf{P}^{-1/2}_{-\frac{1}{2}+i\tau}\left(x\right)" display="inline"><mrow><msubsup><mi href="./14.3#E1" mathvariant="sans-serif">P</mi><mrow><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./14.1#p1.t1.r1">τ</mi></mrow></mrow><mrow><mo>-</mo><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi href="./14.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow></math>, <math class="ltx_Math" altimg="m78.png" altimg-height="20px" altimg-valign="-6px" altimg-width="127px" alttext="\tau=0,1,2,4,8" display="inline"><mrow><mi href="./14.1#p1.t1.r1">τ</mi><mo>=</mo><mrow><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>8</mn></mrow></mrow></math>.
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./14.3#E1" title="(14.3.1) ‣ Ferrers Function of the First Kind ‣ §14.3(i) Interval - 1 < x < 1 ‣ §14.3 Definitions and Hypergeometric Representations ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="23px" altimg-valign="-7px" altimg-width="58px" alttext="\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msubsup><mi href="./14.3#E1" mathvariant="sans-serif">P</mi><mi class="ltx_nvar" href="./14.1#p1.t1.r4">ν</mi><mi class="ltx_nvar" href="./14.1#p1.t1.r4">μ</mi></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./14.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow></math>: Ferrers function of the first kind</a>,
<a href="./14.1#p1.t1.r1" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./14.1#p1.t1.r1">x</mi></math>: real variable</a> and
<a href="./14.1#p1.t1.r1" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m83.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\tau" display="inline"><mi href="./14.1#p1.t1.r1">τ</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</figure>
</td>
<td class="ltx_subfigure">
<figure id="F4" class="ltx_figure"><figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 14.20.4: </span><math class="ltx_Math" altimg="m90.png" altimg-height="37px" altimg-valign="-15px" altimg-width="101px" alttext="\widehat{\mathsf{Q}}^{-1/2}_{-\frac{1}{2}+i\tau}\left(x\right)" display="inline"><mrow><msubsup><mover accent="true"><mi href="./14.20#E2" mathvariant="sans-serif">Q</mi><mo href="./14.20#E2">^</mo></mover><mrow><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./14.1#p1.t1.r1">τ</mi></mrow></mrow><mrow><mo>-</mo><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi href="./14.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow></math>,
<math class="ltx_Math" altimg="m81.png" altimg-height="27px" altimg-valign="-9px" altimg-width="111px" alttext="\tau=\tfrac{1}{2},1,2,4" display="inline"><mrow><mi href="./14.1#p1.t1.r1">τ</mi><mo>=</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>4</mn></mrow></mrow></math>.
(This function does not exist when <math class="ltx_Math" altimg="m80.png" altimg-height="17px" altimg-valign="-2px" altimg-width="52px" alttext="\tau=0" display="inline"><mrow><mi href="./14.1#p1.t1.r1">τ</mi><mo>=</mo><mn>0</mn></mrow></math>.)
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./14.20#E2" title="(14.20.2) ‣ §14.20(i) Definitions and Wronskians ‣ §14.20 Conical (or Mehler) Functions ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m93.png" altimg-height="36px" altimg-valign="-15px" altimg-width="101px" alttext="\widehat{\mathsf{Q}}^{-\NVar{\mu}}_{\NVar{-\frac{1}{2}+i\tau}}\left(\NVar{x}\right)" display="inline"><mrow><msubsup><mover accent="true"><mi href="./14.20#E2" mathvariant="sans-serif">Q</mi><mo href="./14.20#E2">^</mo></mover><mrow><mrow><mo class="ltx_nvar">-</mo><mfrac class="ltx_nvar"><mn class="ltx_nvar">1</mn><mn class="ltx_nvar">2</mn></mfrac></mrow><mo class="ltx_nvar">+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi class="ltx_nvar" href="./14.1#p1.t1.r1">τ</mi></mrow></mrow><mrow><mo>-</mo><mi class="ltx_nvar" href="./14.1#p1.t1.r4">μ</mi></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./14.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow></math>: conical function</a>,
<a href="./14.1#p1.t1.r1" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./14.1#p1.t1.r1">x</mi></math>: real variable</a> and
<a href="./14.1#p1.t1.r1" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m83.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\tau" display="inline"><mi href="./14.1#p1.t1.r1">τ</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</figure>
</td>
</tr>
</table>
</figure>
<div id="SS2.p1" class="ltx_para">
</div>
<figure id="SS2.fig3" class="ltx_figure">
<table style="width:100%;">
<tr>
<td class="ltx_subfigure">
<figure id="F5" class="ltx_figure"><figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 14.20.5: </span><math class="ltx_Math" altimg="m45.png" altimg-height="34px" altimg-valign="-15px" altimg-width="99px" alttext="\mathsf{P}^{-1}_{-\frac{1}{2}+i\tau}\left(x\right)" display="inline"><mrow><msubsup><mi href="./14.3#E1" mathvariant="sans-serif">P</mi><mrow><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./14.1#p1.t1.r1">τ</mi></mrow></mrow><mrow><mo>-</mo><mn>1</mn></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi href="./14.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow></math>, <math class="ltx_Math" altimg="m78.png" altimg-height="20px" altimg-valign="-6px" altimg-width="127px" alttext="\tau=0,1,2,4,8" display="inline"><mrow><mi href="./14.1#p1.t1.r1">τ</mi><mo>=</mo><mrow><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>8</mn></mrow></mrow></math>.
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./14.3#E1" title="(14.3.1) ‣ Ferrers Function of the First Kind ‣ §14.3(i) Interval - 1 < x < 1 ‣ §14.3 Definitions and Hypergeometric Representations ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="23px" altimg-valign="-7px" altimg-width="58px" alttext="\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msubsup><mi href="./14.3#E1" mathvariant="sans-serif">P</mi><mi class="ltx_nvar" href="./14.1#p1.t1.r4">ν</mi><mi class="ltx_nvar" href="./14.1#p1.t1.r4">μ</mi></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./14.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow></math>: Ferrers function of the first kind</a>,
<a href="./14.1#p1.t1.r1" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./14.1#p1.t1.r1">x</mi></math>: real variable</a> and
<a href="./14.1#p1.t1.r1" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m83.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\tau" display="inline"><mi href="./14.1#p1.t1.r1">τ</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</figure>
</td>
<td class="ltx_subfigure">
<figure id="F6" class="ltx_figure"><figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 14.20.6: </span><math class="ltx_Math" altimg="m91.png" altimg-height="36px" altimg-valign="-15px" altimg-width="101px" alttext="\widehat{\mathsf{Q}}^{-1}_{-\frac{1}{2}+i\tau}\left(x\right)" display="inline"><mrow><msubsup><mover accent="true"><mi href="./14.20#E2" mathvariant="sans-serif">Q</mi><mo href="./14.20#E2">^</mo></mover><mrow><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./14.1#p1.t1.r1">τ</mi></mrow></mrow><mrow><mo>-</mo><mn>1</mn></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi href="./14.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow></math>, <math class="ltx_Math" altimg="m79.png" altimg-height="27px" altimg-valign="-9px" altimg-width="130px" alttext="\tau=0,\tfrac{1}{2},1,2,4" display="inline"><mrow><mi href="./14.1#p1.t1.r1">τ</mi><mo>=</mo><mrow><mn>0</mn><mo>,</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>4</mn></mrow></mrow></math>.
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./14.20#E2" title="(14.20.2) ‣ §14.20(i) Definitions and Wronskians ‣ §14.20 Conical (or Mehler) Functions ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m93.png" altimg-height="36px" altimg-valign="-15px" altimg-width="101px" alttext="\widehat{\mathsf{Q}}^{-\NVar{\mu}}_{\NVar{-\frac{1}{2}+i\tau}}\left(\NVar{x}\right)" display="inline"><mrow><msubsup><mover accent="true"><mi href="./14.20#E2" mathvariant="sans-serif">Q</mi><mo href="./14.20#E2">^</mo></mover><mrow><mrow><mo class="ltx_nvar">-</mo><mfrac class="ltx_nvar"><mn class="ltx_nvar">1</mn><mn class="ltx_nvar">2</mn></mfrac></mrow><mo class="ltx_nvar">+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi class="ltx_nvar" href="./14.1#p1.t1.r1">τ</mi></mrow></mrow><mrow><mo>-</mo><mi class="ltx_nvar" href="./14.1#p1.t1.r4">μ</mi></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./14.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow></math>: conical function</a>,
<a href="./14.1#p1.t1.r1" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./14.1#p1.t1.r1">x</mi></math>: real variable</a> and
<a href="./14.1#p1.t1.r1" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m83.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\tau" display="inline"><mi href="./14.1#p1.t1.r1">τ</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</figure>
</td>
</tr>
</table>
</figure>
<figure id="SS2.fig4" class="ltx_figure">
<table style="width:100%;">
<tr>
<td class="ltx_subfigure">
<figure id="F7" class="ltx_figure"><figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 14.20.7: </span><math class="ltx_Math" altimg="m46.png" altimg-height="34px" altimg-valign="-15px" altimg-width="234px" alttext="\mathsf{P}^{-2}_{-\frac{1}{2}+i\tau}\left(x\right),\tau=0,1,2,4,8" display="inline"><mrow><mrow><mrow><mrow><msubsup><mi href="./14.3#E1" mathvariant="sans-serif">P</mi><mrow><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./14.1#p1.t1.r1">τ</mi></mrow></mrow><mrow><mo>-</mo><mn>2</mn></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi href="./14.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow><mo>,</mo><mi href="./14.1#p1.t1.r1">τ</mi></mrow><mo>=</mo><mn>0</mn></mrow><mo>,</mo><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>8</mn></mrow></mrow></math>.
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./14.3#E1" title="(14.3.1) ‣ Ferrers Function of the First Kind ‣ §14.3(i) Interval - 1 < x < 1 ‣ §14.3 Definitions and Hypergeometric Representations ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="23px" altimg-valign="-7px" altimg-width="58px" alttext="\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msubsup><mi href="./14.3#E1" mathvariant="sans-serif">P</mi><mi class="ltx_nvar" href="./14.1#p1.t1.r4">ν</mi><mi class="ltx_nvar" href="./14.1#p1.t1.r4">μ</mi></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./14.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow></math>: Ferrers function of the first kind</a>,
<a href="./14.1#p1.t1.r1" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./14.1#p1.t1.r1">x</mi></math>: real variable</a> and
<a href="./14.1#p1.t1.r1" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m83.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\tau" display="inline"><mi href="./14.1#p1.t1.r1">τ</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</figure>
</td>
<td class="ltx_subfigure">
<figure id="F8" class="ltx_figure"><figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 14.20.8: </span><math class="ltx_Math" altimg="m92.png" altimg-height="36px" altimg-valign="-15px" altimg-width="101px" alttext="\widehat{\mathsf{Q}}^{-2}_{-\frac{1}{2}+i\tau}\left(x\right)" display="inline"><mrow><msubsup><mover accent="true"><mi href="./14.20#E2" mathvariant="sans-serif">Q</mi><mo href="./14.20#E2">^</mo></mover><mrow><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./14.1#p1.t1.r1">τ</mi></mrow></mrow><mrow><mo>-</mo><mn>2</mn></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi href="./14.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow></math>, <math class="ltx_Math" altimg="m79.png" altimg-height="27px" altimg-valign="-9px" altimg-width="130px" alttext="\tau=0,\tfrac{1}{2},1,2,4" display="inline"><mrow><mi href="./14.1#p1.t1.r1">τ</mi><mo>=</mo><mrow><mn>0</mn><mo>,</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>4</mn></mrow></mrow></math>.
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./14.20#E2" title="(14.20.2) ‣ §14.20(i) Definitions and Wronskians ‣ §14.20 Conical (or Mehler) Functions ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m93.png" altimg-height="36px" altimg-valign="-15px" altimg-width="101px" alttext="\widehat{\mathsf{Q}}^{-\NVar{\mu}}_{\NVar{-\frac{1}{2}+i\tau}}\left(\NVar{x}\right)" display="inline"><mrow><msubsup><mover accent="true"><mi href="./14.20#E2" mathvariant="sans-serif">Q</mi><mo href="./14.20#E2">^</mo></mover><mrow><mrow><mo class="ltx_nvar">-</mo><mfrac class="ltx_nvar"><mn class="ltx_nvar">1</mn><mn class="ltx_nvar">2</mn></mfrac></mrow><mo class="ltx_nvar">+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi class="ltx_nvar" href="./14.1#p1.t1.r1">τ</mi></mrow></mrow><mrow><mo>-</mo><mi class="ltx_nvar" href="./14.1#p1.t1.r4">μ</mi></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./14.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow></math>: conical function</a>,
<a href="./14.1#p1.t1.r1" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./14.1#p1.t1.r1">x</mi></math>: real variable</a> and
<a href="./14.1#p1.t1.r1" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m83.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\tau" display="inline"><mi href="./14.1#p1.t1.r1">τ</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</figure>
</td>
</tr>
</table>
</figure>
</section>
<section id="iii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§14.20(iii) </span>Behavior as <math class="ltx_Math" altimg="m107.png" altimg-height="17px" altimg-valign="-2px" altimg-width="57px" alttext="x\to 1" display="inline"><mrow><mi href="./14.1#p1.t1.r1">x</mi><mo>→</mo><mn>1</mn></mrow></math>
</h2>
<div id="SS3.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="SS3.p1" class="ltx_para">
<p class="ltx_p">The behavior of <math class="ltx_Math" altimg="m50.png" altimg-height="35px" altimg-valign="-15px" altimg-width="115px" alttext="\mathsf{P}^{-\mu}_{-\frac{1}{2}+i\tau}\left(\pm x\right)" display="inline"><mrow><msubsup><mi href="./14.3#E1" mathvariant="sans-serif">P</mi><mrow><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./14.1#p1.t1.r1">τ</mi></mrow></mrow><mrow><mo>-</mo><mi href="./14.1#p1.t1.r4">μ</mi></mrow></msubsup><mo></mo><mrow><mo>(</mo><mrow><mo>±</mo><mi href="./14.1#p1.t1.r1">x</mi></mrow><mo>)</mo></mrow></mrow></math> as <math class="ltx_Math" altimg="m106.png" altimg-height="18px" altimg-valign="-4px" altimg-width="72px" alttext="x\to 1-" display="inline"><mrow><mi href="./14.1#p1.t1.r1">x</mi><mo>→</mo><mrow><mn>1</mn><mo>-</mo></mrow></mrow></math> is
given in §. For <math class="ltx_Math" altimg="m60.png" altimg-height="20px" altimg-valign="-6px" altimg-width="53px" alttext="\mu>0" display="inline"><mrow><mi href="./14.1#p1.t1.r4">μ</mi><mo>></mo><mn>0</mn></mrow></math> and <math class="ltx_Math" altimg="m106.png" altimg-height="18px" altimg-valign="-4px" altimg-width="72px" alttext="x\to 1-" display="inline"><mrow><mi href="./14.1#p1.t1.r1">x</mi><mo>→</mo><mrow><mn>1</mn><mo>-</mo></mrow></mrow></math>,</p>
<table id="E7" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">14.20.7</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E7.png" altimg-height="60px" altimg-valign="-21px" altimg-width="298px" alttext="\widehat{\mathsf{Q}}^{\mu}_{-\frac{1}{2}+i\tau}\left(x\right)\sim\tfrac{1}{2}%
\Gamma\left(\mu\right)\left(\frac{2}{1-x}\right)^{\mu/2}," display="block"><mrow><mrow><mrow><msubsup><mover accent="true"><mi href="./14.20#E2" mathvariant="sans-serif">Q</mi><mo href="./14.20#E2">^</mo></mover><mrow><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./14.1#p1.t1.r1">τ</mi></mrow></mrow><mi href="./14.1#p1.t1.r4">μ</mi></msubsup><mo></mo><mrow><mo>(</mo><mi href="./14.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow><mo href="./2.1#E1">∼</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi href="./14.1#p1.t1.r4">μ</mi><mo>)</mo></mrow></mrow><mo></mo><msup><mrow><mo>(</mo><mfrac><mn>2</mn><mrow><mn>1</mn><mo>-</mo><mi href="./14.1#p1.t1.r1">x</mi></mrow></mfrac><mo>)</mo></mrow><mrow><mi href="./14.1#p1.t1.r4">μ</mi><mo>/</mo><mn>2</mn></mrow></msup></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E7.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m27.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./14.20#E2" title="(14.20.2) ‣ §14.20(i) Definitions and Wronskians ‣ §14.20 Conical (or Mehler) Functions ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m93.png" altimg-height="36px" altimg-valign="-15px" altimg-width="101px" alttext="\widehat{\mathsf{Q}}^{-\NVar{\mu}}_{\NVar{-\frac{1}{2}+i\tau}}\left(\NVar{x}\right)" display="inline"><mrow><msubsup><mover accent="true"><mi href="./14.20#E2" mathvariant="sans-serif">Q</mi><mo href="./14.20#E2">^</mo></mover><mrow><mrow><mo class="ltx_nvar">-</mo><mfrac class="ltx_nvar"><mn class="ltx_nvar">1</mn><mn class="ltx_nvar">2</mn></mfrac></mrow><mo class="ltx_nvar">+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi class="ltx_nvar" href="./14.1#p1.t1.r1">τ</mi></mrow></mrow><mrow><mo>-</mo><mi class="ltx_nvar" href="./14.1#p1.t1.r4">μ</mi></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./14.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow></math>: conical function</a>,
<a href="./2.1#E1" title="(2.1.1) ‣ §2.1(i) Asymptotic and Order Symbols ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="12px" altimg-valign="-2px" altimg-width="20px" alttext="\sim" display="inline"><mo href="./2.1#E1">∼</mo></math>: asymptotic equality</a>,
<a href="./14.1#p1.t1.r1" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./14.1#p1.t1.r1">x</mi></math>: real variable</a>,
<a href="./14.1#p1.t1.r1" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m83.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\tau" display="inline"><mi href="./14.1#p1.t1.r1">τ</mi></math>: real variable</a> and
<a href="./14.1#p1.t1.r4" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m61.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\mu" display="inline"><mi href="./14.1#p1.t1.r4">μ</mi></math>: general order</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E8" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">14.20.8</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E8.png" altimg-height="66px" altimg-valign="-28px" altimg-width="611px" alttext="\widehat{\mathsf{Q}}^{-\mu}_{-\frac{1}{2}+i\tau}\left(x\right)\sim\frac{\pi%
\Gamma\left(\mu\right)(e^{-\tau\pi}{\cos^{2}}\left(\mu\pi\right)+\sinh\left(%
\tau\pi\right))}{2({\cosh^{2}}\left(\tau\pi\right)-{\sin^{2}}\left(\mu\pi%
\right)){\left|\Gamma\left(\mu+\frac{1}{2}+\mathrm{i}\tau\right)\right|^{2}}}%
\*\left(\frac{2}{1-x}\right)^{\mu/2}." display="block"><mrow><mrow><mrow><msubsup><mover accent="true"><mi href="./14.20#E2" mathvariant="sans-serif">Q</mi><mo href="./14.20#E2">^</mo></mover><mrow><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./14.1#p1.t1.r1">τ</mi></mrow></mrow><mrow><mo>-</mo><mi href="./14.1#p1.t1.r4">μ</mi></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi href="./14.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow><mo href="./2.1#E1">∼</mo><mrow><mfrac><mrow><mi href="./3.12#E1">π</mi><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi href="./14.1#p1.t1.r4">μ</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mi href="./14.1#p1.t1.r1">τ</mi><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow></msup><mo></mo><mrow><msup><mi href="./4.14#E2">cos</mi><mn>2</mn></msup><mo></mo><mrow><mo>(</mo><mrow><mi href="./14.1#p1.t1.r4">μ</mi><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mi href="./4.28#E1">sinh</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./14.1#p1.t1.r1">τ</mi><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow><mrow><mn>2</mn><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><msup><mi href="./4.28#E2">cosh</mi><mn>2</mn></msup><mo></mo><mrow><mo>(</mo><mrow><mi href="./14.1#p1.t1.r1">τ</mi><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo>)</mo></mrow></mrow><mo>-</mo><mrow><msup><mi href="./4.14#E1">sin</mi><mn>2</mn></msup><mo></mo><mrow><mo>(</mo><mrow><mi href="./14.1#p1.t1.r4">μ</mi><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msup><mrow><mo>|</mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./14.1#p1.t1.r4">μ</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./14.1#p1.t1.r1">τ</mi></mrow></mrow><mo>)</mo></mrow></mrow><mo>|</mo></mrow><mn>2</mn></msup></mrow></mfrac><mo></mo><msup><mrow><mo>(</mo><mfrac><mn>2</mn><mrow><mn>1</mn><mo>-</mo><mi href="./14.1#p1.t1.r1">x</mi></mrow></mfrac><mo>)</mo></mrow><mrow><mi href="./14.1#p1.t1.r4">μ</mi><mo>/</mo><mn>2</mn></mrow></msup></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E8.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m27.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./14.20#E2" title="(14.20.2) ‣ §14.20(i) Definitions and Wronskians ‣ §14.20 Conical (or Mehler) Functions ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m93.png" altimg-height="36px" altimg-valign="-15px" altimg-width="101px" alttext="\widehat{\mathsf{Q}}^{-\NVar{\mu}}_{\NVar{-\frac{1}{2}+i\tau}}\left(\NVar{x}\right)" display="inline"><mrow><msubsup><mover accent="true"><mi href="./14.20#E2" mathvariant="sans-serif">Q</mi><mo href="./14.20#E2">^</mo></mover><mrow><mrow><mo class="ltx_nvar">-</mo><mfrac class="ltx_nvar"><mn class="ltx_nvar">1</mn><mn class="ltx_nvar">2</mn></mfrac></mrow><mo class="ltx_nvar">+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi class="ltx_nvar" href="./14.1#p1.t1.r1">τ</mi></mrow></mrow><mrow><mo>-</mo><mi class="ltx_nvar" href="./14.1#p1.t1.r4">μ</mi></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./14.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow></math>: conical function</a>,
<a href="./2.1#E1" title="(2.1.1) ‣ §2.1(i) Asymptotic and Order Symbols ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="12px" altimg-valign="-2px" altimg-width="20px" alttext="\sim" display="inline"><mo href="./2.1#E1">∼</mo></math>: asymptotic equality</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m68.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m31.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m42.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./4.28#E2" title="(4.28.2) ‣ §4.28 Definitions and Periodicity ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m32.png" altimg-height="18px" altimg-valign="-2px" altimg-width="56px" alttext="\cosh\NVar{z}" display="inline"><mrow><mi href="./4.28#E2">cosh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: hyperbolic cosine function</a>,
<a href="./4.28#E1" title="(4.28.1) ‣ §4.28 Definitions and Periodicity ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m76.png" altimg-height="18px" altimg-valign="-2px" altimg-width="53px" alttext="\sinh\NVar{z}" display="inline"><mrow><mi href="./4.28#E1">sinh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: hyperbolic sine function</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m75.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./14.1#p1.t1.r1" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./14.1#p1.t1.r1">x</mi></math>: real variable</a>,
<a href="./14.1#p1.t1.r1" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m83.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\tau" display="inline"><mi href="./14.1#p1.t1.r1">τ</mi></math>: real variable</a> and
<a href="./14.1#p1.t1.r4" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m61.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\mu" display="inline"><mi href="./14.1#p1.t1.r4">μ</mi></math>: general order</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="iv" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§14.20(iv) </span>Integral Representation</h2>
<div id="SS4.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="SS4.p1" class="ltx_para">
<p class="ltx_p">When <math class="ltx_Math" altimg="m10.png" altimg-height="18px" altimg-valign="-3px" altimg-width="89px" alttext="0<\theta<\pi" display="inline"><mrow><mn>0</mn><mo><</mo><mi href="./14.20#SS4.p1">θ</mi><mo><</mo><mi href="./3.12#E1">π</mi></mrow></math>,
</p>
<table id="E9" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">14.20.9</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E9.png" altimg-height="60px" altimg-valign="-25px" altimg-width="393px" alttext="\mathsf{P}_{-\frac{1}{2}+i\tau}\left(\cos\theta\right)=\frac{2}{\pi}\int_{0}^{%
\theta}\frac{\cosh\left(\tau\phi\right)}{\sqrt{2(\cos\phi-\cos\theta)}}\mathrm%
{d}\phi." display="block"><mrow><mrow><mrow><msub><mi href="./14.2#SS2.p2" mathvariant="sans-serif">P</mi><mrow><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./14.1#p1.t1.r1">τ</mi></mrow></mrow></msub><mo></mo><mrow><mo>(</mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi href="./14.20#SS4.p1">θ</mi></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mfrac><mn>2</mn><mi href="./3.12#E1">π</mi></mfrac><mo></mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi href="./14.20#SS4.p1">θ</mi></msubsup><mrow><mfrac><mrow><mi href="./4.28#E2">cosh</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./14.1#p1.t1.r1">τ</mi><mo></mo><mi>ϕ</mi></mrow><mo>)</mo></mrow></mrow><msqrt><mrow><mn>2</mn><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi>ϕ</mi></mrow><mo>-</mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi href="./14.20#SS4.p1">θ</mi></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></msqrt></mfrac><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>ϕ</mi></mrow></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E9.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m68.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m31.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m41.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m103.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.28#E2" title="(4.28.2) ‣ §4.28 Definitions and Periodicity ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m32.png" altimg-height="18px" altimg-valign="-2px" altimg-width="56px" alttext="\cosh\NVar{z}" display="inline"><mrow><mi href="./4.28#E2">cosh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: hyperbolic cosine function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./14.2#SS2.p2" title="§14.2(ii) Associated Legendre Equation ‣ §14.2 Differential Equations ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m57.png" altimg-height="25px" altimg-valign="-7px" altimg-width="137px" alttext="\mathsf{P}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{P}^{0}_{\nu}\left(x\right)" display="inline"><mrow><mrow><msub><mi href="./14.2#SS2.p2" mathvariant="sans-serif">P</mi><mi class="ltx_nvar" href="./14.1#p1.t1.r4">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./14.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msubsup><mi href="./14.3#E1" mathvariant="sans-serif">P</mi><mi href="./14.1#p1.t1.r4">ν</mi><mn>0</mn></msubsup><mo></mo><mrow><mo>(</mo><mi href="./14.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow></mrow></math>: Ferrers function of the first kind</a>,
<a href="./14.1#p1.t1.r1" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m83.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\tau" display="inline"><mi href="./14.1#p1.t1.r1">τ</mi></math>: real variable</a> and
<a href="./14.20#SS4.p1" title="§14.20(iv) Integral Representation ‣ §14.20 Conical (or Mehler) Functions ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m10.png" altimg-height="18px" altimg-valign="-3px" altimg-width="89px" alttext="0<\theta<\pi" display="inline"><mrow><mn>0</mn><mo><</mo><mi href="./14.20#SS4.p1">θ</mi><mo><</mo><mi href="./3.12#E1">π</mi></mrow></math>: variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">8.12.3</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="v" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§14.20(v) </span>Trigonometric Expansion</h2>
<div id="SS5.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="SS5.p1" class="ltx_para">
<table id="E10" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">14.20.10</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E10.png" altimg-height="52px" altimg-valign="-16px" altimg-width="695px" alttext="\mathsf{P}_{-\frac{1}{2}+i\tau}\left(\cos\theta\right)=1+\frac{4\tau^{2}+1^{2}%
}{2^{2}}{\sin^{2}}\left(\tfrac{1}{2}\theta\right)+\frac{\left(4\tau^{2}+1^{2}%
\right)\left(4\tau^{2}+3^{2}\right)}{2^{2}\cdot 4^{2}}{\sin^{4}}\left(\tfrac{1%
}{2}\theta\right)+\cdots," display="block"><mrow><mrow><mrow><msub><mi href="./14.2#SS2.p2" mathvariant="sans-serif">P</mi><mrow><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./14.1#p1.t1.r1">τ</mi></mrow></mrow></msub><mo></mo><mrow><mo>(</mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi>θ</mi></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mn>1</mn><mo>+</mo><mrow><mfrac><mrow><mrow><mn>4</mn><mo></mo><msup><mi href="./14.1#p1.t1.r1">τ</mi><mn>2</mn></msup></mrow><mo>+</mo><msup><mn>1</mn><mn>2</mn></msup></mrow><msup><mn>2</mn><mn>2</mn></msup></mfrac><mo></mo><mrow><msup><mi href="./4.14#E1">sin</mi><mn>2</mn></msup><mo></mo><mrow><mo>(</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi>θ</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mfrac><mrow><mrow><mo>(</mo><mrow><mrow><mn>4</mn><mo></mo><msup><mi href="./14.1#p1.t1.r1">τ</mi><mn>2</mn></msup></mrow><mo>+</mo><msup><mn>1</mn><mn>2</mn></msup></mrow><mo>)</mo></mrow><mo></mo><mrow><mo>(</mo><mrow><mrow><mn>4</mn><mo></mo><msup><mi href="./14.1#p1.t1.r1">τ</mi><mn>2</mn></msup></mrow><mo>+</mo><msup><mn>3</mn><mn>2</mn></msup></mrow><mo>)</mo></mrow></mrow><mrow><msup><mn>2</mn><mn>2</mn></msup><mo>⋅</mo><msup><mn>4</mn><mn>2</mn></msup></mrow></mfrac><mo></mo><mrow><msup><mi href="./4.14#E1">sin</mi><mn>4</mn></msup><mo></mo><mrow><mo>(</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi>θ</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>+</mo><mi mathvariant="normal">⋯</mi></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m13.png" altimg-height="20px" altimg-valign="-5px" altimg-width="89px" alttext="0\leq\theta\leq\pi" display="inline"><mrow><mn>0</mn><mo>≤</mo><mi>θ</mi><mo>≤</mo><mi href="./3.12#E1">π</mi></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E10.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m68.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m31.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./14.2#SS2.p2" title="§14.2(ii) Associated Legendre Equation ‣ §14.2 Differential Equations ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m57.png" altimg-height="25px" altimg-valign="-7px" altimg-width="137px" alttext="\mathsf{P}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{P}^{0}_{\nu}\left(x\right)" display="inline"><mrow><mrow><msub><mi href="./14.2#SS2.p2" mathvariant="sans-serif">P</mi><mi class="ltx_nvar" href="./14.1#p1.t1.r4">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./14.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msubsup><mi href="./14.3#E1" mathvariant="sans-serif">P</mi><mi href="./14.1#p1.t1.r4">ν</mi><mn>0</mn></msubsup><mo></mo><mrow><mo>(</mo><mi href="./14.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow></mrow></math>: Ferrers function of the first kind</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m75.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a> and
<a href="./14.1#p1.t1.r1" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m83.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\tau" display="inline"><mi href="./14.1#p1.t1.r1">τ</mi></math>: real variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">8.12.1</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">From () it is evident
that <math class="ltx_Math" altimg="m55.png" altimg-height="28px" altimg-valign="-12px" altimg-width="128px" alttext="\mathsf{P}_{-\frac{1}{2}+i\tau}\left(\cos\theta\right)" display="inline"><mrow><msub><mi href="./14.2#SS2.p2" mathvariant="sans-serif">P</mi><mrow><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./14.1#p1.t1.r1">τ</mi></mrow></mrow></msub><mo></mo><mrow><mo>(</mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi>θ</mi></mrow><mo>)</mo></mrow></mrow></math> is positive for real <math class="ltx_Math" altimg="m88.png" altimg-height="18px" altimg-valign="-2px" altimg-width="14px" alttext="\theta" display="inline"><mi>θ</mi></math>.
</p>
</div>
</section>
<section id="vi" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§14.20(vi) </span>Generalized Mehler–Fock Transformation</h2>
<div id="SS6.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="SS6.p1" class="ltx_para">
<table id="E11" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">14.20.11</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E11.png" altimg-height="53px" altimg-valign="-20px" altimg-width="636px" alttext="f(\tau)=\frac{\tau}{\pi}\sinh\left(\tau\pi\right)\Gamma\left(\tfrac{1}{2}-\mu+%
i\tau\right)\*\Gamma\left(\tfrac{1}{2}-\mu-i\tau\right)\int_{1}^{\infty}P^{\mu%
}_{-\frac{1}{2}+i\tau}\left(x\right)g(x)\mathrm{d}x," display="block"><mrow><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./14.1#p1.t1.r1">τ</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mfrac><mi href="./14.1#p1.t1.r1">τ</mi><mi href="./3.12#E1">π</mi></mfrac><mo></mo><mrow><mi href="./4.28#E1">sinh</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./14.1#p1.t1.r1">τ</mi><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo>-</mo><mi href="./14.1#p1.t1.r4">μ</mi></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./14.1#p1.t1.r1">τ</mi></mrow></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo>-</mo><mi href="./14.1#p1.t1.r4">μ</mi><mo>-</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./14.1#p1.t1.r1">τ</mi></mrow></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>1</mn><mi mathvariant="normal">∞</mi></msubsup><mrow><mrow><msubsup><mi href="./14.21#SS1.p1">P</mi><mrow><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./14.1#p1.t1.r1">τ</mi></mrow></mrow><mi href="./14.1#p1.t1.r4">μ</mi></msubsup><mo></mo><mrow><mo>(</mo><mi href="./14.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mi>g</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./14.1#p1.t1.r1">x</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./14.1#p1.t1.r1">x</mi></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E11.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m27.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./14.21#SS1.p1" title="§14.21(i) Associated Legendre Equation ‣ §14.21 Definitions and Basic Properties ‣ Complex Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m23.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msubsup><mi href="./14.21#SS1.p1">P</mi><mi class="ltx_nvar" href="./14.1#p1.t1.r4">ν</mi><mi class="ltx_nvar" href="./14.1#p1.t1.r4">μ</mi></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./14.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: associated Legendre function of the first kind</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m68.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m41.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m103.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.28#E1" title="(4.28.1) ‣ §4.28 Definitions and Periodicity ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m76.png" altimg-height="18px" altimg-valign="-2px" altimg-width="53px" alttext="\sinh\NVar{z}" display="inline"><mrow><mi href="./4.28#E1">sinh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: hyperbolic sine function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./14.1#p1.t1.r1" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./14.1#p1.t1.r1">x</mi></math>: real variable</a>,
<a href="./14.1#p1.t1.r1" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m83.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\tau" display="inline"><mi href="./14.1#p1.t1.r1">τ</mi></math>: real variable</a> and
<a href="./14.1#p1.t1.r4" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m61.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\mu" display="inline"><mi href="./14.1#p1.t1.r4">μ</mi></math>: general order</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where</p>
<table id="E12" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">14.20.12</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E12.png" altimg-height="53px" altimg-valign="-20px" altimg-width="273px" alttext="g(x)=\int_{0}^{\infty}P^{\mu}_{-\frac{1}{2}+i\tau}\left(x\right)f(\tau)\mathrm%
{d}\tau." display="block"><mrow><mrow><mrow><mi>g</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./14.1#p1.t1.r1">x</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><mrow><mrow><msubsup><mi href="./14.21#SS1.p1">P</mi><mrow><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./14.1#p1.t1.r1">τ</mi></mrow></mrow><mi href="./14.1#p1.t1.r4">μ</mi></msubsup><mo></mo><mrow><mo>(</mo><mi href="./14.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./14.1#p1.t1.r1">τ</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./14.1#p1.t1.r1">τ</mi></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E12.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./14.21#SS1.p1" title="§14.21(i) Associated Legendre Equation ‣ §14.21 Definitions and Basic Properties ‣ Complex Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m23.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msubsup><mi href="./14.21#SS1.p1">P</mi><mi class="ltx_nvar" href="./14.1#p1.t1.r4">ν</mi><mi class="ltx_nvar" href="./14.1#p1.t1.r4">μ</mi></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./14.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: associated Legendre function of the first kind</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m41.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m103.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./14.1#p1.t1.r1" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./14.1#p1.t1.r1">x</mi></math>: real variable</a>,
<a href="./14.1#p1.t1.r1" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m83.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\tau" display="inline"><mi href="./14.1#p1.t1.r1">τ</mi></math>: real variable</a> and
<a href="./14.1#p1.t1.r4" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m61.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\mu" display="inline"><mi href="./14.1#p1.t1.r4">μ</mi></math>: general order</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Special cases:</p>
<table id="E13" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">14.20.13</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E13.png" altimg-height="56px" altimg-valign="-20px" altimg-width="375px" alttext="P_{-\frac{1}{2}+i\tau}\left(x\right)=\frac{\cosh\left(\tau\pi\right)}{\pi}\int%
_{1}^{\infty}\frac{P_{-\frac{1}{2}+i\tau}\left(t\right)}{x+t}\mathrm{d}t," display="block"><mrow><mrow><mrow><msub><mi href="./14.2#SS2.p2">P</mi><mrow><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./14.1#p1.t1.r1">τ</mi></mrow></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./14.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mfrac><mrow><mi href="./4.28#E2">cosh</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./14.1#p1.t1.r1">τ</mi><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo>)</mo></mrow></mrow><mi href="./3.12#E1">π</mi></mfrac><mo></mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>1</mn><mi mathvariant="normal">∞</mi></msubsup><mrow><mfrac><mrow><msub><mi href="./14.2#SS2.p2">P</mi><mrow><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./14.1#p1.t1.r1">τ</mi></mrow></mrow></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow><mrow><mi href="./14.1#p1.t1.r1">x</mi><mo>+</mo><mi>t</mi></mrow></mfrac><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E13.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m68.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m41.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m103.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.28#E2" title="(4.28.2) ‣ §4.28 Definitions and Periodicity ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m32.png" altimg-height="18px" altimg-valign="-2px" altimg-width="56px" alttext="\cosh\NVar{z}" display="inline"><mrow><mi href="./4.28#E2">cosh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: hyperbolic cosine function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./14.2#SS2.p2" title="§14.2(ii) Associated Legendre Equation ‣ §14.2 Differential Equations ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m24.png" altimg-height="25px" altimg-valign="-7px" altimg-width="136px" alttext="P_{\NVar{\nu}}\left(\NVar{z}\right)=P^{0}_{\nu}\left(z\right)" display="inline"><mrow><mrow><msub><mi href="./14.2#SS2.p2">P</mi><mi class="ltx_nvar" href="./14.1#p1.t1.r4">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./14.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msubsup><mi href="./14.21#SS1.p1">P</mi><mi href="./14.1#p1.t1.r4">ν</mi><mn>0</mn></msubsup><mo></mo><mrow><mo>(</mo><mi href="./14.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow></math>: Legendre function of the first kind</a>,
<a href="./14.1#p1.t1.r1" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./14.1#p1.t1.r1">x</mi></math>: real variable</a> and
<a href="./14.1#p1.t1.r1" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m83.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\tau" display="inline"><mi href="./14.1#p1.t1.r1">τ</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E14" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">14.20.14</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E14.png" altimg-height="53px" altimg-valign="-21px" altimg-width="463px" alttext="\pi\int_{0}^{\infty}\frac{\tau\tanh\left(\tau\pi\right)}{\cosh\left(\tau\pi%
\right)}P_{-\frac{1}{2}+i\tau}\left(x\right)P_{-\frac{1}{2}+i\tau}\left(y%
\right)\mathrm{d}\tau=\frac{1}{y+x}." display="block"><mrow><mrow><mrow><mi href="./3.12#E1">π</mi><mo></mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><mrow><mfrac><mrow><mi href="./14.1#p1.t1.r1">τ</mi><mo></mo><mrow><mi href="./4.28#E4">tanh</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./14.1#p1.t1.r1">τ</mi><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo>)</mo></mrow></mrow></mrow><mrow><mi href="./4.28#E2">cosh</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./14.1#p1.t1.r1">τ</mi><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo>)</mo></mrow></mrow></mfrac><mo></mo><mrow><msub><mi href="./14.2#SS2.p2">P</mi><mrow><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./14.1#p1.t1.r1">τ</mi></mrow></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./14.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./14.2#SS2.p2">P</mi><mrow><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./14.1#p1.t1.r1">τ</mi></mrow></mrow></msub><mo></mo><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./14.1#p1.t1.r1">τ</mi></mrow></mrow></mrow></mrow><mo>=</mo><mfrac><mn>1</mn><mrow><mi>y</mi><mo>+</mo><mi href="./14.1#p1.t1.r1">x</mi></mrow></mfrac></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E14.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m68.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m41.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m103.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.28#E2" title="(4.28.2) ‣ §4.28 Definitions and Periodicity ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m32.png" altimg-height="18px" altimg-valign="-2px" altimg-width="56px" alttext="\cosh\NVar{z}" display="inline"><mrow><mi href="./4.28#E2">cosh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: hyperbolic cosine function</a>,
<a href="./4.28#E4" title="(4.28.4) ‣ §4.28 Definitions and Periodicity ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="18px" altimg-valign="-2px" altimg-width="58px" alttext="\tanh\NVar{z}" display="inline"><mrow><mi href="./4.28#E4">tanh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: hyperbolic tangent function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./14.2#SS2.p2" title="§14.2(ii) Associated Legendre Equation ‣ §14.2 Differential Equations ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m24.png" altimg-height="25px" altimg-valign="-7px" altimg-width="136px" alttext="P_{\NVar{\nu}}\left(\NVar{z}\right)=P^{0}_{\nu}\left(z\right)" display="inline"><mrow><mrow><msub><mi href="./14.2#SS2.p2">P</mi><mi class="ltx_nvar" href="./14.1#p1.t1.r4">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./14.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msubsup><mi href="./14.21#SS1.p1">P</mi><mi href="./14.1#p1.t1.r4">ν</mi><mn>0</mn></msubsup><mo></mo><mrow><mo>(</mo><mi href="./14.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow></math>: Legendre function of the first kind</a>,
<a href="./14.1#p1.t1.r1" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./14.1#p1.t1.r1">x</mi></math>: real variable</a> and
<a href="./14.1#p1.t1.r1" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m83.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\tau" display="inline"><mi href="./14.1#p1.t1.r1">τ</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="vii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§14.20(vii) </span>Asymptotic Approximations: Large <math class="ltx_Math" altimg="m83.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\tau" display="inline"><mi href="./14.1#p1.t1.r1">τ</mi></math>, Fixed
<math class="ltx_Math" altimg="m61.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\mu" display="inline"><mi href="./14.1#p1.t1.r4">μ</mi></math>
</h2>
<div id="SS7.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="SS7.p1" class="ltx_para">
<p class="ltx_p">For <math class="ltx_Math" altimg="m87.png" altimg-height="13px" altimg-valign="-2px" altimg-width="66px" alttext="\tau\to\infty" display="inline"><mrow><mi href="./14.1#p1.t1.r1">τ</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></math> and fixed <math class="ltx_Math" altimg="m61.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\mu" display="inline"><mi href="./14.1#p1.t1.r4">μ</mi></math>,</p>
<table id="EGx1" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E15">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">14.20.15</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m3.png" altimg-height="37px" altimg-valign="-15px" altimg-width="129px" alttext="\displaystyle\mathsf{P}^{-\mu}_{-\frac{1}{2}+i\tau}\left(\cos\theta\right)" display="inline"><mrow><msubsup><mi href="./14.3#E1" mathvariant="sans-serif">P</mi><mrow><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./14.1#p1.t1.r1">τ</mi></mrow></mrow><mrow><mo>-</mo><mi href="./14.1#p1.t1.r4">μ</mi></mrow></msubsup><mo></mo><mrow><mo>(</mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi href="./14.20#SS7.p1">θ</mi></mrow><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m1.png" altimg-height="60px" altimg-valign="-21px" altimg-width="345px" alttext="\displaystyle=\frac{1}{\tau^{\mu}}\left(\frac{\theta}{\sin\theta}\right)^{1/2}%
I_{\mu}\left(\tau\theta\right)\*\left(1+O\left(\ifrac{1}{\tau}\right)\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><mfrac><mn>1</mn><msup><mi href="./14.1#p1.t1.r1">τ</mi><mi href="./14.1#p1.t1.r4">μ</mi></msup></mfrac></mstyle><mo></mo><msup><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac><mi href="./14.20#SS7.p1">θ</mi><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi href="./14.20#SS7.p1">θ</mi></mrow></mfrac></mstyle><mo>)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo></mo><mrow><msub><mi href="./10.25#E2">I</mi><mi href="./14.1#p1.t1.r4">μ</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi href="./14.1#p1.t1.r1">τ</mi><mo></mo><mi href="./14.20#SS7.p1">θ</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mrow><mi href="./2.1#E3">O</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>/</mo><mi href="./14.1#p1.t1.r1">τ</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E15.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./2.1#E3" title="(2.1.3) ‣ §2.1(i) Asymptotic and Order Symbols ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m21.png" altimg-height="23px" altimg-valign="-7px" altimg-width="50px" alttext="O\left(\NVar{x}\right)" display="inline"><mrow><mi href="./2.1#E3">O</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>)</mo></mrow></mrow></math>: order not exceeding</a>,
<a href="./14.3#E1" title="(14.3.1) ‣ Ferrers Function of the First Kind ‣ §14.3(i) Interval - 1 < x < 1 ‣ §14.3 Definitions and Hypergeometric Representations ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="23px" altimg-valign="-7px" altimg-width="58px" alttext="\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msubsup><mi href="./14.3#E1" mathvariant="sans-serif">P</mi><mi class="ltx_nvar" href="./14.1#p1.t1.r4">ν</mi><mi class="ltx_nvar" href="./14.1#p1.t1.r4">μ</mi></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./14.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow></math>: Ferrers function of the first kind</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m31.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./10.25#E2" title="(10.25.2) ‣ §10.25(ii) Standard Solutions ‣ §10.25 Definitions ‣ Modified Bessel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m18.png" altimg-height="23px" altimg-valign="-7px" altimg-width="52px" alttext="I_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.25#E2">I</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: modified Bessel function of the first kind</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m75.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./14.1#p1.t1.r1" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m83.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\tau" display="inline"><mi href="./14.1#p1.t1.r1">τ</mi></math>: real variable</a>,
<a href="./14.1#p1.t1.r4" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m61.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\mu" display="inline"><mi href="./14.1#p1.t1.r4">μ</mi></math>: general order</a> and
<a href="./14.20#SS7.p1" title="§14.20(vii) Asymptotic Approximations: Large τ , Fixed μ ‣ §14.20 Conical (or Mehler) Functions ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m10.png" altimg-height="18px" altimg-valign="-3px" altimg-width="89px" alttext="0<\theta<\pi" display="inline"><mrow><mn>0</mn><mo><</mo><mi href="./14.20#SS7.p1">θ</mi><mo><</mo><mi href="./3.12#E1">π</mi></mrow></math>: variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E16">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">14.20.16</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m4.png" altimg-height="38px" altimg-valign="-15px" altimg-width="131px" alttext="\displaystyle\widehat{\mathsf{Q}}^{-\mu}_{-\frac{1}{2}+i\tau}\left(\cos\theta\right)" display="inline"><mrow><msubsup><mover accent="true"><mi href="./14.20#E2" mathvariant="sans-serif">Q</mi><mo href="./14.20#E2">^</mo></mover><mrow><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./14.1#p1.t1.r1">τ</mi></mrow></mrow><mrow><mo>-</mo><mi href="./14.1#p1.t1.r4">μ</mi></mrow></msubsup><mo></mo><mrow><mo>(</mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi href="./14.20#SS7.p1">θ</mi></mrow><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m2.png" altimg-height="60px" altimg-valign="-21px" altimg-width="354px" alttext="\displaystyle=\frac{1}{\tau^{\mu}}\left(\frac{\theta}{\sin\theta}\right)^{1/2}%
K_{\mu}\left(\tau\theta\right)\*\left(1+O\left(\ifrac{1}{\tau}\right)\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><mfrac><mn>1</mn><msup><mi href="./14.1#p1.t1.r1">τ</mi><mi href="./14.1#p1.t1.r4">μ</mi></msup></mfrac></mstyle><mo></mo><msup><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac><mi href="./14.20#SS7.p1">θ</mi><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi href="./14.20#SS7.p1">θ</mi></mrow></mfrac></mstyle><mo>)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo></mo><mrow><msub><mi href="./10.25#E3">K</mi><mi href="./14.1#p1.t1.r4">μ</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi href="./14.1#p1.t1.r1">τ</mi><mo></mo><mi href="./14.20#SS7.p1">θ</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mrow><mi href="./2.1#E3">O</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>/</mo><mi href="./14.1#p1.t1.r1">τ</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E16.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./2.1#E3" title="(2.1.3) ‣ §2.1(i) Asymptotic and Order Symbols ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m21.png" altimg-height="23px" altimg-valign="-7px" altimg-width="50px" alttext="O\left(\NVar{x}\right)" display="inline"><mrow><mi href="./2.1#E3">O</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>)</mo></mrow></mrow></math>: order not exceeding</a>,
<a href="./14.20#E2" title="(14.20.2) ‣ §14.20(i) Definitions and Wronskians ‣ §14.20 Conical (or Mehler) Functions ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m93.png" altimg-height="36px" altimg-valign="-15px" altimg-width="101px" alttext="\widehat{\mathsf{Q}}^{-\NVar{\mu}}_{\NVar{-\frac{1}{2}+i\tau}}\left(\NVar{x}\right)" display="inline"><mrow><msubsup><mover accent="true"><mi href="./14.20#E2" mathvariant="sans-serif">Q</mi><mo href="./14.20#E2">^</mo></mover><mrow><mrow><mo class="ltx_nvar">-</mo><mfrac class="ltx_nvar"><mn class="ltx_nvar">1</mn><mn class="ltx_nvar">2</mn></mfrac></mrow><mo class="ltx_nvar">+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi class="ltx_nvar" href="./14.1#p1.t1.r1">τ</mi></mrow></mrow><mrow><mo>-</mo><mi class="ltx_nvar" href="./14.1#p1.t1.r4">μ</mi></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./14.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow></math>: conical function</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m31.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./10.25#E3" title="(10.25.3) ‣ §10.25(ii) Standard Solutions ‣ §10.25 Definitions ‣ Modified Bessel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="K_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.25#E3">K</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: modified Bessel function of the second kind</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m75.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./14.1#p1.t1.r1" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m83.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\tau" display="inline"><mi href="./14.1#p1.t1.r1">τ</mi></math>: real variable</a>,
<a href="./14.1#p1.t1.r4" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m61.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\mu" display="inline"><mi href="./14.1#p1.t1.r4">μ</mi></math>: general order</a> and
<a href="./14.20#SS7.p1" title="§14.20(vii) Asymptotic Approximations: Large τ , Fixed μ ‣ §14.20 Conical (or Mehler) Functions ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m10.png" altimg-height="18px" altimg-valign="-3px" altimg-width="89px" alttext="0<\theta<\pi" display="inline"><mrow><mn>0</mn><mo><</mo><mi href="./14.20#SS7.p1">θ</mi><mo><</mo><mi href="./3.12#E1">π</mi></mrow></math>: variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
<p class="ltx_p">uniformly for <math class="ltx_Math" altimg="m89.png" altimg-height="23px" altimg-valign="-7px" altimg-width="117px" alttext="\theta\in(0,\pi-\delta]" display="inline"><mrow><mi href="./14.20#SS7.p1">θ</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mrow><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">(</mo><mn>0</mn><mo href="./front/introduction#Sx4.p2.t1.r1">,</mo><mrow><mi href="./3.12#E1">π</mi><mo>-</mo><mi href="./14.20#SS8.p1">δ</mi></mrow><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">]</mo></mrow></mrow></math>, where <math class="ltx_Math" altimg="m17.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="I" display="inline"><mi href="./10.25#E2">I</mi></math> and <math class="ltx_Math" altimg="m19.png" altimg-height="18px" altimg-valign="-2px" altimg-width="23px" alttext="K" display="inline"><mi href="./10.25#E3">K</mi></math>
are the modified Bessel functions (§) and <math class="ltx_Math" altimg="m33.png" altimg-height="18px" altimg-valign="-2px" altimg-width="14px" alttext="\delta" display="inline"><mi href="./14.20#SS8.p1">δ</mi></math> is an
arbitrary constant such that <math class="ltx_Math" altimg="m8.png" altimg-height="18px" altimg-valign="-3px" altimg-width="89px" alttext="0<\delta<\pi" display="inline"><mrow><mn>0</mn><mo><</mo><mi href="./14.20#SS8.p1">δ</mi><mo><</mo><mi href="./3.12#E1">π</mi></mrow></math>. For asymptotic expansions and
explicit error bounds, see <cite class="ltx_cite ltx_citemacro_citet">Olver ()</cite>.</p>
</div>
</section>
<section id="viii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§14.20(viii) </span>Asymptotic Approximations: Large <math class="ltx_Math" altimg="m83.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\tau" display="inline"><mi href="./14.1#p1.t1.r1">τ</mi></math>,
<math class="ltx_Math" altimg="m11.png" altimg-height="21px" altimg-valign="-6px" altimg-width="105px" alttext="0\leq\mu\leq A\tau" display="inline"><mrow><mn>0</mn><mo>≤</mo><mi href="./14.1#p1.t1.r4">μ</mi><mo>≤</mo><mrow><mi href="./14.20#SS8.p1">A</mi><mo></mo><mi href="./14.1#p1.t1.r1">τ</mi></mrow></mrow></math>
</h2>
<div id="SS8.info" class="ltx_metadata ltx_info">
, <math class="ltx_Math" altimg="m16.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="A" display="inline"><mi href="./14.20#SS8.p1">A</mi></math> and <math class="ltx_Math" altimg="m33.png" altimg-height="18px" altimg-valign="-2px" altimg-width="14px" alttext="\delta" display="inline"><mi href="./14.20#SS8.p1">δ</mi></math> denote
arbitrary constants such that <math class="ltx_Math" altimg="m15.png" altimg-height="18px" altimg-valign="-3px" altimg-width="56px" alttext="A>0" display="inline"><mrow><mi href="./14.20#SS8.p1">A</mi><mo>></mo><mn>0</mn></mrow></math> and <math class="ltx_Math" altimg="m7.png" altimg-height="18px" altimg-valign="-3px" altimg-width="87px" alttext="0<\delta<2" display="inline"><mrow><mn>0</mn><mo><</mo><mi href="./14.20#SS8.p1">δ</mi><mo><</mo><mn>2</mn></mrow></math>.</p>
</div>
<div id="SS8.p2" class="ltx_para">
<p class="ltx_p">As <math class="ltx_Math" altimg="m87.png" altimg-height="13px" altimg-valign="-2px" altimg-width="66px" alttext="\tau\to\infty" display="inline"><mrow><mi href="./14.1#p1.t1.r1">τ</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></math>,</p>
<table id="E17" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">14.20.17</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E17.png" altimg-height="60px" altimg-valign="-21px" altimg-width="576px" alttext="\mathsf{P}^{-\mu}_{-\frac{1}{2}+i\tau}\left(x\right)=\sigma(\mu,\tau)\left(%
\frac{\alpha^{2}+\eta}{1+\alpha^{2}-x^{2}}\right)^{1/4}I_{\mu}\left(\tau\eta^{%
1/2}\right)\*\left(1+O\left(\ifrac{1}{\tau}\right)\right)," display="block"><mrow><mrow><mrow><msubsup><mi href="./14.3#E1" mathvariant="sans-serif">P</mi><mrow><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./14.1#p1.t1.r1">τ</mi></mrow></mrow><mrow><mo>-</mo><mi href="./14.1#p1.t1.r4">μ</mi></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi href="./14.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mi href="./14.20#SS8.p2">σ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./14.1#p1.t1.r4">μ</mi><mo>,</mo><mi href="./14.1#p1.t1.r1">τ</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><msup><mrow><mo>(</mo><mfrac><mrow><msup><mi href="./14.20#SS8.p2">α</mi><mn>2</mn></msup><mo>+</mo><mi href="./14.20#SS8.p3">η</mi></mrow><mrow><mrow><mn>1</mn><mo>+</mo><msup><mi href="./14.20#SS8.p2">α</mi><mn>2</mn></msup></mrow><mo>-</mo><msup><mi href="./14.1#p1.t1.r1">x</mi><mn>2</mn></msup></mrow></mfrac><mo>)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>4</mn></mrow></msup><mo></mo><mrow><msub><mi href="./10.25#E2">I</mi><mi href="./14.1#p1.t1.r4">μ</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi href="./14.1#p1.t1.r1">τ</mi><mo></mo><msup><mi href="./14.20#SS8.p3">η</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mrow><mi href="./2.1#E3">O</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>/</mo><mi href="./14.1#p1.t1.r1">τ</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E17.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./2.1#E3" title="(2.1.3) ‣ §2.1(i) Asymptotic and Order Symbols ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m21.png" altimg-height="23px" altimg-valign="-7px" altimg-width="50px" alttext="O\left(\NVar{x}\right)" display="inline"><mrow><mi href="./2.1#E3">O</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>)</mo></mrow></mrow></math>: order not exceeding</a>,
<a href="./14.3#E1" title="(14.3.1) ‣ Ferrers Function of the First Kind ‣ §14.3(i) Interval - 1 < x < 1 ‣ §14.3 Definitions and Hypergeometric Representations ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="23px" altimg-valign="-7px" altimg-width="58px" alttext="\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msubsup><mi href="./14.3#E1" mathvariant="sans-serif">P</mi><mi class="ltx_nvar" href="./14.1#p1.t1.r4">ν</mi><mi class="ltx_nvar" href="./14.1#p1.t1.r4">μ</mi></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./14.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow></math>: Ferrers function of the first kind</a>,
<a href="./10.25#E2" title="(10.25.2) ‣ §10.25(ii) Standard Solutions ‣ §10.25 Definitions ‣ Modified Bessel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m18.png" altimg-height="23px" altimg-valign="-7px" altimg-width="52px" alttext="I_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.25#E2">I</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: modified Bessel function of the first kind</a>,
<a href="./14.1#p1.t1.r1" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./14.1#p1.t1.r1">x</mi></math>: real variable</a>,
<a href="./14.1#p1.t1.r1" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m83.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\tau" display="inline"><mi href="./14.1#p1.t1.r1">τ</mi></math>: real variable</a>,
<a href="./14.1#p1.t1.r4" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m61.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\mu" display="inline"><mi href="./14.1#p1.t1.r4">μ</mi></math>: general order</a>,
<a href="./14.20#SS8.p2" title="§14.20(viii) Asymptotic Approximations: Large τ , 0 ≤ μ ≤ A τ ‣ §14.20 Conical (or Mehler) Functions ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="13px" altimg-valign="-2px" altimg-width="17px" alttext="\alpha" display="inline"><mi href="./14.20#SS8.p2">α</mi></math></a>,
<a href="./14.20#SS8.p2" title="§14.20(viii) Asymptotic Approximations: Large τ , 0 ≤ μ ≤ A τ ‣ §14.20 Conical (or Mehler) Functions ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m73.png" altimg-height="23px" altimg-valign="-7px" altimg-width="64px" alttext="\sigma(\mu,\tau)" display="inline"><mrow><mi href="./14.20#SS8.p2">σ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./14.1#p1.t1.r4">μ</mi><mo>,</mo><mi href="./14.1#p1.t1.r1">τ</mi><mo stretchy="false">)</mo></mrow></mrow></math></a> and
<a href="./14.20#SS8.p3" title="§14.20(viii) Asymptotic Approximations: Large τ , 0 ≤ μ ≤ A τ ‣ §14.20 Conical (or Mehler) Functions ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m36.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="\eta" display="inline"><mi href="./14.20#SS8.p3">η</mi></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E18" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">14.20.18</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E18.png" altimg-height="60px" altimg-valign="-21px" altimg-width="586px" alttext="\widehat{\mathsf{Q}}^{-\mu}_{-\frac{1}{2}+i\tau}\left(x\right)=\sigma(\mu,\tau%
)\left(\frac{\alpha^{2}+\eta}{1+\alpha^{2}-x^{2}}\right)^{1/4}K_{\mu}\left(%
\tau\eta^{1/2}\right)\*\left(1+O\left(\ifrac{1}{\tau}\right)\right)," display="block"><mrow><mrow><mrow><msubsup><mover accent="true"><mi href="./14.20#E2" mathvariant="sans-serif">Q</mi><mo href="./14.20#E2">^</mo></mover><mrow><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./14.1#p1.t1.r1">τ</mi></mrow></mrow><mrow><mo>-</mo><mi href="./14.1#p1.t1.r4">μ</mi></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi href="./14.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mi href="./14.20#SS8.p2">σ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./14.1#p1.t1.r4">μ</mi><mo>,</mo><mi href="./14.1#p1.t1.r1">τ</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><msup><mrow><mo>(</mo><mfrac><mrow><msup><mi href="./14.20#SS8.p2">α</mi><mn>2</mn></msup><mo>+</mo><mi href="./14.20#SS8.p3">η</mi></mrow><mrow><mrow><mn>1</mn><mo>+</mo><msup><mi href="./14.20#SS8.p2">α</mi><mn>2</mn></msup></mrow><mo>-</mo><msup><mi href="./14.1#p1.t1.r1">x</mi><mn>2</mn></msup></mrow></mfrac><mo>)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>4</mn></mrow></msup><mo></mo><mrow><msub><mi href="./10.25#E3">K</mi><mi href="./14.1#p1.t1.r4">μ</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi href="./14.1#p1.t1.r1">τ</mi><mo></mo><msup><mi href="./14.20#SS8.p3">η</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mrow><mi href="./2.1#E3">O</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>/</mo><mi href="./14.1#p1.t1.r1">τ</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E18.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./2.1#E3" title="(2.1.3) ‣ §2.1(i) Asymptotic and Order Symbols ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m21.png" altimg-height="23px" altimg-valign="-7px" altimg-width="50px" alttext="O\left(\NVar{x}\right)" display="inline"><mrow><mi href="./2.1#E3">O</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>)</mo></mrow></mrow></math>: order not exceeding</a>,
<a href="./14.20#E2" title="(14.20.2) ‣ §14.20(i) Definitions and Wronskians ‣ §14.20 Conical (or Mehler) Functions ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m93.png" altimg-height="36px" altimg-valign="-15px" altimg-width="101px" alttext="\widehat{\mathsf{Q}}^{-\NVar{\mu}}_{\NVar{-\frac{1}{2}+i\tau}}\left(\NVar{x}\right)" display="inline"><mrow><msubsup><mover accent="true"><mi href="./14.20#E2" mathvariant="sans-serif">Q</mi><mo href="./14.20#E2">^</mo></mover><mrow><mrow><mo class="ltx_nvar">-</mo><mfrac class="ltx_nvar"><mn class="ltx_nvar">1</mn><mn class="ltx_nvar">2</mn></mfrac></mrow><mo class="ltx_nvar">+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi class="ltx_nvar" href="./14.1#p1.t1.r1">τ</mi></mrow></mrow><mrow><mo>-</mo><mi class="ltx_nvar" href="./14.1#p1.t1.r4">μ</mi></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./14.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow></math>: conical function</a>,
<a href="./10.25#E3" title="(10.25.3) ‣ §10.25(ii) Standard Solutions ‣ §10.25 Definitions ‣ Modified Bessel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="K_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.25#E3">K</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: modified Bessel function of the second kind</a>,
<a href="./14.1#p1.t1.r1" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./14.1#p1.t1.r1">x</mi></math>: real variable</a>,
<a href="./14.1#p1.t1.r1" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m83.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\tau" display="inline"><mi href="./14.1#p1.t1.r1">τ</mi></math>: real variable</a>,
<a href="./14.1#p1.t1.r4" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m61.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\mu" display="inline"><mi href="./14.1#p1.t1.r4">μ</mi></math>: general order</a>,
<a href="./14.20#SS8.p2" title="§14.20(viii) Asymptotic Approximations: Large τ , 0 ≤ μ ≤ A τ ‣ §14.20 Conical (or Mehler) Functions ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="13px" altimg-valign="-2px" altimg-width="17px" alttext="\alpha" display="inline"><mi href="./14.20#SS8.p2">α</mi></math></a>,
<a href="./14.20#SS8.p2" title="§14.20(viii) Asymptotic Approximations: Large τ , 0 ≤ μ ≤ A τ ‣ §14.20 Conical (or Mehler) Functions ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m73.png" altimg-height="23px" altimg-valign="-7px" altimg-width="64px" alttext="\sigma(\mu,\tau)" display="inline"><mrow><mi href="./14.20#SS8.p2">σ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./14.1#p1.t1.r4">μ</mi><mo>,</mo><mi href="./14.1#p1.t1.r1">τ</mi><mo stretchy="false">)</mo></mrow></mrow></math></a> and
<a href="./14.20#SS8.p3" title="§14.20(viii) Asymptotic Approximations: Large τ , 0 ≤ μ ≤ A τ ‣ §14.20 Conical (or Mehler) Functions ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m36.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="\eta" display="inline"><mi href="./14.20#SS8.p3">η</mi></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">uniformly for <math class="ltx_Math" altimg="m105.png" altimg-height="23px" altimg-valign="-7px" altimg-width="130px" alttext="x\in[-1+\delta,1)" display="inline"><mrow><mi href="./14.1#p1.t1.r1">x</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mrow><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">[</mo><mrow><mrow><mo>-</mo><mn>1</mn></mrow><mo>+</mo><mi href="./14.20#SS8.p1">δ</mi></mrow><mo href="./front/introduction#Sx4.p2.t1.r1">,</mo><mn>1</mn><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">)</mo></mrow></mrow></math> and <math class="ltx_Math" altimg="m63.png" altimg-height="23px" altimg-valign="-7px" altimg-width="96px" alttext="\mu\in[0,A\tau]" display="inline"><mrow><mi href="./14.1#p1.t1.r4">μ</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">[</mo><mn>0</mn><mo href="./front/introduction#Sx4.p1.t1.r29">,</mo><mrow><mi href="./14.20#SS8.p1">A</mi><mo></mo><mi href="./14.1#p1.t1.r1">τ</mi></mrow><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">]</mo></mrow></mrow></math>. Here
</p>
<table id="E19" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">14.20.19</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E19.png" altimg-height="25px" altimg-valign="-7px" altimg-width="83px" alttext="\alpha=\mu/\tau," display="block"><mrow><mrow><mi href="./14.20#SS8.p2">α</mi><mo>=</mo><mrow><mi href="./14.1#p1.t1.r4">μ</mi><mo>/</mo><mi href="./14.1#p1.t1.r1">τ</mi></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E19.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./14.1#p1.t1.r1" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m83.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\tau" display="inline"><mi href="./14.1#p1.t1.r1">τ</mi></math>: real variable</a>,
<a href="./14.1#p1.t1.r4" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m61.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\mu" display="inline"><mi href="./14.1#p1.t1.r4">μ</mi></math>: general order</a> and
<a href="./14.20#SS8.p2" title="§14.20(viii) Asymptotic Approximations: Large τ , 0 ≤ μ ≤ A τ ‣ §14.20 Conical (or Mehler) Functions ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="13px" altimg-valign="-2px" altimg-width="17px" alttext="\alpha" display="inline"><mi href="./14.20#SS8.p2">α</mi></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E20" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">14.20.20</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E20.png" altimg-height="57px" altimg-valign="-25px" altimg-width="274px" alttext="\sigma(\mu,\tau)=\frac{\exp\left(\mu-\tau\operatorname{arctan}\alpha\right)}{%
\left(\mu^{2}+\tau^{2}\right)^{\mu/2}}." display="block"><mrow><mrow><mrow><mi href="./14.20#SS8.p2">σ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./14.1#p1.t1.r4">μ</mi><mo>,</mo><mi href="./14.1#p1.t1.r1">τ</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mfrac><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./14.1#p1.t1.r4">μ</mi><mo>-</mo><mrow><mi href="./14.1#p1.t1.r1">τ</mi><mo></mo><mrow><mi href="./4.23#SS2.p1">arctan</mi><mo></mo><mi href="./14.20#SS8.p2">α</mi></mrow></mrow></mrow><mo>)</mo></mrow></mrow><msup><mrow><mo>(</mo><mrow><msup><mi href="./14.1#p1.t1.r4">μ</mi><mn>2</mn></msup><mo>+</mo><msup><mi href="./14.1#p1.t1.r1">τ</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow><mrow><mi href="./14.1#p1.t1.r4">μ</mi><mo>/</mo><mn>2</mn></mrow></msup></mfrac></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E20.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.2#E19" title="(4.2.19) ‣ §4.2(iii) The Exponential Function ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="16px" altimg-valign="-6px" altimg-width="48px" alttext="\exp\NVar{z}" display="inline"><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: exponential function</a>,
<a href="./4.23#SS2.p1" title="§4.23(ii) Principal Values ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m67.png" altimg-height="17px" altimg-valign="-2px" altimg-width="73px" alttext="\operatorname{arctan}\NVar{z}" display="inline"><mrow><mi href="./4.23#SS2.p1">arctan</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: arctangent function</a>,
<a href="./14.1#p1.t1.r1" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m83.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\tau" display="inline"><mi href="./14.1#p1.t1.r1">τ</mi></math>: real variable</a>,
<a href="./14.1#p1.t1.r4" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m61.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\mu" display="inline"><mi href="./14.1#p1.t1.r4">μ</mi></math>: general order</a>,
<a href="./14.20#SS8.p2" title="§14.20(viii) Asymptotic Approximations: Large τ , 0 ≤ μ ≤ A τ ‣ §14.20 Conical (or Mehler) Functions ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="13px" altimg-valign="-2px" altimg-width="17px" alttext="\alpha" display="inline"><mi href="./14.20#SS8.p2">α</mi></math></a> and
<a href="./14.20#SS8.p2" title="§14.20(viii) Asymptotic Approximations: Large τ , 0 ≤ μ ≤ A τ ‣ §14.20 Conical (or Mehler) Functions ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m73.png" altimg-height="23px" altimg-valign="-7px" altimg-width="64px" alttext="\sigma(\mu,\tau)" display="inline"><mrow><mi href="./14.20#SS8.p2">σ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./14.1#p1.t1.r4">μ</mi><mo>,</mo><mi href="./14.1#p1.t1.r1">τ</mi><mo stretchy="false">)</mo></mrow></mrow></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS8.p3" class="ltx_para">
<p class="ltx_p">The variable <math class="ltx_Math" altimg="m36.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="\eta" display="inline"><mi href="./14.20#SS8.p3">η</mi></math> is defined implicitly by
</p>
<table id="E21" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">14.20.21</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_Math ltx_eqn_cell ltx_align_center"><math class="ltx_Math" alttext="{\left(\alpha^{2}+\eta\right)^{1/2}+\tfrac{1}{2}\alpha\ln\eta-\alpha\ln\left(%
\left(\alpha^{2}+\eta\right)^{1/2}+\alpha\right)}={\operatorname{arccos}\left(%
\frac{x}{\left(1+\alpha^{2}\right)^{1/2}}\right)+\frac{\alpha}{2}\ln\left(%
\frac{1+\alpha^{2}+\left(\alpha^{2}-1\right)x^{2}-2\alpha x\left(1+\alpha^{2}-%
x^{2}\right)^{1/2}}{\left(1+\alpha^{2}\right)\left(1-x^{2}\right)}\right)}," display="block"><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mrow><mrow><msup><mrow><mo>(</mo><mrow><msup><mi href="./14.20#SS8.p2">α</mi><mn>2</mn></msup><mo>+</mo><mi href="./14.20#SS8.p3">η</mi></mrow><mo>)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>+</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./14.20#SS8.p2">α</mi><mo></mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi href="./14.20#SS8.p3">η</mi></mrow></mrow></mrow><mo>-</mo><mrow><mi href="./14.20#SS8.p2">α</mi><mo></mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><mo>(</mo><mrow><msup><mrow><mo>(</mo><mrow><msup><mi href="./14.20#SS8.p2">α</mi><mn>2</mn></msup><mo>+</mo><mi href="./14.20#SS8.p3">η</mi></mrow><mo>)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>+</mo><mi href="./14.20#SS8.p2">α</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>=</mo><mrow><mrow><mi href="./4.23#SS2.p1">arccos</mi><mo></mo><mrow><mo>(</mo><mfrac><mi href="./14.1#p1.t1.r1">x</mi><msup><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><msup><mi href="./14.20#SS8.p2">α</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mfrac><mo>)</mo></mrow></mrow><mo>+</mo><mrow><mfrac><mi href="./14.20#SS8.p2">α</mi><mn>2</mn></mfrac><mo></mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><mo>(</mo><mfrac><mrow><mrow><mn>1</mn><mo>+</mo><msup><mi href="./14.20#SS8.p2">α</mi><mn>2</mn></msup><mo>+</mo><mrow><mrow><mo>(</mo><mrow><msup><mi href="./14.20#SS8.p2">α</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo></mo><msup><mi href="./14.1#p1.t1.r1">x</mi><mn>2</mn></msup></mrow></mrow><mo>-</mo><mrow><mn>2</mn><mo></mo><mi href="./14.20#SS8.p2">α</mi><mo></mo><mi href="./14.1#p1.t1.r1">x</mi><mo></mo><msup><mrow><mo>(</mo><mrow><mrow><mn>1</mn><mo>+</mo><msup><mi href="./14.20#SS8.p2">α</mi><mn>2</mn></msup></mrow><mo>-</mo><msup><mi href="./14.1#p1.t1.r1">x</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></mrow><mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><msup><mi href="./14.20#SS8.p2">α</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><msup><mi href="./14.1#p1.t1.r1">x</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow></mrow></mfrac><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mtd></mtr></mtable></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E21.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.23#SS2.p1" title="§4.23(ii) Principal Values ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="13px" altimg-valign="-2px" altimg-width="71px" alttext="\operatorname{arccos}\NVar{z}" display="inline"><mrow><mi href="./4.23#SS2.p1">arccos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: arccosine function</a>,
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m40.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a>,
<a href="./14.1#p1.t1.r1" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./14.1#p1.t1.r1">x</mi></math>: real variable</a>,
<a href="./14.20#SS8.p2" title="§14.20(viii) Asymptotic Approximations: Large τ , 0 ≤ μ ≤ A τ ‣ §14.20 Conical (or Mehler) Functions ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="13px" altimg-valign="-2px" altimg-width="17px" alttext="\alpha" display="inline"><mi href="./14.20#SS8.p2">α</mi></math></a> and
<a href="./14.20#SS8.p3" title="§14.20(viii) Asymptotic Approximations: Large τ , 0 ≤ μ ≤ A τ ‣ §14.20 Conical (or Mehler) Functions ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m36.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="\eta" display="inline"><mi href="./14.20#SS8.p3">η</mi></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where the inverse trigonometric functions take their principal values. The
interval <math class="ltx_Math" altimg="m5.png" altimg-height="18px" altimg-valign="-4px" altimg-width="104px" alttext="-1<x<1" display="inline"><mrow><mrow><mo>-</mo><mn>1</mn></mrow><mo><</mo><mi href="./14.1#p1.t1.r1">x</mi><mo><</mo><mn>1</mn></mrow></math> is mapped one-to-one to the interval
<math class="ltx_Math" altimg="m9.png" altimg-height="20px" altimg-valign="-6px" altimg-width="98px" alttext="0<\eta<\infty" display="inline"><mrow><mn>0</mn><mo><</mo><mi href="./14.20#SS8.p3">η</mi><mo><</mo><mi mathvariant="normal">∞</mi></mrow></math>, with the points <math class="ltx_Math" altimg="m100.png" altimg-height="18px" altimg-valign="-4px" altimg-width="68px" alttext="x=-1" display="inline"><mrow><mi href="./14.1#p1.t1.r1">x</mi><mo>=</mo><mrow><mo>-</mo><mn>1</mn></mrow></mrow></math> and <math class="ltx_Math" altimg="m102.png" altimg-height="17px" altimg-valign="-2px" altimg-width="52px" alttext="x=1" display="inline"><mrow><mi href="./14.1#p1.t1.r1">x</mi><mo>=</mo><mn>1</mn></mrow></math> corresponding to
<math class="ltx_Math" altimg="m35.png" altimg-height="16px" altimg-valign="-6px" altimg-width="61px" alttext="\eta=\infty" display="inline"><mrow><mi href="./14.20#SS8.p3">η</mi><mo>=</mo><mi mathvariant="normal">∞</mi></mrow></math> and <math class="ltx_Math" altimg="m34.png" altimg-height="20px" altimg-valign="-6px" altimg-width="51px" alttext="\eta=0" display="inline"><mrow><mi href="./14.20#SS8.p3">η</mi><mo>=</mo><mn>0</mn></mrow></math>, respectively.</p>
</div>
</section>
<section id="ix" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§14.20(ix) </span>Asymptotic Approximations: Large <math class="ltx_Math" altimg="m61.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\mu" display="inline"><mi href="./14.1#p1.t1.r4">μ</mi></math>,
<math class="ltx_Math" altimg="m12.png" altimg-height="21px" altimg-valign="-6px" altimg-width="105px" alttext="0\leq\tau\leq A\mu" display="inline"><mrow><mn>0</mn><mo>≤</mo><mi href="./14.1#p1.t1.r1">τ</mi><mo>≤</mo><mrow><mi href="./14.20#SS8.p1">A</mi><mo></mo><mi href="./14.1#p1.t1.r4">μ</mi></mrow></mrow></math>
</h2>
<div id="SS9.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="SS9.p1" class="ltx_para">
<p class="ltx_p">As <math class="ltx_Math" altimg="m64.png" altimg-height="16px" altimg-valign="-6px" altimg-width="67px" alttext="\mu\to\infty" display="inline"><mrow><mi href="./14.1#p1.t1.r4">μ</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></math>,</p>
<table id="E22" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">14.20.22</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E22.png" altimg-height="58px" altimg-valign="-25px" altimg-width="619px" alttext="\mathsf{P}^{-\mu}_{-\frac{1}{2}+i\tau}\left(x\right)=\frac{\beta\exp\left(\mu%
\beta\operatorname{arctan}\beta\right)}{\Gamma\left(\mu+1\right)\left(1+\beta^%
{2}\right)^{\mu/2}}\frac{e^{-\mu\rho}}{\left(1+\beta^{2}-x^{2}\beta^{2}\right)%
^{1/4}}\left(1+O\left(\frac{1}{\mu}\right)\right)," display="block"><mrow><mrow><mrow><msubsup><mi href="./14.3#E1" mathvariant="sans-serif">P</mi><mrow><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./14.1#p1.t1.r1">τ</mi></mrow></mrow><mrow><mo>-</mo><mi href="./14.1#p1.t1.r4">μ</mi></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi href="./14.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mfrac><mrow><mi href="./14.20#SS9.p1">β</mi><mo></mo><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./14.1#p1.t1.r4">μ</mi><mo></mo><mi href="./14.20#SS9.p1">β</mi><mo></mo><mrow><mi href="./4.23#SS2.p1">arctan</mi><mo></mo><mi href="./14.20#SS9.p1">β</mi></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./14.1#p1.t1.r4">μ</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mo></mo><msup><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><msup><mi href="./14.20#SS9.p1">β</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow><mrow><mi href="./14.1#p1.t1.r4">μ</mi><mo>/</mo><mn>2</mn></mrow></msup></mrow></mfrac><mo></mo><mfrac><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mi href="./14.1#p1.t1.r4">μ</mi><mo></mo><mi href="./14.20#SS9.p1">ρ</mi></mrow></mrow></msup><msup><mrow><mo>(</mo><mrow><mrow><mn>1</mn><mo>+</mo><msup><mi href="./14.20#SS9.p1">β</mi><mn>2</mn></msup></mrow><mo>-</mo><mrow><msup><mi href="./14.1#p1.t1.r1">x</mi><mn>2</mn></msup><mo></mo><msup><mi href="./14.20#SS9.p1">β</mi><mn>2</mn></msup></mrow></mrow><mo>)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>4</mn></mrow></msup></mfrac><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mrow><mi href="./2.1#E3">O</mi><mo></mo><mrow><mo>(</mo><mfrac><mn>1</mn><mi href="./14.1#p1.t1.r4">μ</mi></mfrac><mo>)</mo></mrow></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E22.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./2.1#E3" title="(2.1.3) ‣ §2.1(i) Asymptotic and Order Symbols ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m21.png" altimg-height="23px" altimg-valign="-7px" altimg-width="50px" alttext="O\left(\NVar{x}\right)" display="inline"><mrow><mi href="./2.1#E3">O</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar">x</mi><mo>)</mo></mrow></mrow></math>: order not exceeding</a>,
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m27.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./14.3#E1" title="(14.3.1) ‣ Ferrers Function of the First Kind ‣ §14.3(i) Interval - 1 < x < 1 ‣ §14.3 Definitions and Hypergeometric Representations ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="23px" altimg-valign="-7px" altimg-width="58px" alttext="\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)" display="inline"><mrow><msubsup><mi href="./14.3#E1" mathvariant="sans-serif">P</mi><mi class="ltx_nvar" href="./14.1#p1.t1.r4">ν</mi><mi class="ltx_nvar" href="./14.1#p1.t1.r4">μ</mi></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./14.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow></math>: Ferrers function of the first kind</a>,
<a href="./4.2#E19" title="(4.2.19) ‣ §4.2(iii) The Exponential Function ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="16px" altimg-valign="-6px" altimg-width="48px" alttext="\exp\NVar{z}" display="inline"><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: exponential function</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m42.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./4.23#SS2.p1" title="§4.23(ii) Principal Values ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m67.png" altimg-height="17px" altimg-valign="-2px" altimg-width="73px" alttext="\operatorname{arctan}\NVar{z}" display="inline"><mrow><mi href="./4.23#SS2.p1">arctan</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: arctangent function</a>,
<a href="./14.1#p1.t1.r1" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./14.1#p1.t1.r1">x</mi></math>: real variable</a>,
<a href="./14.1#p1.t1.r1" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m83.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\tau" display="inline"><mi href="./14.1#p1.t1.r1">τ</mi></math>: real variable</a>,
<a href="./14.1#p1.t1.r4" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m61.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\mu" display="inline"><mi href="./14.1#p1.t1.r4">μ</mi></math>: general order</a>,
<a href="./14.20#SS9.p1" title="§14.20(ix) Asymptotic Approximations: Large μ , 0 ≤ τ ≤ A μ ‣ §14.20 Conical (or Mehler) Functions ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m30.png" altimg-height="21px" altimg-valign="-6px" altimg-width="17px" alttext="\beta" display="inline"><mi href="./14.20#SS9.p1">β</mi></math></a> and
<a href="./14.20#SS9.p1" title="§14.20(ix) Asymptotic Approximations: Large μ , 0 ≤ τ ≤ A μ ‣ §14.20 Conical (or Mehler) Functions ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="\rho" display="inline"><mi href="./14.20#SS9.p1">ρ</mi></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">uniformly for <math class="ltx_Math" altimg="m104.png" altimg-height="23px" altimg-valign="-7px" altimg-width="100px" alttext="x\in(-1,1)" display="inline"><mrow><mi href="./14.1#p1.t1.r1">x</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mn>1</mn><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></mrow></math> and <math class="ltx_Math" altimg="m85.png" altimg-height="23px" altimg-valign="-7px" altimg-width="96px" alttext="\tau\in[0,A\mu]" display="inline"><mrow><mi href="./14.1#p1.t1.r1">τ</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">[</mo><mn>0</mn><mo href="./front/introduction#Sx4.p1.t1.r29">,</mo><mrow><mi href="./14.20#SS8.p1">A</mi><mo></mo><mi href="./14.1#p1.t1.r4">μ</mi></mrow><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">]</mo></mrow></mrow></math>. Here
</p>
<table id="E23" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">14.20.23</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E23.png" altimg-height="25px" altimg-valign="-7px" altimg-width="83px" alttext="\beta=\tau/\mu," display="block"><mrow><mrow><mi href="./14.20#SS9.p1">β</mi><mo>=</mo><mrow><mi href="./14.1#p1.t1.r1">τ</mi><mo>/</mo><mi href="./14.1#p1.t1.r4">μ</mi></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E23.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./14.1#p1.t1.r1" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m83.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\tau" display="inline"><mi href="./14.1#p1.t1.r1">τ</mi></math>: real variable</a>,
<a href="./14.1#p1.t1.r4" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m61.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\mu" display="inline"><mi href="./14.1#p1.t1.r4">μ</mi></math>: general order</a> and
<a href="./14.20#SS9.p1" title="§14.20(ix) Asymptotic Approximations: Large μ , 0 ≤ τ ≤ A μ ‣ §14.20 Conical (or Mehler) Functions ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m30.png" altimg-height="21px" altimg-valign="-6px" altimg-width="17px" alttext="\beta" display="inline"><mi href="./14.20#SS9.p1">β</mi></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and the variable <math class="ltx_Math" altimg="m72.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="\rho" display="inline"><mi href="./14.20#SS9.p1">ρ</mi></math> is defined by
</p>
<table id="E24" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">14.20.24</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_Math ltx_eqn_cell ltx_align_center"><math class="ltx_Math" alttext="\rho=\frac{1}{2}\ln\left(\frac{\left(1-\beta^{2}\right)x^{2}+1+\beta^{2}+2x%
\left(1+\beta^{2}-\beta^{2}x^{2}\right)^{1/2}}{1-x^{2}}\right)+\beta%
\operatorname{arctan}\left(\frac{\beta x}{\sqrt{1+\beta^{2}-\beta^{2}x^{2}}}%
\right)-\frac{1}{2}\ln\left(1+\beta^{2}\right)," display="block"><mrow><mi href="./14.20#SS9.p1">ρ</mi><mo>=</mo><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><mo>(</mo><mfrac><mrow><mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><msup><mi href="./14.20#SS9.p1">β</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow><mo></mo><msup><mi href="./14.1#p1.t1.r1">x</mi><mn>2</mn></msup></mrow><mo>+</mo><mn>1</mn><mo>+</mo><msup><mi href="./14.20#SS9.p1">β</mi><mn>2</mn></msup><mo>+</mo><mrow><mn>2</mn><mo></mo><mi href="./14.1#p1.t1.r1">x</mi><mo></mo><msup><mrow><mo>(</mo><mrow><mrow><mn>1</mn><mo>+</mo><msup><mi href="./14.20#SS9.p1">β</mi><mn>2</mn></msup></mrow><mo>-</mo><mrow><msup><mi href="./14.20#SS9.p1">β</mi><mn>2</mn></msup><mo></mo><msup><mi href="./14.1#p1.t1.r1">x</mi><mn>2</mn></msup></mrow></mrow><mo>)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></mrow><mrow><mn>1</mn><mo>-</mo><msup><mi href="./14.1#p1.t1.r1">x</mi><mn>2</mn></msup></mrow></mfrac><mo>)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mi href="./14.20#SS9.p1">β</mi><mo></mo><mrow><mi href="./4.23#SS2.p1">arctan</mi><mo></mo><mrow><mo>(</mo><mfrac><mrow><mi href="./14.20#SS9.p1">β</mi><mo></mo><mi href="./14.1#p1.t1.r1">x</mi></mrow><msqrt><mrow><mrow><mn>1</mn><mo>+</mo><msup><mi href="./14.20#SS9.p1">β</mi><mn>2</mn></msup></mrow><mo>-</mo><mrow><msup><mi href="./14.20#SS9.p1">β</mi><mn>2</mn></msup><mo></mo><msup><mi href="./14.1#p1.t1.r1">x</mi><mn>2</mn></msup></mrow></mrow></msqrt></mfrac><mo>)</mo></mrow></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><msup><mi href="./14.20#SS9.p1">β</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mtd></mtr></mtable></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E24.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.23#SS2.p1" title="§4.23(ii) Principal Values ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m67.png" altimg-height="17px" altimg-valign="-2px" altimg-width="73px" alttext="\operatorname{arctan}\NVar{z}" display="inline"><mrow><mi href="./4.23#SS2.p1">arctan</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: arctangent function</a>,
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m40.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a>,
<a href="./14.1#p1.t1.r1" title="§14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./14.1#p1.t1.r1">x</mi></math>: real variable</a>,
<a href="./14.20#SS9.p1" title="§14.20(ix) Asymptotic Approximations: Large μ , 0 ≤ τ ≤ A μ ‣ §14.20 Conical (or Mehler) Functions ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m30.png" altimg-height="21px" altimg-valign="-6px" altimg-width="17px" alttext="\beta" display="inline"><mi href="./14.20#SS9.p1">β</mi></math></a> and
<a href="./14.20#SS9.p1" title="§14.20(ix) Asymptotic Approximations: Large μ , 0 ≤ τ ≤ A μ ‣ §14.20 Conical (or Mehler) Functions ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="\rho" display="inline"><mi href="./14.20#SS9.p1">ρ</mi></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS9.p2" class="ltx_para">
<p class="ltx_p">with the inverse tangent taking its principal value. The interval <math class="ltx_Math" altimg="m5.png" altimg-height="18px" altimg-valign="-4px" altimg-width="104px" alttext="-1<x<1" display="inline"><mrow><mrow><mo>-</mo><mn>1</mn></mrow><mo><</mo><mi href="./14.1#p1.t1.r1">x</mi><mo><</mo><mn>1</mn></mrow></math>
is mapped one-to-one to the interval <math class="ltx_Math" altimg="m6.png" altimg-height="19px" altimg-valign="-6px" altimg-width="123px" alttext="-\infty<\rho<\infty" display="inline"><mrow><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mo><</mo><mi href="./14.20#SS9.p1">ρ</mi><mo><</mo><mi mathvariant="normal">∞</mi></mrow></math>, with the points
<math class="ltx_Math" altimg="m100.png" altimg-height="18px" altimg-valign="-4px" altimg-width="68px" alttext="x=-1" display="inline"><mrow><mi href="./14.1#p1.t1.r1">x</mi><mo>=</mo><mrow><mo>-</mo><mn>1</mn></mrow></mrow></math>, <math class="ltx_Math" altimg="m101.png" altimg-height="17px" altimg-valign="-2px" altimg-width="52px" alttext="x=0" display="inline"><mrow><mi href="./14.1#p1.t1.r1">x</mi><mo>=</mo><mn>0</mn></mrow></math>, and <math class="ltx_Math" altimg="m102.png" altimg-height="17px" altimg-valign="-2px" altimg-width="52px" alttext="x=1" display="inline"><mrow><mi href="./14.1#p1.t1.r1">x</mi><mo>=</mo><mn>1</mn></mrow></math> corresponding to <math class="ltx_Math" altimg="m69.png" altimg-height="19px" altimg-valign="-6px" altimg-width="77px" alttext="\rho=-\infty" display="inline"><mrow><mi href="./14.20#SS9.p1">ρ</mi><mo>=</mo><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow></mrow></math>, <math class="ltx_Math" altimg="m70.png" altimg-height="20px" altimg-valign="-6px" altimg-width="51px" alttext="\rho=0" display="inline"><mrow><mi href="./14.20#SS9.p1">ρ</mi><mo>=</mo><mn>0</mn></mrow></math>,
and <math class="ltx_Math" altimg="m71.png" altimg-height="16px" altimg-valign="-6px" altimg-width="61px" alttext="\rho=\infty" display="inline"><mrow><mi href="./14.20#SS9.p1">ρ</mi><mo>=</mo><mi mathvariant="normal">∞</mi></mrow></math>, respectively.</p>
</div>
<div id="SS9.p3" class="ltx_para">
<p class="ltx_p">With the same conditions, the corresponding approximation for
<math class="ltx_Math" altimg="m49.png" altimg-height="35px" altimg-valign="-15px" altimg-width="115px" alttext="\mathsf{P}^{-\mu}_{-\frac{1}{2}+i\tau}\left(-x\right)" display="inline"><mrow><msubsup><mi href="./14.3#E1" mathvariant="sans-serif">P</mi><mrow><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./14.1#p1.t1.r1">τ</mi></mrow></mrow><mrow><mo>-</mo><mi href="./14.1#p1.t1.r4">μ</mi></mrow></msubsup><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mi href="./14.1#p1.t1.r1">x</mi></mrow><mo>)</mo></mrow></mrow></math> is obtainable by replacing
<math class="ltx_Math" altimg="m97.png" altimg-height="19px" altimg-valign="-2px" altimg-width="45px" alttext="e^{-\mu\rho}" display="inline"><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mi href="./14.1#p1.t1.r4">μ</mi><mo></mo><mi href="./14.20#SS9.p1">ρ</mi></mrow></mrow></msup></math> by <math class="ltx_Math" altimg="m98.png" altimg-height="18px" altimg-valign="-2px" altimg-width="33px" alttext="e^{\mu\rho}" display="inline"><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mi href="./14.1#p1.t1.r4">μ</mi><mo></mo><mi href="./14.20#SS9.p1">ρ</mi></mrow></msup></math> on the right-hand side of
(). Approximations for
<math class="ltx_Math" altimg="m54.png" altimg-height="32px" altimg-valign="-15px" altimg-width="99px" alttext="\mathsf{P}^{\mu}_{-\frac{1}{2}+i\tau}\left(x\right)" display="inline"><mrow><msubsup><mi href="./14.3#E1" mathvariant="sans-serif">P</mi><mrow><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./14.1#p1.t1.r1">τ</mi></mrow></mrow><mi href="./14.1#p1.t1.r4">μ</mi></msubsup><mo></mo><mrow><mo>(</mo><mi href="./14.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow></math> and
<math class="ltx_Math" altimg="m95.png" altimg-height="36px" altimg-valign="-15px" altimg-width="101px" alttext="\widehat{\mathsf{Q}}^{-\mu}_{-\frac{1}{2}+i\tau}\left(x\right)" display="inline"><mrow><msubsup><mover accent="true"><mi href="./14.20#E2" mathvariant="sans-serif">Q</mi><mo href="./14.20#E2">^</mo></mover><mrow><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./14.1#p1.t1.r1">τ</mi></mrow></mrow><mrow><mo>-</mo><mi href="./14.1#p1.t1.r4">μ</mi></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi href="./14.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow></math> can then be achieved via
().</p>
</div>
<div id="SS9.p4" class="ltx_para">
<p class="ltx_p">For extensions to complex arguments (including the range <math class="ltx_Math" altimg="m14.png" altimg-height="17px" altimg-valign="-3px" altimg-width="99px" alttext="1<x<\infty" display="inline"><mrow><mn>1</mn><mo><</mo><mi href="./14.1#p1.t1.r1">x</mi><mo><</mo><mi mathvariant="normal">∞</mi></mrow></math>),
asymptotic expansions, and explicit error bounds, see <cite class="ltx_cite ltx_citemacro_citet">Dunster (</dd>
</dl>
</div>
</div>
<div id="SS10.p1" class="ltx_para">
<p class="ltx_p">For zeros of <math class="ltx_Math" altimg="m56.png" altimg-height="28px" altimg-valign="-12px" altimg-width="99px" alttext="\mathsf{P}_{-\frac{1}{2}+i\tau}\left(x\right)" display="inline"><mrow><msub><mi href="./14.2#SS2.p2" mathvariant="sans-serif">P</mi><mrow><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./14.1#p1.t1.r1">τ</mi></mrow></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./14.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow></math> see
<cite class="ltx_cite ltx_citemacro_citet">Hobson (, §237)</cite>.</p>
</div>
<div id="SS10.p2" class="ltx_para">
<p class="ltx_p">For integrals with respect to <math class="ltx_Math" altimg="m83.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\tau" display="inline"><mi href="./14.1#p1.t1.r1">τ</mi></math> involving
<math class="ltx_Math" altimg="m56.png" altimg-height="28px" altimg-valign="-12px" altimg-width="99px" alttext="\mathsf{P}_{-\frac{1}{2}+i\tau}\left(x\right)" display="inline"><mrow><msub><mi href="./14.2#SS2.p2" mathvariant="sans-serif">P</mi><mrow><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./14.1#p1.t1.r1">τ</mi></mrow></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./14.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow></math>, see
<cite class="ltx_cite ltx_citemacro_citet">Prudnikov<span class="ltx_text ltx_bib_etal"> et al.</span> (</div>
</div>
</body></text>
</html>
</page>
<page>
<!DOCTYPE html><html>
<head>
<title>DLMF: 21.7 Riemann Surfaces</title>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
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<div class="ltx_page_navbar">
<div class="ltx_page_navlogo">)</cite>). Consider the set of points in <math class="ltx_Math" altimg="m87.png" altimg-height="20px" altimg-valign="-2px" altimg-width="28px" alttext="{\mathbb{C}^{2}}" display="inline"><msup><mi href="./front/introduction#Sx4.p1.t1.r1">ℂ</mi><mn>2</mn></msup></math> that
satisfy the equation</p>
<table id="E1" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">21.7.1</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E1.png" altimg-height="25px" altimg-valign="-7px" altimg-width="112px" alttext="P(\lambda,\mu)=0," display="block"><mrow><mrow><mrow><mi href="./21.7#SS1.p1">P</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>λ</mi><mo>,</mo><mi>μ</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mn>0</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./21.7#SS1.p1" title="§21.7(i) Connection of Riemann Theta Functions to Riemann Surfaces ‣ §21.7 Riemann Surfaces ‣ Applications ‣ Chapter 21 Multidimensional Theta Functions" class="ltx_ref"><math class="ltx_Math" altimg="m16.png" altimg-height="23px" altimg-valign="-7px" altimg-width="68px" alttext="P(\lambda,\mu)" display="inline"><mrow><mi href="./21.7#SS1.p1">P</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>λ</mi><mo>,</mo><mi>μ</mi><mo stretchy="false">)</mo></mrow></mrow></math>: polynomial</a></dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m16.png" altimg-height="23px" altimg-valign="-7px" altimg-width="68px" alttext="P(\lambda,\mu)" display="inline"><mrow><mi href="./21.7#SS1.p1">P</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>λ</mi><mo>,</mo><mi>μ</mi><mo stretchy="false">)</mo></mrow></mrow></math> is a polynomial in <math class="ltx_Math" altimg="m53.png" altimg-height="18px" altimg-valign="-2px" altimg-width="16px" alttext="\lambda" display="inline"><mi>λ</mi></math> and <math class="ltx_Math" altimg="m60.png" altimg-height="16px" altimg-valign="-6px" altimg-width="16px" alttext="\mu" display="inline"><mi>μ</mi></math> that does not
factor over <math class="ltx_Math" altimg="m87.png" altimg-height="20px" altimg-valign="-2px" altimg-width="28px" alttext="{\mathbb{C}^{2}}" display="inline"><msup><mi href="./front/introduction#Sx4.p1.t1.r1">ℂ</mi><mn>2</mn></msup></math>. Equation () determines a plane
algebraic curve in <math class="ltx_Math" altimg="m87.png" altimg-height="20px" altimg-valign="-2px" altimg-width="28px" alttext="{\mathbb{C}^{2}}" display="inline"><msup><mi href="./front/introduction#Sx4.p1.t1.r1">ℂ</mi><mn>2</mn></msup></math>, which is made compact by adding its points at
infinity. To accomplish this we write () in terms of
homogeneous coordinates:</p>
<table id="E2" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">21.7.2</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E2.png" altimg-height="29px" altimg-valign="-7px" altimg-width="131px" alttext="\tilde{P}(\tilde{\lambda},\tilde{\mu},\tilde{\eta})=0," display="block"><mrow><mrow><mrow><mover accent="true"><mi href="./21.7#SS1.p1">P</mi><mo stretchy="false">~</mo></mover><mo></mo><mrow><mo stretchy="false">(</mo><mover accent="true"><mi>λ</mi><mo stretchy="false">~</mo></mover><mo>,</mo><mover accent="true"><mi>μ</mi><mo stretchy="false">~</mo></mover><mo>,</mo><mover accent="true"><mi>η</mi><mo stretchy="false">~</mo></mover><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mn>0</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./21.7#SS1.p1" title="§21.7(i) Connection of Riemann Theta Functions to Riemann Surfaces ‣ §21.7 Riemann Surfaces ‣ Applications ‣ Chapter 21 Multidimensional Theta Functions" class="ltx_ref"><math class="ltx_Math" altimg="m16.png" altimg-height="23px" altimg-valign="-7px" altimg-width="68px" alttext="P(\lambda,\mu)" display="inline"><mrow><mi href="./21.7#SS1.p1">P</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>λ</mi><mo>,</mo><mi>μ</mi><mo stretchy="false">)</mo></mrow></mrow></math>: polynomial</a></dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">by setting <math class="ltx_Math" altimg="m52.png" altimg-height="27px" altimg-valign="-7px" altimg-width="75px" alttext="\lambda=\tilde{\lambda}/\tilde{\eta}" display="inline"><mrow><mi>λ</mi><mo>=</mo><mrow><mover accent="true"><mi>λ</mi><mo stretchy="false">~</mo></mover><mo>/</mo><mover accent="true"><mi>η</mi><mo stretchy="false">~</mo></mover></mrow></mrow></math>,
<math class="ltx_Math" altimg="m59.png" altimg-height="23px" altimg-valign="-7px" altimg-width="75px" alttext="\mu=\tilde{\mu}/\tilde{\eta}" display="inline"><mrow><mi>μ</mi><mo>=</mo><mrow><mover accent="true"><mi>μ</mi><mo stretchy="false">~</mo></mover><mo>/</mo><mover accent="true"><mi>η</mi><mo stretchy="false">~</mo></mover></mrow></mrow></math>, and then clearing fractions. This compact
curve may have singular points, that is, points at which the gradient of
<math class="ltx_Math" altimg="m72.png" altimg-height="22px" altimg-valign="-2px" altimg-width="20px" alttext="\tilde{P}" display="inline"><mover accent="true"><mi href="./21.7#SS1.p1">P</mi><mo stretchy="false">~</mo></mover></math> vanishes. Removing the singularities of this curve gives rise to a
two-dimensional connected manifold with a complex-analytic structure, that is, a <em class="ltx_emph ltx_font_italic">Riemann
surface. All compact Riemann surfaces can be obtained this
way.</em></p>
</div>
<div id="SS1.p2" class="ltx_para">
<p class="ltx_p">Since a Riemann surface <math class="ltx_Math" altimg="m29.png" altimg-height="18px" altimg-valign="-2px" altimg-width="17px" alttext="\Gamma" display="inline"><mi href="./21.7#i" mathvariant="normal">Γ</mi></math> is a two-dimensional manifold that is
orientable (owing to its analytic structure), its only topological invariant is
its <em class="ltx_emph ltx_font_italic">genus</em> <math class="ltx_Math" altimg="m80.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="g" display="inline"><mi href="./21.1#p2.t1.r1">g</mi></math> (the number of <em class="ltx_emph ltx_font_italic">handles</em> in the surface). On this
surface, we choose <math class="ltx_Math" altimg="m10.png" altimg-height="20px" altimg-valign="-6px" altimg-width="24px" alttext="2g" display="inline"><mrow><mn>2</mn><mo></mo><mi href="./21.1#p2.t1.r1">g</mi></mrow></math> <em class="ltx_emph ltx_font_italic">cycles</em> (that is, closed oriented curves, each
with at most a finite number of singular points) <math class="ltx_Math" altimg="m73.png" altimg-height="18px" altimg-valign="-8px" altimg-width="23px" alttext="a_{j}" display="inline"><msub><mi href="./21.7#SS1.p2">a</mi><mi>j</mi></msub></math>, <math class="ltx_Math" altimg="m74.png" altimg-height="23px" altimg-valign="-8px" altimg-width="21px" alttext="b_{j}" display="inline"><msub><mi href="./21.7#SS1.p2">b</mi><mi>j</mi></msub></math>,
<math class="ltx_Math" altimg="m84.png" altimg-height="20px" altimg-valign="-6px" altimg-width="123px" alttext="j=1,2,\dots,g" display="inline"><mrow><mi>j</mi><mo>=</mo><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><mi href="./21.1#p2.t1.r1">g</mi></mrow></mrow></math>, such that their
<em class="ltx_emph ltx_font_italic">intersection indices</em> satisfy</p>
<table id="E3" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex1" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="3" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">21.7.3</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m3.png" altimg-height="20px" altimg-valign="-8px" altimg-width="64px" alttext="\displaystyle a_{j}\circ a_{k}" display="inline"><mrow><msub><mi href="./21.7#SS1.p2">a</mi><mi>j</mi></msub><mo>∘</mo><msub><mi href="./21.7#SS1.p2">a</mi><mi>k</mi></msub></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m1.png" altimg-height="22px" altimg-valign="-6px" altimg-width="43px" alttext="\displaystyle=0," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mn>0</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex2" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m5.png" altimg-height="25px" altimg-valign="-8px" altimg-width="60px" alttext="\displaystyle b_{j}\circ b_{k}" display="inline"><mrow><msub><mi href="./21.7#SS1.p2">b</mi><mi>j</mi></msub><mo>∘</mo><msub><mi href="./21.7#SS1.p2">b</mi><mi>k</mi></msub></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m1.png" altimg-height="22px" altimg-valign="-6px" altimg-width="43px" alttext="\displaystyle=0," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mn>0</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex3" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m4.png" altimg-height="25px" altimg-valign="-8px" altimg-width="62px" alttext="\displaystyle a_{j}\circ b_{k}" display="inline"><mrow><msub><mi href="./21.7#SS1.p2">a</mi><mi>j</mi></msub><mo>∘</mo><msub><mi href="./21.7#SS1.p2">b</mi><mi>k</mi></msub></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m2.png" altimg-height="25px" altimg-valign="-8px" altimg-width="63px" alttext="\displaystyle=\delta_{j,k}." display="inline"><mrow><mrow><mi></mi><mo>=</mo><msub><mi href="./front/introduction#Sx4.p1.t1.r4">δ</mi><mrow><mi>j</mi><mo href="./front/introduction#Sx4.p1.t1.r4">,</mo><mi>k</mi></mrow></msub></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./front/introduction#Sx4.p1.t1.r4" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="23px" altimg-valign="-8px" altimg-width="35px" alttext="\delta_{\NVar{j},\NVar{k}}" display="inline"><msub><mi href="./front/introduction#Sx4.p1.t1.r4">δ</mi><mrow><mi class="ltx_nvar">j</mi><mo href="./front/introduction#Sx4.p1.t1.r4">,</mo><mi class="ltx_nvar">k</mi></mrow></msub></math>: Kronecker delta</a>,
<a href="./21.7#SS1.p2" title="§21.7(i) Connection of Riemann Theta Functions to Riemann Surfaces ‣ §21.7 Riemann Surfaces ‣ Applications ‣ Chapter 21 Multidimensional Theta Functions" class="ltx_ref"><math class="ltx_Math" altimg="m73.png" altimg-height="18px" altimg-valign="-8px" altimg-width="23px" alttext="a_{j}" display="inline"><msub><mi href="./21.7#SS1.p2">a</mi><mi>j</mi></msub></math>: cycles</a> and
<a href="./21.7#SS1.p2" title="§21.7(i) Connection of Riemann Theta Functions to Riemann Surfaces ‣ §21.7 Riemann Surfaces ‣ Applications ‣ Chapter 21 Multidimensional Theta Functions" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="23px" altimg-valign="-8px" altimg-width="21px" alttext="b_{j}" display="inline"><msub><mi href="./21.7#SS1.p2">b</mi><mi>j</mi></msub></math>: cycles</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">For example, Figure </dd>
</dl>
</div>
</div>
</figure>
<div id="SS1.p3" class="ltx_para">
<p class="ltx_p">On a Riemann surface of genus <math class="ltx_Math" altimg="m80.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="g" display="inline"><mi href="./21.1#p2.t1.r1">g</mi></math>, there are <math class="ltx_Math" altimg="m80.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="g" display="inline"><mi href="./21.1#p2.t1.r1">g</mi></math> linearly independent
<em class="ltx_emph ltx_font_italic">holomorphic differentials</em> <math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-8px" altimg-width="25px" alttext="\omega_{j}" display="inline"><msub><mi href="./21.1#p2.t1.r20">ω</mi><mi>j</mi></msub></math>, <math class="ltx_Math" altimg="m84.png" altimg-height="20px" altimg-valign="-6px" altimg-width="123px" alttext="j=1,2,\dots,g" display="inline"><mrow><mi>j</mi><mo>=</mo><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><mi href="./21.1#p2.t1.r1">g</mi></mrow></mrow></math>. If a local
coordinate <math class="ltx_Math" altimg="m86.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi>z</mi></math> is chosen on the Riemann surface, then the local coordinate
representation of these holomorphic differentials is given by</p>
<table id="E4" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">21.7.4</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E4.png" altimg-height="26px" altimg-valign="-8px" altimg-width="124px" alttext="\omega_{j}=f_{j}(z)\mathrm{d}z," display="block"><mrow><mrow><msub><mi href="./21.1#p2.t1.r20">ω</mi><mi>j</mi></msub><mo>=</mo><mrow><mrow><msub><mi href="./21.7#SS1.p3">f</mi><mi>j</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>z</mi></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m84.png" altimg-height="20px" altimg-valign="-6px" altimg-width="123px" alttext="j=1,2,\dots,g" display="inline"><mrow><mi>j</mi><mo>=</mo><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><mi href="./21.1#p2.t1.r1">g</mi></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E4.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./21.1#p2.t1.r1" title="§21.1 Special Notation ‣ Notation ‣ Chapter 21 Multidimensional Theta Functions" class="ltx_ref"><math class="ltx_Math" altimg="m80.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="g" display="inline"><mi href="./21.1#p2.t1.r1">g</mi></math>: positive integer</a>,
<a href="./21.1#p2.t1.r20" title="§21.1 Special Notation ‣ Notation ‣ Chapter 21 Multidimensional Theta Functions" class="ltx_ref"><math class="ltx_Math" altimg="m62.png" altimg-height="13px" altimg-valign="-2px" altimg-width="17px" alttext="\omega" display="inline"><mi href="./21.1#p2.t1.r20">ω</mi></math>: differential</a> and
<a href="./21.7#SS1.p3" title="§21.7(i) Connection of Riemann Theta Functions to Riemann Surfaces ‣ §21.7 Riemann Surfaces ‣ Applications ‣ Chapter 21 Multidimensional Theta Functions" class="ltx_ref"><math class="ltx_Math" altimg="m76.png" altimg-height="24px" altimg-valign="-8px" altimg-width="48px" alttext="f_{j}(z)" display="inline"><mrow><msub><mi href="./21.7#SS1.p3">f</mi><mi>j</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: analytic functions</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m76.png" altimg-height="24px" altimg-valign="-8px" altimg-width="48px" alttext="f_{j}(z)" display="inline"><mrow><msub><mi href="./21.7#SS1.p3">f</mi><mi>j</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math>, <math class="ltx_Math" altimg="m84.png" altimg-height="20px" altimg-valign="-6px" altimg-width="123px" alttext="j=1,2,\dots,g" display="inline"><mrow><mi>j</mi><mo>=</mo><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><mi href="./21.1#p2.t1.r1">g</mi></mrow></mrow></math> are analytic functions. Thus the
differentials <math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-8px" altimg-width="25px" alttext="\omega_{j}" display="inline"><msub><mi href="./21.1#p2.t1.r20">ω</mi><mi>j</mi></msub></math>, <math class="ltx_Math" altimg="m84.png" altimg-height="20px" altimg-valign="-6px" altimg-width="123px" alttext="j=1,2,\dots,g" display="inline"><mrow><mi>j</mi><mo>=</mo><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><mi href="./21.1#p2.t1.r1">g</mi></mrow></mrow></math> have no singularities on <math class="ltx_Math" altimg="m29.png" altimg-height="18px" altimg-valign="-2px" altimg-width="17px" alttext="\Gamma" display="inline"><mi href="./21.7#i" mathvariant="normal">Γ</mi></math>.
Note that for the purposes of integrating these holomorphic differentials, all
cycles on the surface are a linear combination of the cycles <math class="ltx_Math" altimg="m73.png" altimg-height="18px" altimg-valign="-8px" altimg-width="23px" alttext="a_{j}" display="inline"><msub><mi href="./21.7#SS1.p2">a</mi><mi>j</mi></msub></math>, <math class="ltx_Math" altimg="m74.png" altimg-height="23px" altimg-valign="-8px" altimg-width="21px" alttext="b_{j}" display="inline"><msub><mi href="./21.7#SS1.p2">b</mi><mi>j</mi></msub></math>,
<math class="ltx_Math" altimg="m84.png" altimg-height="20px" altimg-valign="-6px" altimg-width="123px" alttext="j=1,2,\dots,g" display="inline"><mrow><mi>j</mi><mo>=</mo><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><mi href="./21.1#p2.t1.r1">g</mi></mrow></mrow></math>. The <math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-8px" altimg-width="25px" alttext="\omega_{j}" display="inline"><msub><mi href="./21.1#p2.t1.r20">ω</mi><mi>j</mi></msub></math> are normalized so that</p>
<table id="E5" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">21.7.5</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E5.png" altimg-height="53px" altimg-valign="-22px" altimg-width="122px" alttext="\oint_{a_{k}}\omega_{j}=\delta_{j,k}," display="block"><mrow><mrow><mrow><msub><mo largeop="true" symmetric="true">∮</mo><msub><mi href="./21.7#SS1.p2">a</mi><mi>k</mi></msub></msub><msub><mi href="./21.1#p2.t1.r20">ω</mi><mi>j</mi></msub></mrow><mo>=</mo><msub><mi href="./front/introduction#Sx4.p1.t1.r4">δ</mi><mrow><mi>j</mi><mo href="./front/introduction#Sx4.p1.t1.r4">,</mo><mi>k</mi></mrow></msub></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m81.png" altimg-height="21px" altimg-valign="-6px" altimg-width="142px" alttext="j,k=1,2,\dots,g" display="inline"><mrow><mrow><mrow><mi>j</mi><mo>,</mo><mi>k</mi></mrow><mo>=</mo><mn>1</mn></mrow><mo>,</mo><mrow><mn>2</mn><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><mi href="./21.1#p2.t1.r1">g</mi></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E5.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./front/introduction#Sx4.p1.t1.r4" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m47.png" altimg-height="23px" altimg-valign="-8px" altimg-width="35px" alttext="\delta_{\NVar{j},\NVar{k}}" display="inline"><msub><mi href="./front/introduction#Sx4.p1.t1.r4">δ</mi><mrow><mi class="ltx_nvar">j</mi><mo href="./front/introduction#Sx4.p1.t1.r4">,</mo><mi class="ltx_nvar">k</mi></mrow></msub></math>: Kronecker delta</a>,
<a href="./21.1#p2.t1.r1" title="§21.1 Special Notation ‣ Notation ‣ Chapter 21 Multidimensional Theta Functions" class="ltx_ref"><math class="ltx_Math" altimg="m80.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="g" display="inline"><mi href="./21.1#p2.t1.r1">g</mi></math>: positive integer</a>,
<a href="./21.1#p2.t1.r20" title="§21.1 Special Notation ‣ Notation ‣ Chapter 21 Multidimensional Theta Functions" class="ltx_ref"><math class="ltx_Math" altimg="m62.png" altimg-height="13px" altimg-valign="-2px" altimg-width="17px" alttext="\omega" display="inline"><mi href="./21.1#p2.t1.r20">ω</mi></math>: differential</a> and
<a href="./21.7#SS1.p2" title="§21.7(i) Connection of Riemann Theta Functions to Riemann Surfaces ‣ §21.7 Riemann Surfaces ‣ Applications ‣ Chapter 21 Multidimensional Theta Functions" class="ltx_ref"><math class="ltx_Math" altimg="m73.png" altimg-height="18px" altimg-valign="-8px" altimg-width="23px" alttext="a_{j}" display="inline"><msub><mi href="./21.7#SS1.p2">a</mi><mi>j</mi></msub></math>: cycles</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Then the matrix defined by
</p>
<table id="E6" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">21.7.6</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E6.png" altimg-height="53px" altimg-valign="-22px" altimg-width="122px" alttext="\Omega_{jk}=\oint_{b_{k}}\omega_{j}," display="block"><mrow><mrow><msub><mi mathvariant="normal">Ω</mi><mrow><mi>j</mi><mo></mo><mi>k</mi></mrow></msub><mo>=</mo><mrow><msub><mo largeop="true" symmetric="true">∮</mo><msub><mi href="./21.7#SS1.p2">b</mi><mi>k</mi></msub></msub><msub><mi href="./21.1#p2.t1.r20">ω</mi><mi>j</mi></msub></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m81.png" altimg-height="21px" altimg-valign="-6px" altimg-width="142px" alttext="j,k=1,2,\dots,g" display="inline"><mrow><mrow><mrow><mi>j</mi><mo>,</mo><mi>k</mi></mrow><mo>=</mo><mn>1</mn></mrow><mo>,</mo><mrow><mn>2</mn><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><mi href="./21.1#p2.t1.r1">g</mi></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E6.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./21.1#p2.t1.r1" title="§21.1 Special Notation ‣ Notation ‣ Chapter 21 Multidimensional Theta Functions" class="ltx_ref"><math class="ltx_Math" altimg="m80.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="g" display="inline"><mi href="./21.1#p2.t1.r1">g</mi></math>: positive integer</a>,
<a href="./21.1#p2.t1.r20" title="§21.1 Special Notation ‣ Notation ‣ Chapter 21 Multidimensional Theta Functions" class="ltx_ref"><math class="ltx_Math" altimg="m62.png" altimg-height="13px" altimg-valign="-2px" altimg-width="17px" alttext="\omega" display="inline"><mi href="./21.1#p2.t1.r20">ω</mi></math>: differential</a> and
<a href="./21.7#SS1.p2" title="§21.7(i) Connection of Riemann Theta Functions to Riemann Surfaces ‣ §21.7 Riemann Surfaces ‣ Applications ‣ Chapter 21 Multidimensional Theta Functions" class="ltx_ref"><math class="ltx_Math" altimg="m74.png" altimg-height="23px" altimg-valign="-8px" altimg-width="21px" alttext="b_{j}" display="inline"><msub><mi href="./21.7#SS1.p2">b</mi><mi>j</mi></msub></math>: cycles</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">is a Riemann matrix and it is used to define the corresponding Riemann theta
function. <em class="ltx_emph ltx_font_italic">In this way, we associate a Riemann theta function with every
compact Riemann surface <math class="ltx_Math" altimg="m29.png" altimg-height="18px" altimg-valign="-2px" altimg-width="17px" alttext="\Gamma" display="inline"><mi href="./21.7#i" mathvariant="normal">Γ</mi></math>.</em></p>
</div>
<div id="SS1.p4" class="ltx_para">
<p class="ltx_p">Riemann theta functions originating from Riemann surfaces are special in the
sense that a general <math class="ltx_Math" altimg="m80.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="g" display="inline"><mi href="./21.1#p2.t1.r1">g</mi></math>-dimensional Riemann theta function depends on
<math class="ltx_Math" altimg="m77.png" altimg-height="23px" altimg-valign="-7px" altimg-width="94px" alttext="g(g+1)/2" display="inline"><mrow><mrow><mi href="./21.1#p2.t1.r1">g</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./21.1#p2.t1.r1">g</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>/</mo><mn>2</mn></mrow></math> complex parameters. In contrast, a <math class="ltx_Math" altimg="m80.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="g" display="inline"><mi href="./21.1#p2.t1.r1">g</mi></math>-dimensional Riemann theta
function arising from a compact Riemann surface of genus <math class="ltx_Math" altimg="m80.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="g" display="inline"><mi href="./21.1#p2.t1.r1">g</mi></math> (<math class="ltx_Math" altimg="m12.png" altimg-height="17px" altimg-valign="-3px" altimg-width="35px" alttext=">1" display="inline"><mrow><mi></mi><mo>></mo><mn>1</mn></mrow></math>)
depends on at most <math class="ltx_Math" altimg="m11.png" altimg-height="20px" altimg-valign="-6px" altimg-width="59px" alttext="3g-3" display="inline"><mrow><mrow><mn>3</mn><mo></mo><mi href="./21.1#p2.t1.r1">g</mi></mrow><mo>-</mo><mn>3</mn></mrow></math> complex parameters (one complex parameter for the
case <math class="ltx_Math" altimg="m79.png" altimg-height="20px" altimg-valign="-6px" altimg-width="51px" alttext="g=1" display="inline"><mrow><mi href="./21.1#p2.t1.r1">g</mi><mo>=</mo><mn>1</mn></mrow></math>). These special Riemann theta functions satisfy many special
identities, two of which appear in the following subsections. For more
information, see <cite class="ltx_cite ltx_citemacro_citet">Dubrovin (</dd>
</dl>
</div>
</div>
<div id="SS2.p1" class="ltx_para">
<p class="ltx_p">Let <math class="ltx_Math" altimg="m31.png" altimg-height="13px" altimg-valign="-2px" altimg-width="19px" alttext="\boldsymbol{{\alpha}}" display="inline"><mi href="./21.1#p2.t1.r6" mathvariant="bold-italic">α</mi></math>, <math class="ltx_Math" altimg="m34.png" altimg-height="21px" altimg-valign="-6px" altimg-width="18px" alttext="\boldsymbol{{\beta}}" display="inline"><mi href="./21.1#p2.t1.r6" mathvariant="bold-italic">β</mi></math> be such that</p>
</div>
<div id="SS2.p2" class="ltx_para">
<table id="E7" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">21.7.7</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E7.png" altimg-height="55px" altimg-valign="-21px" altimg-width="468px" alttext="\left(\frac{\partial}{\partial z_{1}}\theta\genfrac{[}{]}{0.0pt}{}{\boldsymbol%
{{\alpha}}}{\boldsymbol{{\beta}}}\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}%
\right)\Big{|}_{\mathbf{z}=\boldsymbol{{0}}},\dots,\frac{\partial}{\partial z_%
{g}}\theta\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}}{\boldsymbol{{\beta}}}%
\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)\Big{|}_{\mathbf{z}=%
\boldsymbol{{0}}}\right)\neq\boldsymbol{{0}}." display="block"><mrow><mrow><mrow><mo>(</mo><msub><mrow><mrow><mfrac><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><msub><mi>z</mi><mn>1</mn></msub></mrow></mfrac><mo></mo><mrow><mrow><mi href="./21.2#E5">θ</mi><mo href="./21.2#E5"></mo><mrow><mo href="./21.2#E5">[</mo><mfrac linethickness="0.0pt"><mi href="./21.1#p2.t1.r6" mathvariant="bold-italic">α</mi><mi href="./21.1#p2.t1.r6" mathvariant="bold-italic">β</mi></mfrac><mo href="./21.2#E5">]</mo></mrow></mrow><mo></mo><mrow><mo>(</mo><mi mathvariant="bold">z</mi><mo>|</mo><mi href="./21.1#p2.t1.r5" mathvariant="bold">Ω</mi><mo>)</mo></mrow></mrow></mrow><mo fence="true" maxsize="160%" minsize="160%">|</mo></mrow><mrow><mi mathvariant="bold">z</mi><mo>=</mo><mn mathvariant="bold">0</mn></mrow></msub><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><msub><mrow><mrow><mfrac><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><msub><mi>z</mi><mi href="./21.1#p2.t1.r1">g</mi></msub></mrow></mfrac><mo></mo><mrow><mrow><mi href="./21.2#E5">θ</mi><mo href="./21.2#E5"></mo><mrow><mo href="./21.2#E5">[</mo><mfrac linethickness="0.0pt"><mi href="./21.1#p2.t1.r6" mathvariant="bold-italic">α</mi><mi href="./21.1#p2.t1.r6" mathvariant="bold-italic">β</mi></mfrac><mo href="./21.2#E5">]</mo></mrow></mrow><mo></mo><mrow><mo>(</mo><mi mathvariant="bold">z</mi><mo>|</mo><mi href="./21.1#p2.t1.r5" mathvariant="bold">Ω</mi><mo>)</mo></mrow></mrow></mrow><mo fence="true" maxsize="160%" minsize="160%">|</mo></mrow><mrow><mi mathvariant="bold">z</mi><mo>=</mo><mn mathvariant="bold">0</mn></mrow></msub><mo>)</mo></mrow><mo>≠</mo><mn mathvariant="bold">0</mn></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E7.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./21.2#E5" title="(21.2.5) ‣ §21.2(ii) Riemann Theta Functions with Characteristics ‣ §21.2 Definitions ‣ Properties ‣ Chapter 21 Multidimensional Theta Functions" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="30px" altimg-valign="-12px" altimg-width="94px" alttext="\theta\genfrac{[}{]}{0.0pt}{}{\NVar{\boldsymbol{{\alpha}}}}{\NVar{\boldsymbol{%
{\beta}}}}\left(\NVar{\mathbf{z}}\middle|\NVar{\boldsymbol{{\Omega}}}\right)" display="inline"><mrow><mrow><mi href="./21.2#E5">θ</mi><mo href="./21.2#E5"></mo><mrow><mo href="./21.2#E5">[</mo><mfrac linethickness="0.0pt"><mi class="ltx_nvar" href="./21.1#p2.t1.r6" mathvariant="bold-italic">α</mi><mi class="ltx_nvar" href="./21.1#p2.t1.r6" mathvariant="bold-italic">β</mi></mfrac><mo href="./21.2#E5">]</mo></mrow></mrow><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" mathvariant="bold">z</mi><mo>|</mo><mi class="ltx_nvar" href="./21.1#p2.t1.r5" mathvariant="bold">Ω</mi><mo>)</mo></mrow></mrow></math>: Riemann theta function with characteristics</a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m75.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m64.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./21.1#p2.t1.r1" title="§21.1 Special Notation ‣ Notation ‣ Chapter 21 Multidimensional Theta Functions" class="ltx_ref"><math class="ltx_Math" altimg="m80.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="g" display="inline"><mi href="./21.1#p2.t1.r1">g</mi></math>: positive integer</a>,
<a href="./21.1#p2.t1.r5" title="§21.1 Special Notation ‣ Notation ‣ Chapter 21 Multidimensional Theta Functions" class="ltx_ref"><math class="ltx_Math" altimg="m30.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="\boldsymbol{{\Omega}}" display="inline"><mi href="./21.1#p2.t1.r5" mathvariant="bold">Ω</mi></math>: a Riemann matrix</a>,
<a href="./21.1#p2.t1.r6" title="§21.1 Special Notation ‣ Notation ‣ Chapter 21 Multidimensional Theta Functions" class="ltx_ref"><math class="ltx_Math" altimg="m31.png" altimg-height="13px" altimg-valign="-2px" altimg-width="19px" alttext="\boldsymbol{{\alpha}}" display="inline"><mi href="./21.1#p2.t1.r6" mathvariant="bold-italic">α</mi></math>: <math class="ltx_Math" altimg="m80.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="g" display="inline"><mi href="./21.1#p2.t1.r1">g</mi></math>-dimensional vector</a> and
<a href="./21.1#p2.t1.r6" title="§21.1 Special Notation ‣ Notation ‣ Chapter 21 Multidimensional Theta Functions" class="ltx_ref"><math class="ltx_Math" altimg="m34.png" altimg-height="21px" altimg-valign="-6px" altimg-width="18px" alttext="\boldsymbol{{\beta}}" display="inline"><mi href="./21.1#p2.t1.r6" mathvariant="bold-italic">β</mi></math>: <math class="ltx_Math" altimg="m80.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="g" display="inline"><mi href="./21.1#p2.t1.r1">g</mi></math>-dimensional vector</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Define the holomorphic differential</p>
<table id="E8" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">21.7.8</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E8.png" altimg-height="68px" altimg-valign="-30px" altimg-width="273px" alttext="\zeta=\sum_{j=1}^{g}\omega_{j}\frac{\partial}{\partial z_{j}}\theta\genfrac{[}%
{]}{0.0pt}{}{\boldsymbol{{\alpha}}}{\boldsymbol{{\beta}}}\left(\mathbf{z}%
\middle|\boldsymbol{{\Omega}}\right)\Big{|}_{\mathbf{z}=\boldsymbol{{0}}}." display="block"><mrow><mrow><mi>ζ</mi><mo>=</mo><msub><mrow><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi href="./21.1#p2.t1.r1">g</mi></munderover><mrow><msub><mi href="./21.1#p2.t1.r20">ω</mi><mi>j</mi></msub><mo></mo><mrow><mfrac><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><msub><mi>z</mi><mi>j</mi></msub></mrow></mfrac><mo></mo><mrow><mrow><mi href="./21.2#E5">θ</mi><mo href="./21.2#E5"></mo><mrow><mo href="./21.2#E5">[</mo><mfrac linethickness="0.0pt"><mi href="./21.1#p2.t1.r6" mathvariant="bold-italic">α</mi><mi href="./21.1#p2.t1.r6" mathvariant="bold-italic">β</mi></mfrac><mo href="./21.2#E5">]</mo></mrow></mrow><mo></mo><mrow><mo>(</mo><mi mathvariant="bold">z</mi><mo>|</mo><mi href="./21.1#p2.t1.r5" mathvariant="bold">Ω</mi><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo fence="true" maxsize="160%" minsize="160%">|</mo></mrow><mrow><mi mathvariant="bold">z</mi><mo>=</mo><mn mathvariant="bold">0</mn></mrow></msub></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E8.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./21.2#E5" title="(21.2.5) ‣ §21.2(ii) Riemann Theta Functions with Characteristics ‣ §21.2 Definitions ‣ Properties ‣ Chapter 21 Multidimensional Theta Functions" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="30px" altimg-valign="-12px" altimg-width="94px" alttext="\theta\genfrac{[}{]}{0.0pt}{}{\NVar{\boldsymbol{{\alpha}}}}{\NVar{\boldsymbol{%
{\beta}}}}\left(\NVar{\mathbf{z}}\middle|\NVar{\boldsymbol{{\Omega}}}\right)" display="inline"><mrow><mrow><mi href="./21.2#E5">θ</mi><mo href="./21.2#E5"></mo><mrow><mo href="./21.2#E5">[</mo><mfrac linethickness="0.0pt"><mi class="ltx_nvar" href="./21.1#p2.t1.r6" mathvariant="bold-italic">α</mi><mi class="ltx_nvar" href="./21.1#p2.t1.r6" mathvariant="bold-italic">β</mi></mfrac><mo href="./21.2#E5">]</mo></mrow></mrow><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" mathvariant="bold">z</mi><mo>|</mo><mi class="ltx_nvar" href="./21.1#p2.t1.r5" mathvariant="bold">Ω</mi><mo>)</mo></mrow></mrow></math>: Riemann theta function with characteristics</a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m75.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m64.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./21.1#p2.t1.r1" title="§21.1 Special Notation ‣ Notation ‣ Chapter 21 Multidimensional Theta Functions" class="ltx_ref"><math class="ltx_Math" altimg="m80.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="g" display="inline"><mi href="./21.1#p2.t1.r1">g</mi></math>: positive integer</a>,
<a href="./21.1#p2.t1.r5" title="§21.1 Special Notation ‣ Notation ‣ Chapter 21 Multidimensional Theta Functions" class="ltx_ref"><math class="ltx_Math" altimg="m30.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="\boldsymbol{{\Omega}}" display="inline"><mi href="./21.1#p2.t1.r5" mathvariant="bold">Ω</mi></math>: a Riemann matrix</a>,
<a href="./21.1#p2.t1.r6" title="§21.1 Special Notation ‣ Notation ‣ Chapter 21 Multidimensional Theta Functions" class="ltx_ref"><math class="ltx_Math" altimg="m31.png" altimg-height="13px" altimg-valign="-2px" altimg-width="19px" alttext="\boldsymbol{{\alpha}}" display="inline"><mi href="./21.1#p2.t1.r6" mathvariant="bold-italic">α</mi></math>: <math class="ltx_Math" altimg="m80.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="g" display="inline"><mi href="./21.1#p2.t1.r1">g</mi></math>-dimensional vector</a>,
<a href="./21.1#p2.t1.r6" title="§21.1 Special Notation ‣ Notation ‣ Chapter 21 Multidimensional Theta Functions" class="ltx_ref"><math class="ltx_Math" altimg="m34.png" altimg-height="21px" altimg-valign="-6px" altimg-width="18px" alttext="\boldsymbol{{\beta}}" display="inline"><mi href="./21.1#p2.t1.r6" mathvariant="bold-italic">β</mi></math>: <math class="ltx_Math" altimg="m80.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="g" display="inline"><mi href="./21.1#p2.t1.r1">g</mi></math>-dimensional vector</a> and
<a href="./21.1#p2.t1.r20" title="§21.1 Special Notation ‣ Notation ‣ Chapter 21 Multidimensional Theta Functions" class="ltx_ref"><math class="ltx_Math" altimg="m62.png" altimg-height="13px" altimg-valign="-2px" altimg-width="17px" alttext="\omega" display="inline"><mi href="./21.1#p2.t1.r20">ω</mi></math>: differential</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Then the <em class="ltx_emph ltx_font_italic">prime form</em> on the corresponding compact Riemann
surface <math class="ltx_Math" altimg="m29.png" altimg-height="18px" altimg-valign="-2px" altimg-width="17px" alttext="\Gamma" display="inline"><mi href="./21.7#i" mathvariant="normal">Γ</mi></math> is defined by
</p>
<table id="E9" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">21.7.9</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E9.png" altimg-height="66px" altimg-valign="-27px" altimg-width="477px" alttext="E(P_{1},P_{2})=\theta\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}}{%
\boldsymbol{{\beta}}}\left(\int_{P_{1}}^{P_{2}}\boldsymbol{{\omega}}\middle|%
\boldsymbol{{\Omega}}\right)\Bigg{/}\left(\sqrt{\zeta(P_{1})}\sqrt{\zeta(P_{2}%
)}\right)," display="block"><mrow><mrow><mrow><mi href="./21.7#E9">E</mi><mo></mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><msub><mi href="./21.7#SS2.p2">P</mi><mn>1</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><msub><mi href="./21.7#SS2.p2">P</mi><mn>2</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mrow><mi href="./21.2#E5">θ</mi><mo href="./21.2#E5"></mo><mrow><mo href="./21.2#E5">[</mo><mfrac linethickness="0.0pt"><mi href="./21.1#p2.t1.r6" mathvariant="bold-italic">α</mi><mi href="./21.1#p2.t1.r6" mathvariant="bold-italic">β</mi></mfrac><mo href="./21.2#E5">]</mo></mrow></mrow><mo></mo><mrow><mo>(</mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><msub><mi href="./21.7#SS2.p2">P</mi><mn>1</mn></msub><msub><mi href="./21.7#SS2.p2">P</mi><mn>2</mn></msub></msubsup><mi href="./21.7#SS2.p2" mathvariant="bold-italic">ω</mi></mrow><mo>|</mo><mi href="./21.1#p2.t1.r5" mathvariant="bold">Ω</mi><mo>)</mo></mrow></mrow><mo mathsize="260%" stretchy="false">/</mo><mrow><mo>(</mo><mrow><msqrt><mrow><mi>ζ</mi><mo></mo><mrow><mo stretchy="false">(</mo><msub><mi href="./21.7#SS2.p2">P</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mrow></mrow></msqrt><mo></mo><msqrt><mrow><mi>ζ</mi><mo></mo><mrow><mo stretchy="false">(</mo><msub><mi href="./21.7#SS2.p2">P</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mrow></mrow></msqrt></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E9.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text"><math class="ltx_Math" altimg="m15.png" altimg-height="23px" altimg-valign="-7px" altimg-width="88px" alttext="E(P_{1},P_{2})" display="inline"><mrow><mi href="./21.7#E9">E</mi><mo></mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><msub><mi href="./21.7#SS2.p2">P</mi><mn>1</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><msub><mi href="./21.7#SS2.p2">P</mi><mn>2</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></mrow></math>: prime form (locally)</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./21.2#E5" title="(21.2.5) ‣ §21.2(ii) Riemann Theta Functions with Characteristics ‣ §21.2 Definitions ‣ Properties ‣ Chapter 21 Multidimensional Theta Functions" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="30px" altimg-valign="-12px" altimg-width="94px" alttext="\theta\genfrac{[}{]}{0.0pt}{}{\NVar{\boldsymbol{{\alpha}}}}{\NVar{\boldsymbol{%
{\beta}}}}\left(\NVar{\mathbf{z}}\middle|\NVar{\boldsymbol{{\Omega}}}\right)" display="inline"><mrow><mrow><mi href="./21.2#E5">θ</mi><mo href="./21.2#E5"></mo><mrow><mo href="./21.2#E5">[</mo><mfrac linethickness="0.0pt"><mi class="ltx_nvar" href="./21.1#p2.t1.r6" mathvariant="bold-italic">α</mi><mi class="ltx_nvar" href="./21.1#p2.t1.r6" mathvariant="bold-italic">β</mi></mfrac><mo href="./21.2#E5">]</mo></mrow></mrow><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" mathvariant="bold">z</mi><mo>|</mo><mi class="ltx_nvar" href="./21.1#p2.t1.r5" mathvariant="bold">Ω</mi><mo>)</mo></mrow></mrow></math>: Riemann theta function with characteristics</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./front/introduction#Sx4.p1.t1.r28" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m6.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="(\NVar{a},\NVar{b})" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi class="ltx_nvar">a</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi class="ltx_nvar">b</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math>: open interval</a>,
<a href="./21.1#p2.t1.r5" title="§21.1 Special Notation ‣ Notation ‣ Chapter 21 Multidimensional Theta Functions" class="ltx_ref"><math class="ltx_Math" altimg="m30.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="\boldsymbol{{\Omega}}" display="inline"><mi href="./21.1#p2.t1.r5" mathvariant="bold">Ω</mi></math>: a Riemann matrix</a>,
<a href="./21.1#p2.t1.r6" title="§21.1 Special Notation ‣ Notation ‣ Chapter 21 Multidimensional Theta Functions" class="ltx_ref"><math class="ltx_Math" altimg="m31.png" altimg-height="13px" altimg-valign="-2px" altimg-width="19px" alttext="\boldsymbol{{\alpha}}" display="inline"><mi href="./21.1#p2.t1.r6" mathvariant="bold-italic">α</mi></math>: <math class="ltx_Math" altimg="m80.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="g" display="inline"><mi href="./21.1#p2.t1.r1">g</mi></math>-dimensional vector</a>,
<a href="./21.1#p2.t1.r6" title="§21.1 Special Notation ‣ Notation ‣ Chapter 21 Multidimensional Theta Functions" class="ltx_ref"><math class="ltx_Math" altimg="m34.png" altimg-height="21px" altimg-valign="-6px" altimg-width="18px" alttext="\boldsymbol{{\beta}}" display="inline"><mi href="./21.1#p2.t1.r6" mathvariant="bold-italic">β</mi></math>: <math class="ltx_Math" altimg="m80.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="g" display="inline"><mi href="./21.1#p2.t1.r1">g</mi></math>-dimensional vector</a>,
<a href="./21.7#SS2.p2" title="§21.7(ii) Fay’s Trisecant Identity ‣ §21.7 Riemann Surfaces ‣ Applications ‣ Chapter 21 Multidimensional Theta Functions" class="ltx_ref"><math class="ltx_Math" altimg="m18.png" altimg-height="21px" altimg-valign="-6px" altimg-width="57px" alttext="P_{1},P_{2}" display="inline"><mrow><msub><mi href="./21.7#SS2.p2">P</mi><mn>1</mn></msub><mo>,</mo><msub><mi href="./21.7#SS2.p2">P</mi><mn>2</mn></msub></mrow></math>: points</a> and
<a href="./21.7#SS2.p2" title="§21.7(ii) Fay’s Trisecant Identity ‣ §21.7 Riemann Surfaces ‣ Applications ‣ Chapter 21 Multidimensional Theta Functions" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="13px" altimg-valign="-2px" altimg-width="19px" alttext="\boldsymbol{{\omega}}" display="inline"><mi href="./21.7#SS2.p2" mathvariant="bold-italic">ω</mi></math>: vector of differentials</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m19.png" altimg-height="21px" altimg-valign="-5px" altimg-width="26px" alttext="P_{1}" display="inline"><msub><mi href="./21.7#SS2.p2">P</mi><mn>1</mn></msub></math> and <math class="ltx_Math" altimg="m20.png" altimg-height="21px" altimg-valign="-5px" altimg-width="26px" alttext="P_{2}" display="inline"><msub><mi href="./21.7#SS2.p2">P</mi><mn>2</mn></msub></math> are points on <math class="ltx_Math" altimg="m29.png" altimg-height="18px" altimg-valign="-2px" altimg-width="17px" alttext="\Gamma" display="inline"><mi href="./21.7#i" mathvariant="normal">Γ</mi></math>,
<math class="ltx_Math" altimg="m43.png" altimg-height="24px" altimg-valign="-8px" altimg-width="179px" alttext="\boldsymbol{{\omega}}=(\omega_{1},\omega_{2},\dots,\omega_{g})" display="inline"><mrow><mi href="./21.7#SS2.p2" mathvariant="bold-italic">ω</mi><mo>=</mo><mrow><mo stretchy="false">(</mo><msub><mi href="./21.1#p2.t1.r20">ω</mi><mn>1</mn></msub><mo>,</mo><msub><mi href="./21.1#p2.t1.r20">ω</mi><mn>2</mn></msub><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><msub><mi href="./21.1#p2.t1.r20">ω</mi><mi href="./21.1#p2.t1.r1">g</mi></msub><mo stretchy="false">)</mo></mrow></mrow></math>, and the path of
integration on <math class="ltx_Math" altimg="m29.png" altimg-height="18px" altimg-valign="-2px" altimg-width="17px" alttext="\Gamma" display="inline"><mi href="./21.7#i" mathvariant="normal">Γ</mi></math> from <math class="ltx_Math" altimg="m19.png" altimg-height="21px" altimg-valign="-5px" altimg-width="26px" alttext="P_{1}" display="inline"><msub><mi href="./21.7#SS2.p2">P</mi><mn>1</mn></msub></math> to <math class="ltx_Math" altimg="m20.png" altimg-height="21px" altimg-valign="-5px" altimg-width="26px" alttext="P_{2}" display="inline"><msub><mi href="./21.7#SS2.p2">P</mi><mn>2</mn></msub></math> is identical for all components.
Here <math class="ltx_Math" altimg="m66.png" altimg-height="28px" altimg-valign="-8px" altimg-width="65px" alttext="\sqrt{\zeta(P)}" display="inline"><msqrt><mrow><mi>ζ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./21.7#SS2.p2">P</mi><mo stretchy="false">)</mo></mrow></mrow></msqrt></math> is such that <math class="ltx_Math" altimg="m67.png" altimg-height="32px" altimg-valign="-8px" altimg-width="142px" alttext="\sqrt{\zeta(P)}^{2}=\zeta(P)" display="inline"><mrow><msup><msqrt><mrow><mi>ζ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./21.7#SS2.p2">P</mi><mo stretchy="false">)</mo></mrow></mrow></msqrt><mn>2</mn></msup><mo>=</mo><mrow><mi>ζ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./21.7#SS2.p2">P</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></math>,
<math class="ltx_Math" altimg="m17.png" altimg-height="18px" altimg-valign="-3px" altimg-width="57px" alttext="P\in\Gamma" display="inline"><mrow><mi href="./21.7#SS2.p2">P</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mi href="./21.7#i" mathvariant="normal">Γ</mi></mrow></math>. Either branch of the square roots may be chosen, as
long as the branch is consistent across <math class="ltx_Math" altimg="m29.png" altimg-height="18px" altimg-valign="-2px" altimg-width="17px" alttext="\Gamma" display="inline"><mi href="./21.7#i" mathvariant="normal">Γ</mi></math>. For all
<math class="ltx_Math" altimg="m55.png" altimg-height="18px" altimg-valign="-3px" altimg-width="62px" alttext="\mathbf{z}\in{\mathbb{C}^{g}}" display="inline"><mrow><mi mathvariant="bold">z</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><msup><mi href="./front/introduction#Sx4.p1.t1.r1">ℂ</mi><mi href="./21.1#p2.t1.r1">g</mi></msup></mrow></math>, and all <math class="ltx_Math" altimg="m19.png" altimg-height="21px" altimg-valign="-5px" altimg-width="26px" alttext="P_{1}" display="inline"><msub><mi href="./21.7#SS2.p2">P</mi><mn>1</mn></msub></math>, <math class="ltx_Math" altimg="m20.png" altimg-height="21px" altimg-valign="-5px" altimg-width="26px" alttext="P_{2}" display="inline"><msub><mi href="./21.7#SS2.p2">P</mi><mn>2</mn></msub></math>, <math class="ltx_Math" altimg="m21.png" altimg-height="21px" altimg-valign="-5px" altimg-width="26px" alttext="P_{3}" display="inline"><msub><mi href="./21.7#SS2.p2">P</mi><mn>3</mn></msub></math>, <math class="ltx_Math" altimg="m22.png" altimg-height="21px" altimg-valign="-5px" altimg-width="26px" alttext="P_{4}" display="inline"><msub><mi href="./21.7#SS2.p2">P</mi><mn>4</mn></msub></math> on <math class="ltx_Math" altimg="m29.png" altimg-height="18px" altimg-valign="-2px" altimg-width="17px" alttext="\Gamma" display="inline"><mi href="./21.7#i" mathvariant="normal">Γ</mi></math>,
Fay’s identity is given by</p>
</div>
<div id="SS2.p3" class="ltx_para">
<table id="E10" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">21.7.10</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_Math ltx_eqn_cell ltx_align_center"><math class="ltx_Math" alttext="\theta\left(\mathbf{z}+\int_{P_{1}}^{P_{3}}\boldsymbol{{\omega}}\middle|%
\boldsymbol{{\Omega}}\right)\theta\left(\mathbf{z}+\int_{P_{2}}^{P_{4}}%
\boldsymbol{{\omega}}\middle|\boldsymbol{{\Omega}}\right)E(P_{3},P_{2})E(P_{1}%
,P_{4})+\theta\left(\mathbf{z}+\int_{P_{2}}^{P_{3}}\boldsymbol{{\omega}}%
\middle|\boldsymbol{{\Omega}}\right)\theta\left(\mathbf{z}+\int_{P_{1}}^{P_{4}%
}\boldsymbol{{\omega}}\middle|\boldsymbol{{\Omega}}\right)E(P_{3},P_{1})E(P_{4%
},P_{2})=\theta\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)\theta\left%
(\mathbf{z}+\int_{P_{1}}^{P_{3}}\boldsymbol{{\omega}}+\int_{P_{2}}^{P_{4}}%
\boldsymbol{{\omega}}\middle|\boldsymbol{{\Omega}}\right)E(P_{1},P_{2})E(P_{3}%
,P_{4})," display="block"><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mrow><mrow><mi href="./21.2#E1">θ</mi><mo></mo><mrow><mo>(</mo><mrow><mi mathvariant="bold">z</mi><mo>+</mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><msub><mi href="./21.7#SS2.p2">P</mi><mn>1</mn></msub><msub><mi href="./21.7#SS2.p2">P</mi><mn>3</mn></msub></msubsup><mi href="./21.7#SS2.p2" mathvariant="bold-italic">ω</mi></mrow></mrow><mo>|</mo><mi href="./21.1#p2.t1.r5" mathvariant="bold">Ω</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./21.2#E1">θ</mi><mo></mo><mrow><mo>(</mo><mrow><mi mathvariant="bold">z</mi><mo>+</mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><msub><mi href="./21.7#SS2.p2">P</mi><mn>2</mn></msub><msub><mi href="./21.7#SS2.p2">P</mi><mn>4</mn></msub></msubsup><mi href="./21.7#SS2.p2" mathvariant="bold-italic">ω</mi></mrow></mrow><mo>|</mo><mi href="./21.1#p2.t1.r5" mathvariant="bold">Ω</mi><mo>)</mo></mrow></mrow><mo></mo><mi href="./21.7#E9">E</mi><mo></mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><msub><mi href="./21.7#SS2.p2">P</mi><mn>3</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><msub><mi href="./21.7#SS2.p2">P</mi><mn>2</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow><mo></mo><mi href="./21.7#E9">E</mi><mo></mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><msub><mi href="./21.7#SS2.p2">P</mi><mn>1</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><msub><mi href="./21.7#SS2.p2">P</mi><mn>4</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></mrow></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>+</mo><mrow><mrow><mi href="./21.2#E1">θ</mi><mo></mo><mrow><mo>(</mo><mrow><mi mathvariant="bold">z</mi><mo>+</mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><msub><mi href="./21.7#SS2.p2">P</mi><mn>2</mn></msub><msub><mi href="./21.7#SS2.p2">P</mi><mn>3</mn></msub></msubsup><mi href="./21.7#SS2.p2" mathvariant="bold-italic">ω</mi></mrow></mrow><mo>|</mo><mi href="./21.1#p2.t1.r5" mathvariant="bold">Ω</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./21.2#E1">θ</mi><mo></mo><mrow><mo>(</mo><mrow><mi mathvariant="bold">z</mi><mo>+</mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><msub><mi href="./21.7#SS2.p2">P</mi><mn>1</mn></msub><msub><mi href="./21.7#SS2.p2">P</mi><mn>4</mn></msub></msubsup><mi href="./21.7#SS2.p2" mathvariant="bold-italic">ω</mi></mrow></mrow><mo>|</mo><mi href="./21.1#p2.t1.r5" mathvariant="bold">Ω</mi><mo>)</mo></mrow></mrow><mo></mo><mi href="./21.7#E9">E</mi><mo></mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><msub><mi href="./21.7#SS2.p2">P</mi><mn>3</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><msub><mi href="./21.7#SS2.p2">P</mi><mn>1</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow><mo></mo><mi href="./21.7#E9">E</mi><mo></mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><msub><mi href="./21.7#SS2.p2">P</mi><mn>4</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><msub><mi href="./21.7#SS2.p2">P</mi><mn>2</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></mrow></mtd></mtr></mtable></mrow></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>=</mo><mrow><mrow><mi href="./21.2#E1">θ</mi><mo></mo><mrow><mo>(</mo><mi mathvariant="bold">z</mi><mo>|</mo><mi href="./21.1#p2.t1.r5" mathvariant="bold">Ω</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./21.2#E1">θ</mi><mo></mo><mrow><mo>(</mo><mrow><mi mathvariant="bold">z</mi><mo>+</mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><msub><mi href="./21.7#SS2.p2">P</mi><mn>1</mn></msub><msub><mi href="./21.7#SS2.p2">P</mi><mn>3</mn></msub></msubsup><mi href="./21.7#SS2.p2" mathvariant="bold-italic">ω</mi></mrow><mo>+</mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><msub><mi href="./21.7#SS2.p2">P</mi><mn>2</mn></msub><msub><mi href="./21.7#SS2.p2">P</mi><mn>4</mn></msub></msubsup><mi href="./21.7#SS2.p2" mathvariant="bold-italic">ω</mi></mrow></mrow><mo>|</mo><mi href="./21.1#p2.t1.r5" mathvariant="bold">Ω</mi><mo>)</mo></mrow></mrow><mo></mo><mi href="./21.7#E9">E</mi><mo></mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><msub><mi href="./21.7#SS2.p2">P</mi><mn>1</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><msub><mi href="./21.7#SS2.p2">P</mi><mn>2</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow><mo></mo><mi href="./21.7#E9">E</mi><mo></mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><msub><mi href="./21.7#SS2.p2">P</mi><mn>3</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><msub><mi href="./21.7#SS2.p2">P</mi><mn>4</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></mrow><mo>,</mo></mtd></mtr></mtable></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E10.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./21.2#E1" title="(21.2.1) ‣ §21.2(i) Riemann Theta Functions ‣ §21.2 Definitions ‣ Properties ‣ Chapter 21 Multidimensional Theta Functions" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="23px" altimg-valign="-7px" altimg-width="65px" alttext="\theta\left(\NVar{\mathbf{z}}\middle|\NVar{\boldsymbol{{\Omega}}}\right)" display="inline"><mrow><mi href="./21.2#E1">θ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" mathvariant="bold">z</mi><mo>|</mo><mi class="ltx_nvar" href="./21.1#p2.t1.r5" mathvariant="bold">Ω</mi><mo>)</mo></mrow></mrow></math>: Riemann theta function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./front/introduction#Sx4.p1.t1.r28" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m6.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="(\NVar{a},\NVar{b})" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi class="ltx_nvar">a</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi class="ltx_nvar">b</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math>: open interval</a>,
<a href="./21.1#p2.t1.r5" title="§21.1 Special Notation ‣ Notation ‣ Chapter 21 Multidimensional Theta Functions" class="ltx_ref"><math class="ltx_Math" altimg="m30.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="\boldsymbol{{\Omega}}" display="inline"><mi href="./21.1#p2.t1.r5" mathvariant="bold">Ω</mi></math>: a Riemann matrix</a>,
<a href="./21.7#E9" title="(21.7.9) ‣ §21.7(ii) Fay’s Trisecant Identity ‣ §21.7 Riemann Surfaces ‣ Applications ‣ Chapter 21 Multidimensional Theta Functions" class="ltx_ref"><math class="ltx_Math" altimg="m15.png" altimg-height="23px" altimg-valign="-7px" altimg-width="88px" alttext="E(P_{1},P_{2})" display="inline"><mrow><mi href="./21.7#E9">E</mi><mo></mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><msub><mi href="./21.7#SS2.p2">P</mi><mn>1</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><msub><mi href="./21.7#SS2.p2">P</mi><mn>2</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></mrow></math>: prime form</a>,
<a href="./21.7#SS2.p2" title="§21.7(ii) Fay’s Trisecant Identity ‣ §21.7 Riemann Surfaces ‣ Applications ‣ Chapter 21 Multidimensional Theta Functions" class="ltx_ref"><math class="ltx_Math" altimg="m18.png" altimg-height="21px" altimg-valign="-6px" altimg-width="57px" alttext="P_{1},P_{2}" display="inline"><mrow><msub><mi href="./21.7#SS2.p2">P</mi><mn>1</mn></msub><mo>,</mo><msub><mi href="./21.7#SS2.p2">P</mi><mn>2</mn></msub></mrow></math>: points</a> and
<a href="./21.7#SS2.p2" title="§21.7(ii) Fay’s Trisecant Identity ‣ §21.7 Riemann Surfaces ‣ Applications ‣ Chapter 21 Multidimensional Theta Functions" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="13px" altimg-valign="-2px" altimg-width="19px" alttext="\boldsymbol{{\omega}}" display="inline"><mi href="./21.7#SS2.p2" mathvariant="bold-italic">ω</mi></math>: vector of differentials</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS2.p4" class="ltx_para">
<p class="ltx_p">where again all integration paths are identical for all components.
Generalizations of this identity are given in <cite class="ltx_cite ltx_citemacro_citet">Fay (</dd>
</dl>
</div>
</div>
<div id="SS3.p1" class="ltx_para">
<p class="ltx_p">Let <math class="ltx_Math" altimg="m29.png" altimg-height="18px" altimg-valign="-2px" altimg-width="17px" alttext="\Gamma" display="inline"><mi href="./21.7#i" mathvariant="normal">Γ</mi></math> be a <em class="ltx_emph ltx_font_italic">hyperelliptic Riemann surface</em>. These are Riemann
surfaces that may be obtained from algebraic curves of the form</p>
<table id="E11" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">21.7.11</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E11.png" altimg-height="28px" altimg-valign="-7px" altimg-width="102px" alttext="\mu^{2}=Q(\lambda)," display="block"><mrow><mrow><msup><mi>μ</mi><mn>2</mn></msup><mo>=</mo><mrow><mi href="./21.7#SS3.p1">Q</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E11.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./21.7#SS3.p1" title="§21.7(iii) Frobenius’ Identity ‣ §21.7 Riemann Surfaces ‣ Applications ‣ Chapter 21 Multidimensional Theta Functions" class="ltx_ref"><math class="ltx_Math" altimg="m25.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="Q(\lambda)" display="inline"><mrow><mi href="./21.7#SS3.p1">Q</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">)</mo></mrow></mrow></math>: polynomial</a></dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m25.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="Q(\lambda)" display="inline"><mrow><mi href="./21.7#SS3.p1">Q</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">)</mo></mrow></mrow></math> is a polynomial in <math class="ltx_Math" altimg="m53.png" altimg-height="18px" altimg-valign="-2px" altimg-width="16px" alttext="\lambda" display="inline"><mi>λ</mi></math> of odd degree <math class="ltx_Math" altimg="m9.png" altimg-height="20px" altimg-valign="-6px" altimg-width="59px" alttext="2g+1" display="inline"><mrow><mrow><mn>2</mn><mo></mo><mi href="./21.1#p2.t1.r1">g</mi></mrow><mo>+</mo><mn>1</mn></mrow></math> <math class="ltx_Math" altimg="m7.png" altimg-height="23px" altimg-valign="-7px" altimg-width="51px" alttext="(\geq 5)" display="inline"><mrow><mo stretchy="false">(</mo><mrow><mi></mi><mo>≥</mo><mn>5</mn></mrow><mo stretchy="false">)</mo></mrow></math>.
The genus of this surface is <math class="ltx_Math" altimg="m80.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="g" display="inline"><mi href="./21.1#p2.t1.r1">g</mi></math>. The zeros <math class="ltx_Math" altimg="m54.png" altimg-height="23px" altimg-valign="-8px" altimg-width="24px" alttext="\lambda_{j}" display="inline"><msub><mi>λ</mi><mi>j</mi></msub></math>, <math class="ltx_Math" altimg="m83.png" altimg-height="20px" altimg-valign="-6px" altimg-width="168px" alttext="j=1,2,\dots,2g+1" display="inline"><mrow><mi>j</mi><mo>=</mo><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><mrow><mrow><mn>2</mn><mo></mo><mi href="./21.1#p2.t1.r1">g</mi></mrow><mo>+</mo><mn>1</mn></mrow></mrow></mrow></math> of
<math class="ltx_Math" altimg="m25.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="Q(\lambda)" display="inline"><mrow><mi href="./21.7#SS3.p1">Q</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">)</mo></mrow></mrow></math> specify the finite branch points <math class="ltx_Math" altimg="m24.png" altimg-height="23px" altimg-valign="-8px" altimg-width="25px" alttext="P_{j}" display="inline"><msub><mi href="./21.7#SS3.p1">P</mi><mi>j</mi></msub></math>, that is, points at which
<math class="ltx_Math" altimg="m61.png" altimg-height="22px" altimg-valign="-8px" altimg-width="61px" alttext="\mu_{j}=0" display="inline"><mrow><msub><mi>μ</mi><mi>j</mi></msub><mo>=</mo><mn>0</mn></mrow></math>, on the Riemann surface. Denote the set of all branch
points by <math class="ltx_Math" altimg="m13.png" altimg-height="24px" altimg-valign="-8px" altimg-width="252px" alttext="B=\{P_{1},P_{2},\dots,P_{2g+1},P_{\infty}\}" display="inline"><mrow><mi href="./21.7#SS3.p1">B</mi><mo>=</mo><mrow><mo stretchy="false">{</mo><msub><mi href="./21.7#SS3.p1">P</mi><mn>1</mn></msub><mo>,</mo><msub><mi href="./21.7#SS3.p1">P</mi><mn>2</mn></msub><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><msub><mi href="./21.7#SS3.p1">P</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./21.1#p2.t1.r1">g</mi></mrow><mo>+</mo><mn>1</mn></mrow></msub><mo>,</mo><msub><mi href="./21.7#SS3.p1">P</mi><mi mathvariant="normal">∞</mi></msub><mo stretchy="false">}</mo></mrow></mrow></math>. Consider a fixed
subset <math class="ltx_Math" altimg="m28.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="U" display="inline"><mi href="./21.7#SS3.p1">U</mi></math> of <math class="ltx_Math" altimg="m14.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="B" display="inline"><mi href="./21.7#SS3.p1">B</mi></math>, such that the number of elements <math class="ltx_Math" altimg="m88.png" altimg-height="23px" altimg-valign="-7px" altimg-width="31px" alttext="|U|" display="inline"><mrow><mo stretchy="false">|</mo><mi href="./21.7#SS3.p1">U</mi><mo stretchy="false">|</mo></mrow></math> in the set <math class="ltx_Math" altimg="m28.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="U" display="inline"><mi href="./21.7#SS3.p1">U</mi></math> is
<math class="ltx_Math" altimg="m78.png" altimg-height="20px" altimg-valign="-6px" altimg-width="49px" alttext="g+1" display="inline"><mrow><mi href="./21.1#p2.t1.r1">g</mi><mo>+</mo><mn>1</mn></mrow></math>, and <math class="ltx_Math" altimg="m23.png" altimg-height="21px" altimg-valign="-5px" altimg-width="74px" alttext="P_{\infty}\notin U" display="inline"><mrow><msub><mi href="./21.7#SS3.p1">P</mi><mi mathvariant="normal">∞</mi></msub><mo href="./front/introduction#Sx4.p1.t1.r10">∉</mo><mi href="./21.7#SS3.p1">U</mi></mrow></math>. Next, define an isomorphism <math class="ltx_Math" altimg="m40.png" altimg-height="16px" altimg-valign="-6px" altimg-width="17px" alttext="\boldsymbol{{\eta}}" display="inline"><mi mathvariant="bold-italic">η</mi></math> which maps every
subset <math class="ltx_Math" altimg="m26.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="T" display="inline"><mi href="./21.7#SS3.p1">T</mi></math> of <math class="ltx_Math" altimg="m14.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="B" display="inline"><mi href="./21.7#SS3.p1">B</mi></math> with an even number of elements to a <math class="ltx_Math" altimg="m10.png" altimg-height="20px" altimg-valign="-6px" altimg-width="24px" alttext="2g" display="inline"><mrow><mn>2</mn><mo></mo><mi href="./21.1#p2.t1.r1">g</mi></mrow></math>-dimensional vector
<math class="ltx_Math" altimg="m39.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="\boldsymbol{{\eta}}(T)" display="inline"><mrow><mi mathvariant="bold-italic">η</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./21.7#SS3.p1">T</mi><mo stretchy="false">)</mo></mrow></mrow></math> with elements either <math class="ltx_Math" altimg="m8.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math> or <math class="ltx_Math" altimg="m68.png" altimg-height="27px" altimg-valign="-9px" altimg-width="17px" alttext="\tfrac{1}{2}" display="inline"><mfrac><mn>1</mn><mn>2</mn></mfrac></math>. Define the
operation</p>
<table id="E12" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">21.7.12</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E12.png" altimg-height="25px" altimg-valign="-7px" altimg-width="280px" alttext="T_{1}\ominus T_{2}=(T_{1}\cup T_{2})\setminus(T_{1}\cap T_{2})." display="block"><mrow><mrow><mrow><msub><mi href="./21.7#SS3.p1">T</mi><mn>1</mn></msub><mo>⊖</mo><msub><mi href="./21.7#SS3.p1">T</mi><mn>2</mn></msub></mrow><mo>=</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><msub><mi href="./21.7#SS3.p1">T</mi><mn>1</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r27">∪</mo><msub><mi href="./21.7#SS3.p1">T</mi><mn>2</mn></msub></mrow><mo stretchy="false">)</mo></mrow><mo href="./front/introduction#Sx4.p2.t1.r19">∖</mo><mrow><mo stretchy="false">(</mo><mrow><msub><mi href="./21.7#SS3.p1">T</mi><mn>1</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r26">∩</mo><msub><mi href="./21.7#SS3.p1">T</mi><mn>2</mn></msub></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E12.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./front/introduction#Sx4.p1.t1.r26" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m45.png" altimg-height="15px" altimg-valign="-2px" altimg-width="18px" alttext="\cap" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r26">∩</mo></math>: intersection</a>,
<a href="./front/introduction#Sx4.p2.t1.r19" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m65.png" altimg-height="23px" altimg-valign="-7px" altimg-width="14px" alttext="\setminus" display="inline"><mo href="./front/introduction#Sx4.p2.t1.r19">∖</mo></math>: set subtraction</a>,
<a href="./front/introduction#Sx4.p1.t1.r27" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m46.png" altimg-height="15px" altimg-valign="-2px" altimg-width="18px" alttext="\cup" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r27">∪</mo></math>: union</a> and
<a href="./21.7#SS3.p1" title="§21.7(iii) Frobenius’ Identity ‣ §21.7 Riemann Surfaces ‣ Applications ‣ Chapter 21 Multidimensional Theta Functions" class="ltx_ref"><math class="ltx_Math" altimg="m26.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="T" display="inline"><mi href="./21.7#SS3.p1">T</mi></math>: subset of <math class="ltx_Math" altimg="m14.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="B" display="inline"><mi href="./21.7#SS3.p1">B</mi></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Also, <math class="ltx_Math" altimg="m27.png" altimg-height="23px" altimg-valign="-7px" altimg-width="103px" alttext="T^{c}=B\setminus T" display="inline"><mrow><msup><mi href="./21.7#SS3.p1">T</mi><mi>c</mi></msup><mo>=</mo><mrow><mi href="./21.7#SS3.p1">B</mi><mo href="./front/introduction#Sx4.p2.t1.r19">∖</mo><mi href="./21.7#SS3.p1">T</mi></mrow></mrow></math>,
<math class="ltx_Math" altimg="m41.png" altimg-height="25px" altimg-valign="-8px" altimg-width="298px" alttext="\boldsymbol{{\eta}}^{1}(T)=(\eta_{1}(T),\eta_{2}(T),\dots,\eta_{g}(T))" display="inline"><mrow><mrow><msup><mi mathvariant="bold-italic">η</mi><mn>1</mn></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./21.7#SS3.p1">T</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mo stretchy="false">(</mo><mrow><msub><mi>η</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./21.7#SS3.p1">T</mi><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo><mrow><msub><mi>η</mi><mn>2</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./21.7#SS3.p1">T</mi><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><mrow><msub><mi>η</mi><mi href="./21.1#p2.t1.r1">g</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./21.7#SS3.p1">T</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></math>, and
<math class="ltx_Math" altimg="m42.png" altimg-height="25px" altimg-valign="-8px" altimg-width="347px" alttext="\boldsymbol{{\eta}}^{2}(T)=(\eta_{g+1}(T),\eta_{g+2}(T),\dots,\eta_{2g}(T))" display="inline"><mrow><mrow><msup><mi mathvariant="bold-italic">η</mi><mn>2</mn></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./21.7#SS3.p1">T</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mo stretchy="false">(</mo><mrow><msub><mi>η</mi><mrow><mi href="./21.1#p2.t1.r1">g</mi><mo>+</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./21.7#SS3.p1">T</mi><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo><mrow><msub><mi>η</mi><mrow><mi href="./21.1#p2.t1.r1">g</mi><mo>+</mo><mn>2</mn></mrow></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./21.7#SS3.p1">T</mi><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><mrow><msub><mi>η</mi><mrow><mn>2</mn><mo></mo><mi href="./21.1#p2.t1.r1">g</mi></mrow></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./21.7#SS3.p1">T</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></math>. Then the
isomorphism is determined completely by:
</p>
<table id="E13" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">21.7.13</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E13.png" altimg-height="25px" altimg-valign="-7px" altimg-width="131px" alttext="\boldsymbol{{\eta}}(T)=\boldsymbol{{\eta}}(T^{c})," display="block"><mrow><mrow><mrow><mi mathvariant="bold-italic">η</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./21.7#SS3.p1">T</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mi mathvariant="bold-italic">η</mi><mo></mo><mrow><mo stretchy="false">(</mo><msup><mi href="./21.7#SS3.p1">T</mi><mi>c</mi></msup><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E13.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./21.7#SS3.p1" title="§21.7(iii) Frobenius’ Identity ‣ §21.7 Riemann Surfaces ‣ Applications ‣ Chapter 21 Multidimensional Theta Functions" class="ltx_ref"><math class="ltx_Math" altimg="m26.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="T" display="inline"><mi href="./21.7#SS3.p1">T</mi></math>: subset of <math class="ltx_Math" altimg="m14.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="B" display="inline"><mi href="./21.7#SS3.p1">B</mi></math></a></dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E14" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">21.7.14</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E14.png" altimg-height="25px" altimg-valign="-7px" altimg-width="254px" alttext="\boldsymbol{{\eta}}(T_{1}\ominus T_{2})=\boldsymbol{{\eta}}(T_{1})+\boldsymbol%
{{\eta}}(T_{2})," display="block"><mrow><mrow><mrow><mi mathvariant="bold-italic">η</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><msub><mi href="./21.7#SS3.p1">T</mi><mn>1</mn></msub><mo>⊖</mo><msub><mi href="./21.7#SS3.p1">T</mi><mn>2</mn></msub></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mi mathvariant="bold-italic">η</mi><mo></mo><mrow><mo stretchy="false">(</mo><msub><mi href="./21.7#SS3.p1">T</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mrow></mrow><mo>+</mo><mrow><mi mathvariant="bold-italic">η</mi><mo></mo><mrow><mo stretchy="false">(</mo><msub><mi href="./21.7#SS3.p1">T</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E14.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./21.7#SS3.p1" title="§21.7(iii) Frobenius’ Identity ‣ §21.7 Riemann Surfaces ‣ Applications ‣ Chapter 21 Multidimensional Theta Functions" class="ltx_ref"><math class="ltx_Math" altimg="m26.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="T" display="inline"><mi href="./21.7#SS3.p1">T</mi></math>: subset of <math class="ltx_Math" altimg="m14.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="B" display="inline"><mi href="./21.7#SS3.p1">B</mi></math></a></dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E15" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">21.7.15</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E15.png" altimg-height="30px" altimg-valign="-9px" altimg-width="425px" alttext="4\boldsymbol{{\eta}}^{1}(T)\cdot\boldsymbol{{\eta}}^{2}(T)=\tfrac{1}{2}\left(|%
T\ominus U|-g-1\right)\pmod{2}," display="block"><mrow><mrow><mrow><mrow><mn>4</mn><mo></mo><mrow><msup><mi mathvariant="bold-italic">η</mi><mn>1</mn></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./21.7#SS3.p1">T</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>⋅</mo><mrow><msup><mi mathvariant="bold-italic">η</mi><mn>2</mn></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./21.7#SS3.p1">T</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>=</mo><mrow><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mrow><mo>(</mo><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./21.7#SS3.p1">T</mi><mo>⊖</mo><mi href="./21.7#SS3.p1">U</mi></mrow><mo stretchy="false">|</mo></mrow><mo>-</mo><mi href="./21.1#p2.t1.r1">g</mi><mo>-</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mspace width="veryverythickmathspace"></mspace><mrow><mo lspace="8.1pt" stretchy="false">(</mo><mrow><mo movablelimits="false">mod</mo><mn>2</mn></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E15.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./21.1#p2.t1.r1" title="§21.1 Special Notation ‣ Notation ‣ Chapter 21 Multidimensional Theta Functions" class="ltx_ref"><math class="ltx_Math" altimg="m80.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="g" display="inline"><mi href="./21.1#p2.t1.r1">g</mi></math>: positive integer</a>,
<a href="./21.7#SS3.p1" title="§21.7(iii) Frobenius’ Identity ‣ §21.7 Riemann Surfaces ‣ Applications ‣ Chapter 21 Multidimensional Theta Functions" class="ltx_ref"><math class="ltx_Math" altimg="m28.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="U" display="inline"><mi href="./21.7#SS3.p1">U</mi></math>: subset of <math class="ltx_Math" altimg="m14.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="B" display="inline"><mi href="./21.7#SS3.p1">B</mi></math></a> and
<a href="./21.7#SS3.p1" title="§21.7(iii) Frobenius’ Identity ‣ §21.7 Riemann Surfaces ‣ Applications ‣ Chapter 21 Multidimensional Theta Functions" class="ltx_ref"><math class="ltx_Math" altimg="m26.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="T" display="inline"><mi href="./21.7#SS3.p1">T</mi></math>: subset of <math class="ltx_Math" altimg="m14.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="B" display="inline"><mi href="./21.7#SS3.p1">B</mi></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E16" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">21.7.16</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E16.png" altimg-height="28px" altimg-valign="-7px" altimg-width="512px" alttext="4(\boldsymbol{{\eta}}^{1}(T_{1})\cdot\boldsymbol{{\eta}}^{2}(T_{2})-%
\boldsymbol{{\eta}}^{2}(T_{1})\cdot\boldsymbol{{\eta}}^{1}(T_{2}))=|T_{1}\cap T%
_{2}|\pmod{2}." display="block"><mrow><mrow><mrow><mn>4</mn><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mrow><msup><mi mathvariant="bold-italic">η</mi><mn>1</mn></msup><mo></mo><mrow><mo stretchy="false">(</mo><msub><mi href="./21.7#SS3.p1">T</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mrow></mrow><mo>⋅</mo><mrow><msup><mi mathvariant="bold-italic">η</mi><mn>2</mn></msup><mo></mo><mrow><mo stretchy="false">(</mo><msub><mi href="./21.7#SS3.p1">T</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>-</mo><mrow><mrow><msup><mi mathvariant="bold-italic">η</mi><mn>2</mn></msup><mo></mo><mrow><mo stretchy="false">(</mo><msub><mi href="./21.7#SS3.p1">T</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mrow></mrow><mo>⋅</mo><mrow><msup><mi mathvariant="bold-italic">η</mi><mn>1</mn></msup><mo></mo><mrow><mo stretchy="false">(</mo><msub><mi href="./21.7#SS3.p1">T</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mo stretchy="false">|</mo><mrow><msub><mi href="./21.7#SS3.p1">T</mi><mn>1</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r26">∩</mo><msub><mi href="./21.7#SS3.p1">T</mi><mn>2</mn></msub></mrow><mo stretchy="false">|</mo></mrow><mspace width="veryverythickmathspace"></mspace><mrow><mo lspace="8.1pt" stretchy="false">(</mo><mrow><mo movablelimits="false">mod</mo><mn>2</mn></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E16.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./front/introduction#Sx4.p1.t1.r26" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m45.png" altimg-height="15px" altimg-valign="-2px" altimg-width="18px" alttext="\cap" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r26">∩</mo></math>: intersection</a> and
<a href="./21.7#SS3.p1" title="§21.7(iii) Frobenius’ Identity ‣ §21.7 Riemann Surfaces ‣ Applications ‣ Chapter 21 Multidimensional Theta Functions" class="ltx_ref"><math class="ltx_Math" altimg="m26.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="T" display="inline"><mi href="./21.7#SS3.p1">T</mi></math>: subset of <math class="ltx_Math" altimg="m14.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="B" display="inline"><mi href="./21.7#SS3.p1">B</mi></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS3.p2" class="ltx_para">
<p class="ltx_p">Furthermore, let <math class="ltx_Math" altimg="m37.png" altimg-height="23px" altimg-valign="-7px" altimg-width="100px" alttext="\boldsymbol{{\eta}}(P_{\infty})=\boldsymbol{{0}}" display="inline"><mrow><mrow><mi mathvariant="bold-italic">η</mi><mo></mo><mrow><mo stretchy="false">(</mo><msub><mi href="./21.7#SS3.p1">P</mi><mi mathvariant="normal">∞</mi></msub><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mn mathvariant="bold">0</mn></mrow></math> and
<math class="ltx_Math" altimg="m38.png" altimg-height="24px" altimg-valign="-8px" altimg-width="188px" alttext="\boldsymbol{{\eta}}(P_{j})=\boldsymbol{{\eta}}(\{P_{j},P_{\infty}\})" display="inline"><mrow><mrow><mi mathvariant="bold-italic">η</mi><mo></mo><mrow><mo stretchy="false">(</mo><msub><mi href="./21.7#SS3.p1">P</mi><mi>j</mi></msub><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mi mathvariant="bold-italic">η</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mo stretchy="false">{</mo><msub><mi href="./21.7#SS3.p1">P</mi><mi>j</mi></msub><mo>,</mo><msub><mi href="./21.7#SS3.p1">P</mi><mi mathvariant="normal">∞</mi></msub><mo stretchy="false">}</mo></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></math>. Then for all
<math class="ltx_Math" altimg="m57.png" altimg-height="23px" altimg-valign="-8px" altimg-width="71px" alttext="\mathbf{z}_{j}\in{\mathbb{C}^{g}}" display="inline"><mrow><msub><mi mathvariant="bold">z</mi><mi>j</mi></msub><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><msup><mi href="./front/introduction#Sx4.p1.t1.r1">ℂ</mi><mi href="./21.1#p2.t1.r1">g</mi></msup></mrow></math>, <math class="ltx_Math" altimg="m82.png" altimg-height="20px" altimg-valign="-6px" altimg-width="107px" alttext="j=1,2,3,4" display="inline"><mrow><mi>j</mi><mo>=</mo><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>4</mn></mrow></mrow></math>, such that
<math class="ltx_Math" altimg="m56.png" altimg-height="20px" altimg-valign="-5px" altimg-width="190px" alttext="\mathbf{z}_{1}+\mathbf{z}_{2}+\mathbf{z}_{3}+\mathbf{z}_{4}=0" display="inline"><mrow><mrow><msub><mi mathvariant="bold">z</mi><mn>1</mn></msub><mo>+</mo><msub><mi mathvariant="bold">z</mi><mn>2</mn></msub><mo>+</mo><msub><mi mathvariant="bold">z</mi><mn>3</mn></msub><mo>+</mo><msub><mi mathvariant="bold">z</mi><mn>4</mn></msub></mrow><mo>=</mo><mn>0</mn></mrow></math>, and for all
<math class="ltx_Math" altimg="m33.png" altimg-height="18px" altimg-valign="-8px" altimg-width="28px" alttext="\boldsymbol{{\alpha}}_{j}" display="inline"><msub><mi href="./21.1#p2.t1.r6" mathvariant="bold-italic">α</mi><mi>j</mi></msub></math>, <math class="ltx_Math" altimg="m36.png" altimg-height="25px" altimg-valign="-10px" altimg-width="27px" alttext="\boldsymbol{{\beta}}_{j}" display="inline"><msub><mi href="./21.1#p2.t1.r6" mathvariant="bold-italic">β</mi><mi>j</mi></msub></math> <math class="ltx_Math" altimg="m51.png" altimg-height="18px" altimg-valign="-3px" altimg-width="47px" alttext="\in{\mathbb{R}^{g}}" display="inline"><mrow><mi></mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><msup><mi href="./front/introduction#Sx4.p2.t1.r14">ℝ</mi><mi href="./21.1#p2.t1.r1">g</mi></msup></mrow></math>, such that
<math class="ltx_Math" altimg="m32.png" altimg-height="20px" altimg-valign="-5px" altimg-width="210px" alttext="\boldsymbol{{\alpha}}_{1}+\boldsymbol{{\alpha}}_{2}+\boldsymbol{{\alpha}}_{3}+%
\boldsymbol{{\alpha}}_{4}=0" display="inline"><mrow><mrow><msub><mi href="./21.1#p2.t1.r6" mathvariant="bold-italic">α</mi><mn>1</mn></msub><mo>+</mo><msub><mi href="./21.1#p2.t1.r6" mathvariant="bold-italic">α</mi><mn>2</mn></msub><mo>+</mo><msub><mi href="./21.1#p2.t1.r6" mathvariant="bold-italic">α</mi><mn>3</mn></msub><mo>+</mo><msub><mi href="./21.1#p2.t1.r6" mathvariant="bold-italic">α</mi><mn>4</mn></msub></mrow><mo>=</mo><mn>0</mn></mrow></math> and
<math class="ltx_Math" altimg="m35.png" altimg-height="22px" altimg-valign="-7px" altimg-width="205px" alttext="\boldsymbol{{\beta}}_{1}+\boldsymbol{{\beta}}_{2}+\boldsymbol{{\beta}}_{3}+%
\boldsymbol{{\beta}}_{4}=0" display="inline"><mrow><mrow><msub><mi href="./21.1#p2.t1.r6" mathvariant="bold-italic">β</mi><mn>1</mn></msub><mo>+</mo><msub><mi href="./21.1#p2.t1.r6" mathvariant="bold-italic">β</mi><mn>2</mn></msub><mo>+</mo><msub><mi href="./21.1#p2.t1.r6" mathvariant="bold-italic">β</mi><mn>3</mn></msub><mo>+</mo><msub><mi href="./21.1#p2.t1.r6" mathvariant="bold-italic">β</mi><mn>4</mn></msub></mrow><mo>=</mo><mn>0</mn></mrow></math>, we have <em class="ltx_emph ltx_font_italic">Frobenius’ identity</em>:</p>
<table id="E17" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">21.7.17</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E17.png" altimg-height="71px" altimg-valign="-32px" altimg-width="610px" alttext="\sum_{P_{j}\in U}\prod_{k=1}^{4}\theta\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{%
\alpha}}_{k}+\boldsymbol{{\eta}}^{1}(P_{j})}{\boldsymbol{{\beta}}_{k}+%
\boldsymbol{{\eta}}^{2}(P_{j})}\left(\mathbf{z}_{k}\middle|\boldsymbol{{\Omega%
}}\right)=\sum_{P_{j}\in U^{c}}\prod_{k=1}^{4}\theta\genfrac{[}{]}{0.0pt}{}{%
\boldsymbol{{\alpha}}_{k}+\boldsymbol{{\eta}}^{1}(P_{j})}{\boldsymbol{{\beta}}%
_{k}+\boldsymbol{{\eta}}^{2}(P_{j})}\left(\mathbf{z}_{k}\middle|\boldsymbol{{%
\Omega}}\right)." display="block"><mrow><mrow><mrow><munder><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><msub><mi href="./21.7#SS3.p1">P</mi><mi>j</mi></msub><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mi href="./21.7#SS3.p1">U</mi></mrow></munder><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∏</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mn>4</mn></munderover><mrow><mrow><mi href="./21.2#E5">θ</mi><mo href="./21.2#E5"></mo><mrow><mo href="./21.2#E5">[</mo><mfrac linethickness="0.0pt"><mrow><msub><mi href="./21.1#p2.t1.r6" mathvariant="bold-italic">α</mi><mi>k</mi></msub><mo>+</mo><mrow><msup><mi mathvariant="bold-italic">η</mi><mn>1</mn></msup><mo></mo><mrow><mo stretchy="false">(</mo><msub><mi href="./21.7#SS3.p1">P</mi><mi>j</mi></msub><mo stretchy="false">)</mo></mrow></mrow></mrow><mrow><msub><mi href="./21.1#p2.t1.r6" mathvariant="bold-italic">β</mi><mi>k</mi></msub><mo>+</mo><mrow><msup><mi mathvariant="bold-italic">η</mi><mn>2</mn></msup><mo></mo><mrow><mo stretchy="false">(</mo><msub><mi href="./21.7#SS3.p1">P</mi><mi>j</mi></msub><mo stretchy="false">)</mo></mrow></mrow></mrow></mfrac><mo href="./21.2#E5">]</mo></mrow></mrow><mo></mo><mrow><mo>(</mo><msub><mi mathvariant="bold">z</mi><mi>k</mi></msub><mo>|</mo><mi href="./21.1#p2.t1.r5" mathvariant="bold">Ω</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>=</mo><mrow><munder><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><msub><mi href="./21.7#SS3.p1">P</mi><mi>j</mi></msub><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><msup><mi href="./21.7#SS3.p1">U</mi><mi>c</mi></msup></mrow></munder><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∏</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mn>4</mn></munderover><mrow><mrow><mi href="./21.2#E5">θ</mi><mo href="./21.2#E5"></mo><mrow><mo href="./21.2#E5">[</mo><mfrac linethickness="0.0pt"><mrow><msub><mi href="./21.1#p2.t1.r6" mathvariant="bold-italic">α</mi><mi>k</mi></msub><mo>+</mo><mrow><msup><mi mathvariant="bold-italic">η</mi><mn>1</mn></msup><mo></mo><mrow><mo stretchy="false">(</mo><msub><mi href="./21.7#SS3.p1">P</mi><mi>j</mi></msub><mo stretchy="false">)</mo></mrow></mrow></mrow><mrow><msub><mi href="./21.1#p2.t1.r6" mathvariant="bold-italic">β</mi><mi>k</mi></msub><mo>+</mo><mrow><msup><mi mathvariant="bold-italic">η</mi><mn>2</mn></msup><mo></mo><mrow><mo stretchy="false">(</mo><msub><mi href="./21.7#SS3.p1">P</mi><mi>j</mi></msub><mo stretchy="false">)</mo></mrow></mrow></mrow></mfrac><mo href="./21.2#E5">]</mo></mrow></mrow><mo></mo><mrow><mo>(</mo><msub><mi mathvariant="bold">z</mi><mi>k</mi></msub><mo>|</mo><mi href="./21.1#p2.t1.r5" mathvariant="bold">Ω</mi><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E17.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./21.2#E5" title="(21.2.5) ‣ §21.2(ii) Riemann Theta Functions with Characteristics ‣ §21.2 Definitions ‣ Properties ‣ Chapter 21 Multidimensional Theta Functions" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="30px" altimg-valign="-12px" altimg-width="94px" alttext="\theta\genfrac{[}{]}{0.0pt}{}{\NVar{\boldsymbol{{\alpha}}}}{\NVar{\boldsymbol{%
{\beta}}}}\left(\NVar{\mathbf{z}}\middle|\NVar{\boldsymbol{{\Omega}}}\right)" display="inline"><mrow><mrow><mi href="./21.2#E5">θ</mi><mo href="./21.2#E5"></mo><mrow><mo href="./21.2#E5">[</mo><mfrac linethickness="0.0pt"><mi class="ltx_nvar" href="./21.1#p2.t1.r6" mathvariant="bold-italic">α</mi><mi class="ltx_nvar" href="./21.1#p2.t1.r6" mathvariant="bold-italic">β</mi></mfrac><mo href="./21.2#E5">]</mo></mrow></mrow><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" mathvariant="bold">z</mi><mo>|</mo><mi class="ltx_nvar" href="./21.1#p2.t1.r5" mathvariant="bold">Ω</mi><mo>)</mo></mrow></mrow></math>: Riemann theta function with characteristics</a>,
<a href="./front/introduction#Sx4.p1.t1.r9" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="15px" altimg-valign="-3px" altimg-width="18px" alttext="\in" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo></math>: element of</a>,
<a href="./21.1#p2.t1.r5" title="§21.1 Special Notation ‣ Notation ‣ Chapter 21 Multidimensional Theta Functions" class="ltx_ref"><math class="ltx_Math" altimg="m30.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="\boldsymbol{{\Omega}}" display="inline"><mi href="./21.1#p2.t1.r5" mathvariant="bold">Ω</mi></math>: a Riemann matrix</a>,
<a href="./21.1#p2.t1.r6" title="§21.1 Special Notation ‣ Notation ‣ Chapter 21 Multidimensional Theta Functions" class="ltx_ref"><math class="ltx_Math" altimg="m31.png" altimg-height="13px" altimg-valign="-2px" altimg-width="19px" alttext="\boldsymbol{{\alpha}}" display="inline"><mi href="./21.1#p2.t1.r6" mathvariant="bold-italic">α</mi></math>: <math class="ltx_Math" altimg="m80.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="g" display="inline"><mi href="./21.1#p2.t1.r1">g</mi></math>-dimensional vector</a>,
<a href="./21.1#p2.t1.r6" title="§21.1 Special Notation ‣ Notation ‣ Chapter 21 Multidimensional Theta Functions" class="ltx_ref"><math class="ltx_Math" altimg="m34.png" altimg-height="21px" altimg-valign="-6px" altimg-width="18px" alttext="\boldsymbol{{\beta}}" display="inline"><mi href="./21.1#p2.t1.r6" mathvariant="bold-italic">β</mi></math>: <math class="ltx_Math" altimg="m80.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="g" display="inline"><mi href="./21.1#p2.t1.r1">g</mi></math>-dimensional vector</a>,
<a href="./21.7#SS3.p1" title="§21.7(iii) Frobenius’ Identity ‣ §21.7 Riemann Surfaces ‣ Applications ‣ Chapter 21 Multidimensional Theta Functions" class="ltx_ref"><math class="ltx_Math" altimg="m24.png" altimg-height="23px" altimg-valign="-8px" altimg-width="25px" alttext="P_{j}" display="inline"><msub><mi href="./21.7#SS3.p1">P</mi><mi>j</mi></msub></math>: branch points</a> and
<a href="./21.7#SS3.p1" title="§21.7(iii) Frobenius’ Identity ‣ §21.7 Riemann Surfaces ‣ Applications ‣ Chapter 21 Multidimensional Theta Functions" class="ltx_ref"><math class="ltx_Math" altimg="m28.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="U" display="inline"><mi href="./21.7#SS3.p1">U</mi></math>: subset of <math class="ltx_Math" altimg="m14.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="B" display="inline"><mi href="./21.7#SS3.p1">B</mi></math></a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
</section>
</div>
<div class="ltx_page_footer">
<span></div>
</div>
</body></text>
</html>
</page>
<page>
<!DOCTYPE html><html>
<head>
<title>DLMF: 28.31 Equations of Whittaker–Hill and Ince</title>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<!--GOOGLE BOOTSTRAP--></head>
<text><body class="color_default textfont_default titlefont_default fontsize_default navbar_default">
<div class="ltx_page_navbar">
<div class="ltx_page_navlogo"></dd>
</dl>
</div>
</div>
<div id="SS1.p1" class="ltx_para">
<p class="ltx_p">Hill’s equation with three terms
</p>
<table id="E1" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">28.31.1</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E1.png" altimg-height="30px" altimg-valign="-9px" altimg-width="416px" alttext="W^{\prime\prime}+\left(A+B\cos\left(2z\right)-\tfrac{1}{2}(kc)^{2}\cos\left(4z%
\right)\right)W=0" display="block"><mrow><mrow><msup><mi href="./28.31#SS1.p1">W</mi><mo>′′</mo></msup><mo>+</mo><mrow><mrow><mo>(</mo><mrow><mrow><mi>A</mi><mo>+</mo><mrow><mi>B</mi><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><mi href="./28.1#p2.t1.r3">z</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>-</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi>k</mi><mo></mo><mi>c</mi></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mn>4</mn><mo></mo><mi href="./28.1#p2.t1.r3">z</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>)</mo></mrow><mo></mo><mi href="./28.31#SS1.p1">W</mi></mrow></mrow><mo>=</mo><mn>0</mn></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./28.1#p2.t1.r3" title="§28.1 Special Notation ‣ Notation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m114.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./28.1#p2.t1.r3">z</mi></math>: complex variable</a> and
<a href="./28.31#SS1.p1" title="§28.31(i) Whittaker–Hill Equation ‣ §28.31 Equations of Whittaker–Hill and Ince ‣ Hill’s Equation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="23px" altimg-valign="-7px" altimg-width="52px" alttext="W(z)" display="inline"><mrow><mi href="./28.31#SS1.p1">W</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.1#p2.t1.r3">z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: solution</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and constant values of <math class="ltx_Math" altimg="m49.png" altimg-height="21px" altimg-valign="-6px" altimg-width="64px" alttext="A,B,k" display="inline"><mrow><mi>A</mi><mo>,</mo><mi>B</mi><mo>,</mo><mi>k</mi></mrow></math>, and <math class="ltx_Math" altimg="m92.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="c" display="inline"><mi>c</mi></math>, is called the <em class="ltx_emph ltx_font_italic">Equation of
Whittaker–Hill</em>. It has been discussed in detail by <cite class="ltx_cite ltx_citemacro_citet">Arscott ()</cite> for
<math class="ltx_Math" altimg="m94.png" altimg-height="21px" altimg-valign="-3px" altimg-width="61px" alttext="k^{2}<0" display="inline"><mrow><msup><mi>k</mi><mn>2</mn></msup><mo><</mo><mn>0</mn></mrow></math>, and by <cite class="ltx_cite ltx_citemacro_citet">Urwin and Arscott ()</cite> for <math class="ltx_Math" altimg="m95.png" altimg-height="21px" altimg-valign="-3px" altimg-width="61px" alttext="k^{2}>0" display="inline"><mrow><msup><mi>k</mi><mn>2</mn></msup><mo>></mo><mn>0</mn></mrow></math>.</p>
</div>
</section>
<section id="ii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§28.31(ii) </span>Equation of Ince; Ince Polynomials</h2>
<div id="SS2.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="SS2.p1" class="ltx_para">
<p class="ltx_p">When <math class="ltx_Math" altimg="m94.png" altimg-height="21px" altimg-valign="-3px" altimg-width="61px" alttext="k^{2}<0" display="inline"><mrow><msup><mi href="./28.6#SS1.p6">k</mi><mn>2</mn></msup><mo><</mo><mn>0</mn></mrow></math>, we substitute
</p>
<table id="E2" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex1" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="4" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">28.31.2</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m38.png" altimg-height="27px" altimg-valign="-6px" altimg-width="25px" alttext="\displaystyle\xi^{2}" display="inline"><msup><mi href="./28.31#SS2.p1">ξ</mi><mn>2</mn></msup></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m12.png" altimg-height="27px" altimg-valign="-6px" altimg-width="96px" alttext="\displaystyle=-4k^{2}c^{2}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo>-</mo><mrow><mn>4</mn><mo></mo><msup><mi href="./28.6#SS1.p6">k</mi><mn>2</mn></msup><mo></mo><msup><mi>c</mi><mn>2</mn></msup></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex2" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m18.png" altimg-height="19px" altimg-valign="-2px" altimg-width="21px" alttext="\displaystyle A" display="inline"><mi href="./28.31#SS2.p1">A</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m14.png" altimg-height="30px" altimg-valign="-9px" altimg-width="99px" alttext="\displaystyle=\eta-\tfrac{1}{8}\xi^{2}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./28.31#SS2.p1">η</mi><mo>-</mo><mrow><mfrac><mn>1</mn><mn>8</mn></mfrac><mo></mo><msup><mi href="./28.31#SS2.p1">ξ</mi><mn>2</mn></msup></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex3" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m19.png" altimg-height="19px" altimg-valign="-2px" altimg-width="22px" alttext="\displaystyle B" display="inline"><mi href="./28.31#SS2.p1">B</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m11.png" altimg-height="25px" altimg-valign="-7px" altimg-width="118px" alttext="\displaystyle=-(p+1)\xi," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo>-</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi href="./28.31#SS2.p1">ξ</mi></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex4" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m23.png" altimg-height="25px" altimg-valign="-7px" altimg-width="53px" alttext="\displaystyle W(z)" display="inline"><mrow><mi href="./28.31#SS1.p1">W</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.1#p2.t1.r3">z</mi><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m17.png" altimg-height="29px" altimg-valign="-9px" altimg-width="238px" alttext="\displaystyle=w(z)\exp\left(-\tfrac{1}{4}\xi\cos\left(2z\right)\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mi href="./28.2#SS1.p1">w</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.1#p2.t1.r3">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac><mo></mo><mi href="./28.31#SS2.p1">ξ</mi><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><mi href="./28.1#p2.t1.r3">z</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./4.2#E19" title="(4.2.19) ‣ §4.2(iii) The Exponential Function ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="16px" altimg-valign="-6px" altimg-width="48px" alttext="\exp\NVar{z}" display="inline"><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: exponential function</a>,
<a href="./28.1#p2.t1.r3" title="§28.1 Special Notation ‣ Notation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m114.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./28.1#p2.t1.r3">z</mi></math>: complex variable</a>,
<a href="./28.2#SS1.p1" title="§28.2(i) Mathieu’s Equation ‣ §28.2 Definitions and Basic Properties ‣ Mathieu Functions of Integer Order ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m110.png" altimg-height="23px" altimg-valign="-7px" altimg-width="45px" alttext="w(z)" display="inline"><mrow><mi href="./28.2#SS1.p1">w</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.1#p2.t1.r3">z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: Mathieu’s equation solution</a>,
<a href="./28.31#SS1.p1" title="§28.31(i) Whittaker–Hill Equation ‣ §28.31 Equations of Whittaker–Hill and Ince ‣ Hill’s Equation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m58.png" altimg-height="23px" altimg-valign="-7px" altimg-width="52px" alttext="W(z)" display="inline"><mrow><mi href="./28.31#SS1.p1">W</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.1#p2.t1.r3">z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: solution</a>,
<a href="./28.31#SS2.p1" title="§28.31(ii) Equation of Ince; Ince Polynomials ‣ §28.31 Equations of Whittaker–Hill and Ince ‣ Hill’s Equation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="A" display="inline"><mi href="./28.31#SS2.p1">A</mi></math>: constant</a>,
<a href="./28.31#SS2.p1" title="§28.31(ii) Equation of Ince; Ince Polynomials ‣ §28.31 Equations of Whittaker–Hill and Ince ‣ Hill’s Equation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="B" display="inline"><mi href="./28.31#SS2.p1">B</mi></math>: constant</a>,
<a href="./28.31#SS2.p1" title="§28.31(ii) Equation of Ince; Ince Polynomials ‣ §28.31 Equations of Whittaker–Hill and Ince ‣ Hill’s Equation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m65.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="\eta" display="inline"><mi href="./28.31#SS2.p1">η</mi></math>: variable</a>,
<a href="./28.31#SS2.p1" title="§28.31(ii) Equation of Ince; Ince Polynomials ‣ §28.31 Equations of Whittaker–Hill and Ince ‣ Hill’s Equation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="21px" altimg-valign="-6px" altimg-width="14px" alttext="\xi" display="inline"><mi href="./28.31#SS2.p1">ξ</mi></math>: variable</a> and
<a href="./28.6#SS1.p6" title="§28.6(i) Eigenvalues ‣ §28.6 Expansions for Small q ‣ Mathieu Functions of Integer Order ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m93.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./28.6#SS1.p6">k</mi></math>: root</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">in (). The result is the <em class="ltx_emph ltx_font_italic">Equation of Ince</em>:</p>
<table id="E3" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">28.31.3</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E3.png" altimg-height="26px" altimg-valign="-7px" altimg-width="376px" alttext="w^{\prime\prime}+\xi\sin\left(2z\right)w^{\prime}+(\eta-p\xi\cos\left(2z\right%
))w=0." display="block"><mrow><mrow><mrow><msup><mi href="./28.2#SS1.p1">w</mi><mo>′′</mo></msup><mo>+</mo><mrow><mi href="./28.31#SS2.p1">ξ</mi><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><mi href="./28.1#p2.t1.r3">z</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><msup><mi href="./28.2#SS1.p1">w</mi><mo>′</mo></msup></mrow><mo>+</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./28.31#SS2.p1">η</mi><mo>-</mo><mrow><mi>p</mi><mo></mo><mi href="./28.31#SS2.p1">ξ</mi><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><mi href="./28.1#p2.t1.r3">z</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi href="./28.2#SS1.p1">w</mi></mrow></mrow><mo>=</mo><mn>0</mn></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m84.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./28.1#p2.t1.r3" title="§28.1 Special Notation ‣ Notation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m114.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./28.1#p2.t1.r3">z</mi></math>: complex variable</a>,
<a href="./28.2#SS1.p1" title="§28.2(i) Mathieu’s Equation ‣ §28.2 Definitions and Basic Properties ‣ Mathieu Functions of Integer Order ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m110.png" altimg-height="23px" altimg-valign="-7px" altimg-width="45px" alttext="w(z)" display="inline"><mrow><mi href="./28.2#SS1.p1">w</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.1#p2.t1.r3">z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: Mathieu’s equation solution</a>,
<a href="./28.31#SS2.p1" title="§28.31(ii) Equation of Ince; Ince Polynomials ‣ §28.31 Equations of Whittaker–Hill and Ince ‣ Hill’s Equation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m65.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="\eta" display="inline"><mi href="./28.31#SS2.p1">η</mi></math>: variable</a> and
<a href="./28.31#SS2.p1" title="§28.31(ii) Equation of Ince; Ince Polynomials ‣ §28.31 Equations of Whittaker–Hill and Ince ‣ Hill’s Equation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="21px" altimg-valign="-6px" altimg-width="14px" alttext="\xi" display="inline"><mi href="./28.31#SS2.p1">ξ</mi></math>: variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS2.p2" class="ltx_para">
<p class="ltx_p">Formal <math class="ltx_Math" altimg="m48.png" altimg-height="17px" altimg-valign="-2px" altimg-width="26px" alttext="2\pi" display="inline"><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi></mrow></math>-periodic solutions can be constructed as Fourier series; compare
§:</p>
<table id="EGx1" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E4">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">28.31.4</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m43.png" altimg-height="26px" altimg-valign="-8px" altimg-width="68px" alttext="\displaystyle w_{\mathit{e},s}(z)" display="inline"><mrow><msub><mi href="./28.2#SS1.p1">w</mi><mrow><mi>e</mi><mo>,</mo><mi>s</mi></mrow></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.1#p2.t1.r3">z</mi><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m15.png" altimg-height="64px" altimg-valign="-28px" altimg-width="223px" alttext="\displaystyle=\sum_{\ell=0}^{\infty}A_{2\ell+s}\cos(2\ell+s)z," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi mathvariant="normal">ℓ</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover></mstyle><mrow><msub><mi href="./28.31#SS2.p1">A</mi><mrow><mrow><mn>2</mn><mo></mo><mi mathvariant="normal">ℓ</mi></mrow><mo>+</mo><mi>s</mi></mrow></msub><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>2</mn><mo></mo><mi mathvariant="normal">ℓ</mi></mrow><mo>+</mo><mi>s</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi href="./28.1#p2.t1.r3">z</mi></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m108.png" altimg-height="20px" altimg-valign="-6px" altimg-width="69px" alttext="s=0,1" display="inline"><mrow><mi>s</mi><mo>=</mo><mrow><mn>0</mn><mo>,</mo><mn>1</mn></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E4.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./28.1#p2.t1.r3" title="§28.1 Special Notation ‣ Notation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m114.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./28.1#p2.t1.r3">z</mi></math>: complex variable</a>,
<a href="./28.2#SS1.p1" title="§28.2(i) Mathieu’s Equation ‣ §28.2 Definitions and Basic Properties ‣ Mathieu Functions of Integer Order ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m110.png" altimg-height="23px" altimg-valign="-7px" altimg-width="45px" alttext="w(z)" display="inline"><mrow><mi href="./28.2#SS1.p1">w</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.1#p2.t1.r3">z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: Mathieu’s equation solution</a> and
<a href="./28.31#SS2.p1" title="§28.31(ii) Equation of Ince; Ince Polynomials ‣ §28.31 Equations of Whittaker–Hill and Ince ‣ Hill’s Equation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="A" display="inline"><mi href="./28.31#SS2.p1">A</mi></math>: constant</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E5">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">28.31.5</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m44.png" altimg-height="26px" altimg-valign="-8px" altimg-width="68px" alttext="\displaystyle w_{\mathit{o},s}(z)" display="inline"><mrow><msub><mi href="./28.2#SS1.p1">w</mi><mrow><mi>o</mi><mo>,</mo><mi>s</mi></mrow></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.1#p2.t1.r3">z</mi><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m16.png" altimg-height="64px" altimg-valign="-28px" altimg-width="221px" alttext="\displaystyle=\sum_{\ell=0}^{\infty}B_{2\ell+s}\sin(2\ell+s)z," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi mathvariant="normal">ℓ</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover></mstyle><mrow><msub><mi href="./28.31#SS2.p1">B</mi><mrow><mrow><mn>2</mn><mo></mo><mi mathvariant="normal">ℓ</mi></mrow><mo>+</mo><mi>s</mi></mrow></msub><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>2</mn><mo></mo><mi mathvariant="normal">ℓ</mi></mrow><mo>+</mo><mi>s</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi href="./28.1#p2.t1.r3">z</mi></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m109.png" altimg-height="20px" altimg-valign="-6px" altimg-width="69px" alttext="s=1,2" display="inline"><mrow><mi>s</mi><mo>=</mo><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E5.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m84.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./28.1#p2.t1.r3" title="§28.1 Special Notation ‣ Notation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m114.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./28.1#p2.t1.r3">z</mi></math>: complex variable</a>,
<a href="./28.2#SS1.p1" title="§28.2(i) Mathieu’s Equation ‣ §28.2 Definitions and Basic Properties ‣ Mathieu Functions of Integer Order ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m110.png" altimg-height="23px" altimg-valign="-7px" altimg-width="45px" alttext="w(z)" display="inline"><mrow><mi href="./28.2#SS1.p1">w</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.1#p2.t1.r3">z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: Mathieu’s equation solution</a> and
<a href="./28.31#SS2.p1" title="§28.31(ii) Equation of Ince; Ince Polynomials ‣ §28.31 Equations of Whittaker–Hill and Ince ‣ Hill’s Equation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="B" display="inline"><mi href="./28.31#SS2.p1">B</mi></math>: constant</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
<p class="ltx_p">where the coefficients satisfy</p>
</div>
<div id="SS2.p3" class="ltx_para">
<table id="E6" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex5" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="4" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">28.31.6</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m6.png" altimg-height="25px" altimg-valign="-7px" altimg-width="184px" alttext="\displaystyle-2\eta A_{0}+(2+p)\xi A_{2}" display="inline"><mrow><mrow><mo>-</mo><mrow><mn>2</mn><mo></mo><mi href="./28.31#SS2.p1">η</mi><mo></mo><msub><mi href="./28.31#SS2.p1">A</mi><mn>0</mn></msub></mrow></mrow><mo>+</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mn>2</mn><mo>+</mo><mi>p</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi href="./28.31#SS2.p1">ξ</mi><mo></mo><msub><mi href="./28.31#SS2.p1">A</mi><mn>2</mn></msub></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m13.png" altimg-height="22px" altimg-valign="-6px" altimg-width="43px" alttext="\displaystyle=0," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mn>0</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex6" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m42.png" altimg-height="29px" altimg-valign="-9px" altimg-width="295px" alttext="\displaystyle p\xi A_{0}+(4-\eta)A_{2}+\left(\tfrac{1}{2}p+2\right)\xi A_{4}" display="inline"><mrow><mrow><mi>p</mi><mo></mo><mi href="./28.31#SS2.p1">ξ</mi><mo></mo><msub><mi href="./28.31#SS2.p1">A</mi><mn>0</mn></msub></mrow><mo>+</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mn>4</mn><mo>-</mo><mi href="./28.31#SS2.p1">η</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msub><mi href="./28.31#SS2.p1">A</mi><mn>2</mn></msub></mrow><mo>+</mo><mrow><mrow><mo>(</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi>p</mi></mrow><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow><mo></mo><mi href="./28.31#SS2.p1">ξ</mi><mo></mo><msub><mi href="./28.31#SS2.p1">A</mi><mn>4</mn></msub></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m13.png" altimg-height="22px" altimg-valign="-6px" altimg-width="43px" alttext="\displaystyle=0," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mn>0</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex7" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m2.png" altimg-height="30px" altimg-valign="-9px" altimg-width="500px" alttext="\displaystyle(\tfrac{1}{2}p-\ell+1)\xi A_{2\ell-2}+\left(4\ell^{2}-\eta\right)%
A_{2\ell}+(\tfrac{1}{2}p+\ell+1)\xi A_{2\ell+2}" display="inline"><mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi>p</mi></mrow><mo>-</mo><mi mathvariant="normal">ℓ</mi></mrow><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi href="./28.31#SS2.p1">ξ</mi><mo></mo><msub><mi href="./28.31#SS2.p1">A</mi><mrow><mrow><mn>2</mn><mo></mo><mi mathvariant="normal">ℓ</mi></mrow><mo>-</mo><mn>2</mn></mrow></msub></mrow><mo>+</mo><mrow><mrow><mo>(</mo><mrow><mrow><mn>4</mn><mo></mo><msup><mi mathvariant="normal">ℓ</mi><mn>2</mn></msup></mrow><mo>-</mo><mi href="./28.31#SS2.p1">η</mi></mrow><mo>)</mo></mrow><mo></mo><msub><mi href="./28.31#SS2.p1">A</mi><mrow><mn>2</mn><mo></mo><mi mathvariant="normal">ℓ</mi></mrow></msub></mrow><mo>+</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi>p</mi></mrow><mo>+</mo><mi mathvariant="normal">ℓ</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi href="./28.31#SS2.p1">ξ</mi><mo></mo><msub><mi href="./28.31#SS2.p1">A</mi><mrow><mrow><mn>2</mn><mo></mo><mi mathvariant="normal">ℓ</mi></mrow><mo>+</mo><mn>2</mn></mrow></msub></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m13.png" altimg-height="22px" altimg-valign="-6px" altimg-width="43px" alttext="\displaystyle=0," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mn>0</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m62.png" altimg-height="20px" altimg-valign="-5px" altimg-width="49px" alttext="\ell\geq 2" display="inline"><mrow><mi mathvariant="normal">ℓ</mi><mo>≥</mo><mn>2</mn></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E6.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./28.31#SS2.p1" title="§28.31(ii) Equation of Ince; Ince Polynomials ‣ §28.31 Equations of Whittaker–Hill and Ince ‣ Hill’s Equation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="A" display="inline"><mi href="./28.31#SS2.p1">A</mi></math>: constant</a>,
<a href="./28.31#SS2.p1" title="§28.31(ii) Equation of Ince; Ince Polynomials ‣ §28.31 Equations of Whittaker–Hill and Ince ‣ Hill’s Equation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m65.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="\eta" display="inline"><mi href="./28.31#SS2.p1">η</mi></math>: variable</a> and
<a href="./28.31#SS2.p1" title="§28.31(ii) Equation of Ince; Ince Polynomials ‣ §28.31 Equations of Whittaker–Hill and Ince ‣ Hill’s Equation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="21px" altimg-valign="-6px" altimg-width="14px" alttext="\xi" display="inline"><mi href="./28.31#SS2.p1">ξ</mi></math>: variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E7" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex8" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="3" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">28.31.7</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m24.png" altimg-height="29px" altimg-valign="-9px" altimg-width="351px" alttext="\displaystyle\left(1-\eta+\left(\tfrac{1}{2}p+\tfrac{1}{2}\right)\xi\right)A_{%
1}+\left(\tfrac{1}{2}p+\tfrac{3}{2}\right)\xi A_{3}" display="inline"><mrow><mrow><mrow><mo>(</mo><mrow><mrow><mn>1</mn><mo>-</mo><mi href="./28.31#SS2.p1">η</mi></mrow><mo>+</mo><mrow><mrow><mo>(</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi>p</mi></mrow><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow><mo></mo><mi href="./28.31#SS2.p1">ξ</mi></mrow></mrow><mo>)</mo></mrow><mo></mo><msub><mi href="./28.31#SS2.p1">A</mi><mn>1</mn></msub></mrow><mo>+</mo><mrow><mrow><mo>(</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi>p</mi></mrow><mo>+</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow><mo></mo><mi href="./28.31#SS2.p1">ξ</mi><mo></mo><msub><mi href="./28.31#SS2.p1">A</mi><mn>3</mn></msub></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m13.png" altimg-height="22px" altimg-valign="-6px" altimg-width="43px" alttext="\displaystyle=0," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mn>0</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex9" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m4.png" altimg-height="30px" altimg-valign="-9px" altimg-width="576px" alttext="\displaystyle(\tfrac{1}{2}p-\ell+\tfrac{1}{2})\xi A_{2\ell-1}+\left((2\ell+1)^%
{2}-\eta\right)A_{2\ell+1}+(\tfrac{1}{2}p+\ell+\tfrac{3}{2})\xi A_{2\ell+3}" display="inline"><mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi>p</mi></mrow><mo>-</mo><mi mathvariant="normal">ℓ</mi></mrow><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi href="./28.31#SS2.p1">ξ</mi><mo></mo><msub><mi href="./28.31#SS2.p1">A</mi><mrow><mrow><mn>2</mn><mo></mo><mi mathvariant="normal">ℓ</mi></mrow><mo>-</mo><mn>1</mn></mrow></msub></mrow><mo>+</mo><mrow><mrow><mo>(</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>2</mn><mo></mo><mi mathvariant="normal">ℓ</mi></mrow><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>-</mo><mi href="./28.31#SS2.p1">η</mi></mrow><mo>)</mo></mrow><mo></mo><msub><mi href="./28.31#SS2.p1">A</mi><mrow><mrow><mn>2</mn><mo></mo><mi mathvariant="normal">ℓ</mi></mrow><mo>+</mo><mn>1</mn></mrow></msub></mrow><mo>+</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi>p</mi></mrow><mo>+</mo><mi mathvariant="normal">ℓ</mi><mo>+</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi href="./28.31#SS2.p1">ξ</mi><mo></mo><msub><mi href="./28.31#SS2.p1">A</mi><mrow><mrow><mn>2</mn><mo></mo><mi mathvariant="normal">ℓ</mi></mrow><mo>+</mo><mn>3</mn></mrow></msub></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m13.png" altimg-height="22px" altimg-valign="-6px" altimg-width="43px" alttext="\displaystyle=0," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mn>0</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m61.png" altimg-height="20px" altimg-valign="-5px" altimg-width="49px" alttext="\ell\geq 1" display="inline"><mrow><mi mathvariant="normal">ℓ</mi><mo>≥</mo><mn>1</mn></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E7.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./28.31#SS2.p1" title="§28.31(ii) Equation of Ince; Ince Polynomials ‣ §28.31 Equations of Whittaker–Hill and Ince ‣ Hill’s Equation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="A" display="inline"><mi href="./28.31#SS2.p1">A</mi></math>: constant</a>,
<a href="./28.31#SS2.p1" title="§28.31(ii) Equation of Ince; Ince Polynomials ‣ §28.31 Equations of Whittaker–Hill and Ince ‣ Hill’s Equation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m65.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="\eta" display="inline"><mi href="./28.31#SS2.p1">η</mi></math>: variable</a> and
<a href="./28.31#SS2.p1" title="§28.31(ii) Equation of Ince; Ince Polynomials ‣ §28.31 Equations of Whittaker–Hill and Ince ‣ Hill’s Equation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="21px" altimg-valign="-6px" altimg-width="14px" alttext="\xi" display="inline"><mi href="./28.31#SS2.p1">ξ</mi></math>: variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E8" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex10" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="3" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">28.31.8</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m25.png" altimg-height="29px" altimg-valign="-9px" altimg-width="351px" alttext="\displaystyle\left(1-\eta-\left(\tfrac{1}{2}p+\tfrac{1}{2}\right)\xi\right)B_{%
1}+\left(\tfrac{1}{2}p+\tfrac{3}{2}\right)\xi B_{3}" display="inline"><mrow><mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mi href="./28.31#SS2.p1">η</mi><mo>-</mo><mrow><mrow><mo>(</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi>p</mi></mrow><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow><mo></mo><mi href="./28.31#SS2.p1">ξ</mi></mrow></mrow><mo>)</mo></mrow><mo></mo><msub><mi href="./28.31#SS2.p1">B</mi><mn>1</mn></msub></mrow><mo>+</mo><mrow><mrow><mo>(</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi>p</mi></mrow><mo>+</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow><mo></mo><mi href="./28.31#SS2.p1">ξ</mi><mo></mo><msub><mi href="./28.31#SS2.p1">B</mi><mn>3</mn></msub></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m13.png" altimg-height="22px" altimg-valign="-6px" altimg-width="43px" alttext="\displaystyle=0," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mn>0</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex11" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m5.png" altimg-height="30px" altimg-valign="-9px" altimg-width="576px" alttext="\displaystyle(\tfrac{1}{2}p-\ell+\tfrac{1}{2})\xi B_{2\ell-1}+\left((2\ell+1)^%
{2}-\eta\right)B_{2\ell+1}+(\tfrac{1}{2}p+\ell+\tfrac{3}{2})\xi B_{2\ell+3}" display="inline"><mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi>p</mi></mrow><mo>-</mo><mi mathvariant="normal">ℓ</mi></mrow><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi href="./28.31#SS2.p1">ξ</mi><mo></mo><msub><mi href="./28.31#SS2.p1">B</mi><mrow><mrow><mn>2</mn><mo></mo><mi mathvariant="normal">ℓ</mi></mrow><mo>-</mo><mn>1</mn></mrow></msub></mrow><mo>+</mo><mrow><mrow><mo>(</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>2</mn><mo></mo><mi mathvariant="normal">ℓ</mi></mrow><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>-</mo><mi href="./28.31#SS2.p1">η</mi></mrow><mo>)</mo></mrow><mo></mo><msub><mi href="./28.31#SS2.p1">B</mi><mrow><mrow><mn>2</mn><mo></mo><mi mathvariant="normal">ℓ</mi></mrow><mo>+</mo><mn>1</mn></mrow></msub></mrow><mo>+</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi>p</mi></mrow><mo>+</mo><mi mathvariant="normal">ℓ</mi><mo>+</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi href="./28.31#SS2.p1">ξ</mi><mo></mo><msub><mi href="./28.31#SS2.p1">B</mi><mrow><mrow><mn>2</mn><mo></mo><mi mathvariant="normal">ℓ</mi></mrow><mo>+</mo><mn>3</mn></mrow></msub></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m13.png" altimg-height="22px" altimg-valign="-6px" altimg-width="43px" alttext="\displaystyle=0," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mn>0</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m61.png" altimg-height="20px" altimg-valign="-5px" altimg-width="49px" alttext="\ell\geq 1" display="inline"><mrow><mi mathvariant="normal">ℓ</mi><mo>≥</mo><mn>1</mn></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E8.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./28.31#SS2.p1" title="§28.31(ii) Equation of Ince; Ince Polynomials ‣ §28.31 Equations of Whittaker–Hill and Ince ‣ Hill’s Equation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="B" display="inline"><mi href="./28.31#SS2.p1">B</mi></math>: constant</a>,
<a href="./28.31#SS2.p1" title="§28.31(ii) Equation of Ince; Ince Polynomials ‣ §28.31 Equations of Whittaker–Hill and Ince ‣ Hill’s Equation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m65.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="\eta" display="inline"><mi href="./28.31#SS2.p1">η</mi></math>: variable</a> and
<a href="./28.31#SS2.p1" title="§28.31(ii) Equation of Ince; Ince Polynomials ‣ §28.31 Equations of Whittaker–Hill and Ince ‣ Hill’s Equation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="21px" altimg-valign="-6px" altimg-width="14px" alttext="\xi" display="inline"><mi href="./28.31#SS2.p1">ξ</mi></math>: variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E9" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex12" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="3" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">28.31.9</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m1.png" altimg-height="29px" altimg-valign="-9px" altimg-width="227px" alttext="\displaystyle(4-\eta)B_{2}+\left(\tfrac{1}{2}p+2\right)\xi B_{4}" display="inline"><mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mn>4</mn><mo>-</mo><mi href="./28.31#SS2.p1">η</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msub><mi href="./28.31#SS2.p1">B</mi><mn>2</mn></msub></mrow><mo>+</mo><mrow><mrow><mo>(</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi>p</mi></mrow><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow><mo></mo><mi href="./28.31#SS2.p1">ξ</mi><mo></mo><msub><mi href="./28.31#SS2.p1">B</mi><mn>4</mn></msub></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m13.png" altimg-height="22px" altimg-valign="-6px" altimg-width="43px" alttext="\displaystyle=0," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mn>0</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex13" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m3.png" altimg-height="30px" altimg-valign="-9px" altimg-width="495px" alttext="\displaystyle(\tfrac{1}{2}p-\ell+1)\xi B_{2\ell-2}+(4\ell^{2}-\eta)B_{2\ell}+(%
\tfrac{1}{2}p+\ell+1)\xi B_{2\ell+2}" display="inline"><mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi>p</mi></mrow><mo>-</mo><mi mathvariant="normal">ℓ</mi></mrow><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi href="./28.31#SS2.p1">ξ</mi><mo></mo><msub><mi href="./28.31#SS2.p1">B</mi><mrow><mrow><mn>2</mn><mo></mo><mi mathvariant="normal">ℓ</mi></mrow><mo>-</mo><mn>2</mn></mrow></msub></mrow><mo>+</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>4</mn><mo></mo><msup><mi mathvariant="normal">ℓ</mi><mn>2</mn></msup></mrow><mo>-</mo><mi href="./28.31#SS2.p1">η</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msub><mi href="./28.31#SS2.p1">B</mi><mrow><mn>2</mn><mo></mo><mi mathvariant="normal">ℓ</mi></mrow></msub></mrow><mo>+</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi>p</mi></mrow><mo>+</mo><mi mathvariant="normal">ℓ</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi href="./28.31#SS2.p1">ξ</mi><mo></mo><msub><mi href="./28.31#SS2.p1">B</mi><mrow><mrow><mn>2</mn><mo></mo><mi mathvariant="normal">ℓ</mi></mrow><mo>+</mo><mn>2</mn></mrow></msub></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m13.png" altimg-height="22px" altimg-valign="-6px" altimg-width="43px" alttext="\displaystyle=0," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mn>0</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m62.png" altimg-height="20px" altimg-valign="-5px" altimg-width="49px" alttext="\ell\geq 2" display="inline"><mrow><mi mathvariant="normal">ℓ</mi><mo>≥</mo><mn>2</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E9.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./28.31#SS2.p1" title="§28.31(ii) Equation of Ince; Ince Polynomials ‣ §28.31 Equations of Whittaker–Hill and Ince ‣ Hill’s Equation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="B" display="inline"><mi href="./28.31#SS2.p1">B</mi></math>: constant</a>,
<a href="./28.31#SS2.p1" title="§28.31(ii) Equation of Ince; Ince Polynomials ‣ §28.31 Equations of Whittaker–Hill and Ince ‣ Hill’s Equation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m65.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="\eta" display="inline"><mi href="./28.31#SS2.p1">η</mi></math>: variable</a> and
<a href="./28.31#SS2.p1" title="§28.31(ii) Equation of Ince; Ince Polynomials ‣ §28.31 Equations of Whittaker–Hill and Ince ‣ Hill’s Equation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="21px" altimg-valign="-6px" altimg-width="14px" alttext="\xi" display="inline"><mi href="./28.31#SS2.p1">ξ</mi></math>: variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS2.p4" class="ltx_para">
<p class="ltx_p">When <math class="ltx_Math" altimg="m101.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="p" display="inline"><mi>p</mi></math> is a nonnegative integer, the parameter <math class="ltx_Math" altimg="m65.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="\eta" display="inline"><mi href="./28.31#SS2.p1">η</mi></math> can be chosen so that
solutions of () are trigonometric polynomials, called
<em class="ltx_emph ltx_font_italic">Ince polynomials</em>.
They are denoted by
</p>
<table id="E10" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">28.31.10</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E10.png" altimg-height="54px" altimg-valign="-21px" altimg-width="293px" alttext="\begin{array}[]{cl}C_{2n}^{2m}(z,\xi)&\mbox{with $p=2n$},\\
C_{2n+1}^{2m+1}(z,\xi)&\mbox{with $p=2n+1$},\end{array}" display="block"><mtable columnspacing="5pt" displaystyle="true" rowspacing="2.0pt"><mtr><mtd columnalign="center"><mrow><msubsup><mi href="./28.31#E10">C</mi><mrow><mn>2</mn><mo></mo><mi href="./28.1#p2.t1.r1">n</mi></mrow><mrow><mn>2</mn><mo></mo><mi href="./28.1#p2.t1.r1">m</mi></mrow></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.1#p2.t1.r3">z</mi><mo>,</mo><mi href="./28.31#SS2.p1">ξ</mi><mo stretchy="false">)</mo></mrow></mrow></mtd><mtd columnalign="left"><mrow><mrow><mtext>with </mtext><mrow><mi>p</mi><mo>=</mo><mrow><mn>2</mn><mo></mo><mi href="./28.1#p2.t1.r1">n</mi></mrow></mrow></mrow><mo>,</mo></mrow></mtd></mtr><mtr><mtd columnalign="center"><mrow><msubsup><mi href="./28.31#E10">C</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./28.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mn>1</mn></mrow><mrow><mrow><mn>2</mn><mo></mo><mi href="./28.1#p2.t1.r1">m</mi></mrow><mo>+</mo><mn>1</mn></mrow></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.1#p2.t1.r3">z</mi><mo>,</mo><mi href="./28.31#SS2.p1">ξ</mi><mo stretchy="false">)</mo></mrow></mrow></mtd><mtd columnalign="left"><mrow><mrow><mtext>with </mtext><mrow><mi>p</mi><mo>=</mo><mrow><mrow><mn>2</mn><mo></mo><mi href="./28.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mn>1</mn></mrow></mrow></mrow><mo>,</mo></mrow></mtd></mtr></mtable></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E10.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text"><math class="ltx_Math" altimg="m54.png" altimg-height="26px" altimg-valign="-10px" altimg-width="79px" alttext="C_{p}^{m}(z,\xi)" display="inline"><mrow><msubsup><mi href="./28.31#E10">C</mi><mi>p</mi><mi href="./28.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.1#p2.t1.r3">z</mi><mo>,</mo><mi href="./28.31#SS2.p1">ξ</mi><mo stretchy="false">)</mo></mrow></mrow></math>: Ince polynomials (locally)</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./28.1#p2.t1.r1" title="§28.1 Special Notation ‣ Notation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m97.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./28.1#p2.t1.r1">m</mi></math>: integer</a>,
<a href="./28.1#p2.t1.r1" title="§28.1 Special Notation ‣ Notation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m100.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./28.1#p2.t1.r1">n</mi></math>: integer</a>,
<a href="./28.1#p2.t1.r3" title="§28.1 Special Notation ‣ Notation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m114.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./28.1#p2.t1.r3">z</mi></math>: complex variable</a> and
<a href="./28.31#SS2.p1" title="§28.31(ii) Equation of Ince; Ince Polynomials ‣ §28.31 Equations of Whittaker–Hill and Ince ‣ Hill’s Equation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="21px" altimg-valign="-6px" altimg-width="14px" alttext="\xi" display="inline"><mi href="./28.31#SS2.p1">ξ</mi></math>: variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E11" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">28.31.11</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E11.png" altimg-height="55px" altimg-valign="-22px" altimg-width="290px" alttext="\begin{array}[]{cl}S_{2n+1}^{2m+1}(z,\xi)&\mbox{with $p=2n+1$},\\
S_{2n+2}^{2m+2}(z,\xi)&\mbox{with $p=2n+2$},\end{array}" display="block"><mtable columnspacing="5pt" displaystyle="true" rowspacing="2.0pt"><mtr><mtd columnalign="center"><mrow><msubsup><mi href="./28.31#E11">S</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./28.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mn>1</mn></mrow><mrow><mrow><mn>2</mn><mo></mo><mi href="./28.1#p2.t1.r1">m</mi></mrow><mo>+</mo><mn>1</mn></mrow></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.1#p2.t1.r3">z</mi><mo>,</mo><mi href="./28.31#SS2.p1">ξ</mi><mo stretchy="false">)</mo></mrow></mrow></mtd><mtd columnalign="left"><mrow><mrow><mtext>with </mtext><mrow><mi>p</mi><mo>=</mo><mrow><mrow><mn>2</mn><mo></mo><mi href="./28.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mn>1</mn></mrow></mrow></mrow><mo>,</mo></mrow></mtd></mtr><mtr><mtd columnalign="center"><mrow><msubsup><mi href="./28.31#E11">S</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./28.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mn>2</mn></mrow><mrow><mrow><mn>2</mn><mo></mo><mi href="./28.1#p2.t1.r1">m</mi></mrow><mo>+</mo><mn>2</mn></mrow></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.1#p2.t1.r3">z</mi><mo>,</mo><mi href="./28.31#SS2.p1">ξ</mi><mo stretchy="false">)</mo></mrow></mrow></mtd><mtd columnalign="left"><mrow><mrow><mtext>with </mtext><mrow><mi>p</mi><mo>=</mo><mrow><mrow><mn>2</mn><mo></mo><mi href="./28.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mn>2</mn></mrow></mrow></mrow><mo>,</mo></mrow></mtd></mtr></mtable></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E11.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text"><math class="ltx_Math" altimg="m56.png" altimg-height="26px" altimg-valign="-10px" altimg-width="140px" alttext="S_{\NVar{p}}^{\NVar{m}}(\NVar{z},\\
NVar{xi})" display="inline"><mrow><msubsup><mi href="./28.31#E11">S</mi><mi class="ltx_nvar">p</mi><mi class="ltx_nvar" href="./28.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi class="ltx_nvar" href="./28.1#p2.t1.r3">z</mi><mo>,</mo><mrow><mi>N</mi><mo></mo><mi>V</mi><mo></mo><mi href="./28.1#p2.t1.r6">a</mi><mo></mo><mi>r</mi><mo></mo><mi href="./28.1#p2.t1.r2">x</mi><mo></mo><mi>i</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></math>: Ince polynomials (locally)</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./28.1#p2.t1.r1" title="§28.1 Special Notation ‣ Notation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m97.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./28.1#p2.t1.r1">m</mi></math>: integer</a>,
<a href="./28.1#p2.t1.r1" title="§28.1 Special Notation ‣ Notation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m100.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./28.1#p2.t1.r1">n</mi></math>: integer</a>,
<a href="./28.1#p2.t1.r2" title="§28.1 Special Notation ‣ Notation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m112.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./28.1#p2.t1.r2">x</mi></math>: real variable</a>,
<a href="./28.1#p2.t1.r3" title="§28.1 Special Notation ‣ Notation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m114.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./28.1#p2.t1.r3">z</mi></math>: complex variable</a>,
<a href="./28.1#p2.t1.r6" title="§28.1 Special Notation ‣ Notation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m89.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./28.1#p2.t1.r6">a</mi></math>: parameter</a> and
<a href="./28.31#SS2.p1" title="§28.31(ii) Equation of Ince; Ince Polynomials ‣ §28.31 Equations of Whittaker–Hill and Ince ‣ Hill’s Equation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="21px" altimg-valign="-6px" altimg-width="14px" alttext="\xi" display="inline"><mi href="./28.31#SS2.p1">ξ</mi></math>: variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and <math class="ltx_Math" altimg="m96.png" altimg-height="20px" altimg-valign="-6px" altimg-width="133px" alttext="m=0,1,\dots,n" display="inline"><mrow><mi href="./28.1#p2.t1.r1">m</mi><mo>=</mo><mrow><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><mi href="./28.1#p2.t1.r1">n</mi></mrow></mrow></math> in all cases.
</p>
</div>
<div id="SS2.p5" class="ltx_para">
<p class="ltx_p">The values of <math class="ltx_Math" altimg="m65.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="\eta" display="inline"><mi href="./28.31#SS2.p1">η</mi></math> corresponding to <math class="ltx_Math" altimg="m54.png" altimg-height="26px" altimg-valign="-10px" altimg-width="79px" alttext="C_{p}^{m}(z,\xi)" display="inline"><mrow><msubsup><mi href="./28.31#E10">C</mi><mi>p</mi><mi href="./28.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.1#p2.t1.r3">z</mi><mo>,</mo><mi href="./28.31#SS2.p1">ξ</mi><mo stretchy="false">)</mo></mrow></mrow></math>, <math class="ltx_Math" altimg="m57.png" altimg-height="26px" altimg-valign="-10px" altimg-width="77px" alttext="S_{p}^{m}(z,\xi)" display="inline"><mrow><msubsup><mi href="./28.31#E11">S</mi><mi>p</mi><mi href="./28.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.1#p2.t1.r3">z</mi><mo>,</mo><mi href="./28.31#SS2.p1">ξ</mi><mo stretchy="false">)</mo></mrow></mrow></math> are
denoted by <math class="ltx_Math" altimg="m90.png" altimg-height="26px" altimg-valign="-10px" altimg-width="55px" alttext="a_{p}^{m}(\xi)" display="inline"><mrow><msubsup><mi>a</mi><mi>p</mi><mi href="./28.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.31#SS2.p1">ξ</mi><mo stretchy="false">)</mo></mrow></mrow></math>, <math class="ltx_Math" altimg="m91.png" altimg-height="26px" altimg-valign="-10px" altimg-width="53px" alttext="b_{p}^{m}(\xi)" display="inline"><mrow><msubsup><mi>b</mi><mi>p</mi><mi href="./28.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.31#SS2.p1">ξ</mi><mo stretchy="false">)</mo></mrow></mrow></math>, respectively. They are real and
distinct, and can be ordered so that <math class="ltx_Math" altimg="m54.png" altimg-height="26px" altimg-valign="-10px" altimg-width="79px" alttext="C_{p}^{m}(z,\xi)" display="inline"><mrow><msubsup><mi href="./28.31#E10">C</mi><mi>p</mi><mi href="./28.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.1#p2.t1.r3">z</mi><mo>,</mo><mi href="./28.31#SS2.p1">ξ</mi><mo stretchy="false">)</mo></mrow></mrow></math> and <math class="ltx_Math" altimg="m57.png" altimg-height="26px" altimg-valign="-10px" altimg-width="77px" alttext="S_{p}^{m}(z,\xi)" display="inline"><mrow><msubsup><mi href="./28.31#E11">S</mi><mi>p</mi><mi href="./28.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.1#p2.t1.r3">z</mi><mo>,</mo><mi href="./28.31#SS2.p1">ξ</mi><mo stretchy="false">)</mo></mrow></mrow></math> have
precisely <math class="ltx_Math" altimg="m97.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./28.1#p2.t1.r1">m</mi></math> zeros, all simple, in <math class="ltx_Math" altimg="m47.png" altimg-height="19px" altimg-valign="-5px" altimg-width="90px" alttext="0\leq z<\pi" display="inline"><mrow><mn>0</mn><mo>≤</mo><mi href="./28.1#p2.t1.r3">z</mi><mo><</mo><mi href="./3.12#E1">π</mi></mrow></math>.
The normalization is given by
</p>
<table id="E12" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">28.31.12</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E12.png" altimg-height="56px" altimg-valign="-20px" altimg-width="450px" alttext="\dfrac{1}{\pi}\int_{0}^{2\pi}\left(C_{p}^{m}(x,\xi)\right)^{2}\mathrm{d}x=%
\dfrac{1}{\pi}\int_{0}^{2\pi}\left(S_{p}^{m}(x,\xi)\right)^{2}\mathrm{d}x=1," display="block"><mrow><mrow><mrow><mfrac><mn>1</mn><mi href="./3.12#E1">π</mi></mfrac><mo></mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi></mrow></msubsup><mrow><msup><mrow><mo>(</mo><mrow><msubsup><mi href="./28.31#E10">C</mi><mi>p</mi><mi href="./28.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.1#p2.t1.r2">x</mi><mo>,</mo><mi href="./28.31#SS2.p1">ξ</mi><mo stretchy="false">)</mo></mrow></mrow><mo>)</mo></mrow><mn>2</mn></msup><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./28.1#p2.t1.r2">x</mi></mrow></mrow></mrow></mrow><mo>=</mo><mrow><mfrac><mn>1</mn><mi href="./3.12#E1">π</mi></mfrac><mo></mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi></mrow></msubsup><mrow><msup><mrow><mo>(</mo><mrow><msubsup><mi href="./28.31#E11">S</mi><mi>p</mi><mi href="./28.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.1#p2.t1.r2">x</mi><mo>,</mo><mi href="./28.31#SS2.p1">ξ</mi><mo stretchy="false">)</mo></mrow></mrow><mo>)</mo></mrow><mn>2</mn></msup><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./28.1#p2.t1.r2">x</mi></mrow></mrow></mrow></mrow><mo>=</mo><mn>1</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E12.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m83.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m112.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./28.1#p2.t1.r1" title="§28.1 Special Notation ‣ Notation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m97.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./28.1#p2.t1.r1">m</mi></math>: integer</a>,
<a href="./28.1#p2.t1.r2" title="§28.1 Special Notation ‣ Notation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m112.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./28.1#p2.t1.r2">x</mi></math>: real variable</a>,
<a href="./28.31#SS2.p1" title="§28.31(ii) Equation of Ince; Ince Polynomials ‣ §28.31 Equations of Whittaker–Hill and Ince ‣ Hill’s Equation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="21px" altimg-valign="-6px" altimg-width="14px" alttext="\xi" display="inline"><mi href="./28.31#SS2.p1">ξ</mi></math>: variable</a>,
<a href="./28.31#E10" title="(28.31.10) ‣ §28.31(ii) Equation of Ince; Ince Polynomials ‣ §28.31 Equations of Whittaker–Hill and Ince ‣ Hill’s Equation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="26px" altimg-valign="-10px" altimg-width="79px" alttext="C_{p}^{m}(z,\xi)" display="inline"><mrow><msubsup><mi href="./28.31#E10">C</mi><mi>p</mi><mi href="./28.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.1#p2.t1.r3">z</mi><mo>,</mo><mi href="./28.31#SS2.p1">ξ</mi><mo stretchy="false">)</mo></mrow></mrow></math>: Ince polynomials</a> and
<a href="./28.31#E11" title="(28.31.11) ‣ §28.31(ii) Equation of Ince; Ince Polynomials ‣ §28.31 Equations of Whittaker–Hill and Ince ‣ Hill’s Equation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="26px" altimg-valign="-10px" altimg-width="140px" alttext="S_{\NVar{p}}^{\NVar{m}}(\NVar{z},\\
NVar{xi})" display="inline"><mrow><msubsup><mi href="./28.31#E11">S</mi><mi class="ltx_nvar">p</mi><mi class="ltx_nvar" href="./28.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi class="ltx_nvar" href="./28.1#p2.t1.r3">z</mi><mo>,</mo><mrow><mi>N</mi><mo></mo><mi>V</mi><mo></mo><mi href="./28.1#p2.t1.r6">a</mi><mo></mo><mi>r</mi><mo></mo><mi href="./28.1#p2.t1.r2">x</mi><mo></mo><mi>i</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></math>: Ince polynomials</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">ambiguities in sign being resolved by requiring <math class="ltx_Math" altimg="m53.png" altimg-height="26px" altimg-valign="-10px" altimg-width="80px" alttext="C_{p}^{m}(x,\xi)" display="inline"><mrow><msubsup><mi href="./28.31#E10">C</mi><mi>p</mi><mi href="./28.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.1#p2.t1.r2">x</mi><mo>,</mo><mi href="./28.31#SS2.p1">ξ</mi><mo stretchy="false">)</mo></mrow></mrow></math> and
<math class="ltx_Math" altimg="m117.png" altimg-height="28px" altimg-valign="-10px" altimg-width="84px" alttext="{S_{p}^{m}}^{\prime}(x,\xi)" display="inline"><mrow><mmultiscripts><mi href="./28.31#E11">S</mi><mi>p</mi><mi href="./28.1#p2.t1.r1">m</mi><none></none><mo>′</mo></mmultiscripts><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.1#p2.t1.r2">x</mi><mo>,</mo><mi href="./28.31#SS2.p1">ξ</mi><mo stretchy="false">)</mo></mrow></mrow></math> to be continuous functions of <math class="ltx_Math" altimg="m112.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./28.1#p2.t1.r2">x</mi></math> and positive when <math class="ltx_Math" altimg="m111.png" altimg-height="17px" altimg-valign="-2px" altimg-width="52px" alttext="x=0" display="inline"><mrow><mi href="./28.1#p2.t1.r2">x</mi><mo>=</mo><mn>0</mn></mrow></math>.</p>
</div>
<div id="SS2.p6" class="ltx_para">
<p class="ltx_p">For <math class="ltx_Math" altimg="m88.png" altimg-height="21px" altimg-valign="-6px" altimg-width="55px" alttext="\xi\to 0" display="inline"><mrow><mi href="./28.31#SS2.p1">ξ</mi><mo>→</mo><mn>0</mn></mrow></math>, with <math class="ltx_Math" altimg="m112.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./28.1#p2.t1.r2">x</mi></math> fixed,
</p>
<table id="E13" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex14" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="5" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">28.31.13</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m20.png" altimg-height="31px" altimg-valign="-10px" altimg-width="76px" alttext="\displaystyle C_{p}^{0}(x,\xi)" display="inline"><mrow><msubsup><mi href="./28.31#E10">C</mi><mi>p</mi><mn>0</mn></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.1#p2.t1.r2">x</mi><mo>,</mo><mi href="./28.31#SS2.p1">ξ</mi><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m30.png" altimg-height="29px" altimg-valign="-7px" altimg-width="84px" alttext="\displaystyle\to 1/{\sqrt{2}}," display="inline"><mrow><mrow><mi></mi><mo>→</mo><mrow><mn>1</mn><mo>/</mo><msqrt><mn>2</mn></msqrt></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex15" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m21.png" altimg-height="28px" altimg-valign="-10px" altimg-width="82px" alttext="\displaystyle C_{p}^{m}(x,\xi)" display="inline"><mrow><msubsup><mi href="./28.31#E10">C</mi><mi>p</mi><mi href="./28.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.1#p2.t1.r2">x</mi><mo>,</mo><mi href="./28.31#SS2.p1">ξ</mi><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m34.png" altimg-height="25px" altimg-valign="-7px" altimg-width="115px" alttext="\displaystyle\to\cos\left(mx\right)," display="inline"><mrow><mrow><mi></mi><mo>→</mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./28.1#p2.t1.r1">m</mi><mo></mo><mi href="./28.1#p2.t1.r2">x</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex16" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m22.png" altimg-height="28px" altimg-valign="-10px" altimg-width="80px" alttext="\displaystyle S_{p}^{m}(x,\xi)" display="inline"><mrow><msubsup><mi href="./28.31#E11">S</mi><mi>p</mi><mi href="./28.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.1#p2.t1.r2">x</mi><mo>,</mo><mi href="./28.31#SS2.p1">ξ</mi><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m37.png" altimg-height="25px" altimg-valign="-7px" altimg-width="113px" alttext="\displaystyle\to\sin\left(mx\right)," display="inline"><mrow><mrow><mi></mi><mo>→</mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./28.1#p2.t1.r1">m</mi><mo></mo><mi href="./28.1#p2.t1.r2">x</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m98.png" altimg-height="21px" altimg-valign="-6px" altimg-width="58px" alttext="m\neq 0" display="inline"><mrow><mi href="./28.1#p2.t1.r1">m</mi><mo>≠</mo><mn>0</mn></mrow></math>;</span></td></tr>
<tr id="Ex17" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m39.png" altimg-height="28px" altimg-valign="-10px" altimg-width="118px" alttext="\displaystyle a_{p}^{m}(\xi),\,b_{p}^{m}(\xi)" display="inline"><mrow><mrow><msubsup><mi>a</mi><mi>p</mi><mi href="./28.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.31#SS2.p1">ξ</mi><mo stretchy="false">)</mo></mrow></mrow><mo rspace="4.2pt">,</mo><mrow><msubsup><mi>b</mi><mi>p</mi><mi href="./28.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.31#SS2.p1">ξ</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m33.png" altimg-height="23px" altimg-valign="-2px" altimg-width="63px" alttext="\displaystyle\to m^{2}." display="inline"><mrow><mrow><mi></mi><mo>→</mo><msup><mi href="./28.1#p2.t1.r1">m</mi><mn>2</mn></msup></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E13.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m84.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./28.1#p2.t1.r1" title="§28.1 Special Notation ‣ Notation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m97.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./28.1#p2.t1.r1">m</mi></math>: integer</a>,
<a href="./28.1#p2.t1.r2" title="§28.1 Special Notation ‣ Notation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m112.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./28.1#p2.t1.r2">x</mi></math>: real variable</a>,
<a href="./28.31#SS2.p1" title="§28.31(ii) Equation of Ince; Ince Polynomials ‣ §28.31 Equations of Whittaker–Hill and Ince ‣ Hill’s Equation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="21px" altimg-valign="-6px" altimg-width="14px" alttext="\xi" display="inline"><mi href="./28.31#SS2.p1">ξ</mi></math>: variable</a>,
<a href="./28.31#E10" title="(28.31.10) ‣ §28.31(ii) Equation of Ince; Ince Polynomials ‣ §28.31 Equations of Whittaker–Hill and Ince ‣ Hill’s Equation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="26px" altimg-valign="-10px" altimg-width="79px" alttext="C_{p}^{m}(z,\xi)" display="inline"><mrow><msubsup><mi href="./28.31#E10">C</mi><mi>p</mi><mi href="./28.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.1#p2.t1.r3">z</mi><mo>,</mo><mi href="./28.31#SS2.p1">ξ</mi><mo stretchy="false">)</mo></mrow></mrow></math>: Ince polynomials</a> and
<a href="./28.31#E11" title="(28.31.11) ‣ §28.31(ii) Equation of Ince; Ince Polynomials ‣ §28.31 Equations of Whittaker–Hill and Ince ‣ Hill’s Equation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="26px" altimg-valign="-10px" altimg-width="140px" alttext="S_{\NVar{p}}^{\NVar{m}}(\NVar{z},\\
NVar{xi})" display="inline"><mrow><msubsup><mi href="./28.31#E11">S</mi><mi class="ltx_nvar">p</mi><mi class="ltx_nvar" href="./28.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi class="ltx_nvar" href="./28.1#p2.t1.r3">z</mi><mo>,</mo><mrow><mi>N</mi><mo></mo><mi>V</mi><mo></mo><mi href="./28.1#p2.t1.r6">a</mi><mo></mo><mi>r</mi><mo></mo><mi href="./28.1#p2.t1.r2">x</mi><mo></mo><mi>i</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></math>: Ince polynomials</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS2.p7" class="ltx_para">
<p class="ltx_p">If <math class="ltx_Math" altimg="m102.png" altimg-height="16px" altimg-valign="-6px" altimg-width="65px" alttext="p\to\infty" display="inline"><mrow><mi>p</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></math> and <math class="ltx_Math" altimg="m88.png" altimg-height="21px" altimg-valign="-6px" altimg-width="55px" alttext="\xi\to 0" display="inline"><mrow><mi href="./28.31#SS2.p1">ξ</mi><mo>→</mo><mn>0</mn></mrow></math> in such a way that <math class="ltx_Math" altimg="m103.png" altimg-height="21px" altimg-valign="-6px" altimg-width="75px" alttext="p\xi\to 2q" display="inline"><mrow><mrow><mi>p</mi><mo></mo><mi href="./28.31#SS2.p1">ξ</mi></mrow><mo>→</mo><mrow><mn>2</mn><mo></mo><mi href="./28.1#p2.t1.r6">q</mi></mrow></mrow></math>, then in the
notation of §§</p>
<table id="E14" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex18" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">28.31.14</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m21.png" altimg-height="28px" altimg-valign="-10px" altimg-width="82px" alttext="\displaystyle C_{p}^{m}(x,\xi)" display="inline"><mrow><msubsup><mi href="./28.31#E10">C</mi><mi>p</mi><mi href="./28.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.1#p2.t1.r2">x</mi><mo>,</mo><mi href="./28.31#SS2.p1">ξ</mi><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m35.png" altimg-height="25px" altimg-valign="-7px" altimg-width="122px" alttext="\displaystyle\to\mathrm{ce}_{m}\left(x,q\right)," display="inline"><mrow><mrow><mi></mi><mo>→</mo><mrow><msub><mi href="./28.2#SS6.p1">ce</mi><mi href="./28.1#p2.t1.r1">m</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./28.1#p2.t1.r2">x</mi><mo>,</mo><mi href="./28.1#p2.t1.r6">q</mi><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex19" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m22.png" altimg-height="28px" altimg-valign="-10px" altimg-width="80px" alttext="\displaystyle S_{p}^{m}(x,\xi)" display="inline"><mrow><msubsup><mi href="./28.31#E11">S</mi><mi>p</mi><mi href="./28.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.1#p2.t1.r2">x</mi><mo>,</mo><mi href="./28.31#SS2.p1">ξ</mi><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m36.png" altimg-height="25px" altimg-valign="-7px" altimg-width="121px" alttext="\displaystyle\to\mathrm{se}_{m}\left(x,q\right)," display="inline"><mrow><mrow><mi></mi><mo>→</mo><mrow><msub><mi href="./28.2#SS6.p1">se</mi><mi href="./28.1#p2.t1.r1">m</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./28.1#p2.t1.r2">x</mi><mo>,</mo><mi href="./28.1#p2.t1.r6">q</mi><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E14.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./28.2#SS6.p1" title="§28.2(vi) Eigenfunctions ‣ §28.2 Definitions and Basic Properties ‣ Mathieu Functions of Integer Order ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m76.png" altimg-height="23px" altimg-valign="-7px" altimg-width="80px" alttext="\mathrm{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)" display="inline"><mrow><msub><mi href="./28.2#SS6.p1">ce</mi><mi class="ltx_nvar" href="./28.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./28.1#p2.t1.r3">z</mi><mo>,</mo><mi class="ltx_nvar" href="./28.1#p2.t1.r6">q</mi><mo>)</mo></mrow></mrow></math>: Mathieu function</a>,
<a href="./28.2#SS6.p1" title="§28.2(vi) Eigenfunctions ‣ §28.2 Definitions and Basic Properties ‣ Mathieu Functions of Integer Order ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="23px" altimg-valign="-7px" altimg-width="79px" alttext="\mathrm{se}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)" display="inline"><mrow><msub><mi href="./28.2#SS6.p1">se</mi><mi class="ltx_nvar" href="./28.1#p2.t1.r1">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./28.1#p2.t1.r3">z</mi><mo>,</mo><mi class="ltx_nvar" href="./28.1#p2.t1.r6">q</mi><mo>)</mo></mrow></mrow></math>: Mathieu function</a>,
<a href="./28.1#p2.t1.r1" title="§28.1 Special Notation ‣ Notation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m97.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./28.1#p2.t1.r1">m</mi></math>: integer</a>,
<a href="./28.1#p2.t1.r6" title="§28.1 Special Notation ‣ Notation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m106.png" altimg-height="24px" altimg-valign="-6px" altimg-width="61px" alttext="q=h^{2}" display="inline"><mrow><mi href="./28.1#p2.t1.r6">q</mi><mo>=</mo><msup><mi href="./28.1#p2.t1.r6">h</mi><mn>2</mn></msup></mrow></math>: parameter</a>,
<a href="./28.1#p2.t1.r2" title="§28.1 Special Notation ‣ Notation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m112.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./28.1#p2.t1.r2">x</mi></math>: real variable</a>,
<a href="./28.31#SS2.p1" title="§28.31(ii) Equation of Ince; Ince Polynomials ‣ §28.31 Equations of Whittaker–Hill and Ince ‣ Hill’s Equation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="21px" altimg-valign="-6px" altimg-width="14px" alttext="\xi" display="inline"><mi href="./28.31#SS2.p1">ξ</mi></math>: variable</a>,
<a href="./28.31#E10" title="(28.31.10) ‣ §28.31(ii) Equation of Ince; Ince Polynomials ‣ §28.31 Equations of Whittaker–Hill and Ince ‣ Hill’s Equation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="26px" altimg-valign="-10px" altimg-width="79px" alttext="C_{p}^{m}(z,\xi)" display="inline"><mrow><msubsup><mi href="./28.31#E10">C</mi><mi>p</mi><mi href="./28.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.1#p2.t1.r3">z</mi><mo>,</mo><mi href="./28.31#SS2.p1">ξ</mi><mo stretchy="false">)</mo></mrow></mrow></math>: Ince polynomials</a> and
<a href="./28.31#E11" title="(28.31.11) ‣ §28.31(ii) Equation of Ince; Ince Polynomials ‣ §28.31 Equations of Whittaker–Hill and Ince ‣ Hill’s Equation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="26px" altimg-valign="-10px" altimg-width="140px" alttext="S_{\NVar{p}}^{\NVar{m}}(\NVar{z},\\
NVar{xi})" display="inline"><mrow><msubsup><mi href="./28.31#E11">S</mi><mi class="ltx_nvar">p</mi><mi class="ltx_nvar" href="./28.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi class="ltx_nvar" href="./28.1#p2.t1.r3">z</mi><mo>,</mo><mrow><mi>N</mi><mo></mo><mi>V</mi><mo></mo><mi href="./28.1#p2.t1.r6">a</mi><mo></mo><mi>r</mi><mo></mo><mi href="./28.1#p2.t1.r2">x</mi><mo></mo><mi>i</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></math>: Ince polynomials</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E15" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex20" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">28.31.15</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m40.png" altimg-height="28px" altimg-valign="-10px" altimg-width="57px" alttext="\displaystyle a_{p}^{m}(\xi)" display="inline"><mrow><msubsup><mi>a</mi><mi>p</mi><mi href="./28.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.31#SS2.p1">ξ</mi><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m31.png" altimg-height="25px" altimg-valign="-7px" altimg-width="88px" alttext="\displaystyle\to a_{m}(q)," display="inline"><mrow><mrow><mi></mi><mo>→</mo><mrow><msub><mi>a</mi><mi href="./28.1#p2.t1.r1">m</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.1#p2.t1.r6">q</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex21" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m41.png" altimg-height="28px" altimg-valign="-10px" altimg-width="55px" alttext="\displaystyle b_{p}^{m}(\xi)" display="inline"><mrow><msubsup><mi>b</mi><mi>p</mi><mi href="./28.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.31#SS2.p1">ξ</mi><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m32.png" altimg-height="25px" altimg-valign="-7px" altimg-width="86px" alttext="\displaystyle\to b_{m}(q)." display="inline"><mrow><mrow><mi></mi><mo>→</mo><mrow><msub><mi>b</mi><mi href="./28.1#p2.t1.r1">m</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.1#p2.t1.r6">q</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E15.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./28.1#p2.t1.r1" title="§28.1 Special Notation ‣ Notation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m97.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./28.1#p2.t1.r1">m</mi></math>: integer</a>,
<a href="./28.1#p2.t1.r6" title="§28.1 Special Notation ‣ Notation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m106.png" altimg-height="24px" altimg-valign="-6px" altimg-width="61px" alttext="q=h^{2}" display="inline"><mrow><mi href="./28.1#p2.t1.r6">q</mi><mo>=</mo><msup><mi href="./28.1#p2.t1.r6">h</mi><mn>2</mn></msup></mrow></math>: parameter</a> and
<a href="./28.31#SS2.p1" title="§28.31(ii) Equation of Ince; Ince Polynomials ‣ §28.31 Equations of Whittaker–Hill and Ince ‣ Hill’s Equation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="21px" altimg-valign="-6px" altimg-width="14px" alttext="\xi" display="inline"><mi href="./28.31#SS2.p1">ξ</mi></math>: variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS2.p8" class="ltx_para">
<p class="ltx_p">For proofs and further information, including convergence of the series
(),
</p>
<table id="E16" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">28.31.16</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E16.png" altimg-height="33px" altimg-valign="-10px" altimg-width="287px" alttext="\mathit{hc}_{p}^{m}(z,\xi)=e^{-\frac{1}{4}\xi\cos\left(2z\right)}C_{p}^{m}(z,%
\xi)," display="block"><mrow><mrow><mrow><msubsup><mi href="./28.31#E16" mathvariant="italic">hc</mi><mi>p</mi><mi href="./28.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.1#p2.t1.r3">z</mi><mo>,</mo><mi>ξ</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac><mo></mo><mi>ξ</mi><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><mi href="./28.1#p2.t1.r3">z</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow></msup><mo></mo><mrow><msubsup><mi href="./28.31#E10">C</mi><mi>p</mi><mi href="./28.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.1#p2.t1.r3">z</mi><mo>,</mo><mi>ξ</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E16.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text"><math class="ltx_Math" altimg="m72.png" altimg-height="26px" altimg-valign="-10px" altimg-width="84px" alttext="\mathit{hc}_{p}^{m}(z,\xi)" display="inline"><mrow><msubsup><mi href="./28.31#E16" mathvariant="italic">hc</mi><mi>p</mi><mi href="./28.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.1#p2.t1.r3">z</mi><mo>,</mo><mi>ξ</mi><mo stretchy="false">)</mo></mrow></mrow></math>: paraboloidal wave function (locally)</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./28.1#p2.t1.r1" title="§28.1 Special Notation ‣ Notation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m97.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./28.1#p2.t1.r1">m</mi></math>: integer</a>,
<a href="./28.1#p2.t1.r3" title="§28.1 Special Notation ‣ Notation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m114.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./28.1#p2.t1.r3">z</mi></math>: complex variable</a> and
<a href="./28.31#E10" title="(28.31.10) ‣ §28.31(ii) Equation of Ince; Ince Polynomials ‣ §28.31 Equations of Whittaker–Hill and Ince ‣ Hill’s Equation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m54.png" altimg-height="26px" altimg-valign="-10px" altimg-width="79px" alttext="C_{p}^{m}(z,\xi)" display="inline"><mrow><msubsup><mi href="./28.31#E10">C</mi><mi>p</mi><mi href="./28.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.1#p2.t1.r3">z</mi><mo>,</mo><mi href="./28.31#SS2.p1">ξ</mi><mo stretchy="false">)</mo></mrow></mrow></math>: Ince polynomials</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E17" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">28.31.17</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E17.png" altimg-height="33px" altimg-valign="-10px" altimg-width="285px" alttext="\mathit{hs}_{p}^{m}(z,\xi)=e^{-\frac{1}{4}\xi\cos\left(2z\right)}S_{p}^{m}(z,%
\xi)," display="block"><mrow><mrow><mrow><msubsup><mi href="./28.31#E17" mathvariant="italic">hs</mi><mi>p</mi><mi href="./28.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.1#p2.t1.r3">z</mi><mo>,</mo><mi>ξ</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac><mo></mo><mi>ξ</mi><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><mi href="./28.1#p2.t1.r3">z</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow></msup><mo></mo><mrow><msubsup><mi href="./28.31#E11">S</mi><mi>p</mi><mi href="./28.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.1#p2.t1.r3">z</mi><mo>,</mo><mi>ξ</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E17.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text"><math class="ltx_Math" altimg="m75.png" altimg-height="26px" altimg-valign="-10px" altimg-width="84px" alttext="\mathit{hs}_{p}^{m}(z,\xi)" display="inline"><mrow><msubsup><mi href="./28.31#E17" mathvariant="italic">hs</mi><mi>p</mi><mi href="./28.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.1#p2.t1.r3">z</mi><mo>,</mo><mi>ξ</mi><mo stretchy="false">)</mo></mrow></mrow></math>: paraboloidal wave function (locally)</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./28.1#p2.t1.r1" title="§28.1 Special Notation ‣ Notation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m97.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./28.1#p2.t1.r1">m</mi></math>: integer</a>,
<a href="./28.1#p2.t1.r3" title="§28.1 Special Notation ‣ Notation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m114.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./28.1#p2.t1.r3">z</mi></math>: complex variable</a> and
<a href="./28.31#E11" title="(28.31.11) ‣ §28.31(ii) Equation of Ince; Ince Polynomials ‣ §28.31 Equations of Whittaker–Hill and Ince ‣ Hill’s Equation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="26px" altimg-valign="-10px" altimg-width="140px" alttext="S_{\NVar{p}}^{\NVar{m}}(\NVar{z},\\
NVar{xi})" display="inline"><mrow><msubsup><mi href="./28.31#E11">S</mi><mi class="ltx_nvar">p</mi><mi class="ltx_nvar" href="./28.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi class="ltx_nvar" href="./28.1#p2.t1.r3">z</mi><mo>,</mo><mrow><mi>N</mi><mo></mo><mi>V</mi><mo></mo><mi href="./28.1#p2.t1.r6">a</mi><mo></mo><mi>r</mi><mo></mo><mi href="./28.1#p2.t1.r2">x</mi><mo></mo><mi>i</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></math>: Ince polynomials</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">are called <em class="ltx_emph ltx_font_italic">paraboloidal wave functions</em>. They satisfy the differential
equation</p>
<table id="E18" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">28.31.18</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E18.png" altimg-height="30px" altimg-valign="-9px" altimg-width="487px" alttext="w^{\prime\prime}+\left(\eta-\tfrac{1}{8}\xi^{2}-(p+1)\xi\cos\left(2z\right)+%
\tfrac{1}{8}\xi^{2}\cos\left(4z\right)\right)w=0," display="block"><mrow><mrow><mrow><msup><mi href="./28.2#SS1.p1">w</mi><mo>′′</mo></msup><mo>+</mo><mrow><mrow><mo>(</mo><mrow><mrow><mi href="./28.31#SS3.p1">η</mi><mo>-</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>8</mn></mfrac></mstyle><mo></mo><msup><mi>ξ</mi><mn>2</mn></msup></mrow><mo>-</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi>ξ</mi><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><mi href="./28.1#p2.t1.r3">z</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>+</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>8</mn></mfrac></mstyle><mo></mo><msup><mi>ξ</mi><mn>2</mn></msup><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mn>4</mn><mo></mo><mi href="./28.1#p2.t1.r3">z</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>)</mo></mrow><mo></mo><mi href="./28.2#SS1.p1">w</mi></mrow></mrow><mo>=</mo><mn>0</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E18.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./28.1#p2.t1.r3" title="§28.1 Special Notation ‣ Notation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m114.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./28.1#p2.t1.r3">z</mi></math>: complex variable</a>,
<a href="./28.2#SS1.p1" title="§28.2(i) Mathieu’s Equation ‣ §28.2 Definitions and Basic Properties ‣ Mathieu Functions of Integer Order ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m110.png" altimg-height="23px" altimg-valign="-7px" altimg-width="45px" alttext="w(z)" display="inline"><mrow><mi href="./28.2#SS1.p1">w</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.1#p2.t1.r3">z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: Mathieu’s equation solution</a> and
<a href="./28.31#SS3.p1" title="§28.31(iii) Paraboloidal Wave Functions ‣ §28.31 Equations of Whittaker–Hill and Ince ‣ Hill’s Equation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m65.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="\eta" display="inline"><mi href="./28.31#SS3.p1">η</mi></math>: change of variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">with <math class="ltx_Math" altimg="m63.png" altimg-height="26px" altimg-valign="-10px" altimg-width="92px" alttext="\eta=a_{p}^{m}(\xi)" display="inline"><mrow><mi href="./28.31#SS3.p1">η</mi><mo>=</mo><mrow><msubsup><mi href="./28.31#SS2.p5">a</mi><mi>p</mi><mi href="./28.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi>ξ</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></math>, <math class="ltx_Math" altimg="m64.png" altimg-height="26px" altimg-valign="-10px" altimg-width="90px" alttext="\eta=b_{p}^{m}(\xi)" display="inline"><mrow><mi href="./28.31#SS3.p1">η</mi><mo>=</mo><mrow><msubsup><mi href="./28.31#SS2.p5">b</mi><mi>p</mi><mi href="./28.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi>ξ</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></math>, respectively.</p>
</div>
<div id="SS3.p2" class="ltx_para">
<p class="ltx_p">For change of sign of <math class="ltx_Math" altimg="m87.png" altimg-height="21px" altimg-valign="-6px" altimg-width="14px" alttext="\xi" display="inline"><mi>ξ</mi></math>,
</p>
<table id="E19" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex22" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">28.31.19</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m27.png" altimg-height="28px" altimg-valign="-7px" altimg-width="109px" alttext="\displaystyle\mathit{hc}_{2n}^{2m}(z,-\xi)" display="inline"><mrow><msubsup><mi href="./28.31#E16" mathvariant="italic">hc</mi><mrow><mn>2</mn><mo></mo><mi href="./28.1#p2.t1.r1">n</mi></mrow><mrow><mn>2</mn><mo></mo><mi href="./28.1#p2.t1.r1">m</mi></mrow></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.1#p2.t1.r3">z</mi><mo>,</mo><mrow><mo>-</mo><mi>ξ</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m8.png" altimg-height="30px" altimg-valign="-9px" altimg-width="226px" alttext="\displaystyle=(-1)^{m}\mathit{hc}_{2n}^{2m}(\tfrac{1}{2}\pi-z,\xi)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./28.1#p2.t1.r1">m</mi></msup><mo></mo><mrow><msubsup><mi href="./28.31#E16" mathvariant="italic">hc</mi><mrow><mn>2</mn><mo></mo><mi href="./28.1#p2.t1.r1">n</mi></mrow><mrow><mn>2</mn><mo></mo><mi href="./28.1#p2.t1.r1">m</mi></mrow></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo>-</mo><mi href="./28.1#p2.t1.r3">z</mi></mrow><mo>,</mo><mi>ξ</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex23" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m26.png" altimg-height="30px" altimg-valign="-9px" altimg-width="129px" alttext="\displaystyle\mathit{hc}_{2n+1}^{2m+1}(z,-\xi)" display="inline"><mrow><msubsup><mi href="./28.31#E16" mathvariant="italic">hc</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./28.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mn>1</mn></mrow><mrow><mrow><mn>2</mn><mo></mo><mi href="./28.1#p2.t1.r1">m</mi></mrow><mo>+</mo><mn>1</mn></mrow></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.1#p2.t1.r3">z</mi><mo>,</mo><mrow><mo>-</mo><mi>ξ</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m9.png" altimg-height="30px" altimg-valign="-9px" altimg-width="245px" alttext="\displaystyle=(-1)^{m}\mathit{hs}_{2n+1}^{2m+1}(\tfrac{1}{2}\pi-z,\xi)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./28.1#p2.t1.r1">m</mi></msup><mo></mo><mrow><msubsup><mi href="./28.31#E17" mathvariant="italic">hs</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./28.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mn>1</mn></mrow><mrow><mrow><mn>2</mn><mo></mo><mi href="./28.1#p2.t1.r1">m</mi></mrow><mo>+</mo><mn>1</mn></mrow></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo>-</mo><mi href="./28.1#p2.t1.r3">z</mi></mrow><mo>,</mo><mi>ξ</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E19.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m83.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./28.1#p2.t1.r1" title="§28.1 Special Notation ‣ Notation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m97.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./28.1#p2.t1.r1">m</mi></math>: integer</a>,
<a href="./28.1#p2.t1.r1" title="§28.1 Special Notation ‣ Notation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m100.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./28.1#p2.t1.r1">n</mi></math>: integer</a>,
<a href="./28.1#p2.t1.r3" title="§28.1 Special Notation ‣ Notation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m114.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./28.1#p2.t1.r3">z</mi></math>: complex variable</a>,
<a href="./28.31#E16" title="(28.31.16) ‣ §28.31(iii) Paraboloidal Wave Functions ‣ §28.31 Equations of Whittaker–Hill and Ince ‣ Hill’s Equation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="26px" altimg-valign="-10px" altimg-width="84px" alttext="\mathit{hc}_{p}^{m}(z,\xi)" display="inline"><mrow><msubsup><mi href="./28.31#E16" mathvariant="italic">hc</mi><mi>p</mi><mi href="./28.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.1#p2.t1.r3">z</mi><mo>,</mo><mi>ξ</mi><mo stretchy="false">)</mo></mrow></mrow></math>: paraboloidal wave function</a> and
<a href="./28.31#E17" title="(28.31.17) ‣ §28.31(iii) Paraboloidal Wave Functions ‣ §28.31 Equations of Whittaker–Hill and Ince ‣ Hill’s Equation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m75.png" altimg-height="26px" altimg-valign="-10px" altimg-width="84px" alttext="\mathit{hs}_{p}^{m}(z,\xi)" display="inline"><mrow><msubsup><mi href="./28.31#E17" mathvariant="italic">hs</mi><mi>p</mi><mi href="./28.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.1#p2.t1.r3">z</mi><mo>,</mo><mi>ξ</mi><mo stretchy="false">)</mo></mrow></mrow></math>: paraboloidal wave function</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and</p>
<table id="E20" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex24" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">28.31.20</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m28.png" altimg-height="30px" altimg-valign="-9px" altimg-width="129px" alttext="\displaystyle\mathit{hs}_{2n+1}^{2m+1}(z,-\xi)" display="inline"><mrow><msubsup><mi href="./28.31#E17" mathvariant="italic">hs</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./28.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mn>1</mn></mrow><mrow><mrow><mn>2</mn><mo></mo><mi href="./28.1#p2.t1.r1">m</mi></mrow><mo>+</mo><mn>1</mn></mrow></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.1#p2.t1.r3">z</mi><mo>,</mo><mrow><mo>-</mo><mi>ξ</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m7.png" altimg-height="30px" altimg-valign="-9px" altimg-width="246px" alttext="\displaystyle=(-1)^{m}\mathit{hc}_{2n+1}^{2m+1}(\tfrac{1}{2}\pi-z,\xi)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./28.1#p2.t1.r1">m</mi></msup><mo></mo><mrow><msubsup><mi href="./28.31#E16" mathvariant="italic">hc</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./28.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mn>1</mn></mrow><mrow><mrow><mn>2</mn><mo></mo><mi href="./28.1#p2.t1.r1">m</mi></mrow><mo>+</mo><mn>1</mn></mrow></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo>-</mo><mi href="./28.1#p2.t1.r3">z</mi></mrow><mo>,</mo><mi>ξ</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex25" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m29.png" altimg-height="30px" altimg-valign="-9px" altimg-width="129px" alttext="\displaystyle\mathit{hs}_{2n+2}^{2m+2}(z,-\xi)" display="inline"><mrow><msubsup><mi href="./28.31#E17" mathvariant="italic">hs</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./28.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mn>2</mn></mrow><mrow><mrow><mn>2</mn><mo></mo><mi href="./28.1#p2.t1.r1">m</mi></mrow><mo>+</mo><mn>2</mn></mrow></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.1#p2.t1.r3">z</mi><mo>,</mo><mrow><mo>-</mo><mi>ξ</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m10.png" altimg-height="30px" altimg-valign="-9px" altimg-width="245px" alttext="\displaystyle=(-1)^{m}\mathit{hs}_{2n+2}^{2m+2}(\tfrac{1}{2}\pi-z,\xi)." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./28.1#p2.t1.r1">m</mi></msup><mo></mo><mrow><msubsup><mi href="./28.31#E17" mathvariant="italic">hs</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./28.1#p2.t1.r1">n</mi></mrow><mo>+</mo><mn>2</mn></mrow><mrow><mrow><mn>2</mn><mo></mo><mi href="./28.1#p2.t1.r1">m</mi></mrow><mo>+</mo><mn>2</mn></mrow></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo>-</mo><mi href="./28.1#p2.t1.r3">z</mi></mrow><mo>,</mo><mi>ξ</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E20.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m83.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./28.1#p2.t1.r1" title="§28.1 Special Notation ‣ Notation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m97.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./28.1#p2.t1.r1">m</mi></math>: integer</a>,
<a href="./28.1#p2.t1.r1" title="§28.1 Special Notation ‣ Notation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m100.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./28.1#p2.t1.r1">n</mi></math>: integer</a>,
<a href="./28.1#p2.t1.r3" title="§28.1 Special Notation ‣ Notation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m114.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./28.1#p2.t1.r3">z</mi></math>: complex variable</a>,
<a href="./28.31#E16" title="(28.31.16) ‣ §28.31(iii) Paraboloidal Wave Functions ‣ §28.31 Equations of Whittaker–Hill and Ince ‣ Hill’s Equation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="26px" altimg-valign="-10px" altimg-width="84px" alttext="\mathit{hc}_{p}^{m}(z,\xi)" display="inline"><mrow><msubsup><mi href="./28.31#E16" mathvariant="italic">hc</mi><mi>p</mi><mi href="./28.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.1#p2.t1.r3">z</mi><mo>,</mo><mi>ξ</mi><mo stretchy="false">)</mo></mrow></mrow></math>: paraboloidal wave function</a> and
<a href="./28.31#E17" title="(28.31.17) ‣ §28.31(iii) Paraboloidal Wave Functions ‣ §28.31 Equations of Whittaker–Hill and Ince ‣ Hill’s Equation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m75.png" altimg-height="26px" altimg-valign="-10px" altimg-width="84px" alttext="\mathit{hs}_{p}^{m}(z,\xi)" display="inline"><mrow><msubsup><mi href="./28.31#E17" mathvariant="italic">hs</mi><mi>p</mi><mi href="./28.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.1#p2.t1.r3">z</mi><mo>,</mo><mi>ξ</mi><mo stretchy="false">)</mo></mrow></mrow></math>: paraboloidal wave function</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS3.p3" class="ltx_para">
<p class="ltx_p">For <math class="ltx_Math" altimg="m99.png" altimg-height="21px" altimg-valign="-6px" altimg-width="84px" alttext="m_{1}\neq m_{2}" display="inline"><mrow><msub><mi href="./28.1#p2.t1.r1">m</mi><mn>1</mn></msub><mo>≠</mo><msub><mi href="./28.1#p2.t1.r1">m</mi><mn>2</mn></msub></mrow></math>,</p>
<table id="E21" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">28.31.21</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E21.png" altimg-height="56px" altimg-valign="-20px" altimg-width="558px" alttext="\int_{0}^{2\pi}\mathit{hc}_{p}^{m_{1}}(x,\xi)\mathit{hc}_{p}^{m_{2}}(x,\xi)%
\mathrm{d}x=\int_{0}^{2\pi}\mathit{hs}_{p}^{m_{1}}(x,\xi)\mathit{hs}_{p}^{m_{2%
}}(x,\xi)\mathrm{d}x=0." display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi></mrow></msubsup><mrow><mrow><msubsup><mi href="./28.31#E16" mathvariant="italic">hc</mi><mi>p</mi><msub><mi href="./28.1#p2.t1.r1">m</mi><mn>1</mn></msub></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.1#p2.t1.r2">x</mi><mo>,</mo><mi>ξ</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><msubsup><mi href="./28.31#E16" mathvariant="italic">hc</mi><mi>p</mi><msub><mi href="./28.1#p2.t1.r1">m</mi><mn>2</mn></msub></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.1#p2.t1.r2">x</mi><mo>,</mo><mi>ξ</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./28.1#p2.t1.r2">x</mi></mrow></mrow></mrow><mo>=</mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi></mrow></msubsup><mrow><mrow><msubsup><mi href="./28.31#E17" mathvariant="italic">hs</mi><mi>p</mi><msub><mi href="./28.1#p2.t1.r1">m</mi><mn>1</mn></msub></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.1#p2.t1.r2">x</mi><mo>,</mo><mi>ξ</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><msubsup><mi href="./28.31#E17" mathvariant="italic">hs</mi><mi>p</mi><msub><mi href="./28.1#p2.t1.r1">m</mi><mn>2</mn></msub></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.1#p2.t1.r2">x</mi><mo>,</mo><mi>ξ</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./28.1#p2.t1.r2">x</mi></mrow></mrow></mrow><mo>=</mo><mn>0</mn></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E21.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m83.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m112.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./28.1#p2.t1.r1" title="§28.1 Special Notation ‣ Notation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m97.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./28.1#p2.t1.r1">m</mi></math>: integer</a>,
<a href="./28.1#p2.t1.r2" title="§28.1 Special Notation ‣ Notation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m112.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./28.1#p2.t1.r2">x</mi></math>: real variable</a>,
<a href="./28.31#E16" title="(28.31.16) ‣ §28.31(iii) Paraboloidal Wave Functions ‣ §28.31 Equations of Whittaker–Hill and Ince ‣ Hill’s Equation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="26px" altimg-valign="-10px" altimg-width="84px" alttext="\mathit{hc}_{p}^{m}(z,\xi)" display="inline"><mrow><msubsup><mi href="./28.31#E16" mathvariant="italic">hc</mi><mi>p</mi><mi href="./28.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.1#p2.t1.r3">z</mi><mo>,</mo><mi>ξ</mi><mo stretchy="false">)</mo></mrow></mrow></math>: paraboloidal wave function</a> and
<a href="./28.31#E17" title="(28.31.17) ‣ §28.31(iii) Paraboloidal Wave Functions ‣ §28.31 Equations of Whittaker–Hill and Ince ‣ Hill’s Equation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m75.png" altimg-height="26px" altimg-valign="-10px" altimg-width="84px" alttext="\mathit{hs}_{p}^{m}(z,\xi)" display="inline"><mrow><msubsup><mi href="./28.31#E17" mathvariant="italic">hs</mi><mi>p</mi><mi href="./28.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.1#p2.t1.r3">z</mi><mo>,</mo><mi>ξ</mi><mo stretchy="false">)</mo></mrow></mrow></math>: paraboloidal wave function</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">More important are the <em class="ltx_emph ltx_font_italic">double orthogonality relations</em> for <math class="ltx_Math" altimg="m105.png" altimg-height="21px" altimg-valign="-6px" altimg-width="69px" alttext="p_{1}\neq p_{2}" display="inline"><mrow><msub><mi>p</mi><mn>1</mn></msub><mo>≠</mo><msub><mi>p</mi><mn>2</mn></msub></mrow></math>
or <math class="ltx_Math" altimg="m99.png" altimg-height="21px" altimg-valign="-6px" altimg-width="84px" alttext="m_{1}\neq m_{2}" display="inline"><mrow><msub><mi href="./28.1#p2.t1.r1">m</mi><mn>1</mn></msub><mo>≠</mo><msub><mi href="./28.1#p2.t1.r1">m</mi><mn>2</mn></msub></mrow></math> or both, given by
</p>
<table id="E22" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">28.31.22</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_Math ltx_eqn_cell ltx_align_center"><math class="ltx_Math" alttext="\int_{u_{0}}^{u_{\infty}}\int_{0}^{2\pi}\mathit{hc}_{p_{1}}^{m_{1}}(u,\xi)%
\mathit{hc}_{p_{1}}^{m_{1}}(v,\xi)\mathit{hc}_{p_{2}}^{m_{2}}(u,\xi)\mathit{hc%
}_{p_{2}}^{m_{2}}(v,\xi)\*\left(\cos\left(2u\right)-\cos\left(2v\right)\right)%
\mathrm{d}v\mathrm{d}u=0," display="block"><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><msub><mi>u</mi><mn>0</mn></msub><msub><mi>u</mi><mi mathvariant="normal">∞</mi></msub></msubsup><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi></mrow></msubsup><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mrow><msubsup><mi href="./28.31#E16" mathvariant="italic">hc</mi><msub><mi>p</mi><mn>1</mn></msub><msub><mi href="./28.1#p2.t1.r1">m</mi><mn>1</mn></msub></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi>u</mi><mo>,</mo><mi>ξ</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><msubsup><mi href="./28.31#E16" mathvariant="italic">hc</mi><msub><mi>p</mi><mn>1</mn></msub><msub><mi href="./28.1#p2.t1.r1">m</mi><mn>1</mn></msub></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><mi>ξ</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><msubsup><mi href="./28.31#E16" mathvariant="italic">hc</mi><msub><mi>p</mi><mn>2</mn></msub><msub><mi href="./28.1#p2.t1.r1">m</mi><mn>2</mn></msub></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi>u</mi><mo>,</mo><mi>ξ</mi><mo stretchy="false">)</mo></mrow></mrow></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>×</mo><mrow><msubsup><mi href="./28.31#E16" mathvariant="italic">hc</mi><msub><mi>p</mi><mn>2</mn></msub><msub><mi href="./28.1#p2.t1.r1">m</mi><mn>2</mn></msub></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><mi>ξ</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo>(</mo><mrow><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><mi>u</mi></mrow><mo>)</mo></mrow></mrow><mo>-</mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><mi>v</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>)</mo></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>v</mi></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>u</mi></mrow></mtd></mtr></mtable></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>=</mo><mn>0</mn><mo>,</mo></mtd></mtr></mtable></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E22.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m83.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m112.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./28.1#p2.t1.r1" title="§28.1 Special Notation ‣ Notation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m97.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./28.1#p2.t1.r1">m</mi></math>: integer</a> and
<a href="./28.31#E16" title="(28.31.16) ‣ §28.31(iii) Paraboloidal Wave Functions ‣ §28.31 Equations of Whittaker–Hill and Ince ‣ Hill’s Equation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="26px" altimg-valign="-10px" altimg-width="84px" alttext="\mathit{hc}_{p}^{m}(z,\xi)" display="inline"><mrow><msubsup><mi href="./28.31#E16" mathvariant="italic">hc</mi><mi>p</mi><mi href="./28.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.1#p2.t1.r3">z</mi><mo>,</mo><mi>ξ</mi><mo stretchy="false">)</mo></mrow></mrow></math>: paraboloidal wave function</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and
</p>
<table id="E23" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">28.31.23</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_Math ltx_eqn_cell ltx_align_center"><math class="ltx_Math" alttext="\int_{u_{0}}^{u_{\infty}}\int_{0}^{2\pi}\mathit{hs}_{p_{1}}^{m_{1}}(u,\xi)%
\mathit{hs}_{p_{1}}^{m_{1}}(v,\xi)\mathit{hs}_{p_{2}}^{m_{2}}(u,\xi)\mathit{hs%
}_{p_{2}}^{m_{2}}(v,\xi)\*\left(\cos\left(2u\right)-\cos\left(2v\right)\right)%
\mathrm{d}v\mathrm{d}u=0," display="block"><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><msub><mi>u</mi><mn>0</mn></msub><msub><mi>u</mi><mi mathvariant="normal">∞</mi></msub></msubsup><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi></mrow></msubsup><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mrow><msubsup><mi href="./28.31#E17" mathvariant="italic">hs</mi><msub><mi>p</mi><mn>1</mn></msub><msub><mi href="./28.1#p2.t1.r1">m</mi><mn>1</mn></msub></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi>u</mi><mo>,</mo><mi>ξ</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><msubsup><mi href="./28.31#E17" mathvariant="italic">hs</mi><msub><mi>p</mi><mn>1</mn></msub><msub><mi href="./28.1#p2.t1.r1">m</mi><mn>1</mn></msub></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><mi>ξ</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><msubsup><mi href="./28.31#E17" mathvariant="italic">hs</mi><msub><mi>p</mi><mn>2</mn></msub><msub><mi href="./28.1#p2.t1.r1">m</mi><mn>2</mn></msub></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi>u</mi><mo>,</mo><mi>ξ</mi><mo stretchy="false">)</mo></mrow></mrow></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>×</mo><mrow><msubsup><mi href="./28.31#E17" mathvariant="italic">hs</mi><msub><mi>p</mi><mn>2</mn></msub><msub><mi href="./28.1#p2.t1.r1">m</mi><mn>2</mn></msub></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><mi>ξ</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo>(</mo><mrow><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><mi>u</mi></mrow><mo>)</mo></mrow></mrow><mo>-</mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><mi>v</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>)</mo></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>v</mi></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>u</mi></mrow></mtd></mtr></mtable></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>=</mo><mn>0</mn><mo>,</mo></mtd></mtr></mtable></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E23.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m83.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m112.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./28.1#p2.t1.r1" title="§28.1 Special Notation ‣ Notation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m97.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./28.1#p2.t1.r1">m</mi></math>: integer</a> and
<a href="./28.31#E17" title="(28.31.17) ‣ §28.31(iii) Paraboloidal Wave Functions ‣ §28.31 Equations of Whittaker–Hill and Ince ‣ Hill’s Equation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m75.png" altimg-height="26px" altimg-valign="-10px" altimg-width="84px" alttext="\mathit{hs}_{p}^{m}(z,\xi)" display="inline"><mrow><msubsup><mi href="./28.31#E17" mathvariant="italic">hs</mi><mi>p</mi><mi href="./28.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.1#p2.t1.r3">z</mi><mo>,</mo><mi>ξ</mi><mo stretchy="false">)</mo></mrow></mrow></math>: paraboloidal wave function</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and also for all <math class="ltx_Math" altimg="m104.png" altimg-height="16px" altimg-valign="-6px" altimg-width="122px" alttext="p_{1},p_{2},m_{1},m_{2}" display="inline"><mrow><msub><mi>p</mi><mn>1</mn></msub><mo>,</mo><msub><mi>p</mi><mn>2</mn></msub><mo>,</mo><msub><mi href="./28.1#p2.t1.r1">m</mi><mn>1</mn></msub><mo>,</mo><msub><mi href="./28.1#p2.t1.r1">m</mi><mn>2</mn></msub></mrow></math>, given by</p>
<table id="E24" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">28.31.24</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_Math ltx_eqn_cell ltx_align_center"><math class="ltx_Math" alttext="\int_{u_{0}}^{u_{\infty}}\int_{0}^{2\pi}\mathit{hc}_{p_{1}}^{m_{1}}(u,\xi)%
\mathit{hc}_{p_{1}}^{m_{1}}(v,\xi)\mathit{hs}_{p_{2}}^{m_{2}}(u,\xi)\mathit{hs%
}_{p_{2}}^{m_{2}}(v,\xi)\*\left(\cos\left(2u\right)-\cos\left(2v\right)\right)%
\mathrm{d}v\mathrm{d}u=0," display="block"><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><msub><mi>u</mi><mn>0</mn></msub><msub><mi>u</mi><mi mathvariant="normal">∞</mi></msub></msubsup><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi></mrow></msubsup><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mrow><msubsup><mi href="./28.31#E16" mathvariant="italic">hc</mi><msub><mi>p</mi><mn>1</mn></msub><msub><mi href="./28.1#p2.t1.r1">m</mi><mn>1</mn></msub></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi>u</mi><mo>,</mo><mi>ξ</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><msubsup><mi href="./28.31#E16" mathvariant="italic">hc</mi><msub><mi>p</mi><mn>1</mn></msub><msub><mi href="./28.1#p2.t1.r1">m</mi><mn>1</mn></msub></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><mi>ξ</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><msubsup><mi href="./28.31#E17" mathvariant="italic">hs</mi><msub><mi>p</mi><mn>2</mn></msub><msub><mi href="./28.1#p2.t1.r1">m</mi><mn>2</mn></msub></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi>u</mi><mo>,</mo><mi>ξ</mi><mo stretchy="false">)</mo></mrow></mrow></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>×</mo><mrow><msubsup><mi href="./28.31#E17" mathvariant="italic">hs</mi><msub><mi>p</mi><mn>2</mn></msub><msub><mi href="./28.1#p2.t1.r1">m</mi><mn>2</mn></msub></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><mi>ξ</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo>(</mo><mrow><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><mi>u</mi></mrow><mo>)</mo></mrow></mrow><mo>-</mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><mi>v</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>)</mo></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>v</mi></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>u</mi></mrow></mtd></mtr></mtable></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>=</mo><mn>0</mn><mo>,</mo></mtd></mtr></mtable></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E24.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m83.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m112.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./28.1#p2.t1.r1" title="§28.1 Special Notation ‣ Notation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m97.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./28.1#p2.t1.r1">m</mi></math>: integer</a>,
<a href="./28.31#E16" title="(28.31.16) ‣ §28.31(iii) Paraboloidal Wave Functions ‣ §28.31 Equations of Whittaker–Hill and Ince ‣ Hill’s Equation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="26px" altimg-valign="-10px" altimg-width="84px" alttext="\mathit{hc}_{p}^{m}(z,\xi)" display="inline"><mrow><msubsup><mi href="./28.31#E16" mathvariant="italic">hc</mi><mi>p</mi><mi href="./28.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.1#p2.t1.r3">z</mi><mo>,</mo><mi>ξ</mi><mo stretchy="false">)</mo></mrow></mrow></math>: paraboloidal wave function</a> and
<a href="./28.31#E17" title="(28.31.17) ‣ §28.31(iii) Paraboloidal Wave Functions ‣ §28.31 Equations of Whittaker–Hill and Ince ‣ Hill’s Equation ‣ Chapter 28 Mathieu Functions and Hill’s Equation" class="ltx_ref"><math class="ltx_Math" altimg="m75.png" altimg-height="26px" altimg-valign="-10px" altimg-width="84px" alttext="\mathit{hs}_{p}^{m}(z,\xi)" display="inline"><mrow><msubsup><mi href="./28.31#E17" mathvariant="italic">hs</mi><mi>p</mi><mi href="./28.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.1#p2.t1.r3">z</mi><mo>,</mo><mi>ξ</mi><mo stretchy="false">)</mo></mrow></mrow></math>: paraboloidal wave function</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m45.png" altimg-height="23px" altimg-valign="-7px" altimg-width="164px" alttext="(u_{0},u_{\infty})=(0,\mathrm{i}\infty)" display="inline"><mrow><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><msub><mi>u</mi><mn>0</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><msub><mi>u</mi><mi mathvariant="normal">∞</mi></msub><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow><mo>=</mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mn>0</mn><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi mathvariant="normal">∞</mi></mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></mrow></math> when <math class="ltx_Math" altimg="m86.png" altimg-height="21px" altimg-valign="-6px" altimg-width="50px" alttext="\xi>0" display="inline"><mrow><mi>ξ</mi><mo>></mo><mn>0</mn></mrow></math>, and
<math class="ltx_Math" altimg="m46.png" altimg-height="27px" altimg-valign="-9px" altimg-width="228px" alttext="(u_{0},u_{\infty})=(\tfrac{1}{2}\pi,\tfrac{1}{2}\pi+\mathrm{i}\infty)" display="inline"><mrow><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><msub><mi>u</mi><mn>0</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><msub><mi>u</mi><mi mathvariant="normal">∞</mi></msub><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow><mo>=</mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi mathvariant="normal">∞</mi></mrow></mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></mrow></math> when
<math class="ltx_Math" altimg="m85.png" altimg-height="21px" altimg-valign="-6px" altimg-width="50px" alttext="\xi<0" display="inline"><mrow><mi>ξ</mi><mo><</mo><mn>0</mn></mrow></math>.</p>
</div>
<div id="SS3.p4" class="ltx_para">
<p class="ltx_p">For proofs and further integral equations see <cite class="ltx_cite ltx_citemacro_citet">Urwin (</dd>
</dl>
</div>
</div>
<div id="Px1.p1" class="ltx_para">
<p class="ltx_p">For <math class="ltx_Math" altimg="m86.png" altimg-height="21px" altimg-valign="-6px" altimg-width="50px" alttext="\xi>0" display="inline"><mrow><mi>ξ</mi><mo>></mo><mn>0</mn></mrow></math>, the functions <math class="ltx_Math" altimg="m72.png" altimg-height="26px" altimg-valign="-10px" altimg-width="84px" alttext="\mathit{hc}_{p}^{m}(z,\xi)" display="inline"><mrow><msubsup><mi href="./28.31#E16" mathvariant="italic">hc</mi><mi>p</mi><mi href="./28.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.1#p2.t1.r3">z</mi><mo>,</mo><mi>ξ</mi><mo stretchy="false">)</mo></mrow></mrow></math>, <math class="ltx_Math" altimg="m75.png" altimg-height="26px" altimg-valign="-10px" altimg-width="84px" alttext="\mathit{hs}_{p}^{m}(z,\xi)" display="inline"><mrow><msubsup><mi href="./28.31#E17" mathvariant="italic">hs</mi><mi>p</mi><mi href="./28.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.1#p2.t1.r3">z</mi><mo>,</mo><mi>ξ</mi><mo stretchy="false">)</mo></mrow></mrow></math> behave
asymptotically as multiples of <math class="ltx_Math" altimg="m67.png" altimg-height="27px" altimg-valign="-9px" altimg-width="231px" alttext="\exp\left(-\tfrac{1}{4}\xi\cos\left(2z\right)\right)\left(\cos z\right)^{p}" display="inline"><mrow><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac><mo></mo><mi>ξ</mi><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><mi href="./28.1#p2.t1.r3">z</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>)</mo></mrow></mrow><mo></mo><msup><mrow><mo>(</mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi href="./28.1#p2.t1.r3">z</mi></mrow><mo>)</mo></mrow><mi>p</mi></msup></mrow></math>
as <math class="ltx_Math" altimg="m115.png" altimg-height="19px" altimg-valign="-4px" altimg-width="86px" alttext="z\to\pm\mathrm{i}\infty" display="inline"><mrow><mi href="./28.1#p2.t1.r3">z</mi><mo>→</mo><mrow><mo>±</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi mathvariant="normal">∞</mi></mrow></mrow></mrow></math>. All other periodic solutions behave as multiples
of <math class="ltx_Math" altimg="m68.png" altimg-height="27px" altimg-valign="-9px" altimg-width="249px" alttext="\exp\left(\tfrac{1}{4}\xi\cos\left(2z\right)\right)(\cos z)^{-p-2}" display="inline"><mrow><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac><mo></mo><mi>ξ</mi><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><mi href="./28.1#p2.t1.r3">z</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>)</mo></mrow></mrow><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi href="./28.1#p2.t1.r3">z</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mrow><mo>-</mo><mi>p</mi></mrow><mo>-</mo><mn>2</mn></mrow></msup></mrow></math>.</p>
</div>
<div id="Px1.p2" class="ltx_para">
<p class="ltx_p">For <math class="ltx_Math" altimg="m86.png" altimg-height="21px" altimg-valign="-6px" altimg-width="50px" alttext="\xi>0" display="inline"><mrow><mi>ξ</mi><mo>></mo><mn>0</mn></mrow></math>, the functions <math class="ltx_Math" altimg="m71.png" altimg-height="26px" altimg-valign="-10px" altimg-width="100px" alttext="\mathit{hc}_{p}^{m}(z,-\xi)" display="inline"><mrow><msubsup><mi href="./28.31#E16" mathvariant="italic">hc</mi><mi>p</mi><mi href="./28.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.1#p2.t1.r3">z</mi><mo>,</mo><mrow><mo>-</mo><mi>ξ</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></math>, <math class="ltx_Math" altimg="m74.png" altimg-height="26px" altimg-valign="-10px" altimg-width="99px" alttext="\mathit{hs}_{p}^{m}(z,-\xi)" display="inline"><mrow><msubsup><mi href="./28.31#E17" mathvariant="italic">hs</mi><mi>p</mi><mi href="./28.1#p2.t1.r1">m</mi></msubsup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./28.1#p2.t1.r3">z</mi><mo>,</mo><mrow><mo>-</mo><mi>ξ</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></math> behave
asymptotically as multiples of
<math class="ltx_Math" altimg="m68.png" altimg-height="27px" altimg-valign="-9px" altimg-width="249px" alttext="\exp\left(\tfrac{1}{4}\xi\cos\left(2z\right)\right)(\cos z)^{-p-2}" display="inline"><mrow><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac><mo></mo><mi>ξ</mi><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><mi href="./28.1#p2.t1.r3">z</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>)</mo></mrow></mrow><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi href="./28.1#p2.t1.r3">z</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mrow><mo>-</mo><mi>p</mi></mrow><mo>-</mo><mn>2</mn></mrow></msup></mrow></math> as <math class="ltx_Math" altimg="m116.png" altimg-height="27px" altimg-valign="-9px" altimg-width="120px" alttext="z\to\tfrac{1}{2}\pi\pm\mathrm{i}\infty" display="inline"><mrow><mi href="./28.1#p2.t1.r3">z</mi><mo>→</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo>±</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi mathvariant="normal">∞</mi></mrow></mrow></mrow></math>. All other periodic solutions behave as
multiples of <math class="ltx_Math" altimg="m67.png" altimg-height="27px" altimg-valign="-9px" altimg-width="231px" alttext="\exp\left(-\tfrac{1}{4}\xi\cos\left(2z\right)\right)\left(\cos z\right)^{p}" display="inline"><mrow><mrow><mi href="./4.2#E19">exp</mi><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac><mo></mo><mi>ξ</mi><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><mi href="./28.1#p2.t1.r3">z</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>)</mo></mrow></mrow><mo></mo><msup><mrow><mo>(</mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi href="./28.1#p2.t1.r3">z</mi></mrow><mo>)</mo></mrow><mi>p</mi></msup></mrow></math>.</p>
</div>
</section>
</section>
</section>
</div>
<div class="ltx_page_footer">
<span></div>
</div>
</body></text>
</html>
</page>
<page>
<!DOCTYPE html><html>
<head>
<title>DLMF: 27.2 Functions</title>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<!--GOOGLE BOOTSTRAP--></head>
<text><body class="color_default textfont_default titlefont_default fontsize_default navbar_default">
<div class="ltx_page_navbar">
<div class="ltx_page_navlogo"><div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd>
<span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m39.png" altimg-height="23px" altimg-valign="-7px" altimg-width="55px" alttext="d_{\NVar{k}}\left(\NVar{n}\right)" display="inline"><mrow><msub><mi href="./27.2#SS1.p4">d</mi><mi class="ltx_nvar" href="./27.1#p2.t1.r1">k</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow></math>: divisor function</span> and
<span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m18.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\nu\left(\NVar{n}\right)" display="inline"><mrow><mi href="./27.2#SS1.p1">ν</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow></math>: number of distinct primes dividing <math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./27.1#p2.t1.r1">n</mi></math></span>
</dd>
<dt>Keywords:</dt>
<dd>
</dd>
</dl>
</div>
</div>
<div id="SS1.p1" class="ltx_para">
<p class="ltx_p">Functions in this section derive their properties from the <em class="ltx_emph ltx_font_italic">fundamental
theorem of arithmetic</em>, which states that every integer <math class="ltx_Math" altimg="m48.png" altimg-height="17px" altimg-valign="-3px" altimg-width="53px" alttext="n>1" display="inline"><mrow><mi href="./27.1#p2.t1.r1">n</mi><mo>></mo><mn>1</mn></mrow></math> can be
represented uniquely as a product of prime powers,
</p>
<table id="E1" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">27.2.1</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E1.png" altimg-height="71px" altimg-valign="-27px" altimg-width="112px" alttext="n=\prod_{r=1}^{\nu\left(n\right)}p^{a_{r}}_{r}," display="block"><mrow><mrow><mi href="./27.1#p2.t1.r1">n</mi><mo>=</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∏</mo><mrow><mi>r</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi href="./27.2#SS1.p1">ν</mi><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow></munderover><msubsup><mi href="./27.1#p2.t1.r7">p</mi><mi>r</mi><msub><mi href="./27.2#SS1.p4">a</mi><mi>r</mi></msub></msubsup></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./27.2#SS1.p1" title="§27.2(i) Definitions ‣ §27.2 Functions ‣ Multiplicative Number Theory ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m18.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\nu\left(\NVar{n}\right)" display="inline"><mrow><mi href="./27.2#SS1.p1">ν</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow></math>: number of distinct primes dividing <math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./27.1#p2.t1.r1">n</mi></math></a>,
<a href="./27.1#p2.t1.r1" title="§27.1 Special Notation ‣ Notation ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./27.1#p2.t1.r1">n</mi></math>: positive integer</a>,
<a href="./27.1#p2.t1.r7" title="§27.1 Special Notation ‣ Notation ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="16px" altimg-valign="-6px" altimg-width="74px" alttext="p,p_{1},\ldots" display="inline"><mrow><mi href="./27.1#p2.t1.r7">p</mi><mo>,</mo><msub><mi href="./27.1#p2.t1.r7">p</mi><mn>1</mn></msub><mo>,</mo><mi mathvariant="normal">…</mi></mrow></math>: prime numbers</a> and
<a href="./27.2#SS1.p4" title="§27.2(i) Definitions ‣ §27.2 Functions ‣ Multiplicative Number Theory ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m31.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./27.2#SS1.p4">a</mi></math>: primitive root</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m55.png" altimg-height="20px" altimg-valign="-9px" altimg-width="138px" alttext="p_{1},p_{2},\dots,p_{\nu\left(n\right)}" display="inline"><mrow><msub><mi href="./27.1#p2.t1.r7">p</mi><mn>1</mn></msub><mo>,</mo><msub><mi href="./27.1#p2.t1.r7">p</mi><mn>2</mn></msub><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><msub><mi href="./27.1#p2.t1.r7">p</mi><mrow><mi href="./27.2#SS1.p1">ν</mi><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow></msub></mrow></math> are the distinct prime factors of
<math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./27.1#p2.t1.r1">n</mi></math>, each exponent <math class="ltx_Math" altimg="m35.png" altimg-height="16px" altimg-valign="-5px" altimg-width="24px" alttext="a_{r}" display="inline"><msub><mi href="./27.2#SS1.p4">a</mi><mi>r</mi></msub></math> is positive, and <math class="ltx_Math" altimg="m19.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\nu\left(n\right)" display="inline"><mrow><mi href="./27.2#SS1.p1">ν</mi><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow></math> is the number of
distinct primes dividing <math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./27.1#p2.t1.r1">n</mi></math>. (<math class="ltx_Math" altimg="m17.png" altimg-height="23px" altimg-valign="-7px" altimg-width="44px" alttext="\nu\left(1\right)" display="inline"><mrow><mi href="./27.2#SS1.p1">ν</mi><mo></mo><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></math> is defined to be 0.)
Euclid’s Elements (<cite class="ltx_cite ltx_citemacro_citet">Euclid () reveal
great irregularity in their distribution. They tend to thin out among the large
integers, but this thinning out is not completely regular. There is great
interest in the function <math class="ltx_Math" altimg="m24.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="\pi\left(x\right)" display="inline"><mrow><mi href="./27.2#E2">π</mi><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r9">x</mi><mo>)</mo></mrow></mrow></math> that counts the number of
primes not exceeding <math class="ltx_Math" altimg="m67.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./27.1#p2.t1.r9">x</mi></math>. It can be expressed as a sum over all primes
<math class="ltx_Math" altimg="m53.png" altimg-height="20px" altimg-valign="-6px" altimg-width="52px" alttext="p\leq x" display="inline"><mrow><mi href="./27.1#p2.t1.r7">p</mi><mo>≤</mo><mi href="./27.1#p2.t1.r9">x</mi></mrow></math>:</p>
<table id="E2" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">27.2.2</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E2.png" altimg-height="55px" altimg-valign="-31px" altimg-width="123px" alttext="\pi\left(x\right)=\sum_{p\leq x}1." display="block"><mrow><mrow><mrow><mi href="./27.2#E2">π</mi><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r9">x</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><munder><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./27.1#p2.t1.r7">p</mi><mo>≤</mo><mi href="./27.1#p2.t1.r9">x</mi></mrow></munder><mn>1</mn></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m23.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="\pi\left(\NVar{x}\right)" display="inline"><mrow><mi href="./27.2#E2">π</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./27.1#p2.t1.r9">x</mi><mo>)</mo></mrow></mrow></math>: number of primes not exceeding <math class="ltx_Math" altimg="m67.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./27.1#p2.t1.r9">x</mi></math></span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./27.1#p2.t1.r7" title="§27.1 Special Notation ‣ Notation ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="16px" altimg-valign="-6px" altimg-width="74px" alttext="p,p_{1},\ldots" display="inline"><mrow><mi href="./27.1#p2.t1.r7">p</mi><mo>,</mo><msub><mi href="./27.1#p2.t1.r7">p</mi><mn>1</mn></msub><mo>,</mo><mi mathvariant="normal">…</mi></mrow></math>: prime numbers</a> and
<a href="./27.1#p2.t1.r9" title="§27.1 Special Notation ‣ Notation ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m67.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./27.1#p2.t1.r9">x</mi></math>: real number</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS1.p2" class="ltx_para">
<p class="ltx_p">Gauss and Legendre conjectured that <math class="ltx_Math" altimg="m24.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="\pi\left(x\right)" display="inline"><mrow><mi href="./27.2#E2">π</mi><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r9">x</mi><mo>)</mo></mrow></mrow></math> is asymptotic to
<math class="ltx_Math" altimg="m66.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="x/\ln x" display="inline"><mrow><mi href="./27.1#p2.t1.r9">x</mi><mo>/</mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi href="./27.1#p2.t1.r9">x</mi></mrow></mrow></math> as <math class="ltx_Math" altimg="m68.png" altimg-height="13px" altimg-valign="-2px" altimg-width="67px" alttext="x\to\infty" display="inline"><mrow><mi href="./27.1#p2.t1.r9">x</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></math>:
</p>
<table id="E3" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">27.2.3</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E3.png" altimg-height="41px" altimg-valign="-16px" altimg-width="117px" alttext="\pi\left(x\right)\sim\frac{x}{\ln x}." display="block"><mrow><mrow><mrow><mi href="./27.2#E2">π</mi><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r9">x</mi><mo>)</mo></mrow></mrow><mo href="./2.1#E1">∼</mo><mfrac><mi href="./27.1#p2.t1.r9">x</mi><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi href="./27.1#p2.t1.r9">x</mi></mrow></mfrac></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./2.1#E1" title="(2.1.1) ‣ §2.1(i) Asymptotic and Order Symbols ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="12px" altimg-valign="-2px" altimg-width="20px" alttext="\sim" display="inline"><mo href="./2.1#E1">∼</mo></math>: asymptotic equality</a>,
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m13.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a>,
<a href="./27.2#E2" title="(27.2.2) ‣ §27.2(i) Definitions ‣ §27.2 Functions ‣ Multiplicative Number Theory ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m23.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="\pi\left(\NVar{x}\right)" display="inline"><mrow><mi href="./27.2#E2">π</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./27.1#p2.t1.r9">x</mi><mo>)</mo></mrow></mrow></math>: number of primes not exceeding <math class="ltx_Math" altimg="m67.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./27.1#p2.t1.r9">x</mi></math></a> and
<a href="./27.1#p2.t1.r9" title="§27.1 Special Notation ‣ Notation ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m67.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./27.1#p2.t1.r9">x</mi></math>: real number</a>
</dd>
<dt>Change of Notation (effective with 1.0.10):</dt>
<dd>
The notation for logarithm has been changed to <math class="ltx_Math" altimg="m14.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="\ln" display="inline"><mi href="./4.2#E2">ln</mi></math> from <math class="ltx_Math" altimg="m15.png" altimg-height="21px" altimg-valign="-6px" altimg-width="30px" alttext="\mathrm{log}" display="inline"><mi>log</mi></math>.
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">(See <cite class="ltx_cite ltx_citemacro_citet">Gauss ()</cite>, is known as
the <em class="ltx_emph ltx_font_italic">prime number theorem</em>. An equivalent form states that the <math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./27.1#p2.t1.r1">n</mi></math>th prime
<math class="ltx_Math" altimg="m65.png" altimg-height="16px" altimg-valign="-6px" altimg-width="25px" alttext="p_{n}" display="inline"><msub><mi href="./27.1#p2.t1.r7">p</mi><mi href="./27.1#p2.t1.r1">n</mi></msub></math> (when the primes are listed in increasing order) is asymptotic to
<math class="ltx_Math" altimg="m50.png" altimg-height="18px" altimg-valign="-2px" altimg-width="51px" alttext="n\ln n" display="inline"><mrow><mi href="./27.1#p2.t1.r1">n</mi><mo></mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi href="./27.1#p2.t1.r1">n</mi></mrow></mrow></math> as <math class="ltx_Math" altimg="m51.png" altimg-height="13px" altimg-valign="-2px" altimg-width="67px" alttext="n\to\infty" display="inline"><mrow><mi href="./27.1#p2.t1.r1">n</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></math>:</p>
<table id="E4" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">27.2.4</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E4.png" altimg-height="23px" altimg-valign="-6px" altimg-width="106px" alttext="p_{n}\sim n\ln n." display="block"><mrow><mrow><msub><mi href="./27.1#p2.t1.r7">p</mi><mi href="./27.1#p2.t1.r1">n</mi></msub><mo href="./2.1#E1">∼</mo><mrow><mi href="./27.1#p2.t1.r1">n</mi><mo></mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi href="./27.1#p2.t1.r1">n</mi></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E4.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./2.1#E1" title="(2.1.1) ‣ §2.1(i) Asymptotic and Order Symbols ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="12px" altimg-valign="-2px" altimg-width="20px" alttext="\sim" display="inline"><mo href="./2.1#E1">∼</mo></math>: asymptotic equality</a>,
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m13.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a>,
<a href="./27.1#p2.t1.r1" title="§27.1 Special Notation ‣ Notation ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./27.1#p2.t1.r1">n</mi></math>: positive integer</a> and
<a href="./27.1#p2.t1.r7" title="§27.1 Special Notation ‣ Notation ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="16px" altimg-valign="-6px" altimg-width="74px" alttext="p,p_{1},\ldots" display="inline"><mrow><mi href="./27.1#p2.t1.r7">p</mi><mo>,</mo><msub><mi href="./27.1#p2.t1.r7">p</mi><mn>1</mn></msub><mo>,</mo><mi mathvariant="normal">…</mi></mrow></math>: prime numbers</a>
</dd>
<dt>Change of Notation (effective with 1.0.10):</dt>
<dd>
The notation for logarithm has been changed to <math class="ltx_Math" altimg="m14.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="\ln" display="inline"><mi href="./4.2#E2">ln</mi></math> from <math class="ltx_Math" altimg="m15.png" altimg-height="21px" altimg-valign="-6px" altimg-width="30px" alttext="\mathrm{log}" display="inline"><mi>log</mi></math>.
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">(See also §.) Other examples of number-theoretic functions
treated in this chapter are as follows.</p>
<table id="E5" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">27.2.5</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E5.png" altimg-height="65px" altimg-valign="-27px" altimg-width="180px" alttext="\left\lfloor\frac{1}{n}\right\rfloor=\begin{cases}1,&n=1,\\
0,&n>1.\end{cases}" display="block"><mrow><mrow><mo href="./front/introduction#Sx4.p1.t1.r16">⌊</mo><mfrac><mn>1</mn><mi href="./27.1#p2.t1.r1">n</mi></mfrac><mo href="./front/introduction#Sx4.p1.t1.r16">⌋</mo></mrow><mo>=</mo><mrow><mo>{</mo><mtable columnspacing="5pt" displaystyle="true" rowspacing="0pt"><mtr><mtd columnalign="left"><mrow><mn>1</mn><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mrow><mi href="./27.1#p2.t1.r1">n</mi><mo>=</mo><mn>1</mn></mrow><mo>,</mo></mrow></mtd></mtr><mtr><mtd columnalign="left"><mrow><mn>0</mn><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mrow><mi href="./27.1#p2.t1.r1">n</mi><mo>></mo><mn>1</mn></mrow><mo>.</mo></mrow></mtd></mtr></mtable></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E5.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./front/introduction#Sx4.p1.t1.r16" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m11.png" altimg-height="23px" altimg-valign="-7px" altimg-width="33px" alttext="\left\lfloor\NVar{x}\right\rfloor" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r16">⌊</mo><mi class="ltx_nvar">x</mi><mo href="./front/introduction#Sx4.p1.t1.r16">⌋</mo></mrow></math>: floor of <math class="ltx_Math" altimg="m67.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./27.1#p2.t1.r1" title="§27.1 Special Notation ‣ Notation ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./27.1#p2.t1.r1">n</mi></math>: positive integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E6" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">27.2.6</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E6.png" altimg-height="57px" altimg-valign="-32px" altimg-width="183px" alttext="\phi_{k}\left(n\right)=\sum_{\left(m,n\right)=1}m^{k}," display="block"><mrow><mrow><mrow><msub><mi href="./27.2#E6">ϕ</mi><mi href="./27.1#p2.t1.r1">k</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><munder><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mrow><mo href="./27.1#p2.t1.r3">(</mo><mrow><mi href="./27.1#p2.t1.r1">m</mi><mo>,</mo><mi href="./27.1#p2.t1.r1">n</mi></mrow><mo href="./27.1#p2.t1.r3">)</mo></mrow><mo>=</mo><mn>1</mn></mrow></munder><msup><mi href="./27.1#p2.t1.r1">m</mi><mi href="./27.1#p2.t1.r1">k</mi></msup></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E6.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m22.png" altimg-height="23px" altimg-valign="-7px" altimg-width="57px" alttext="\phi_{\NVar{k}}\left(\NVar{n}\right)" display="inline"><mrow><msub><mi href="./27.2#E6">ϕ</mi><mi class="ltx_nvar" href="./27.1#p2.t1.r1">k</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow></math>: sum of powers of integers relatively prime to <math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./27.1#p2.t1.r1">n</mi></math></span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./27.1#p2.t1.r3" title="§27.1 Special Notation ‣ Notation ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m9.png" altimg-height="23px" altimg-valign="-7px" altimg-width="58px" alttext="\left(\NVar{m},\NVar{n}\right)" display="inline"><mrow><mo href="./27.1#p2.t1.r3">(</mo><mrow><mi class="ltx_nvar" href="./27.1#p2.t1.r1">m</mi><mo>,</mo><mi class="ltx_nvar" href="./27.1#p2.t1.r1">n</mi></mrow><mo href="./27.1#p2.t1.r3">)</mo></mrow></math>: greatest common divisor (gcd)</a>,
<a href="./27.1#p2.t1.r1" title="§27.1 Special Notation ‣ Notation ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./27.1#p2.t1.r1">k</mi></math>: positive integer</a>,
<a href="./27.1#p2.t1.r1" title="§27.1 Special Notation ‣ Notation ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./27.1#p2.t1.r1">m</mi></math>: positive integer</a> and
<a href="./27.1#p2.t1.r1" title="§27.1 Special Notation ‣ Notation ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./27.1#p2.t1.r1">n</mi></math>: positive integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">the sum of the <math class="ltx_Math" altimg="m43.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./27.1#p2.t1.r1">k</mi></math>th powers of the positive integers <math class="ltx_Math" altimg="m45.png" altimg-height="19px" altimg-valign="-5px" altimg-width="60px" alttext="m\leq n" display="inline"><mrow><mi href="./27.1#p2.t1.r1">m</mi><mo>≤</mo><mi href="./27.1#p2.t1.r1">n</mi></mrow></math> that are
relatively prime to <math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./27.1#p2.t1.r1">n</mi></math>.</p>
<table id="E7" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">27.2.7</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E7.png" altimg-height="25px" altimg-valign="-7px" altimg-width="136px" alttext="\phi\left(n\right)=\phi_{0}\left(n\right)." display="block"><mrow><mrow><mrow><mi href="./27.2#E7">ϕ</mi><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msub><mi href="./27.2#E6">ϕ</mi><mn>0</mn></msub><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E7.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m20.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="\phi\left(\NVar{n}\right)" display="inline"><mrow><mi href="./27.2#E7">ϕ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow></math>: Euler’s totient</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./27.2#E6" title="(27.2.6) ‣ §27.2(i) Definitions ‣ §27.2 Functions ‣ Multiplicative Number Theory ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="23px" altimg-valign="-7px" altimg-width="57px" alttext="\phi_{\NVar{k}}\left(\NVar{n}\right)" display="inline"><mrow><msub><mi href="./27.2#E6">ϕ</mi><mi class="ltx_nvar" href="./27.1#p2.t1.r1">k</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow></math>: sum of powers of integers relatively prime to <math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./27.1#p2.t1.r1">n</mi></math></a> and
<a href="./27.1#p2.t1.r1" title="§27.1 Special Notation ‣ Notation ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./27.1#p2.t1.r1">n</mi></math>: positive integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">This is the number of positive integers <math class="ltx_Math" altimg="m12.png" altimg-height="19px" altimg-valign="-5px" altimg-width="37px" alttext="\leq n" display="inline"><mrow><mi></mi><mo>≤</mo><mi href="./27.1#p2.t1.r1">n</mi></mrow></math> that are relatively prime to
<math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./27.1#p2.t1.r1">n</mi></math>; <math class="ltx_Math" altimg="m21.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="\phi\left(n\right)" display="inline"><mrow><mi href="./27.2#E7">ϕ</mi><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow></math> is <em class="ltx_emph ltx_font_italic">Euler’s totient</em>.</p>
</div>
<div id="SS1.p4" class="ltx_para">
<p class="ltx_p">If <math class="ltx_Math" altimg="m10.png" altimg-height="23px" altimg-valign="-7px" altimg-width="88px" alttext="\left(a,n\right)=1" display="inline"><mrow><mrow><mo href="./27.1#p2.t1.r3">(</mo><mrow><mi href="./27.2#SS1.p4">a</mi><mo>,</mo><mi href="./27.1#p2.t1.r1">n</mi></mrow><mo href="./27.1#p2.t1.r3">)</mo></mrow><mo>=</mo><mn>1</mn></mrow></math>, then the <em class="ltx_emph ltx_font_italic">Euler–Fermat theorem</em> states that</p>
<table id="E8" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">27.2.8</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E8.png" altimg-height="29px" altimg-valign="-7px" altimg-width="184px" alttext="a^{\phi\left(n\right)}\equiv 1\pmod{n}," display="block"><mrow><mrow><msup><mi href="./27.2#SS1.p4">a</mi><mrow><mi href="./27.2#E7">ϕ</mi><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow></msup><mo>≡</mo><mrow><mn>1</mn><mspace width="veryverythickmathspace"></mspace><mrow><mo lspace="8.1pt" stretchy="false">(</mo><mrow><mo movablelimits="false">mod</mo><mi href="./27.1#p2.t1.r1">n</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E8.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./27.2#E7" title="(27.2.7) ‣ §27.2(i) Definitions ‣ §27.2 Functions ‣ Multiplicative Number Theory ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="\phi\left(\NVar{n}\right)" display="inline"><mrow><mi href="./27.2#E7">ϕ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow></math>: Euler’s totient</a>,
<a href="./27.1#p2.t1.r1" title="§27.1 Special Notation ‣ Notation ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./27.1#p2.t1.r1">n</mi></math>: positive integer</a> and
<a href="./27.2#SS1.p4" title="§27.2(i) Definitions ‣ §27.2 Functions ‣ Multiplicative Number Theory ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m31.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./27.2#SS1.p4">a</mi></math>: primitive root</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">24.3.2 II.B</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and if <math class="ltx_Math" altimg="m21.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="\phi\left(n\right)" display="inline"><mrow><mi href="./27.2#E7">ϕ</mi><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow></math> is the smallest positive integer <math class="ltx_Math" altimg="m41.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> such that
<math class="ltx_Math" altimg="m33.png" altimg-height="25px" altimg-valign="-7px" altimg-width="143px" alttext="a^{f}\equiv 1\pmod{n}" display="inline"><mrow><msup><mi href="./27.2#SS1.p4">a</mi><mi>f</mi></msup><mo>≡</mo><mrow><mn>1</mn><mspace width="veryverythickmathspace"></mspace><mrow><mo lspace="8.1pt" stretchy="false">(</mo><mrow><mo>mod</mo><mi href="./27.1#p2.t1.r1">n</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></math>, then <math class="ltx_Math" altimg="m31.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./27.2#SS1.p4">a</mi></math> is a <em class="ltx_emph ltx_font_italic">primitive root</em> mod <math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./27.1#p2.t1.r1">n</mi></math>. The
<math class="ltx_Math" altimg="m21.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="\phi\left(n\right)" display="inline"><mrow><mi href="./27.2#E7">ϕ</mi><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow></math> numbers <math class="ltx_Math" altimg="m30.png" altimg-height="25px" altimg-valign="-6px" altimg-width="131px" alttext="a,a^{2},\dots,a^{\phi\left(n\right)}" display="inline"><mrow><mi href="./27.2#SS1.p4">a</mi><mo>,</mo><msup><mi href="./27.2#SS1.p4">a</mi><mn>2</mn></msup><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><msup><mi href="./27.2#SS1.p4">a</mi><mrow><mi href="./27.2#E7">ϕ</mi><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow></msup></mrow></math> are
relatively prime to <math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./27.1#p2.t1.r1">n</mi></math> and distinct (mod <math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./27.1#p2.t1.r1">n</mi></math>). Such a set is a <em class="ltx_emph ltx_font_italic">reduced
residue system</em> modulo <math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./27.1#p2.t1.r1">n</mi></math>.
</p>
<table id="E9" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">27.2.9</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E9.png" altimg-height="57px" altimg-valign="-32px" altimg-width="116px" alttext="d\left(n\right)=\sum_{d\mathbin{|}n}1" display="block"><mrow><mrow><mi href="./27.2#SS1.p4">d</mi><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><munder><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./27.1#p2.t1.r1">d</mi><mo stretchy="false">|</mo><mi href="./27.1#p2.t1.r1">n</mi></mrow></munder><mn>1</mn></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E9.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./27.2#SS1.p4" title="§27.2(i) Definitions ‣ §27.2 Functions ‣ Multiplicative Number Theory ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="23px" altimg-valign="-7px" altimg-width="55px" alttext="d_{\NVar{k}}\left(\NVar{n}\right)" display="inline"><mrow><msub><mi href="./27.2#SS1.p4">d</mi><mi class="ltx_nvar" href="./27.1#p2.t1.r1">k</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow></math>: divisor function</a>,
<a href="./27.1#p2.t1.r1" title="§27.1 Special Notation ‣ Notation ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m36.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="d" display="inline"><mi href="./27.1#p2.t1.r1">d</mi></math>: positive integer</a> and
<a href="./27.1#p2.t1.r1" title="§27.1 Special Notation ‣ Notation ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./27.1#p2.t1.r1">n</mi></math>: positive integer</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">24.3.3 I.A</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">is the number of divisors of <math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./27.1#p2.t1.r1">n</mi></math> and is the <em class="ltx_emph ltx_font_italic">divisor function</em>. It is the
special case <math class="ltx_Math" altimg="m42.png" altimg-height="18px" altimg-valign="-2px" altimg-width="52px" alttext="k=2" display="inline"><mrow><mi href="./27.1#p2.t1.r1">k</mi><mo>=</mo><mn>2</mn></mrow></math> of the function <math class="ltx_Math" altimg="m40.png" altimg-height="23px" altimg-valign="-7px" altimg-width="55px" alttext="d_{k}\left(n\right)" display="inline"><mrow><msub><mi href="./27.2#SS1.p4">d</mi><mi href="./27.1#p2.t1.r1">k</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow></math> that counts the
number of ways of expressing <math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./27.1#p2.t1.r1">n</mi></math> as the product of <math class="ltx_Math" altimg="m43.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./27.1#p2.t1.r1">k</mi></math> factors, with the order
of factors taken into account.</p>
<table id="E10" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">27.2.10</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E10.png" altimg-height="57px" altimg-valign="-32px" altimg-width="145px" alttext="\sigma_{\alpha}\left(n\right)=\sum_{d\mathbin{|}n}d^{\alpha}," display="block"><mrow><mrow><mrow><msub><mi href="./27.2#E10">σ</mi><mi href="./27.2#SS1.p4">α</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><munder><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./27.1#p2.t1.r1">d</mi><mo stretchy="false">|</mo><mi href="./27.1#p2.t1.r1">n</mi></mrow></munder><msup><mi href="./27.1#p2.t1.r1">d</mi><mi href="./27.2#SS1.p4">α</mi></msup></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E10.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m28.png" altimg-height="23px" altimg-valign="-7px" altimg-width="58px" alttext="\sigma_{\NVar{\alpha}}\left(\NVar{n}\right)" display="inline"><mrow><msub><mi href="./27.2#E10">σ</mi><mi class="ltx_nvar" href="./27.2#SS1.p4">α</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow></math>: sum of powers of divisors of <math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./27.1#p2.t1.r1">n</mi></math></span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./27.1#p2.t1.r1" title="§27.1 Special Notation ‣ Notation ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m36.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="d" display="inline"><mi href="./27.1#p2.t1.r1">d</mi></math>: positive integer</a>,
<a href="./27.1#p2.t1.r1" title="§27.1 Special Notation ‣ Notation ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./27.1#p2.t1.r1">n</mi></math>: positive integer</a> and
<a href="./27.2#SS1.p4" title="§27.2(i) Definitions ‣ §27.2 Functions ‣ Multiplicative Number Theory ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m7.png" altimg-height="13px" altimg-valign="-2px" altimg-width="17px" alttext="\alpha" display="inline"><mi href="./27.2#SS1.p4">α</mi></math>: parameter</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">24.3.3 I.A</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">is the sum of the <math class="ltx_Math" altimg="m7.png" altimg-height="13px" altimg-valign="-2px" altimg-width="17px" alttext="\alpha" display="inline"><mi href="./27.2#SS1.p4">α</mi></math>th powers of the divisors of <math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./27.1#p2.t1.r1">n</mi></math>, where the exponent
<math class="ltx_Math" altimg="m7.png" altimg-height="13px" altimg-valign="-2px" altimg-width="17px" alttext="\alpha" display="inline"><mi href="./27.2#SS1.p4">α</mi></math> can be real or complex. Note that
<math class="ltx_Math" altimg="m27.png" altimg-height="23px" altimg-valign="-7px" altimg-width="123px" alttext="\sigma_{0}\left(n\right)=d\left(n\right)" display="inline"><mrow><mrow><msub><mi href="./27.2#E10">σ</mi><mn>0</mn></msub><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mi href="./27.2#SS1.p4">d</mi><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow></mrow></math>.</p>
<table id="E11" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">27.2.11</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E11.png" altimg-height="57px" altimg-valign="-32px" altimg-width="219px" alttext="J_{k}\left(n\right)=\sum_{\left(\left(d_{1},\dots,d_{k}\right),n\right)=1}1," display="block"><mrow><mrow><mrow><msub><mi href="./27.2#E11">J</mi><mi href="./27.1#p2.t1.r1">k</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><munder><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mrow><mo href="./27.1#p2.t1.r3">(</mo><mrow><mrow><mo href="./27.1#p2.t1.r3">(</mo><mrow><msub><mi href="./27.1#p2.t1.r1">d</mi><mn>1</mn></msub><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><msub><mi href="./27.1#p2.t1.r1">d</mi><mi href="./27.1#p2.t1.r1">k</mi></msub></mrow><mo href="./27.1#p2.t1.r3">)</mo></mrow><mo>,</mo><mi href="./27.1#p2.t1.r1">n</mi></mrow><mo href="./27.1#p2.t1.r3">)</mo></mrow><mo>=</mo><mn>1</mn></mrow></munder><mn>1</mn></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E11.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m4.png" altimg-height="23px" altimg-valign="-7px" altimg-width="56px" alttext="J_{\NVar{k}}\left(\NVar{n}\right)" display="inline"><mrow><msub><mi href="./27.2#E11">J</mi><mi class="ltx_nvar" href="./27.1#p2.t1.r1">k</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow></math>: Jordan’s function</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./27.1#p2.t1.r3" title="§27.1 Special Notation ‣ Notation ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m9.png" altimg-height="23px" altimg-valign="-7px" altimg-width="58px" alttext="\left(\NVar{m},\NVar{n}\right)" display="inline"><mrow><mo href="./27.1#p2.t1.r3">(</mo><mrow><mi class="ltx_nvar" href="./27.1#p2.t1.r1">m</mi><mo>,</mo><mi class="ltx_nvar" href="./27.1#p2.t1.r1">n</mi></mrow><mo href="./27.1#p2.t1.r3">)</mo></mrow></math>: greatest common divisor (gcd)</a>,
<a href="./27.1#p2.t1.r1" title="§27.1 Special Notation ‣ Notation ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m36.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="d" display="inline"><mi href="./27.1#p2.t1.r1">d</mi></math>: positive integer</a>,
<a href="./27.1#p2.t1.r1" title="§27.1 Special Notation ‣ Notation ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m43.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./27.1#p2.t1.r1">k</mi></math>: positive integer</a> and
<a href="./27.1#p2.t1.r1" title="§27.1 Special Notation ‣ Notation ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./27.1#p2.t1.r1">n</mi></math>: positive integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">is the number of <math class="ltx_Math" altimg="m43.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./27.1#p2.t1.r1">k</mi></math>-tuples of integers <math class="ltx_Math" altimg="m12.png" altimg-height="19px" altimg-valign="-5px" altimg-width="37px" alttext="\leq n" display="inline"><mrow><mi></mi><mo>≤</mo><mi href="./27.1#p2.t1.r1">n</mi></mrow></math> whose greatest common divisor
is relatively prime to <math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./27.1#p2.t1.r1">n</mi></math>. This is <em class="ltx_emph ltx_font_italic">Jordan’s function</em>. Note that
<math class="ltx_Math" altimg="m3.png" altimg-height="23px" altimg-valign="-7px" altimg-width="124px" alttext="J_{1}\left(n\right)=\phi\left(n\right)" display="inline"><mrow><mrow><msub><mi href="./27.2#E11">J</mi><mn>1</mn></msub><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mi href="./27.2#E7">ϕ</mi><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow></mrow></math>.</p>
</div>
<div id="SS1.p5" class="ltx_para">
<p class="ltx_p">In the following examples, <math class="ltx_Math" altimg="m34.png" altimg-height="20px" altimg-valign="-9px" altimg-width="111px" alttext="a_{1},\dots,a_{\nu\left(n\right)}" display="inline"><mrow><msub><mi href="./27.2#SS1.p4">a</mi><mn>1</mn></msub><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><msub><mi href="./27.2#SS1.p4">a</mi><mrow><mi href="./27.2#SS1.p1">ν</mi><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow></msub></mrow></math> are the exponents
in the factorization of <math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./27.1#p2.t1.r1">n</mi></math> in ().</p>
</div>
<div id="SS1.p6" class="ltx_para">
<table id="E12" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">27.2.12</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E12.png" altimg-height="92px" altimg-valign="-40px" altimg-width="421px" alttext="\mu\left(n\right)=\begin{cases}1,&n=1,\\
(-1)^{\nu\left(n\right)},&a_{1}=a_{2}=\dots=a_{\nu\left(n\right)}=1,\\
0,&\mbox{otherwise}.\end{cases}" display="block"><mrow><mrow><mi href="./27.2#E12">μ</mi><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mo>{</mo><mtable columnspacing="5pt" displaystyle="true" rowspacing="0pt"><mtr><mtd columnalign="left"><mrow><mn>1</mn><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mrow><mi href="./27.1#p2.t1.r1">n</mi><mo>=</mo><mn>1</mn></mrow><mo>,</mo></mrow></mtd></mtr><mtr><mtd columnalign="left"><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mrow><mi href="./27.2#SS1.p1">ν</mi><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow></msup><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mrow><msub><mi href="./27.2#SS1.p4">a</mi><mn>1</mn></msub><mo>=</mo><msub><mi href="./27.2#SS1.p4">a</mi><mn>2</mn></msub><mo>=</mo><mi mathvariant="normal">…</mi><mo>=</mo><msub><mi href="./27.2#SS1.p4">a</mi><mrow><mi href="./27.2#SS1.p1">ν</mi><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow></msub><mo>=</mo><mn>1</mn></mrow><mo>,</mo></mrow></mtd></mtr><mtr><mtd columnalign="left"><mrow><mn>0</mn><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mtext>otherwise</mtext><mo>.</mo></mrow></mtd></mtr></mtable></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E12.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m16.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="\mu\left(\NVar{n}\right)" display="inline"><mrow><mi href="./27.2#E12">μ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow></math>: Möbius function</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./27.2#SS1.p1" title="§27.2(i) Definitions ‣ §27.2 Functions ‣ Multiplicative Number Theory ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m18.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\nu\left(\NVar{n}\right)" display="inline"><mrow><mi href="./27.2#SS1.p1">ν</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow></math>: number of distinct primes dividing <math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./27.1#p2.t1.r1">n</mi></math></a>,
<a href="./27.1#p2.t1.r1" title="§27.1 Special Notation ‣ Notation ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./27.1#p2.t1.r1">n</mi></math>: positive integer</a> and
<a href="./27.2#SS1.p4" title="§27.2(i) Definitions ‣ §27.2 Functions ‣ Multiplicative Number Theory ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m31.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./27.2#SS1.p4">a</mi></math>: primitive root</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">24.3.1 I.A</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">This is the <em class="ltx_emph ltx_font_italic">Möbius function</em>.</p>
</div>
<div id="SS1.p7" class="ltx_para">
<table id="E13" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">27.2.13</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E13.png" altimg-height="65px" altimg-valign="-27px" altimg-width="307px" alttext="\lambda\left(n\right)=\begin{cases}1,&n=1,\\
(-1)^{a_{1}+\dots+a_{\nu\left(n\right)}},&n>1.\end{cases}" display="block"><mrow><mrow><mi href="./27.2#E13">λ</mi><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mo>{</mo><mtable columnspacing="5pt" displaystyle="true" rowspacing="0pt"><mtr><mtd columnalign="left"><mrow><mn>1</mn><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mrow><mi href="./27.1#p2.t1.r1">n</mi><mo>=</mo><mn>1</mn></mrow><mo>,</mo></mrow></mtd></mtr><mtr><mtd columnalign="left"><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mrow><msub><mi href="./27.2#SS1.p4">a</mi><mn>1</mn></msub><mo>+</mo><mi mathvariant="normal">…</mi><mo>+</mo><msub><mi href="./27.2#SS1.p4">a</mi><mrow><mi href="./27.2#SS1.p1">ν</mi><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow></msub></mrow></msup><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mrow><mi href="./27.1#p2.t1.r1">n</mi><mo>></mo><mn>1</mn></mrow><mo>.</mo></mrow></mtd></mtr></mtable></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E13.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m8.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="\lambda\left(\NVar{n}\right)" display="inline"><mrow><mi href="./27.2#E13">λ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow></math>: Liouville’s function</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./27.2#SS1.p1" title="§27.2(i) Definitions ‣ §27.2 Functions ‣ Multiplicative Number Theory ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m18.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\nu\left(\NVar{n}\right)" display="inline"><mrow><mi href="./27.2#SS1.p1">ν</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow></math>: number of distinct primes dividing <math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./27.1#p2.t1.r1">n</mi></math></a>,
<a href="./27.1#p2.t1.r1" title="§27.1 Special Notation ‣ Notation ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./27.1#p2.t1.r1">n</mi></math>: positive integer</a> and
<a href="./27.2#SS1.p4" title="§27.2(i) Definitions ‣ §27.2 Functions ‣ Multiplicative Number Theory ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m31.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./27.2#SS1.p4">a</mi></math>: primitive root</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">This is <em class="ltx_emph ltx_font_italic">Liouville’s function</em>.</p>
</div>
<div id="SS1.p8" class="ltx_para">
<table id="E14" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">27.2.14</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E14.png" altimg-height="25px" altimg-valign="-7px" altimg-width="113px" alttext="\Lambda\left(n\right)=\ln p," display="block"><mrow><mrow><mrow><mi href="./27.2#E14" mathvariant="normal">Λ</mi><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi href="./27.1#p2.t1.r7">p</mi></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m47.png" altimg-height="21px" altimg-valign="-6px" altimg-width="62px" alttext="n=p^{a}" display="inline"><mrow><mi href="./27.1#p2.t1.r1">n</mi><mo>=</mo><msup><mi href="./27.1#p2.t1.r7">p</mi><mi href="./27.2#SS1.p4">a</mi></msup></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E14.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m5.png" altimg-height="23px" altimg-valign="-7px" altimg-width="49px" alttext="\Lambda\left(\NVar{n}\right)" display="inline"><mrow><mi href="./27.2#E14" mathvariant="normal">Λ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow></math>: Mangoldt’s function</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m13.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a>,
<a href="./27.1#p2.t1.r1" title="§27.1 Special Notation ‣ Notation ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./27.1#p2.t1.r1">n</mi></math>: positive integer</a>,
<a href="./27.1#p2.t1.r7" title="§27.1 Special Notation ‣ Notation ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="16px" altimg-valign="-6px" altimg-width="74px" alttext="p,p_{1},\ldots" display="inline"><mrow><mi href="./27.1#p2.t1.r7">p</mi><mo>,</mo><msub><mi href="./27.1#p2.t1.r7">p</mi><mn>1</mn></msub><mo>,</mo><mi mathvariant="normal">…</mi></mrow></math>: prime numbers</a> and
<a href="./27.2#SS1.p4" title="§27.2(i) Definitions ‣ §27.2 Functions ‣ Multiplicative Number Theory ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m31.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi href="./27.2#SS1.p4">a</mi></math>: primitive root</a>
</dd>
<dt>Change of Notation (effective with 1.0.10):</dt>
<dd>
The notation for logarithm has been changed to <math class="ltx_Math" altimg="m14.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="\ln" display="inline"><mi href="./4.2#E2">ln</mi></math> from <math class="ltx_Math" altimg="m15.png" altimg-height="21px" altimg-valign="-6px" altimg-width="30px" alttext="\mathrm{log}" display="inline"><mi>log</mi></math>.
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m54.png" altimg-height="21px" altimg-valign="-6px" altimg-width="24px" alttext="p^{a}" display="inline"><msup><mi href="./27.1#p2.t1.r7">p</mi><mi href="./27.2#SS1.p4">a</mi></msup></math> is a prime power with <math class="ltx_Math" altimg="m32.png" altimg-height="19px" altimg-valign="-5px" altimg-width="51px" alttext="a\geq 1" display="inline"><mrow><mi href="./27.2#SS1.p4">a</mi><mo>≥</mo><mn>1</mn></mrow></math>; otherwise
<math class="ltx_Math" altimg="m6.png" altimg-height="23px" altimg-valign="-7px" altimg-width="85px" alttext="\Lambda\left(n\right)=0" display="inline"><mrow><mrow><mi href="./27.2#E14" mathvariant="normal">Λ</mi><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow><mo>=</mo><mn>0</mn></mrow></math>. This is <em class="ltx_emph ltx_font_italic">Mangoldt’s function</em>.</p>
</div>
</section>
<section id="ii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§27.2(ii) </span>Tables</h2>
<div id="SS2.info" class="ltx_metadata ltx_info">
lists the first 100 prime numbers <math class="ltx_Math" altimg="m65.png" altimg-height="16px" altimg-valign="-6px" altimg-width="25px" alttext="p_{n}" display="inline"><msub><mi href="./27.1#p2.t1.r7">p</mi><mi href="./27.1#p2.t1.r1">n</mi></msub></math>.
Table tabulates the Euler totient function
<math class="ltx_Math" altimg="m21.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="\phi\left(n\right)" display="inline"><mrow><mi href="./27.2#E7">ϕ</mi><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow></math>, the divisor function <math class="ltx_Math" altimg="m38.png" altimg-height="23px" altimg-valign="-7px" altimg-width="45px" alttext="d\left(n\right)" display="inline"><mrow><mi href="./27.2#SS1.p4">d</mi><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow></math>
(<math class="ltx_Math" altimg="m1.png" altimg-height="23px" altimg-valign="-7px" altimg-width="73px" alttext="=\sigma_{0}(n)" display="inline"><mrow><mi></mi><mo>=</mo><mrow><msub><mi href="./27.2#E10">σ</mi><mn>0</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./27.1#p2.t1.r1">n</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></math>), and the sum of the divisors
<math class="ltx_Math" altimg="m25.png" altimg-height="23px" altimg-valign="-7px" altimg-width="44px" alttext="\sigma(n)" display="inline"><mrow><mi href="./27.2#E10">σ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./27.1#p2.t1.r1">n</mi><mo stretchy="false">)</mo></mrow></mrow></math> (<math class="ltx_Math" altimg="m2.png" altimg-height="23px" altimg-valign="-7px" altimg-width="73px" alttext="=\sigma_{1}(n)" display="inline"><mrow><mi></mi><mo>=</mo><mrow><msub><mi href="./27.2#E10">σ</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./27.1#p2.t1.r1">n</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></math>), for <math class="ltx_Math" altimg="m46.png" altimg-height="23px" altimg-valign="-7px" altimg-width="98px" alttext="n=1(1)52" display="inline"><mrow><mi href="./27.1#p2.t1.r1">n</mi><mo>=</mo><mrow><mn>1</mn><mo></mo><mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mo></mo><mn>52</mn></mrow></mrow></math>.</p>
</div>
<figure id="T1" class="ltx_table">
<figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_table">Table 27.2.1: </span>Primes.
</figcaption>
<table id="T1.t1" class="ltx_tabular ltx_centering ltx_guessed_headers ltx_align_middle">
<thead class="ltx_thead">
<tr id="T1.t1.r1" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_border_tt" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;"><math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./27.1#p2.t1.r1">n</mi></math></th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_border_tt" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;"><math class="ltx_Math" altimg="m65.png" altimg-height="16px" altimg-valign="-6px" altimg-width="25px" alttext="p_{n}" display="inline"><msub><mi href="./27.1#p2.t1.r7">p</mi><mi href="./27.1#p2.t1.r1">n</mi></msub></math></th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_border_tt" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;"><math class="ltx_Math" altimg="m56.png" altimg-height="17px" altimg-valign="-7px" altimg-width="53px" alttext="p_{n+10}" display="inline"><msub><mi href="./27.1#p2.t1.r7">p</mi><mrow><mi href="./27.1#p2.t1.r1">n</mi><mo>+</mo><mn>10</mn></mrow></msub></math></th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_border_tt" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;"><math class="ltx_Math" altimg="m57.png" altimg-height="17px" altimg-valign="-7px" altimg-width="53px" alttext="p_{n+20}" display="inline"><msub><mi href="./27.1#p2.t1.r7">p</mi><mrow><mi href="./27.1#p2.t1.r1">n</mi><mo>+</mo><mn>20</mn></mrow></msub></math></th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_border_tt" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;"><math class="ltx_Math" altimg="m58.png" altimg-height="17px" altimg-valign="-7px" altimg-width="53px" alttext="p_{n+30}" display="inline"><msub><mi href="./27.1#p2.t1.r7">p</mi><mrow><mi href="./27.1#p2.t1.r1">n</mi><mo>+</mo><mn>30</mn></mrow></msub></math></th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_border_tt" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;"><math class="ltx_Math" altimg="m59.png" altimg-height="17px" altimg-valign="-7px" altimg-width="53px" alttext="p_{n+40}" display="inline"><msub><mi href="./27.1#p2.t1.r7">p</mi><mrow><mi href="./27.1#p2.t1.r1">n</mi><mo>+</mo><mn>40</mn></mrow></msub></math></th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_border_tt" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;"><math class="ltx_Math" altimg="m60.png" altimg-height="17px" altimg-valign="-7px" altimg-width="53px" alttext="p_{n+50}" display="inline"><msub><mi href="./27.1#p2.t1.r7">p</mi><mrow><mi href="./27.1#p2.t1.r1">n</mi><mo>+</mo><mn>50</mn></mrow></msub></math></th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_border_tt" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;"><math class="ltx_Math" altimg="m61.png" altimg-height="17px" altimg-valign="-7px" altimg-width="53px" alttext="p_{n+60}" display="inline"><msub><mi href="./27.1#p2.t1.r7">p</mi><mrow><mi href="./27.1#p2.t1.r1">n</mi><mo>+</mo><mn>60</mn></mrow></msub></math></th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_border_tt" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;"><math class="ltx_Math" altimg="m62.png" altimg-height="17px" altimg-valign="-7px" altimg-width="53px" alttext="p_{n+70}" display="inline"><msub><mi href="./27.1#p2.t1.r7">p</mi><mrow><mi href="./27.1#p2.t1.r1">n</mi><mo>+</mo><mn>70</mn></mrow></msub></math></th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_border_tt" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;"><math class="ltx_Math" altimg="m63.png" altimg-height="17px" altimg-valign="-7px" altimg-width="53px" alttext="p_{n+80}" display="inline"><msub><mi href="./27.1#p2.t1.r7">p</mi><mrow><mi href="./27.1#p2.t1.r1">n</mi><mo>+</mo><mn>80</mn></mrow></msub></math></th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_border_tt" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;"><math class="ltx_Math" altimg="m64.png" altimg-height="17px" altimg-valign="-7px" altimg-width="53px" alttext="p_{n+90}" display="inline"><msub><mi href="./27.1#p2.t1.r7">p</mi><mrow><mi href="./27.1#p2.t1.r1">n</mi><mo>+</mo><mn>90</mn></mrow></msub></math></th>
</tr>
</thead>
<tbody class="ltx_tbody">
<tr id="T1.t1.r2" class="ltx_tr">
<td class="ltx_td ltx_align_right ltx_border_t" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">1</td>
<td class="ltx_td ltx_align_right ltx_border_t" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">2</td>
<td class="ltx_td ltx_align_right ltx_border_t" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">31</td>
<td class="ltx_td ltx_align_right ltx_border_t" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">73</td>
<td class="ltx_td ltx_align_right ltx_border_t" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">127</td>
<td class="ltx_td ltx_align_right ltx_border_t" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">179</td>
<td class="ltx_td ltx_align_right ltx_border_t" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">233</td>
<td class="ltx_td ltx_align_right ltx_border_t" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">283</td>
<td class="ltx_td ltx_align_right ltx_border_t" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">353</td>
<td class="ltx_td ltx_align_right ltx_border_t" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">419</td>
<td class="ltx_td ltx_align_right ltx_border_t" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">467</td>
</tr>
<tr id="T1.t1.r3" class="ltx_tr">
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">2</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">3</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">37</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">79</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">131</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">181</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">239</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">293</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">359</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">421</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">479</td>
</tr>
<tr id="T1.t1.r4" class="ltx_tr">
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">3</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">5</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">41</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">83</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">137</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">191</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">241</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">307</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">367</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">431</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">487</td>
</tr>
<tr id="T1.t1.r5" class="ltx_tr">
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">4</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">7</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">43</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">89</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">139</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">193</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">251</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">311</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">373</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">433</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">491</td>
</tr>
<tr id="T1.t1.r6" class="ltx_tr">
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">5</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">11</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">47</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">97</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">149</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">197</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">257</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">313</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">379</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">439</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">499</td>
</tr>
<tr id="T1.t1.r7" class="ltx_tr">
<td class="ltx_td ltx_align_right ltx_border_T" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">6</td>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">13</td>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">53</td>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">101</td>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">151</td>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">199</td>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">263</td>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">317</td>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">383</td>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">443</td>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">503</td>
</tr>
<tr id="T1.t1.r8" class="ltx_tr">
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">7</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">17</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">59</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">103</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">157</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">211</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">269</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">331</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">389</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">449</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">509</td>
</tr>
<tr id="T1.t1.r9" class="ltx_tr">
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">8</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">19</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">61</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">107</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">163</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">223</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">271</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">337</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">397</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">457</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">521</td>
</tr>
<tr id="T1.t1.r10" class="ltx_tr">
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">9</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">23</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">67</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">109</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">167</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">227</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">277</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">347</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">401</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">461</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">523</td>
</tr>
<tr id="T1.t1.r11" class="ltx_tr">
<td class="ltx_td ltx_align_right ltx_border_B" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">10</td>
<td class="ltx_td ltx_align_right ltx_border_B" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">29</td>
<td class="ltx_td ltx_align_right ltx_border_B" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">71</td>
<td class="ltx_td ltx_align_right ltx_border_B" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">113</td>
<td class="ltx_td ltx_align_right ltx_border_B" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">173</td>
<td class="ltx_td ltx_align_right ltx_border_B" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">229</td>
<td class="ltx_td ltx_align_right ltx_border_B" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">281</td>
<td class="ltx_td ltx_align_right ltx_border_B" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">349</td>
<td class="ltx_td ltx_align_right ltx_border_B" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">409</td>
<td class="ltx_td ltx_align_right ltx_border_B" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">463</td>
<td class="ltx_td ltx_align_right ltx_border_B" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">541</td>
</tr>
</tbody>
</table>
<div id="T1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./27.1#p2.t1.r1" title="§27.1 Special Notation ‣ Notation ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./27.1#p2.t1.r1">n</mi></math>: positive integer</a> and
<a href="./27.1#p2.t1.r7" title="§27.1 Special Notation ‣ Notation ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="16px" altimg-valign="-6px" altimg-width="74px" alttext="p,p_{1},\ldots" display="inline"><mrow><mi href="./27.1#p2.t1.r7">p</mi><mo>,</mo><msub><mi href="./27.1#p2.t1.r7">p</mi><mn>1</mn></msub><mo>,</mo><mi mathvariant="normal">…</mi></mrow></math>: prime numbers</a>
</dd>
</dl>
</div>
</div>
</figure>
<figure id="T2" class="ltx_table">
<figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_table">Table 27.2.2: </span>Functions related to division.
</figcaption>
<table id="T2.t1" class="ltx_tabular ltx_guessed_headers ltx_align_middle">
<thead class="ltx_thead">
<tr id="T2.t1.r1" class="ltx_tr">
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_border_tt" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;"><math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./27.1#p2.t1.r1">n</mi></math></th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_border_tt" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;"><math class="ltx_Math" altimg="m21.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="\phi\left(n\right)" display="inline"><mrow><mi href="./27.2#E7">ϕ</mi><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow></math></th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_border_tt" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;"><math class="ltx_Math" altimg="m38.png" altimg-height="23px" altimg-valign="-7px" altimg-width="45px" alttext="d\left(n\right)" display="inline"><mrow><mi href="./27.2#SS1.p4">d</mi><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow></math></th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_border_r ltx_border_tt" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;"><math class="ltx_Math" altimg="m26.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="\sigma\left(n\right)" display="inline"><mrow><mi href="./27.2#E10">σ</mi><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow></math></th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_border_tt" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;"><math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./27.1#p2.t1.r1">n</mi></math></th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_border_tt" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;"><math class="ltx_Math" altimg="m21.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="\phi\left(n\right)" display="inline"><mrow><mi href="./27.2#E7">ϕ</mi><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow></math></th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_border_tt" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;"><math class="ltx_Math" altimg="m38.png" altimg-height="23px" altimg-valign="-7px" altimg-width="45px" alttext="d\left(n\right)" display="inline"><mrow><mi href="./27.2#SS1.p4">d</mi><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow></math></th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_border_r ltx_border_tt" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;"><math class="ltx_Math" altimg="m26.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="\sigma\left(n\right)" display="inline"><mrow><mi href="./27.2#E10">σ</mi><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow></math></th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_border_tt" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;"><math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./27.1#p2.t1.r1">n</mi></math></th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_border_tt" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;"><math class="ltx_Math" altimg="m21.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="\phi\left(n\right)" display="inline"><mrow><mi href="./27.2#E7">ϕ</mi><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow></math></th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_border_tt" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;"><math class="ltx_Math" altimg="m38.png" altimg-height="23px" altimg-valign="-7px" altimg-width="45px" alttext="d\left(n\right)" display="inline"><mrow><mi href="./27.2#SS1.p4">d</mi><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow></math></th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_border_r ltx_border_tt" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;"><math class="ltx_Math" altimg="m26.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="\sigma\left(n\right)" display="inline"><mrow><mi href="./27.2#E10">σ</mi><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow></math></th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_border_tt" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;"><math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./27.1#p2.t1.r1">n</mi></math></th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_border_tt" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;"><math class="ltx_Math" altimg="m21.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="\phi\left(n\right)" display="inline"><mrow><mi href="./27.2#E7">ϕ</mi><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow></math></th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_border_tt" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;"><math class="ltx_Math" altimg="m38.png" altimg-height="23px" altimg-valign="-7px" altimg-width="45px" alttext="d\left(n\right)" display="inline"><mrow><mi href="./27.2#SS1.p4">d</mi><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow></math></th>
<th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_border_tt" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;"><math class="ltx_Math" altimg="m26.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="\sigma\left(n\right)" display="inline"><mrow><mi href="./27.2#E10">σ</mi><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow></math></th>
</tr>
</thead>
<tbody class="ltx_tbody">
<tr id="T2.t1.r2" class="ltx_tr">
<td class="ltx_td ltx_align_right ltx_border_t" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">1</td>
<td class="ltx_td ltx_align_right ltx_border_t" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">1</td>
<td class="ltx_td ltx_align_right ltx_border_t" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">1</td>
<td class="ltx_td ltx_align_right ltx_border_r ltx_border_t" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">1</td>
<td class="ltx_td ltx_align_right ltx_border_t" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">14</td>
<td class="ltx_td ltx_align_right ltx_border_t" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">6</td>
<td class="ltx_td ltx_align_right ltx_border_t" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">4</td>
<td class="ltx_td ltx_align_right ltx_border_r ltx_border_t" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">24</td>
<td class="ltx_td ltx_align_right ltx_border_t" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">27</td>
<td class="ltx_td ltx_align_right ltx_border_t" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">18</td>
<td class="ltx_td ltx_align_right ltx_border_t" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">4</td>
<td class="ltx_td ltx_align_right ltx_border_r ltx_border_t" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">40</td>
<td class="ltx_td ltx_align_right ltx_border_t" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">40</td>
<td class="ltx_td ltx_align_right ltx_border_t" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">16</td>
<td class="ltx_td ltx_align_right ltx_border_t" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">8</td>
<td class="ltx_td ltx_align_right ltx_border_t" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">90</td>
</tr>
<tr id="T2.t1.r3" class="ltx_tr">
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">2</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">1</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">2</td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">3</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">15</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">8</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">4</td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">24</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">28</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">12</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">6</td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">56</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">41</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">40</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">2</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">42</td>
</tr>
<tr id="T2.t1.r4" class="ltx_tr">
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">3</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">2</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">2</td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">4</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">16</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">8</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">5</td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">31</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">29</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">28</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">2</td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">30</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">42</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">12</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">8</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">96</td>
</tr>
<tr id="T2.t1.r5" class="ltx_tr">
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">4</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">2</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">3</td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">7</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">17</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">16</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">2</td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">18</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">30</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">8</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">8</td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">72</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">43</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">42</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">2</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">44</td>
</tr>
<tr id="T2.t1.r6" class="ltx_tr">
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">5</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">4</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">2</td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">6</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">18</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">6</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">6</td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">39</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">31</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">30</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">2</td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">32</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">44</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">20</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">6</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">84</td>
</tr>
<tr id="T2.t1.r7" class="ltx_tr">
<td class="ltx_td ltx_align_right ltx_border_T" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">6</td>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">2</td>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">4</td>
<td class="ltx_td ltx_align_right ltx_border_T ltx_border_r" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">12</td>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">19</td>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">18</td>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">2</td>
<td class="ltx_td ltx_align_right ltx_border_T ltx_border_r" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">20</td>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">32</td>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">16</td>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">6</td>
<td class="ltx_td ltx_align_right ltx_border_T ltx_border_r" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">63</td>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">45</td>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">24</td>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">6</td>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">78</td>
</tr>
<tr id="T2.t1.r8" class="ltx_tr">
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">7</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">6</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">2</td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">8</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">20</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">8</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">6</td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">42</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">33</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">20</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">4</td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">48</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">46</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">22</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">4</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">72</td>
</tr>
<tr id="T2.t1.r9" class="ltx_tr">
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">8</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">4</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">4</td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">15</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">21</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">12</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">4</td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">32</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">34</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">16</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">4</td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">54</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">47</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">46</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">2</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">48</td>
</tr>
<tr id="T2.t1.r10" class="ltx_tr">
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">9</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">6</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">3</td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">13</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">22</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">10</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">4</td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">36</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">35</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">24</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">4</td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">48</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">48</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">16</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">10</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">124</td>
</tr>
<tr id="T2.t1.r11" class="ltx_tr">
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">10</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">4</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">4</td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">18</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">23</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">22</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">2</td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">24</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">36</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">12</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">9</td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">91</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">49</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">42</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">3</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">57</td>
</tr>
<tr id="T2.t1.r12" class="ltx_tr">
<td class="ltx_td ltx_align_right ltx_border_T" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">11</td>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">10</td>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">2</td>
<td class="ltx_td ltx_align_right ltx_border_T ltx_border_r" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">12</td>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">24</td>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">8</td>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">8</td>
<td class="ltx_td ltx_align_right ltx_border_T ltx_border_r" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">60</td>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">37</td>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">36</td>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">2</td>
<td class="ltx_td ltx_align_right ltx_border_T ltx_border_r" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">38</td>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">50</td>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">20</td>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">6</td>
<td class="ltx_td ltx_align_right ltx_border_T" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">93</td>
</tr>
<tr id="T2.t1.r13" class="ltx_tr">
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">12</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">4</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">6</td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">28</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">25</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">20</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">3</td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">31</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">38</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">18</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">4</td>
<td class="ltx_td ltx_align_right ltx_border_r" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">60</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">51</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">32</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">4</td>
<td class="ltx_td ltx_align_right" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">72</td>
</tr>
<tr id="T2.t1.r14" class="ltx_tr">
<td class="ltx_td ltx_align_right ltx_border_b" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">13</td>
<td class="ltx_td ltx_align_right ltx_border_b" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">12</td>
<td class="ltx_td ltx_align_right ltx_border_b" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">2</td>
<td class="ltx_td ltx_align_right ltx_border_b ltx_border_r" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">14</td>
<td class="ltx_td ltx_align_right ltx_border_b" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">26</td>
<td class="ltx_td ltx_align_right ltx_border_b" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">12</td>
<td class="ltx_td ltx_align_right ltx_border_b" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">4</td>
<td class="ltx_td ltx_align_right ltx_border_b ltx_border_r" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">42</td>
<td class="ltx_td ltx_align_right ltx_border_b" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">39</td>
<td class="ltx_td ltx_align_right ltx_border_b" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">24</td>
<td class="ltx_td ltx_align_right ltx_border_b" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">4</td>
<td class="ltx_td ltx_align_right ltx_border_b ltx_border_r" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">56</td>
<td class="ltx_td ltx_align_right ltx_border_b" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">52</td>
<td class="ltx_td ltx_align_right ltx_border_b" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">24</td>
<td class="ltx_td ltx_align_right ltx_border_b" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">6</td>
<td class="ltx_td ltx_align_right ltx_border_b" style="padding-top:1.388888888888889px;padding-bottom:1.388888888888889px;">98</td>
</tr>
</tbody>
</table>
<div id="T2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./27.2#E7" title="(27.2.7) ‣ §27.2(i) Definitions ‣ §27.2 Functions ‣ Multiplicative Number Theory ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="\phi\left(\NVar{n}\right)" display="inline"><mrow><mi href="./27.2#E7">ϕ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow></math>: Euler’s totient</a>,
<a href="./27.2#SS1.p4" title="§27.2(i) Definitions ‣ §27.2 Functions ‣ Multiplicative Number Theory ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="23px" altimg-valign="-7px" altimg-width="55px" alttext="d_{\NVar{k}}\left(\NVar{n}\right)" display="inline"><mrow><msub><mi href="./27.2#SS1.p4">d</mi><mi class="ltx_nvar" href="./27.1#p2.t1.r1">k</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow></math>: divisor function</a>,
<a href="./27.2#E10" title="(27.2.10) ‣ §27.2(i) Definitions ‣ §27.2 Functions ‣ Multiplicative Number Theory ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m28.png" altimg-height="23px" altimg-valign="-7px" altimg-width="58px" alttext="\sigma_{\NVar{\alpha}}\left(\NVar{n}\right)" display="inline"><mrow><msub><mi href="./27.2#E10">σ</mi><mi class="ltx_nvar" href="./27.2#SS1.p4">α</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow></math>: sum of powers of divisors of <math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./27.1#p2.t1.r1">n</mi></math></a> and
<a href="./27.1#p2.t1.r1" title="§27.1 Special Notation ‣ Notation ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./27.1#p2.t1.r1">n</mi></math>: positive integer</a>
</dd>
</dl>
</div>
</div>
</figure>
</section>
</section>
</div>
<div class="ltx_page_footer">
<span></div>
</div>
</body></text>
</html>
</page>
<page>
<!DOCTYPE html><html>
<head>
<title>DLMF: 27.4 Euler Products and Dirichlet Series</title>
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<text><body class="color_default textfont_default titlefont_default fontsize_default navbar_default">
<div class="ltx_page_navbar">
<div class="ltx_page_navlogo"></dd>
</dl>
</div>
</div>
<div id="p1" class="ltx_para">
<p class="ltx_p">The fundamental theorem of arithmetic is linked to analysis through the concept
of the Euler product. Every multiplicative <math class="ltx_Math" altimg="m37.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi href="./27.4#p1">f</mi></math> satisfies the identity
</p>
<table id="E1" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">27.4.1</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E1.png" altimg-height="68px" altimg-valign="-30px" altimg-width="292px" alttext="\sum_{n=1}^{\infty}f(n)=\prod_{p}\left(1+\sum_{r=1}^{\infty}f(p^{r})\right)," display="block"><mrow><mrow><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./27.1#p2.t1.r1">n</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><mi href="./27.4#p1">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./27.1#p2.t1.r1">n</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>=</mo><mrow><munder><mo largeop="true" movablelimits="false" symmetric="true">∏</mo><mi href="./27.1#p2.t1.r7">p</mi></munder><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi>r</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><mi href="./27.4#p1">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><msup><mi href="./27.1#p2.t1.r7">p</mi><mi>r</mi></msup><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./27.1#p2.t1.r1" title="§27.1 Special Notation ‣ Notation ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./27.1#p2.t1.r1">n</mi></math>: positive integer</a>,
<a href="./27.1#p2.t1.r7" title="§27.1 Special Notation ‣ Notation ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m40.png" altimg-height="16px" altimg-valign="-6px" altimg-width="74px" alttext="p,p_{1},\ldots" display="inline"><mrow><mi href="./27.1#p2.t1.r7">p</mi><mo>,</mo><msub><mi href="./27.1#p2.t1.r7">p</mi><mn>1</mn></msub><mo>,</mo><mi mathvariant="normal">…</mi></mrow></math>: prime numbers</a> and
<a href="./27.4#p1" title="§27.4 Euler Products and Dirichlet Series ‣ Multiplicative Number Theory ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi href="./27.4#p1">f</mi></math>: multiplicative function</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">if the series on the left is absolutely convergent. In this case the infinite
product on the right (extended over all primes <math class="ltx_Math" altimg="m41.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="p" display="inline"><mi href="./27.1#p2.t1.r7">p</mi></math>) is also absolutely
convergent and is called the <em class="ltx_emph ltx_font_italic">Euler product</em> of the series. If <math class="ltx_Math" altimg="m36.png" altimg-height="23px" altimg-valign="-7px" altimg-width="44px" alttext="f(n)" display="inline"><mrow><mi href="./27.4#p1">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./27.1#p2.t1.r1">n</mi><mo stretchy="false">)</mo></mrow></mrow></math> is
completely multiplicative, then each factor in the product is a geometric
series and the Euler product becomes</p>
<table id="E2" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">27.4.2</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E2.png" altimg-height="66px" altimg-valign="-30px" altimg-width="249px" alttext="\sum_{n=1}^{\infty}f(n)=\prod_{p}(1-f(p))^{-1}." display="block"><mrow><mrow><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./27.1#p2.t1.r1">n</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><mi href="./27.4#p1">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./27.1#p2.t1.r1">n</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>=</mo><mrow><munder><mo largeop="true" movablelimits="false" symmetric="true">∏</mo><mi href="./27.1#p2.t1.r7">p</mi></munder><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><mrow><mi href="./27.4#p1">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./27.1#p2.t1.r7">p</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mn>1</mn></mrow></msup></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./27.1#p2.t1.r1" title="§27.1 Special Notation ‣ Notation ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./27.1#p2.t1.r1">n</mi></math>: positive integer</a>,
<a href="./27.1#p2.t1.r7" title="§27.1 Special Notation ‣ Notation ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m40.png" altimg-height="16px" altimg-valign="-6px" altimg-width="74px" alttext="p,p_{1},\ldots" display="inline"><mrow><mi href="./27.1#p2.t1.r7">p</mi><mo>,</mo><msub><mi href="./27.1#p2.t1.r7">p</mi><mn>1</mn></msub><mo>,</mo><mi mathvariant="normal">…</mi></mrow></math>: prime numbers</a> and
<a href="./27.4#p1" title="§27.4 Euler Products and Dirichlet Series ‣ Multiplicative Number Theory ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi href="./27.4#p1">f</mi></math>: multiplicative function</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="p2" class="ltx_para">
<p class="ltx_p">Euler products are used to find series that generate many functions of
multiplicative number theory. The completely multiplicative function
<math class="ltx_Math" altimg="m35.png" altimg-height="24px" altimg-valign="-7px" altimg-width="103px" alttext="f(n)=n^{-s}" display="inline"><mrow><mrow><mi href="./27.4#p1">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./27.1#p2.t1.r1">n</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><msup><mi href="./27.1#p2.t1.r1">n</mi><mrow><mo>-</mo><mi>s</mi></mrow></msup></mrow></math> gives the Euler product representation of the
<em class="ltx_emph ltx_font_italic">Riemann zeta function</em> <math class="ltx_Math" altimg="m33.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="\zeta\left(s\right)" display="inline"><mrow><mi href="./25.2#E1">ζ</mi><mo></mo><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></mrow></math> (§):
</p>
<table id="E3" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">27.4.3</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E3.png" altimg-height="66px" altimg-valign="-30px" altimg-width="297px" alttext="\zeta\left(s\right)=\sum_{n=1}^{\infty}n^{-s}=\prod_{p}(1-p^{-s})^{-1}," display="block"><mrow><mrow><mrow><mi href="./25.2#E1">ζ</mi><mo></mo><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./27.1#p2.t1.r1">n</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover><msup><mi href="./27.1#p2.t1.r1">n</mi><mrow><mo>-</mo><mi>s</mi></mrow></msup></mrow><mo>=</mo><mrow><munder><mo largeop="true" movablelimits="false" symmetric="true">∏</mo><mi href="./27.1#p2.t1.r7">p</mi></munder><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><msup><mi href="./27.1#p2.t1.r7">p</mi><mrow><mo>-</mo><mi>s</mi></mrow></msup></mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mn>1</mn></mrow></msup></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m19.png" altimg-height="18px" altimg-valign="-3px" altimg-width="65px" alttext="\Re s>1" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>s</mi></mrow><mo>></mo><mn>1</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./25.2#E1" title="(25.2.1) ‣ §25.2(i) Definition ‣ §25.2 Definition and Expansions ‣ Riemann Zeta Function ‣ Chapter 25 Zeta and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m32.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="\zeta\left(\NVar{s}\right)" display="inline"><mrow><mi href="./25.2#E1">ζ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./25.1#p2.t1.r5">s</mi><mo>)</mo></mrow></mrow></math>: Riemann zeta function</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./27.1#p2.t1.r1" title="§27.1 Special Notation ‣ Notation ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./27.1#p2.t1.r1">n</mi></math>: positive integer</a> and
<a href="./27.1#p2.t1.r7" title="§27.1 Special Notation ‣ Notation ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m40.png" altimg-height="16px" altimg-valign="-6px" altimg-width="74px" alttext="p,p_{1},\ldots" display="inline"><mrow><mi href="./27.1#p2.t1.r7">p</mi><mo>,</mo><msub><mi href="./27.1#p2.t1.r7">p</mi><mn>1</mn></msub><mo>,</mo><mi mathvariant="normal">…</mi></mrow></math>: prime numbers</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">23.2.1 and 23.2.2</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="p3" class="ltx_para">
<p class="ltx_p">The Riemann zeta function is the prototype of series of the form
</p>
<table id="E4" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">27.4.4</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E4.png" altimg-height="64px" altimg-valign="-27px" altimg-width="184px" alttext="F(s)=\sum_{n=1}^{\infty}f(n)n^{-s}," display="block"><mrow><mrow><mrow><mi href="./27.4#p3">F</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./27.1#p2.t1.r1">n</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><mrow><mi href="./27.4#p1">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./27.1#p2.t1.r1">n</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><msup><mi href="./27.1#p2.t1.r1">n</mi><mrow><mo>-</mo><mi>s</mi></mrow></msup></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E4.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./27.1#p2.t1.r1" title="§27.1 Special Notation ‣ Notation ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./27.1#p2.t1.r1">n</mi></math>: positive integer</a>,
<a href="./27.4#p1" title="§27.4 Euler Products and Dirichlet Series ‣ Multiplicative Number Theory ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi href="./27.4#p1">f</mi></math>: multiplicative function</a> and
<a href="./27.4#p3" title="§27.4 Euler Products and Dirichlet Series ‣ Multiplicative Number Theory ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m17.png" altimg-height="23px" altimg-valign="-7px" altimg-width="45px" alttext="F(s)" display="inline"><mrow><mi href="./27.4#p3">F</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mrow></mrow></math>: generating function</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">called <em class="ltx_emph ltx_font_italic">Dirichlet series</em> with coefficients <math class="ltx_Math" altimg="m36.png" altimg-height="23px" altimg-valign="-7px" altimg-width="44px" alttext="f(n)" display="inline"><mrow><mi href="./27.4#p1">f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./27.1#p2.t1.r1">n</mi><mo stretchy="false">)</mo></mrow></mrow></math>. The function <math class="ltx_Math" altimg="m17.png" altimg-height="23px" altimg-valign="-7px" altimg-width="45px" alttext="F(s)" display="inline"><mrow><mi href="./27.4#p3">F</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mrow></mrow></math> is
a <em class="ltx_emph ltx_font_italic">generating function</em>, or more precisely, a <em class="ltx_emph ltx_font_italic">Dirichlet generating
function</em>, for the coefficients. The following examples have generating
functions related to the zeta function:</p>
<table id="EGx1" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E5">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">27.4.5</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m12.png" altimg-height="64px" altimg-valign="-27px" altimg-width="123px" alttext="\displaystyle\sum_{n=1}^{\infty}\mu\left(n\right)n^{-s}" display="inline"><mrow><mstyle displaystyle="true"><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./27.1#p2.t1.r1">n</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover></mstyle><mrow><mrow><mi href="./27.2#E12">μ</mi><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow><mo></mo><msup><mi href="./27.1#p2.t1.r1">n</mi><mrow><mo>-</mo><mi>s</mi></mrow></msup></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m5.png" altimg-height="51px" altimg-valign="-21px" altimg-width="76px" alttext="\displaystyle=\frac{1}{\zeta\left(s\right)}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mstyle displaystyle="true"><mfrac><mn>1</mn><mrow><mi href="./25.2#E1">ζ</mi><mo></mo><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></mrow></mfrac></mstyle></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m19.png" altimg-height="18px" altimg-valign="-3px" altimg-width="65px" alttext="\Re s>1" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>s</mi></mrow><mo>></mo><mn>1</mn></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E5.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./27.2#E12" title="(27.2.12) ‣ §27.2(i) Definitions ‣ §27.2 Functions ‣ Multiplicative Number Theory ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m27.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="\mu\left(\NVar{n}\right)" display="inline"><mrow><mi href="./27.2#E12">μ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow></math>: Möbius function</a>,
<a href="./25.2#E1" title="(25.2.1) ‣ §25.2(i) Definition ‣ §25.2 Definition and Expansions ‣ Riemann Zeta Function ‣ Chapter 25 Zeta and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m32.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="\zeta\left(\NVar{s}\right)" display="inline"><mrow><mi href="./25.2#E1">ζ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./25.1#p2.t1.r5">s</mi><mo>)</mo></mrow></mrow></math>: Riemann zeta function</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a> and
<a href="./27.1#p2.t1.r1" title="§27.1 Special Notation ‣ Notation ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./27.1#p2.t1.r1">n</mi></math>: positive integer</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">24.3.1 I.B</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E6">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">27.4.6</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m13.png" altimg-height="64px" altimg-valign="-27px" altimg-width="123px" alttext="\displaystyle\sum_{n=1}^{\infty}\phi\left(n\right)n^{-s}" display="inline"><mrow><mstyle displaystyle="true"><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./27.1#p2.t1.r1">n</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover></mstyle><mrow><mrow><mi href="./27.2#E7">ϕ</mi><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow><mo></mo><msup><mi href="./27.1#p2.t1.r1">n</mi><mrow><mo>-</mo><mi>s</mi></mrow></msup></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m7.png" altimg-height="53px" altimg-valign="-21px" altimg-width="110px" alttext="\displaystyle=\frac{\zeta\left(s-1\right)}{\zeta\left(s\right)}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mstyle displaystyle="true"><mfrac><mrow><mi href="./25.2#E1">ζ</mi><mo></mo><mrow><mo>(</mo><mrow><mi>s</mi><mo>-</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mrow><mi href="./25.2#E1">ζ</mi><mo></mo><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></mrow></mfrac></mstyle></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m20.png" altimg-height="18px" altimg-valign="-3px" altimg-width="65px" alttext="\Re s>2" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>s</mi></mrow><mo>></mo><mn>2</mn></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E6.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./27.2#E7" title="(27.2.7) ‣ §27.2(i) Definitions ‣ §27.2 Functions ‣ Multiplicative Number Theory ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="\phi\left(\NVar{n}\right)" display="inline"><mrow><mi href="./27.2#E7">ϕ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow></math>: Euler’s totient</a>,
<a href="./25.2#E1" title="(25.2.1) ‣ §25.2(i) Definition ‣ §25.2 Definition and Expansions ‣ Riemann Zeta Function ‣ Chapter 25 Zeta and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m32.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="\zeta\left(\NVar{s}\right)" display="inline"><mrow><mi href="./25.2#E1">ζ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./25.1#p2.t1.r5">s</mi><mo>)</mo></mrow></mrow></math>: Riemann zeta function</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a> and
<a href="./27.1#p2.t1.r1" title="§27.1 Special Notation ‣ Notation ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./27.1#p2.t1.r1">n</mi></math>: positive integer</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">24.3.2 I.B</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E7">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">27.4.7</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m11.png" altimg-height="64px" altimg-valign="-27px" altimg-width="123px" alttext="\displaystyle\sum_{n=1}^{\infty}\lambda\left(n\right)n^{-s}" display="inline"><mrow><mstyle displaystyle="true"><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./27.1#p2.t1.r1">n</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover></mstyle><mrow><mrow><mi href="./27.2#E13">λ</mi><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow><mo></mo><msup><mi href="./27.1#p2.t1.r1">n</mi><mrow><mo>-</mo><mi>s</mi></mrow></msup></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m6.png" altimg-height="53px" altimg-valign="-21px" altimg-width="86px" alttext="\displaystyle=\frac{\zeta\left(2s\right)}{\zeta\left(s\right)}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mstyle displaystyle="true"><mfrac><mrow><mi href="./25.2#E1">ζ</mi><mo></mo><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><mi>s</mi></mrow><mo>)</mo></mrow></mrow><mrow><mi href="./25.2#E1">ζ</mi><mo></mo><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></mrow></mfrac></mstyle></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m19.png" altimg-height="18px" altimg-valign="-3px" altimg-width="65px" alttext="\Re s>1" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>s</mi></mrow><mo>></mo><mn>1</mn></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E7.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./27.2#E13" title="(27.2.13) ‣ §27.2(i) Definitions ‣ §27.2 Functions ‣ Multiplicative Number Theory ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m23.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="\lambda\left(\NVar{n}\right)" display="inline"><mrow><mi href="./27.2#E13">λ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow></math>: Liouville’s function</a>,
<a href="./25.2#E1" title="(25.2.1) ‣ §25.2(i) Definition ‣ §25.2 Definition and Expansions ‣ Riemann Zeta Function ‣ Chapter 25 Zeta and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m32.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="\zeta\left(\NVar{s}\right)" display="inline"><mrow><mi href="./25.2#E1">ζ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./25.1#p2.t1.r5">s</mi><mo>)</mo></mrow></mrow></math>: Riemann zeta function</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a> and
<a href="./27.1#p2.t1.r1" title="§27.1 Special Notation ‣ Notation ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./27.1#p2.t1.r1">n</mi></math>: positive integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E8">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">27.4.8</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m15.png" altimg-height="64px" altimg-valign="-27px" altimg-width="134px" alttext="\displaystyle\sum_{n=1}^{\infty}|\mu\left(n\right)|n^{-s}" display="inline"><mrow><mstyle displaystyle="true"><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./27.1#p2.t1.r1">n</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover></mstyle><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./27.2#E12">μ</mi><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow><mo></mo><msup><mi href="./27.1#p2.t1.r1">n</mi><mrow><mo>-</mo><mi>s</mi></mrow></msup></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m8.png" altimg-height="53px" altimg-valign="-21px" altimg-width="86px" alttext="\displaystyle=\frac{\zeta\left(s\right)}{\zeta\left(2s\right)}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mstyle displaystyle="true"><mfrac><mrow><mi href="./25.2#E1">ζ</mi><mo></mo><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></mrow><mrow><mi href="./25.2#E1">ζ</mi><mo></mo><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><mi>s</mi></mrow><mo>)</mo></mrow></mrow></mfrac></mstyle></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m19.png" altimg-height="18px" altimg-valign="-3px" altimg-width="65px" alttext="\Re s>1" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>s</mi></mrow><mo>></mo><mn>1</mn></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E8.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./27.2#E12" title="(27.2.12) ‣ §27.2(i) Definitions ‣ §27.2 Functions ‣ Multiplicative Number Theory ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m27.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="\mu\left(\NVar{n}\right)" display="inline"><mrow><mi href="./27.2#E12">μ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow></math>: Möbius function</a>,
<a href="./25.2#E1" title="(25.2.1) ‣ §25.2(i) Definition ‣ §25.2 Definition and Expansions ‣ Riemann Zeta Function ‣ Chapter 25 Zeta and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m32.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="\zeta\left(\NVar{s}\right)" display="inline"><mrow><mi href="./25.2#E1">ζ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./25.1#p2.t1.r5">s</mi><mo>)</mo></mrow></mrow></math>: Riemann zeta function</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a> and
<a href="./27.1#p2.t1.r1" title="§27.1 Special Notation ‣ Notation ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./27.1#p2.t1.r1">n</mi></math>: positive integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E9">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">27.4.9</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m9.png" altimg-height="64px" altimg-valign="-27px" altimg-width="119px" alttext="\displaystyle\sum_{n=1}^{\infty}2^{\nu\left(n\right)}n^{-s}" display="inline"><mrow><mstyle displaystyle="true"><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./27.1#p2.t1.r1">n</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover></mstyle><mrow><msup><mn>2</mn><mrow><mi href="./27.2#SS1.p1">ν</mi><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow></msup><mo></mo><msup><mi href="./27.1#p2.t1.r1">n</mi><mrow><mo>-</mo><mi>s</mi></mrow></msup></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m4.png" altimg-height="55px" altimg-valign="-21px" altimg-width="100px" alttext="\displaystyle=\frac{(\zeta\left(s\right))^{2}}{\zeta\left(2s\right)}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mstyle displaystyle="true"><mfrac><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./25.2#E1">ζ</mi><mo></mo><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mrow><mi href="./25.2#E1">ζ</mi><mo></mo><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><mi>s</mi></mrow><mo>)</mo></mrow></mrow></mfrac></mstyle></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m19.png" altimg-height="18px" altimg-valign="-3px" altimg-width="65px" alttext="\Re s>1" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>s</mi></mrow><mo>></mo><mn>1</mn></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E9.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./25.2#E1" title="(25.2.1) ‣ §25.2(i) Definition ‣ §25.2 Definition and Expansions ‣ Riemann Zeta Function ‣ Chapter 25 Zeta and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m32.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="\zeta\left(\NVar{s}\right)" display="inline"><mrow><mi href="./25.2#E1">ζ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./25.1#p2.t1.r5">s</mi><mo>)</mo></mrow></mrow></math>: Riemann zeta function</a>,
<a href="./27.2#SS1.p1" title="§27.2(i) Definitions ‣ §27.2 Functions ‣ Multiplicative Number Theory ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m28.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\nu\left(\NVar{n}\right)" display="inline"><mrow><mi href="./27.2#SS1.p1">ν</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow></math>: number of distinct primes dividing <math class="ltx_Math" altimg="m39.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./27.1#p2.t1.r1">n</mi></math></a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a> and
<a href="./27.1#p2.t1.r1" title="§27.1 Special Notation ‣ Notation ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./27.1#p2.t1.r1">n</mi></math>: positive integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E10">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">27.4.10</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m14.png" altimg-height="64px" altimg-valign="-27px" altimg-width="131px" alttext="\displaystyle\sum_{n=1}^{\infty}d_{k}\left(n\right)n^{-s}" display="inline"><mrow><mstyle displaystyle="true"><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./27.1#p2.t1.r1">n</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover></mstyle><mrow><mrow><msub><mi href="./27.2#SS1.p4">d</mi><mi href="./27.1#p2.t1.r1">k</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow><mo></mo><msup><mi href="./27.1#p2.t1.r1">n</mi><mrow><mo>-</mo><mi>s</mi></mrow></msup></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m1.png" altimg-height="28px" altimg-valign="-7px" altimg-width="96px" alttext="\displaystyle=(\zeta\left(s\right))^{k}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./25.2#E1">ζ</mi><mo></mo><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow><mi href="./27.1#p2.t1.r1">k</mi></msup></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m19.png" altimg-height="18px" altimg-valign="-3px" altimg-width="65px" alttext="\Re s>1" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>s</mi></mrow><mo>></mo><mn>1</mn></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E10.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./25.2#E1" title="(25.2.1) ‣ §25.2(i) Definition ‣ §25.2 Definition and Expansions ‣ Riemann Zeta Function ‣ Chapter 25 Zeta and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m32.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="\zeta\left(\NVar{s}\right)" display="inline"><mrow><mi href="./25.2#E1">ζ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./25.1#p2.t1.r5">s</mi><mo>)</mo></mrow></mrow></math>: Riemann zeta function</a>,
<a href="./27.2#SS1.p4" title="§27.2(i) Definitions ‣ §27.2 Functions ‣ Multiplicative Number Theory ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m34.png" altimg-height="23px" altimg-valign="-7px" altimg-width="55px" alttext="d_{\NVar{k}}\left(\NVar{n}\right)" display="inline"><mrow><msub><mi href="./27.2#SS1.p4">d</mi><mi class="ltx_nvar" href="./27.1#p2.t1.r1">k</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow></math>: divisor function</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./27.1#p2.t1.r1" title="§27.1 Special Notation ‣ Notation ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m38.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./27.1#p2.t1.r1">k</mi></math>: positive integer</a> and
<a href="./27.1#p2.t1.r1" title="§27.1 Special Notation ‣ Notation ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./27.1#p2.t1.r1">n</mi></math>: positive integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
<table id="E11" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">27.4.11</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E11.png" altimg-height="64px" altimg-valign="-27px" altimg-width="286px" alttext="\sum_{n=1}^{\infty}\sigma_{\alpha}\left(n\right)n^{-s}=\zeta\left(s\right)%
\zeta\left(s-\alpha\right)," display="block"><mrow><mrow><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./27.1#p2.t1.r1">n</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><mrow><msub><mi href="./27.2#E10">σ</mi><mi>α</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow><mo></mo><msup><mi href="./27.1#p2.t1.r1">n</mi><mrow><mo>-</mo><mi>s</mi></mrow></msup></mrow></mrow><mo>=</mo><mrow><mrow><mi href="./25.2#E1">ζ</mi><mo></mo><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./25.2#E1">ζ</mi><mo></mo><mrow><mo>(</mo><mrow><mi>s</mi><mo>-</mo><mi>α</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m21.png" altimg-height="23px" altimg-valign="-7px" altimg-width="188px" alttext="\Re s>\max(1,1+\Re\alpha)" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>s</mi></mrow><mo>></mo><mrow><mi>max</mi><mo></mo><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mrow><mn>1</mn><mo>+</mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>α</mi></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E11.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./25.2#E1" title="(25.2.1) ‣ §25.2(i) Definition ‣ §25.2 Definition and Expansions ‣ Riemann Zeta Function ‣ Chapter 25 Zeta and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m32.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="\zeta\left(\NVar{s}\right)" display="inline"><mrow><mi href="./25.2#E1">ζ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./25.1#p2.t1.r5">s</mi><mo>)</mo></mrow></mrow></math>: Riemann zeta function</a>,
<a href="./27.2#E10" title="(27.2.10) ‣ §27.2(i) Definitions ‣ §27.2 Functions ‣ Multiplicative Number Theory ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m30.png" altimg-height="23px" altimg-valign="-7px" altimg-width="58px" alttext="\sigma_{\NVar{\alpha}}\left(\NVar{n}\right)" display="inline"><mrow><msub><mi href="./27.2#E10">σ</mi><mi class="ltx_nvar" href="./27.2#SS1.p4">α</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow></math>: sum of powers of divisors of <math class="ltx_Math" altimg="m39.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./27.1#p2.t1.r1">n</mi></math></a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a> and
<a href="./27.1#p2.t1.r1" title="§27.1 Special Notation ‣ Notation ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./27.1#p2.t1.r1">n</mi></math>: positive integer</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">24.3.3 I.B</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="EGx2" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E12">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">27.4.12</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m10.png" altimg-height="64px" altimg-valign="-27px" altimg-width="125px" alttext="\displaystyle\sum_{n=1}^{\infty}\Lambda\left(n\right)n^{-s}" display="inline"><mrow><mstyle displaystyle="true"><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./27.1#p2.t1.r1">n</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover></mstyle><mrow><mrow><mi href="./27.2#E14" mathvariant="normal">Λ</mi><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow><mo></mo><msup><mi href="./27.1#p2.t1.r1">n</mi><mrow><mo>-</mo><mi>s</mi></mrow></msup></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m2.png" altimg-height="53px" altimg-valign="-21px" altimg-width="97px" alttext="\displaystyle=-\frac{\zeta'\left(s\right)}{\zeta\left(s\right)}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo>-</mo><mstyle displaystyle="true"><mfrac><mrow><msup><mi href="./25.2#E1">ζ</mi><mo>′</mo></msup><mo></mo><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></mrow><mrow><mi href="./25.2#E1">ζ</mi><mo></mo><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></mrow></mfrac></mstyle></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m19.png" altimg-height="18px" altimg-valign="-3px" altimg-width="65px" alttext="\Re s>1" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>s</mi></mrow><mo>></mo><mn>1</mn></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E12.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./27.2#E14" title="(27.2.14) ‣ §27.2(i) Definitions ‣ §27.2 Functions ‣ Multiplicative Number Theory ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m18.png" altimg-height="23px" altimg-valign="-7px" altimg-width="49px" alttext="\Lambda\left(\NVar{n}\right)" display="inline"><mrow><mi href="./27.2#E14" mathvariant="normal">Λ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow></math>: Mangoldt’s function</a>,
<a href="./25.2#E1" title="(25.2.1) ‣ §25.2(i) Definition ‣ §25.2 Definition and Expansions ‣ Riemann Zeta Function ‣ Chapter 25 Zeta and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m32.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="\zeta\left(\NVar{s}\right)" display="inline"><mrow><mi href="./25.2#E1">ζ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./25.1#p2.t1.r5">s</mi><mo>)</mo></mrow></mrow></math>: Riemann zeta function</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a> and
<a href="./27.1#p2.t1.r1" title="§27.1 Special Notation ‣ Notation ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./27.1#p2.t1.r1">n</mi></math>: positive integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E13">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">27.4.13</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m16.png" altimg-height="64px" altimg-valign="-27px" altimg-width="121px" alttext="\displaystyle\sum_{n=2}^{\infty}(\ln n)n^{-s}" display="inline"><mrow><mstyle displaystyle="true"><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./27.1#p2.t1.r1">n</mi><mo>=</mo><mn>2</mn></mrow><mi mathvariant="normal">∞</mi></munderover></mstyle><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi href="./27.1#p2.t1.r1">n</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msup><mi href="./27.1#p2.t1.r1">n</mi><mrow><mo>-</mo><mi>s</mi></mrow></msup></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m3.png" altimg-height="26px" altimg-valign="-7px" altimg-width="95px" alttext="\displaystyle=-\zeta'\left(s\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo>-</mo><mrow><msup><mi href="./25.2#E1">ζ</mi><mo>′</mo></msup><mo></mo><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m19.png" altimg-height="18px" altimg-valign="-3px" altimg-width="65px" alttext="\Re s>1" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>s</mi></mrow><mo>></mo><mn>1</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E13.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./25.2#E1" title="(25.2.1) ‣ §25.2(i) Definition ‣ §25.2 Definition and Expansions ‣ Riemann Zeta Function ‣ Chapter 25 Zeta and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m32.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="\zeta\left(\NVar{s}\right)" display="inline"><mrow><mi href="./25.2#E1">ζ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./25.1#p2.t1.r5">s</mi><mo>)</mo></mrow></mrow></math>: Riemann zeta function</a>,
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m24.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a> and
<a href="./27.1#p2.t1.r1" title="§27.1 Special Notation ‣ Notation ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./27.1#p2.t1.r1">n</mi></math>: positive integer</a>
</dd>
<dt>Change of Notation (effective with 1.0.10):</dt>
<dd>
The notation for logarithm has been changed to <math class="ltx_Math" altimg="m25.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="\ln" display="inline"><mi href="./4.2#E2">ln</mi></math> from <math class="ltx_Math" altimg="m26.png" altimg-height="21px" altimg-valign="-6px" altimg-width="30px" alttext="\mathrm{log}" display="inline"><mi>log</mi></math>.
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
<p class="ltx_p">In () <math class="ltx_Math" altimg="m31.png" altimg-height="24px" altimg-valign="-7px" altimg-width="48px" alttext="\zeta'\left(s\right)" display="inline"><mrow><msup><mi href="./25.2#E1">ζ</mi><mo>′</mo></msup><mo></mo><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></mrow></math> is the
derivative of <math class="ltx_Math" altimg="m33.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="\zeta\left(s\right)" display="inline"><mrow><mi href="./25.2#E1">ζ</mi><mo></mo><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></mrow></math>.</p>
</div>
</section>
</div>
<div class="ltx_page_footer">
<span></div>
</div>
</body></text>
</html>
</page>
<page>
<!DOCTYPE html><html>
<head>
<title>DLMF: 27.13 Functions</title>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<!--GOOGLE BOOTSTRAP--></head>
<text><body class="color_default textfont_default titlefont_default fontsize_default navbar_default">
<div class="ltx_page_navbar">
<div class="ltx_page_navlogo"></dd>
</dl>
</div>
</div>
<div id="SS1.p1" class="ltx_para">
<p class="ltx_p">Whereas multiplicative number theory is concerned with functions arising from
prime factorization, additive number theory treats functions related to
addition of integers. The basic problem is that of expressing a given positive
integer <math class="ltx_Math" altimg="m55.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./27.1#p2.t1.r1">n</mi></math> as a sum of integers from some prescribed set <math class="ltx_Math" altimg="m25.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi href="./27.13#SS1.p1">S</mi></math> whose members are
primes, squares, cubes, or other special integers. Each representation of <math class="ltx_Math" altimg="m55.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./27.1#p2.t1.r1">n</mi></math>
as a sum of elements of <math class="ltx_Math" altimg="m25.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi href="./27.13#SS1.p1">S</mi></math> is called a <em class="ltx_emph ltx_font_italic">partition</em> of <math class="ltx_Math" altimg="m55.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./27.1#p2.t1.r1">n</mi></math>, and the number
<math class="ltx_Math" altimg="m23.png" altimg-height="23px" altimg-valign="-7px" altimg-width="45px" alttext="S(n)" display="inline"><mrow><mi href="./27.13#SS1.p1">S</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./27.1#p2.t1.r1">n</mi><mo stretchy="false">)</mo></mrow></mrow></math> of such partitions is often of great interest. The subsections that
follow describe problems from additive number theory. See also
<cite class="ltx_cite ltx_citemacro_citet">Apostol (</dd>
</dl>
</div>
</div>
<div id="SS2.p1" class="ltx_para">
<p class="ltx_p"><em class="ltx_emph ltx_font_italic">Every even integer <math class="ltx_Math" altimg="m54.png" altimg-height="17px" altimg-valign="-3px" altimg-width="53px" alttext="n>4" display="inline"><mrow><mi href="./27.1#p2.t1.r1">n</mi><mo mathvariant="normal">></mo><mn mathvariant="normal">4</mn></mrow></math> is the sum of two odd primes.</em> In this case,
<math class="ltx_Math" altimg="m23.png" altimg-height="23px" altimg-valign="-7px" altimg-width="45px" alttext="S(n)" display="inline"><mrow><mi href="./27.13#SS2.p1">S</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./27.1#p2.t1.r1">n</mi><mo stretchy="false">)</mo></mrow></mrow></math> is the number of solutions of the equation <math class="ltx_Math" altimg="m53.png" altimg-height="19px" altimg-valign="-6px" altimg-width="87px" alttext="n=p+q" display="inline"><mrow><mi href="./27.1#p2.t1.r1">n</mi><mo>=</mo><mrow><mi href="./27.13#SS2.p1">p</mi><mo>+</mo><mi href="./27.13#SS2.p1">q</mi></mrow></mrow></math>, where <math class="ltx_Math" altimg="m57.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="p" display="inline"><mi href="./27.13#SS2.p1">p</mi></math> and <math class="ltx_Math" altimg="m59.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="q" display="inline"><mi href="./27.13#SS2.p1">q</mi></math>
are odd primes. Goldbach’s assertion is that <math class="ltx_Math" altimg="m24.png" altimg-height="23px" altimg-valign="-7px" altimg-width="82px" alttext="S(n)\geq 1" display="inline"><mrow><mrow><mi href="./27.13#SS2.p1">S</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./27.1#p2.t1.r1">n</mi><mo stretchy="false">)</mo></mrow></mrow><mo>≥</mo><mn>1</mn></mrow></math> for all even
<math class="ltx_Math" altimg="m54.png" altimg-height="17px" altimg-valign="-3px" altimg-width="53px" alttext="n>4" display="inline"><mrow><mi href="./27.1#p2.t1.r1">n</mi><mo>></mo><mn>4</mn></mrow></math>. This conjecture dates back to 1742 and was undecided in 2009,
although it has been confirmed numerically up to very
large numbers. <cite class="ltx_cite ltx_citemacro_citet">Vinogradov (<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd>
<span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m17.png" altimg-height="23px" altimg-valign="-7px" altimg-width="50px" alttext="G\left(\NVar{k}\right)" display="inline"><mrow><mi href="./27.13#SS3.p1">G</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./27.1#p2.t1.r1">k</mi><mo>)</mo></mrow></mrow></math>: Waring’s function</span> and
<span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m42.png" altimg-height="23px" altimg-valign="-7px" altimg-width="44px" alttext="g\left(\NVar{k}\right)" display="inline"><mrow><mi href="./27.13#SS3.p1">g</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./27.1#p2.t1.r1">k</mi><mo>)</mo></mrow></mrow></math>: Waring’s function</span>
</dd>
<dt>Keywords:</dt>
<dd>
</dd>
</dl>
</div>
</div>
<div id="SS3.p1" class="ltx_para">
<p class="ltx_p">This problem is named after Edward Waring who, in 1770, stated without proof
and with limited numerical evidence, that every positive integer <math class="ltx_Math" altimg="m55.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./27.1#p2.t1.r1">n</mi></math> is the
sum of four squares, of nine cubes, of nineteen fourth powers, and so on. Waring’s
problem is to find, for each positive integer <math class="ltx_Math" altimg="m48.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./27.1#p2.t1.r1">k</mi></math>, whether there is an integer
<math class="ltx_Math" altimg="m52.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./27.1#p2.t1.r1">m</mi></math> (depending only on <math class="ltx_Math" altimg="m48.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./27.1#p2.t1.r1">k</mi></math>) such that the equation</p>
<table id="E1" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">27.13.1</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E1.png" altimg-height="28px" altimg-valign="-7px" altimg-width="210px" alttext="n=x_{1}^{k}+x_{2}^{k}+\dots+x_{m}^{k}" display="block"><mrow><mi href="./27.1#p2.t1.r1">n</mi><mo>=</mo><mrow><msubsup><mi href="./27.1#p2.t1.r9">x</mi><mn>1</mn><mi href="./27.1#p2.t1.r1">k</mi></msubsup><mo>+</mo><msubsup><mi href="./27.1#p2.t1.r9">x</mi><mn>2</mn><mi href="./27.1#p2.t1.r1">k</mi></msubsup><mo>+</mo><mi mathvariant="normal">…</mi><mo>+</mo><msubsup><mi href="./27.1#p2.t1.r9">x</mi><mi href="./27.1#p2.t1.r1">m</mi><mi href="./27.1#p2.t1.r1">k</mi></msubsup></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./27.1#p2.t1.r1" title="§27.1 Special Notation ‣ Notation ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./27.1#p2.t1.r1">k</mi></math>: positive integer</a>,
<a href="./27.1#p2.t1.r1" title="§27.1 Special Notation ‣ Notation ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./27.1#p2.t1.r1">m</mi></math>: positive integer</a>,
<a href="./27.1#p2.t1.r1" title="§27.1 Special Notation ‣ Notation ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m55.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./27.1#p2.t1.r1">n</mi></math>: positive integer</a> and
<a href="./27.1#p2.t1.r9" title="§27.1 Special Notation ‣ Notation ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m67.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./27.1#p2.t1.r9">x</mi></math>: real number</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">has nonnegative integer solutions for all <math class="ltx_Math" altimg="m56.png" altimg-height="19px" altimg-valign="-5px" altimg-width="53px" alttext="n\geq 1" display="inline"><mrow><mi href="./27.1#p2.t1.r1">n</mi><mo>≥</mo><mn>1</mn></mrow></math>. The smallest <math class="ltx_Math" altimg="m52.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./27.1#p2.t1.r1">m</mi></math> that
exists for a given <math class="ltx_Math" altimg="m48.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./27.1#p2.t1.r1">k</mi></math> is denoted by <math class="ltx_Math" altimg="m43.png" altimg-height="23px" altimg-valign="-7px" altimg-width="44px" alttext="g\left(k\right)" display="inline"><mrow><mi href="./27.13#SS3.p1">g</mi><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r1">k</mi><mo>)</mo></mrow></mrow></math>. Similarly, <math class="ltx_Math" altimg="m20.png" altimg-height="23px" altimg-valign="-7px" altimg-width="50px" alttext="G\left(k\right)" display="inline"><mrow><mi href="./27.13#SS3.p1">G</mi><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r1">k</mi><mo>)</mo></mrow></mrow></math>
denotes the smallest <math class="ltx_Math" altimg="m52.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./27.1#p2.t1.r1">m</mi></math> for which () has nonnegative
integer solutions for all sufficiently large <math class="ltx_Math" altimg="m55.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./27.1#p2.t1.r1">n</mi></math>.</p>
</div>
<div id="SS3.p2" class="ltx_para">
<p class="ltx_p"><cite class="ltx_cite ltx_citemacro_citet">Lagrange ()</cite> proves that <math class="ltx_Math" altimg="m35.png" altimg-height="23px" altimg-valign="-7px" altimg-width="80px" alttext="g\left(2\right)=4" display="inline"><mrow><mrow><mi href="./27.13#SS3.p1">g</mi><mo></mo><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow></mrow><mo>=</mo><mn>4</mn></mrow></math>, and during the next
139 years the existence of <math class="ltx_Math" altimg="m43.png" altimg-height="23px" altimg-valign="-7px" altimg-width="44px" alttext="g\left(k\right)" display="inline"><mrow><mi href="./27.13#SS3.p1">g</mi><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r1">k</mi><mo>)</mo></mrow></mrow></math> was shown for <math class="ltx_Math" altimg="m44.png" altimg-height="21px" altimg-valign="-6px" altimg-width="175px" alttext="k=3,4,5,6,7,8,10" display="inline"><mrow><mi href="./27.1#p2.t1.r1">k</mi><mo>=</mo><mrow><mn>3</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>5</mn><mo>,</mo><mn>6</mn><mo>,</mo><mn>7</mn><mo>,</mo><mn>8</mn><mo>,</mo><mn>10</mn></mrow></mrow></math>.
<cite class="ltx_cite ltx_citemacro_citet">Hilbert ()</cite> proves the existence of <math class="ltx_Math" altimg="m43.png" altimg-height="23px" altimg-valign="-7px" altimg-width="44px" alttext="g\left(k\right)" display="inline"><mrow><mi href="./27.13#SS3.p1">g</mi><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r1">k</mi><mo>)</mo></mrow></mrow></math> for every <math class="ltx_Math" altimg="m48.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./27.1#p2.t1.r1">k</mi></math>
but does not determine its corresponding numerical value. The exact value of
<math class="ltx_Math" altimg="m43.png" altimg-height="23px" altimg-valign="-7px" altimg-width="44px" alttext="g\left(k\right)" display="inline"><mrow><mi href="./27.13#SS3.p1">g</mi><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r1">k</mi><mo>)</mo></mrow></mrow></math> is now known for every <math class="ltx_Math" altimg="m50.png" altimg-height="21px" altimg-valign="-6px" altimg-width="110px" alttext="k\leq 200,000" display="inline"><mrow><mi href="./27.1#p2.t1.r1">k</mi><mo>≤</mo><mrow><mn>200</mn><mo>,</mo><mn>000</mn></mrow></mrow></math>. For example,
<math class="ltx_Math" altimg="m36.png" altimg-height="23px" altimg-valign="-7px" altimg-width="80px" alttext="g\left(3\right)=9" display="inline"><mrow><mrow><mi href="./27.13#SS3.p1">g</mi><mo></mo><mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow></mrow><mo>=</mo><mn>9</mn></mrow></math>, <math class="ltx_Math" altimg="m37.png" altimg-height="23px" altimg-valign="-7px" altimg-width="90px" alttext="g\left(4\right)=19" display="inline"><mrow><mrow><mi href="./27.13#SS3.p1">g</mi><mo></mo><mrow><mo>(</mo><mn>4</mn><mo>)</mo></mrow></mrow><mo>=</mo><mn>19</mn></mrow></math>, <math class="ltx_Math" altimg="m38.png" altimg-height="23px" altimg-valign="-7px" altimg-width="90px" alttext="g\left(5\right)=37" display="inline"><mrow><mrow><mi href="./27.13#SS3.p1">g</mi><mo></mo><mrow><mo>(</mo><mn>5</mn><mo>)</mo></mrow></mrow><mo>=</mo><mn>37</mn></mrow></math>,
<math class="ltx_Math" altimg="m39.png" altimg-height="23px" altimg-valign="-7px" altimg-width="90px" alttext="g\left(6\right)=73" display="inline"><mrow><mrow><mi href="./27.13#SS3.p1">g</mi><mo></mo><mrow><mo>(</mo><mn>6</mn><mo>)</mo></mrow></mrow><mo>=</mo><mn>73</mn></mrow></math>, <math class="ltx_Math" altimg="m40.png" altimg-height="23px" altimg-valign="-7px" altimg-width="100px" alttext="g\left(7\right)=143" display="inline"><mrow><mrow><mi href="./27.13#SS3.p1">g</mi><mo></mo><mrow><mo>(</mo><mn>7</mn><mo>)</mo></mrow></mrow><mo>=</mo><mn>143</mn></mrow></math>, and <math class="ltx_Math" altimg="m41.png" altimg-height="23px" altimg-valign="-7px" altimg-width="100px" alttext="g\left(8\right)=279" display="inline"><mrow><mrow><mi href="./27.13#SS3.p1">g</mi><mo></mo><mrow><mo>(</mo><mn>8</mn><mo>)</mo></mrow></mrow><mo>=</mo><mn>279</mn></mrow></math>. A general
formula states that</p>
<table id="E2" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">27.13.2</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E2.png" altimg-height="56px" altimg-valign="-21px" altimg-width="204px" alttext="g\left(k\right)\geq 2^{k}+\left\lfloor\frac{3^{k}}{2^{k}}\right\rfloor-2," display="block"><mrow><mrow><mrow><mi href="./27.13#SS3.p1">g</mi><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r1">k</mi><mo>)</mo></mrow></mrow><mo>≥</mo><mrow><mrow><msup><mn>2</mn><mi href="./27.1#p2.t1.r1">k</mi></msup><mo>+</mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r16">⌊</mo><mfrac><msup><mn>3</mn><mi href="./27.1#p2.t1.r1">k</mi></msup><msup><mn>2</mn><mi href="./27.1#p2.t1.r1">k</mi></msup></mfrac><mo href="./front/introduction#Sx4.p1.t1.r16">⌋</mo></mrow></mrow><mo>-</mo><mn>2</mn></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./27.13#SS3.p1" title="§27.13(iii) Waring’s Problem ‣ §27.13 Functions ‣ Additive Number Theory ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m42.png" altimg-height="23px" altimg-valign="-7px" altimg-width="44px" alttext="g\left(\NVar{k}\right)" display="inline"><mrow><mi href="./27.13#SS3.p1">g</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./27.1#p2.t1.r1">k</mi><mo>)</mo></mrow></mrow></math>: Waring’s function</a>,
<a href="./front/introduction#Sx4.p1.t1.r16" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="23px" altimg-valign="-7px" altimg-width="33px" alttext="\left\lfloor\NVar{x}\right\rfloor" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r16">⌊</mo><mi class="ltx_nvar">x</mi><mo href="./front/introduction#Sx4.p1.t1.r16">⌋</mo></mrow></math>: floor of <math class="ltx_Math" altimg="m67.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./27.1#p2.t1.r1" title="§27.1 Special Notation ‣ Notation ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./27.1#p2.t1.r1">k</mi></math>: positive integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">for all <math class="ltx_Math" altimg="m49.png" altimg-height="20px" altimg-valign="-5px" altimg-width="52px" alttext="k\geq 2" display="inline"><mrow><mi href="./27.1#p2.t1.r1">k</mi><mo>≥</mo><mn>2</mn></mrow></math>, with equality if <math class="ltx_Math" altimg="m6.png" altimg-height="21px" altimg-valign="-6px" altimg-width="147px" alttext="4\leq k\leq 200,000" display="inline"><mrow><mrow><mn>4</mn><mo>≤</mo><mi href="./27.1#p2.t1.r1">k</mi><mo>≤</mo><mn>200</mn></mrow><mo>,</mo><mn>000</mn></mrow></math>. If
<math class="ltx_Math" altimg="m5.png" altimg-height="24px" altimg-valign="-6px" altimg-width="114px" alttext="3^{k}=q2^{k}+r" display="inline"><mrow><msup><mn>3</mn><mi href="./27.1#p2.t1.r1">k</mi></msup><mo>=</mo><mrow><mrow><mi>q</mi><mo></mo><msup><mn>2</mn><mi href="./27.1#p2.t1.r1">k</mi></msup></mrow><mo>+</mo><mi>r</mi></mrow></mrow></math> with <math class="ltx_Math" altimg="m1.png" altimg-height="21px" altimg-valign="-3px" altimg-width="97px" alttext="0<r<2^{k}" display="inline"><mrow><mn>0</mn><mo><</mo><mi>r</mi><mo><</mo><msup><mn>2</mn><mi href="./27.1#p2.t1.r1">k</mi></msup></mrow></math>, then equality holds in
() provided <math class="ltx_Math" altimg="m60.png" altimg-height="24px" altimg-valign="-6px" altimg-width="94px" alttext="r+q\leq 2^{k}" display="inline"><mrow><mrow><mi>r</mi><mo>+</mo><mi>q</mi></mrow><mo>≤</mo><msup><mn>2</mn><mi href="./27.1#p2.t1.r1">k</mi></msup></mrow></math>, a condition that is satisfied
with at most a finite number of exceptions.</p>
</div>
<div id="SS3.p3" class="ltx_para">
<p class="ltx_p">The existence of <math class="ltx_Math" altimg="m20.png" altimg-height="23px" altimg-valign="-7px" altimg-width="50px" alttext="G\left(k\right)" display="inline"><mrow><mi href="./27.13#SS3.p1">G</mi><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r1">k</mi><mo>)</mo></mrow></mrow></math> follows from that of <math class="ltx_Math" altimg="m43.png" altimg-height="23px" altimg-valign="-7px" altimg-width="44px" alttext="g\left(k\right)" display="inline"><mrow><mi href="./27.13#SS3.p1">g</mi><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r1">k</mi><mo>)</mo></mrow></mrow></math> because
<math class="ltx_Math" altimg="m22.png" altimg-height="23px" altimg-valign="-7px" altimg-width="116px" alttext="G\left(k\right)\leq g\left(k\right)" display="inline"><mrow><mrow><mi href="./27.13#SS3.p1">G</mi><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r1">k</mi><mo>)</mo></mrow></mrow><mo>≤</mo><mrow><mi href="./27.13#SS3.p1">g</mi><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r1">k</mi><mo>)</mo></mrow></mrow></mrow></math>, but only the values <math class="ltx_Math" altimg="m10.png" altimg-height="23px" altimg-valign="-7px" altimg-width="85px" alttext="G\left(2\right)=4" display="inline"><mrow><mrow><mi href="./27.13#SS3.p1">G</mi><mo></mo><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow></mrow><mo>=</mo><mn>4</mn></mrow></math> and
<math class="ltx_Math" altimg="m12.png" altimg-height="23px" altimg-valign="-7px" altimg-width="95px" alttext="G\left(4\right)=16" display="inline"><mrow><mrow><mi href="./27.13#SS3.p1">G</mi><mo></mo><mrow><mo>(</mo><mn>4</mn><mo>)</mo></mrow></mrow><mo>=</mo><mn>16</mn></mrow></math> are known exactly. Some upper bounds smaller than
<math class="ltx_Math" altimg="m43.png" altimg-height="23px" altimg-valign="-7px" altimg-width="44px" alttext="g\left(k\right)" display="inline"><mrow><mi href="./27.13#SS3.p1">g</mi><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r1">k</mi><mo>)</mo></mrow></mrow></math> are known. For example, <math class="ltx_Math" altimg="m11.png" altimg-height="23px" altimg-valign="-7px" altimg-width="85px" alttext="G\left(3\right)\leq 7" display="inline"><mrow><mrow><mi href="./27.13#SS3.p1">G</mi><mo></mo><mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow></mrow><mo>≤</mo><mn>7</mn></mrow></math>,
<math class="ltx_Math" altimg="m13.png" altimg-height="23px" altimg-valign="-7px" altimg-width="95px" alttext="G\left(5\right)\leq 23" display="inline"><mrow><mrow><mi href="./27.13#SS3.p1">G</mi><mo></mo><mrow><mo>(</mo><mn>5</mn><mo>)</mo></mrow></mrow><mo>≤</mo><mn>23</mn></mrow></math>, <math class="ltx_Math" altimg="m14.png" altimg-height="23px" altimg-valign="-7px" altimg-width="95px" alttext="G\left(6\right)\leq 36" display="inline"><mrow><mrow><mi href="./27.13#SS3.p1">G</mi><mo></mo><mrow><mo>(</mo><mn>6</mn><mo>)</mo></mrow></mrow><mo>≤</mo><mn>36</mn></mrow></math>, <math class="ltx_Math" altimg="m15.png" altimg-height="23px" altimg-valign="-7px" altimg-width="95px" alttext="G\left(7\right)\leq 53" display="inline"><mrow><mrow><mi href="./27.13#SS3.p1">G</mi><mo></mo><mrow><mo>(</mo><mn>7</mn><mo>)</mo></mrow></mrow><mo>≤</mo><mn>53</mn></mrow></math>, and
<math class="ltx_Math" altimg="m16.png" altimg-height="23px" altimg-valign="-7px" altimg-width="95px" alttext="G\left(8\right)\leq 73" display="inline"><mrow><mrow><mi href="./27.13#SS3.p1">G</mi><mo></mo><mrow><mo>(</mo><mn>8</mn><mo>)</mo></mrow></mrow><mo>≤</mo><mn>73</mn></mrow></math>. <cite class="ltx_cite ltx_citemacro_citet">Hardy and Littlewood ()</cite> conjectures that
<math class="ltx_Math" altimg="m18.png" altimg-height="23px" altimg-valign="-7px" altimg-width="132px" alttext="G\left(k\right)<2k+1" display="inline"><mrow><mrow><mi href="./27.13#SS3.p1">G</mi><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r1">k</mi><mo>)</mo></mrow></mrow><mo><</mo><mrow><mrow><mn>2</mn><mo></mo><mi href="./27.1#p2.t1.r1">k</mi></mrow><mo>+</mo><mn>1</mn></mrow></mrow></math> when <math class="ltx_Math" altimg="m48.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./27.1#p2.t1.r1">k</mi></math> is not a power of 2, and that
<math class="ltx_Math" altimg="m21.png" altimg-height="23px" altimg-valign="-7px" altimg-width="97px" alttext="G\left(k\right)\leq 4k" display="inline"><mrow><mrow><mi href="./27.13#SS3.p1">G</mi><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r1">k</mi><mo>)</mo></mrow></mrow><mo>≤</mo><mrow><mn>4</mn><mo></mo><mi href="./27.1#p2.t1.r1">k</mi></mrow></mrow></math> when <math class="ltx_Math" altimg="m48.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./27.1#p2.t1.r1">k</mi></math> is a power of 2, but the most that is known (in
2009) is <math class="ltx_Math" altimg="m19.png" altimg-height="23px" altimg-valign="-7px" altimg-width="130px" alttext="G\left(k\right)<ck\ln k" display="inline"><mrow><mrow><mi href="./27.13#SS3.p1">G</mi><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r1">k</mi><mo>)</mo></mrow></mrow><mo><</mo><mrow><mi>c</mi><mo></mo><mi href="./27.1#p2.t1.r1">k</mi><mo></mo><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi href="./27.1#p2.t1.r1">k</mi></mrow></mrow></mrow></math> for some constant <math class="ltx_Math" altimg="m34.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="c" display="inline"><mi>c</mi></math>. A survey is
given in <cite class="ltx_cite ltx_citemacro_citet">Ellison (<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m65.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="r_{\NVar{k}}\left(\NVar{n}\right)" display="inline"><mrow><msub><mi href="./27.13#SS4.p1">r</mi><mi class="ltx_nvar" href="./27.1#p2.t1.r1">k</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow></math>: number of squares</span></dd>
<dt>Keywords:</dt>
<dd>
</dd>
</dl>
</div>
</div>
<div id="SS4.p1" class="ltx_para">
<p class="ltx_p">For a given integer <math class="ltx_Math" altimg="m49.png" altimg-height="20px" altimg-valign="-5px" altimg-width="52px" alttext="k\geq 2" display="inline"><mrow><mi href="./27.1#p2.t1.r1">k</mi><mo>≥</mo><mn>2</mn></mrow></math> the function <math class="ltx_Math" altimg="m66.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="r_{k}\left(n\right)" display="inline"><mrow><msub><mi href="./27.13#SS4.p1">r</mi><mi href="./27.1#p2.t1.r1">k</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow></math> is defined as
the number of solutions of the equation
</p>
<table id="E3" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">27.13.3</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E3.png" altimg-height="28px" altimg-valign="-7px" altimg-width="208px" alttext="n=x_{1}^{2}+x_{2}^{2}+\dots+x_{k}^{2}," display="block"><mrow><mrow><mi href="./27.1#p2.t1.r1">n</mi><mo>=</mo><mrow><msubsup><mi href="./27.1#p2.t1.r9">x</mi><mn>1</mn><mn>2</mn></msubsup><mo>+</mo><msubsup><mi href="./27.1#p2.t1.r9">x</mi><mn>2</mn><mn>2</mn></msubsup><mo>+</mo><mi mathvariant="normal">…</mi><mo>+</mo><msubsup><mi href="./27.1#p2.t1.r9">x</mi><mi href="./27.1#p2.t1.r1">k</mi><mn>2</mn></msubsup></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./27.1#p2.t1.r1" title="§27.1 Special Notation ‣ Notation ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./27.1#p2.t1.r1">k</mi></math>: positive integer</a>,
<a href="./27.1#p2.t1.r1" title="§27.1 Special Notation ‣ Notation ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m55.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./27.1#p2.t1.r1">n</mi></math>: positive integer</a> and
<a href="./27.1#p2.t1.r9" title="§27.1 Special Notation ‣ Notation ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m67.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./27.1#p2.t1.r9">x</mi></math>: real number</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where the <math class="ltx_Math" altimg="m70.png" altimg-height="18px" altimg-valign="-8px" altimg-width="24px" alttext="x_{j}" display="inline"><msub><mi href="./27.1#p2.t1.r9">x</mi><mi>j</mi></msub></math> are integers, positive, negative, or zero, and the order of
the summands is taken into account.</p>
</div>
<div id="SS4.p2" class="ltx_para">
<p class="ltx_p"><cite class="ltx_cite ltx_citemacro_citet">Jacobi ()</cite> notes that <math class="ltx_Math" altimg="m62.png" altimg-height="23px" altimg-valign="-7px" altimg-width="53px" alttext="r_{2}\left(n\right)" display="inline"><mrow><msub><mi href="./27.13#SS4.p1">r</mi><mn>2</mn></msub><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow></math> is the coefficient of
<math class="ltx_Math" altimg="m68.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="x^{n}" display="inline"><msup><mi href="./27.1#p2.t1.r9">x</mi><mi href="./27.1#p2.t1.r1">n</mi></msup></math> in the square of the theta function <math class="ltx_Math" altimg="m33.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\vartheta\left(x\right)" display="inline"><mrow><mi href="./27.13#E4">ϑ</mi><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r9">x</mi><mo>)</mo></mrow></mrow></math>:</p>
<table id="E4" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">27.13.4</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E4.png" altimg-height="64px" altimg-valign="-27px" altimg-width="200px" alttext="\vartheta\left(x\right)=1+2\sum_{m=1}^{\infty}x^{m^{2}}," display="block"><mrow><mrow><mrow><mi href="./27.13#E4">ϑ</mi><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r9">x</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mn>1</mn><mo>+</mo><mrow><mn>2</mn><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./27.1#p2.t1.r1">m</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover><msup><mi href="./27.1#p2.t1.r9">x</mi><msup><mi href="./27.1#p2.t1.r1">m</mi><mn>2</mn></msup></msup></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m72.png" altimg-height="23px" altimg-valign="-7px" altimg-width="63px" alttext="|x|<1" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mi href="./27.1#p2.t1.r9">x</mi><mo stretchy="false">|</mo></mrow><mo><</mo><mn>1</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E4.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m32.png" altimg-height="23px" altimg-valign="-7px" altimg-width="140px" alttext="\vartheta\left(\NVar{x}\right)=\theta_{3}\left(0,x\right)" display="inline"><mrow><mrow><mi href="./27.13#E4">ϑ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./27.1#p2.t1.r9">x</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msub><mi href="./20.2#i">θ</mi><mn>3</mn></msub><mo></mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi href="./27.1#p2.t1.r9">x</mi><mo>)</mo></mrow></mrow></mrow></math>: alternative notation</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./20.2#i" title="§20.2(i) Fourier Series ‣ §20.2 Definitions and Periodic Properties ‣ Properties ‣ Chapter 20 Theta Functions" class="ltx_ref"><math class="ltx_Math" altimg="m31.png" altimg-height="24px" altimg-valign="-8px" altimg-width="69px" alttext="\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)" display="inline"><mrow><msub><mi href="./20.2#i">θ</mi><mi class="ltx_nvar">j</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./20.1#p2.t1.r3">z</mi><mo>,</mo><mi class="ltx_nvar" href="./20.1#p2.t1.r4">q</mi><mo>)</mo></mrow></mrow></math>: theta function</a>,
<a href="./27.1#p2.t1.r1" title="§27.1 Special Notation ‣ Notation ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./27.1#p2.t1.r1">m</mi></math>: positive integer</a> and
<a href="./27.1#p2.t1.r9" title="§27.1 Special Notation ‣ Notation ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m67.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./27.1#p2.t1.r9">x</mi></math>: real number</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">16.27.3</span> (with <math class="ltx_Math" altimg="m71.png" altimg-height="17px" altimg-valign="-2px" altimg-width="51px" alttext="z=0" display="inline"><mrow><mi>z</mi><mo>=</mo><mn>0</mn></mrow></math>, <math class="ltx_Math" altimg="m58.png" altimg-height="16px" altimg-valign="-6px" altimg-width="52px" alttext="q=x" display="inline"><mrow><mi>q</mi><mo>=</mo><mi href="./27.1#p2.t1.r9">x</mi></mrow></math>)</span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">(In §, <math class="ltx_Math" altimg="m33.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\vartheta\left(x\right)" display="inline"><mrow><mi href="./27.13#E4">ϑ</mi><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r9">x</mi><mo>)</mo></mrow></mrow></math> is denoted by
<math class="ltx_Math" altimg="m30.png" altimg-height="23px" altimg-valign="-7px" altimg-width="72px" alttext="\theta_{3}\left(0,x\right)" display="inline"><mrow><msub><mi href="./20.2#i">θ</mi><mn>3</mn></msub><mo></mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi href="./27.1#p2.t1.r9">x</mi><mo>)</mo></mrow></mrow></math>.) Thus,</p>
<table id="E5" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">27.13.5</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E5.png" altimg-height="64px" altimg-valign="-27px" altimg-width="246px" alttext="(\vartheta\left(x\right))^{2}=1+\sum_{n=1}^{\infty}r_{2}\left(n\right)x^{n}." display="block"><mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./27.13#E4">ϑ</mi><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r9">x</mi><mo>)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>=</mo><mrow><mn>1</mn><mo>+</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./27.1#p2.t1.r1">n</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><mrow><msub><mi href="./27.13#SS4.p1">r</mi><mn>2</mn></msub><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow><mo></mo><msup><mi href="./27.1#p2.t1.r9">x</mi><mi href="./27.1#p2.t1.r1">n</mi></msup></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E5.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./27.13#E4" title="(27.13.4) ‣ §27.13(iv) Representation by Squares ‣ §27.13 Functions ‣ Additive Number Theory ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m32.png" altimg-height="23px" altimg-valign="-7px" altimg-width="140px" alttext="\vartheta\left(\NVar{x}\right)=\theta_{3}\left(0,x\right)" display="inline"><mrow><mrow><mi href="./27.13#E4">ϑ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./27.1#p2.t1.r9">x</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msub><mi href="./20.2#i">θ</mi><mn>3</mn></msub><mo></mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi href="./27.1#p2.t1.r9">x</mi><mo>)</mo></mrow></mrow></mrow></math>: alternative notation</a>,
<a href="./27.13#SS4.p1" title="§27.13(iv) Representation by Squares ‣ §27.13 Functions ‣ Additive Number Theory ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m65.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="r_{\NVar{k}}\left(\NVar{n}\right)" display="inline"><mrow><msub><mi href="./27.13#SS4.p1">r</mi><mi class="ltx_nvar" href="./27.1#p2.t1.r1">k</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow></math>: number of squares</a>,
<a href="./27.1#p2.t1.r1" title="§27.1 Special Notation ‣ Notation ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m55.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./27.1#p2.t1.r1">n</mi></math>: positive integer</a> and
<a href="./27.1#p2.t1.r9" title="§27.1 Special Notation ‣ Notation ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m67.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./27.1#p2.t1.r9">x</mi></math>: real number</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS4.p3" class="ltx_para">
<p class="ltx_p">One of Jacobi’s identities implies that</p>
<table id="E6" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">27.13.6</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E6.png" altimg-height="64px" altimg-valign="-27px" altimg-width="341px" alttext="(\vartheta\left(x\right))^{2}=1+4\sum_{n=1}^{\infty}\left(\delta_{1}(n)-\delta%
_{3}(n)\right)x^{n}," display="block"><mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./27.13#E4">ϑ</mi><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r9">x</mi><mo>)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>=</mo><mrow><mn>1</mn><mo>+</mo><mrow><mn>4</mn><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./27.1#p2.t1.r1">n</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><mrow><mo>(</mo><mrow><mrow><msub><mi href="./27.13#SS4.p3">δ</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./27.1#p2.t1.r1">n</mi><mo stretchy="false">)</mo></mrow></mrow><mo>-</mo><mrow><msub><mi href="./27.13#SS4.p3">δ</mi><mn>3</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./27.1#p2.t1.r1">n</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>)</mo></mrow><mo></mo><msup><mi href="./27.1#p2.t1.r9">x</mi><mi href="./27.1#p2.t1.r1">n</mi></msup></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E6.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./27.13#E4" title="(27.13.4) ‣ §27.13(iv) Representation by Squares ‣ §27.13 Functions ‣ Additive Number Theory ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m32.png" altimg-height="23px" altimg-valign="-7px" altimg-width="140px" alttext="\vartheta\left(\NVar{x}\right)=\theta_{3}\left(0,x\right)" display="inline"><mrow><mrow><mi href="./27.13#E4">ϑ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./27.1#p2.t1.r9">x</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msub><mi href="./20.2#i">θ</mi><mn>3</mn></msub><mo></mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi href="./27.1#p2.t1.r9">x</mi><mo>)</mo></mrow></mrow></mrow></math>: alternative notation</a>,
<a href="./27.1#p2.t1.r1" title="§27.1 Special Notation ‣ Notation ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m55.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./27.1#p2.t1.r1">n</mi></math>: positive integer</a>,
<a href="./27.1#p2.t1.r9" title="§27.1 Special Notation ‣ Notation ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m67.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./27.1#p2.t1.r9">x</mi></math>: real number</a> and
<a href="./27.13#SS4.p3" title="§27.13(iv) Representation by Squares ‣ §27.13 Functions ‣ Additive Number Theory ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m28.png" altimg-height="24px" altimg-valign="-8px" altimg-width="49px" alttext="\delta_{j}(n)" display="inline"><mrow><msub><mi href="./27.13#SS4.p3">δ</mi><mi>j</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./27.1#p2.t1.r1">n</mi><mo stretchy="false">)</mo></mrow></mrow></math>: number of divisors</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m26.png" altimg-height="23px" altimg-valign="-7px" altimg-width="50px" alttext="\delta_{1}(n)" display="inline"><mrow><msub><mi href="./27.13#SS4.p3">δ</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./27.1#p2.t1.r1">n</mi><mo stretchy="false">)</mo></mrow></mrow></math> and <math class="ltx_Math" altimg="m27.png" altimg-height="23px" altimg-valign="-7px" altimg-width="50px" alttext="\delta_{3}(n)" display="inline"><mrow><msub><mi href="./27.13#SS4.p3">δ</mi><mn>3</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./27.1#p2.t1.r1">n</mi><mo stretchy="false">)</mo></mrow></mrow></math> are the number of divisors of <math class="ltx_Math" altimg="m55.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./27.1#p2.t1.r1">n</mi></math>
congruent respectively to 1 and 3 (mod 4), and by equating coefficients
in () Jacobi deduced that
</p>
<table id="E7" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">27.13.7</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E7.png" altimg-height="25px" altimg-valign="-7px" altimg-width="234px" alttext="r_{2}\left(n\right)=4\left(\delta_{1}(n)-\delta_{3}(n)\right)." display="block"><mrow><mrow><mrow><msub><mi href="./27.13#SS4.p1">r</mi><mn>2</mn></msub><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mn>4</mn><mo></mo><mrow><mo>(</mo><mrow><mrow><msub><mi href="./27.13#SS4.p3">δ</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./27.1#p2.t1.r1">n</mi><mo stretchy="false">)</mo></mrow></mrow><mo>-</mo><mrow><msub><mi href="./27.13#SS4.p3">δ</mi><mn>3</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./27.1#p2.t1.r1">n</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E7.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./27.13#SS4.p1" title="§27.13(iv) Representation by Squares ‣ §27.13 Functions ‣ Additive Number Theory ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m65.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="r_{\NVar{k}}\left(\NVar{n}\right)" display="inline"><mrow><msub><mi href="./27.13#SS4.p1">r</mi><mi class="ltx_nvar" href="./27.1#p2.t1.r1">k</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow></math>: number of squares</a>,
<a href="./27.1#p2.t1.r1" title="§27.1 Special Notation ‣ Notation ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m55.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./27.1#p2.t1.r1">n</mi></math>: positive integer</a> and
<a href="./27.13#SS4.p3" title="§27.13(iv) Representation by Squares ‣ §27.13 Functions ‣ Additive Number Theory ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><math class="ltx_Math" altimg="m28.png" altimg-height="24px" altimg-valign="-8px" altimg-width="49px" alttext="\delta_{j}(n)" display="inline"><mrow><msub><mi href="./27.13#SS4.p3">δ</mi><mi>j</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./27.1#p2.t1.r1">n</mi><mo stretchy="false">)</mo></mrow></mrow></math>: number of divisors</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Hence <math class="ltx_Math" altimg="m61.png" altimg-height="23px" altimg-valign="-7px" altimg-width="88px" alttext="r_{2}\left(5\right)=8" display="inline"><mrow><mrow><msub><mi href="./27.13#SS4.p1">r</mi><mn>2</mn></msub><mo></mo><mrow><mo>(</mo><mn>5</mn><mo>)</mo></mrow></mrow><mo>=</mo><mn>8</mn></mrow></math> because both divisors, <math class="ltx_Math" altimg="m3.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="1" display="inline"><mn>1</mn></math> and <math class="ltx_Math" altimg="m8.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="5" display="inline"><mn>5</mn></math>, are
congruent to <math class="ltx_Math" altimg="m4.png" altimg-height="23px" altimg-valign="-7px" altimg-width="93px" alttext="1\pmod{4}" display="inline"><mrow><mn>1</mn><mspace width="veryverythickmathspace"></mspace><mrow><mo lspace="8.1pt" stretchy="false">(</mo><mrow><mo>mod</mo><mn>4</mn></mrow><mo stretchy="false">)</mo></mrow></mrow></math>. In fact, there are four representations, given by
<math class="ltx_Math" altimg="m7.png" altimg-height="25px" altimg-valign="-7px" altimg-width="493px" alttext="5=2^{2}+1^{2}=2^{2}+(-1)^{2}=(-2)^{2}+1^{2}=(-2)^{2}+(-1)^{2}" display="inline"><mrow><mn>5</mn><mo>=</mo><mrow><msup><mn>2</mn><mn>2</mn></msup><mo>+</mo><msup><mn>1</mn><mn>2</mn></msup></mrow><mo>=</mo><mrow><msup><mn>2</mn><mn>2</mn></msup><mo>+</mo><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow><mo>=</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>2</mn></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>+</mo><msup><mn>1</mn><mn>2</mn></msup></mrow><mo>=</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>2</mn></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow></mrow></math>, and four more
with the order of summands reversed.</p>
</div>
<div id="SS4.p4" class="ltx_para">
<p class="ltx_p">By similar methods Jacobi proved that
<math class="ltx_Math" altimg="m64.png" altimg-height="23px" altimg-valign="-7px" altimg-width="137px" alttext="r_{4}\left(n\right)=8\!\sigma_{1}\left(n\right)" display="inline"><mrow><mrow><msub><mi href="./27.13#SS4.p1">r</mi><mn>4</mn></msub><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mpadded width="-1.7pt"><mn>8</mn></mpadded><mo></mo><mrow><msub><mi href="./27.2#E10">σ</mi><mn>1</mn></msub><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow></mrow></mrow></math> if <math class="ltx_Math" altimg="m55.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./27.1#p2.t1.r1">n</mi></math> is odd, whereas, if <math class="ltx_Math" altimg="m55.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./27.1#p2.t1.r1">n</mi></math>
is even, <math class="ltx_Math" altimg="m63.png" altimg-height="23px" altimg-valign="-7px" altimg-width="99px" alttext="r_{4}\left(n\right)=24" display="inline"><mrow><mrow><msub><mi href="./27.13#SS4.p1">r</mi><mn>4</mn></msub><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow><mo>=</mo><mn>24</mn></mrow></math> times the sum of the odd divisors of <math class="ltx_Math" altimg="m55.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./27.1#p2.t1.r1">n</mi></math>.
<cite class="ltx_cite ltx_citemacro_citet">Mordell ()</cite> notes that <math class="ltx_Math" altimg="m66.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="r_{k}\left(n\right)" display="inline"><mrow><msub><mi href="./27.13#SS4.p1">r</mi><mi href="./27.1#p2.t1.r1">k</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow></math> is the coefficient of
<math class="ltx_Math" altimg="m68.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="x^{n}" display="inline"><msup><mi href="./27.1#p2.t1.r9">x</mi><mi href="./27.1#p2.t1.r1">n</mi></msup></math> in the power-series expansion of the <math class="ltx_Math" altimg="m48.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./27.1#p2.t1.r1">k</mi></math>th power of the series for
<math class="ltx_Math" altimg="m33.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\vartheta\left(x\right)" display="inline"><mrow><mi href="./27.13#E4">ϑ</mi><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r9">x</mi><mo>)</mo></mrow></mrow></math>. Explicit formulas for <math class="ltx_Math" altimg="m66.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="r_{k}\left(n\right)" display="inline"><mrow><msub><mi href="./27.13#SS4.p1">r</mi><mi href="./27.1#p2.t1.r1">k</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow></math> have been
obtained by similar methods for <math class="ltx_Math" altimg="m46.png" altimg-height="21px" altimg-valign="-6px" altimg-width="99px" alttext="k=6,8,10" display="inline"><mrow><mi href="./27.1#p2.t1.r1">k</mi><mo>=</mo><mrow><mn>6</mn><mo>,</mo><mn>8</mn><mo>,</mo><mn>10</mn></mrow></mrow></math>, and <math class="ltx_Math" altimg="m2.png" altimg-height="17px" altimg-valign="-2px" altimg-width="24px" alttext="12" display="inline"><mn>12</mn></math>, but they are more
complicated. Exact formulas for <math class="ltx_Math" altimg="m66.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="r_{k}\left(n\right)" display="inline"><mrow><msub><mi href="./27.13#SS4.p1">r</mi><mi href="./27.1#p2.t1.r1">k</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow></math> have also been found for
<math class="ltx_Math" altimg="m45.png" altimg-height="21px" altimg-valign="-6px" altimg-width="71px" alttext="k=3,5" display="inline"><mrow><mi href="./27.1#p2.t1.r1">k</mi><mo>=</mo><mrow><mn>3</mn><mo>,</mo><mn>5</mn></mrow></mrow></math>, and <math class="ltx_Math" altimg="m9.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="7" display="inline"><mn>7</mn></math>, and for all even <math class="ltx_Math" altimg="m51.png" altimg-height="20px" altimg-valign="-5px" altimg-width="62px" alttext="k\leq 24" display="inline"><mrow><mi href="./27.1#p2.t1.r1">k</mi><mo>≤</mo><mn>24</mn></mrow></math>. For values of <math class="ltx_Math" altimg="m47.png" altimg-height="18px" altimg-valign="-3px" altimg-width="62px" alttext="k>24" display="inline"><mrow><mi href="./27.1#p2.t1.r1">k</mi><mo>></mo><mn>24</mn></mrow></math> the
analysis of <math class="ltx_Math" altimg="m66.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="r_{k}\left(n\right)" display="inline"><mrow><msub><mi href="./27.13#SS4.p1">r</mi><mi href="./27.1#p2.t1.r1">k</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./27.1#p2.t1.r1">n</mi><mo>)</mo></mrow></mrow></math> is considerably more complicated (see
<cite class="ltx_cite ltx_citemacro_citet">Hardy (</div>
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<title>DLMF: 18.5 Explicit Representations</title>
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<div id="Px1.p1" class="ltx_para">
<p class="ltx_p">With <math class="ltx_Math" altimg="m140.png" altimg-height="18px" altimg-valign="-2px" altimg-width="82px" alttext="x=\cos\theta" display="inline"><mrow><mi href="./18.2#Px1.p2">x</mi><mo>=</mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi>θ</mi></mrow></mrow></math>,</p>
<table id="EGx1" class="ltx_equationgroup ltx_eqn_table">
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<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">18.5.1</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m67.png" altimg-height="25px" altimg-valign="-7px" altimg-width="59px" alttext="\displaystyle T_{n}\left(x\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r4">T</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m21.png" altimg-height="25px" altimg-valign="-7px" altimg-width="103px" alttext="\displaystyle=\cos\left(n\theta\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo></mo><mi>θ</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
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<tr><td class="ltx_align_right" colspan="5">
<div id="E1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./18.3#T1.t1.r4" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m105.png" altimg-height="23px" altimg-valign="-7px" altimg-width="57px" alttext="T_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r4">T</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Chebyshev polynomial of the first kind</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m116.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./18.1#SS1.p2.t1.r5" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m136.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m141.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E2">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">18.5.2</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m75.png" altimg-height="25px" altimg-valign="-7px" altimg-width="61px" alttext="\displaystyle U_{n}\left(x\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r5">U</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m23.png" altimg-height="25px" altimg-valign="-7px" altimg-width="200px" alttext="\displaystyle=\ifrac{(\sin(n+1)\theta)}{\sin\theta}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi>θ</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mo>/</mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi>θ</mi></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./18.3#T1.t1.r5" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m107.png" altimg-height="23px" altimg-valign="-7px" altimg-width="59px" alttext="U_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r5">U</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Chebyshev polynomial of the second kind</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m130.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./18.1#SS1.p2.t1.r5" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m136.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m141.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E3">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">18.5.3</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m76.png" altimg-height="25px" altimg-valign="-7px" altimg-width="59px" alttext="\displaystyle V_{n}\left(x\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r6">V</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m24.png" altimg-height="29px" altimg-valign="-9px" altimg-width="235px" alttext="\displaystyle=\ifrac{(\sin(n+\tfrac{1}{2})\theta)}{\sin\left(\tfrac{1}{2}%
\theta\right)}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi>θ</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mo>/</mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi>θ</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./18.3#T1.t1.r6" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m109.png" altimg-height="23px" altimg-valign="-7px" altimg-width="57px" alttext="V_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r6">V</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Chebyshev polynomial of the third kind</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m130.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./18.1#SS1.p2.t1.r5" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m136.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m141.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E4">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">18.5.4</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m77.png" altimg-height="25px" altimg-valign="-7px" altimg-width="66px" alttext="\displaystyle W_{n}\left(x\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r7">W</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m22.png" altimg-height="29px" altimg-valign="-9px" altimg-width="240px" alttext="\displaystyle=\ifrac{(\cos(n+\tfrac{1}{2})\theta)}{\cos\left(\tfrac{1}{2}%
\theta\right)}." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi>θ</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mo>/</mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi>θ</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E4.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./18.3#T1.t1.r7" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m111.png" altimg-height="23px" altimg-valign="-7px" altimg-width="64px" alttext="W_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r7">W</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Chebyshev polynomial of the fourth kind</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m116.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./18.1#SS1.p2.t1.r5" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m136.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m141.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
</div>
</section>
</section>
<section id="ii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§18.5(ii) </span>Rodrigues Formulas</h2>
<div id="SS2.info" class="ltx_metadata ltx_info">
, second row;
Row 8 is the case <math class="ltx_Math" altimg="m113.png" altimg-height="21px" altimg-valign="-6px" altimg-width="93px" alttext="\alpha=\beta=0" display="inline"><mrow><mi>α</mi><mo>=</mo><mi>β</mi><mo>=</mo><mn>0</mn></mrow></math> of Row 2.
For (</dd>
</dl>
</div>
</div>
<div id="SS2.p1" class="ltx_para">
<table id="E5" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">18.5.5</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E5.png" altimg-height="51px" altimg-valign="-21px" altimg-width="325px" alttext="p_{n}(x)=\frac{1}{\kappa_{n}w(x)}\frac{{\mathrm{d}}^{n}}{{\mathrm{d}x}^{n}}%
\left(w(x)(F(x))^{n}\right)." display="block"><mrow><mrow><mrow><msub><mi href="./18.1#SS1.p2.t1.r9">p</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./18.2#Px1.p2">x</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mfrac><mn>1</mn><mrow><msub><mi>κ</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mi href="./18.1#SS1.p2.t1.r11">w</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./18.2#Px1.p2">x</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></mfrac><mo></mo><mrow><mfrac><mpadded lspace="-1.7pt" width="-4.5pt"><msup><mo href="./1.4#E4">d</mo><mi href="./18.1#SS1.p2.t1.r5">n</mi></msup></mpadded><msup><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi href="./18.2#Px1.p2">x</mi></mrow><mi href="./1.4#E4">n</mi></msup></mfrac><mo></mo><mrow><mo>(</mo><mrow><mrow><mi href="./18.1#SS1.p2.t1.r11">w</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./18.2#Px1.p2">x</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi>F</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./18.2#Px1.p2">x</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi></msup></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E5.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#E4" title="(1.4.4) ‣ §1.4(iii) Derivatives ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m121.png" altimg-height="29px" altimg-valign="-9px" altimg-width="27px" alttext="\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: derivative of <math class="ltx_Math" altimg="m132.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m141.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./18.1#SS1.p2.t1.r11" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m139.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="w(x)" display="inline"><mrow><mi href="./18.1#SS1.p2.t1.r11">w</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./18.2#Px1.p2">x</mi><mo stretchy="false">)</mo></mrow></mrow></math>: weight function</a>,
<a href="./18.1#SS1.p2.t1.r5" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m136.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math>: nonnegative integer</a>,
<a href="./18.1#SS1.p2.t1.r9" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m137.png" altimg-height="23px" altimg-valign="-7px" altimg-width="52px" alttext="p_{n}(x)" display="inline"><mrow><msub><mi href="./18.1#SS1.p2.t1.r9">p</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./18.2#Px1.p2">x</mi><mo stretchy="false">)</mo></mrow></mrow></math>: polynomial of degree <math class="ltx_Math" altimg="m136.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math></a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m141.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">22.1.6</span> (incorrectly stated as property of general OP’s)</span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">In this equation <math class="ltx_Math" altimg="m139.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="w(x)" display="inline"><mrow><mi href="./18.1#SS1.p2.t1.r11">w</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./18.2#Px1.p2">x</mi><mo stretchy="false">)</mo></mrow></mrow></math> is as in Table , and <math class="ltx_Math" altimg="m95.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="F(x)" display="inline"><mrow><mi>F</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./18.2#Px1.p2">x</mi><mo stretchy="false">)</mo></mrow></mrow></math>,
<math class="ltx_Math" altimg="m123.png" altimg-height="16px" altimg-valign="-5px" altimg-width="27px" alttext="\kappa_{n}" display="inline"><msub><mi>κ</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi></msub></math> are as in Table ).
</figcaption>
<table id="T1.t1" class="ltx_tabular ltx_centering ltx_guessed_headers ltx_align_middle">
<thead class="ltx_thead">
<tr id="T1.t1.r1" class="ltx_tr">
<th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" style="padding-top:4.166666666666667px;padding-bottom:4.166666666666667px;"><math class="ltx_Math" altimg="m137.png" altimg-height="23px" altimg-valign="-7px" altimg-width="52px" alttext="p_{n}(x)" display="inline"><mrow><msub><mi href="./18.1#SS1.p2.t1.r9">p</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./18.2#Px1.p2">x</mi><mo stretchy="false">)</mo></mrow></mrow></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" style="padding-top:4.166666666666667px;padding-bottom:4.166666666666667px;"><math class="ltx_Math" altimg="m95.png" altimg-height="23px" altimg-valign="-7px" altimg-width="47px" alttext="F(x)" display="inline"><mrow><mi>F</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./18.2#Px1.p2">x</mi><mo stretchy="false">)</mo></mrow></mrow></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" style="padding-top:4.166666666666667px;padding-bottom:4.166666666666667px;"><math class="ltx_Math" altimg="m123.png" altimg-height="16px" altimg-valign="-5px" altimg-width="27px" alttext="\kappa_{n}" display="inline"><msub><mi>κ</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi></msub></math></th>
</tr>
</thead>
<tbody class="ltx_tbody">
<tr id="T1.t1.r2" class="ltx_tr">
<td class="ltx_td ltx_align_center ltx_border_t" style="padding-top:4.166666666666667px;padding-bottom:4.166666666666667px;"><math class="ltx_Math" altimg="m102.png" altimg-height="29px" altimg-valign="-7px" altimg-width="88px" alttext="P^{(\alpha,\beta)}_{n}\left(x\right)" display="inline"><mrow><msubsup><mi href="./18.3#T1.t1.r2">P</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r2" stretchy="false">(</mo><mi>α</mi><mo href="./18.3#T1.t1.r2">,</mo><mi>β</mi><mo href="./18.3#T1.t1.r2" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_center ltx_border_t" style="padding-top:4.166666666666667px;padding-bottom:4.166666666666667px;"><math class="ltx_Math" altimg="m89.png" altimg-height="21px" altimg-valign="-4px" altimg-width="59px" alttext="1-x^{2}" display="inline"><mrow><mn>1</mn><mo>-</mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>2</mn></msup></mrow></math></td>
<td class="ltx_td ltx_align_center ltx_border_t" style="padding-top:4.166666666666667px;padding-bottom:4.166666666666667px;"><math class="ltx_Math" altimg="m87.png" altimg-height="23px" altimg-valign="-7px" altimg-width="74px" alttext="(-2)^{n}n!" display="inline"><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>2</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi></msup><mo></mo><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mrow></math></td>
</tr>
<tr id="T1.t1.r3" class="ltx_tr">
<td class="ltx_td ltx_align_center" style="padding-top:4.166666666666667px;padding-bottom:4.166666666666667px;"><math class="ltx_Math" altimg="m94.png" altimg-height="29px" altimg-valign="-7px" altimg-width="73px" alttext="C^{(\lambda)}_{n}\left(x\right)" display="inline"><mrow><msubsup><mi href="./18.3#T1.t1.r3">C</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r3" stretchy="false">(</mo><mi>λ</mi><mo href="./18.3#T1.t1.r3" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_center" style="padding-top:4.166666666666667px;padding-bottom:4.166666666666667px;"><math class="ltx_Math" altimg="m89.png" altimg-height="21px" altimg-valign="-4px" altimg-width="59px" alttext="1-x^{2}" display="inline"><mrow><mn>1</mn><mo>-</mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>2</mn></msup></mrow></math></td>
<td class="ltx_td ltx_align_center" style="padding-top:4.166666666666667px;padding-bottom:4.166666666666667px;"><math class="ltx_Math" altimg="m119.png" altimg-height="56px" altimg-valign="-22px" altimg-width="156px" alttext="\dfrac{(-2)^{n}{\left(\lambda+\frac{1}{2}\right)_{n}}n!}{{\left(2\lambda\right%
)_{n}}}" display="inline"><mstyle displaystyle="true"><mfrac><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>2</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi></msup><mo></mo><msub><mrow><mo href="./5.2#iii">(</mo><mrow><mi>λ</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo href="./5.2#iii">)</mo></mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mrow><msub><mrow><mo href="./5.2#iii">(</mo><mrow><mn>2</mn><mo></mo><mi>λ</mi></mrow><mo href="./5.2#iii">)</mo></mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi></msub></mfrac></mstyle></math></td>
</tr>
<tr id="T1.t1.r4" class="ltx_tr">
<td class="ltx_td ltx_align_center" style="padding-top:4.166666666666667px;padding-bottom:4.166666666666667px;"><math class="ltx_Math" altimg="m106.png" altimg-height="23px" altimg-valign="-7px" altimg-width="57px" alttext="T_{n}\left(x\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r4">T</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_center" style="padding-top:4.166666666666667px;padding-bottom:4.166666666666667px;"><math class="ltx_Math" altimg="m89.png" altimg-height="21px" altimg-valign="-4px" altimg-width="59px" alttext="1-x^{2}" display="inline"><mrow><mn>1</mn><mo>-</mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>2</mn></msup></mrow></math></td>
<td class="ltx_td ltx_align_center" style="padding-top:4.166666666666667px;padding-bottom:4.166666666666667px;"><math class="ltx_Math" altimg="m88.png" altimg-height="28px" altimg-valign="-10px" altimg-width="98px" alttext="(-2)^{n}{\left(\frac{1}{2}\right)_{n}}" display="inline"><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>2</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi></msup><mo></mo><msub><mrow><mo href="./5.2#iii">(</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo href="./5.2#iii">)</mo></mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi></msub></mrow></math></td>
</tr>
<tr id="T1.t1.r5" class="ltx_tr">
<td class="ltx_td ltx_align_center" style="padding-top:4.166666666666667px;padding-bottom:4.166666666666667px;"><math class="ltx_Math" altimg="m108.png" altimg-height="23px" altimg-valign="-7px" altimg-width="59px" alttext="U_{n}\left(x\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r5">U</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_center" style="padding-top:4.166666666666667px;padding-bottom:4.166666666666667px;"><math class="ltx_Math" altimg="m89.png" altimg-height="21px" altimg-valign="-4px" altimg-width="59px" alttext="1-x^{2}" display="inline"><mrow><mn>1</mn><mo>-</mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>2</mn></msup></mrow></math></td>
<td class="ltx_td ltx_align_center" style="padding-top:4.166666666666667px;padding-bottom:4.166666666666667px;"><math class="ltx_Math" altimg="m118.png" altimg-height="52px" altimg-valign="-17px" altimg-width="103px" alttext="\dfrac{(-2)^{n}{\left(\frac{3}{2}\right)_{n}}}{n+1}" display="inline"><mstyle displaystyle="true"><mfrac><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>2</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi></msup><mo></mo><msub><mrow><mo href="./5.2#iii">(</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mo href="./5.2#iii">)</mo></mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi></msub></mrow><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mstyle></math></td>
</tr>
<tr id="T1.t1.r6" class="ltx_tr">
<td class="ltx_td ltx_align_center" style="padding-top:4.166666666666667px;padding-bottom:4.166666666666667px;"><math class="ltx_Math" altimg="m110.png" altimg-height="23px" altimg-valign="-7px" altimg-width="57px" alttext="V_{n}\left(x\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r6">V</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_center" style="padding-top:4.166666666666667px;padding-bottom:4.166666666666667px;"><math class="ltx_Math" altimg="m89.png" altimg-height="21px" altimg-valign="-4px" altimg-width="59px" alttext="1-x^{2}" display="inline"><mrow><mn>1</mn><mo>-</mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>2</mn></msup></mrow></math></td>
<td class="ltx_td ltx_align_center" style="padding-top:4.166666666666667px;padding-bottom:4.166666666666667px;"><math class="ltx_Math" altimg="m117.png" altimg-height="52px" altimg-valign="-17px" altimg-width="103px" alttext="\dfrac{(-2)^{n}{\left(\frac{3}{2}\right)_{n}}}{2n+1}" display="inline"><mstyle displaystyle="true"><mfrac><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>2</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi></msup><mo></mo><msub><mrow><mo href="./5.2#iii">(</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mo href="./5.2#iii">)</mo></mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi></msub></mrow><mrow><mrow><mn>2</mn><mo></mo><mi href="./18.1#SS1.p2.t1.r5">n</mi></mrow><mo>+</mo><mn>1</mn></mrow></mfrac></mstyle></math></td>
</tr>
<tr id="T1.t1.r7" class="ltx_tr">
<td class="ltx_td ltx_align_center" style="padding-top:4.166666666666667px;padding-bottom:4.166666666666667px;"><math class="ltx_Math" altimg="m112.png" altimg-height="23px" altimg-valign="-7px" altimg-width="64px" alttext="W_{n}\left(x\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r7">W</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_center" style="padding-top:4.166666666666667px;padding-bottom:4.166666666666667px;"><math class="ltx_Math" altimg="m89.png" altimg-height="21px" altimg-valign="-4px" altimg-width="59px" alttext="1-x^{2}" display="inline"><mrow><mn>1</mn><mo>-</mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>2</mn></msup></mrow></math></td>
<td class="ltx_td ltx_align_center" style="padding-top:4.166666666666667px;padding-bottom:4.166666666666667px;"><math class="ltx_Math" altimg="m88.png" altimg-height="28px" altimg-valign="-10px" altimg-width="98px" alttext="(-2)^{n}{\left(\frac{1}{2}\right)_{n}}" display="inline"><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>2</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi></msup><mo></mo><msub><mrow><mo href="./5.2#iii">(</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo href="./5.2#iii">)</mo></mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi></msub></mrow></math></td>
</tr>
<tr id="T1.t1.r8" class="ltx_tr">
<td class="ltx_td ltx_align_center" style="padding-top:4.166666666666667px;padding-bottom:4.166666666666667px;"><math class="ltx_Math" altimg="m104.png" altimg-height="23px" altimg-valign="-7px" altimg-width="58px" alttext="P_{n}\left(x\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r10">P</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_center" style="padding-top:4.166666666666667px;padding-bottom:4.166666666666667px;"><math class="ltx_Math" altimg="m89.png" altimg-height="21px" altimg-valign="-4px" altimg-width="59px" alttext="1-x^{2}" display="inline"><mrow><mn>1</mn><mo>-</mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>2</mn></msup></mrow></math></td>
<td class="ltx_td ltx_align_center" style="padding-top:4.166666666666667px;padding-bottom:4.166666666666667px;"><math class="ltx_Math" altimg="m87.png" altimg-height="23px" altimg-valign="-7px" altimg-width="74px" alttext="(-2)^{n}n!" display="inline"><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>2</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi></msup><mo></mo><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mrow></math></td>
</tr>
<tr id="T1.t1.r9" class="ltx_tr">
<td class="ltx_td ltx_align_center" style="padding-top:4.166666666666667px;padding-bottom:4.166666666666667px;"><math class="ltx_Math" altimg="m99.png" altimg-height="29px" altimg-valign="-7px" altimg-width="72px" alttext="L^{(\alpha)}_{n}\left(x\right)" display="inline"><mrow><msubsup><mi href="./18.3#T1.t1.r12">L</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r12" stretchy="false">(</mo><mi>α</mi><mo href="./18.3#T1.t1.r12" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_center" style="padding-top:4.166666666666667px;padding-bottom:4.166666666666667px;"><math class="ltx_Math" altimg="m141.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math></td>
<td class="ltx_td ltx_align_center" style="padding-top:4.166666666666667px;padding-bottom:4.166666666666667px;"><math class="ltx_Math" altimg="m134.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math></td>
</tr>
<tr id="T1.t1.r10" class="ltx_tr">
<td class="ltx_td ltx_align_center" style="padding-top:4.166666666666667px;padding-bottom:4.166666666666667px;"><math class="ltx_Math" altimg="m97.png" altimg-height="23px" altimg-valign="-7px" altimg-width="62px" alttext="H_{n}\left(x\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r13">H</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_center" style="padding-top:4.166666666666667px;padding-bottom:4.166666666666667px;"><math class="ltx_Math" altimg="m90.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="1" display="inline"><mn>1</mn></math></td>
<td class="ltx_td ltx_align_center" style="padding-top:4.166666666666667px;padding-bottom:4.166666666666667px;"><math class="ltx_Math" altimg="m86.png" altimg-height="23px" altimg-valign="-7px" altimg-width="56px" alttext="(-1)^{n}" display="inline"><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi></msup></math></td>
</tr>
<tr id="T1.t1.r11" class="ltx_tr">
<td class="ltx_td ltx_align_center ltx_border_b" style="padding-top:4.166666666666667px;padding-bottom:4.166666666666667px;"><math class="ltx_Math" altimg="m128.png" altimg-height="23px" altimg-valign="-7px" altimg-width="71px" alttext="\mathit{He}_{n}\left(x\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r14" mathvariant="italic">He</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_center ltx_border_b" style="padding-top:4.166666666666667px;padding-bottom:4.166666666666667px;"><math class="ltx_Math" altimg="m90.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="1" display="inline"><mn>1</mn></math></td>
<td class="ltx_td ltx_align_center ltx_border_b" style="padding-top:4.166666666666667px;padding-bottom:4.166666666666667px;"><math class="ltx_Math" altimg="m86.png" altimg-height="23px" altimg-valign="-7px" altimg-width="56px" alttext="(-1)^{n}" display="inline"><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi></msup></math></td>
</tr>
</tbody>
</table>
<div id="T1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./18.3#T1.t1.r4" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m105.png" altimg-height="23px" altimg-valign="-7px" altimg-width="57px" alttext="T_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r4">T</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Chebyshev polynomial of the first kind</a>,
<a href="./18.3#T1.t1.r7" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m111.png" altimg-height="23px" altimg-valign="-7px" altimg-width="64px" alttext="W_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r7">W</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Chebyshev polynomial of the fourth kind</a>,
<a href="./18.3#T1.t1.r5" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m107.png" altimg-height="23px" altimg-valign="-7px" altimg-width="59px" alttext="U_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r5">U</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Chebyshev polynomial of the second kind</a>,
<a href="./18.3#T1.t1.r6" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m109.png" altimg-height="23px" altimg-valign="-7px" altimg-width="57px" alttext="V_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r6">V</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Chebyshev polynomial of the third kind</a>,
<a href="./18.3#T1.t1.r13" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m96.png" altimg-height="23px" altimg-valign="-7px" altimg-width="62px" alttext="H_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r13">H</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Hermite polynomial</a>,
<a href="./18.3#T1.t1.r14" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m126.png" altimg-height="23px" altimg-valign="-7px" altimg-width="71px" alttext="\mathit{He}_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r14" mathvariant="italic">He</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Hermite polynomial</a>,
<a href="./18.3#T1.t1.r2" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m101.png" altimg-height="29px" altimg-valign="-7px" altimg-width="88px" alttext="P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msubsup><mi href="./18.3#T1.t1.r2">P</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r2" stretchy="false">(</mo><mi class="ltx_nvar">α</mi><mo href="./18.3#T1.t1.r2">,</mo><mi class="ltx_nvar">β</mi><mo href="./18.3#T1.t1.r2" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Jacobi polynomial</a>,
<a href="./18.3#T1.t1.r12" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m98.png" altimg-height="29px" altimg-valign="-7px" altimg-width="72px" alttext="L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msubsup><mi href="./18.3#T1.t1.r12">L</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r12" stretchy="false">(</mo><mi class="ltx_nvar">α</mi><mo href="./18.3#T1.t1.r12" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Laguerre (or generalized Laguerre) polynomial</a>,
<a href="./18.3#T1.t1.r10" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="23px" altimg-valign="-7px" altimg-width="58px" alttext="P_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r10">P</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Legendre polynomial</a>,
<a href="./5.2#iii" title="§5.2(iii) Pochhammer’s Symbol ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m142.png" altimg-height="24px" altimg-valign="-8px" altimg-width="41px" alttext="{\left(\NVar{a}\right)_{\NVar{n}}}" display="inline"><msub><mrow><mo href="./5.2#iii">(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r5">a</mi><mo href="./5.2#iii">)</mo></mrow><mi class="ltx_nvar" href="./5.1#p2.t1.r1">n</mi></msub></math>: Pochhammer’s symbol (or shifted factorial)</a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m134.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./18.3#T1.t1.r3" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m93.png" altimg-height="29px" altimg-valign="-7px" altimg-width="73px" alttext="C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msubsup><mi href="./18.3#T1.t1.r3">C</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r3" stretchy="false">(</mo><mi class="ltx_nvar">λ</mi><mo href="./18.3#T1.t1.r3" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: ultraspherical (or Gegenbauer) polynomial</a>,
<a href="./18.1#SS1.p2.t1.r5" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m136.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math>: nonnegative integer</a>,
<a href="./18.1#SS1.p2.t1.r9" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m137.png" altimg-height="23px" altimg-valign="-7px" altimg-width="52px" alttext="p_{n}(x)" display="inline"><mrow><msub><mi href="./18.1#SS1.p2.t1.r9">p</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./18.2#Px1.p2">x</mi><mo stretchy="false">)</mo></mrow></mrow></math>: polynomial of degree <math class="ltx_Math" altimg="m136.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math></a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m141.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
<dt>Keywords:</dt>
<dd>
</dd>
</dl>
</div>
</div>
</figure>
<div id="SS2.p2" class="ltx_para">
<p class="ltx_p">Related formula:</p>
<table id="E6" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">18.5.6</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E6.png" altimg-height="53px" altimg-valign="-21px" altimg-width="462px" alttext="L^{(\alpha)}_{n}\left(\frac{1}{x}\right)=\frac{(-1)^{n}}{n!}x^{n+\alpha+1}e^{%
\ifrac{1}{x}}\frac{{\mathrm{d}}^{n}}{{\mathrm{d}x}^{n}}\left(x^{-\alpha-1}e^{-%
\ifrac{1}{x}}\right)." display="block"><mrow><mrow><mrow><msubsup><mi href="./18.3#T1.t1.r12">L</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r12" stretchy="false">(</mo><mi>α</mi><mo href="./18.3#T1.t1.r12" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mfrac><mn>1</mn><mi href="./18.2#Px1.p2">x</mi></mfrac><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mfrac><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi></msup><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mfrac><mo></mo><msup><mi href="./18.2#Px1.p2">x</mi><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>+</mo><mi>α</mi><mo>+</mo><mn>1</mn></mrow></msup><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mn>1</mn><mo>/</mo><mi href="./18.2#Px1.p2">x</mi></mrow></msup><mo></mo><mrow><mfrac><mpadded lspace="-1.7pt" width="-4.5pt"><msup><mo href="./1.4#E4">d</mo><mi href="./18.1#SS1.p2.t1.r5">n</mi></msup></mpadded><msup><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi href="./18.2#Px1.p2">x</mi></mrow><mi href="./1.4#E4">n</mi></msup></mfrac><mo></mo><mrow><mo>(</mo><mrow><msup><mi href="./18.2#Px1.p2">x</mi><mrow><mrow><mo>-</mo><mi>α</mi></mrow><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mn>1</mn><mo>/</mo><mi href="./18.2#Px1.p2">x</mi></mrow></mrow></msup></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E6.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./18.3#T1.t1.r12" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m98.png" altimg-height="29px" altimg-valign="-7px" altimg-width="72px" alttext="L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msubsup><mi href="./18.3#T1.t1.r12">L</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r12" stretchy="false">(</mo><mi class="ltx_nvar">α</mi><mo href="./18.3#T1.t1.r12" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Laguerre (or generalized Laguerre) polynomial</a>,
<a href="./1.4#E4" title="(1.4.4) ‣ §1.4(iii) Derivatives ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m121.png" altimg-height="29px" altimg-valign="-9px" altimg-width="27px" alttext="\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: derivative of <math class="ltx_Math" altimg="m132.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m141.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m129.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m134.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./18.1#SS1.p2.t1.r5" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m136.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m141.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="iii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§18.5(iii) </span>Finite Power Series, the Hypergeometric Function, and Generalized
Hypergeometric Functions</h2>
<div id="SS3.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="SS3.p1" class="ltx_para">
<p class="ltx_p">For the definitions of <math class="ltx_Math" altimg="m146.png" altimg-height="21px" altimg-valign="-5px" altimg-width="35px" alttext="{{}_{2}F_{1}}" display="inline"><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts></math>, <math class="ltx_Math" altimg="m143.png" altimg-height="21px" altimg-valign="-5px" altimg-width="35px" alttext="{{}_{1}F_{1}}" display="inline"><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>1</mn><none></none></mmultiscripts></math>, and <math class="ltx_Math" altimg="m145.png" altimg-height="21px" altimg-valign="-5px" altimg-width="35px" alttext="{{}_{2}F_{0}}" display="inline"><mmultiscripts><mi href="./16.2">F</mi><mn>0</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts></math>
see §</dd>
</dl>
</div>
</div>
<div id="Px2.p1" class="ltx_para">
<table id="E7" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">18.5.7</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E7.png" altimg-height="65px" altimg-valign="-28px" altimg-width="904px" alttext="P^{(\alpha,\beta)}_{n}\left(x\right)=\sum_{\ell=0}^{n}\frac{{\left(n+\alpha+%
\beta+1\right)_{\ell}}{\left(\alpha+\ell+1\right)_{n-\ell}}}{\ell!\;(n-\ell)!}%
\left(\frac{x-1}{2}\right)^{\ell}=\frac{{\left(\alpha+1\right)_{n}}}{n!}{{}_{2%
}F_{1}}\left({-n,n+\alpha+\beta+1\atop\alpha+1};\frac{1-x}{2}\right)," display="block"><mrow><mrow><mrow><msubsup><mi href="./18.3#T1.t1.r2">P</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r2" stretchy="false">(</mo><mi>α</mi><mo href="./18.3#T1.t1.r2">,</mo><mi>β</mi><mo href="./18.3#T1.t1.r2" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./18.1#SS1.p2.t1.r4" mathvariant="normal">ℓ</mi><mo>=</mo><mn>0</mn></mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi></munderover><mrow><mfrac><mrow><msub><mrow><mo href="./5.2#iii">(</mo><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>+</mo><mi>α</mi><mo>+</mo><mi>β</mi><mo>+</mo><mn>1</mn></mrow><mo href="./5.2#iii">)</mo></mrow><mi href="./18.1#SS1.p2.t1.r4" mathvariant="normal">ℓ</mi></msub><mo></mo><msub><mrow><mo href="./5.2#iii">(</mo><mrow><mi>α</mi><mo>+</mo><mi href="./18.1#SS1.p2.t1.r4" mathvariant="normal">ℓ</mi><mo>+</mo><mn>1</mn></mrow><mo href="./5.2#iii">)</mo></mrow><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>-</mo><mi href="./18.1#SS1.p2.t1.r4" mathvariant="normal">ℓ</mi></mrow></msub></mrow><mrow><mrow><mi href="./18.1#SS1.p2.t1.r4" mathvariant="normal">ℓ</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow><mo></mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>-</mo><mi href="./18.1#SS1.p2.t1.r4" mathvariant="normal">ℓ</mi></mrow><mo stretchy="false">)</mo></mrow><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mrow></mfrac><mo></mo><msup><mrow><mo>(</mo><mfrac><mrow><mi href="./18.2#Px1.p2">x</mi><mo>-</mo><mn>1</mn></mrow><mn>2</mn></mfrac><mo>)</mo></mrow><mi href="./18.1#SS1.p2.t1.r4" mathvariant="normal">ℓ</mi></msup></mrow></mrow><mo>=</mo><mrow><mfrac><msub><mrow><mo href="./5.2#iii">(</mo><mrow><mi>α</mi><mo>+</mo><mn>1</mn></mrow><mo href="./5.2#iii">)</mo></mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi></msub><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mfrac><mo></mo><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mrow><mo>-</mo><mi href="./18.1#SS1.p2.t1.r5">n</mi></mrow><mo>,</mo><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>+</mo><mi>α</mi><mo>+</mo><mi>β</mi><mo>+</mo><mn>1</mn></mrow></mrow><mrow><mi>α</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mo>;</mo><mfrac><mrow><mn>1</mn><mo>-</mo><mi href="./18.2#Px1.p2">x</mi></mrow><mn>2</mn></mfrac><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E7.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./16.2" title="§16.2 Definition and Analytic Properties ‣ Generalized Hypergeometric Functions ‣ Chapter 16 Generalized Hypergeometric Functions and Meijer G -Function" class="ltx_ref"><math class="ltx_Math" altimg="m147.png" altimg-height="23px" altimg-valign="-7px" altimg-width="118px" alttext="{{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mrow><mi class="ltx_nvar" href="./16.1#p2.t1.r5">a</mi><mo>,</mo><mi class="ltx_nvar" href="./16.1#p2.t1.r5">b</mi></mrow><mo>;</mo><mi class="ltx_nvar">c</mi><mo>;</mo><mi class="ltx_nvar" href="./16.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: <math class="ltx_Math" altimg="m91.png" altimg-height="23px" altimg-valign="-7px" altimg-width="124px" alttext="=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./16.1#p2.t1.r5">a</mi><mo>,</mo><mi class="ltx_nvar" href="./16.1#p2.t1.r5">b</mi><mo>;</mo><mi class="ltx_nvar">c</mi><mo>;</mo><mi class="ltx_nvar" href="./16.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></math> notation for Gauss’ hypergeometric function</a>,
<a href="./18.3#T1.t1.r2" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m101.png" altimg-height="29px" altimg-valign="-7px" altimg-width="88px" alttext="P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msubsup><mi href="./18.3#T1.t1.r2">P</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r2" stretchy="false">(</mo><mi class="ltx_nvar">α</mi><mo href="./18.3#T1.t1.r2">,</mo><mi class="ltx_nvar">β</mi><mo href="./18.3#T1.t1.r2" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Jacobi polynomial</a>,
<a href="./5.2#iii" title="§5.2(iii) Pochhammer’s Symbol ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m142.png" altimg-height="24px" altimg-valign="-8px" altimg-width="41px" alttext="{\left(\NVar{a}\right)_{\NVar{n}}}" display="inline"><msub><mrow><mo href="./5.2#iii">(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r5">a</mi><mo href="./5.2#iii">)</mo></mrow><mi class="ltx_nvar" href="./5.1#p2.t1.r1">n</mi></msub></math>: Pochhammer’s symbol (or shifted factorial)</a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m134.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./18.1#SS1.p2.t1.r4" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m120.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="\ell" display="inline"><mi href="./18.1#SS1.p2.t1.r4" mathvariant="normal">ℓ</mi></math>: nonnegative integer</a>,
<a href="./18.1#SS1.p2.t1.r5" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m136.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m141.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">22.5.42</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E8" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">18.5.8</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E8.png" altimg-height="94px" altimg-valign="-28px" altimg-width="861px" alttext="P^{(\alpha,\beta)}_{n}\left(x\right)=2^{-n}\sum_{\ell=0}^{n}\genfrac{(}{)}{0.0%
pt}{}{n+\alpha}{\ell}\genfrac{(}{)}{0.0pt}{}{n+\beta}{n-\ell}(x-1)^{n-\ell}(x+%
1)^{\ell}=\frac{{\left(\alpha+1\right)_{n}}}{n!}\left(\frac{x+1}{2}\right)^{n}%
{{}_{2}F_{1}}\left({-n,-n-\beta\atop\alpha+1};\frac{x-1}{x+1}\right)," display="block"><mrow><mrow><mrow><msubsup><mi href="./18.3#T1.t1.r2">P</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r2" stretchy="false">(</mo><mi>α</mi><mo href="./18.3#T1.t1.r2">,</mo><mi>β</mi><mo href="./18.3#T1.t1.r2" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msup><mn>2</mn><mrow><mo>-</mo><mi href="./18.1#SS1.p2.t1.r5">n</mi></mrow></msup><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./18.1#SS1.p2.t1.r4" mathvariant="normal">ℓ</mi><mo>=</mo><mn>0</mn></mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi></munderover><mrow><mrow><mo href="./1.2#i">(</mo><mfrac linethickness="0.0pt"><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>+</mo><mi>α</mi></mrow><mi href="./18.1#SS1.p2.t1.r4" mathvariant="normal">ℓ</mi></mfrac><mo href="./1.2#i">)</mo></mrow><mo></mo><mrow><mo href="./1.2#i">(</mo><mfrac linethickness="0.0pt"><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>+</mo><mi>β</mi></mrow><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>-</mo><mi href="./18.1#SS1.p2.t1.r4" mathvariant="normal">ℓ</mi></mrow></mfrac><mo href="./1.2#i">)</mo></mrow><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./18.2#Px1.p2">x</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>-</mo><mi href="./18.1#SS1.p2.t1.r4" mathvariant="normal">ℓ</mi></mrow></msup><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./18.2#Px1.p2">x</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./18.1#SS1.p2.t1.r4" mathvariant="normal">ℓ</mi></msup></mrow></mrow></mrow><mo>=</mo><mrow><mfrac><msub><mrow><mo href="./5.2#iii">(</mo><mrow><mi>α</mi><mo>+</mo><mn>1</mn></mrow><mo href="./5.2#iii">)</mo></mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi></msub><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mfrac><mo></mo><msup><mrow><mo>(</mo><mfrac><mrow><mi href="./18.2#Px1.p2">x</mi><mo>+</mo><mn>1</mn></mrow><mn>2</mn></mfrac><mo>)</mo></mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi></msup><mo></mo><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mrow><mo>-</mo><mi href="./18.1#SS1.p2.t1.r5">n</mi></mrow><mo>,</mo><mrow><mrow><mo>-</mo><mi href="./18.1#SS1.p2.t1.r5">n</mi></mrow><mo>-</mo><mi>β</mi></mrow></mrow><mrow><mi>α</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mo>;</mo><mfrac><mrow><mi href="./18.2#Px1.p2">x</mi><mo>-</mo><mn>1</mn></mrow><mrow><mi href="./18.2#Px1.p2">x</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E8.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./16.2" title="§16.2 Definition and Analytic Properties ‣ Generalized Hypergeometric Functions ‣ Chapter 16 Generalized Hypergeometric Functions and Meijer G -Function" class="ltx_ref"><math class="ltx_Math" altimg="m147.png" altimg-height="23px" altimg-valign="-7px" altimg-width="118px" alttext="{{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mrow><mi class="ltx_nvar" href="./16.1#p2.t1.r5">a</mi><mo>,</mo><mi class="ltx_nvar" href="./16.1#p2.t1.r5">b</mi></mrow><mo>;</mo><mi class="ltx_nvar">c</mi><mo>;</mo><mi class="ltx_nvar" href="./16.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: <math class="ltx_Math" altimg="m91.png" altimg-height="23px" altimg-valign="-7px" altimg-width="124px" alttext="=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./16.1#p2.t1.r5">a</mi><mo>,</mo><mi class="ltx_nvar" href="./16.1#p2.t1.r5">b</mi><mo>;</mo><mi class="ltx_nvar">c</mi><mo>;</mo><mi class="ltx_nvar" href="./16.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></math> notation for Gauss’ hypergeometric function</a>,
<a href="./18.3#T1.t1.r2" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m101.png" altimg-height="29px" altimg-valign="-7px" altimg-width="88px" alttext="P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msubsup><mi href="./18.3#T1.t1.r2">P</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r2" stretchy="false">(</mo><mi class="ltx_nvar">α</mi><mo href="./18.3#T1.t1.r2">,</mo><mi class="ltx_nvar">β</mi><mo href="./18.3#T1.t1.r2" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Jacobi polynomial</a>,
<a href="./5.2#iii" title="§5.2(iii) Pochhammer’s Symbol ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m142.png" altimg-height="24px" altimg-valign="-8px" altimg-width="41px" alttext="{\left(\NVar{a}\right)_{\NVar{n}}}" display="inline"><msub><mrow><mo href="./5.2#iii">(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r5">a</mi><mo href="./5.2#iii">)</mo></mrow><mi class="ltx_nvar" href="./5.1#p2.t1.r1">n</mi></msub></math>: Pochhammer’s symbol (or shifted factorial)</a>,
<a href="./1.2#i" title="§1.2(i) Binomial Coefficients ‣ §1.2 Elementary Algebra ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m122.png" altimg-height="27px" altimg-valign="-9px" altimg-width="37px" alttext="\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}" display="inline"><mrow><mo href="./1.2#i">(</mo><mfrac linethickness="0.0pt"><mi class="ltx_nvar" href="./1.1#p2.t1.r5">m</mi><mi class="ltx_nvar" href="./1.1#p2.t1.r5">n</mi></mfrac><mo href="./1.2#i">)</mo></mrow></math>: binomial coefficient</a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m134.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./18.1#SS1.p2.t1.r4" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m120.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="\ell" display="inline"><mi href="./18.1#SS1.p2.t1.r4" mathvariant="normal">ℓ</mi></math>: nonnegative integer</a>,
<a href="./18.1#SS1.p2.t1.r5" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m136.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m141.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">22.3.1</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and two similar formulas by symmetry; compare the second row in Table
</dd>
</dl>
</div>
</div>
<div id="Px3.p1" class="ltx_para">
<table id="E9" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">18.5.9</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E9.png" altimg-height="55px" altimg-valign="-23px" altimg-width="385px" alttext="C^{(\lambda)}_{n}\left(x\right)=\frac{{\left(2\lambda\right)_{n}}}{n!}{{}_{2}F%
_{1}}\left({-n,n+2\lambda\atop\lambda+\tfrac{1}{2}};\frac{1-x}{2}\right)," display="block"><mrow><mrow><mrow><msubsup><mi href="./18.3#T1.t1.r3">C</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r3" stretchy="false">(</mo><mi>λ</mi><mo href="./18.3#T1.t1.r3" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mfrac><msub><mrow><mo href="./5.2#iii">(</mo><mrow><mn>2</mn><mo></mo><mi>λ</mi></mrow><mo href="./5.2#iii">)</mo></mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi></msub><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mfrac><mo></mo><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mrow><mo>-</mo><mi href="./18.1#SS1.p2.t1.r5">n</mi></mrow><mo>,</mo><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>+</mo><mrow><mn>2</mn><mo></mo><mi>λ</mi></mrow></mrow></mrow><mrow><mi>λ</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mfrac><mo>;</mo><mfrac><mrow><mn>1</mn><mo>-</mo><mi href="./18.2#Px1.p2">x</mi></mrow><mn>2</mn></mfrac><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E9.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./16.2" title="§16.2 Definition and Analytic Properties ‣ Generalized Hypergeometric Functions ‣ Chapter 16 Generalized Hypergeometric Functions and Meijer G -Function" class="ltx_ref"><math class="ltx_Math" altimg="m147.png" altimg-height="23px" altimg-valign="-7px" altimg-width="118px" alttext="{{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mrow><mi class="ltx_nvar" href="./16.1#p2.t1.r5">a</mi><mo>,</mo><mi class="ltx_nvar" href="./16.1#p2.t1.r5">b</mi></mrow><mo>;</mo><mi class="ltx_nvar">c</mi><mo>;</mo><mi class="ltx_nvar" href="./16.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: <math class="ltx_Math" altimg="m91.png" altimg-height="23px" altimg-valign="-7px" altimg-width="124px" alttext="=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./16.1#p2.t1.r5">a</mi><mo>,</mo><mi class="ltx_nvar" href="./16.1#p2.t1.r5">b</mi><mo>;</mo><mi class="ltx_nvar">c</mi><mo>;</mo><mi class="ltx_nvar" href="./16.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></math> notation for Gauss’ hypergeometric function</a>,
<a href="./5.2#iii" title="§5.2(iii) Pochhammer’s Symbol ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m142.png" altimg-height="24px" altimg-valign="-8px" altimg-width="41px" alttext="{\left(\NVar{a}\right)_{\NVar{n}}}" display="inline"><msub><mrow><mo href="./5.2#iii">(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r5">a</mi><mo href="./5.2#iii">)</mo></mrow><mi class="ltx_nvar" href="./5.1#p2.t1.r1">n</mi></msub></math>: Pochhammer’s symbol (or shifted factorial)</a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m134.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./18.3#T1.t1.r3" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m93.png" altimg-height="29px" altimg-valign="-7px" altimg-width="73px" alttext="C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msubsup><mi href="./18.3#T1.t1.r3">C</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r3" stretchy="false">(</mo><mi class="ltx_nvar">λ</mi><mo href="./18.3#T1.t1.r3" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: ultraspherical (or Gegenbauer) polynomial</a>,
<a href="./18.1#SS1.p2.t1.r5" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m136.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m141.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">22.5.46</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E10" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">18.5.10</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E10.png" altimg-height="71px" altimg-valign="-28px" altimg-width="685px" alttext="C^{(\lambda)}_{n}\left(x\right)=\sum_{\ell=0}^{\left\lfloor n/2\right\rfloor}%
\frac{(-1)^{\ell}{\left(\lambda\right)_{n-\ell}}}{\ell!\;(n-2\ell)!}(2x)^{n-2%
\ell}=(2x)^{n}\frac{{\left(\lambda\right)_{n}}}{n!}{{}_{2}F_{1}}\left({-\tfrac%
{1}{2}n,-\tfrac{1}{2}n+\tfrac{1}{2}\atop 1-\lambda-n};\frac{1}{x^{2}}\right)," display="block"><mrow><mrow><mrow><msubsup><mi href="./18.3#T1.t1.r3">C</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r3" stretchy="false">(</mo><mi>λ</mi><mo href="./18.3#T1.t1.r3" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./18.1#SS1.p2.t1.r4" mathvariant="normal">ℓ</mi><mo>=</mo><mn>0</mn></mrow><mrow><mo href="./front/introduction#Sx4.p1.t1.r16">⌊</mo><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>/</mo><mn>2</mn></mrow><mo href="./front/introduction#Sx4.p1.t1.r16">⌋</mo></mrow></munderover><mrow><mfrac><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./18.1#SS1.p2.t1.r4" mathvariant="normal">ℓ</mi></msup><mo></mo><msub><mrow><mo href="./5.2#iii">(</mo><mi>λ</mi><mo href="./5.2#iii">)</mo></mrow><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>-</mo><mi href="./18.1#SS1.p2.t1.r4" mathvariant="normal">ℓ</mi></mrow></msub></mrow><mrow><mrow><mi href="./18.1#SS1.p2.t1.r4" mathvariant="normal">ℓ</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow><mo></mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>-</mo><mrow><mn>2</mn><mo></mo><mi href="./18.1#SS1.p2.t1.r4" mathvariant="normal">ℓ</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mrow></mfrac><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mn>2</mn><mo></mo><mi href="./18.2#Px1.p2">x</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>-</mo><mrow><mn>2</mn><mo></mo><mi href="./18.1#SS1.p2.t1.r4" mathvariant="normal">ℓ</mi></mrow></mrow></msup></mrow></mrow><mo>=</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mn>2</mn><mo></mo><mi href="./18.2#Px1.p2">x</mi></mrow><mo stretchy="false">)</mo></mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi></msup><mo></mo><mfrac><msub><mrow><mo href="./5.2#iii">(</mo><mi>λ</mi><mo href="./5.2#iii">)</mo></mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi></msub><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mfrac><mo></mo><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./18.1#SS1.p2.t1.r5">n</mi></mrow></mrow><mo>,</mo><mrow><mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./18.1#SS1.p2.t1.r5">n</mi></mrow></mrow><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow><mrow><mn>1</mn><mo>-</mo><mi>λ</mi><mo>-</mo><mi href="./18.1#SS1.p2.t1.r5">n</mi></mrow></mfrac><mo>;</mo><mfrac><mn>1</mn><msup><mi href="./18.2#Px1.p2">x</mi><mn>2</mn></msup></mfrac><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E10.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./16.2" title="§16.2 Definition and Analytic Properties ‣ Generalized Hypergeometric Functions ‣ Chapter 16 Generalized Hypergeometric Functions and Meijer G -Function" class="ltx_ref"><math class="ltx_Math" altimg="m147.png" altimg-height="23px" altimg-valign="-7px" altimg-width="118px" alttext="{{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mrow><mi class="ltx_nvar" href="./16.1#p2.t1.r5">a</mi><mo>,</mo><mi class="ltx_nvar" href="./16.1#p2.t1.r5">b</mi></mrow><mo>;</mo><mi class="ltx_nvar">c</mi><mo>;</mo><mi class="ltx_nvar" href="./16.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: <math class="ltx_Math" altimg="m91.png" altimg-height="23px" altimg-valign="-7px" altimg-width="124px" alttext="=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./16.1#p2.t1.r5">a</mi><mo>,</mo><mi class="ltx_nvar" href="./16.1#p2.t1.r5">b</mi><mo>;</mo><mi class="ltx_nvar">c</mi><mo>;</mo><mi class="ltx_nvar" href="./16.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></math> notation for Gauss’ hypergeometric function</a>,
<a href="./5.2#iii" title="§5.2(iii) Pochhammer’s Symbol ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m142.png" altimg-height="24px" altimg-valign="-8px" altimg-width="41px" alttext="{\left(\NVar{a}\right)_{\NVar{n}}}" display="inline"><msub><mrow><mo href="./5.2#iii">(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r5">a</mi><mo href="./5.2#iii">)</mo></mrow><mi class="ltx_nvar" href="./5.1#p2.t1.r1">n</mi></msub></math>: Pochhammer’s symbol (or shifted factorial)</a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m134.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./front/introduction#Sx4.p1.t1.r16" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m125.png" altimg-height="23px" altimg-valign="-7px" altimg-width="33px" alttext="\left\lfloor\NVar{x}\right\rfloor" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r16">⌊</mo><mi class="ltx_nvar">x</mi><mo href="./front/introduction#Sx4.p1.t1.r16">⌋</mo></mrow></math>: floor of <math class="ltx_Math" altimg="m141.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./18.3#T1.t1.r3" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m93.png" altimg-height="29px" altimg-valign="-7px" altimg-width="73px" alttext="C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msubsup><mi href="./18.3#T1.t1.r3">C</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r3" stretchy="false">(</mo><mi class="ltx_nvar">λ</mi><mo href="./18.3#T1.t1.r3" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: ultraspherical (or Gegenbauer) polynomial</a>,
<a href="./18.1#SS1.p2.t1.r4" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m120.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="\ell" display="inline"><mi href="./18.1#SS1.p2.t1.r4" mathvariant="normal">ℓ</mi></math>: nonnegative integer</a>,
<a href="./18.1#SS1.p2.t1.r5" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m136.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m141.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">22.3.4</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E11" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">18.5.11</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E11.png" altimg-height="64px" altimg-valign="-28px" altimg-width="701px" alttext="C^{(\lambda)}_{n}\left(\cos\theta\right)=\sum_{\ell=0}^{n}\frac{{\left(\lambda%
\right)_{\ell}}{\left(\lambda\right)_{n-\ell}}}{\ell!\;(n-\ell)!}\cos\left((n-%
2\ell)\theta\right)=e^{\mathrm{i}n\theta}\frac{{\left(\lambda\right)_{n}}}{n!}%
{{}_{2}F_{1}}\left({-n,\lambda\atop 1-\lambda-n};e^{-2\mathrm{i}\theta}\right)." display="block"><mrow><mrow><mrow><msubsup><mi href="./18.3#T1.t1.r3">C</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r3" stretchy="false">(</mo><mi>λ</mi><mo href="./18.3#T1.t1.r3" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi>θ</mi></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./18.1#SS1.p2.t1.r4" mathvariant="normal">ℓ</mi><mo>=</mo><mn>0</mn></mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi></munderover><mrow><mfrac><mrow><msub><mrow><mo href="./5.2#iii">(</mo><mi>λ</mi><mo href="./5.2#iii">)</mo></mrow><mi href="./18.1#SS1.p2.t1.r4" mathvariant="normal">ℓ</mi></msub><mo></mo><msub><mrow><mo href="./5.2#iii">(</mo><mi>λ</mi><mo href="./5.2#iii">)</mo></mrow><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>-</mo><mi href="./18.1#SS1.p2.t1.r4" mathvariant="normal">ℓ</mi></mrow></msub></mrow><mrow><mrow><mi href="./18.1#SS1.p2.t1.r4" mathvariant="normal">ℓ</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow><mo></mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>-</mo><mi href="./18.1#SS1.p2.t1.r4" mathvariant="normal">ℓ</mi></mrow><mo stretchy="false">)</mo></mrow><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mrow></mfrac><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>-</mo><mrow><mn>2</mn><mo></mo><mi href="./18.1#SS1.p2.t1.r4" mathvariant="normal">ℓ</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi>θ</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>=</mo><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo></mo><mi>θ</mi></mrow></msup><mo></mo><mfrac><msub><mrow><mo href="./5.2#iii">(</mo><mi>λ</mi><mo href="./5.2#iii">)</mo></mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi></msub><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mfrac><mo></mo><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mrow><mo>-</mo><mi href="./18.1#SS1.p2.t1.r5">n</mi></mrow><mo>,</mo><mi>λ</mi></mrow><mrow><mn>1</mn><mo>-</mo><mi>λ</mi><mo>-</mo><mi href="./18.1#SS1.p2.t1.r5">n</mi></mrow></mfrac><mo>;</mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mn>2</mn><mo></mo><mi mathvariant="normal">i</mi><mo></mo><mi>θ</mi></mrow></mrow></msup><mo>)</mo></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E11.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./16.2" title="§16.2 Definition and Analytic Properties ‣ Generalized Hypergeometric Functions ‣ Chapter 16 Generalized Hypergeometric Functions and Meijer G -Function" class="ltx_ref"><math class="ltx_Math" altimg="m147.png" altimg-height="23px" altimg-valign="-7px" altimg-width="118px" alttext="{{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mrow><mi class="ltx_nvar" href="./16.1#p2.t1.r5">a</mi><mo>,</mo><mi class="ltx_nvar" href="./16.1#p2.t1.r5">b</mi></mrow><mo>;</mo><mi class="ltx_nvar">c</mi><mo>;</mo><mi class="ltx_nvar" href="./16.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: <math class="ltx_Math" altimg="m91.png" altimg-height="23px" altimg-valign="-7px" altimg-width="124px" alttext="=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./16.1#p2.t1.r5">a</mi><mo>,</mo><mi class="ltx_nvar" href="./16.1#p2.t1.r5">b</mi><mo>;</mo><mi class="ltx_nvar">c</mi><mo>;</mo><mi class="ltx_nvar" href="./16.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></math> notation for Gauss’ hypergeometric function</a>,
<a href="./5.2#iii" title="§5.2(iii) Pochhammer’s Symbol ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m142.png" altimg-height="24px" altimg-valign="-8px" altimg-width="41px" alttext="{\left(\NVar{a}\right)_{\NVar{n}}}" display="inline"><msub><mrow><mo href="./5.2#iii">(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r5">a</mi><mo href="./5.2#iii">)</mo></mrow><mi class="ltx_nvar" href="./5.1#p2.t1.r1">n</mi></msub></math>: Pochhammer’s symbol (or shifted factorial)</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m116.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m129.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m134.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./18.3#T1.t1.r3" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m93.png" altimg-height="29px" altimg-valign="-7px" altimg-width="73px" alttext="C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msubsup><mi href="./18.3#T1.t1.r3">C</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r3" stretchy="false">(</mo><mi class="ltx_nvar">λ</mi><mo href="./18.3#T1.t1.r3" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: ultraspherical (or Gegenbauer) polynomial</a>,
<a href="./18.1#SS1.p2.t1.r4" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m120.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="\ell" display="inline"><mi href="./18.1#SS1.p2.t1.r4" mathvariant="normal">ℓ</mi></math>: nonnegative integer</a> and
<a href="./18.1#SS1.p2.t1.r5" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m136.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math>: nonnegative integer</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">22.3.12</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div class="ltx_pagination ltx_role_newpage"></div>
</section>
<section id="Px4" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Laguerre</h3>
<div id="Px4.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px4.p1" class="ltx_para">
<table id="E12" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">18.5.12</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E12.png" altimg-height="64px" altimg-valign="-28px" altimg-width="562px" alttext="L^{(\alpha)}_{n}\left(x\right)=\sum_{\ell=0}^{n}\frac{{\left(\alpha+\ell+1%
\right)_{n-\ell}}}{(n-\ell)!\;\ell!}(-x)^{\ell}=\frac{{\left(\alpha+1\right)_{%
n}}}{n!}{{}_{1}F_{1}}\left({-n\atop\alpha+1};x\right)." display="block"><mrow><mrow><mrow><msubsup><mi href="./18.3#T1.t1.r12">L</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r12" stretchy="false">(</mo><mi>α</mi><mo href="./18.3#T1.t1.r12" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./18.1#SS1.p2.t1.r4" mathvariant="normal">ℓ</mi><mo>=</mo><mn>0</mn></mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi></munderover><mrow><mfrac><msub><mrow><mo href="./5.2#iii">(</mo><mrow><mi>α</mi><mo>+</mo><mi href="./18.1#SS1.p2.t1.r4" mathvariant="normal">ℓ</mi><mo>+</mo><mn>1</mn></mrow><mo href="./5.2#iii">)</mo></mrow><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>-</mo><mi href="./18.1#SS1.p2.t1.r4" mathvariant="normal">ℓ</mi></mrow></msub><mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>-</mo><mi href="./18.1#SS1.p2.t1.r4" mathvariant="normal">ℓ</mi></mrow><mo stretchy="false">)</mo></mrow><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow><mo></mo><mrow><mi href="./18.1#SS1.p2.t1.r4" mathvariant="normal">ℓ</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mrow></mfrac><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mi href="./18.2#Px1.p2">x</mi></mrow><mo stretchy="false">)</mo></mrow><mi href="./18.1#SS1.p2.t1.r4" mathvariant="normal">ℓ</mi></msup></mrow></mrow><mo>=</mo><mrow><mfrac><msub><mrow><mo href="./5.2#iii">(</mo><mrow><mi>α</mi><mo>+</mo><mn>1</mn></mrow><mo href="./5.2#iii">)</mo></mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi></msub><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mfrac><mo></mo><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>1</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mo>-</mo><mi href="./18.1#SS1.p2.t1.r5">n</mi></mrow><mrow><mi>α</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mo>;</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E12.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./16.2" title="§16.2 Definition and Analytic Properties ‣ Generalized Hypergeometric Functions ‣ Chapter 16 Generalized Hypergeometric Functions and Meijer G -Function" class="ltx_ref"><math class="ltx_Math" altimg="m144.png" altimg-height="23px" altimg-valign="-7px" altimg-width="101px" alttext="{{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)" display="inline"><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>1</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./16.1#p2.t1.r5">a</mi><mo>;</mo><mi class="ltx_nvar" href="./16.1#p2.t1.r5">b</mi><mo>;</mo><mi class="ltx_nvar" href="./16.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: <math class="ltx_Math" altimg="m92.png" altimg-height="23px" altimg-valign="-7px" altimg-width="113px" alttext="=M\left(\NVar{a},\NVar{b},\NVar{z}\right)" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mi href="./13.2#E2">M</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./16.1#p2.t1.r5">a</mi><mo>,</mo><mi class="ltx_nvar" href="./16.1#p2.t1.r5">b</mi><mo>,</mo><mi class="ltx_nvar" href="./16.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></math> notation for the Kummer confluent hypergeometric function</a>,
<a href="./18.3#T1.t1.r12" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m98.png" altimg-height="29px" altimg-valign="-7px" altimg-width="72px" alttext="L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msubsup><mi href="./18.3#T1.t1.r12">L</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r12" stretchy="false">(</mo><mi class="ltx_nvar">α</mi><mo href="./18.3#T1.t1.r12" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Laguerre (or generalized Laguerre) polynomial</a>,
<a href="./5.2#iii" title="§5.2(iii) Pochhammer’s Symbol ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m142.png" altimg-height="24px" altimg-valign="-8px" altimg-width="41px" alttext="{\left(\NVar{a}\right)_{\NVar{n}}}" display="inline"><msub><mrow><mo href="./5.2#iii">(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r5">a</mi><mo href="./5.2#iii">)</mo></mrow><mi class="ltx_nvar" href="./5.1#p2.t1.r1">n</mi></msub></math>: Pochhammer’s symbol (or shifted factorial)</a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m134.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./18.1#SS1.p2.t1.r4" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m120.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="\ell" display="inline"><mi href="./18.1#SS1.p2.t1.r4" mathvariant="normal">ℓ</mi></math>: nonnegative integer</a>,
<a href="./18.1#SS1.p2.t1.r5" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m136.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m141.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">22.5.54</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px5" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Hermite</h3>
<div id="Px5.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px5.p1" class="ltx_para">
<table id="E13" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">18.5.13</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E13.png" altimg-height="71px" altimg-valign="-28px" altimg-width="610px" alttext="H_{n}\left(x\right)=n!\sum_{\ell=0}^{\left\lfloor n/2\right\rfloor}\frac{(-1)^%
{\ell}(2x)^{n-2\ell}}{\ell!\;(n-2\ell)!}=(2x)^{n}{{}_{2}F_{0}}\left({-\tfrac{1%
}{2}n,-\tfrac{1}{2}n+\tfrac{1}{2}\atop-};-\frac{1}{x^{2}}\right)." display="block"><mrow><mrow><mrow><msub><mi href="./18.3#T1.t1.r13">H</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow><mo></mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./18.1#SS1.p2.t1.r4" mathvariant="normal">ℓ</mi><mo>=</mo><mn>0</mn></mrow><mrow><mo href="./front/introduction#Sx4.p1.t1.r16">⌊</mo><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>/</mo><mn>2</mn></mrow><mo href="./front/introduction#Sx4.p1.t1.r16">⌋</mo></mrow></munderover><mfrac><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./18.1#SS1.p2.t1.r4" mathvariant="normal">ℓ</mi></msup><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mn>2</mn><mo></mo><mi href="./18.2#Px1.p2">x</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>-</mo><mrow><mn>2</mn><mo></mo><mi href="./18.1#SS1.p2.t1.r4" mathvariant="normal">ℓ</mi></mrow></mrow></msup></mrow><mrow><mrow><mi href="./18.1#SS1.p2.t1.r4" mathvariant="normal">ℓ</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow><mo></mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>-</mo><mrow><mn>2</mn><mo></mo><mi href="./18.1#SS1.p2.t1.r4" mathvariant="normal">ℓ</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mrow></mfrac></mrow></mrow><mo>=</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mn>2</mn><mo></mo><mi href="./18.2#Px1.p2">x</mi></mrow><mo stretchy="false">)</mo></mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi></msup><mo></mo><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>0</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./18.1#SS1.p2.t1.r5">n</mi></mrow></mrow><mo>,</mo><mrow><mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./18.1#SS1.p2.t1.r5">n</mi></mrow></mrow><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow><mo>-</mo></mfrac><mo>;</mo><mrow><mo>-</mo><mfrac><mn>1</mn><msup><mi href="./18.2#Px1.p2">x</mi><mn>2</mn></msup></mfrac></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E13.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./16.2" title="§16.2 Definition and Analytic Properties ‣ Generalized Hypergeometric Functions ‣ Chapter 16 Generalized Hypergeometric Functions and Meijer G -Function" class="ltx_ref"><math class="ltx_Math" altimg="m149.png" altimg-height="24px" altimg-valign="-8px" altimg-width="244px" alttext="{{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_%
{q}};\NVar{z}\right)" display="inline"><mrow><mmultiscripts><mi href="./16.2">F</mi><mi class="ltx_nvar" href="./16.1#p2.t1.r1">q</mi><none></none><mprescripts></mprescripts><mi class="ltx_nvar" href="./16.1#p2.t1.r1">p</mi><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mrow><msub><mi class="ltx_nvar" href="./16.1#p2.t1.r5">a</mi><mn class="ltx_nvar">1</mn></msub><mo class="ltx_nvar">,</mo><mi class="ltx_nvar" mathvariant="normal">…</mi><mo class="ltx_nvar">,</mo><msub><mi class="ltx_nvar" href="./16.1#p2.t1.r5">a</mi><mi class="ltx_nvar" href="./16.1#p2.t1.r1">p</mi></msub></mrow><mo>;</mo><mrow><msub><mi class="ltx_nvar" href="./16.1#p2.t1.r5">b</mi><mn class="ltx_nvar">1</mn></msub><mo class="ltx_nvar">,</mo><mi class="ltx_nvar" mathvariant="normal">…</mi><mo class="ltx_nvar">,</mo><msub><mi class="ltx_nvar" href="./16.1#p2.t1.r5">b</mi><mi class="ltx_nvar" href="./16.1#p2.t1.r1">q</mi></msub></mrow><mo>;</mo><mi class="ltx_nvar" href="./16.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>
or <math class="ltx_Math" altimg="m151.png" altimg-height="39px" altimg-valign="-15px" altimg-width="143px" alttext="{{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},%
\dots,b_{q}}};\NVar{z}\right)" display="inline"><mrow><mmultiscripts><mi href="./16.2">F</mi><mi class="ltx_nvar" href="./16.1#p2.t1.r1">q</mi><none></none><mprescripts></mprescripts><mi class="ltx_nvar" href="./16.1#p2.t1.r1">p</mi><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><msub><mi class="ltx_nvar" href="./16.1#p2.t1.r5">a</mi><mn class="ltx_nvar">1</mn></msub><mo class="ltx_nvar">,</mo><mi class="ltx_nvar" mathvariant="normal">…</mi><mo class="ltx_nvar">,</mo><msub><mi class="ltx_nvar" href="./16.1#p2.t1.r5">a</mi><mi class="ltx_nvar" href="./16.1#p2.t1.r1">p</mi></msub></mrow><mrow><msub><mi class="ltx_nvar" href="./16.1#p2.t1.r5">b</mi><mn class="ltx_nvar">1</mn></msub><mo class="ltx_nvar">,</mo><mi class="ltx_nvar" mathvariant="normal">…</mi><mo class="ltx_nvar">,</mo><msub><mi class="ltx_nvar" href="./16.1#p2.t1.r5">b</mi><mi class="ltx_nvar" href="./16.1#p2.t1.r1">q</mi></msub></mrow></mfrac><mo>;</mo><mi class="ltx_nvar" href="./16.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: alternatively <math class="ltx_Math" altimg="m148.png" altimg-height="24px" altimg-valign="-8px" altimg-width="106px" alttext="{{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)" display="inline"><mrow><mmultiscripts><mi href="./16.2">F</mi><mi class="ltx_nvar" href="./16.1#p2.t1.r1">q</mi><none></none><mprescripts></mprescripts><mi class="ltx_nvar" href="./16.1#p2.t1.r1">p</mi><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" mathvariant="bold">a</mi><mo>;</mo><mi class="ltx_nvar" mathvariant="bold">b</mi><mo>;</mo><mi class="ltx_nvar" href="./16.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math> or
<math class="ltx_Math" altimg="m150.png" altimg-height="27px" altimg-valign="-9px" altimg-width="91px" alttext="{{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};%
\NVar{z}\right)" display="inline"><mrow><mmultiscripts><mi href="./16.2">F</mi><mi class="ltx_nvar" href="./16.1#p2.t1.r1">q</mi><none></none><mprescripts></mprescripts><mi class="ltx_nvar" href="./16.1#p2.t1.r1">p</mi><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mi class="ltx_nvar" mathvariant="bold">a</mi><mi class="ltx_nvar" mathvariant="bold">b</mi></mfrac><mo>;</mo><mi class="ltx_nvar" href="./16.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>
<br class="ltx_break">generalized hypergeometric function</a>,
<a href="./18.3#T1.t1.r13" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m96.png" altimg-height="23px" altimg-valign="-7px" altimg-width="62px" alttext="H_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r13">H</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Hermite polynomial</a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m134.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./front/introduction#Sx4.p1.t1.r16" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m125.png" altimg-height="23px" altimg-valign="-7px" altimg-width="33px" alttext="\left\lfloor\NVar{x}\right\rfloor" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r16">⌊</mo><mi class="ltx_nvar">x</mi><mo href="./front/introduction#Sx4.p1.t1.r16">⌋</mo></mrow></math>: floor of <math class="ltx_Math" altimg="m141.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./18.1#SS1.p2.t1.r4" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m120.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="\ell" display="inline"><mi href="./18.1#SS1.p2.t1.r4" mathvariant="normal">ℓ</mi></math>: nonnegative integer</a>,
<a href="./18.1#SS1.p2.t1.r5" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m136.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m141.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">22.3.10</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="Px5.p2" class="ltx_para">
<p class="ltx_p">For corresponding formulas for Chebyshev, Legendre, and the Hermite
<math class="ltx_Math" altimg="m127.png" altimg-height="21px" altimg-valign="-5px" altimg-width="41px" alttext="\mathit{He}_{n}" display="inline"><msub><mi href="./18.3#T1.t1.r14" mathvariant="italic">He</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi></msub></math> polynomials apply () can be regarded as definitions of
<math class="ltx_Math" altimg="m102.png" altimg-height="29px" altimg-valign="-7px" altimg-width="88px" alttext="P^{(\alpha,\beta)}_{n}\left(x\right)" display="inline"><mrow><msubsup><mi href="./18.3#T1.t1.r2">P</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r2" stretchy="false">(</mo><mi>α</mi><mo href="./18.3#T1.t1.r2">,</mo><mi>β</mi><mo href="./18.3#T1.t1.r2" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math> when the conditions <math class="ltx_Math" altimg="m114.png" altimg-height="18px" altimg-valign="-4px" altimg-width="69px" alttext="\alpha>-1" display="inline"><mrow><mi>α</mi><mo>></mo><mrow><mo>-</mo><mn>1</mn></mrow></mrow></math> and
<math class="ltx_Math" altimg="m115.png" altimg-height="21px" altimg-valign="-6px" altimg-width="69px" alttext="\beta>-1" display="inline"><mrow><mi>β</mi><mo>></mo><mrow><mo>-</mo><mn>1</mn></mrow></mrow></math> are not satisfied. However, in these circumstances the
orthogonality property () disappears. For this reason, and
also in the interest of simplicity, in the case of the Jacobi polynomials
<math class="ltx_Math" altimg="m102.png" altimg-height="29px" altimg-valign="-7px" altimg-width="88px" alttext="P^{(\alpha,\beta)}_{n}\left(x\right)" display="inline"><mrow><msubsup><mi href="./18.3#T1.t1.r2">P</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r2" stretchy="false">(</mo><mi>α</mi><mo href="./18.3#T1.t1.r2">,</mo><mi>β</mi><mo href="./18.3#T1.t1.r2" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math> we assume throughout this chapter that
<math class="ltx_Math" altimg="m114.png" altimg-height="18px" altimg-valign="-4px" altimg-width="69px" alttext="\alpha>-1" display="inline"><mrow><mi>α</mi><mo>></mo><mrow><mo>-</mo><mn>1</mn></mrow></mrow></math> and <math class="ltx_Math" altimg="m115.png" altimg-height="21px" altimg-valign="-6px" altimg-width="69px" alttext="\beta>-1" display="inline"><mrow><mi>β</mi><mo>></mo><mrow><mo>-</mo><mn>1</mn></mrow></mrow></math>, <em class="ltx_emph ltx_font_italic">unless stated otherwise</em>. Similarly in
the cases of the ultraspherical polynomials <math class="ltx_Math" altimg="m94.png" altimg-height="29px" altimg-valign="-7px" altimg-width="73px" alttext="C^{(\lambda)}_{n}\left(x\right)" display="inline"><mrow><msubsup><mi href="./18.3#T1.t1.r3">C</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r3" stretchy="false">(</mo><mi>λ</mi><mo href="./18.3#T1.t1.r3" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>
and the Laguerre polynomials <math class="ltx_Math" altimg="m99.png" altimg-height="29px" altimg-valign="-7px" altimg-width="72px" alttext="L^{(\alpha)}_{n}\left(x\right)" display="inline"><mrow><msubsup><mi href="./18.3#T1.t1.r12">L</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r12" stretchy="false">(</mo><mi>α</mi><mo href="./18.3#T1.t1.r12" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math> we assume that
<math class="ltx_Math" altimg="m124.png" altimg-height="27px" altimg-valign="-9px" altimg-width="128px" alttext="\lambda>-\tfrac{1}{2},\lambda\neq 0" display="inline"><mrow><mrow><mi>λ</mi><mo>></mo><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow><mo>,</mo><mrow><mi>λ</mi><mo>≠</mo><mn>0</mn></mrow></mrow></math>, and <math class="ltx_Math" altimg="m114.png" altimg-height="18px" altimg-valign="-4px" altimg-width="69px" alttext="\alpha>-1" display="inline"><mrow><mi>α</mi><mo>></mo><mrow><mo>-</mo><mn>1</mn></mrow></mrow></math>, <em class="ltx_emph ltx_font_italic">unless
stated otherwise</em>.
</p>
</div>
</section>
</section>
<section id="iv" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§18.5(iv) </span>Numerical Coefficients</h2>
<div id="SS4.info" class="ltx_metadata ltx_info">
), with initial values
obtained from the values of <math class="ltx_Math" altimg="m133.png" altimg-height="21px" altimg-valign="-5px" altimg-width="26px" alttext="k_{n}" display="inline"><msub><mi href="./18.2#E7">k</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi></msub></math> and <math class="ltx_Math" altimg="m131.png" altimg-height="27px" altimg-valign="-7px" altimg-width="57px" alttext="\tilde{k}_{n}/k_{n}" display="inline"><mrow><msub><mover accent="true"><mi>k</mi><mo stretchy="false">~</mo></mover><mi href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo>/</mo><msub><mi href="./18.2#E7">k</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi></msub></mrow></math> given in Table with <math class="ltx_Math" altimg="m135.png" altimg-height="20px" altimg-valign="-6px" altimg-width="72px" alttext="n=0,1" display="inline"><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>=</mo><mrow><mn>0</mn><mo>,</mo><mn>1</mn></mrow></mrow></math>.</dd>
</dl>
</div>
</div>
<section id="Px6" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Chebyshev</h3>
<div id="Px6.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px6.p1" class="ltx_para">
<table id="E14" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex1" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="7" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">18.5.14</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m60.png" altimg-height="25px" altimg-valign="-7px" altimg-width="57px" alttext="\displaystyle T_{0}\left(x\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r4">T</mi><mn>0</mn></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m4.png" altimg-height="22px" altimg-valign="-6px" altimg-width="43px" alttext="\displaystyle=1," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mn>1</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex2" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m61.png" altimg-height="25px" altimg-valign="-7px" altimg-width="57px" alttext="\displaystyle T_{1}\left(x\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r4">T</mi><mn>1</mn></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m33.png" altimg-height="18px" altimg-valign="-6px" altimg-width="44px" alttext="\displaystyle=x," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mi href="./18.2#Px1.p2">x</mi></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex3" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m62.png" altimg-height="25px" altimg-valign="-7px" altimg-width="57px" alttext="\displaystyle T_{2}\left(x\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r4">T</mi><mn>2</mn></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m9.png" altimg-height="27px" altimg-valign="-6px" altimg-width="97px" alttext="\displaystyle=2x^{2}-1," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mn>2</mn><mo></mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>2</mn></msup></mrow><mo>-</mo><mn>1</mn></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex4" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m63.png" altimg-height="25px" altimg-valign="-7px" altimg-width="57px" alttext="\displaystyle T_{3}\left(x\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r4">T</mi><mn>3</mn></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m15.png" altimg-height="27px" altimg-valign="-6px" altimg-width="109px" alttext="\displaystyle=4x^{3}-3x," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mn>4</mn><mo></mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>3</mn></msup></mrow><mo>-</mo><mrow><mn>3</mn><mo></mo><mi href="./18.2#Px1.p2">x</mi></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex5" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m64.png" altimg-height="25px" altimg-valign="-7px" altimg-width="57px" alttext="\displaystyle T_{4}\left(x\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r4">T</mi><mn>4</mn></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m20.png" altimg-height="27px" altimg-valign="-6px" altimg-width="152px" alttext="\displaystyle=8x^{4}-8x^{2}+1," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mrow><mn>8</mn><mo></mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>4</mn></msup></mrow><mo>-</mo><mrow><mn>8</mn><mo></mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>2</mn></msup></mrow></mrow><mo>+</mo><mn>1</mn></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex6" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m65.png" altimg-height="25px" altimg-valign="-7px" altimg-width="57px" alttext="\displaystyle T_{5}\left(x\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r4">T</mi><mn>5</mn></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m7.png" altimg-height="27px" altimg-valign="-6px" altimg-width="183px" alttext="\displaystyle=16x^{5}-20x^{3}+5x," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mrow><mn>16</mn><mo></mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>5</mn></msup></mrow><mo>-</mo><mrow><mn>20</mn><mo></mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>3</mn></msup></mrow></mrow><mo>+</mo><mrow><mn>5</mn><mo></mo><mi href="./18.2#Px1.p2">x</mi></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex7" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m66.png" altimg-height="25px" altimg-valign="-7px" altimg-width="57px" alttext="\displaystyle T_{6}\left(x\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r4">T</mi><mn>6</mn></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m12.png" altimg-height="25px" altimg-valign="-4px" altimg-width="236px" alttext="\displaystyle=32x^{6}-48x^{4}+18x^{2}-1." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mrow><mrow><mn>32</mn><mo></mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>6</mn></msup></mrow><mo>-</mo><mrow><mn>48</mn><mo></mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>4</mn></msup></mrow></mrow><mo>+</mo><mrow><mn>18</mn><mo></mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>2</mn></msup></mrow></mrow><mo>-</mo><mn>1</mn></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E14.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./18.3#T1.t1.r4" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m105.png" altimg-height="23px" altimg-valign="-7px" altimg-width="57px" alttext="T_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r4">T</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Chebyshev polynomial of the first kind</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m141.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E15" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex8" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="7" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">18.5.15</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m68.png" altimg-height="25px" altimg-valign="-7px" altimg-width="59px" alttext="\displaystyle U_{0}\left(x\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r5">U</mi><mn>0</mn></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m4.png" altimg-height="22px" altimg-valign="-6px" altimg-width="43px" alttext="\displaystyle=1," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mn>1</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex9" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m69.png" altimg-height="25px" altimg-valign="-7px" altimg-width="59px" alttext="\displaystyle U_{1}\left(x\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r5">U</mi><mn>1</mn></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m8.png" altimg-height="22px" altimg-valign="-6px" altimg-width="54px" alttext="\displaystyle=2x," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mn>2</mn><mo></mo><mi href="./18.2#Px1.p2">x</mi></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex10" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m70.png" altimg-height="25px" altimg-valign="-7px" altimg-width="59px" alttext="\displaystyle U_{2}\left(x\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r5">U</mi><mn>2</mn></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m13.png" altimg-height="27px" altimg-valign="-6px" altimg-width="97px" alttext="\displaystyle=4x^{2}-1," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mn>4</mn><mo></mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>2</mn></msup></mrow><mo>-</mo><mn>1</mn></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex11" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m71.png" altimg-height="25px" altimg-valign="-7px" altimg-width="59px" alttext="\displaystyle U_{3}\left(x\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r5">U</mi><mn>3</mn></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m19.png" altimg-height="27px" altimg-valign="-6px" altimg-width="109px" alttext="\displaystyle=8x^{3}-4x," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mn>8</mn><mo></mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>3</mn></msup></mrow><mo>-</mo><mrow><mn>4</mn><mo></mo><mi href="./18.2#Px1.p2">x</mi></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex12" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m72.png" altimg-height="25px" altimg-valign="-7px" altimg-width="59px" alttext="\displaystyle U_{4}\left(x\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r5">U</mi><mn>4</mn></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m5.png" altimg-height="27px" altimg-valign="-6px" altimg-width="172px" alttext="\displaystyle=16x^{4}-12x^{2}+1," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mrow><mn>16</mn><mo></mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>4</mn></msup></mrow><mo>-</mo><mrow><mn>12</mn><mo></mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>2</mn></msup></mrow></mrow><mo>+</mo><mn>1</mn></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex13" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m73.png" altimg-height="25px" altimg-valign="-7px" altimg-width="59px" alttext="\displaystyle U_{5}\left(x\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r5">U</mi><mn>5</mn></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m11.png" altimg-height="27px" altimg-valign="-6px" altimg-width="183px" alttext="\displaystyle=32x^{5}-32x^{3}+6x," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mrow><mn>32</mn><mo></mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>5</mn></msup></mrow><mo>-</mo><mrow><mn>32</mn><mo></mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>3</mn></msup></mrow></mrow><mo>+</mo><mrow><mn>6</mn><mo></mo><mi href="./18.2#Px1.p2">x</mi></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex14" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m74.png" altimg-height="25px" altimg-valign="-7px" altimg-width="59px" alttext="\displaystyle U_{6}\left(x\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r5">U</mi><mn>6</mn></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m17.png" altimg-height="25px" altimg-valign="-4px" altimg-width="236px" alttext="\displaystyle=64x^{6}-80x^{4}+24x^{2}-1." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mrow><mrow><mn>64</mn><mo></mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>6</mn></msup></mrow><mo>-</mo><mrow><mn>80</mn><mo></mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>4</mn></msup></mrow></mrow><mo>+</mo><mrow><mn>24</mn><mo></mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>2</mn></msup></mrow></mrow><mo>-</mo><mn>1</mn></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E15.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./18.3#T1.t1.r5" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m107.png" altimg-height="23px" altimg-valign="-7px" altimg-width="59px" alttext="U_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r5">U</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Chebyshev polynomial of the second kind</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m141.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px7" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Legendre</h3>
<div id="Px7.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px7.p1" class="ltx_para">
<table id="E16" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex15" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="7" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">18.5.16</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m53.png" altimg-height="25px" altimg-valign="-7px" altimg-width="58px" alttext="\displaystyle P_{0}\left(x\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r10">P</mi><mn>0</mn></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m4.png" altimg-height="22px" altimg-valign="-6px" altimg-width="43px" alttext="\displaystyle=1," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mn>1</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex16" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m54.png" altimg-height="25px" altimg-valign="-7px" altimg-width="58px" alttext="\displaystyle P_{1}\left(x\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r10">P</mi><mn>1</mn></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m33.png" altimg-height="18px" altimg-valign="-6px" altimg-width="44px" alttext="\displaystyle=x," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mi href="./18.2#Px1.p2">x</mi></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex17" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m55.png" altimg-height="25px" altimg-valign="-7px" altimg-width="58px" alttext="\displaystyle P_{2}\left(x\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r10">P</mi><mn>2</mn></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m30.png" altimg-height="30px" altimg-valign="-9px" altimg-width="103px" alttext="\displaystyle=\tfrac{3}{2}x^{2}-\tfrac{1}{2}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mfrac><mn>3</mn><mn>2</mn></mfrac><mo></mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>2</mn></msup></mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex18" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m56.png" altimg-height="25px" altimg-valign="-7px" altimg-width="58px" alttext="\displaystyle P_{3}\left(x\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r10">P</mi><mn>3</mn></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m31.png" altimg-height="30px" altimg-valign="-9px" altimg-width="114px" alttext="\displaystyle=\tfrac{5}{2}x^{3}-\tfrac{3}{2}x," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mfrac><mn>5</mn><mn>2</mn></mfrac><mo></mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>3</mn></msup></mrow><mo>-</mo><mrow><mfrac><mn>3</mn><mn>2</mn></mfrac><mo></mo><mi href="./18.2#Px1.p2">x</mi></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex19" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m57.png" altimg-height="25px" altimg-valign="-7px" altimg-width="58px" alttext="\displaystyle P_{4}\left(x\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r10">P</mi><mn>4</mn></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m29.png" altimg-height="30px" altimg-valign="-9px" altimg-width="176px" alttext="\displaystyle=\tfrac{35}{8}x^{4}-\tfrac{15}{4}x^{2}+\tfrac{3}{8}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mrow><mfrac><mn>35</mn><mn>8</mn></mfrac><mo></mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>4</mn></msup></mrow><mo>-</mo><mrow><mfrac><mn>15</mn><mn>4</mn></mfrac><mo></mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>2</mn></msup></mrow></mrow><mo>+</mo><mfrac><mn>3</mn><mn>8</mn></mfrac></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex20" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m58.png" altimg-height="25px" altimg-valign="-7px" altimg-width="58px" alttext="\displaystyle P_{5}\left(x\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r10">P</mi><mn>5</mn></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m32.png" altimg-height="30px" altimg-valign="-9px" altimg-width="195px" alttext="\displaystyle=\tfrac{63}{8}x^{5}-\tfrac{35}{4}x^{3}+\tfrac{15}{8}x," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mrow><mfrac><mn>63</mn><mn>8</mn></mfrac><mo></mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>5</mn></msup></mrow><mo>-</mo><mrow><mfrac><mn>35</mn><mn>4</mn></mfrac><mo></mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>3</mn></msup></mrow></mrow><mo>+</mo><mrow><mfrac><mn>15</mn><mn>8</mn></mfrac><mo></mo><mi href="./18.2#Px1.p2">x</mi></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex21" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m59.png" altimg-height="25px" altimg-valign="-7px" altimg-width="58px" alttext="\displaystyle P_{6}\left(x\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r10">P</mi><mn>6</mn></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m28.png" altimg-height="30px" altimg-valign="-9px" altimg-width="273px" alttext="\displaystyle=\tfrac{231}{16}x^{6}-\tfrac{315}{16}x^{4}+\tfrac{105}{16}x^{2}-%
\tfrac{5}{16}." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mrow><mrow><mfrac><mn>231</mn><mn>16</mn></mfrac><mo></mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>6</mn></msup></mrow><mo>-</mo><mrow><mfrac><mn>315</mn><mn>16</mn></mfrac><mo></mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>4</mn></msup></mrow></mrow><mo>+</mo><mrow><mfrac><mn>105</mn><mn>16</mn></mfrac><mo></mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>2</mn></msup></mrow></mrow><mo>-</mo><mfrac><mn>5</mn><mn>16</mn></mfrac></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E16.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./18.3#T1.t1.r10" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="23px" altimg-valign="-7px" altimg-width="58px" alttext="P_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r10">P</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Legendre polynomial</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m141.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px8" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Laguerre</h3>
<div id="Px8.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px8.p1" class="ltx_para">
<table id="E17" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex22" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="7" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">18.5.17</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m46.png" altimg-height="25px" altimg-valign="-7px" altimg-width="59px" alttext="\displaystyle L_{0}\left(x\right)" display="inline"><mrow><msub><mi href="./18.1#I1.ix7.p1">L</mi><mn>0</mn></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m4.png" altimg-height="22px" altimg-valign="-6px" altimg-width="43px" alttext="\displaystyle=1," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mn>1</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex23" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m47.png" altimg-height="25px" altimg-valign="-7px" altimg-width="59px" alttext="\displaystyle L_{1}\left(x\right)" display="inline"><mrow><msub><mi href="./18.1#I1.ix7.p1">L</mi><mn>1</mn></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m3.png" altimg-height="22px" altimg-valign="-6px" altimg-width="94px" alttext="\displaystyle=-x+1," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mo>-</mo><mi href="./18.2#Px1.p2">x</mi></mrow><mo>+</mo><mn>1</mn></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex24" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m48.png" altimg-height="25px" altimg-valign="-7px" altimg-width="59px" alttext="\displaystyle L_{2}\left(x\right)" display="inline"><mrow><msub><mi href="./18.1#I1.ix7.p1">L</mi><mn>2</mn></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m26.png" altimg-height="30px" altimg-valign="-9px" altimg-width="146px" alttext="\displaystyle=\tfrac{1}{2}x^{2}-2x+1," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>2</mn></msup></mrow><mo>-</mo><mrow><mn>2</mn><mo></mo><mi href="./18.2#Px1.p2">x</mi></mrow></mrow><mo>+</mo><mn>1</mn></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex25" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m49.png" altimg-height="25px" altimg-valign="-7px" altimg-width="59px" alttext="\displaystyle L_{3}\left(x\right)" display="inline"><mrow><msub><mi href="./18.1#I1.ix7.p1">L</mi><mn>3</mn></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m2.png" altimg-height="30px" altimg-valign="-9px" altimg-width="219px" alttext="\displaystyle=-\tfrac{1}{6}x^{3}+\tfrac{3}{2}x^{2}-3x+1," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mrow><mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>6</mn></mfrac><mo></mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>3</mn></msup></mrow></mrow><mo>+</mo><mrow><mfrac><mn>3</mn><mn>2</mn></mfrac><mo></mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>2</mn></msup></mrow></mrow><mo>-</mo><mrow><mn>3</mn><mo></mo><mi href="./18.2#Px1.p2">x</mi></mrow></mrow><mo>+</mo><mn>1</mn></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex26" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m50.png" altimg-height="25px" altimg-valign="-7px" altimg-width="59px" alttext="\displaystyle L_{4}\left(x\right)" display="inline"><mrow><msub><mi href="./18.1#I1.ix7.p1">L</mi><mn>4</mn></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m25.png" altimg-height="30px" altimg-valign="-9px" altimg-width="266px" alttext="\displaystyle=\tfrac{1}{24}x^{4}-\tfrac{2}{3}x^{3}+3x^{2}-4x+1," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mrow><mrow><mrow><mfrac><mn>1</mn><mn>24</mn></mfrac><mo></mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>4</mn></msup></mrow><mo>-</mo><mrow><mfrac><mn>2</mn><mn>3</mn></mfrac><mo></mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>3</mn></msup></mrow></mrow><mo>+</mo><mrow><mn>3</mn><mo></mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>2</mn></msup></mrow></mrow><mo>-</mo><mrow><mn>4</mn><mo></mo><mi href="./18.2#Px1.p2">x</mi></mrow></mrow><mo>+</mo><mn>1</mn></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex27" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m51.png" altimg-height="25px" altimg-valign="-7px" altimg-width="59px" alttext="\displaystyle L_{5}\left(x\right)" display="inline"><mrow><msub><mi href="./18.1#I1.ix7.p1">L</mi><mn>5</mn></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m1.png" altimg-height="30px" altimg-valign="-9px" altimg-width="354px" alttext="\displaystyle=-\tfrac{1}{120}x^{5}+\tfrac{5}{24}x^{4}-\tfrac{5}{3}x^{3}+5x^{2}%
-5x+1," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mrow><mrow><mrow><mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>120</mn></mfrac><mo></mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>5</mn></msup></mrow></mrow><mo>+</mo><mrow><mfrac><mn>5</mn><mn>24</mn></mfrac><mo></mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>4</mn></msup></mrow></mrow><mo>-</mo><mrow><mfrac><mn>5</mn><mn>3</mn></mfrac><mo></mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>3</mn></msup></mrow></mrow><mo>+</mo><mrow><mn>5</mn><mo></mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>2</mn></msup></mrow></mrow><mo>-</mo><mrow><mn>5</mn><mo></mo><mi href="./18.2#Px1.p2">x</mi></mrow></mrow><mo>+</mo><mn>1</mn></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex28" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m52.png" altimg-height="25px" altimg-valign="-7px" altimg-width="59px" alttext="\displaystyle L_{6}\left(x\right)" display="inline"><mrow><msub><mi href="./18.1#I1.ix7.p1">L</mi><mn>6</mn></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m27.png" altimg-height="30px" altimg-valign="-9px" altimg-width="415px" alttext="\displaystyle=\tfrac{1}{720}x^{6}-\tfrac{1}{20}x^{5}+\tfrac{5}{8}x^{4}-\tfrac{%
10}{3}x^{3}+\tfrac{15}{2}x^{2}-6x+1." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mrow><mrow><mrow><mrow><mrow><mfrac><mn>1</mn><mn>720</mn></mfrac><mo></mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>6</mn></msup></mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>20</mn></mfrac><mo></mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>5</mn></msup></mrow></mrow><mo>+</mo><mrow><mfrac><mn>5</mn><mn>8</mn></mfrac><mo></mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>4</mn></msup></mrow></mrow><mo>-</mo><mrow><mfrac><mn>10</mn><mn>3</mn></mfrac><mo></mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>3</mn></msup></mrow></mrow><mo>+</mo><mrow><mfrac><mn>15</mn><mn>2</mn></mfrac><mo></mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>2</mn></msup></mrow></mrow><mo>-</mo><mrow><mn>6</mn><mo></mo><mi href="./18.2#Px1.p2">x</mi></mrow></mrow><mo>+</mo><mn>1</mn></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E17.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./18.1#I1.ix7.p1" title="Classical OP’s ‣ §18.1(ii) Main Functions ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m100.png" altimg-height="29px" altimg-valign="-7px" altimg-width="151px" alttext="L_{\NVar{n}}\left(\NVar{x}\right)=L^{(0)}_{n}\left(x\right)" display="inline"><mrow><mrow><msub><mi href="./18.1#I1.ix7.p1">L</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msubsup><mi href="./18.3#T1.t1.r12">L</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r12" stretchy="false">(</mo><mn>0</mn><mo href="./18.3#T1.t1.r12" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></mrow></math>: Laguerre polynomial</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m141.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px9" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Hermite</h3>
<div id="Px9.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px9.p1" class="ltx_para">
<table id="E18" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex29" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="7" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">18.5.18</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m39.png" altimg-height="25px" altimg-valign="-7px" altimg-width="62px" alttext="\displaystyle H_{0}\left(x\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r13">H</mi><mn>0</mn></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m4.png" altimg-height="22px" altimg-valign="-6px" altimg-width="43px" alttext="\displaystyle=1," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mn>1</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex30" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m40.png" altimg-height="25px" altimg-valign="-7px" altimg-width="62px" alttext="\displaystyle H_{1}\left(x\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r13">H</mi><mn>1</mn></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m8.png" altimg-height="22px" altimg-valign="-6px" altimg-width="54px" alttext="\displaystyle=2x," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mn>2</mn><mo></mo><mi href="./18.2#Px1.p2">x</mi></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex31" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m41.png" altimg-height="25px" altimg-valign="-7px" altimg-width="62px" alttext="\displaystyle H_{2}\left(x\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r13">H</mi><mn>2</mn></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m14.png" altimg-height="27px" altimg-valign="-6px" altimg-width="97px" alttext="\displaystyle=4x^{2}-2," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mn>4</mn><mo></mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>2</mn></msup></mrow><mo>-</mo><mn>2</mn></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex32" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m42.png" altimg-height="25px" altimg-valign="-7px" altimg-width="62px" alttext="\displaystyle H_{3}\left(x\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r13">H</mi><mn>3</mn></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m18.png" altimg-height="27px" altimg-valign="-6px" altimg-width="119px" alttext="\displaystyle=8x^{3}-12x," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mn>8</mn><mo></mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>3</mn></msup></mrow><mo>-</mo><mrow><mn>12</mn><mo></mo><mi href="./18.2#Px1.p2">x</mi></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex33" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m43.png" altimg-height="25px" altimg-valign="-7px" altimg-width="62px" alttext="\displaystyle H_{4}\left(x\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r13">H</mi><mn>4</mn></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m6.png" altimg-height="27px" altimg-valign="-6px" altimg-width="182px" alttext="\displaystyle=16x^{4}-48x^{2}+12," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mrow><mn>16</mn><mo></mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>4</mn></msup></mrow><mo>-</mo><mrow><mn>48</mn><mo></mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>2</mn></msup></mrow></mrow><mo>+</mo><mn>12</mn></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex34" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m44.png" altimg-height="25px" altimg-valign="-7px" altimg-width="62px" alttext="\displaystyle H_{5}\left(x\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r13">H</mi><mn>5</mn></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m10.png" altimg-height="27px" altimg-valign="-6px" altimg-width="213px" alttext="\displaystyle=32x^{5}-160x^{3}+120x," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mrow><mn>32</mn><mo></mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>5</mn></msup></mrow><mo>-</mo><mrow><mn>160</mn><mo></mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>3</mn></msup></mrow></mrow><mo>+</mo><mrow><mn>120</mn><mo></mo><mi href="./18.2#Px1.p2">x</mi></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex35" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m45.png" altimg-height="25px" altimg-valign="-7px" altimg-width="62px" alttext="\displaystyle H_{6}\left(x\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r13">H</mi><mn>6</mn></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m16.png" altimg-height="25px" altimg-valign="-4px" altimg-width="276px" alttext="\displaystyle=64x^{6}-480x^{4}+720x^{2}-120." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mrow><mrow><mn>64</mn><mo></mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>6</mn></msup></mrow><mo>-</mo><mrow><mn>480</mn><mo></mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>4</mn></msup></mrow></mrow><mo>+</mo><mrow><mn>720</mn><mo></mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>2</mn></msup></mrow></mrow><mo>-</mo><mn>120</mn></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E18.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./18.3#T1.t1.r13" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m96.png" altimg-height="23px" altimg-valign="-7px" altimg-width="62px" alttext="H_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r13">H</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Hermite polynomial</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m141.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E19" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex36" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="7" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">18.5.19</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m78.png" altimg-height="25px" altimg-valign="-7px" altimg-width="71px" alttext="\displaystyle\mathit{He}_{0}\left(x\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r14" mathvariant="italic">He</mi><mn>0</mn></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m4.png" altimg-height="22px" altimg-valign="-6px" altimg-width="43px" alttext="\displaystyle=1," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mn>1</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex37" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m79.png" altimg-height="25px" altimg-valign="-7px" altimg-width="71px" alttext="\displaystyle\mathit{He}_{1}\left(x\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r14" mathvariant="italic">He</mi><mn>1</mn></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m33.png" altimg-height="18px" altimg-valign="-6px" altimg-width="44px" alttext="\displaystyle=x," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mi href="./18.2#Px1.p2">x</mi></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex38" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m80.png" altimg-height="25px" altimg-valign="-7px" altimg-width="71px" alttext="\displaystyle\mathit{He}_{2}\left(x\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r14" mathvariant="italic">He</mi><mn>2</mn></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m34.png" altimg-height="27px" altimg-valign="-6px" altimg-width="87px" alttext="\displaystyle=x^{2}-1," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msup><mi href="./18.2#Px1.p2">x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex39" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m81.png" altimg-height="25px" altimg-valign="-7px" altimg-width="71px" alttext="\displaystyle\mathit{He}_{3}\left(x\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r14" mathvariant="italic">He</mi><mn>3</mn></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m35.png" altimg-height="27px" altimg-valign="-6px" altimg-width="99px" alttext="\displaystyle=x^{3}-3x," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msup><mi href="./18.2#Px1.p2">x</mi><mn>3</mn></msup><mo>-</mo><mrow><mn>3</mn><mo></mo><mi href="./18.2#Px1.p2">x</mi></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex40" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m82.png" altimg-height="25px" altimg-valign="-7px" altimg-width="71px" alttext="\displaystyle\mathit{He}_{4}\left(x\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r14" mathvariant="italic">He</mi><mn>4</mn></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m36.png" altimg-height="27px" altimg-valign="-6px" altimg-width="142px" alttext="\displaystyle=x^{4}-6x^{2}+3," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><msup><mi href="./18.2#Px1.p2">x</mi><mn>4</mn></msup><mo>-</mo><mrow><mn>6</mn><mo></mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>2</mn></msup></mrow></mrow><mo>+</mo><mn>3</mn></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex41" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m83.png" altimg-height="25px" altimg-valign="-7px" altimg-width="71px" alttext="\displaystyle\mathit{He}_{5}\left(x\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r14" mathvariant="italic">He</mi><mn>5</mn></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m37.png" altimg-height="27px" altimg-valign="-6px" altimg-width="173px" alttext="\displaystyle=x^{5}-10x^{3}+15x," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><msup><mi href="./18.2#Px1.p2">x</mi><mn>5</mn></msup><mo>-</mo><mrow><mn>10</mn><mo></mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>3</mn></msup></mrow></mrow><mo>+</mo><mrow><mn>15</mn><mo></mo><mi href="./18.2#Px1.p2">x</mi></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex42" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m84.png" altimg-height="25px" altimg-valign="-7px" altimg-width="71px" alttext="\displaystyle\mathit{He}_{6}\left(x\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r14" mathvariant="italic">He</mi><mn>6</mn></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m38.png" altimg-height="25px" altimg-valign="-4px" altimg-width="226px" alttext="\displaystyle=x^{6}-15x^{4}+45x^{2}-15." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mrow><msup><mi href="./18.2#Px1.p2">x</mi><mn>6</mn></msup><mo>-</mo><mrow><mn>15</mn><mo></mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>4</mn></msup></mrow></mrow><mo>+</mo><mrow><mn>45</mn><mo></mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>2</mn></msup></mrow></mrow><mo>-</mo><mn>15</mn></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E19.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./18.3#T1.t1.r14" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m126.png" altimg-height="23px" altimg-valign="-7px" altimg-width="71px" alttext="\mathit{He}_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r14" mathvariant="italic">He</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Hermite polynomial</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m141.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="Px9.p2" class="ltx_para">
<p class="ltx_p">For the corresponding polynomials of degrees 7 through 12 see <cite class="ltx_cite ltx_citemacro_citet">Abramowitz and Stegun (</div>
</div>
</body></text>
</html>
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<!DOCTYPE html><html>
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<title>DLMF: 18.14 Inequalities</title>
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<div id="Px1.p1" class="ltx_para">
<table id="E1" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">18.14.1</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E1.png" altimg-height="49px" altimg-valign="-16px" altimg-width="324px" alttext="|P^{(\alpha,\beta)}_{n}\left(x\right)|\leq P^{(\alpha,\beta)}_{n}\left(1\right%
)=\frac{{\left(\alpha+1\right)_{n}}}{n!}," display="block"><mrow><mrow><mrow><mo stretchy="false">|</mo><mrow><msubsup><mi href="./18.3#T1.t1.r2">P</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r2" stretchy="false">(</mo><mi>α</mi><mo href="./18.3#T1.t1.r2">,</mo><mi>β</mi><mo href="./18.3#T1.t1.r2" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow><mo>≤</mo><mrow><msubsup><mi href="./18.3#T1.t1.r2">P</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r2" stretchy="false">(</mo><mi>α</mi><mo href="./18.3#T1.t1.r2">,</mo><mi>β</mi><mo href="./18.3#T1.t1.r2" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow><mo>=</mo><mfrac><msub><mrow><mo href="./5.2#iii">(</mo><mrow><mi>α</mi><mo>+</mo><mn>1</mn></mrow><mo href="./5.2#iii">)</mo></mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi></msub><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mfrac></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m16.png" altimg-height="19px" altimg-valign="-5px" altimg-width="104px" alttext="-1\leq x\leq 1" display="inline"><mrow><mrow><mo>-</mo><mn>1</mn></mrow><mo>≤</mo><mi href="./18.2#Px1.p2">x</mi><mo>≤</mo><mn>1</mn></mrow></math>,
<math class="ltx_Math" altimg="m38.png" altimg-height="21px" altimg-valign="-6px" altimg-width="108px" alttext="\alpha\geq\beta>-1" display="inline"><mrow><mi>α</mi><mo>≥</mo><mi>β</mi><mo>></mo><mrow><mo>-</mo><mn>1</mn></mrow></mrow></math>,
<math class="ltx_Math" altimg="m37.png" altimg-height="27px" altimg-valign="-9px" altimg-width="72px" alttext="\alpha\geq-\tfrac{1}{2}" display="inline"><mrow><mi>α</mi><mo>≥</mo><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./18.3#T1.t1.r2" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m26.png" altimg-height="29px" altimg-valign="-7px" altimg-width="88px" alttext="P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msubsup><mi href="./18.3#T1.t1.r2">P</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r2" stretchy="false">(</mo><mi class="ltx_nvar">α</mi><mo href="./18.3#T1.t1.r2">,</mo><mi class="ltx_nvar">β</mi><mo href="./18.3#T1.t1.r2" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Jacobi polynomial</a>,
<a href="./5.2#iii" title="§5.2(iii) Pochhammer’s Symbol ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m57.png" altimg-height="24px" altimg-valign="-8px" altimg-width="41px" alttext="{\left(\NVar{a}\right)_{\NVar{n}}}" display="inline"><msub><mrow><mo href="./5.2#iii">(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r5">a</mi><mo href="./5.2#iii">)</mo></mrow><mi class="ltx_nvar" href="./5.1#p2.t1.r1">n</mi></msub></math>: Pochhammer’s symbol (or shifted factorial)</a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m11.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m50.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./18.1#SS1.p2.t1.r5" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">22.14.1</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E2" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">18.14.2</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E2.png" altimg-height="49px" altimg-valign="-16px" altimg-width="353px" alttext="|P^{(\alpha,\beta)}_{n}\left(x\right)|\leq|P^{(\alpha,\beta)}_{n}\left(-1%
\right)|=\frac{{\left(\beta+1\right)_{n}}}{n!}," display="block"><mrow><mrow><mrow><mo stretchy="false">|</mo><mrow><msubsup><mi href="./18.3#T1.t1.r2">P</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r2" stretchy="false">(</mo><mi>α</mi><mo href="./18.3#T1.t1.r2">,</mo><mi>β</mi><mo href="./18.3#T1.t1.r2" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow><mo>≤</mo><mrow><mo stretchy="false">|</mo><mrow><msubsup><mi href="./18.3#T1.t1.r2">P</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r2" stretchy="false">(</mo><mi>α</mi><mo href="./18.3#T1.t1.r2">,</mo><mi>β</mi><mo href="./18.3#T1.t1.r2" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow><mo>=</mo><mfrac><msub><mrow><mo href="./5.2#iii">(</mo><mrow><mi>β</mi><mo>+</mo><mn>1</mn></mrow><mo href="./5.2#iii">)</mo></mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi></msub><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mfrac></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m16.png" altimg-height="19px" altimg-valign="-5px" altimg-width="104px" alttext="-1\leq x\leq 1" display="inline"><mrow><mrow><mo>-</mo><mn>1</mn></mrow><mo>≤</mo><mi href="./18.2#Px1.p2">x</mi><mo>≤</mo><mn>1</mn></mrow></math>,
<math class="ltx_Math" altimg="m42.png" altimg-height="21px" altimg-valign="-6px" altimg-width="108px" alttext="\beta\geq\alpha>-1" display="inline"><mrow><mi>β</mi><mo>≥</mo><mi>α</mi><mo>></mo><mrow><mo>-</mo><mn>1</mn></mrow></mrow></math>,
<math class="ltx_Math" altimg="m41.png" altimg-height="27px" altimg-valign="-9px" altimg-width="71px" alttext="\beta\geq-\tfrac{1}{2}" display="inline"><mrow><mi>β</mi><mo>≥</mo><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./18.3#T1.t1.r2" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m26.png" altimg-height="29px" altimg-valign="-7px" altimg-width="88px" alttext="P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msubsup><mi href="./18.3#T1.t1.r2">P</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r2" stretchy="false">(</mo><mi class="ltx_nvar">α</mi><mo href="./18.3#T1.t1.r2">,</mo><mi class="ltx_nvar">β</mi><mo href="./18.3#T1.t1.r2" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Jacobi polynomial</a>,
<a href="./5.2#iii" title="§5.2(iii) Pochhammer’s Symbol ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m57.png" altimg-height="24px" altimg-valign="-8px" altimg-width="41px" alttext="{\left(\NVar{a}\right)_{\NVar{n}}}" display="inline"><msub><mrow><mo href="./5.2#iii">(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r5">a</mi><mo href="./5.2#iii">)</mo></mrow><mi class="ltx_nvar" href="./5.1#p2.t1.r1">n</mi></msub></math>: Pochhammer’s symbol (or shifted factorial)</a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m11.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m50.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./18.1#SS1.p2.t1.r5" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">22.14.1</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E3" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">18.14.3</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E3.png" altimg-height="63px" altimg-valign="-31px" altimg-width="724px" alttext="\left(\tfrac{1}{2}(1-x)\right)^{\frac{1}{2}\alpha+\frac{1}{4}}\left(\tfrac{1}{%
2}(1+x)\right)^{\frac{1}{2}\beta+\frac{1}{4}}|P^{(\alpha,\beta)}_{n}\left(x%
\right)|\leq\frac{\Gamma\left(\max(\alpha,\beta)+n+1\right)}{\pi^{\frac{1}{2}}%
n!\left(n+\tfrac{1}{2}(\alpha+\beta+1)\right)^{\max(\alpha,\beta)+\frac{1}{2}}}," display="block"><mrow><mrow><mrow><msup><mrow><mo>(</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><mi href="./18.2#Px1.p2">x</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>)</mo></mrow><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi>α</mi></mrow><mo>+</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></msup><mo></mo><msup><mrow><mo>(</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>+</mo><mi href="./18.2#Px1.p2">x</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>)</mo></mrow><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi>β</mi></mrow><mo>+</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></msup><mo></mo><mrow><mo stretchy="false">|</mo><mrow><msubsup><mi href="./18.3#T1.t1.r2">P</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r2" stretchy="false">(</mo><mi>α</mi><mo href="./18.3#T1.t1.r2">,</mo><mi>β</mi><mo href="./18.3#T1.t1.r2" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow></mrow><mo>≤</mo><mfrac><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mi>max</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo stretchy="false">)</mo></mrow></mrow><mo>+</mo><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mrow><msup><mi href="./3.12#E1">π</mi><mfrac><mn>1</mn><mn>2</mn></mfrac></msup><mo></mo><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow><mo></mo><msup><mrow><mo>(</mo><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>+</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>α</mi><mo>+</mo><mi>β</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>)</mo></mrow><mrow><mrow><mi>max</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo stretchy="false">)</mo></mrow></mrow><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup></mrow></mfrac></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m16.png" altimg-height="19px" altimg-valign="-5px" altimg-width="104px" alttext="-1\leq x\leq 1" display="inline"><mrow><mrow><mo>-</mo><mn>1</mn></mrow><mo>≤</mo><mi href="./18.2#Px1.p2">x</mi><mo>≤</mo><mn>1</mn></mrow></math>,
<math class="ltx_Math" altimg="m19.png" altimg-height="27px" altimg-valign="-9px" altimg-width="111px" alttext="-\tfrac{1}{2}\leq\alpha\leq\tfrac{1}{2}" display="inline"><mrow><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>≤</mo><mi>α</mi><mo>≤</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math>,
<math class="ltx_Math" altimg="m20.png" altimg-height="27px" altimg-valign="-9px" altimg-width="111px" alttext="-\tfrac{1}{2}\leq\beta\leq\tfrac{1}{2}" display="inline"><mrow><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>≤</mo><mi>β</mi><mo>≤</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m31.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./18.3#T1.t1.r2" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m26.png" altimg-height="29px" altimg-valign="-7px" altimg-width="88px" alttext="P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msubsup><mi href="./18.3#T1.t1.r2">P</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r2" stretchy="false">(</mo><mi class="ltx_nvar">α</mi><mo href="./18.3#T1.t1.r2">,</mo><mi class="ltx_nvar">β</mi><mo href="./18.3#T1.t1.r2" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Jacobi polynomial</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m45.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m11.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m50.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./18.1#SS1.p2.t1.r5" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px2" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Ultraspherical</h3>
<div id="Px2.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px2.p1" class="ltx_para">
<table id="E4" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">18.14.4</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E4.png" altimg-height="49px" altimg-valign="-16px" altimg-width="268px" alttext="|C^{(\lambda)}_{n}\left(x\right)|\leq C^{(\lambda)}_{n}\left(1\right)=\frac{{%
\left(2\lambda\right)_{n}}}{n!}," display="block"><mrow><mrow><mrow><mo stretchy="false">|</mo><mrow><msubsup><mi href="./18.3#T1.t1.r3">C</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r3" stretchy="false">(</mo><mi>λ</mi><mo href="./18.3#T1.t1.r3" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow><mo>≤</mo><mrow><msubsup><mi href="./18.3#T1.t1.r3">C</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r3" stretchy="false">(</mo><mi>λ</mi><mo href="./18.3#T1.t1.r3" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow><mo>=</mo><mfrac><msub><mrow><mo href="./5.2#iii">(</mo><mrow><mn>2</mn><mo></mo><mi>λ</mi></mrow><mo href="./5.2#iii">)</mo></mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi></msub><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mfrac></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m16.png" altimg-height="19px" altimg-valign="-5px" altimg-width="104px" alttext="-1\leq x\leq 1" display="inline"><mrow><mrow><mo>-</mo><mn>1</mn></mrow><mo>≤</mo><mi href="./18.2#Px1.p2">x</mi><mo>≤</mo><mn>1</mn></mrow></math>, <math class="ltx_Math" altimg="m43.png" altimg-height="18px" altimg-valign="-3px" altimg-width="52px" alttext="\lambda>0" display="inline"><mrow><mi>λ</mi><mo>></mo><mn>0</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E4.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#iii" title="§5.2(iii) Pochhammer’s Symbol ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m57.png" altimg-height="24px" altimg-valign="-8px" altimg-width="41px" alttext="{\left(\NVar{a}\right)_{\NVar{n}}}" display="inline"><msub><mrow><mo href="./5.2#iii">(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r5">a</mi><mo href="./5.2#iii">)</mo></mrow><mi class="ltx_nvar" href="./5.1#p2.t1.r1">n</mi></msub></math>: Pochhammer’s symbol (or shifted factorial)</a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m11.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m50.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./18.3#T1.t1.r3" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m23.png" altimg-height="29px" altimg-valign="-7px" altimg-width="73px" alttext="C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msubsup><mi href="./18.3#T1.t1.r3">C</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r3" stretchy="false">(</mo><mi class="ltx_nvar">λ</mi><mo href="./18.3#T1.t1.r3" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: ultraspherical (or Gegenbauer) polynomial</a>,
<a href="./18.1#SS1.p2.t1.r5" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">22.14.2</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E5" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">18.14.5</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E5.png" altimg-height="53px" altimg-valign="-21px" altimg-width="293px" alttext="|C^{(\lambda)}_{2m}\left(x\right)|\leq|C^{(\lambda)}_{2m}\left(0\right)|=\left%
|\frac{{\left(\lambda\right)_{m}}}{m!}\right|," display="block"><mrow><mrow><mrow><mo stretchy="false">|</mo><mrow><msubsup><mi href="./18.3#T1.t1.r3">C</mi><mrow><mn>2</mn><mo></mo><mi href="./18.1#SS1.p2.t1.r4">m</mi></mrow><mrow><mo href="./18.3#T1.t1.r3" stretchy="false">(</mo><mi>λ</mi><mo href="./18.3#T1.t1.r3" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow><mo>≤</mo><mrow><mo stretchy="false">|</mo><mrow><msubsup><mi href="./18.3#T1.t1.r3">C</mi><mrow><mn>2</mn><mo></mo><mi href="./18.1#SS1.p2.t1.r4">m</mi></mrow><mrow><mo href="./18.3#T1.t1.r3" stretchy="false">(</mo><mi>λ</mi><mo href="./18.3#T1.t1.r3" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow><mo>=</mo><mrow><mo>|</mo><mfrac><msub><mrow><mo href="./5.2#iii">(</mo><mi>λ</mi><mo href="./5.2#iii">)</mo></mrow><mi href="./18.1#SS1.p2.t1.r4">m</mi></msub><mrow><mi href="./18.1#SS1.p2.t1.r4">m</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mfrac><mo>|</mo></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m16.png" altimg-height="19px" altimg-valign="-5px" altimg-width="104px" alttext="-1\leq x\leq 1" display="inline"><mrow><mrow><mo>-</mo><mn>1</mn></mrow><mo>≤</mo><mi href="./18.2#Px1.p2">x</mi><mo>≤</mo><mn>1</mn></mrow></math>, <math class="ltx_Math" altimg="m18.png" altimg-height="27px" altimg-valign="-9px" altimg-width="107px" alttext="-\tfrac{1}{2}<\lambda<0" display="inline"><mrow><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo><</mo><mi>λ</mi><mo><</mo><mn>0</mn></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E5.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#iii" title="§5.2(iii) Pochhammer’s Symbol ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m57.png" altimg-height="24px" altimg-valign="-8px" altimg-width="41px" alttext="{\left(\NVar{a}\right)_{\NVar{n}}}" display="inline"><msub><mrow><mo href="./5.2#iii">(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r5">a</mi><mo href="./5.2#iii">)</mo></mrow><mi class="ltx_nvar" href="./5.1#p2.t1.r1">n</mi></msub></math>: Pochhammer’s symbol (or shifted factorial)</a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m11.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m50.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./18.3#T1.t1.r3" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m23.png" altimg-height="29px" altimg-valign="-7px" altimg-width="73px" alttext="C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msubsup><mi href="./18.3#T1.t1.r3">C</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r3" stretchy="false">(</mo><mi class="ltx_nvar">λ</mi><mo href="./18.3#T1.t1.r3" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: ultraspherical (or Gegenbauer) polynomial</a>,
<a href="./18.1#SS1.p2.t1.r4" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./18.1#SS1.p2.t1.r4">m</mi></math>: nonnegative integer</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">22.14.2</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E6" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">18.14.6</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E6.png" altimg-height="60px" altimg-valign="-27px" altimg-width="400px" alttext="|C^{(\lambda)}_{2m+1}\left(x\right)|<\frac{-2{\left(\lambda\right)_{m+1}}}{%
\left((2m+1)(2\lambda+2m+1)\right)^{\frac{1}{2}}m!}," display="block"><mrow><mrow><mrow><mo stretchy="false">|</mo><mrow><msubsup><mi href="./18.3#T1.t1.r3">C</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./18.1#SS1.p2.t1.r4">m</mi></mrow><mo>+</mo><mn>1</mn></mrow><mrow><mo href="./18.3#T1.t1.r3" stretchy="false">(</mo><mi>λ</mi><mo href="./18.3#T1.t1.r3" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow><mo><</mo><mfrac><mrow><mo>-</mo><mrow><mn>2</mn><mo></mo><msub><mrow><mo href="./5.2#iii">(</mo><mi>λ</mi><mo href="./5.2#iii">)</mo></mrow><mrow><mi href="./18.1#SS1.p2.t1.r4">m</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></mrow><mrow><msup><mrow><mo>(</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>2</mn><mo></mo><mi href="./18.1#SS1.p2.t1.r4">m</mi></mrow><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>2</mn><mo></mo><mi>λ</mi></mrow><mo>+</mo><mrow><mn>2</mn><mo></mo><mi href="./18.1#SS1.p2.t1.r4">m</mi></mrow><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>)</mo></mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></msup><mo></mo><mrow><mi href="./18.1#SS1.p2.t1.r4">m</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mrow></mfrac></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m16.png" altimg-height="19px" altimg-valign="-5px" altimg-width="104px" alttext="-1\leq x\leq 1" display="inline"><mrow><mrow><mo>-</mo><mn>1</mn></mrow><mo>≤</mo><mi href="./18.2#Px1.p2">x</mi><mo>≤</mo><mn>1</mn></mrow></math>, <math class="ltx_Math" altimg="m18.png" altimg-height="27px" altimg-valign="-9px" altimg-width="107px" alttext="-\tfrac{1}{2}<\lambda<0" display="inline"><mrow><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo><</mo><mi>λ</mi><mo><</mo><mn>0</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E6.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#iii" title="§5.2(iii) Pochhammer’s Symbol ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m57.png" altimg-height="24px" altimg-valign="-8px" altimg-width="41px" alttext="{\left(\NVar{a}\right)_{\NVar{n}}}" display="inline"><msub><mrow><mo href="./5.2#iii">(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r5">a</mi><mo href="./5.2#iii">)</mo></mrow><mi class="ltx_nvar" href="./5.1#p2.t1.r1">n</mi></msub></math>: Pochhammer’s symbol (or shifted factorial)</a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m11.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m50.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./18.3#T1.t1.r3" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m23.png" altimg-height="29px" altimg-valign="-7px" altimg-width="73px" alttext="C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msubsup><mi href="./18.3#T1.t1.r3">C</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r3" stretchy="false">(</mo><mi class="ltx_nvar">λ</mi><mo href="./18.3#T1.t1.r3" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: ultraspherical (or Gegenbauer) polynomial</a>,
<a href="./18.1#SS1.p2.t1.r4" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./18.1#SS1.p2.t1.r4">m</mi></math>: nonnegative integer</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E7" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">18.14.7</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E7.png" altimg-height="55px" altimg-valign="-21px" altimg-width="361px" alttext="{(n+\lambda)^{1-\lambda}(1-x^{2})^{\frac{1}{2}\lambda}|C^{(\lambda)}_{n}\left(%
x\right)|<\frac{2^{1-\lambda}}{\Gamma\left(\lambda\right)}}," display="block"><mrow><mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>+</mo><mi>λ</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>-</mo><mi>λ</mi></mrow></msup><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi>λ</mi></mrow></msup><mo></mo><mrow><mo stretchy="false">|</mo><mrow><msubsup><mi href="./18.3#T1.t1.r3">C</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r3" stretchy="false">(</mo><mi>λ</mi><mo href="./18.3#T1.t1.r3" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow></mrow><mo><</mo><mfrac><msup><mn>2</mn><mrow><mn>1</mn><mo>-</mo><mi>λ</mi></mrow></msup><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi>λ</mi><mo>)</mo></mrow></mrow></mfrac></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m16.png" altimg-height="19px" altimg-valign="-5px" altimg-width="104px" alttext="-1\leq x\leq 1" display="inline"><mrow><mrow><mo>-</mo><mn>1</mn></mrow><mo>≤</mo><mi href="./18.2#Px1.p2">x</mi><mo>≤</mo><mn>1</mn></mrow></math>, <math class="ltx_Math" altimg="m21.png" altimg-height="18px" altimg-valign="-3px" altimg-width="89px" alttext="0<\lambda<1" display="inline"><mrow><mn>0</mn><mo><</mo><mi>λ</mi><mo><</mo><mn>1</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E7.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m31.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./18.3#T1.t1.r3" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m23.png" altimg-height="29px" altimg-valign="-7px" altimg-width="73px" alttext="C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msubsup><mi href="./18.3#T1.t1.r3">C</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r3" stretchy="false">(</mo><mi class="ltx_nvar">λ</mi><mo href="./18.3#T1.t1.r3" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: ultraspherical (or Gegenbauer) polynomial</a>,
<a href="./18.1#SS1.p2.t1.r5" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px3" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Laguerre</h3>
<div id="Px3.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px3.p1" class="ltx_para">
<table id="E8" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">18.14.8</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E8.png" altimg-height="49px" altimg-valign="-16px" altimg-width="337px" alttext="e^{-\frac{1}{2}x}\left|L^{(\alpha)}_{n}\left(x\right)\right|\leq L^{(\alpha)}_%
{n}\left(0\right)=\frac{{\left(\alpha+1\right)_{n}}}{n!}," display="block"><mrow><mrow><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./18.2#Px1.p2">x</mi></mrow></mrow></msup><mo></mo><mrow><mo>|</mo><mrow><msubsup><mi href="./18.3#T1.t1.r12">L</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r12" stretchy="false">(</mo><mi>α</mi><mo href="./18.3#T1.t1.r12" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow><mo>|</mo></mrow></mrow><mo>≤</mo><mrow><msubsup><mi href="./18.3#T1.t1.r12">L</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r12" stretchy="false">(</mo><mi>α</mi><mo href="./18.3#T1.t1.r12" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow></mrow><mo>=</mo><mfrac><msub><mrow><mo href="./5.2#iii">(</mo><mrow><mi>α</mi><mo>+</mo><mn>1</mn></mrow><mo href="./5.2#iii">)</mo></mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi></msub><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mfrac></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m22.png" altimg-height="19px" altimg-valign="-5px" altimg-width="99px" alttext="0\leq x<\infty" display="inline"><mrow><mn>0</mn><mo>≤</mo><mi href="./18.2#Px1.p2">x</mi><mo><</mo><mi mathvariant="normal">∞</mi></mrow></math>, <math class="ltx_Math" altimg="m36.png" altimg-height="19px" altimg-valign="-5px" altimg-width="54px" alttext="\alpha\geq 0" display="inline"><mrow><mi>α</mi><mo>≥</mo><mn>0</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E8.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./18.3#T1.t1.r12" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m25.png" altimg-height="29px" altimg-valign="-7px" altimg-width="72px" alttext="L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msubsup><mi href="./18.3#T1.t1.r12">L</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r12" stretchy="false">(</mo><mi class="ltx_nvar">α</mi><mo href="./18.3#T1.t1.r12" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Laguerre (or generalized Laguerre) polynomial</a>,
<a href="./5.2#iii" title="§5.2(iii) Pochhammer’s Symbol ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m57.png" altimg-height="24px" altimg-valign="-8px" altimg-width="41px" alttext="{\left(\NVar{a}\right)_{\NVar{n}}}" display="inline"><msub><mrow><mo href="./5.2#iii">(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r5">a</mi><mo href="./5.2#iii">)</mo></mrow><mi class="ltx_nvar" href="./5.1#p2.t1.r1">n</mi></msub></math>: Pochhammer’s symbol (or shifted factorial)</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m11.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m50.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./18.1#SS1.p2.t1.r5" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">22.14.13</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px4" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Hermite</h3>
<div id="Px4.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px4.p1" class="ltx_para">
<table id="E9" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">18.14.9</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E9.png" altimg-height="53px" altimg-valign="-24px" altimg-width="242px" alttext="\frac{1}{(2^{n}n!)^{\frac{1}{2}}}e^{-\frac{1}{2}x^{2}}|H_{n}\left(x\right)|%
\leq 1," display="block"><mrow><mrow><mrow><mfrac><mn>1</mn><msup><mrow><mo stretchy="false">(</mo><mrow><msup><mn>2</mn><mi href="./18.1#SS1.p2.t1.r5">n</mi></msup><mo></mo><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mrow><mo stretchy="false">)</mo></mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></msup></mfrac><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>2</mn></msup></mrow></mrow></msup><mo></mo><mrow><mo stretchy="false">|</mo><mrow><msub><mi href="./18.3#T1.t1.r13">H</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow></mrow><mo>≤</mo><mn>1</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m17.png" altimg-height="17px" altimg-valign="-4px" altimg-width="124px" alttext="-\infty<x<\infty" display="inline"><mrow><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mo><</mo><mi href="./18.2#Px1.p2">x</mi><mo><</mo><mi mathvariant="normal">∞</mi></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E9.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./18.3#T1.t1.r13" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m24.png" altimg-height="23px" altimg-valign="-7px" altimg-width="62px" alttext="H_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r13">H</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Hermite polynomial</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m11.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m50.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./18.1#SS1.p2.t1.r5" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">22.14.17</span> ((version here is sharpened))</span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="Px4.p2" class="ltx_para">
<p class="ltx_p">For further inequalities see <cite class="ltx_cite ltx_citemacro_citet">Abramowitz and Stegun (</dd>
</dl>
</div>
</div>
<div id="Px5.p1" class="ltx_para">
<table id="E10" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">18.14.10</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E10.png" altimg-height="28px" altimg-valign="-7px" altimg-width="271px" alttext="(P_{n}\left(x\right))^{2}\geq P_{n-1}\left(x\right)P_{n+1}\left(x\right)," display="block"><mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><msub><mi href="./18.3#T1.t1.r10">P</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>≥</mo><mrow><mrow><msub><mi href="./18.3#T1.t1.r10">P</mi><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./18.3#T1.t1.r10">P</mi><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m16.png" altimg-height="19px" altimg-valign="-5px" altimg-width="104px" alttext="-1\leq x\leq 1" display="inline"><mrow><mrow><mo>-</mo><mn>1</mn></mrow><mo>≤</mo><mi href="./18.2#Px1.p2">x</mi><mo>≤</mo><mn>1</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E10.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./18.3#T1.t1.r10" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m27.png" altimg-height="23px" altimg-valign="-7px" altimg-width="58px" alttext="P_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r10">P</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Legendre polynomial</a>,
<a href="./18.1#SS1.p2.t1.r5" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px6" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Jacobi</h3>
<div id="Px6.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px6.p1" class="ltx_para">
<p class="ltx_p">Let <math class="ltx_Math" altimg="m28.png" altimg-height="29px" altimg-valign="-7px" altimg-width="264px" alttext="R_{n}(x)=P^{(\alpha,\beta)}_{n}\left(x\right)/P^{(\alpha,\beta)}_{n}\left(1\right)" display="inline"><mrow><mrow><msub><mi>R</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./18.2#Px1.p2">x</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mrow><msubsup><mi href="./18.3#T1.t1.r2">P</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r2" stretchy="false">(</mo><mi>α</mi><mo href="./18.3#T1.t1.r2">,</mo><mi>β</mi><mo href="./18.3#T1.t1.r2" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow><mo>/</mo><mrow><msubsup><mi href="./18.3#T1.t1.r2">P</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r2" stretchy="false">(</mo><mi>α</mi><mo href="./18.3#T1.t1.r2">,</mo><mi>β</mi><mo href="./18.3#T1.t1.r2" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></mrow></mrow></math>.
Then</p>
<table id="E11" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">18.14.11</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E11.png" altimg-height="28px" altimg-valign="-7px" altimg-width="262px" alttext="(R_{n}(x))^{2}\geq R_{n-1}(x)R_{n+1}(x)," display="block"><mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><msub><mi>R</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./18.2#Px1.p2">x</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>≥</mo><mrow><mrow><msub><mi>R</mi><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./18.2#Px1.p2">x</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><msub><mi>R</mi><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./18.2#Px1.p2">x</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m16.png" altimg-height="19px" altimg-valign="-5px" altimg-width="104px" alttext="-1\leq x\leq 1" display="inline"><mrow><mrow><mo>-</mo><mn>1</mn></mrow><mo>≤</mo><mi href="./18.2#Px1.p2">x</mi><mo>≤</mo><mn>1</mn></mrow></math>, <math class="ltx_Math" altimg="m42.png" altimg-height="21px" altimg-valign="-6px" altimg-width="108px" alttext="\beta\geq\alpha>-1" display="inline"><mrow><mi>β</mi><mo>≥</mo><mi>α</mi><mo>></mo><mrow><mo>-</mo><mn>1</mn></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E11.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./18.1#SS1.p2.t1.r5" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px7" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Laguerre</h3>
<div id="Px7.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px7.p1" class="ltx_para">
<table id="E12" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">18.14.12</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E12.png" altimg-height="33px" altimg-valign="-9px" altimg-width="287px" alttext="(L^{(\alpha)}_{n}\left(x\right))^{2}\geq L^{(\alpha)}_{n-1}\left(x\right)L^{(%
\alpha)}_{n+1}\left(x\right)," display="block"><mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><msubsup><mi href="./18.3#T1.t1.r12">L</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r12" stretchy="false">(</mo><mi>α</mi><mo href="./18.3#T1.t1.r12" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>≥</mo><mrow><mrow><msubsup><mi href="./18.3#T1.t1.r12">L</mi><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow><mrow><mo href="./18.3#T1.t1.r12" stretchy="false">(</mo><mi>α</mi><mo href="./18.3#T1.t1.r12" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><msubsup><mi href="./18.3#T1.t1.r12">L</mi><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>+</mo><mn>1</mn></mrow><mrow><mo href="./18.3#T1.t1.r12" stretchy="false">(</mo><mi>α</mi><mo href="./18.3#T1.t1.r12" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m22.png" altimg-height="19px" altimg-valign="-5px" altimg-width="99px" alttext="0\leq x<\infty" display="inline"><mrow><mn>0</mn><mo>≤</mo><mi href="./18.2#Px1.p2">x</mi><mo><</mo><mi mathvariant="normal">∞</mi></mrow></math>, <math class="ltx_Math" altimg="m36.png" altimg-height="19px" altimg-valign="-5px" altimg-width="54px" alttext="\alpha\geq 0" display="inline"><mrow><mi>α</mi><mo>≥</mo><mn>0</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E12.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./18.3#T1.t1.r12" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m25.png" altimg-height="29px" altimg-valign="-7px" altimg-width="72px" alttext="L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msubsup><mi href="./18.3#T1.t1.r12">L</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r12" stretchy="false">(</mo><mi class="ltx_nvar">α</mi><mo href="./18.3#T1.t1.r12" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Laguerre (or generalized Laguerre) polynomial</a>,
<a href="./18.1#SS1.p2.t1.r5" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px8" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Hermite</h3>
<div id="Px8.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px8.p1" class="ltx_para">
<table id="E13" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">18.14.13</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E13.png" altimg-height="28px" altimg-valign="-7px" altimg-width="283px" alttext="(H_{n}\left(x\right))^{2}\geq H_{n-1}\left(x\right)H_{n+1}\left(x\right)," display="block"><mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><msub><mi href="./18.3#T1.t1.r13">H</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>≥</mo><mrow><mrow><msub><mi href="./18.3#T1.t1.r13">H</mi><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./18.3#T1.t1.r13">H</mi><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m17.png" altimg-height="17px" altimg-valign="-4px" altimg-width="124px" alttext="-\infty<x<\infty" display="inline"><mrow><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mo><</mo><mi href="./18.2#Px1.p2">x</mi><mo><</mo><mi mathvariant="normal">∞</mi></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E13.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./18.3#T1.t1.r13" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m24.png" altimg-height="23px" altimg-valign="-7px" altimg-width="62px" alttext="H_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r13">H</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Hermite polynomial</a>,
<a href="./18.1#SS1.p2.t1.r5" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
</section>
<section id="iii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§18.14(iii) </span>Local Maxima and Minima</h2>
<div id="SS3.info" class="ltx_metadata ltx_info">
, Theorem 7.6.1)</cite>.
For the last statement about the successive maxima of
<math class="ltx_Math" altimg="m58.png" altimg-height="23px" altimg-valign="-7px" altimg-width="76px" alttext="|H_{n}\left(x\right)|" display="inline"><mrow><mo stretchy="false">|</mo><mrow><msub><mi href="./18.3#T1.t1.r13">H</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow></math> see <cite class="ltx_cite ltx_citemacro_citet">Szegő (</dd>
</dl>
</div>
</div>
<div id="Px9.p1" class="ltx_para">
<p class="ltx_p">Let the maxima <math class="ltx_Math" altimg="m56.png" altimg-height="18px" altimg-valign="-8px" altimg-width="45px" alttext="x_{n,m}" display="inline"><msub><mi href="./18.2#Px1.p2">x</mi><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>,</mo><mi href="./18.1#SS1.p2.t1.r4">m</mi></mrow></msub></math>, <math class="ltx_Math" altimg="m47.png" altimg-height="20px" altimg-valign="-6px" altimg-width="133px" alttext="m=0,1,\dots,n" display="inline"><mrow><mi href="./18.1#SS1.p2.t1.r4">m</mi><mo>=</mo><mrow><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><mi href="./18.1#SS1.p2.t1.r5">n</mi></mrow></mrow></math>, of
<math class="ltx_Math" altimg="m60.png" altimg-height="29px" altimg-valign="-7px" altimg-width="103px" alttext="|P^{(\alpha,\beta)}_{n}\left(x\right)|" display="inline"><mrow><mo stretchy="false">|</mo><mrow><msubsup><mi href="./18.3#T1.t1.r2">P</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r2" stretchy="false">(</mo><mi>α</mi><mo href="./18.3#T1.t1.r2">,</mo><mi>β</mi><mo href="./18.3#T1.t1.r2" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow></math> in <math class="ltx_Math" altimg="m29.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="[-1,1]" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">[</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo href="./front/introduction#Sx4.p1.t1.r29">,</mo><mn>1</mn><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">]</mo></mrow></math> be arranged so that</p>
<table id="E14" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">18.14.14</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E14.png" altimg-height="24px" altimg-valign="-8px" altimg-width="393px" alttext="-1=x_{n,0}<x_{n,1}<\cdots<x_{n,n-1}<x_{n,n}=1." display="block"><mrow><mrow><mrow><mo>-</mo><mn>1</mn></mrow><mo>=</mo><msub><mi href="./18.2#Px1.p2">x</mi><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>,</mo><mn>0</mn></mrow></msub><mo><</mo><msub><mi href="./18.2#Px1.p2">x</mi><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>,</mo><mn>1</mn></mrow></msub><mo><</mo><mi mathvariant="normal">⋯</mi><mo><</mo><msub><mi href="./18.2#Px1.p2">x</mi><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>,</mo><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow></mrow></msub><mo><</mo><msub><mi href="./18.2#Px1.p2">x</mi><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>,</mo><mi href="./18.1#SS1.p2.t1.r5">n</mi></mrow></msub><mo>=</mo><mn>1</mn></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E14.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./18.1#SS1.p2.t1.r5" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="Px9.p2" class="ltx_para">
<p class="ltx_p">When <math class="ltx_Math" altimg="m12.png" altimg-height="27px" altimg-valign="-9px" altimg-width="171px" alttext="(\alpha+\tfrac{1}{2})(\beta+\tfrac{1}{2})>0" display="inline"><mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mi>α</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>β</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>></mo><mn>0</mn></mrow></math> choose <math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./18.1#SS1.p2.t1.r4">m</mi></math> so that</p>
<table id="E15" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">18.14.15</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E15.png" altimg-height="26px" altimg-valign="-8px" altimg-width="341px" alttext="x_{n,m}\leq(\beta-\alpha)/(\alpha+\beta+1)\leq x_{n,m+1}." display="block"><mrow><mrow><msub><mi href="./18.2#Px1.p2">x</mi><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>,</mo><mi href="./18.1#SS1.p2.t1.r4">m</mi></mrow></msub><mo>≤</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mi>β</mi><mo>-</mo><mi>α</mi></mrow><mo stretchy="false">)</mo></mrow><mo>/</mo><mrow><mo stretchy="false">(</mo><mrow><mi>α</mi><mo>+</mo><mi>β</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>≤</mo><msub><mi href="./18.2#Px1.p2">x</mi><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>,</mo><mrow><mi href="./18.1#SS1.p2.t1.r4">m</mi><mo>+</mo><mn>1</mn></mrow></mrow></msub></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E15.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./18.1#SS1.p2.t1.r4" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./18.1#SS1.p2.t1.r4">m</mi></math>: nonnegative integer</a>,
<a href="./18.1#SS1.p2.t1.r5" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Then</p>
</div>
<div id="Px9.p3" class="ltx_para">
<table id="E16" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex1" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="3" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">18.14.16</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m9.png" altimg-height="30px" altimg-valign="-8px" altimg-width="128px" alttext="\displaystyle|P^{(\alpha,\beta)}_{n}\left(x_{n,0}\right)|" display="inline"><mrow><mo stretchy="false">|</mo><mrow><msubsup><mi href="./18.3#T1.t1.r2">P</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r2" stretchy="false">(</mo><mi>α</mi><mo href="./18.3#T1.t1.r2">,</mo><mi>β</mi><mo href="./18.3#T1.t1.r2" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><msub><mi href="./18.2#Px1.p2">x</mi><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>,</mo><mn>0</mn></mrow></msub><mo>)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m5.png" altimg-height="30px" altimg-valign="-8px" altimg-width="359px" alttext="\displaystyle>|P^{(\alpha,\beta)}_{n}\left(x_{n,1}\right)|>\cdots>|P^{(\alpha,%
\beta)}_{n}\left(x_{n,m}\right)|," display="inline"><mrow><mrow><mi></mi><mo>></mo><mrow><mo stretchy="false">|</mo><mrow><msubsup><mi href="./18.3#T1.t1.r2">P</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r2" stretchy="false">(</mo><mi>α</mi><mo href="./18.3#T1.t1.r2">,</mo><mi>β</mi><mo href="./18.3#T1.t1.r2" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><msub><mi href="./18.2#Px1.p2">x</mi><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>,</mo><mn>1</mn></mrow></msub><mo>)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow><mo>></mo><mi mathvariant="normal">⋯</mi><mo>></mo><mrow><mo stretchy="false">|</mo><mrow><msubsup><mi href="./18.3#T1.t1.r2">P</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r2" stretchy="false">(</mo><mi>α</mi><mo href="./18.3#T1.t1.r2">,</mo><mi>β</mi><mo href="./18.3#T1.t1.r2" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><msub><mi href="./18.2#Px1.p2">x</mi><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>,</mo><mi href="./18.1#SS1.p2.t1.r4">m</mi></mrow></msub><mo>)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex2" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m10.png" altimg-height="30px" altimg-valign="-8px" altimg-width="130px" alttext="\displaystyle|P^{(\alpha,\beta)}_{n}\left(x_{n,n}\right)|" display="inline"><mrow><mo stretchy="false">|</mo><mrow><msubsup><mi href="./18.3#T1.t1.r2">P</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r2" stretchy="false">(</mo><mi>α</mi><mo href="./18.3#T1.t1.r2">,</mo><mi>β</mi><mo href="./18.3#T1.t1.r2" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><msub><mi href="./18.2#Px1.p2">x</mi><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>,</mo><mi href="./18.1#SS1.p2.t1.r5">n</mi></mrow></msub><mo>)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m6.png" altimg-height="30px" altimg-valign="-8px" altimg-width="402px" alttext="\displaystyle>|P^{(\alpha,\beta)}_{n}\left(x_{n,n-1}\right)|>\cdots>|P^{(%
\alpha,\beta)}_{n}\left(x_{n,m+1}\right)|," display="inline"><mrow><mrow><mi></mi><mo>></mo><mrow><mo stretchy="false">|</mo><mrow><msubsup><mi href="./18.3#T1.t1.r2">P</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r2" stretchy="false">(</mo><mi>α</mi><mo href="./18.3#T1.t1.r2">,</mo><mi>β</mi><mo href="./18.3#T1.t1.r2" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><msub><mi href="./18.2#Px1.p2">x</mi><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>,</mo><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow></mrow></msub><mo>)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow><mo>></mo><mi mathvariant="normal">⋯</mi><mo>></mo><mrow><mo stretchy="false">|</mo><mrow><msubsup><mi href="./18.3#T1.t1.r2">P</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r2" stretchy="false">(</mo><mi>α</mi><mo href="./18.3#T1.t1.r2">,</mo><mi>β</mi><mo href="./18.3#T1.t1.r2" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><msub><mi href="./18.2#Px1.p2">x</mi><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>,</mo><mrow><mi href="./18.1#SS1.p2.t1.r4">m</mi><mo>+</mo><mn>1</mn></mrow></mrow></msub><mo>)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m34.png" altimg-height="27px" altimg-valign="-9px" altimg-width="153px" alttext="\alpha>-\tfrac{1}{2},\beta>-\tfrac{1}{2}." display="inline"><mrow><mrow><mrow><mi>α</mi><mo>></mo><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow><mo>,</mo><mrow><mi>β</mi><mo>></mo><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow></mrow><mo>.</mo></mrow></math></span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E16.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./18.3#T1.t1.r2" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m26.png" altimg-height="29px" altimg-valign="-7px" altimg-width="88px" alttext="P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msubsup><mi href="./18.3#T1.t1.r2">P</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r2" stretchy="false">(</mo><mi class="ltx_nvar">α</mi><mo href="./18.3#T1.t1.r2">,</mo><mi class="ltx_nvar">β</mi><mo href="./18.3#T1.t1.r2" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Jacobi polynomial</a>,
<a href="./18.1#SS1.p2.t1.r4" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./18.1#SS1.p2.t1.r4">m</mi></math>: nonnegative integer</a>,
<a href="./18.1#SS1.p2.t1.r5" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E17" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex3" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="3" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">18.14.17</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m9.png" altimg-height="30px" altimg-valign="-8px" altimg-width="128px" alttext="\displaystyle|P^{(\alpha,\beta)}_{n}\left(x_{n,0}\right)|" display="inline"><mrow><mo stretchy="false">|</mo><mrow><msubsup><mi href="./18.3#T1.t1.r2">P</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r2" stretchy="false">(</mo><mi>α</mi><mo href="./18.3#T1.t1.r2">,</mo><mi>β</mi><mo href="./18.3#T1.t1.r2" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><msub><mi href="./18.2#Px1.p2">x</mi><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>,</mo><mn>0</mn></mrow></msub><mo>)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m1.png" altimg-height="30px" altimg-valign="-8px" altimg-width="359px" alttext="\displaystyle<|P^{(\alpha,\beta)}_{n}\left(x_{n,1}\right)|<\cdots<|P^{(\alpha,%
\beta)}_{n}\left(x_{n,m}\right)|," display="inline"><mrow><mrow><mi></mi><mo><</mo><mrow><mo stretchy="false">|</mo><mrow><msubsup><mi href="./18.3#T1.t1.r2">P</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r2" stretchy="false">(</mo><mi>α</mi><mo href="./18.3#T1.t1.r2">,</mo><mi>β</mi><mo href="./18.3#T1.t1.r2" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><msub><mi href="./18.2#Px1.p2">x</mi><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>,</mo><mn>1</mn></mrow></msub><mo>)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow><mo><</mo><mi mathvariant="normal">⋯</mi><mo><</mo><mrow><mo stretchy="false">|</mo><mrow><msubsup><mi href="./18.3#T1.t1.r2">P</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r2" stretchy="false">(</mo><mi>α</mi><mo href="./18.3#T1.t1.r2">,</mo><mi>β</mi><mo href="./18.3#T1.t1.r2" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><msub><mi href="./18.2#Px1.p2">x</mi><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>,</mo><mi href="./18.1#SS1.p2.t1.r4">m</mi></mrow></msub><mo>)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex4" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m10.png" altimg-height="30px" altimg-valign="-8px" altimg-width="130px" alttext="\displaystyle|P^{(\alpha,\beta)}_{n}\left(x_{n,n}\right)|" display="inline"><mrow><mo stretchy="false">|</mo><mrow><msubsup><mi href="./18.3#T1.t1.r2">P</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r2" stretchy="false">(</mo><mi>α</mi><mo href="./18.3#T1.t1.r2">,</mo><mi>β</mi><mo href="./18.3#T1.t1.r2" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><msub><mi href="./18.2#Px1.p2">x</mi><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>,</mo><mi href="./18.1#SS1.p2.t1.r5">n</mi></mrow></msub><mo>)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m2.png" altimg-height="30px" altimg-valign="-8px" altimg-width="402px" alttext="\displaystyle<|P^{(\alpha,\beta)}_{n}\left(x_{n,n-1}\right)|<\cdots<|P^{(%
\alpha,\beta)}_{n}\left(x_{n,m+1}\right)|," display="inline"><mrow><mrow><mi></mi><mo><</mo><mrow><mo stretchy="false">|</mo><mrow><msubsup><mi href="./18.3#T1.t1.r2">P</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r2" stretchy="false">(</mo><mi>α</mi><mo href="./18.3#T1.t1.r2">,</mo><mi>β</mi><mo href="./18.3#T1.t1.r2" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><msub><mi href="./18.2#Px1.p2">x</mi><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>,</mo><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow></mrow></msub><mo>)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow><mo><</mo><mi mathvariant="normal">⋯</mi><mo><</mo><mrow><mo stretchy="false">|</mo><mrow><msubsup><mi href="./18.3#T1.t1.r2">P</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r2" stretchy="false">(</mo><mi>α</mi><mo href="./18.3#T1.t1.r2">,</mo><mi>β</mi><mo href="./18.3#T1.t1.r2" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><msub><mi href="./18.2#Px1.p2">x</mi><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>,</mo><mrow><mi href="./18.1#SS1.p2.t1.r4">m</mi><mo>+</mo><mn>1</mn></mrow></mrow></msub><mo>)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m13.png" altimg-height="27px" altimg-valign="-9px" altimg-width="257px" alttext="-1<\alpha<-\tfrac{1}{2},-1<\beta<-\tfrac{1}{2}." display="inline"><mrow><mrow><mrow><mrow><mo>-</mo><mn>1</mn></mrow><mo><</mo><mi>α</mi><mo><</mo><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow><mo>,</mo><mrow><mrow><mo>-</mo><mn>1</mn></mrow><mo><</mo><mi>β</mi><mo><</mo><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow></mrow><mo>.</mo></mrow></math></span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E17.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./18.3#T1.t1.r2" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m26.png" altimg-height="29px" altimg-valign="-7px" altimg-width="88px" alttext="P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msubsup><mi href="./18.3#T1.t1.r2">P</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r2" stretchy="false">(</mo><mi class="ltx_nvar">α</mi><mo href="./18.3#T1.t1.r2">,</mo><mi class="ltx_nvar">β</mi><mo href="./18.3#T1.t1.r2" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Jacobi polynomial</a>,
<a href="./18.1#SS1.p2.t1.r4" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./18.1#SS1.p2.t1.r4">m</mi></math>: nonnegative integer</a>,
<a href="./18.1#SS1.p2.t1.r5" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="Px9.p4" class="ltx_para">
<p class="ltx_p">Also,</p>
<table id="E18" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">18.14.18</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E18.png" altimg-height="30px" altimg-valign="-8px" altimg-width="482px" alttext="|P^{(\alpha,\beta)}_{n}\left(x_{n,0}\right)|<|P^{(\alpha,\beta)}_{n}\left(x_{n%
,1}\right)|<\cdots<|P^{(\alpha,\beta)}_{n}\left(x_{n,n}\right)|," display="block"><mrow><mrow><mrow><mo stretchy="false">|</mo><mrow><msubsup><mi href="./18.3#T1.t1.r2">P</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r2" stretchy="false">(</mo><mi>α</mi><mo href="./18.3#T1.t1.r2">,</mo><mi>β</mi><mo href="./18.3#T1.t1.r2" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><msub><mi href="./18.2#Px1.p2">x</mi><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>,</mo><mn>0</mn></mrow></msub><mo>)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow><mo><</mo><mrow><mo stretchy="false">|</mo><mrow><msubsup><mi href="./18.3#T1.t1.r2">P</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r2" stretchy="false">(</mo><mi>α</mi><mo href="./18.3#T1.t1.r2">,</mo><mi>β</mi><mo href="./18.3#T1.t1.r2" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><msub><mi href="./18.2#Px1.p2">x</mi><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>,</mo><mn>1</mn></mrow></msub><mo>)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow><mo><</mo><mi mathvariant="normal">⋯</mi><mo><</mo><mrow><mo stretchy="false">|</mo><mrow><msubsup><mi href="./18.3#T1.t1.r2">P</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r2" stretchy="false">(</mo><mi>α</mi><mo href="./18.3#T1.t1.r2">,</mo><mi>β</mi><mo href="./18.3#T1.t1.r2" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><msub><mi href="./18.2#Px1.p2">x</mi><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>,</mo><mi href="./18.1#SS1.p2.t1.r5">n</mi></mrow></msub><mo>)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m37.png" altimg-height="27px" altimg-valign="-9px" altimg-width="72px" alttext="\alpha\geq-\tfrac{1}{2}" display="inline"><mrow><mi>α</mi><mo>≥</mo><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow></math>, <math class="ltx_Math" altimg="m15.png" altimg-height="27px" altimg-valign="-9px" altimg-width="123px" alttext="-1<\beta\leq-\tfrac{1}{2}" display="inline"><mrow><mrow><mo>-</mo><mn>1</mn></mrow><mo><</mo><mi>β</mi><mo>≤</mo><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E18.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./18.3#T1.t1.r2" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m26.png" altimg-height="29px" altimg-valign="-7px" altimg-width="88px" alttext="P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msubsup><mi href="./18.3#T1.t1.r2">P</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r2" stretchy="false">(</mo><mi class="ltx_nvar">α</mi><mo href="./18.3#T1.t1.r2">,</mo><mi class="ltx_nvar">β</mi><mo href="./18.3#T1.t1.r2" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Jacobi polynomial</a>,
<a href="./18.1#SS1.p2.t1.r5" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E19" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">18.14.19</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E19.png" altimg-height="30px" altimg-valign="-8px" altimg-width="482px" alttext="|P^{(\alpha,\beta)}_{n}\left(x_{n,0}\right)|>|P^{(\alpha,\beta)}_{n}\left(x_{n%
,1}\right)|>\cdots>|P^{(\alpha,\beta)}_{n}\left(x_{n,n}\right)|," display="block"><mrow><mrow><mrow><mo stretchy="false">|</mo><mrow><msubsup><mi href="./18.3#T1.t1.r2">P</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r2" stretchy="false">(</mo><mi>α</mi><mo href="./18.3#T1.t1.r2">,</mo><mi>β</mi><mo href="./18.3#T1.t1.r2" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><msub><mi href="./18.2#Px1.p2">x</mi><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>,</mo><mn>0</mn></mrow></msub><mo>)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow><mo>></mo><mrow><mo stretchy="false">|</mo><mrow><msubsup><mi href="./18.3#T1.t1.r2">P</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r2" stretchy="false">(</mo><mi>α</mi><mo href="./18.3#T1.t1.r2">,</mo><mi>β</mi><mo href="./18.3#T1.t1.r2" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><msub><mi href="./18.2#Px1.p2">x</mi><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>,</mo><mn>1</mn></mrow></msub><mo>)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow><mo>></mo><mi mathvariant="normal">⋯</mi><mo>></mo><mrow><mo stretchy="false">|</mo><mrow><msubsup><mi href="./18.3#T1.t1.r2">P</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r2" stretchy="false">(</mo><mi>α</mi><mo href="./18.3#T1.t1.r2">,</mo><mi>β</mi><mo href="./18.3#T1.t1.r2" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><msub><mi href="./18.2#Px1.p2">x</mi><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>,</mo><mi href="./18.1#SS1.p2.t1.r5">n</mi></mrow></msub><mo>)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m40.png" altimg-height="27px" altimg-valign="-9px" altimg-width="71px" alttext="\beta\geq-\frac{1}{2}" display="inline"><mrow><mi>β</mi><mo>≥</mo><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow></math>, <math class="ltx_Math" altimg="m14.png" altimg-height="27px" altimg-valign="-9px" altimg-width="124px" alttext="-1<\alpha\leq-\tfrac{1}{2}" display="inline"><mrow><mrow><mo>-</mo><mn>1</mn></mrow><mo><</mo><mi>α</mi><mo>≤</mo><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E19.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./18.3#T1.t1.r2" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m26.png" altimg-height="29px" altimg-valign="-7px" altimg-width="88px" alttext="P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msubsup><mi href="./18.3#T1.t1.r2">P</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r2" stretchy="false">(</mo><mi class="ltx_nvar">α</mi><mo href="./18.3#T1.t1.r2">,</mo><mi class="ltx_nvar">β</mi><mo href="./18.3#T1.t1.r2" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Jacobi polynomial</a>,
<a href="./18.1#SS1.p2.t1.r5" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">except that when <math class="ltx_Math" altimg="m32.png" altimg-height="27px" altimg-valign="-9px" altimg-width="111px" alttext="\alpha=\beta=-\tfrac{1}{2}" display="inline"><mrow><mi>α</mi><mo>=</mo><mi>β</mi><mo>=</mo><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow></math> (Chebyshev case)
<math class="ltx_Math" altimg="m61.png" altimg-height="30px" altimg-valign="-8px" altimg-width="132px" alttext="|P^{(\alpha,\beta)}_{n}\left(x_{n,m}\right)|" display="inline"><mrow><mo stretchy="false">|</mo><mrow><msubsup><mi href="./18.3#T1.t1.r2">P</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r2" stretchy="false">(</mo><mi>α</mi><mo href="./18.3#T1.t1.r2">,</mo><mi>β</mi><mo href="./18.3#T1.t1.r2" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><msub><mi href="./18.2#Px1.p2">x</mi><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>,</mo><mi href="./18.1#SS1.p2.t1.r4">m</mi></mrow></msub><mo>)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow></math> is constant.
</p>
</div>
</section>
<section id="Px10" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Szegő–Szász Inequality</h3>
<div id="Px10.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px10.p1" class="ltx_para">
<table id="E20" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">18.14.20</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E20.png" altimg-height="67px" altimg-valign="-28px" altimg-width="390px" alttext="\left|\frac{P^{(\alpha,\beta)}_{n}\left(x_{n,n-m}\right)}{P^{(\alpha,\beta)}_{%
n}\left(1\right)}\right|>\left|\frac{P^{(\alpha,\beta)}_{n+1}\left(x_{n+1,n-m+%
1}\right)}{P^{(\alpha,\beta)}_{n+1}\left(1\right)}\right|," display="block"><mrow><mrow><mrow><mo>|</mo><mfrac><mrow><msubsup><mi href="./18.3#T1.t1.r2">P</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r2" stretchy="false">(</mo><mi>α</mi><mo href="./18.3#T1.t1.r2">,</mo><mi>β</mi><mo href="./18.3#T1.t1.r2" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><msub><mi href="./18.2#Px1.p2">x</mi><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>,</mo><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>-</mo><mi href="./18.1#SS1.p2.t1.r4">m</mi></mrow></mrow></msub><mo>)</mo></mrow></mrow><mrow><msubsup><mi href="./18.3#T1.t1.r2">P</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r2" stretchy="false">(</mo><mi>α</mi><mo href="./18.3#T1.t1.r2">,</mo><mi>β</mi><mo href="./18.3#T1.t1.r2" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></mfrac><mo>|</mo></mrow><mo>></mo><mrow><mo>|</mo><mfrac><mrow><msubsup><mi href="./18.3#T1.t1.r2">P</mi><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>+</mo><mn>1</mn></mrow><mrow><mo href="./18.3#T1.t1.r2" stretchy="false">(</mo><mi>α</mi><mo href="./18.3#T1.t1.r2">,</mo><mi>β</mi><mo href="./18.3#T1.t1.r2" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><msub><mi href="./18.2#Px1.p2">x</mi><mrow><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>+</mo><mn>1</mn></mrow><mo>,</mo><mrow><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>-</mo><mi href="./18.1#SS1.p2.t1.r4">m</mi></mrow><mo>+</mo><mn>1</mn></mrow></mrow></msub><mo>)</mo></mrow></mrow><mrow><msubsup><mi href="./18.3#T1.t1.r2">P</mi><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>+</mo><mn>1</mn></mrow><mrow><mo href="./18.3#T1.t1.r2" stretchy="false">(</mo><mi>α</mi><mo href="./18.3#T1.t1.r2">,</mo><mi>β</mi><mo href="./18.3#T1.t1.r2" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></mfrac><mo>|</mo></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m33.png" altimg-height="27px" altimg-valign="-9px" altimg-width="111px" alttext="\alpha=\beta>-\tfrac{1}{2}" display="inline"><mrow><mi>α</mi><mo>=</mo><mi>β</mi><mo>></mo><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow></math>, <math class="ltx_Math" altimg="m48.png" altimg-height="20px" altimg-valign="-6px" altimg-width="133px" alttext="m=1,2,\dots,n" display="inline"><mrow><mi href="./18.1#SS1.p2.t1.r4">m</mi><mo>=</mo><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><mi href="./18.1#SS1.p2.t1.r5">n</mi></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E20.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./18.3#T1.t1.r2" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m26.png" altimg-height="29px" altimg-valign="-7px" altimg-width="88px" alttext="P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msubsup><mi href="./18.3#T1.t1.r2">P</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r2" stretchy="false">(</mo><mi class="ltx_nvar">α</mi><mo href="./18.3#T1.t1.r2">,</mo><mi class="ltx_nvar">β</mi><mo href="./18.3#T1.t1.r2" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Jacobi polynomial</a>,
<a href="./18.1#SS1.p2.t1.r4" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./18.1#SS1.p2.t1.r4">m</mi></math>: nonnegative integer</a>,
<a href="./18.1#SS1.p2.t1.r5" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">For extensions of (</dd>
</dl>
</div>
</div>
<div id="Px11.p1" class="ltx_para">
<p class="ltx_p">Let the maxima <math class="ltx_Math" altimg="m56.png" altimg-height="18px" altimg-valign="-8px" altimg-width="45px" alttext="x_{n,m}" display="inline"><msub><mi href="./18.2#Px1.p2">x</mi><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>,</mo><mi href="./18.1#SS1.p2.t1.r4">m</mi></mrow></msub></math>, <math class="ltx_Math" altimg="m46.png" altimg-height="20px" altimg-valign="-6px" altimg-width="168px" alttext="m=0,1,\dots,n-1" display="inline"><mrow><mi href="./18.1#SS1.p2.t1.r4">m</mi><mo>=</mo><mrow><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow></mrow></mrow></math>, of <math class="ltx_Math" altimg="m59.png" altimg-height="29px" altimg-valign="-7px" altimg-width="86px" alttext="|L^{(\alpha)}_{n}\left(x\right)|" display="inline"><mrow><mo stretchy="false">|</mo><mrow><msubsup><mi href="./18.3#T1.t1.r12">L</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r12" stretchy="false">(</mo><mi>α</mi><mo href="./18.3#T1.t1.r12" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow></math>
in <math class="ltx_Math" altimg="m30.png" altimg-height="23px" altimg-valign="-7px" altimg-width="56px" alttext="[0,\infty)" display="inline"><mrow><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">[</mo><mn>0</mn><mo href="./front/introduction#Sx4.p2.t1.r1">,</mo><mi mathvariant="normal">∞</mi><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">)</mo></mrow></math> be arranged so that</p>
<table id="E21" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">18.14.21</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E21.png" altimg-height="24px" altimg-valign="-8px" altimg-width="388px" alttext="0=x_{n,0}<x_{n,1}<\cdots<x_{n,n-1}<x_{n,n}=\infty." display="block"><mrow><mrow><mn>0</mn><mo>=</mo><msub><mi href="./18.2#Px1.p2">x</mi><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>,</mo><mn>0</mn></mrow></msub><mo><</mo><msub><mi href="./18.2#Px1.p2">x</mi><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>,</mo><mn>1</mn></mrow></msub><mo><</mo><mi mathvariant="normal">⋯</mi><mo><</mo><msub><mi href="./18.2#Px1.p2">x</mi><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>,</mo><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow></mrow></msub><mo><</mo><msub><mi href="./18.2#Px1.p2">x</mi><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>,</mo><mi href="./18.1#SS1.p2.t1.r5">n</mi></mrow></msub><mo>=</mo><mi mathvariant="normal">∞</mi></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E21.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./18.1#SS1.p2.t1.r5" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="Px11.p2" class="ltx_para">
<p class="ltx_p">When <math class="ltx_Math" altimg="m35.png" altimg-height="27px" altimg-valign="-9px" altimg-width="72px" alttext="\alpha>-\tfrac{1}{2}" display="inline"><mrow><mi>α</mi><mo>></mo><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow></math> choose <math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./18.1#SS1.p2.t1.r4">m</mi></math> so that</p>
<table id="E22" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">18.14.22</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E22.png" altimg-height="29px" altimg-valign="-9px" altimg-width="217px" alttext="x_{n,m}\leq\alpha+\tfrac{1}{2}\leq x_{n,m+1}." display="block"><mrow><mrow><msub><mi href="./18.2#Px1.p2">x</mi><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>,</mo><mi href="./18.1#SS1.p2.t1.r4">m</mi></mrow></msub><mo>≤</mo><mrow><mi>α</mi><mo>+</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle></mrow><mo>≤</mo><msub><mi href="./18.2#Px1.p2">x</mi><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>,</mo><mrow><mi href="./18.1#SS1.p2.t1.r4">m</mi><mo>+</mo><mn>1</mn></mrow></mrow></msub></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E22.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./18.1#SS1.p2.t1.r4" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./18.1#SS1.p2.t1.r4">m</mi></math>: nonnegative integer</a>,
<a href="./18.1#SS1.p2.t1.r5" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Then </p>
<table id="E23" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex5" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">18.14.23</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m7.png" altimg-height="30px" altimg-valign="-8px" altimg-width="112px" alttext="\displaystyle|L^{(\alpha)}_{n}\left(x_{n,0}\right)|" display="inline"><mrow><mo stretchy="false">|</mo><mrow><msubsup><mi href="./18.3#T1.t1.r12">L</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r12" stretchy="false">(</mo><mi>α</mi><mo href="./18.3#T1.t1.r12" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><msub><mi href="./18.2#Px1.p2">x</mi><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>,</mo><mn>0</mn></mrow></msub><mo>)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m3.png" altimg-height="30px" altimg-valign="-8px" altimg-width="326px" alttext="\displaystyle>|L^{(\alpha)}_{n}\left(x_{n,1}\right)|>\cdots>|L^{(\alpha)}_{n}%
\left(x_{n,m}\right)|," display="inline"><mrow><mrow><mi></mi><mo>></mo><mrow><mo stretchy="false">|</mo><mrow><msubsup><mi href="./18.3#T1.t1.r12">L</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r12" stretchy="false">(</mo><mi>α</mi><mo href="./18.3#T1.t1.r12" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><msub><mi href="./18.2#Px1.p2">x</mi><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>,</mo><mn>1</mn></mrow></msub><mo>)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow><mo>></mo><mi mathvariant="normal">⋯</mi><mo>></mo><mrow><mo stretchy="false">|</mo><mrow><msubsup><mi href="./18.3#T1.t1.r12">L</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r12" stretchy="false">(</mo><mi>α</mi><mo href="./18.3#T1.t1.r12" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><msub><mi href="./18.2#Px1.p2">x</mi><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>,</mo><mi href="./18.1#SS1.p2.t1.r4">m</mi></mrow></msub><mo>)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex6" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m8.png" altimg-height="30px" altimg-valign="-8px" altimg-width="134px" alttext="\displaystyle|L^{(\alpha)}_{n}\left(x_{n,n-1}\right)|" display="inline"><mrow><mo stretchy="false">|</mo><mrow><msubsup><mi href="./18.3#T1.t1.r12">L</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r12" stretchy="false">(</mo><mi>α</mi><mo href="./18.3#T1.t1.r12" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><msub><mi href="./18.2#Px1.p2">x</mi><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>,</mo><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow></mrow></msub><mo>)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m4.png" altimg-height="30px" altimg-valign="-8px" altimg-width="369px" alttext="\displaystyle>|L^{(\alpha)}_{n}\left(x_{n,n-2}\right)|>\cdots>|L^{(\alpha)}_{n%
}\left(x_{n,m+1}\right)|." display="inline"><mrow><mrow><mi></mi><mo>></mo><mrow><mo stretchy="false">|</mo><mrow><msubsup><mi href="./18.3#T1.t1.r12">L</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r12" stretchy="false">(</mo><mi>α</mi><mo href="./18.3#T1.t1.r12" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><msub><mi href="./18.2#Px1.p2">x</mi><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>,</mo><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>-</mo><mn>2</mn></mrow></mrow></msub><mo>)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow><mo>></mo><mi mathvariant="normal">⋯</mi><mo>></mo><mrow><mo stretchy="false">|</mo><mrow><msubsup><mi href="./18.3#T1.t1.r12">L</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r12" stretchy="false">(</mo><mi>α</mi><mo href="./18.3#T1.t1.r12" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><msub><mi href="./18.2#Px1.p2">x</mi><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>,</mo><mrow><mi href="./18.1#SS1.p2.t1.r4">m</mi><mo>+</mo><mn>1</mn></mrow></mrow></msub><mo>)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E23.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./18.3#T1.t1.r12" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m25.png" altimg-height="29px" altimg-valign="-7px" altimg-width="72px" alttext="L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msubsup><mi href="./18.3#T1.t1.r12">L</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r12" stretchy="false">(</mo><mi class="ltx_nvar">α</mi><mo href="./18.3#T1.t1.r12" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Laguerre (or generalized Laguerre) polynomial</a>,
<a href="./18.1#SS1.p2.t1.r4" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./18.1#SS1.p2.t1.r4">m</mi></math>: nonnegative integer</a>,
<a href="./18.1#SS1.p2.t1.r5" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="Px11.p3" class="ltx_para">
<p class="ltx_p">Also, when <math class="ltx_Math" altimg="m39.png" altimg-height="27px" altimg-valign="-9px" altimg-width="72px" alttext="\alpha\leq-\tfrac{1}{2}" display="inline"><mrow><mi>α</mi><mo>≤</mo><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow></math></p>
<table id="E24" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">18.14.24</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E24.png" altimg-height="30px" altimg-valign="-8px" altimg-width="453px" alttext="|L^{(\alpha)}_{n}\left(x_{n,0}\right)|<|L^{(\alpha)}_{n}\left(x_{n,1}\right)|<%
\cdots<|L^{(\alpha)}_{n}\left(x_{n,n-1}\right)|." display="block"><mrow><mrow><mrow><mo stretchy="false">|</mo><mrow><msubsup><mi href="./18.3#T1.t1.r12">L</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r12" stretchy="false">(</mo><mi>α</mi><mo href="./18.3#T1.t1.r12" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><msub><mi href="./18.2#Px1.p2">x</mi><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>,</mo><mn>0</mn></mrow></msub><mo>)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow><mo><</mo><mrow><mo stretchy="false">|</mo><mrow><msubsup><mi href="./18.3#T1.t1.r12">L</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r12" stretchy="false">(</mo><mi>α</mi><mo href="./18.3#T1.t1.r12" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><msub><mi href="./18.2#Px1.p2">x</mi><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>,</mo><mn>1</mn></mrow></msub><mo>)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow><mo><</mo><mi mathvariant="normal">⋯</mi><mo><</mo><mrow><mo stretchy="false">|</mo><mrow><msubsup><mi href="./18.3#T1.t1.r12">L</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r12" stretchy="false">(</mo><mi>α</mi><mo href="./18.3#T1.t1.r12" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><msub><mi href="./18.2#Px1.p2">x</mi><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>,</mo><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow></mrow></msub><mo>)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E24.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./18.3#T1.t1.r12" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m25.png" altimg-height="29px" altimg-valign="-7px" altimg-width="72px" alttext="L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msubsup><mi href="./18.3#T1.t1.r12">L</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r12" stretchy="false">(</mo><mi class="ltx_nvar">α</mi><mo href="./18.3#T1.t1.r12" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Laguerre (or generalized Laguerre) polynomial</a>,
<a href="./18.1#SS1.p2.t1.r5" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px12" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Hermite</h3>
<div id="Px12.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px12.p1" class="ltx_para">
<p class="ltx_p">The successive maxima of <math class="ltx_Math" altimg="m58.png" altimg-height="23px" altimg-valign="-7px" altimg-width="76px" alttext="|H_{n}\left(x\right)|" display="inline"><mrow><mo stretchy="false">|</mo><mrow><msub><mi href="./18.3#T1.t1.r13">H</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow></math> form a decreasing sequence for
<math class="ltx_Math" altimg="m55.png" altimg-height="19px" altimg-valign="-5px" altimg-width="52px" alttext="x\leq 0" display="inline"><mrow><mi href="./18.2#Px1.p2">x</mi><mo>≤</mo><mn>0</mn></mrow></math>, and an increasing sequence for <math class="ltx_Math" altimg="m54.png" altimg-height="19px" altimg-valign="-5px" altimg-width="52px" alttext="x\geq 0" display="inline"><mrow><mi href="./18.2#Px1.p2">x</mi><mo>≥</mo><mn>0</mn></mrow></math>.</p>
</div>
</section>
</section>
</section>
</div>
<div class="ltx_page_footer">
<span></div>
</div>
</body></text>
</html>
</page>
<page>
<!DOCTYPE html><html>
<head>
<title>DLMF: 18.17 Integrals</title>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<!--GOOGLE BOOTSTRAP--></head>
<text><body class="color_default textfont_default titlefont_default fontsize_default navbar_default">
<div class="ltx_page_navbar">
<div class="ltx_page_navlogo">) with <math class="ltx_Math" altimg="m41.png" altimg-height="17px" altimg-valign="-2px" altimg-width="54px" alttext="\alpha=0" display="inline"><mrow><mi>α</mi><mo>=</mo><mn>0</mn></mrow></math>.
For the second equation combine (</dd>
</dl>
</div>
</div>
<div id="Px1.p1" class="ltx_para">
<table id="E1" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">18.17.1</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_Math ltx_eqn_cell ltx_align_center"><math class="ltx_Math" alttext="2n\int_{0}^{x}(1-y)^{\alpha}(1+y)^{\beta}P^{(\alpha,\beta)}_{n}\left(y\right)%
\mathrm{d}y=P^{(\alpha+1,\beta+1)}_{n-1}\left(0\right)-(1-x)^{\alpha+1}(1+x)^{%
\beta+1}P^{(\alpha+1,\beta+1)}_{n-1}\left(x\right)." display="block"><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mrow><mn>2</mn><mo></mo><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo></mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi href="./18.2#Px1.p2">x</mi></msubsup><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><mi href="./18.1#SS1.p2.t1.r1">y</mi></mrow><mo stretchy="false">)</mo></mrow><mi>α</mi></msup><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>+</mo><mi href="./18.1#SS1.p2.t1.r1">y</mi></mrow><mo stretchy="false">)</mo></mrow><mi>β</mi></msup><mo></mo><mrow><msubsup><mi href="./18.3#T1.t1.r2">P</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r2" stretchy="false">(</mo><mi>α</mi><mo href="./18.3#T1.t1.r2">,</mo><mi>β</mi><mo href="./18.3#T1.t1.r2" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi href="./18.1#SS1.p2.t1.r1">y</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./18.1#SS1.p2.t1.r1">y</mi></mrow></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>=</mo><mrow><mrow><msubsup><mi href="./18.3#T1.t1.r2">P</mi><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow><mrow><mo href="./18.3#T1.t1.r2" stretchy="false">(</mo><mrow><mi>α</mi><mo>+</mo><mn>1</mn></mrow><mo href="./18.3#T1.t1.r2">,</mo><mrow><mi>β</mi><mo>+</mo><mn>1</mn></mrow><mo href="./18.3#T1.t1.r2" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow></mrow><mo>-</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><mi href="./18.2#Px1.p2">x</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mi>α</mi><mo>+</mo><mn>1</mn></mrow></msup><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>+</mo><mi href="./18.2#Px1.p2">x</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mi>β</mi><mo>+</mo><mn>1</mn></mrow></msup><mo></mo><mrow><msubsup><mi href="./18.3#T1.t1.r2">P</mi><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow><mrow><mo href="./18.3#T1.t1.r2" stretchy="false">(</mo><mrow><mi>α</mi><mo>+</mo><mn>1</mn></mrow><mo href="./18.3#T1.t1.r2">,</mo><mrow><mi>β</mi><mo>+</mo><mn>1</mn></mrow><mo href="./18.3#T1.t1.r2" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>.</mo></mtd></mtr></mtable></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./18.3#T1.t1.r2" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m27.png" altimg-height="29px" altimg-valign="-7px" altimg-width="88px" alttext="P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msubsup><mi href="./18.3#T1.t1.r2">P</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r2" stretchy="false">(</mo><mi class="ltx_nvar">α</mi><mo href="./18.3#T1.t1.r2">,</mo><mi class="ltx_nvar">β</mi><mo href="./18.3#T1.t1.r2" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Jacobi polynomial</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./18.1#SS1.p2.t1.r1" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./18.1#SS1.p2.t1.r1">y</mi></math>: real variable</a>,
<a href="./18.1#SS1.p2.t1.r5" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m80.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px2" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Laguerre</h3>
<div id="Px2.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px2.p1" class="ltx_para">
<table id="E2" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">18.17.2</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E2.png" altimg-height="53px" altimg-valign="-20px" altimg-width="622px" alttext="\int_{0}^{x}L_{m}\left(y\right)L_{n}\left(x-y\right)\mathrm{d}y=\int_{0}^{x}L_%
{m+n}\left(y\right)\mathrm{d}y=L_{m+n}\left(x\right)-L_{m+n+1}\left(x\right)." display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi href="./18.2#Px1.p2">x</mi></msubsup><mrow><mrow><msub><mi href="./18.1#I1.ix7.p1">L</mi><mi href="./18.1#SS1.p2.t1.r4">m</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./18.1#SS1.p2.t1.r1">y</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./18.1#I1.ix7.p1">L</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi href="./18.2#Px1.p2">x</mi><mo>-</mo><mi href="./18.1#SS1.p2.t1.r1">y</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./18.1#SS1.p2.t1.r1">y</mi></mrow></mrow></mrow><mo>=</mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi href="./18.2#Px1.p2">x</mi></msubsup><mrow><mrow><msub><mi href="./18.1#I1.ix7.p1">L</mi><mrow><mi href="./18.1#SS1.p2.t1.r4">m</mi><mo>+</mo><mi href="./18.1#SS1.p2.t1.r5">n</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./18.1#SS1.p2.t1.r1">y</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./18.1#SS1.p2.t1.r1">y</mi></mrow></mrow></mrow><mo>=</mo><mrow><mrow><msub><mi href="./18.1#I1.ix7.p1">L</mi><mrow><mi href="./18.1#SS1.p2.t1.r4">m</mi><mo>+</mo><mi href="./18.1#SS1.p2.t1.r5">n</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow><mo>-</mo><mrow><msub><mi href="./18.1#I1.ix7.p1">L</mi><mrow><mi href="./18.1#SS1.p2.t1.r4">m</mi><mo>+</mo><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./18.1#I1.ix7.p1" title="Classical OP’s ‣ §18.1(ii) Main Functions ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m26.png" altimg-height="29px" altimg-valign="-7px" altimg-width="151px" alttext="L_{\NVar{n}}\left(\NVar{x}\right)=L^{(0)}_{n}\left(x\right)" display="inline"><mrow><mrow><msub><mi href="./18.1#I1.ix7.p1">L</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msubsup><mi href="./18.3#T1.t1.r12">L</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r12" stretchy="false">(</mo><mn>0</mn><mo href="./18.3#T1.t1.r12" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></mrow></math>: Laguerre polynomial</a>,
<a href="./18.1#SS1.p2.t1.r1" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./18.1#SS1.p2.t1.r1">y</mi></math>: real variable</a>,
<a href="./18.1#SS1.p2.t1.r4" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./18.1#SS1.p2.t1.r4">m</mi></math>: nonnegative integer</a>,
<a href="./18.1#SS1.p2.t1.r5" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m80.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px3" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Hermite</h3>
<div id="Px3.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px3.p1" class="ltx_para">
<table id="E3" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">18.17.3</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E3.png" altimg-height="52px" altimg-valign="-21px" altimg-width="424px" alttext="\int_{0}^{x}H_{n}\left(y\right)\mathrm{d}y=\frac{1}{2(n+1)}(H_{n+1}\left(x%
\right)-H_{n+1}\left(0\right))," display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi href="./18.2#Px1.p2">x</mi></msubsup><mrow><mrow><msub><mi href="./18.3#T1.t1.r13">H</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./18.1#SS1.p2.t1.r1">y</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./18.1#SS1.p2.t1.r1">y</mi></mrow></mrow></mrow><mo>=</mo><mrow><mfrac><mn>1</mn><mrow><mn>2</mn><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow></mfrac><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><msub><mi href="./18.3#T1.t1.r13">H</mi><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow><mo>-</mo><mrow><msub><mi href="./18.3#T1.t1.r13">H</mi><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./18.3#T1.t1.r13" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m21.png" altimg-height="23px" altimg-valign="-7px" altimg-width="62px" alttext="H_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r13">H</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Hermite polynomial</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./18.1#SS1.p2.t1.r1" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./18.1#SS1.p2.t1.r1">y</mi></math>: real variable</a>,
<a href="./18.1#SS1.p2.t1.r5" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m80.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E4" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">18.17.4</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E4.png" altimg-height="53px" altimg-valign="-20px" altimg-width="414px" alttext="\int_{0}^{x}e^{-y^{2}}H_{n}\left(y\right)\mathrm{d}y=H_{n-1}\left(0\right)-e^{%
-x^{2}}H_{n-1}\left(x\right)." display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi href="./18.2#Px1.p2">x</mi></msubsup><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><msup><mi href="./18.1#SS1.p2.t1.r1">y</mi><mn>2</mn></msup></mrow></msup><mo></mo><mrow><msub><mi href="./18.3#T1.t1.r13">H</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./18.1#SS1.p2.t1.r1">y</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./18.1#SS1.p2.t1.r1">y</mi></mrow></mrow></mrow><mo>=</mo><mrow><mrow><msub><mi href="./18.3#T1.t1.r13">H</mi><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow></mrow><mo>-</mo><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>2</mn></msup></mrow></msup><mo></mo><mrow><msub><mi href="./18.3#T1.t1.r13">H</mi><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E4.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./18.3#T1.t1.r13" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m21.png" altimg-height="23px" altimg-valign="-7px" altimg-width="62px" alttext="H_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r13">H</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Hermite polynomial</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./18.1#SS1.p2.t1.r1" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./18.1#SS1.p2.t1.r1">y</mi></math>: real variable</a>,
<a href="./18.1#SS1.p2.t1.r5" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m80.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
</section>
<section id="ii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§18.17(ii) </span>Integral Representations for Products</h2>
<div id="SS2.info" class="ltx_metadata ltx_info">
) is the case <math class="ltx_Math" altimg="m41.png" altimg-height="17px" altimg-valign="-2px" altimg-width="54px" alttext="\alpha=0" display="inline"><mrow><mi>α</mi><mo>=</mo><mn>0</mn></mrow></math> of
(</dd>
</dl>
</div>
</div>
<div id="Px4.p1" class="ltx_para">
<table id="E5" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">18.17.5</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_Math ltx_eqn_cell ltx_align_center"><math class="ltx_Math" alttext="\frac{C^{(\lambda)}_{n}\left(\cos\theta_{1}\right)}{C^{(\lambda)}_{n}\left(1%
\right)}\frac{C^{(\lambda)}_{n}\left(\cos\theta_{2}\right)}{C^{(\lambda)}_{n}%
\left(1\right)}=\frac{\Gamma\left(\lambda+\frac{1}{2}\right)}{\pi^{\frac{1}{2}%
}\Gamma\left(\lambda\right)}\*\int_{0}^{\pi}\frac{C^{(\lambda)}_{n}\left(\cos%
\theta_{1}\cos\theta_{2}+\sin\theta_{1}\sin\theta_{2}\cos\phi\right)}{C^{(%
\lambda)}_{n}\left(1\right)}(\sin\phi)^{2\lambda-1}\mathrm{d}\phi," display="block"><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mrow><mfrac><mrow><msubsup><mi href="./18.3#T1.t1.r3">C</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r3" stretchy="false">(</mo><mi>λ</mi><mo href="./18.3#T1.t1.r3" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><msub><mi>θ</mi><mn>1</mn></msub></mrow><mo>)</mo></mrow></mrow><mrow><msubsup><mi href="./18.3#T1.t1.r3">C</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r3" stretchy="false">(</mo><mi>λ</mi><mo href="./18.3#T1.t1.r3" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></mfrac><mo></mo><mfrac><mrow><msubsup><mi href="./18.3#T1.t1.r3">C</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r3" stretchy="false">(</mo><mi>λ</mi><mo href="./18.3#T1.t1.r3" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><msub><mi>θ</mi><mn>2</mn></msub></mrow><mo>)</mo></mrow></mrow><mrow><msubsup><mi href="./18.3#T1.t1.r3">C</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r3" stretchy="false">(</mo><mi>λ</mi><mo href="./18.3#T1.t1.r3" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></mfrac></mrow></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>=</mo><mrow><mfrac><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi>λ</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mrow><msup><mi href="./3.12#E1">π</mi><mfrac><mn>1</mn><mn>2</mn></mfrac></msup><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi>λ</mi><mo>)</mo></mrow></mrow></mrow></mfrac><mo></mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi href="./3.12#E1">π</mi></msubsup><mrow><mfrac><mrow><msubsup><mi href="./18.3#T1.t1.r3">C</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r3" stretchy="false">(</mo><mi>λ</mi><mo href="./18.3#T1.t1.r3" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mrow><mrow><mrow><mi href="./4.14#E2">cos</mi><mo></mo><msub><mi>θ</mi><mn>1</mn></msub></mrow><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><msub><mi>θ</mi><mn>2</mn></msub></mrow></mrow><mo>+</mo><mrow><mrow><mi href="./4.14#E1">sin</mi><mo></mo><msub><mi>θ</mi><mn>1</mn></msub></mrow><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><msub><mi>θ</mi><mn>2</mn></msub></mrow><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi>ϕ</mi></mrow></mrow></mrow><mo>)</mo></mrow></mrow><mrow><msubsup><mi href="./18.3#T1.t1.r3">C</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r3" stretchy="false">(</mo><mi>λ</mi><mo href="./18.3#T1.t1.r3" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></mfrac><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi>ϕ</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mrow><mn>2</mn><mo></mo><mi>λ</mi></mrow><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>ϕ</mi></mrow></mrow></mrow></mrow><mo>,</mo></mtd></mtr></mtable></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m53.png" altimg-height="18px" altimg-valign="-3px" altimg-width="52px" alttext="\lambda>0" display="inline"><mrow><mi>λ</mi><mo>></mo><mn>0</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E5.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m35.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m71.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./18.3#T1.t1.r3" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="29px" altimg-valign="-7px" altimg-width="73px" alttext="C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msubsup><mi href="./18.3#T1.t1.r3">C</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r3" stretchy="false">(</mo><mi class="ltx_nvar">λ</mi><mo href="./18.3#T1.t1.r3" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: ultraspherical (or Gegenbauer) polynomial</a> and
<a href="./18.1#SS1.p2.t1.r5" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m80.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math>: nonnegative integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px5" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Legendre</h3>
<div id="Px5.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px5.p1" class="ltx_para">
<table id="E6" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">18.17.6</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E6.png" altimg-height="53px" altimg-valign="-20px" altimg-width="611px" alttext="P_{n}\left(\cos\theta_{1}\right)P_{n}\left(\cos\theta_{2}\right)=\frac{1}{\pi}%
\int_{0}^{\pi}P_{n}\left(\cos\theta_{1}\cos\theta_{2}+\sin\theta_{1}\sin\theta%
_{2}\cos\phi\right)\mathrm{d}\phi." display="block"><mrow><mrow><mrow><mrow><msub><mi href="./18.3#T1.t1.r10">P</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><msub><mi>θ</mi><mn>1</mn></msub></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./18.3#T1.t1.r10">P</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><msub><mi>θ</mi><mn>2</mn></msub></mrow><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mrow><mfrac><mn>1</mn><mi href="./3.12#E1">π</mi></mfrac><mo></mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi href="./3.12#E1">π</mi></msubsup><mrow><mrow><msub><mi href="./18.3#T1.t1.r10">P</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mrow><mrow><mi href="./4.14#E2">cos</mi><mo></mo><msub><mi>θ</mi><mn>1</mn></msub></mrow><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><msub><mi>θ</mi><mn>2</mn></msub></mrow></mrow><mo>+</mo><mrow><mrow><mi href="./4.14#E1">sin</mi><mo></mo><msub><mi>θ</mi><mn>1</mn></msub></mrow><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><msub><mi>θ</mi><mn>2</mn></msub></mrow><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi>ϕ</mi></mrow></mrow></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>ϕ</mi></mrow></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E6.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./18.3#T1.t1.r10" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m28.png" altimg-height="23px" altimg-valign="-7px" altimg-width="58px" alttext="P_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r10">P</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Legendre polynomial</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m71.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a> and
<a href="./18.1#SS1.p2.t1.r5" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m80.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math>: nonnegative integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">For formulas for Jacobi and Laguerre polynomials analogous to
(</dd>
</dl>
</div>
</div>
<div id="Px6.p1" class="ltx_para">
<table id="E7" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">18.17.7</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E7.png" altimg-height="53px" altimg-valign="-20px" altimg-width="637px" alttext="\left(P_{n}\left(x\right)\right)^{2}+4\pi^{-2}\left(\mathsf{Q}_{n}\left(x%
\right)\right)^{2}=4\pi^{-2}\*\int_{1}^{\infty}Q_{n}\left(x^{2}+(1-x^{2})t%
\right)(t^{2}-1)^{-\frac{1}{2}}\mathrm{d}t," display="block"><mrow><mrow><mrow><msup><mrow><mo>(</mo><mrow><msub><mi href="./18.3#T1.t1.r10">P</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow><mo>)</mo></mrow><mn>2</mn></msup><mo>+</mo><mrow><mn>4</mn><mo></mo><msup><mi href="./3.12#E1">π</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup><mo></mo><msup><mrow><mo>(</mo><mrow><msub><mi href="./14.2#SS2.p2" mathvariant="sans-serif">Q</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow><mo>)</mo></mrow><mn>2</mn></msup></mrow></mrow><mo>=</mo><mrow><mn>4</mn><mo></mo><msup><mi href="./3.12#E1">π</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup><mo></mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>1</mn><mi mathvariant="normal">∞</mi></msubsup><mrow><mrow><msub><mi href="./14.2#SS2.p2">Q</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mrow><msup><mi href="./18.2#Px1.p2">x</mi><mn>2</mn></msup><mo>+</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi>t</mi></mrow></mrow><mo>)</mo></mrow></mrow><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><msup><mi>t</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m14.png" altimg-height="18px" altimg-valign="-4px" altimg-width="104px" alttext="-1<x<1" display="inline"><mrow><mrow><mo>-</mo><mn>1</mn></mrow><mo><</mo><mi href="./18.2#Px1.p2">x</mi><mo><</mo><mn>1</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E7.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./18.3#T1.t1.r10" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m28.png" altimg-height="23px" altimg-valign="-7px" altimg-width="58px" alttext="P_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r10">P</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Legendre polynomial</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./14.2#SS2.p2" title="§14.2(ii) Associated Legendre Equation ‣ §14.2 Differential Equations ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m61.png" altimg-height="25px" altimg-valign="-7px" altimg-width="140px" alttext="\mathsf{Q}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{Q}^{0}_{\nu}\left(x\right)" display="inline"><mrow><mrow><msub><mi href="./14.2#SS2.p2" mathvariant="sans-serif">Q</mi><mi class="ltx_nvar" href="./14.1#p1.t1.r4">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./14.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msubsup><mi href="./14.3#E2" mathvariant="sans-serif">Q</mi><mi href="./14.1#p1.t1.r4">ν</mi><mn>0</mn></msubsup><mo></mo><mrow><mo>(</mo><mi href="./14.1#p1.t1.r1">x</mi><mo>)</mo></mrow></mrow></mrow></math>: Ferrers function of the second kind</a>,
<a href="./14.2#SS2.p2" title="§14.2(ii) Associated Legendre Equation ‣ §14.2 Differential Equations ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m29.png" altimg-height="25px" altimg-valign="-7px" altimg-width="140px" alttext="Q_{\NVar{\nu}}\left(\NVar{z}\right)=Q^{0}_{\nu}\left(z\right)" display="inline"><mrow><mrow><msub><mi href="./14.2#SS2.p2">Q</mi><mi class="ltx_nvar" href="./14.1#p1.t1.r4">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./14.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msubsup><mi href="./14.21#SS1.p1">Q</mi><mi href="./14.1#p1.t1.r4">ν</mi><mn>0</mn></msubsup><mo></mo><mrow><mo>(</mo><mi href="./14.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow></math>: Legendre function of the second kind</a>,
<a href="./18.1#SS1.p2.t1.r5" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m80.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">For the Ferrers function <math class="ltx_Math" altimg="m62.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="\mathsf{Q}_{n}\left(x\right)" display="inline"><mrow><msub><mi href="./14.2#SS2.p2" mathvariant="sans-serif">Q</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math> and Legendre function <math class="ltx_Math" altimg="m30.png" altimg-height="23px" altimg-valign="-7px" altimg-width="61px" alttext="Q_{n}\left(x\right)" display="inline"><mrow><msub><mi href="./14.2#SS2.p2">Q</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math> see
§§, with <math class="ltx_Math" altimg="m63.png" altimg-height="20px" altimg-valign="-6px" altimg-width="53px" alttext="\mu=0" display="inline"><mrow><mi>μ</mi><mo>=</mo><mn>0</mn></mrow></math> and <math class="ltx_Math" altimg="m66.png" altimg-height="13px" altimg-valign="-2px" altimg-width="54px" alttext="\nu=n" display="inline"><mrow><mi>ν</mi><mo>=</mo><mi href="./18.1#SS1.p2.t1.r5">n</mi></mrow></math>.</p>
</div>
</section>
<section id="Px7" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Hermite</h3>
<div id="Px7.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px7.p1" class="ltx_para">
<table id="E8" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">18.17.8</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_Math ltx_eqn_cell ltx_align_center"><math class="ltx_Math" alttext="\left(H_{n}\left(x\right)\right)^{2}+2^{n}(n!)^{2}e^{x^{2}}\left(V\left(-n-%
\tfrac{1}{2},2^{\frac{1}{2}}x\right)\right)^{2}=\frac{2^{n+\frac{3}{2}}n!\,e^{%
x^{2}}}{\pi}\int_{0}^{\infty}\frac{e^{-(2n+1)t+x^{2}\tanh t}}{(\sinh 2t)^{%
\frac{1}{2}}}\mathrm{d}t." display="block"><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mrow><msup><mrow><mo>(</mo><mrow><msub><mi href="./18.3#T1.t1.r13">H</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow><mo>)</mo></mrow><mn>2</mn></msup><mo>+</mo><mrow><msup><mn>2</mn><mi href="./18.1#SS1.p2.t1.r5">n</mi></msup><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><msup><mi href="./18.2#Px1.p2">x</mi><mn>2</mn></msup></msup><mo></mo><msup><mrow><mo>(</mo><mrow><mi href="./12.2#SS1.p1">V</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mo>-</mo><mi href="./18.1#SS1.p2.t1.r5">n</mi></mrow><mo>-</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle></mrow><mo>,</mo><mrow><msup><mn>2</mn><mfrac><mn>1</mn><mn>2</mn></mfrac></msup><mo></mo><mi href="./18.2#Px1.p2">x</mi></mrow><mo>)</mo></mrow></mrow><mo>)</mo></mrow><mn>2</mn></msup></mrow></mrow></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>=</mo><mrow><mfrac><mrow><msup><mn>2</mn><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>+</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow></msup><mo></mo><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><msup><mi href="./18.2#Px1.p2">x</mi><mn>2</mn></msup></msup></mrow><mi href="./3.12#E1">π</mi></mfrac><mo></mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><mrow><mfrac><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mrow><mo>-</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>2</mn><mo></mo><mi href="./18.1#SS1.p2.t1.r5">n</mi></mrow><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi>t</mi></mrow></mrow><mo>+</mo><mrow><msup><mi href="./18.2#Px1.p2">x</mi><mn>2</mn></msup><mo></mo><mrow><mi href="./4.28#E4">tanh</mi><mo></mo><mi>t</mi></mrow></mrow></mrow></msup><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./4.28#E1">sinh</mi><mo></mo><mrow><mn>2</mn><mo></mo><mi>t</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></msup></mfrac><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow><mo>.</mo></mtd></mtr></mtable></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E8.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./18.3#T1.t1.r13" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m21.png" altimg-height="23px" altimg-valign="-7px" altimg-width="62px" alttext="H_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r13">H</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Hermite polynomial</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m13.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m79.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./4.28#E1" title="(4.28.1) ‣ §4.28 Definitions and Periodicity ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m72.png" altimg-height="18px" altimg-valign="-2px" altimg-width="53px" alttext="\sinh\NVar{z}" display="inline"><mrow><mi href="./4.28#E1">sinh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: hyperbolic sine function</a>,
<a href="./4.28#E4" title="(4.28.4) ‣ §4.28 Definitions and Periodicity ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m73.png" altimg-height="18px" altimg-valign="-2px" altimg-width="58px" alttext="\tanh\NVar{z}" display="inline"><mrow><mi href="./4.28#E4">tanh</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: hyperbolic tangent function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./12.2#SS1.p1" title="§12.2(i) Introduction ‣ §12.2 Differential Equations ‣ Properties ‣ Chapter 12 Parabolic Cylinder Functions" class="ltx_ref"><math class="ltx_Math" altimg="m33.png" altimg-height="23px" altimg-valign="-7px" altimg-width="69px" alttext="V\left(\NVar{a},\NVar{z}\right)" display="inline"><mrow><mi href="./12.2#SS1.p1">V</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./12.2#E1">a</mi><mo>,</mo><mi class="ltx_nvar" href="./12.1#p1.t1.r2">z</mi><mo>)</mo></mrow></mrow></math>: parabolic cylinder function</a>,
<a href="./18.1#SS1.p2.t1.r5" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m80.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">For the parabolic cylinder function <math class="ltx_Math" altimg="m34.png" altimg-height="23px" altimg-valign="-7px" altimg-width="69px" alttext="V\left(a,z\right)" display="inline"><mrow><mi href="./12.2#SS1.p1">V</mi><mo></mo><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi href="./18.1#SS1.p2.t1.r2">z</mi><mo>)</mo></mrow></mrow></math> see §</dd>
</dl>
</div>
</div>
<div id="Px8.p1" class="ltx_para">
<table id="E9" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">18.17.9</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E9.png" altimg-height="58px" altimg-valign="-21px" altimg-width="576px" alttext="\frac{(1-x)^{\alpha+\mu}P^{(\alpha+\mu,\beta-\mu)}_{n}\left(x\right)}{\Gamma%
\left(\alpha+\mu+n+1\right)}=\int_{x}^{1}\frac{(1-y)^{\alpha}P^{(\alpha,\beta)%
}_{n}\left(y\right)}{\Gamma\left(\alpha+n+1\right)}\frac{(y-x)^{\mu-1}}{\Gamma%
\left(\mu\right)}\mathrm{d}y," display="block"><mrow><mrow><mfrac><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><mi href="./18.2#Px1.p2">x</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mi>α</mi><mo>+</mo><mi>μ</mi></mrow></msup><mo></mo><mrow><msubsup><mi href="./18.3#T1.t1.r2">P</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r2" stretchy="false">(</mo><mrow><mi>α</mi><mo>+</mo><mi>μ</mi></mrow><mo href="./18.3#T1.t1.r2">,</mo><mrow><mi>β</mi><mo>-</mo><mi>μ</mi></mrow><mo href="./18.3#T1.t1.r2" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi>α</mi><mo>+</mo><mi>μ</mi><mo>+</mo><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mfrac><mo>=</mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi href="./18.2#Px1.p2">x</mi><mn>1</mn></msubsup><mrow><mfrac><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><mi href="./18.1#SS1.p2.t1.r1">y</mi></mrow><mo stretchy="false">)</mo></mrow><mi>α</mi></msup><mo></mo><mrow><msubsup><mi href="./18.3#T1.t1.r2">P</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r2" stretchy="false">(</mo><mi>α</mi><mo href="./18.3#T1.t1.r2">,</mo><mi>β</mi><mo href="./18.3#T1.t1.r2" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi href="./18.1#SS1.p2.t1.r1">y</mi><mo>)</mo></mrow></mrow></mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi>α</mi><mo>+</mo><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mfrac><mo></mo><mfrac><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./18.1#SS1.p2.t1.r1">y</mi><mo>-</mo><mi href="./18.2#Px1.p2">x</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mi>μ</mi><mo>-</mo><mn>1</mn></mrow></msup><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi>μ</mi><mo>)</mo></mrow></mrow></mfrac><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./18.1#SS1.p2.t1.r1">y</mi></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m65.png" altimg-height="20px" altimg-valign="-6px" altimg-width="53px" alttext="\mu>0" display="inline"><mrow><mi>μ</mi><mo>></mo><mn>0</mn></mrow></math>, <math class="ltx_Math" altimg="m14.png" altimg-height="18px" altimg-valign="-4px" altimg-width="104px" alttext="-1<x<1" display="inline"><mrow><mrow><mo>-</mo><mn>1</mn></mrow><mo><</mo><mi href="./18.2#Px1.p2">x</mi><mo><</mo><mn>1</mn></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E9.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m35.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./18.3#T1.t1.r2" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m27.png" altimg-height="29px" altimg-valign="-7px" altimg-width="88px" alttext="P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msubsup><mi href="./18.3#T1.t1.r2">P</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r2" stretchy="false">(</mo><mi class="ltx_nvar">α</mi><mo href="./18.3#T1.t1.r2">,</mo><mi class="ltx_nvar">β</mi><mo href="./18.3#T1.t1.r2" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Jacobi polynomial</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./18.1#SS1.p2.t1.r1" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./18.1#SS1.p2.t1.r1">y</mi></math>: real variable</a>,
<a href="./18.1#SS1.p2.t1.r5" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m80.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="EGx1" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E10">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">18.17.10</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m10.png" altimg-height="56px" altimg-valign="-21px" altimg-width="320px" alttext="\displaystyle\frac{x^{\beta+\mu}(x+1)^{n}}{\Gamma\left(\beta+\mu+n+1\right)}P^%
{(\alpha,\beta+\mu)}_{n}\left(\frac{x-1}{x+1}\right)" display="inline"><mrow><mstyle displaystyle="true"><mfrac><mrow><msup><mi href="./18.2#Px1.p2">x</mi><mrow><mi>β</mi><mo>+</mo><mi>μ</mi></mrow></msup><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./18.2#Px1.p2">x</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi></msup></mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi>β</mi><mo>+</mo><mi>μ</mi><mo>+</mo><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mfrac></mstyle><mo></mo><mrow><msubsup><mi href="./18.3#T1.t1.r2">P</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r2" stretchy="false">(</mo><mi>α</mi><mo href="./18.3#T1.t1.r2">,</mo><mrow><mi>β</mi><mo>+</mo><mi>μ</mi></mrow><mo href="./18.3#T1.t1.r2" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac><mrow><mi href="./18.2#Px1.p2">x</mi><mo>-</mo><mn>1</mn></mrow><mrow><mi href="./18.2#Px1.p2">x</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mstyle><mo>)</mo></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m2.png" altimg-height="56px" altimg-valign="-21px" altimg-width="447px" alttext="\displaystyle=\int_{0}^{x}\frac{y^{\beta}(y+1)^{n}}{\Gamma\left(\beta+n+1%
\right)}P^{(\alpha,\beta)}_{n}\left(\frac{y-1}{y+1}\right)\*\frac{(x-y)^{\mu-1%
}}{\Gamma\left(\mu\right)}\mathrm{d}y," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi href="./18.2#Px1.p2">x</mi></msubsup></mstyle><mrow><mstyle displaystyle="true"><mfrac><mrow><msup><mi href="./18.1#SS1.p2.t1.r1">y</mi><mi>β</mi></msup><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./18.1#SS1.p2.t1.r1">y</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi></msup></mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi>β</mi><mo>+</mo><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mfrac></mstyle><mo></mo><mrow><msubsup><mi href="./18.3#T1.t1.r2">P</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r2" stretchy="false">(</mo><mi>α</mi><mo href="./18.3#T1.t1.r2">,</mo><mi>β</mi><mo href="./18.3#T1.t1.r2" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac><mrow><mi href="./18.1#SS1.p2.t1.r1">y</mi><mo>-</mo><mn>1</mn></mrow><mrow><mi href="./18.1#SS1.p2.t1.r1">y</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mstyle><mo>)</mo></mrow></mrow><mo></mo><mstyle displaystyle="true"><mfrac><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./18.2#Px1.p2">x</mi><mo>-</mo><mi href="./18.1#SS1.p2.t1.r1">y</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mi>μ</mi><mo>-</mo><mn>1</mn></mrow></msup><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi>μ</mi><mo>)</mo></mrow></mrow></mfrac></mstyle><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./18.1#SS1.p2.t1.r1">y</mi></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m65.png" altimg-height="20px" altimg-valign="-6px" altimg-width="53px" alttext="\mu>0" display="inline"><mrow><mi>μ</mi><mo>></mo><mn>0</mn></mrow></math>, <math class="ltx_Math" altimg="m83.png" altimg-height="17px" altimg-valign="-3px" altimg-width="52px" alttext="x>0" display="inline"><mrow><mi href="./18.2#Px1.p2">x</mi><mo>></mo><mn>0</mn></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E10.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m35.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./18.3#T1.t1.r2" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m27.png" altimg-height="29px" altimg-valign="-7px" altimg-width="88px" alttext="P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msubsup><mi href="./18.3#T1.t1.r2">P</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r2" stretchy="false">(</mo><mi class="ltx_nvar">α</mi><mo href="./18.3#T1.t1.r2">,</mo><mi class="ltx_nvar">β</mi><mo href="./18.3#T1.t1.r2" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Jacobi polynomial</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./18.1#SS1.p2.t1.r1" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./18.1#SS1.p2.t1.r1">y</mi></math>: real variable</a>,
<a href="./18.1#SS1.p2.t1.r5" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m80.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E11">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">18.17.11</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m8.png" altimg-height="48px" altimg-valign="-16px" altimg-width="373px" alttext="\displaystyle\frac{\Gamma\left(n+\alpha+\beta-\mu+1\right)}{x^{n+\alpha+\beta-%
\mu+1}}P^{(\alpha,\beta-\mu)}_{n}\left(1-2x^{-1}\right)" display="inline"><mrow><mstyle displaystyle="true"><mfrac><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>+</mo><mi>α</mi><mo>+</mo><mi>β</mi></mrow><mo>-</mo><mi>μ</mi></mrow><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><msup><mi href="./18.2#Px1.p2">x</mi><mrow><mrow><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>+</mo><mi>α</mi><mo>+</mo><mi>β</mi></mrow><mo>-</mo><mi>μ</mi></mrow><mo>+</mo><mn>1</mn></mrow></msup></mfrac></mstyle><mo></mo><mrow><msubsup><mi href="./18.3#T1.t1.r2">P</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r2" stretchy="false">(</mo><mi>α</mi><mo href="./18.3#T1.t1.r2">,</mo><mrow><mi>β</mi><mo>-</mo><mi>μ</mi></mrow><mo href="./18.3#T1.t1.r2" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mrow><mn>2</mn><mo></mo><msup><mi href="./18.2#Px1.p2">x</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></mrow></mrow><mo>)</mo></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m5.png" altimg-height="55px" altimg-valign="-21px" altimg-width="506px" alttext="\displaystyle=\int_{x}^{\infty}\frac{\Gamma\left(n+\alpha+\beta+1\right)}{y^{n%
+\alpha+\beta+1}}P^{(\alpha,\beta)}_{n}\left(1-2y^{-1}\right)\*\frac{(y-x)^{%
\mu-1}}{\Gamma\left(\mu\right)}\mathrm{d}y," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi href="./18.2#Px1.p2">x</mi><mi mathvariant="normal">∞</mi></msubsup></mstyle><mrow><mstyle displaystyle="true"><mfrac><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>+</mo><mi>α</mi><mo>+</mo><mi>β</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><msup><mi href="./18.1#SS1.p2.t1.r1">y</mi><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>+</mo><mi>α</mi><mo>+</mo><mi>β</mi><mo>+</mo><mn>1</mn></mrow></msup></mfrac></mstyle><mo></mo><mrow><msubsup><mi href="./18.3#T1.t1.r2">P</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r2" stretchy="false">(</mo><mi>α</mi><mo href="./18.3#T1.t1.r2">,</mo><mi>β</mi><mo href="./18.3#T1.t1.r2" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mrow><mn>2</mn><mo></mo><msup><mi href="./18.1#SS1.p2.t1.r1">y</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></mrow></mrow><mo>)</mo></mrow></mrow><mo></mo><mstyle displaystyle="true"><mfrac><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./18.1#SS1.p2.t1.r1">y</mi><mo>-</mo><mi href="./18.2#Px1.p2">x</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mi>μ</mi><mo>-</mo><mn>1</mn></mrow></msup><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi>μ</mi><mo>)</mo></mrow></mrow></mfrac></mstyle><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./18.1#SS1.p2.t1.r1">y</mi></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m40.png" altimg-height="21px" altimg-valign="-6px" altimg-width="163px" alttext="\alpha+\beta+1>\mu>0" display="inline"><mrow><mrow><mi>α</mi><mo>+</mo><mi>β</mi><mo>+</mo><mn>1</mn></mrow><mo>></mo><mi>μ</mi><mo>></mo><mn>0</mn></mrow></math>, <math class="ltx_Math" altimg="m84.png" altimg-height="17px" altimg-valign="-3px" altimg-width="52px" alttext="x>1" display="inline"><mrow><mi href="./18.2#Px1.p2">x</mi><mo>></mo><mn>1</mn></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E11.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m35.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./18.3#T1.t1.r2" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m27.png" altimg-height="29px" altimg-valign="-7px" altimg-width="88px" alttext="P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msubsup><mi href="./18.3#T1.t1.r2">P</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r2" stretchy="false">(</mo><mi class="ltx_nvar">α</mi><mo href="./18.3#T1.t1.r2">,</mo><mi class="ltx_nvar">β</mi><mo href="./18.3#T1.t1.r2" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Jacobi polynomial</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./18.1#SS1.p2.t1.r1" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./18.1#SS1.p2.t1.r1">y</mi></math>: real variable</a>,
<a href="./18.1#SS1.p2.t1.r5" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m80.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
<p class="ltx_p">and three formulas similar to (</dd>
</dl>
</div>
</div>
<div id="Px9.p1" class="ltx_para">
<table id="EGx2" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E12">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">18.17.12</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m7.png" altimg-height="66px" altimg-valign="-19px" altimg-width="218px" alttext="\displaystyle\frac{\Gamma\left(\lambda-\mu\right)C^{(\lambda-\mu)}_{n}\left(x^%
{-\frac{1}{2}}\right)}{x^{\lambda-\mu+\frac{1}{2}n}}" display="inline"><mstyle displaystyle="true"><mfrac><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi>λ</mi><mo>-</mo><mi>μ</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><msubsup><mi href="./18.3#T1.t1.r3">C</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r3" stretchy="false">(</mo><mrow><mi>λ</mi><mo>-</mo><mi>μ</mi></mrow><mo href="./18.3#T1.t1.r3" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><msup><mi href="./18.2#Px1.p2">x</mi><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup><mo>)</mo></mrow></mrow></mrow><msup><mi href="./18.2#Px1.p2">x</mi><mrow><mrow><mi>λ</mi><mo>-</mo><mi>μ</mi></mrow><mo>+</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./18.1#SS1.p2.t1.r5">n</mi></mrow></mrow></msup></mfrac></mstyle></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m4.png" altimg-height="70px" altimg-valign="-22px" altimg-width="344px" alttext="\displaystyle=\int_{x}^{\infty}\frac{\Gamma\left(\lambda\right)C^{(\lambda)}_{%
n}\left(y^{-\frac{1}{2}}\right)}{y^{\lambda+\frac{1}{2}n}}\frac{(y-x)^{\mu-1}}%
{\Gamma\left(\mu\right)}\mathrm{d}y," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi href="./18.2#Px1.p2">x</mi><mi mathvariant="normal">∞</mi></msubsup></mstyle><mrow><mstyle displaystyle="true"><mfrac><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi>λ</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><msubsup><mi href="./18.3#T1.t1.r3">C</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r3" stretchy="false">(</mo><mi>λ</mi><mo href="./18.3#T1.t1.r3" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><msup><mi href="./18.1#SS1.p2.t1.r1">y</mi><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup><mo>)</mo></mrow></mrow></mrow><msup><mi href="./18.1#SS1.p2.t1.r1">y</mi><mrow><mi>λ</mi><mo>+</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./18.1#SS1.p2.t1.r5">n</mi></mrow></mrow></msup></mfrac></mstyle><mo></mo><mstyle displaystyle="true"><mfrac><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./18.1#SS1.p2.t1.r1">y</mi><mo>-</mo><mi href="./18.2#Px1.p2">x</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mi>μ</mi><mo>-</mo><mn>1</mn></mrow></msup><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi>μ</mi><mo>)</mo></mrow></mrow></mfrac></mstyle><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./18.1#SS1.p2.t1.r1">y</mi></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m54.png" altimg-height="21px" altimg-valign="-6px" altimg-width="91px" alttext="\lambda>\mu>0" display="inline"><mrow><mi>λ</mi><mo>></mo><mi>μ</mi><mo>></mo><mn>0</mn></mrow></math>, <math class="ltx_Math" altimg="m83.png" altimg-height="17px" altimg-valign="-3px" altimg-width="52px" alttext="x>0" display="inline"><mrow><mi href="./18.2#Px1.p2">x</mi><mo>></mo><mn>0</mn></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E12.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m35.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./18.3#T1.t1.r3" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="29px" altimg-valign="-7px" altimg-width="73px" alttext="C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msubsup><mi href="./18.3#T1.t1.r3">C</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r3" stretchy="false">(</mo><mi class="ltx_nvar">λ</mi><mo href="./18.3#T1.t1.r3" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: ultraspherical (or Gegenbauer) polynomial</a>,
<a href="./18.1#SS1.p2.t1.r1" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./18.1#SS1.p2.t1.r1">y</mi></math>: real variable</a>,
<a href="./18.1#SS1.p2.t1.r5" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m80.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E13">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">18.17.13</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m11.png" altimg-height="73px" altimg-valign="-26px" altimg-width="291px" alttext="\displaystyle\frac{x^{\frac{1}{2}n}(x-1)^{\lambda+\mu-\frac{1}{2}}}{\Gamma%
\left(\lambda+\mu+\tfrac{1}{2}\right)}\frac{C^{(\lambda+\mu)}_{n}\left(x^{-%
\frac{1}{2}}\right)}{C^{(\lambda+\mu)}_{n}\left(1\right)}" display="inline"><mrow><mstyle displaystyle="true"><mfrac><mrow><msup><mi href="./18.2#Px1.p2">x</mi><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./18.1#SS1.p2.t1.r5">n</mi></mrow></msup><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./18.2#Px1.p2">x</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mrow><mrow><mi>λ</mi><mo>+</mo><mi>μ</mi></mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup></mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi>λ</mi><mo>+</mo><mi>μ</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow></mfrac></mstyle><mo></mo><mstyle displaystyle="true"><mfrac><mrow><msubsup><mi href="./18.3#T1.t1.r3">C</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r3" stretchy="false">(</mo><mrow><mi>λ</mi><mo>+</mo><mi>μ</mi></mrow><mo href="./18.3#T1.t1.r3" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><msup><mi href="./18.2#Px1.p2">x</mi><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup><mo>)</mo></mrow></mrow><mrow><msubsup><mi href="./18.3#T1.t1.r3">C</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r3" stretchy="false">(</mo><mrow><mi>λ</mi><mo>+</mo><mi>μ</mi></mrow><mo href="./18.3#T1.t1.r3" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></mfrac></mstyle></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m3.png" altimg-height="73px" altimg-valign="-26px" altimg-width="423px" alttext="\displaystyle=\int_{1}^{x}\frac{y^{\frac{1}{2}n}(y-1)^{\lambda-\frac{1}{2}}}{%
\Gamma\left(\lambda+\tfrac{1}{2}\right)}\frac{C^{(\lambda)}_{n}\left(y^{-\frac%
{1}{2}}\right)}{C^{(\lambda)}_{n}\left(1\right)}\frac{(x-y)^{\mu-1}}{\Gamma%
\left(\mu\right)}\mathrm{d}y," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>1</mn><mi href="./18.2#Px1.p2">x</mi></msubsup></mstyle><mrow><mstyle displaystyle="true"><mfrac><mrow><msup><mi href="./18.1#SS1.p2.t1.r1">y</mi><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./18.1#SS1.p2.t1.r5">n</mi></mrow></msup><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./18.1#SS1.p2.t1.r1">y</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mrow><mi>λ</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup></mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi>λ</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow></mfrac></mstyle><mo></mo><mstyle displaystyle="true"><mfrac><mrow><msubsup><mi href="./18.3#T1.t1.r3">C</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r3" stretchy="false">(</mo><mi>λ</mi><mo href="./18.3#T1.t1.r3" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><msup><mi href="./18.1#SS1.p2.t1.r1">y</mi><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup><mo>)</mo></mrow></mrow><mrow><msubsup><mi href="./18.3#T1.t1.r3">C</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r3" stretchy="false">(</mo><mi>λ</mi><mo href="./18.3#T1.t1.r3" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></mfrac></mstyle><mo></mo><mstyle displaystyle="true"><mfrac><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./18.2#Px1.p2">x</mi><mo>-</mo><mi href="./18.1#SS1.p2.t1.r1">y</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mi>μ</mi><mo>-</mo><mn>1</mn></mrow></msup><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi>μ</mi><mo>)</mo></mrow></mrow></mfrac></mstyle><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./18.1#SS1.p2.t1.r1">y</mi></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m65.png" altimg-height="20px" altimg-valign="-6px" altimg-width="53px" alttext="\mu>0" display="inline"><mrow><mi>μ</mi><mo>></mo><mn>0</mn></mrow></math>, <math class="ltx_Math" altimg="m84.png" altimg-height="17px" altimg-valign="-3px" altimg-width="52px" alttext="x>1" display="inline"><mrow><mi href="./18.2#Px1.p2">x</mi><mo>></mo><mn>1</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E13.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m35.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./18.3#T1.t1.r3" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="29px" altimg-valign="-7px" altimg-width="73px" alttext="C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msubsup><mi href="./18.3#T1.t1.r3">C</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r3" stretchy="false">(</mo><mi class="ltx_nvar">λ</mi><mo href="./18.3#T1.t1.r3" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: ultraspherical (or Gegenbauer) polynomial</a>,
<a href="./18.1#SS1.p2.t1.r1" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./18.1#SS1.p2.t1.r1">y</mi></math>: real variable</a>,
<a href="./18.1#SS1.p2.t1.r5" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m80.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
</div>
</section>
<section id="Px10" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Laguerre</h3>
<div id="Px10.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px10.p1" class="ltx_para">
<table id="EGx3" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E14">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">18.17.14</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m9.png" altimg-height="58px" altimg-valign="-21px" altimg-width="162px" alttext="\displaystyle\frac{x^{\alpha+\mu}L^{(\alpha+\mu)}_{n}\left(x\right)}{\Gamma%
\left(\alpha+\mu+n+1\right)}" display="inline"><mstyle displaystyle="true"><mfrac><mrow><msup><mi href="./18.2#Px1.p2">x</mi><mrow><mi>α</mi><mo>+</mo><mi>μ</mi></mrow></msup><mo></mo><mrow><msubsup><mi href="./18.3#T1.t1.r12">L</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r12" stretchy="false">(</mo><mrow><mi>α</mi><mo>+</mo><mi>μ</mi></mrow><mo href="./18.3#T1.t1.r12" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi>α</mi><mo>+</mo><mi>μ</mi><mo>+</mo><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mfrac></mstyle></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m1.png" altimg-height="58px" altimg-valign="-21px" altimg-width="305px" alttext="\displaystyle=\int_{0}^{x}\frac{y^{\alpha}L^{(\alpha)}_{n}\left(y\right)}{%
\Gamma\left(\alpha+n+1\right)}\frac{(x-y)^{\mu-1}}{\Gamma\left(\mu\right)}%
\mathrm{d}y," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi href="./18.2#Px1.p2">x</mi></msubsup></mstyle><mrow><mstyle displaystyle="true"><mfrac><mrow><msup><mi href="./18.1#SS1.p2.t1.r1">y</mi><mi>α</mi></msup><mo></mo><mrow><msubsup><mi href="./18.3#T1.t1.r12">L</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r12" stretchy="false">(</mo><mi>α</mi><mo href="./18.3#T1.t1.r12" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi href="./18.1#SS1.p2.t1.r1">y</mi><mo>)</mo></mrow></mrow></mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi>α</mi><mo>+</mo><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mfrac></mstyle><mo></mo><mstyle displaystyle="true"><mfrac><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./18.2#Px1.p2">x</mi><mo>-</mo><mi href="./18.1#SS1.p2.t1.r1">y</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mi>μ</mi><mo>-</mo><mn>1</mn></mrow></msup><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi>μ</mi><mo>)</mo></mrow></mrow></mfrac></mstyle><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./18.1#SS1.p2.t1.r1">y</mi></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m65.png" altimg-height="20px" altimg-valign="-6px" altimg-width="53px" alttext="\mu>0" display="inline"><mrow><mi>μ</mi><mo>></mo><mn>0</mn></mrow></math>, <math class="ltx_Math" altimg="m83.png" altimg-height="17px" altimg-valign="-3px" altimg-width="52px" alttext="x>0" display="inline"><mrow><mi href="./18.2#Px1.p2">x</mi><mo>></mo><mn>0</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E14.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m35.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./18.3#T1.t1.r12" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m24.png" altimg-height="29px" altimg-valign="-7px" altimg-width="72px" alttext="L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msubsup><mi href="./18.3#T1.t1.r12">L</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r12" stretchy="false">(</mo><mi class="ltx_nvar">α</mi><mo href="./18.3#T1.t1.r12" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Laguerre (or generalized Laguerre) polynomial</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./18.1#SS1.p2.t1.r1" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./18.1#SS1.p2.t1.r1">y</mi></math>: real variable</a>,
<a href="./18.1#SS1.p2.t1.r5" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m80.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">22.13.13</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E15">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">18.17.15</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m12.png" altimg-height="29px" altimg-valign="-7px" altimg-width="105px" alttext="\displaystyle e^{-x}L^{(\alpha)}_{n}\left(x\right)" display="inline"><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mi href="./18.2#Px1.p2">x</mi></mrow></msup><mo></mo><mrow><msubsup><mi href="./18.3#T1.t1.r12">L</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r12" stretchy="false">(</mo><mi>α</mi><mo href="./18.3#T1.t1.r12" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m6.png" altimg-height="55px" altimg-valign="-21px" altimg-width="315px" alttext="\displaystyle=\int_{x}^{\infty}e^{-y}L^{(\alpha+\mu)}_{n}\left(y\right)\frac{(%
y-x)^{\mu-1}}{\Gamma\left(\mu\right)}\mathrm{d}y," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi href="./18.2#Px1.p2">x</mi><mi mathvariant="normal">∞</mi></msubsup></mstyle><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mi href="./18.1#SS1.p2.t1.r1">y</mi></mrow></msup><mo></mo><mrow><msubsup><mi href="./18.3#T1.t1.r12">L</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r12" stretchy="false">(</mo><mrow><mi>α</mi><mo>+</mo><mi>μ</mi></mrow><mo href="./18.3#T1.t1.r12" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi href="./18.1#SS1.p2.t1.r1">y</mi><mo>)</mo></mrow></mrow><mo></mo><mstyle displaystyle="true"><mfrac><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./18.1#SS1.p2.t1.r1">y</mi><mo>-</mo><mi href="./18.2#Px1.p2">x</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mi>μ</mi><mo>-</mo><mn>1</mn></mrow></msup><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi>μ</mi><mo>)</mo></mrow></mrow></mfrac></mstyle><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./18.1#SS1.p2.t1.r1">y</mi></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="5"><span class="ltx_constraint"><math class="ltx_Math" altimg="m65.png" altimg-height="20px" altimg-valign="-6px" altimg-width="53px" alttext="\mu>0" display="inline"><mrow><mi>μ</mi><mo>></mo><mn>0</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E15.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m35.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./18.3#T1.t1.r12" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m24.png" altimg-height="29px" altimg-valign="-7px" altimg-width="72px" alttext="L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msubsup><mi href="./18.3#T1.t1.r12">L</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r12" stretchy="false">(</mo><mi class="ltx_nvar">α</mi><mo href="./18.3#T1.t1.r12" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Laguerre (or generalized Laguerre) polynomial</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./18.1#SS1.p2.t1.r1" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./18.1#SS1.p2.t1.r1">y</mi></math>: real variable</a>,
<a href="./18.1#SS1.p2.t1.r5" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m80.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
</div>
</section>
</section>
<section id="v" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§18.17(v) </span>Fourier Transforms</h2>
<div id="SS5.info" class="ltx_metadata ltx_info">
), integrate by parts
<math class="ltx_Math" altimg="m80.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math> times, expand <math class="ltx_Math" altimg="m77.png" altimg-height="21px" altimg-valign="-2px" altimg-width="82px" alttext="e^{-\mathrm{i}y(1-x)}" display="inline"><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./18.1#SS1.p2.t1.r1">y</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><mi href="./18.2#Px1.p2">x</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></msup></math> in a Maclaurin series, and
integrate term by term.
For (), integrate
repeatedly by parts, expand <math class="ltx_Math" altimg="m45.png" altimg-height="23px" altimg-valign="-7px" altimg-width="72px" alttext="\cos\left(xy\right)" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./18.2#Px1.p2">x</mi><mo></mo><mi href="./18.1#SS1.p2.t1.r1">y</mi></mrow><mo>)</mo></mrow></mrow></math> in a Maclaurin series, and
integrate term by term;
the proofs of () expand <math class="ltx_Math" altimg="m45.png" altimg-height="23px" altimg-valign="-7px" altimg-width="72px" alttext="\cos\left(xy\right)" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./18.2#Px1.p2">x</mi><mo></mo><mi href="./18.1#SS1.p2.t1.r1">y</mi></mrow><mo>)</mo></mrow></mrow></math> in a Maclaurin series, make
the change of integration variable <math class="ltx_Math" altimg="m16.png" altimg-height="21px" altimg-valign="-4px" altimg-width="103px" alttext="1-2x^{2}=t" display="inline"><mrow><mrow><mn>1</mn><mo>-</mo><mrow><mn>2</mn><mo></mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>2</mn></msup></mrow></mrow><mo>=</mo><mi>t</mi></mrow></math>, apply
(), integrate by parts <math class="ltx_Math" altimg="m80.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math> times, and use
() and the fact that the
Fourier transform of <math class="ltx_Math" altimg="m75.png" altimg-height="23px" altimg-valign="-2px" altimg-width="55px" alttext="e^{-\frac{1}{2}x^{2}}" display="inline"><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>2</mn></msup></mrow></mrow></msup></math> is <math class="ltx_Math" altimg="m76.png" altimg-height="23px" altimg-valign="-2px" altimg-width="55px" alttext="e^{-\frac{1}{2}y^{2}}" display="inline"><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><msup><mi href="./18.1#SS1.p2.t1.r1">y</mi><mn>2</mn></msup></mrow></mrow></msup></math>;
the proofs of () consider the Fourier transform of this function
instead of the cosine transform, and replace
<math class="ltx_Math" altimg="m25.png" altimg-height="34px" altimg-valign="-9px" altimg-width="120px" alttext="L^{(n-\frac{1}{2})}_{n}\left(\frac{1}{2}x^{2}\right)" display="inline"><mrow><msubsup><mi href="./18.3#T1.t1.r12">L</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r12" stretchy="false">(</mo><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo href="./18.3#T1.t1.r12" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow></mrow></math> by its explicit form
(</dd>
</dl>
</div>
</div>
<div id="SS5.p1" class="ltx_para">
<p class="ltx_p">Throughout this subsection we assume <math class="ltx_Math" altimg="m86.png" altimg-height="20px" altimg-valign="-6px" altimg-width="51px" alttext="y>0" display="inline"><mrow><mi href="./18.1#SS1.p2.t1.r1">y</mi><mo>></mo><mn>0</mn></mrow></math>; sometimes however, this
restriction can be eased by analytic continuation.</p>
</div>
<section id="Px11" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Jacobi</h3>
<div id="Px11.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px11.p1" class="ltx_para">
<table id="E16" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">18.17.16</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_Math ltx_eqn_cell ltx_align_center"><math class="ltx_Math" alttext="\int_{-1}^{1}(1-x)^{\alpha}(1+x)^{\beta}P^{(\alpha,\beta)}_{n}\left(x\right)e^%
{ixy}\mathrm{d}x=\frac{(iy)^{n}e^{iy}}{n!}2^{n+\alpha+\beta+1}\mathrm{B}\left(%
n+\alpha+1,n+\beta+1\right){{}_{1}F_{1}}\left(n+\alpha+1;2n+\alpha+\beta+2;-2%
iy\right)." display="block"><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><mo>-</mo><mn>1</mn></mrow><mn>1</mn></msubsup><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><mi href="./18.2#Px1.p2">x</mi></mrow><mo stretchy="false">)</mo></mrow><mi>α</mi></msup><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>+</mo><mi href="./18.2#Px1.p2">x</mi></mrow><mo stretchy="false">)</mo></mrow><mi>β</mi></msup><mo></mo><mrow><msubsup><mi href="./18.3#T1.t1.r2">P</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r2" stretchy="false">(</mo><mi>α</mi><mo href="./18.3#T1.t1.r2">,</mo><mi>β</mi><mo href="./18.3#T1.t1.r2" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./18.2#Px1.p2">x</mi><mo></mo><mi href="./18.1#SS1.p2.t1.r1">y</mi></mrow></msup><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./18.2#Px1.p2">x</mi></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>=</mo><mrow><mfrac><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./18.1#SS1.p2.t1.r1">y</mi></mrow><mo stretchy="false">)</mo></mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi></msup><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./18.1#SS1.p2.t1.r1">y</mi></mrow></msup></mrow><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mfrac><mo></mo><msup><mn>2</mn><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>+</mo><mi>α</mi><mo>+</mo><mi>β</mi><mo>+</mo><mn>1</mn></mrow></msup><mo></mo><mrow><mi href="./5.12#E1" mathvariant="normal">B</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>+</mo><mi>α</mi><mo>+</mo><mn>1</mn></mrow><mo>,</mo><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>+</mo><mi>β</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>1</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>+</mo><mi>α</mi><mo>+</mo><mn>1</mn></mrow><mo>;</mo><mrow><mrow><mn>2</mn><mo></mo><mi href="./18.1#SS1.p2.t1.r5">n</mi></mrow><mo>+</mo><mi>α</mi><mo>+</mo><mi>β</mi><mo>+</mo><mn>2</mn></mrow><mo>;</mo><mrow><mo>-</mo><mrow><mn>2</mn><mo></mo><mi mathvariant="normal">i</mi><mo></mo><mi href="./18.1#SS1.p2.t1.r1">y</mi></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>.</mo></mtd></mtr></mtable></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E16.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.12#E1" title="(5.12.1) ‣ Euler’s Beta Integral ‣ §5.12 Beta Function ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m57.png" altimg-height="23px" altimg-valign="-7px" altimg-width="65px" alttext="\mathrm{B}\left(\NVar{a},\NVar{b}\right)" display="inline"><mrow><mi href="./5.12#E1" mathvariant="normal">B</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r5">a</mi><mo>,</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r5">b</mi><mo>)</mo></mrow></mrow></math>: beta function</a>,
<a href="./18.3#T1.t1.r2" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m27.png" altimg-height="29px" altimg-valign="-7px" altimg-width="88px" alttext="P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msubsup><mi href="./18.3#T1.t1.r2">P</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r2" stretchy="false">(</mo><mi class="ltx_nvar">α</mi><mo href="./18.3#T1.t1.r2">,</mo><mi class="ltx_nvar">β</mi><mo href="./18.3#T1.t1.r2" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Jacobi polynomial</a>,
<a href="./16.2" title="§16.2 Definition and Analytic Properties ‣ Generalized Hypergeometric Functions ‣ Chapter 16 Generalized Hypergeometric Functions and Meijer G -Function" class="ltx_ref"><math class="ltx_Math" altimg="m92.png" altimg-height="23px" altimg-valign="-7px" altimg-width="101px" alttext="{{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)" display="inline"><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>1</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./16.1#p2.t1.r5">a</mi><mo>;</mo><mi class="ltx_nvar" href="./16.1#p2.t1.r5">b</mi><mo>;</mo><mi class="ltx_nvar" href="./16.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: <math class="ltx_Math" altimg="m19.png" altimg-height="23px" altimg-valign="-7px" altimg-width="113px" alttext="=M\left(\NVar{a},\NVar{b},\NVar{z}\right)" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mi href="./13.2#E2">M</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./16.1#p2.t1.r5">a</mi><mo>,</mo><mi class="ltx_nvar" href="./16.1#p2.t1.r5">b</mi><mo>,</mo><mi class="ltx_nvar" href="./16.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></math> notation for the Kummer confluent hypergeometric function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m13.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m79.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./18.1#SS1.p2.t1.r1" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./18.1#SS1.p2.t1.r1">y</mi></math>: real variable</a>,
<a href="./18.1#SS1.p2.t1.r5" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m80.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">For the beta function <math class="ltx_Math" altimg="m58.png" altimg-height="23px" altimg-valign="-7px" altimg-width="65px" alttext="\mathrm{B}\left(a,b\right)" display="inline"><mrow><mi href="./5.12#E1" mathvariant="normal">B</mi><mo></mo><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></mrow></mrow></math> see §, and for the confluent
hypergeometric function <math class="ltx_Math" altimg="m91.png" altimg-height="21px" altimg-valign="-5px" altimg-width="35px" alttext="{{}_{1}F_{1}}" display="inline"><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>1</mn><none></none></mmultiscripts></math> see (</dd>
</dl>
</div>
</div>
<div id="Px12.p1" class="ltx_para">
<table id="E17" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">18.17.17</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E17.png" altimg-height="55px" altimg-valign="-21px" altimg-width="589px" alttext="\int_{0}^{1}(1-x^{2})^{\lambda-\frac{1}{2}}C^{(\lambda)}_{2n}\left(x\right)%
\cos\left(xy\right)\mathrm{d}x=\frac{(-1)^{n}\pi\Gamma\left(2n+2\lambda\right)%
J_{\lambda+2n}\left(y\right)}{(2n)!\Gamma\left(\lambda\right)(2y)^{\lambda}}," display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mn>1</mn></msubsup><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mrow><mi>λ</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup><mo></mo><mrow><msubsup><mi href="./18.3#T1.t1.r3">C</mi><mrow><mn>2</mn><mo></mo><mi href="./18.1#SS1.p2.t1.r5">n</mi></mrow><mrow><mo href="./18.3#T1.t1.r3" stretchy="false">(</mo><mi>λ</mi><mo href="./18.3#T1.t1.r3" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./18.2#Px1.p2">x</mi><mo></mo><mi href="./18.1#SS1.p2.t1.r1">y</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./18.2#Px1.p2">x</mi></mrow></mrow></mrow><mo>=</mo><mfrac><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi></msup><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mn>2</mn><mo></mo><mi href="./18.1#SS1.p2.t1.r5">n</mi></mrow><mo>+</mo><mrow><mn>2</mn><mo></mo><mi>λ</mi></mrow></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mrow><mi>λ</mi><mo>+</mo><mrow><mn>2</mn><mo></mo><mi href="./18.1#SS1.p2.t1.r5">n</mi></mrow></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./18.1#SS1.p2.t1.r1">y</mi><mo>)</mo></mrow></mrow></mrow><mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mn>2</mn><mo></mo><mi href="./18.1#SS1.p2.t1.r5">n</mi></mrow><mo stretchy="false">)</mo></mrow><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi>λ</mi><mo>)</mo></mrow></mrow><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mn>2</mn><mo></mo><mi href="./18.1#SS1.p2.t1.r1">y</mi></mrow><mo stretchy="false">)</mo></mrow><mi>λ</mi></msup></mrow></mfrac></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E17.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m35.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m13.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m79.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./18.3#T1.t1.r3" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="29px" altimg-valign="-7px" altimg-width="73px" alttext="C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msubsup><mi href="./18.3#T1.t1.r3">C</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r3" stretchy="false">(</mo><mi class="ltx_nvar">λ</mi><mo href="./18.3#T1.t1.r3" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: ultraspherical (or Gegenbauer) polynomial</a>,
<a href="./18.1#SS1.p2.t1.r1" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./18.1#SS1.p2.t1.r1">y</mi></math>: real variable</a>,
<a href="./18.1#SS1.p2.t1.r5" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m80.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E18" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">18.17.18</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E18.png" altimg-height="55px" altimg-valign="-21px" altimg-width="656px" alttext="\int_{0}^{1}(1-x^{2})^{\lambda-\frac{1}{2}}C^{(\lambda)}_{2n+1}\left(x\right)%
\sin\left(xy\right)\mathrm{d}x=\frac{(-1)^{n}\pi\Gamma\left(2n+2\lambda+1%
\right)J_{2n+\lambda+1}\left(y\right)}{(2n+1)!\Gamma\left(\lambda\right)(2y)^{%
\lambda}}." display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mn>1</mn></msubsup><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mrow><mi>λ</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup><mo></mo><mrow><msubsup><mi href="./18.3#T1.t1.r3">C</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./18.1#SS1.p2.t1.r5">n</mi></mrow><mo>+</mo><mn>1</mn></mrow><mrow><mo href="./18.3#T1.t1.r3" stretchy="false">(</mo><mi>λ</mi><mo href="./18.3#T1.t1.r3" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./18.2#Px1.p2">x</mi><mo></mo><mi href="./18.1#SS1.p2.t1.r1">y</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./18.2#Px1.p2">x</mi></mrow></mrow></mrow><mo>=</mo><mfrac><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi></msup><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mn>2</mn><mo></mo><mi href="./18.1#SS1.p2.t1.r5">n</mi></mrow><mo>+</mo><mrow><mn>2</mn><mo></mo><mi>λ</mi></mrow><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./18.1#SS1.p2.t1.r5">n</mi></mrow><mo>+</mo><mi>λ</mi><mo>+</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./18.1#SS1.p2.t1.r1">y</mi><mo>)</mo></mrow></mrow></mrow><mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>2</mn><mo></mo><mi href="./18.1#SS1.p2.t1.r5">n</mi></mrow><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi>λ</mi><mo>)</mo></mrow></mrow><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mn>2</mn><mo></mo><mi href="./18.1#SS1.p2.t1.r1">y</mi></mrow><mo stretchy="false">)</mo></mrow><mi>λ</mi></msup></mrow></mfrac></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E18.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m35.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m13.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m79.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m71.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./18.3#T1.t1.r3" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="29px" altimg-valign="-7px" altimg-width="73px" alttext="C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msubsup><mi href="./18.3#T1.t1.r3">C</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r3" stretchy="false">(</mo><mi class="ltx_nvar">λ</mi><mo href="./18.3#T1.t1.r3" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: ultraspherical (or Gegenbauer) polynomial</a>,
<a href="./18.1#SS1.p2.t1.r1" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./18.1#SS1.p2.t1.r1">y</mi></math>: real variable</a>,
<a href="./18.1#SS1.p2.t1.r5" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m80.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">For the Bessel function <math class="ltx_Math" altimg="m23.png" altimg-height="21px" altimg-valign="-5px" altimg-width="25px" alttext="J_{\nu}" display="inline"><msub><mi href="./10.2#E2">J</mi><mi>ν</mi></msub></math> see §</dd>
</dl>
</div>
</div>
<div id="Px13.p1" class="ltx_para">
<table id="E19" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">18.17.19</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E19.png" altimg-height="57px" altimg-valign="-22px" altimg-width="330px" alttext="\int_{-1}^{1}P_{n}\left(x\right)e^{ixy}\mathrm{d}x=i^{n}\sqrt{\frac{2\pi}{y}}J%
_{n+\frac{1}{2}}\left(y\right)," display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><mo>-</mo><mn>1</mn></mrow><mn>1</mn></msubsup><mrow><mrow><msub><mi href="./18.3#T1.t1.r10">P</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./18.2#Px1.p2">x</mi><mo></mo><mi href="./18.1#SS1.p2.t1.r1">y</mi></mrow></msup><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./18.2#Px1.p2">x</mi></mrow></mrow></mrow><mo>=</mo><mrow><msup><mi mathvariant="normal">i</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi></msup><mo></mo><msqrt><mfrac><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi></mrow><mi href="./18.1#SS1.p2.t1.r1">y</mi></mfrac></msqrt><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./18.1#SS1.p2.t1.r1">y</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E19.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./18.3#T1.t1.r10" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m28.png" altimg-height="23px" altimg-valign="-7px" altimg-width="58px" alttext="P_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r10">P</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Legendre polynomial</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./18.1#SS1.p2.t1.r1" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./18.1#SS1.p2.t1.r1">y</mi></math>: real variable</a>,
<a href="./18.1#SS1.p2.t1.r5" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m80.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E20" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">18.17.20</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E20.png" altimg-height="56px" altimg-valign="-20px" altimg-width="553px" alttext="\int_{0}^{1}P_{n}\left(1-2x^{2}\right)\cos\left(xy\right)\mathrm{d}x=(-1)^{n}%
\tfrac{1}{2}\pi J_{n+\frac{1}{2}}\left(\tfrac{1}{2}y\right)J_{-n-\frac{1}{2}}%
\left(\tfrac{1}{2}y\right)," display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mn>1</mn></msubsup><mrow><mrow><msub><mi href="./18.3#T1.t1.r10">P</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mrow><mn>2</mn><mo></mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>2</mn></msup></mrow></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./18.2#Px1.p2">x</mi><mo></mo><mi href="./18.1#SS1.p2.t1.r1">y</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./18.2#Px1.p2">x</mi></mrow></mrow></mrow><mo>=</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi></msup><mo></mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msub><mo></mo><mrow><mo>(</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./18.1#SS1.p2.t1.r1">y</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./10.2#E2">J</mi><mrow><mrow><mo>-</mo><mi href="./18.1#SS1.p2.t1.r5">n</mi></mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msub><mo></mo><mrow><mo>(</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./18.1#SS1.p2.t1.r1">y</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E20.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./18.3#T1.t1.r10" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m28.png" altimg-height="23px" altimg-valign="-7px" altimg-width="58px" alttext="P_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r10">P</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Legendre polynomial</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./18.1#SS1.p2.t1.r1" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./18.1#SS1.p2.t1.r1">y</mi></math>: real variable</a>,
<a href="./18.1#SS1.p2.t1.r5" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m80.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E21" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">18.17.21</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E21.png" altimg-height="56px" altimg-valign="-20px" altimg-width="425px" alttext="\int_{0}^{1}P_{n}\left(1-2x^{2}\right)\sin\left(xy\right)\mathrm{d}x=\tfrac{1}%
{2}\pi\left(J_{n+\frac{1}{2}}\left(\tfrac{1}{2}y\right)\right)^{2}." display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mn>1</mn></msubsup><mrow><mrow><msub><mi href="./18.3#T1.t1.r10">P</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mrow><mn>2</mn><mo></mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>2</mn></msup></mrow></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./18.2#Px1.p2">x</mi><mo></mo><mi href="./18.1#SS1.p2.t1.r1">y</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./18.2#Px1.p2">x</mi></mrow></mrow></mrow><mo>=</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><msup><mrow><mo>(</mo><mrow><msub><mi href="./10.2#E2">J</mi><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msub><mo></mo><mrow><mo>(</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./18.1#SS1.p2.t1.r1">y</mi></mrow><mo>)</mo></mrow></mrow><mo>)</mo></mrow><mn>2</mn></msup></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E21.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./10.2#E2" title="(10.2.2) ‣ Bessel Function of the First Kind ‣ §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="J_{\NVar{\nu}}\left(\NVar{z}\right)" display="inline"><mrow><msub><mi href="./10.2#E2">J</mi><mi class="ltx_nvar" href="./10.1#p2.t1.r5">ν</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./10.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: Bessel function of the first kind</a>,
<a href="./18.3#T1.t1.r10" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m28.png" altimg-height="23px" altimg-valign="-7px" altimg-width="58px" alttext="P_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r10">P</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Legendre polynomial</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m71.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./18.1#SS1.p2.t1.r1" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./18.1#SS1.p2.t1.r1">y</mi></math>: real variable</a>,
<a href="./18.1#SS1.p2.t1.r5" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m80.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px14" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Hermite</h3>
<div id="Px14.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px14.p1" class="ltx_para">
<table id="E22" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">18.17.22</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E22.png" altimg-height="54px" altimg-valign="-22px" altimg-width="452px" alttext="\frac{1}{2\sqrt{\pi}}\int_{-\infty}^{\infty}e^{-\frac{1}{4}x^{2}}\mathit{He}_{%
n}\left(x\right)e^{\frac{1}{2}\mathrm{i}xy}\mathrm{d}x={\mathrm{i}^{n}}e^{-%
\frac{1}{4}y^{2}}\mathit{He}_{n}\left(y\right)," display="block"><mrow><mrow><mrow><mfrac><mn>1</mn><mrow><mn>2</mn><mo></mo><msqrt><mi href="./3.12#E1">π</mi></msqrt></mrow></mfrac><mo></mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mi mathvariant="normal">∞</mi></msubsup><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac><mo></mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>2</mn></msup></mrow></mrow></msup><mo></mo><mrow><msub><mi href="./18.3#T1.t1.r14" mathvariant="italic">He</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi mathvariant="normal">i</mi><mo></mo><mi href="./18.2#Px1.p2">x</mi><mo></mo><mi href="./18.1#SS1.p2.t1.r1">y</mi></mrow></msup><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./18.2#Px1.p2">x</mi></mrow></mrow></mrow></mrow><mo>=</mo><mrow><msup><mi mathvariant="normal">i</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi></msup><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac><mo></mo><msup><mi href="./18.1#SS1.p2.t1.r1">y</mi><mn>2</mn></msup></mrow></mrow></msup><mo></mo><mrow><msub><mi href="./18.3#T1.t1.r14" mathvariant="italic">He</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./18.1#SS1.p2.t1.r1">y</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E22.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./18.3#T1.t1.r14" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="23px" altimg-valign="-7px" altimg-width="71px" alttext="\mathit{He}_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r14" mathvariant="italic">He</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Hermite polynomial</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./18.1#SS1.p2.t1.r1" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./18.1#SS1.p2.t1.r1">y</mi></math>: real variable</a>,
<a href="./18.1#SS1.p2.t1.r5" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m80.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E23" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">18.17.23</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E23.png" altimg-height="53px" altimg-valign="-20px" altimg-width="477px" alttext="\int_{0}^{\infty}e^{-\frac{1}{2}x^{2}}\mathit{He}_{2n}\left(x\right)\cos\left(%
xy\right)\mathrm{d}x=(-1)^{n}\sqrt{\tfrac{1}{2}\pi}y^{2n}e^{-\frac{1}{2}y^{2}}," display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>2</mn></msup></mrow></mrow></msup><mo></mo><mrow><msub><mi href="./18.3#T1.t1.r14" mathvariant="italic">He</mi><mrow><mn>2</mn><mo></mo><mi href="./18.1#SS1.p2.t1.r5">n</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./18.2#Px1.p2">x</mi><mo></mo><mi href="./18.1#SS1.p2.t1.r1">y</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./18.2#Px1.p2">x</mi></mrow></mrow></mrow><mo>=</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi></msup><mo></mo><msqrt><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./3.12#E1">π</mi></mrow></msqrt><mo></mo><msup><mi href="./18.1#SS1.p2.t1.r1">y</mi><mrow><mn>2</mn><mo></mo><mi href="./18.1#SS1.p2.t1.r5">n</mi></mrow></msup><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><msup><mi href="./18.1#SS1.p2.t1.r1">y</mi><mn>2</mn></msup></mrow></mrow></msup></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E23.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./18.3#T1.t1.r14" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="23px" altimg-valign="-7px" altimg-width="71px" alttext="\mathit{He}_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r14" mathvariant="italic">He</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Hermite polynomial</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./18.1#SS1.p2.t1.r1" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./18.1#SS1.p2.t1.r1">y</mi></math>: real variable</a>,
<a href="./18.1#SS1.p2.t1.r5" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m80.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E24" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">18.17.24</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E24.png" altimg-height="53px" altimg-valign="-20px" altimg-width="519px" alttext="\int_{0}^{\infty}e^{-x^{2}}\mathit{He}_{2n}\left(2x\right)\cos\left(xy\right)%
\mathrm{d}x=(-1)^{n}\tfrac{1}{2}\sqrt{\pi}e^{-\frac{1}{4}y^{2}}\mathit{He}_{2n%
}\left(y\right)." display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>2</mn></msup></mrow></msup><mo></mo><mrow><msub><mi href="./18.3#T1.t1.r14" mathvariant="italic">He</mi><mrow><mn>2</mn><mo></mo><mi href="./18.1#SS1.p2.t1.r5">n</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><mi href="./18.2#Px1.p2">x</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./18.2#Px1.p2">x</mi><mo></mo><mi href="./18.1#SS1.p2.t1.r1">y</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./18.2#Px1.p2">x</mi></mrow></mrow></mrow><mo>=</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi></msup><mo></mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><msqrt><mi href="./3.12#E1">π</mi></msqrt><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac><mo></mo><msup><mi href="./18.1#SS1.p2.t1.r1">y</mi><mn>2</mn></msup></mrow></mrow></msup><mo></mo><mrow><msub><mi href="./18.3#T1.t1.r14" mathvariant="italic">He</mi><mrow><mn>2</mn><mo></mo><mi href="./18.1#SS1.p2.t1.r5">n</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./18.1#SS1.p2.t1.r1">y</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E24.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./18.3#T1.t1.r14" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="23px" altimg-valign="-7px" altimg-width="71px" alttext="\mathit{He}_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r14" mathvariant="italic">He</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Hermite polynomial</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./18.1#SS1.p2.t1.r1" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./18.1#SS1.p2.t1.r1">y</mi></math>: real variable</a>,
<a href="./18.1#SS1.p2.t1.r5" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m80.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E25" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">18.17.25</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_Math ltx_eqn_cell ltx_align_center"><math class="ltx_Math" alttext="\int_{0}^{\infty}e^{-\frac{1}{2}x^{2}}\mathit{He}_{n}\left(x\right)\mathit{He}%
_{n+2m}\left(x\right)\cos\left(xy\right)\mathrm{d}x=(-1)^{m}\sqrt{\tfrac{1}{2}%
\pi}n!\,y^{2m}e^{-\frac{1}{2}y^{2}}L^{(2m)}_{n}\left(y^{2}\right)," display="block"><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>2</mn></msup></mrow></mrow></msup><mo></mo><mrow><msub><mi href="./18.3#T1.t1.r14" mathvariant="italic">He</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./18.3#T1.t1.r14" mathvariant="italic">He</mi><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>+</mo><mrow><mn>2</mn><mo></mo><mi href="./18.1#SS1.p2.t1.r4">m</mi></mrow></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./18.2#Px1.p2">x</mi><mo></mo><mi href="./18.1#SS1.p2.t1.r1">y</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./18.2#Px1.p2">x</mi></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>=</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./18.1#SS1.p2.t1.r4">m</mi></msup><mo></mo><msqrt><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./3.12#E1">π</mi></mrow></msqrt><mo></mo><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow><mo></mo><msup><mi href="./18.1#SS1.p2.t1.r1">y</mi><mrow><mn>2</mn><mo></mo><mi href="./18.1#SS1.p2.t1.r4">m</mi></mrow></msup><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><msup><mi href="./18.1#SS1.p2.t1.r1">y</mi><mn>2</mn></msup></mrow></mrow></msup><mo></mo><mrow><msubsup><mi href="./18.3#T1.t1.r12">L</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r12" stretchy="false">(</mo><mrow><mn>2</mn><mo></mo><mi href="./18.1#SS1.p2.t1.r4">m</mi></mrow><mo href="./18.3#T1.t1.r12" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><msup><mi href="./18.1#SS1.p2.t1.r1">y</mi><mn>2</mn></msup><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mtd></mtr></mtable></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E25.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./18.3#T1.t1.r14" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="23px" altimg-valign="-7px" altimg-width="71px" alttext="\mathit{He}_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r14" mathvariant="italic">He</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Hermite polynomial</a>,
<a href="./18.3#T1.t1.r12" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m24.png" altimg-height="29px" altimg-valign="-7px" altimg-width="72px" alttext="L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msubsup><mi href="./18.3#T1.t1.r12">L</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r12" stretchy="false">(</mo><mi class="ltx_nvar">α</mi><mo href="./18.3#T1.t1.r12" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Laguerre (or generalized Laguerre) polynomial</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m13.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m79.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./18.1#SS1.p2.t1.r1" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./18.1#SS1.p2.t1.r1">y</mi></math>: real variable</a>,
<a href="./18.1#SS1.p2.t1.r4" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./18.1#SS1.p2.t1.r4">m</mi></math>: nonnegative integer</a>,
<a href="./18.1#SS1.p2.t1.r5" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m80.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E26" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">18.17.26</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_Math ltx_eqn_cell ltx_align_center"><math class="ltx_Math" alttext="\int_{0}^{\infty}e^{-\frac{1}{2}x^{2}}\mathit{He}_{n}\left(x\right)\mathit{He}%
_{n+2m+1}\left(x\right)\sin\left(xy\right)\mathrm{d}x=(-1)^{m}\sqrt{\tfrac{1}{%
2}\pi}n!\,y^{2m+1}e^{-\frac{1}{2}y^{2}}L^{(2m+1)}_{n}\left(y^{2}\right)." display="block"><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>2</mn></msup></mrow></mrow></msup><mo></mo><mrow><msub><mi href="./18.3#T1.t1.r14" mathvariant="italic">He</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./18.3#T1.t1.r14" mathvariant="italic">He</mi><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>+</mo><mrow><mn>2</mn><mo></mo><mi href="./18.1#SS1.p2.t1.r4">m</mi></mrow><mo>+</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./18.2#Px1.p2">x</mi><mo></mo><mi href="./18.1#SS1.p2.t1.r1">y</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./18.2#Px1.p2">x</mi></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>=</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./18.1#SS1.p2.t1.r4">m</mi></msup><mo></mo><msqrt><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./3.12#E1">π</mi></mrow></msqrt><mo></mo><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow><mo></mo><msup><mi href="./18.1#SS1.p2.t1.r1">y</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./18.1#SS1.p2.t1.r4">m</mi></mrow><mo>+</mo><mn>1</mn></mrow></msup><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><msup><mi href="./18.1#SS1.p2.t1.r1">y</mi><mn>2</mn></msup></mrow></mrow></msup><mo></mo><mrow><msubsup><mi href="./18.3#T1.t1.r12">L</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r12" stretchy="false">(</mo><mrow><mrow><mn>2</mn><mo></mo><mi href="./18.1#SS1.p2.t1.r4">m</mi></mrow><mo>+</mo><mn>1</mn></mrow><mo href="./18.3#T1.t1.r12" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><msup><mi href="./18.1#SS1.p2.t1.r1">y</mi><mn>2</mn></msup><mo>)</mo></mrow></mrow></mrow><mo>.</mo></mtd></mtr></mtable></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E26.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./18.3#T1.t1.r14" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="23px" altimg-valign="-7px" altimg-width="71px" alttext="\mathit{He}_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r14" mathvariant="italic">He</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Hermite polynomial</a>,
<a href="./18.3#T1.t1.r12" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m24.png" altimg-height="29px" altimg-valign="-7px" altimg-width="72px" alttext="L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msubsup><mi href="./18.3#T1.t1.r12">L</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r12" stretchy="false">(</mo><mi class="ltx_nvar">α</mi><mo href="./18.3#T1.t1.r12" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Laguerre (or generalized Laguerre) polynomial</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m13.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m79.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m71.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./18.1#SS1.p2.t1.r1" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./18.1#SS1.p2.t1.r1">y</mi></math>: real variable</a>,
<a href="./18.1#SS1.p2.t1.r4" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./18.1#SS1.p2.t1.r4">m</mi></math>: nonnegative integer</a>,
<a href="./18.1#SS1.p2.t1.r5" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m80.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E27" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">18.17.27</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E27.png" altimg-height="53px" altimg-valign="-20px" altimg-width="515px" alttext="\int_{0}^{\infty}e^{-\frac{1}{2}x^{2}}\mathit{He}_{2n+1}\left(x\right)\sin%
\left(xy\right)\mathrm{d}x=(-1)^{n}\sqrt{\tfrac{1}{2}\pi}y^{2n+1}e^{-\frac{1}{%
2}y^{2}}," display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>2</mn></msup></mrow></mrow></msup><mo></mo><mrow><msub><mi href="./18.3#T1.t1.r14" mathvariant="italic">He</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./18.1#SS1.p2.t1.r5">n</mi></mrow><mo>+</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./18.2#Px1.p2">x</mi><mo></mo><mi href="./18.1#SS1.p2.t1.r1">y</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./18.2#Px1.p2">x</mi></mrow></mrow></mrow><mo>=</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi></msup><mo></mo><msqrt><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./3.12#E1">π</mi></mrow></msqrt><mo></mo><msup><mi href="./18.1#SS1.p2.t1.r1">y</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./18.1#SS1.p2.t1.r5">n</mi></mrow><mo>+</mo><mn>1</mn></mrow></msup><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><msup><mi href="./18.1#SS1.p2.t1.r1">y</mi><mn>2</mn></msup></mrow></mrow></msup></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E27.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./18.3#T1.t1.r14" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="23px" altimg-valign="-7px" altimg-width="71px" alttext="\mathit{He}_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r14" mathvariant="italic">He</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Hermite polynomial</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m71.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./18.1#SS1.p2.t1.r1" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./18.1#SS1.p2.t1.r1">y</mi></math>: real variable</a>,
<a href="./18.1#SS1.p2.t1.r5" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m80.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E28" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">18.17.28</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E28.png" altimg-height="53px" altimg-valign="-20px" altimg-width="558px" alttext="\int_{0}^{\infty}e^{-x^{2}}\mathit{He}_{2n+1}\left(2x\right)\sin\left(xy\right%
)\mathrm{d}x=(-1)^{n}\tfrac{1}{2}\sqrt{\pi}e^{-\frac{1}{4}y^{2}}\mathit{He}_{2%
n+1}\left(y\right)." display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>2</mn></msup></mrow></msup><mo></mo><mrow><msub><mi href="./18.3#T1.t1.r14" mathvariant="italic">He</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./18.1#SS1.p2.t1.r5">n</mi></mrow><mo>+</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><mi href="./18.2#Px1.p2">x</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./18.2#Px1.p2">x</mi><mo></mo><mi href="./18.1#SS1.p2.t1.r1">y</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./18.2#Px1.p2">x</mi></mrow></mrow></mrow><mo>=</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi></msup><mo></mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><msqrt><mi href="./3.12#E1">π</mi></msqrt><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac><mo></mo><msup><mi href="./18.1#SS1.p2.t1.r1">y</mi><mn>2</mn></msup></mrow></mrow></msup><mo></mo><mrow><msub><mi href="./18.3#T1.t1.r14" mathvariant="italic">He</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./18.1#SS1.p2.t1.r5">n</mi></mrow><mo>+</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./18.1#SS1.p2.t1.r1">y</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E28.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./18.3#T1.t1.r14" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="23px" altimg-valign="-7px" altimg-width="71px" alttext="\mathit{He}_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r14" mathvariant="italic">He</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Hermite polynomial</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m71.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./18.1#SS1.p2.t1.r1" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./18.1#SS1.p2.t1.r1">y</mi></math>: real variable</a>,
<a href="./18.1#SS1.p2.t1.r5" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m80.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px15" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Laguerre</h3>
<div id="Px15.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px15.p1" class="ltx_para">
<table id="E29" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">18.17.29</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_Math ltx_eqn_cell ltx_align_center"><math class="ltx_Math" alttext="\int_{0}^{\infty}x^{2m}e^{-\frac{1}{2}x^{2}}L^{(2m)}_{n}\left(x^{2}\right)\cos%
\left(xy\right)\mathrm{d}x=(-1)^{m}\sqrt{\tfrac{1}{2}\pi}\frac{1}{n!}e^{-\frac%
{1}{2}y^{2}}\mathit{He}_{n}\left(y\right)\mathit{He}_{n+2m}\left(y\right)." display="block"><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><mrow><msup><mi href="./18.2#Px1.p2">x</mi><mrow><mn>2</mn><mo></mo><mi href="./18.1#SS1.p2.t1.r4">m</mi></mrow></msup><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>2</mn></msup></mrow></mrow></msup><mo></mo><mrow><msubsup><mi href="./18.3#T1.t1.r12">L</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r12" stretchy="false">(</mo><mrow><mn>2</mn><mo></mo><mi href="./18.1#SS1.p2.t1.r4">m</mi></mrow><mo href="./18.3#T1.t1.r12" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>2</mn></msup><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./18.2#Px1.p2">x</mi><mo></mo><mi href="./18.1#SS1.p2.t1.r1">y</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./18.2#Px1.p2">x</mi></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>=</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./18.1#SS1.p2.t1.r4">m</mi></msup><mo></mo><msqrt><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./3.12#E1">π</mi></mrow></msqrt><mo></mo><mfrac><mn>1</mn><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mfrac><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><msup><mi href="./18.1#SS1.p2.t1.r1">y</mi><mn>2</mn></msup></mrow></mrow></msup><mo></mo><mrow><msub><mi href="./18.3#T1.t1.r14" mathvariant="italic">He</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./18.1#SS1.p2.t1.r1">y</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi href="./18.3#T1.t1.r14" mathvariant="italic">He</mi><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>+</mo><mrow><mn>2</mn><mo></mo><mi href="./18.1#SS1.p2.t1.r4">m</mi></mrow></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./18.1#SS1.p2.t1.r1">y</mi><mo>)</mo></mrow></mrow></mrow><mo>.</mo></mtd></mtr></mtable></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E29.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./18.3#T1.t1.r14" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="23px" altimg-valign="-7px" altimg-width="71px" alttext="\mathit{He}_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r14" mathvariant="italic">He</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Hermite polynomial</a>,
<a href="./18.3#T1.t1.r12" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m24.png" altimg-height="29px" altimg-valign="-7px" altimg-width="72px" alttext="L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msubsup><mi href="./18.3#T1.t1.r12">L</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r12" stretchy="false">(</mo><mi class="ltx_nvar">α</mi><mo href="./18.3#T1.t1.r12" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Laguerre (or generalized Laguerre) polynomial</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m13.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m79.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./18.1#SS1.p2.t1.r1" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./18.1#SS1.p2.t1.r1">y</mi></math>: real variable</a>,
<a href="./18.1#SS1.p2.t1.r4" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./18.1#SS1.p2.t1.r4">m</mi></math>: nonnegative integer</a>,
<a href="./18.1#SS1.p2.t1.r5" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m80.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E30" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">18.17.30</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E30.png" altimg-height="53px" altimg-valign="-20px" altimg-width="614px" alttext="\int_{0}^{\infty}x^{2n}e^{-\frac{1}{2}x^{2}}L^{(n-\frac{1}{2})}_{n}\left(%
\tfrac{1}{2}x^{2}\right)\cos\left(xy\right)\mathrm{d}x=\sqrt{\tfrac{1}{2}\pi}y%
^{2n}e^{-\frac{1}{2}y^{2}}L^{(n-\frac{1}{2})}_{n}\left(\tfrac{1}{2}y^{2}\right)." display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><mrow><msup><mi href="./18.2#Px1.p2">x</mi><mrow><mn>2</mn><mo></mo><mi href="./18.1#SS1.p2.t1.r5">n</mi></mrow></msup><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>2</mn></msup></mrow></mrow></msup><mo></mo><mrow><msubsup><mi href="./18.3#T1.t1.r12">L</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r12" stretchy="false">(</mo><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo href="./18.3#T1.t1.r12" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./18.2#Px1.p2">x</mi><mo></mo><mi href="./18.1#SS1.p2.t1.r1">y</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./18.2#Px1.p2">x</mi></mrow></mrow></mrow><mo>=</mo><mrow><msqrt><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./3.12#E1">π</mi></mrow></msqrt><mo></mo><msup><mi href="./18.1#SS1.p2.t1.r1">y</mi><mrow><mn>2</mn><mo></mo><mi href="./18.1#SS1.p2.t1.r5">n</mi></mrow></msup><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><msup><mi href="./18.1#SS1.p2.t1.r1">y</mi><mn>2</mn></msup></mrow></mrow></msup><mo></mo><mrow><msubsup><mi href="./18.3#T1.t1.r12">L</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r12" stretchy="false">(</mo><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo href="./18.3#T1.t1.r12" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><msup><mi href="./18.1#SS1.p2.t1.r1">y</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E30.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./18.3#T1.t1.r12" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m24.png" altimg-height="29px" altimg-valign="-7px" altimg-width="72px" alttext="L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msubsup><mi href="./18.3#T1.t1.r12">L</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r12" stretchy="false">(</mo><mi class="ltx_nvar">α</mi><mo href="./18.3#T1.t1.r12" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Laguerre (or generalized Laguerre) polynomial</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./18.1#SS1.p2.t1.r1" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./18.1#SS1.p2.t1.r1">y</mi></math>: real variable</a>,
<a href="./18.1#SS1.p2.t1.r5" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m80.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E31" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">18.17.31</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_Math ltx_eqn_cell ltx_align_center"><math class="ltx_Math" alttext="\int_{0}^{\infty}e^{-ax}x^{\nu-2n}L^{(\nu-2n)}_{2n-1}\left(ax\right)\cos\left(%
xy\right)\mathrm{d}x=i\frac{(-1)^{n}\Gamma\left(\nu\right)}{2(2n-1)!}y^{2n-1}%
\left((a+iy)^{-\nu}-(a-iy)^{-\nu}\right)," display="block"><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mi>a</mi><mo></mo><mi href="./18.2#Px1.p2">x</mi></mrow></mrow></msup><mo></mo><msup><mi href="./18.2#Px1.p2">x</mi><mrow><mi>ν</mi><mo>-</mo><mrow><mn>2</mn><mo></mo><mi href="./18.1#SS1.p2.t1.r5">n</mi></mrow></mrow></msup><mo></mo><mrow><msubsup><mi href="./18.3#T1.t1.r12">L</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./18.1#SS1.p2.t1.r5">n</mi></mrow><mo>-</mo><mn>1</mn></mrow><mrow><mo href="./18.3#T1.t1.r12" stretchy="false">(</mo><mrow><mi>ν</mi><mo>-</mo><mrow><mn>2</mn><mo></mo><mi href="./18.1#SS1.p2.t1.r5">n</mi></mrow></mrow><mo href="./18.3#T1.t1.r12" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mrow><mi>a</mi><mo></mo><mi href="./18.2#Px1.p2">x</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./18.2#Px1.p2">x</mi><mo></mo><mi href="./18.1#SS1.p2.t1.r1">y</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./18.2#Px1.p2">x</mi></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>=</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mfrac><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi></msup><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi>ν</mi><mo>)</mo></mrow></mrow></mrow><mrow><mn>2</mn><mo></mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>2</mn><mo></mo><mi href="./18.1#SS1.p2.t1.r5">n</mi></mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mrow></mfrac><mo></mo><msup><mi href="./18.1#SS1.p2.t1.r1">y</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./18.1#SS1.p2.t1.r5">n</mi></mrow><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mrow><mo>(</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mi>a</mi><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./18.1#SS1.p2.t1.r1">y</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mi>ν</mi></mrow></msup><mo>-</mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi>a</mi><mo>-</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./18.1#SS1.p2.t1.r1">y</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mi>ν</mi></mrow></msup></mrow><mo>)</mo></mrow></mrow><mo>,</mo></mtd></mtr></mtable></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m67.png" altimg-height="18px" altimg-valign="-4px" altimg-width="98px" alttext="\nu>2n-1" display="inline"><mrow><mi>ν</mi><mo>></mo><mrow><mrow><mn>2</mn><mo></mo><mi href="./18.1#SS1.p2.t1.r5">n</mi></mrow><mo>-</mo><mn>1</mn></mrow></mrow></math>, <math class="ltx_Math" altimg="m74.png" altimg-height="17px" altimg-valign="-3px" altimg-width="51px" alttext="a>0" display="inline"><mrow><mi>a</mi><mo>></mo><mn>0</mn></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E31.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m35.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./18.3#T1.t1.r12" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m24.png" altimg-height="29px" altimg-valign="-7px" altimg-width="72px" alttext="L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msubsup><mi href="./18.3#T1.t1.r12">L</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r12" stretchy="false">(</mo><mi class="ltx_nvar">α</mi><mo href="./18.3#T1.t1.r12" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Laguerre (or generalized Laguerre) polynomial</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m13.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m79.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./18.1#SS1.p2.t1.r1" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./18.1#SS1.p2.t1.r1">y</mi></math>: real variable</a>,
<a href="./18.1#SS1.p2.t1.r5" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m80.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E32" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">18.17.32</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_Math ltx_eqn_cell ltx_align_center"><math class="ltx_Math" alttext="\int_{0}^{\infty}e^{-ax}x^{\nu-1-2n}L^{(\nu-1-2n)}_{2n}\left(ax\right)\cos%
\left(xy\right)\mathrm{d}x=\frac{(-1)^{n}\Gamma\left(\nu\right)}{2(2n)!}y^{2n}%
\left((a+iy)^{-\nu}+(a-iy)^{-\nu}\right)," display="block"><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mi>a</mi><mo></mo><mi href="./18.2#Px1.p2">x</mi></mrow></mrow></msup><mo></mo><msup><mi href="./18.2#Px1.p2">x</mi><mrow><mi>ν</mi><mo>-</mo><mn>1</mn><mo>-</mo><mrow><mn>2</mn><mo></mo><mi href="./18.1#SS1.p2.t1.r5">n</mi></mrow></mrow></msup><mo></mo><mrow><msubsup><mi href="./18.3#T1.t1.r12">L</mi><mrow><mn>2</mn><mo></mo><mi href="./18.1#SS1.p2.t1.r5">n</mi></mrow><mrow><mo href="./18.3#T1.t1.r12" stretchy="false">(</mo><mrow><mi>ν</mi><mo>-</mo><mn>1</mn><mo>-</mo><mrow><mn>2</mn><mo></mo><mi href="./18.1#SS1.p2.t1.r5">n</mi></mrow></mrow><mo href="./18.3#T1.t1.r12" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mrow><mi>a</mi><mo></mo><mi href="./18.2#Px1.p2">x</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./18.2#Px1.p2">x</mi><mo></mo><mi href="./18.1#SS1.p2.t1.r1">y</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./18.2#Px1.p2">x</mi></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>=</mo><mrow><mfrac><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi></msup><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi>ν</mi><mo>)</mo></mrow></mrow></mrow><mrow><mn>2</mn><mo></mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mn>2</mn><mo></mo><mi href="./18.1#SS1.p2.t1.r5">n</mi></mrow><mo stretchy="false">)</mo></mrow><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mrow></mfrac><mo></mo><msup><mi href="./18.1#SS1.p2.t1.r1">y</mi><mrow><mn>2</mn><mo></mo><mi href="./18.1#SS1.p2.t1.r5">n</mi></mrow></msup><mo></mo><mrow><mo>(</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mi>a</mi><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./18.1#SS1.p2.t1.r1">y</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mi>ν</mi></mrow></msup><mo>+</mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi>a</mi><mo>-</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./18.1#SS1.p2.t1.r1">y</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mi>ν</mi></mrow></msup></mrow><mo>)</mo></mrow></mrow><mo>,</mo></mtd></mtr></mtable></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m68.png" altimg-height="17px" altimg-valign="-3px" altimg-width="64px" alttext="\nu>2n" display="inline"><mrow><mi>ν</mi><mo>></mo><mrow><mn>2</mn><mo></mo><mi href="./18.1#SS1.p2.t1.r5">n</mi></mrow></mrow></math>, <math class="ltx_Math" altimg="m74.png" altimg-height="17px" altimg-valign="-3px" altimg-width="51px" alttext="a>0" display="inline"><mrow><mi>a</mi><mo>></mo><mn>0</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E32.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m35.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./18.3#T1.t1.r12" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m24.png" altimg-height="29px" altimg-valign="-7px" altimg-width="72px" alttext="L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msubsup><mi href="./18.3#T1.t1.r12">L</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r12" stretchy="false">(</mo><mi class="ltx_nvar">α</mi><mo href="./18.3#T1.t1.r12" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Laguerre (or generalized Laguerre) polynomial</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m13.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m79.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./18.1#SS1.p2.t1.r1" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./18.1#SS1.p2.t1.r1">y</mi></math>: real variable</a>,
<a href="./18.1#SS1.p2.t1.r5" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m80.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
</section>
<section id="vi" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§18.17(vi) </span>Laplace Transforms</h2>
<div id="SS6.info" class="ltx_metadata ltx_info">
) expand the exponential in the integral as a
power series in <math class="ltx_Math" altimg="m88.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./18.1#SS1.p2.t1.r2">z</mi></math> and interchange integration and summation. The
resulting integral can be evaluated by considering the term <math class="ltx_Math" altimg="m48.png" altimg-height="18px" altimg-valign="-2px" altimg-width="49px" alttext="\ell=0" display="inline"><mrow><mi href="./18.1#SS1.p2.t1.r4" mathvariant="normal">ℓ</mi><mo>=</mo><mn>0</mn></mrow></math> in
(</dd>
</dl>
</div>
</div>
<div id="Px16.p1" class="ltx_para">
<table id="E33" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">18.17.33</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_Math ltx_eqn_cell ltx_align_center"><math class="ltx_Math" alttext="\int_{-1}^{1}e^{-(x+1)z}P^{(\alpha,\beta)}_{n}\left(x\right)(1-x)^{\alpha}(1+x%
)^{\beta}\mathrm{d}x=\frac{(-1)^{n}2^{\alpha+\beta+n+1}\Gamma\left(\alpha+n+1%
\right)\Gamma\left(\beta+n+1\right)}{\Gamma\left(\alpha+\beta+2n+2\right)n!}z^%
{n}{{}_{1}F_{1}}\left({\beta+n+1\atop\alpha+\beta+2n+2};-2z\right)," display="block"><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><mo>-</mo><mn>1</mn></mrow><mn>1</mn></msubsup><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./18.2#Px1.p2">x</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi href="./18.1#SS1.p2.t1.r2">z</mi></mrow></mrow></msup><mo></mo><mrow><msubsup><mi href="./18.3#T1.t1.r2">P</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r2" stretchy="false">(</mo><mi>α</mi><mo href="./18.3#T1.t1.r2">,</mo><mi>β</mi><mo href="./18.3#T1.t1.r2" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><mi href="./18.2#Px1.p2">x</mi></mrow><mo stretchy="false">)</mo></mrow><mi>α</mi></msup><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>+</mo><mi href="./18.2#Px1.p2">x</mi></mrow><mo stretchy="false">)</mo></mrow><mi>β</mi></msup><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./18.2#Px1.p2">x</mi></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>=</mo><mrow><mfrac><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi></msup><mo></mo><msup><mn>2</mn><mrow><mi>α</mi><mo>+</mo><mi>β</mi><mo>+</mo><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi>α</mi><mo>+</mo><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi>β</mi><mo>+</mo><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mrow><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi>α</mi><mo>+</mo><mi>β</mi><mo>+</mo><mrow><mn>2</mn><mo></mo><mi href="./18.1#SS1.p2.t1.r5">n</mi></mrow><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mrow></mfrac><mo></mo><msup><mi href="./18.1#SS1.p2.t1.r2">z</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi></msup><mo></mo><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>1</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mi>β</mi><mo>+</mo><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>+</mo><mn>1</mn></mrow><mrow><mi>α</mi><mo>+</mo><mi>β</mi><mo>+</mo><mrow><mn>2</mn><mo></mo><mi href="./18.1#SS1.p2.t1.r5">n</mi></mrow><mo>+</mo><mn>2</mn></mrow></mfrac><mo>;</mo><mrow><mo>-</mo><mrow><mn>2</mn><mo></mo><mi href="./18.1#SS1.p2.t1.r2">z</mi></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mtd></mtr></mtable></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m89.png" altimg-height="18px" altimg-valign="-3px" altimg-width="53px" alttext="z\in\mathbb{C}" display="inline"><mrow><mi href="./18.1#SS1.p2.t1.r2">z</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mi href="./front/introduction#Sx4.p1.t1.r1">ℂ</mi></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E33.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m35.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./18.3#T1.t1.r2" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m27.png" altimg-height="29px" altimg-valign="-7px" altimg-width="88px" alttext="P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msubsup><mi href="./18.3#T1.t1.r2">P</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r2" stretchy="false">(</mo><mi class="ltx_nvar">α</mi><mo href="./18.3#T1.t1.r2">,</mo><mi class="ltx_nvar">β</mi><mo href="./18.3#T1.t1.r2" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Jacobi polynomial</a>,
<a href="./16.2" title="§16.2 Definition and Analytic Properties ‣ Generalized Hypergeometric Functions ‣ Chapter 16 Generalized Hypergeometric Functions and Meijer G -Function" class="ltx_ref"><math class="ltx_Math" altimg="m92.png" altimg-height="23px" altimg-valign="-7px" altimg-width="101px" alttext="{{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)" display="inline"><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>1</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./16.1#p2.t1.r5">a</mi><mo>;</mo><mi class="ltx_nvar" href="./16.1#p2.t1.r5">b</mi><mo>;</mo><mi class="ltx_nvar" href="./16.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: <math class="ltx_Math" altimg="m19.png" altimg-height="23px" altimg-valign="-7px" altimg-width="113px" alttext="=M\left(\NVar{a},\NVar{b},\NVar{z}\right)" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mi href="./13.2#E2">M</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./16.1#p2.t1.r5">a</mi><mo>,</mo><mi class="ltx_nvar" href="./16.1#p2.t1.r5">b</mi><mo>,</mo><mi class="ltx_nvar" href="./16.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></math> notation for the Kummer confluent hypergeometric function</a>,
<a href="./front/introduction#Sx4.p1.t1.r1" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m55.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\mathbb{C}" display="inline"><mi href="./front/introduction#Sx4.p1.t1.r1">ℂ</mi></math>: complex plane</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./front/introduction#Sx4.p1.t1.r9" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="15px" altimg-valign="-3px" altimg-width="18px" alttext="\in" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo></math>: element of</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m13.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m79.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./18.1#SS1.p2.t1.r2" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m88.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./18.1#SS1.p2.t1.r2">z</mi></math>: complex variable</a>,
<a href="./18.1#SS1.p2.t1.r5" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m80.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">For the confluent hypergeometric function <math class="ltx_Math" altimg="m91.png" altimg-height="21px" altimg-valign="-5px" altimg-width="35px" alttext="{{}_{1}F_{1}}" display="inline"><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>1</mn><none></none></mmultiscripts></math> see (</dd>
</dl>
</div>
</div>
<div id="Px17.p1" class="ltx_para">
<table id="E34" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">18.17.34</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E34.png" altimg-height="53px" altimg-valign="-21px" altimg-width="410px" alttext="\int_{0}^{\infty}e^{-xz}L^{(\alpha)}_{n}\left(x\right)e^{-x}x^{\alpha}\mathrm{%
d}x=\frac{\Gamma\left(\alpha+n+1\right)z^{n}}{n!(z+1)^{\alpha+n+1}}," display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mi href="./18.2#Px1.p2">x</mi><mo></mo><mi href="./18.1#SS1.p2.t1.r2">z</mi></mrow></mrow></msup><mo></mo><mrow><msubsup><mi href="./18.3#T1.t1.r12">L</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r12" stretchy="false">(</mo><mi>α</mi><mo href="./18.3#T1.t1.r12" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mi href="./18.2#Px1.p2">x</mi></mrow></msup><mo></mo><msup><mi href="./18.2#Px1.p2">x</mi><mi>α</mi></msup><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./18.2#Px1.p2">x</mi></mrow></mrow></mrow><mo>=</mo><mfrac><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi>α</mi><mo>+</mo><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mo></mo><msup><mi href="./18.1#SS1.p2.t1.r2">z</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi></msup></mrow><mrow><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./18.1#SS1.p2.t1.r2">z</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mrow><mi>α</mi><mo>+</mo><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow></mfrac></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m37.png" altimg-height="19px" altimg-valign="-4px" altimg-width="81px" alttext="\Re z>-1" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./18.1#SS1.p2.t1.r2">z</mi></mrow><mo>></mo><mrow><mo>-</mo><mn>1</mn></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E34.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m35.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./18.3#T1.t1.r12" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m24.png" altimg-height="29px" altimg-valign="-7px" altimg-width="72px" alttext="L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msubsup><mi href="./18.3#T1.t1.r12">L</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r12" stretchy="false">(</mo><mi class="ltx_nvar">α</mi><mo href="./18.3#T1.t1.r12" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Laguerre (or generalized Laguerre) polynomial</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m13.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m79.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./18.1#SS1.p2.t1.r2" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m88.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./18.1#SS1.p2.t1.r2">z</mi></math>: complex variable</a>,
<a href="./18.1#SS1.p2.t1.r5" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m80.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px18" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Hermite</h3>
<div id="Px18.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px18.p1" class="ltx_para">
<table id="E35" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">18.17.35</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E35.png" altimg-height="54px" altimg-valign="-22px" altimg-width="359px" alttext="\int_{-\infty}^{\infty}e^{-xz}H_{n}\left(x\right)e^{-x^{2}}\mathrm{d}x=\pi^{%
\frac{1}{2}}(-z)^{n}e^{\frac{1}{4}z^{2}}," display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mi mathvariant="normal">∞</mi></msubsup><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mi href="./18.2#Px1.p2">x</mi><mo></mo><mi href="./18.1#SS1.p2.t1.r2">z</mi></mrow></mrow></msup><mo></mo><mrow><msub><mi href="./18.3#T1.t1.r13">H</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>2</mn></msup></mrow></msup><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./18.2#Px1.p2">x</mi></mrow></mrow></mrow><mo>=</mo><mrow><msup><mi href="./3.12#E1">π</mi><mfrac><mn>1</mn><mn>2</mn></mfrac></msup><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mi href="./18.1#SS1.p2.t1.r2">z</mi></mrow><mo stretchy="false">)</mo></mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi></msup><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac><mo></mo><msup><mi href="./18.1#SS1.p2.t1.r2">z</mi><mn>2</mn></msup></mrow></msup></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m89.png" altimg-height="18px" altimg-valign="-3px" altimg-width="53px" alttext="z\in\mathbb{C}" display="inline"><mrow><mi href="./18.1#SS1.p2.t1.r2">z</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mi href="./front/introduction#Sx4.p1.t1.r1">ℂ</mi></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E35.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./18.3#T1.t1.r13" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m21.png" altimg-height="23px" altimg-valign="-7px" altimg-width="62px" alttext="H_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r13">H</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Hermite polynomial</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./front/introduction#Sx4.p1.t1.r1" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m55.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\mathbb{C}" display="inline"><mi href="./front/introduction#Sx4.p1.t1.r1">ℂ</mi></math>: complex plane</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./front/introduction#Sx4.p1.t1.r9" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m50.png" altimg-height="15px" altimg-valign="-3px" altimg-width="18px" alttext="\in" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo></math>: element of</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./18.1#SS1.p2.t1.r2" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m88.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./18.1#SS1.p2.t1.r2">z</mi></math>: complex variable</a>,
<a href="./18.1#SS1.p2.t1.r5" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m80.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
</section>
<section id="vii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§18.17(vii) </span>Mellin Transforms</h2>
<div id="SS7.info" class="ltx_metadata ltx_info">
) and integrate by
parts <math class="ltx_Math" altimg="m80.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math> times.
For (), with <math class="ltx_Math" altimg="m52.png" altimg-height="27px" altimg-valign="-9px" altimg-width="55px" alttext="\lambda=\frac{1}{2}" display="inline"><mrow><mi>λ</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math> and <math class="ltx_Math" altimg="m80.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math> replaced
by <math class="ltx_Math" altimg="m17.png" altimg-height="17px" altimg-valign="-2px" altimg-width="26px" alttext="2n" display="inline"><mrow><mn>2</mn><mo></mo><mi href="./18.1#SS1.p2.t1.r5">n</mi></mrow></math>, integrate term by term, then apply (</dd>
</dl>
</div>
</div>
<div id="Px19.p1" class="ltx_para">
<table id="E36" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">18.17.36</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E36.png" altimg-height="57px" altimg-valign="-22px" altimg-width="659px" alttext="\int_{-1}^{1}(1-x)^{z-1}(1+x)^{\beta}P^{(\alpha,\beta)}_{n}\left(x\right)%
\mathrm{d}x=\frac{2^{\beta+z}\Gamma\left(z\right)\Gamma\left(1+\beta+n\right){%
\left(1+\alpha-z\right)_{n}}}{n!\Gamma\left(1+\beta+z+n\right)}," display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><mo>-</mo><mn>1</mn></mrow><mn>1</mn></msubsup><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><mi href="./18.2#Px1.p2">x</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mi href="./18.1#SS1.p2.t1.r2">z</mi><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>+</mo><mi href="./18.2#Px1.p2">x</mi></mrow><mo stretchy="false">)</mo></mrow><mi>β</mi></msup><mo></mo><mrow><msubsup><mi href="./18.3#T1.t1.r2">P</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r2" stretchy="false">(</mo><mi>α</mi><mo href="./18.3#T1.t1.r2">,</mo><mi>β</mi><mo href="./18.3#T1.t1.r2" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./18.2#Px1.p2">x</mi></mrow></mrow></mrow><mo>=</mo><mfrac><mrow><msup><mn>2</mn><mrow><mi>β</mi><mo>+</mo><mi href="./18.1#SS1.p2.t1.r2">z</mi></mrow></msup><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi href="./18.1#SS1.p2.t1.r2">z</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mi>β</mi><mo>+</mo><mi href="./18.1#SS1.p2.t1.r5">n</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><msub><mrow><mo href="./5.2#iii">(</mo><mrow><mrow><mn>1</mn><mo>+</mo><mi>α</mi></mrow><mo>-</mo><mi href="./18.1#SS1.p2.t1.r2">z</mi></mrow><mo href="./5.2#iii">)</mo></mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi></msub></mrow><mrow><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mi>β</mi><mo>+</mo><mi href="./18.1#SS1.p2.t1.r2">z</mi><mo>+</mo><mi href="./18.1#SS1.p2.t1.r5">n</mi></mrow><mo>)</mo></mrow></mrow></mrow></mfrac></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m38.png" altimg-height="18px" altimg-valign="-3px" altimg-width="65px" alttext="\Re z>0" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./18.1#SS1.p2.t1.r2">z</mi></mrow><mo>></mo><mn>0</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E36.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m35.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./18.3#T1.t1.r2" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m27.png" altimg-height="29px" altimg-valign="-7px" altimg-width="88px" alttext="P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msubsup><mi href="./18.3#T1.t1.r2">P</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r2" stretchy="false">(</mo><mi class="ltx_nvar">α</mi><mo href="./18.3#T1.t1.r2">,</mo><mi class="ltx_nvar">β</mi><mo href="./18.3#T1.t1.r2" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Jacobi polynomial</a>,
<a href="./5.2#iii" title="§5.2(iii) Pochhammer’s Symbol ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m90.png" altimg-height="24px" altimg-valign="-8px" altimg-width="41px" alttext="{\left(\NVar{a}\right)_{\NVar{n}}}" display="inline"><msub><mrow><mo href="./5.2#iii">(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r5">a</mi><mo href="./5.2#iii">)</mo></mrow><mi class="ltx_nvar" href="./5.1#p2.t1.r1">n</mi></msub></math>: Pochhammer’s symbol (or shifted factorial)</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m13.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m79.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./18.1#SS1.p2.t1.r2" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m88.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./18.1#SS1.p2.t1.r2">z</mi></math>: complex variable</a>,
<a href="./18.1#SS1.p2.t1.r5" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m80.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px20" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Ultraspherical</h3>
<div id="Px20.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px20.p1" class="ltx_para">
<table id="E37" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">18.17.37</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E37.png" altimg-height="58px" altimg-valign="-24px" altimg-width="701px" alttext="\int_{0}^{1}(1-x^{2})^{\lambda-\frac{1}{2}}C^{(\lambda)}_{n}\left(x\right)x^{z%
-1}\mathrm{d}x=\frac{\pi\,2^{1-2\lambda-z}\Gamma\left(n+2\lambda\right)\Gamma%
\left(z\right)}{n!\Gamma\left(\lambda\right)\Gamma\left(\frac{1}{2}+\frac{1}{2%
}n+\lambda+\frac{1}{2}z\right)\Gamma\left(\frac{1}{2}+\frac{1}{2}z-\frac{1}{2}%
n\right)}," display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mn>1</mn></msubsup><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><msup><mi href="./18.2#Px1.p2">x</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mrow><mi>λ</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup><mo></mo><mrow><msubsup><mi href="./18.3#T1.t1.r3">C</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r3" stretchy="false">(</mo><mi>λ</mi><mo href="./18.3#T1.t1.r3" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow><mo></mo><msup><mi href="./18.2#Px1.p2">x</mi><mrow><mi href="./18.1#SS1.p2.t1.r2">z</mi><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./18.2#Px1.p2">x</mi></mrow></mrow></mrow><mo>=</mo><mfrac><mrow><mi href="./3.12#E1">π</mi><mo></mo><msup><mn> 2</mn><mrow><mn>1</mn><mo>-</mo><mrow><mn>2</mn><mo></mo><mi>λ</mi></mrow><mo>-</mo><mi href="./18.1#SS1.p2.t1.r2">z</mi></mrow></msup><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>+</mo><mrow><mn>2</mn><mo></mo><mi>λ</mi></mrow></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi href="./18.1#SS1.p2.t1.r2">z</mi><mo>)</mo></mrow></mrow></mrow><mrow><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi>λ</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>+</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./18.1#SS1.p2.t1.r5">n</mi></mrow><mo>+</mo><mi>λ</mi><mo>+</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./18.1#SS1.p2.t1.r2">z</mi></mrow></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>+</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./18.1#SS1.p2.t1.r2">z</mi></mrow></mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./18.1#SS1.p2.t1.r5">n</mi></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mfrac></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m38.png" altimg-height="18px" altimg-valign="-3px" altimg-width="65px" alttext="\Re z>0" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./18.1#SS1.p2.t1.r2">z</mi></mrow><mo>></mo><mn>0</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E37.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m35.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m13.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m79.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./18.3#T1.t1.r3" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m20.png" altimg-height="29px" altimg-valign="-7px" altimg-width="73px" alttext="C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msubsup><mi href="./18.3#T1.t1.r3">C</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r3" stretchy="false">(</mo><mi class="ltx_nvar">λ</mi><mo href="./18.3#T1.t1.r3" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: ultraspherical (or Gegenbauer) polynomial</a>,
<a href="./18.1#SS1.p2.t1.r2" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m88.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./18.1#SS1.p2.t1.r2">z</mi></math>: complex variable</a>,
<a href="./18.1#SS1.p2.t1.r5" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m80.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px21" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Legendre</h3>
<div id="Px21.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px21.p1" class="ltx_para">
<table id="E38" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">18.17.38</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E38.png" altimg-height="63px" altimg-valign="-26px" altimg-width="344px" alttext="\int_{0}^{1}P_{2n}\left(x\right)x^{z-1}\mathrm{d}x=\frac{(-1)^{n}{\left(\frac{%
1}{2}-\frac{1}{2}z\right)_{n}}}{2{\left(\frac{1}{2}z\right)_{n+1}}}," display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mn>1</mn></msubsup><mrow><mrow><msub><mi href="./18.3#T1.t1.r10">P</mi><mrow><mn>2</mn><mo></mo><mi href="./18.1#SS1.p2.t1.r5">n</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow><mo></mo><msup><mi href="./18.2#Px1.p2">x</mi><mrow><mi href="./18.1#SS1.p2.t1.r2">z</mi><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./18.2#Px1.p2">x</mi></mrow></mrow></mrow><mo>=</mo><mfrac><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi></msup><mo></mo><msub><mrow><mo href="./5.2#iii">(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./18.1#SS1.p2.t1.r2">z</mi></mrow></mrow><mo href="./5.2#iii">)</mo></mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi></msub></mrow><mrow><mn>2</mn><mo></mo><msub><mrow><mo href="./5.2#iii">(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./18.1#SS1.p2.t1.r2">z</mi></mrow><mo href="./5.2#iii">)</mo></mrow><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></mfrac></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m38.png" altimg-height="18px" altimg-valign="-3px" altimg-width="65px" alttext="\Re z>0" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./18.1#SS1.p2.t1.r2">z</mi></mrow><mo>></mo><mn>0</mn></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E38.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./18.3#T1.t1.r10" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m28.png" altimg-height="23px" altimg-valign="-7px" altimg-width="58px" alttext="P_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r10">P</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Legendre polynomial</a>,
<a href="./5.2#iii" title="§5.2(iii) Pochhammer’s Symbol ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m90.png" altimg-height="24px" altimg-valign="-8px" altimg-width="41px" alttext="{\left(\NVar{a}\right)_{\NVar{n}}}" display="inline"><msub><mrow><mo href="./5.2#iii">(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r5">a</mi><mo href="./5.2#iii">)</mo></mrow><mi class="ltx_nvar" href="./5.1#p2.t1.r1">n</mi></msub></math>: Pochhammer’s symbol (or shifted factorial)</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./18.1#SS1.p2.t1.r2" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m88.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./18.1#SS1.p2.t1.r2">z</mi></math>: complex variable</a>,
<a href="./18.1#SS1.p2.t1.r5" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m80.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E39" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">18.17.39</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E39.png" altimg-height="63px" altimg-valign="-26px" altimg-width="362px" alttext="\int_{0}^{1}P_{2n+1}\left(x\right)x^{z-1}\mathrm{d}x=\frac{(-1)^{n}{\left(1-%
\frac{1}{2}z\right)_{n}}}{2{\left(\frac{1}{2}+\frac{1}{2}z\right)_{n+1}}}," display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mn>1</mn></msubsup><mrow><mrow><msub><mi href="./18.3#T1.t1.r10">P</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./18.1#SS1.p2.t1.r5">n</mi></mrow><mo>+</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow><mo></mo><msup><mi href="./18.2#Px1.p2">x</mi><mrow><mi href="./18.1#SS1.p2.t1.r2">z</mi><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./18.2#Px1.p2">x</mi></mrow></mrow></mrow><mo>=</mo><mfrac><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi></msup><mo></mo><msub><mrow><mo href="./5.2#iii">(</mo><mrow><mn>1</mn><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./18.1#SS1.p2.t1.r2">z</mi></mrow></mrow><mo href="./5.2#iii">)</mo></mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi></msub></mrow><mrow><mn>2</mn><mo></mo><msub><mrow><mo href="./5.2#iii">(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>+</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./18.1#SS1.p2.t1.r2">z</mi></mrow></mrow><mo href="./5.2#iii">)</mo></mrow><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></mfrac></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m37.png" altimg-height="19px" altimg-valign="-4px" altimg-width="81px" alttext="\Re z>-1" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./18.1#SS1.p2.t1.r2">z</mi></mrow><mo>></mo><mrow><mo>-</mo><mn>1</mn></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E39.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./18.3#T1.t1.r10" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m28.png" altimg-height="23px" altimg-valign="-7px" altimg-width="58px" alttext="P_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r10">P</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Legendre polynomial</a>,
<a href="./5.2#iii" title="§5.2(iii) Pochhammer’s Symbol ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m90.png" altimg-height="24px" altimg-valign="-8px" altimg-width="41px" alttext="{\left(\NVar{a}\right)_{\NVar{n}}}" display="inline"><msub><mrow><mo href="./5.2#iii">(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r5">a</mi><mo href="./5.2#iii">)</mo></mrow><mi class="ltx_nvar" href="./5.1#p2.t1.r1">n</mi></msub></math>: Pochhammer’s symbol (or shifted factorial)</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./18.1#SS1.p2.t1.r2" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m88.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./18.1#SS1.p2.t1.r2">z</mi></math>: complex variable</a>,
<a href="./18.1#SS1.p2.t1.r5" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m80.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px22" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Laguerre</h3>
<div id="Px22.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px22.p1" class="ltx_para">
<table id="E40" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">18.17.40</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E40.png" altimg-height="53px" altimg-valign="-21px" altimg-width="721px" alttext="\int_{0}^{\infty}e^{-ax}L^{(\alpha)}_{n}\left(bx\right)x^{z-1}\mathrm{d}x=%
\frac{\Gamma\left(z+n\right)}{n!}\*{(a-b)^{n}}a^{-n-z}\*{{}_{2}F_{1}}\left({-n%
,1+\alpha-z\atop 1-n-z};\frac{a}{a-b}\right)," display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mi>a</mi><mo></mo><mi href="./18.2#Px1.p2">x</mi></mrow></mrow></msup><mo></mo><mrow><msubsup><mi href="./18.3#T1.t1.r12">L</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r12" stretchy="false">(</mo><mi>α</mi><mo href="./18.3#T1.t1.r12" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mrow><mi>b</mi><mo></mo><mi href="./18.2#Px1.p2">x</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><msup><mi href="./18.2#Px1.p2">x</mi><mrow><mi href="./18.1#SS1.p2.t1.r2">z</mi><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./18.2#Px1.p2">x</mi></mrow></mrow></mrow><mo>=</mo><mrow><mfrac><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./18.1#SS1.p2.t1.r2">z</mi><mo>+</mo><mi href="./18.1#SS1.p2.t1.r5">n</mi></mrow><mo>)</mo></mrow></mrow><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mfrac><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi>a</mi><mo>-</mo><mi>b</mi></mrow><mo stretchy="false">)</mo></mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi></msup><mo></mo><msup><mi>a</mi><mrow><mrow><mo>-</mo><mi href="./18.1#SS1.p2.t1.r5">n</mi></mrow><mo>-</mo><mi href="./18.1#SS1.p2.t1.r2">z</mi></mrow></msup><mo></mo><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mrow><mo>-</mo><mi href="./18.1#SS1.p2.t1.r5">n</mi></mrow><mo>,</mo><mrow><mrow><mn>1</mn><mo>+</mo><mi>α</mi></mrow><mo>-</mo><mi href="./18.1#SS1.p2.t1.r2">z</mi></mrow></mrow><mrow><mn>1</mn><mo>-</mo><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>-</mo><mi href="./18.1#SS1.p2.t1.r2">z</mi></mrow></mfrac><mo>;</mo><mfrac><mi>a</mi><mrow><mi>a</mi><mo>-</mo><mi>b</mi></mrow></mfrac><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m36.png" altimg-height="18px" altimg-valign="-3px" altimg-width="66px" alttext="\Re a>0" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>a</mi></mrow><mo>></mo><mn>0</mn></mrow></math>, <math class="ltx_Math" altimg="m38.png" altimg-height="18px" altimg-valign="-3px" altimg-width="65px" alttext="\Re z>0" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./18.1#SS1.p2.t1.r2">z</mi></mrow><mo>></mo><mn>0</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E40.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m35.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./16.2" title="§16.2 Definition and Analytic Properties ‣ Generalized Hypergeometric Functions ‣ Chapter 16 Generalized Hypergeometric Functions and Meijer G -Function" class="ltx_ref"><math class="ltx_Math" altimg="m94.png" altimg-height="23px" altimg-valign="-7px" altimg-width="118px" alttext="{{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mrow><mi class="ltx_nvar" href="./16.1#p2.t1.r5">a</mi><mo>,</mo><mi class="ltx_nvar" href="./16.1#p2.t1.r5">b</mi></mrow><mo>;</mo><mi class="ltx_nvar">c</mi><mo>;</mo><mi class="ltx_nvar" href="./16.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: <math class="ltx_Math" altimg="m18.png" altimg-height="23px" altimg-valign="-7px" altimg-width="124px" alttext="=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mi href="./15.2#E1">F</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./16.1#p2.t1.r5">a</mi><mo>,</mo><mi class="ltx_nvar" href="./16.1#p2.t1.r5">b</mi><mo>;</mo><mi class="ltx_nvar">c</mi><mo>;</mo><mi class="ltx_nvar" href="./16.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></mrow></math> notation for Gauss’ hypergeometric function</a>,
<a href="./18.3#T1.t1.r12" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m24.png" altimg-height="29px" altimg-valign="-7px" altimg-width="72px" alttext="L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msubsup><mi href="./18.3#T1.t1.r12">L</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi><mrow><mo href="./18.3#T1.t1.r12" stretchy="false">(</mo><mi class="ltx_nvar">α</mi><mo href="./18.3#T1.t1.r12" stretchy="false">)</mo></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Laguerre (or generalized Laguerre) polynomial</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m13.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m79.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./18.1#SS1.p2.t1.r2" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m88.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./18.1#SS1.p2.t1.r2">z</mi></math>: complex variable</a>,
<a href="./18.1#SS1.p2.t1.r5" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m80.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">For the hypergeometric function <math class="ltx_Math" altimg="m93.png" altimg-height="21px" altimg-valign="-5px" altimg-width="35px" alttext="{{}_{2}F_{1}}" display="inline"><mmultiscripts><mi href="./16.2">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts></math> see §§</dd>
</dl>
</div>
</div>
<div id="Px23.p1" class="ltx_para">
<table id="E41" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">18.17.41</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E41.png" altimg-height="65px" altimg-valign="-27px" altimg-width="738px" alttext="\int_{0}^{\infty}e^{-ax}\mathit{He}_{n}\left(x\right)x^{z-1}\mathrm{d}x=\Gamma%
\left(z+n\right)a^{-n-2}{{}_{2}F_{2}}\left({-\tfrac{1}{2}n,-\tfrac{1}{2}n+%
\tfrac{1}{2}\atop-\tfrac{1}{2}z-\tfrac{1}{2}n,-\tfrac{1}{2}z-\tfrac{1}{2}n+%
\tfrac{1}{2}};-\tfrac{1}{2}a^{2}\right)," display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mo>-</mo><mrow><mi>a</mi><mo></mo><mi href="./18.2#Px1.p2">x</mi></mrow></mrow></msup><mo></mo><mrow><msub><mi href="./18.3#T1.t1.r14" mathvariant="italic">He</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow><mo></mo><msup><mi href="./18.2#Px1.p2">x</mi><mrow><mi href="./18.1#SS1.p2.t1.r2">z</mi><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./18.2#Px1.p2">x</mi></mrow></mrow></mrow><mo>=</mo><mrow><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mrow><mi href="./18.1#SS1.p2.t1.r2">z</mi><mo>+</mo><mi href="./18.1#SS1.p2.t1.r5">n</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><msup><mi>a</mi><mrow><mrow><mo>-</mo><mi href="./18.1#SS1.p2.t1.r5">n</mi></mrow><mo>-</mo><mn>2</mn></mrow></msup><mo></mo><mrow><mmultiscripts><mi href="./16.2">F</mi><mn>2</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./18.1#SS1.p2.t1.r5">n</mi></mrow></mrow><mo>,</mo><mrow><mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./18.1#SS1.p2.t1.r5">n</mi></mrow></mrow><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow><mrow><mrow><mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./18.1#SS1.p2.t1.r2">z</mi></mrow></mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./18.1#SS1.p2.t1.r5">n</mi></mrow></mrow><mo>,</mo><mrow><mrow><mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./18.1#SS1.p2.t1.r2">z</mi></mrow></mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi href="./18.1#SS1.p2.t1.r5">n</mi></mrow></mrow><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow></mfrac><mo>;</mo><mrow><mo>-</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><msup><mi>a</mi><mn>2</mn></msup></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m36.png" altimg-height="18px" altimg-valign="-3px" altimg-width="66px" alttext="\Re a>0" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi>a</mi></mrow><mo>></mo><mn>0</mn></mrow></math>. Also,
<math class="ltx_Math" altimg="m38.png" altimg-height="18px" altimg-valign="-3px" altimg-width="65px" alttext="\Re z>0" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./18.1#SS1.p2.t1.r2">z</mi></mrow><mo>></mo><mn>0</mn></mrow></math>, <math class="ltx_Math" altimg="m80.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math> even;
<math class="ltx_Math" altimg="m37.png" altimg-height="19px" altimg-valign="-4px" altimg-width="81px" alttext="\Re z>-1" display="inline"><mrow><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./18.1#SS1.p2.t1.r2">z</mi></mrow><mo>></mo><mrow><mo>-</mo><mn>1</mn></mrow></mrow></math>, <math class="ltx_Math" altimg="m80.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math> odd.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E41.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m35.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href="./5.2#E1" mathvariant="normal">Γ</mi><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>,
<a href="./16.2" title="§16.2 Definition and Analytic Properties ‣ Generalized Hypergeometric Functions ‣ Chapter 16 Generalized Hypergeometric Functions and Meijer G -Function" class="ltx_ref"><math class="ltx_Math" altimg="m97.png" altimg-height="24px" altimg-valign="-8px" altimg-width="244px" alttext="{{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_%
{q}};\NVar{z}\right)" display="inline"><mrow><mmultiscripts><mi href="./16.2">F</mi><mi class="ltx_nvar" href="./16.1#p2.t1.r1">q</mi><none></none><mprescripts></mprescripts><mi class="ltx_nvar" href="./16.1#p2.t1.r1">p</mi><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mrow><msub><mi class="ltx_nvar" href="./16.1#p2.t1.r5">a</mi><mn class="ltx_nvar">1</mn></msub><mo class="ltx_nvar">,</mo><mi class="ltx_nvar" mathvariant="normal">…</mi><mo class="ltx_nvar">,</mo><msub><mi class="ltx_nvar" href="./16.1#p2.t1.r5">a</mi><mi class="ltx_nvar" href="./16.1#p2.t1.r1">p</mi></msub></mrow><mo>;</mo><mrow><msub><mi class="ltx_nvar" href="./16.1#p2.t1.r5">b</mi><mn class="ltx_nvar">1</mn></msub><mo class="ltx_nvar">,</mo><mi class="ltx_nvar" mathvariant="normal">…</mi><mo class="ltx_nvar">,</mo><msub><mi class="ltx_nvar" href="./16.1#p2.t1.r5">b</mi><mi class="ltx_nvar" href="./16.1#p2.t1.r1">q</mi></msub></mrow><mo>;</mo><mi class="ltx_nvar" href="./16.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>
or <math class="ltx_Math" altimg="m99.png" altimg-height="39px" altimg-valign="-15px" altimg-width="143px" alttext="{{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},%
\dots,b_{q}}};\NVar{z}\right)" display="inline"><mrow><mmultiscripts><mi href="./16.2">F</mi><mi class="ltx_nvar" href="./16.1#p2.t1.r1">q</mi><none></none><mprescripts></mprescripts><mi class="ltx_nvar" href="./16.1#p2.t1.r1">p</mi><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mrow><msub><mi class="ltx_nvar" href="./16.1#p2.t1.r5">a</mi><mn class="ltx_nvar">1</mn></msub><mo class="ltx_nvar">,</mo><mi class="ltx_nvar" mathvariant="normal">…</mi><mo class="ltx_nvar">,</mo><msub><mi class="ltx_nvar" href="./16.1#p2.t1.r5">a</mi><mi class="ltx_nvar" href="./16.1#p2.t1.r1">p</mi></msub></mrow><mrow><msub><mi class="ltx_nvar" href="./16.1#p2.t1.r5">b</mi><mn class="ltx_nvar">1</mn></msub><mo class="ltx_nvar">,</mo><mi class="ltx_nvar" mathvariant="normal">…</mi><mo class="ltx_nvar">,</mo><msub><mi class="ltx_nvar" href="./16.1#p2.t1.r5">b</mi><mi class="ltx_nvar" href="./16.1#p2.t1.r1">q</mi></msub></mrow></mfrac><mo>;</mo><mi class="ltx_nvar" href="./16.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>: alternatively <math class="ltx_Math" altimg="m96.png" altimg-height="24px" altimg-valign="-8px" altimg-width="106px" alttext="{{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)" display="inline"><mrow><mmultiscripts><mi href="./16.2">F</mi><mi class="ltx_nvar" href="./16.1#p2.t1.r1">q</mi><none></none><mprescripts></mprescripts><mi class="ltx_nvar" href="./16.1#p2.t1.r1">p</mi><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" mathvariant="bold">a</mi><mo>;</mo><mi class="ltx_nvar" mathvariant="bold">b</mi><mo>;</mo><mi class="ltx_nvar" href="./16.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math> or
<math class="ltx_Math" altimg="m98.png" altimg-height="27px" altimg-valign="-9px" altimg-width="91px" alttext="{{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};%
\NVar{z}\right)" display="inline"><mrow><mmultiscripts><mi href="./16.2">F</mi><mi class="ltx_nvar" href="./16.1#p2.t1.r1">q</mi><none></none><mprescripts></mprescripts><mi class="ltx_nvar" href="./16.1#p2.t1.r1">p</mi><none></none></mmultiscripts><mo></mo><mrow><mo>(</mo><mfrac linethickness="0pt"><mi class="ltx_nvar" mathvariant="bold">a</mi><mi class="ltx_nvar" mathvariant="bold">b</mi></mfrac><mo>;</mo><mi class="ltx_nvar" href="./16.1#p2.t1.r4">z</mi><mo>)</mo></mrow></mrow></math>
<br class="ltx_break">generalized hypergeometric function</a>,
<a href="./18.3#T1.t1.r14" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="23px" altimg-valign="-7px" altimg-width="71px" alttext="\mathit{He}_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r14" mathvariant="italic">He</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Hermite polynomial</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m39.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a>,
<a href="./18.1#SS1.p2.t1.r2" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m88.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./18.1#SS1.p2.t1.r2">z</mi></math>: complex variable</a>,
<a href="./18.1#SS1.p2.t1.r5" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m80.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">For the generalized hypergeometric function <math class="ltx_Math" altimg="m95.png" altimg-height="21px" altimg-valign="-5px" altimg-width="35px" alttext="{{}_{2}F_{2}}" display="inline"><mmultiscripts><mi href="./16.2">F</mi><mn>2</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts></math> see
() follow from the
case <math class="ltx_Math" altimg="m43.png" altimg-height="27px" altimg-valign="-9px" altimg-width="111px" alttext="\alpha=\beta=\pm\frac{1}{2}" display="inline"><mrow><mi>α</mi><mo>=</mo><mi>β</mi><mo>=</mo><mrow><mo>±</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mrow></math> of
<cite class="ltx_cite ltx_citemacro_citet">Szegő () for
<math class="ltx_Math" altimg="m42.png" altimg-height="21px" altimg-valign="-6px" altimg-width="93px" alttext="\alpha=\beta=0" display="inline"><mrow><mi>α</mi><mo>=</mo><mi>β</mi><mo>=</mo><mn>0</mn></mrow></math>, <math class="ltx_Math" altimg="m64.png" altimg-height="27px" altimg-valign="-9px" altimg-width="56px" alttext="\mu=\frac{1}{2}" display="inline"><mrow><mi>μ</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math>, together with
(</dd>
</dl>
</div>
</div>
<div id="Px24.p1" class="ltx_para">
<table id="E42" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">18.17.42</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E42.png" altimg-height="59px" altimg-valign="-22px" altimg-width="338px" alttext="\pvint_{-1}^{1}T_{n}\left(y\right)\frac{(1-y^{2})^{-\frac{1}{2}}}{y-x}\mathrm{%
d}y=\pi U_{n-1}\left(x\right)," display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#E24" largeop="true" symmetric="true">⨍</mo><mrow><mo>-</mo><mn>1</mn></mrow><mn>1</mn></msubsup><mrow><mrow><msub><mi href="./18.3#T1.t1.r4">T</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./18.1#SS1.p2.t1.r1">y</mi><mo>)</mo></mrow></mrow><mo></mo><mfrac><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><msup><mi href="./18.1#SS1.p2.t1.r1">y</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup><mrow><mi href="./18.1#SS1.p2.t1.r1">y</mi><mo>-</mo><mi href="./18.2#Px1.p2">x</mi></mrow></mfrac><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./18.1#SS1.p2.t1.r1">y</mi></mrow></mrow></mrow><mo>=</mo><mrow><mi href="./3.12#E1">π</mi><mo></mo><mrow><msub><mi href="./18.3#T1.t1.r5">U</mi><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E42.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./18.3#T1.t1.r4" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m31.png" altimg-height="23px" altimg-valign="-7px" altimg-width="57px" alttext="T_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r4">T</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Chebyshev polynomial of the first kind</a>,
<a href="./18.3#T1.t1.r5" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m32.png" altimg-height="23px" altimg-valign="-7px" altimg-width="59px" alttext="U_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r5">U</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Chebyshev polynomial of the second kind</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#E24" title="(1.4.24) ‣ Cauchy Principal Values ‣ §1.4(v) Definite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="31px" altimg-valign="-9px" altimg-width="23px" alttext="\pvint_{\NVar{a}}^{\NVar{b}}" display="inline"><msubsup><mo href="./1.4#E24" largeop="true" symmetric="true">⨍</mo><mi class="ltx_nvar">a</mi><mi class="ltx_nvar">b</mi></msubsup></math>: Cauchy principal value</a>,
<a href="./18.1#SS1.p2.t1.r1" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./18.1#SS1.p2.t1.r1">y</mi></math>: real variable</a>,
<a href="./18.1#SS1.p2.t1.r5" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m80.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">22.13.3</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E43" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">18.17.43</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E43.png" altimg-height="59px" altimg-valign="-22px" altimg-width="341px" alttext="\pvint_{-1}^{1}U_{n-1}\left(y\right)\frac{(1-y^{2})^{\frac{1}{2}}}{y-x}\mathrm%
{d}y=-\pi T_{n}\left(x\right)." display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#E24" largeop="true" symmetric="true">⨍</mo><mrow><mo>-</mo><mn>1</mn></mrow><mn>1</mn></msubsup><mrow><mrow><msub><mi href="./18.3#T1.t1.r5">U</mi><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./18.1#SS1.p2.t1.r1">y</mi><mo>)</mo></mrow></mrow><mo></mo><mfrac><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><msup><mi href="./18.1#SS1.p2.t1.r1">y</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></msup><mrow><mi href="./18.1#SS1.p2.t1.r1">y</mi><mo>-</mo><mi href="./18.2#Px1.p2">x</mi></mrow></mfrac><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./18.1#SS1.p2.t1.r1">y</mi></mrow></mrow></mrow><mo>=</mo><mrow><mo>-</mo><mrow><mi href="./3.12#E1">π</mi><mo></mo><mrow><msub><mi href="./18.3#T1.t1.r4">T</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E43.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./18.3#T1.t1.r4" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m31.png" altimg-height="23px" altimg-valign="-7px" altimg-width="57px" alttext="T_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r4">T</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Chebyshev polynomial of the first kind</a>,
<a href="./18.3#T1.t1.r5" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m32.png" altimg-height="23px" altimg-valign="-7px" altimg-width="59px" alttext="U_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r5">U</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Chebyshev polynomial of the second kind</a>,
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m69.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#E24" title="(1.4.24) ‣ Cauchy Principal Values ‣ §1.4(v) Definite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m70.png" altimg-height="31px" altimg-valign="-9px" altimg-width="23px" alttext="\pvint_{\NVar{a}}^{\NVar{b}}" display="inline"><msubsup><mo href="./1.4#E24" largeop="true" symmetric="true">⨍</mo><mi class="ltx_nvar">a</mi><mi class="ltx_nvar">b</mi></msubsup></math>: Cauchy principal value</a>,
<a href="./18.1#SS1.p2.t1.r1" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi href="./18.1#SS1.p2.t1.r1">y</mi></math>: real variable</a>,
<a href="./18.1#SS1.p2.t1.r5" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m80.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">22.13.4</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">These integrals are Cauchy principal values (§</dd>
</dl>
</div>
</div>
<div id="Px25.p1" class="ltx_para">
<table id="E44" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">18.17.44</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E44.png" altimg-height="57px" altimg-valign="-22px" altimg-width="450px" alttext="\int_{-1}^{1}\frac{P_{n}\left(x\right)-P_{n}\left(t\right)}{|x-t|}\mathrm{d}t=%
2\left(1+\tfrac{1}{2}+\dots+\tfrac{1}{n}\right)P_{n}\left(x\right)," display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><mo>-</mo><mn>1</mn></mrow><mn>1</mn></msubsup><mrow><mfrac><mrow><mrow><msub><mi href="./18.3#T1.t1.r10">P</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow><mo>-</mo><mrow><msub><mi href="./18.3#T1.t1.r10">P</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./18.2#Px1.p2">x</mi><mo>-</mo><mi>t</mi></mrow><mo stretchy="false">|</mo></mrow></mfrac><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow><mo>=</mo><mrow><mn>2</mn><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo>+</mo><mi mathvariant="normal">…</mi><mo>+</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mi href="./18.1#SS1.p2.t1.r5">n</mi></mfrac></mstyle></mrow><mo>)</mo></mrow><mo></mo><mrow><msub><mi href="./18.3#T1.t1.r10">P</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m15.png" altimg-height="19px" altimg-valign="-5px" altimg-width="104px" alttext="-1\leq x\leq 1" display="inline"><mrow><mrow><mo>-</mo><mn>1</mn></mrow><mo>≤</mo><mi href="./18.2#Px1.p2">x</mi><mo>≤</mo><mn>1</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E44.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./18.3#T1.t1.r10" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m28.png" altimg-height="23px" altimg-valign="-7px" altimg-width="58px" alttext="P_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r10">P</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Legendre polynomial</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./18.1#SS1.p2.t1.r5" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m80.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">The case <math class="ltx_Math" altimg="m82.png" altimg-height="17px" altimg-valign="-2px" altimg-width="52px" alttext="x=1" display="inline"><mrow><mi href="./18.2#Px1.p2">x</mi><mo>=</mo><mn>1</mn></mrow></math> is a limit case of an integral for Jacobi polynomials; see
<cite class="ltx_cite ltx_citemacro_citet">Askey and Razban ()</cite>.
</p>
<table id="E45" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">18.17.45</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E45.png" altimg-height="54px" altimg-valign="-22px" altimg-width="520px" alttext="(n+\tfrac{1}{2})(1+x)^{\frac{1}{2}}\int_{-1}^{x}(x-t)^{-\frac{1}{2}}P_{n}\left%
(t\right)\mathrm{d}t=T_{n}\left(x\right)+T_{n+1}\left(x\right)," display="block"><mrow><mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>+</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>+</mo><mi href="./18.2#Px1.p2">x</mi></mrow><mo stretchy="false">)</mo></mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></msup><mo></mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><mo>-</mo><mn>1</mn></mrow><mi href="./18.2#Px1.p2">x</mi></msubsup><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./18.2#Px1.p2">x</mi><mo>-</mo><mi>t</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup><mo></mo><mrow><msub><mi href="./18.3#T1.t1.r10">P</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow><mo>=</mo><mrow><mrow><msub><mi href="./18.3#T1.t1.r4">T</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow><mo>+</mo><mrow><msub><mi href="./18.3#T1.t1.r4">T</mi><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E45.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./18.3#T1.t1.r4" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m31.png" altimg-height="23px" altimg-valign="-7px" altimg-width="57px" alttext="T_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r4">T</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Chebyshev polynomial of the first kind</a>,
<a href="./18.3#T1.t1.r10" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m28.png" altimg-height="23px" altimg-valign="-7px" altimg-width="58px" alttext="P_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r10">P</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Legendre polynomial</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./18.1#SS1.p2.t1.r5" title="§18.1(i) Special Notation ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m80.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./18.1#SS1.p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./18.2#Px1.p2" title="Orthogonality on Intervals ‣ §18.2(i) Definition ‣ §18.2 General Orthogonal Polynomials ‣ General Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi href="./18.2#Px1.p2">x</mi></math>: real variable</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E46" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">18.17.46</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E46.png" altimg-height="56px" altimg-valign="-20px" altimg-width="517px" alttext="(n+\tfrac{1}{2})(1-x)^{\frac{1}{2}}\int_{x}^{1}(t-x)^{-\frac{1}{2}}P_{n}\left(%
t\right)\mathrm{d}t=T_{n}\left(x\right)-T_{n+1}\left(x\right)." display="block"><mrow><mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>+</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>-</mo><mi href="./18.2#Px1.p2">x</mi></mrow><mo stretchy="false">)</mo></mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></msup><mo></mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi href="./18.2#Px1.p2">x</mi><mn>1</mn></msubsup><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mi>t</mi><mo>-</mo><mi href="./18.2#Px1.p2">x</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup><mo></mo><mrow><msub><mi href="./18.3#T1.t1.r10">P</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow><mo>=</mo><mrow><mrow><msub><mi href="./18.3#T1.t1.r4">T</mi><mi href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow><mo>-</mo><mrow><msub><mi href="./18.3#T1.t1.r4">T</mi><mrow><mi href="./18.1#SS1.p2.t1.r5">n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mi href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E46.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./18.3#T1.t1.r4" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m31.png" altimg-height="23px" altimg-valign="-7px" altimg-width="57px" alttext="T_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r4">T</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)</mo></mrow></mrow></math>: Chebyshev polynomial of the first kind</a>,
<a href="./18.3#T1.t1.r10" title="Table 18.3.1 ‣ §18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" class="ltx_ref"><math class="ltx_Math" altimg="m28.png" altimg-height="23px" altimg-valign="-7px" altimg-width="58px" alttext="P_{\NVar{n}}\left(\NVar{x}\right)" display="inline"><mrow><msub><mi href="./18.3#T1.t1.r10">P</mi><mi class="ltx_nvar" href="./18.1#SS1.p2.t1.r5">n</mi></msub><mo></mo><mrow><mo>(</mo><mi class="ltx_nvar" href="./18.2#Px1.p2">x</mi><mo>)<